appendices - cern · appendices a units the choice of a system of units to use is sometimes...

Appendices

A Units

The choice of a system of units to use is sometimes regarded as an emotionally charged subject Although there are many exceptions experimental papers often use mksa (or SI) units and theoretical papers may use Gaussian units (or perhaps a system in which several fundamental constants are set equal to one)

All theories of physics must be checked by comparison to experiment before they can be accepted For this reason it is convenient to express final equations in the mksa system Of course much of the older literature is still in Gaussian units so one must have some familiarity with it The main thing to do is to settle on a system of units and stick to it Anyone who has reached the graduate level in physics can convert units whenever needed It just may take a little longer than we wish to spend

In this appendix no description of the mksa system will be made An adequate description can be found in practically any sophomore physics book1

In solid-state physics another unit system is often convenient These units are called Hartree atomic units Let e be the charge on the electron and m be the mass of the electron The easiest way to get the Hartree system of units is to start from the Gaussian (cgs) formulas and let |e| = Bohr radius of hydrogen = |m| = 1 The results are summarized in Table A1 The Hartree unit of energy is 272 eV Ex-pressing your answer in terms of the fundamental physical quantities shown in Table A1 and then using Hartree atomic units leads to simple numerical answers for solid-state quantities In such units the solid-state quantities usually do not differ by too many orders of magnitude from one

We also include in Table A2 some other conversion factors and in Table A3 some quantities in units often used

1 Or see ldquoGuide for Metric Practicerdquo by Robert A Nelson at

httpwwwphysicstodayorgguidemetrichtml

652 Appendices

Table A1 Fundamental physical quantities

Quantity Symbol Expression value in mksa units

Expression value in Gaussian units

Value in Hartree units

Charge on electron

e 16 times 10minus19 coulomb 480 times 10minus10 esu 1

Mass of elec-tron

m 091 times 10minus30 kg 091 times 10minus27 g 1

Planckrsquos con-stant

1054 times 10minus34 joule s 1054 times 10minus27 erg s 1

Compton wavelength of electron

λc 2π( mc) 243 times 10minus12 m

2π( mc) 243 times 10minus10 cm

(2π) 1371

Bohr radius of hydrogen

a0 4πε02me2

053 times 10minus10 m

2me2 053 times 10minus8 cm

1

Fine structure constant

α e24πε0 c

1371 (approx)

e2 c

1371

1371

Speed of light c 3 times 108 m sminus1 3 times 1010 cm sminus1 137

Classical electron ra-dius

r0 e24πε0mc2 282 times 10minus15 m

e2mc2 282 times 10minus13 cm

2137

1 )(

Energy of ground state of hydrogen (1 Rydberg)

E0 e4m32(πε0 )2

1361dagger eV me42 2

1361dagger eV 21

Bohr magne-ton (calcu-lated from above)

μB e 2m 0927 times 10minus23 amp meter2

e 2mc 0927 times 10minus20 erg gaussminus1

2741

Cyclotron frequency

ωc or ωh

(μ0e2m)(2H) (e2mc)(2H) 2741 (2H)

The values given are greatly rounded off from the standard values The list of fundamen-tal constants has been updated and published yearly in part B of the August issue of Phys-ics Today See eg Peter J Mohr and Barry N Taylor ldquoThe Fundamental Physical Con-stantsrdquo Physics Today pp BG6-BG13 August 2003 Now see httpwwwphysicstodayorgguidefundconhtml dagger 1 eV = 16 times 10minus12 erg = 16 times 10minus19 joule

Normal Coordinates 653

Table A2 Some other conversion factors

Quantity Conversion

1Aring = 10minus8 cm = 10minus10 m

1 year = π times 107 sec (actually 316 times 107 s)

1 calorie = 419 J

1 T = 1 Wbm2 = 104 gauss

me c2 = 051 MeV

mp c2 = 938 MeV

Table A3 Some other quantities in units often used

Quantity Symbol Value

Gravitational constant G 667 times 10minus11 Nm2kg2

Mass of proton mp 167 times 10minus27 kg

Permeability constant μ0 4π times 10minus7 NA2

Permittivity constant ε0 (4π)minus1 (9 times 109)minus1 Fm

Avogadrorsquos number NA 602 times 1023 molminus1

Boltzmannrsquos constant k 138 times 10minus23 JK

Universal gas constant (NAk) R 831 J(mole K)

Hartree atomic unit of energy (e24πε0aB) 2Ry 27 eV

Magnetic flux quantum (h2|e|) φ0 207 times 10minus15 Wb

B Normal Coordinates

The main purpose of this appendix is to review clearly how the normal coordinate transformation arises and how it leads to a diagonalization of the Hamiltonian Our development will be made for classical systems but a similar development can be made for quantum systems An interesting discussion of normal modes has been given by Starzak2 The use of normal coordinates is important for collective excitations such as encountered in the discussion of lattice vibrations

2 See Starzak [A25 Chap 5]

654 Appendices

We will assume that our mechanical system is described by the Hamiltonian

sum += ji jiijijji xxxx21 )( υδH (B1)

In (B1) the first term is the kinetic energy and the second term is the potential energy of interaction among the particles We consider only the case that each particle has the same mass that has been set equal to one In (B1) it is also as-sumed that υij = υji and that each of the υij is real The coordinates xi in (B2) are measured from equilibrium that is assumed to be stable For a system of N parti-cles in three dimensions one would need 3N xi to describe the vibration of the system The dot of x i of course means differentiation with respect to time x i = dxidt

The Hamiltonian (B1) implies the following equation of motion for the me-chanical system

0)( =+sum j jijjij xx υδ (B2)

The normal coordinate transformation is the transformation that takes us from the coordinates xi to the normal coordinates A normal coordinate describes the motion of the system in a normal mode In a normal mode each of the coordinates vibrates with the same frequency Seeking a normal mode solution is equivalent to seeking solutions of the form

tjj cax ωieminus= (B3)

In (B3) c is a constant that is usually selected so that sumj|xj|2 = 1 and |caj| is the amplitude of vibration of xj in the mode with frequency ω The different frequen-cies ω for the different normal modes are yet to be determined

Equation (B2) has solutions of the form (B3) provided that

0)( 2 =minussum j jijjij aa δωυ (B4)

Equation (B4) has nontrivial solutions for the aj (ie solutions in which all the aj do not vanish) provided that the determinant of the coefficient matrix of the aj vanishes This condition determines the different frequencies corresponding to the different normal modes of the mechanical system If V is the matrix whose elements are given by υij (in the usual notation) then the eigenvalues of V are ω2 determined by (B4) V is a real symmetric matrix hence it is Hermitian hence its eigenvalues must be real

Let us suppose that the eigenvalues ω2 determined by (B4) are denoted by Ωk There will be the same number of eigenvalues as there are coordinates xi Let ajk be the value of aj which has a normalization determined by (B7) when the sys-tem is in the mode corresponding to the kth eigenvalue Ωk In this situation we can write

sumsum = j jkijkj jkij aa δΩυ (B5)


Let A stand for the matrix with elements ajk and Ω be the matrix with elements Ωlk = Ωkδlk Since Ωksumjδijajk = Ωkaik = aikΩk = sumlailΩkδlk = sumlailΩlk we can write (B5) in matrix notation as

ΩAVA = (B6)

It can be shown [2] that the matrix A that is constructed from the eigenvectors is an orthogonal matrix so that

IAAAA == ~~ (B7)

Atilde means the transpose of A Combining (B6) and (B7) we have

Ω=VAA~ (B8)

This equation shows how V is diagonalized by the use of the matrix that is con-structed from the eigenvectors

We still must indicate how the new eigenvectors are related to the old coordi-nates If a column matrix a is constructed from the aj as defined by (B3) then the eigenvectors E (also a column vector each element of which is an eigenvector) are defined by

aAE ~= (B9a)

or

AEa = (B9b)

That (B9) does define the eigenvectors is easy to see because substituting (B9b) into the Hamiltonian reduces the Hamiltonian to diagonal form The kinetic en-ergy is already diagonal so we need consider only the potential energy

)~()~(

)~()~(

~~~~

2

sumsum

sumsumsum

==

==

===

kj jkiijj jjj

kj kjkkjjk kjkj

jiij

EEEE

EEEEEEVAEAEVaaaa

δωΩ

δΩΩΩυ

which tells us that the substitution reduces V to diagonal form For our purposes the essential thing is to notice that a substitution of the form (B9) reduces the Hamiltonian to a much simpler form

An example should clarify these ideas Suppose the eigenvalue condition yielded

032

21det 2

2=⎟

⎟⎠

⎞⎜⎜⎝

⎛

minusminus

ωω (B10)

This implies the two eigenvalues

5221 +=ω (B11a)

5222 minus=ω (B11b)

656 Appendices

Equation (B4) for each of the eigenvalues gives for

51

2 21

21 +

== aaωω (B12a)

and for

51

2 21

22 minus

== aaωω (B12b)

From (B12) we then obtain the matrix A

⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜

⎝

⎛

minus

+=2

2

11

51

2

51

2~

NN

NN

A (B13)

where

21

21

1 1)15(

4)(⎥⎥⎦

⎤

⎢⎢⎣

⎡+

+=minusN (B14a)

and

21

21

2 1)15(

4)(⎥⎥⎦

⎤

⎢⎢⎣

⎡+

minus=minusN (B14b)

The normal coordinates of this system are given by

⎟⎟⎠

⎞⎜⎜⎝

⎛

⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜

⎝

⎛

minus

+=⎟⎟⎠

⎞⎜⎜⎝

⎛=

2

1

22

11

2

1

51

2

51

2

aa

NN

NN

EE

E (B15)

Problems

B1 Show that (B13) satisfies (B7)

B2 Show for A defined by (B13) that

⎟⎟⎠

⎞⎜⎜⎝

⎛

minus+=⎟⎟

⎠

⎞⎜⎜⎝

⎛

520052

3221~ AA

This result checks (B8)

Derivations of Blochrsquos Theorem 657

C Derivations of Blochrsquos Theorem

Blochrsquos theorem concerns itself with the classifications of eigenfunctions and ei-genvalues of Schroumldinger-like equations with a periodic potential It applies equally well to electrons or lattice vibrations In fact Blochrsquos theorem holds for any wave going through a periodic structure We start with a simple one-dimensional derivation

C1 Simple One-Dimensional Derivation3ndash5

This derivation is particularly applicable to the KronigndashPenney model We will write the Schroumldinger wave equation as

0)()(d

)(d2

2=+ xxU

xx ψψ (C1)

where U(x) is periodic with period a ie

)()( xUnaxU =+ (C2)

with n an integer Equation (C1) is a second-order differential equation so that there are two linearly independent solutions ψ1 and ψ2

011 =+primeprime ψψ U (C3) 022 =+primeprime ψψ U (C4)

From (C3) and (C4) we can write

01212 =+primeprime ψψψψ U 02121 =+primeprime ψψψψ U

Subtracting these last two equations we obtain

02112 =primeprimeminusprimeprime ψψψψ (C5)

This last equation is equivalent to writing

0d

d =x

W (C6)

where

21

21ψψψψprimeprime

=W (C7)

3 See Ashcroft and Mermin [A3] 4 See Jones [A10] 5 See Dekker [A4]

658 Appendices

is called the Wronskian For linearly independent solutions the Wronskian is a constant not equal to zero

It is easy to prove one result from the periodicity of the potential By dummy variable change (x) rarr (x + a) in (C1) we can write

0)()(d

)(d2

2=++++ axaxU

xax ψψ

The periodicity of the potential implies

0)()(d

)(d2

2=+++ axxU

xax ψψ (C8)

Equations (C1) and (C8) imply that if ψ(x) is a solution then so is ψ(x + a) Since there are only two linearly independent solutions ψ1 and ψ2 we can write

)()()( 211 xBxAax ψψψ +=+ (C9)

)()()( 212 xDxCax ψψψ +=+ (C10)

The Wronskian W is a constant ne 0 so W(x + a) = W(x) and we can write

21

21

21

21

2121

2121ψψψψ

ψψψψ

ψψψψψψψψ

primeprime=

primeprime=

prime+primeprime+prime++

DBCA

DCBADCBA

or

1=DBCA

or

1=minus BCAD (C11)

We can now prove that it is possible to choose solutions ψ(x) so that

)()( xΔax ψψ =+ (C12)

where Δ is a constant ne 0 We want ψ(x) to be a solution so that

)()()( 21 xxx βψαψψ += (C13a)

or

)()()( 21 axaxax +++=+ βψαψψ (C13b)

Using (C9) (C10) (C12) and (C13) we can write

)()()()()()()(

2121

xxxDBxCAax

βψΔαψΔψβαψβαψ

+=+++=+ (C14)


In other words we have a solution of the form (C12) provided that

αΔβα =+ CA

and

βΔβα =+ DB

For nontrivial solutions for α and β we must have

0=minus

minusΔ

ΔDB

CA (C15)

Equation (C15) is equivalent to using (C11)

DA+=+ minus1ΔΔ (C16)

If we let Δ+ and Δminus be the eigenvalues of the matrix ( AB

CD ) and use the fact that the

trace of a matrix is the sum of the eigenvalues then we readily find from (C16) and the trace condition

DA +=+ minus++

1)(ΔΔ

DA +=+ minusminusminus

1)(ΔΔ (C17)

and

DA +=+ minus+ ΔΔ

Equations (C17) imply that we can write

1)( minusminus+ = ΔΔ (C18)

If we set

be=+Δ (C19)

and

bminusminus = eΔ (C20)

the above implies that we can find linearly independent solutions ψi1 that satisfy

)()( 11

11 xeax bψψ =+ (C21)

and

)()( 12

12 xeax bψψ minus=+ (C22)

660 Appendices

Real b is ruled out for finite wave functions (as x rarr plusmn infin ) so we can write b = ika where k is real Dropping the superscripts we can write

)(e)( i xax kaψψ plusmn=+ (C23)

Finally we note that if

)(e)( i xux kx=ψ (C24)

where

)()( xuaxu =+ (C25)

then (C23) is satisfied (C23) or (C24) and (C25) are different forms of Blochrsquos theorem

C2 Simple Derivation in Three Dimensions

Let

)()( 11 NN xxExx ψψ =H (C26)

be the usual Schroumldinger wave equation Let Tl be a translation operator that trans-lates the lattice by l1a1 + l2a2 + l3a3 where the li are integers and the ai are the primitive translation vectors of the lattice

Since the Hamiltonian is invariant with respect to translations by Tl we have

0][ =lTH (C27)

and

0][ =primell TT (C28)

Now we know that we can always find simultaneous eigenfunctions of com-muting observables Observables are represented by Hermitian operators The Tl are unitary Fortunately the same theorem applies to them (we shall not prove this here) Thus we can write

ll EE Eψψ =H (C29)

llll EE tT ψψ = (C30)

Now certainly we can find a vector k such that

lkmiddotl

ie=t (C31)

Further

d)(d)(d)( 222space all

2intintint =+= τψτψτψ rlrr lt


so that

12 =lt (C32)

This implies that k must be a vector over the real field We thus arrive at Blochrsquos theorem

)(e)()( i rlrr lkmiddotl ψψψ =+=T (C33)

The theorem says we can always choose the eigenfunctions to satisfy (C33) It does not say the eigenfunction must be of this form If periodic boundary condi-tions are applied the usual restrictions on the k are obtained

C3 Derivation of Blochrsquos Theorem by Group Theory

The derivation here is relatively easy once the appropriate group theoretic knowl-edge is acquired We have already discussed in Chaps 1 and 7 the needed results from group theory We simply collect together here the needed facts to establish Blochrsquos theorem

1 It is clear that the group of the Tl is Abelian (ie all the Tl commute)

2 In an Abelian group each element forms a class by itself Therefore the number of classes is O(G) the order of the group

3 The number of irreducible representations (of dimension ni) is the number of classes

4 sumni2 = O(G) and thus by above

)(02)(0

22

21 Gnnn G =+++

This can be satisfied only if each ni = 1 Thus the dimensions of the irreducible representations of the Tl are all one

5 In general sum= j

kj

klij

kil AT ψψ

where the Ail

jk are the matrix elements of the Tl for the kth representation and the

sum over j goes over the dimensionality of the kth representation The ψik are the

basis functions selected as eigenfunctions of H (which is possible since [H Tl] = 0) In our case the sum over j is not necessary and so

kklkl AT ψψ =

As before the Alk can be chosen to be eilmiddotk Also in one dimension we could use the fact that Tl is a cyclic group so that the Alk are automatically the roots of one

662 Appendices

D Density Matrices and Thermodynamics

A few results will be collected here The proofs of these results can be found in any of several books on statistical mechanics

If ψi(x t) is the wave function of system (in an ensemble of N systems where 1 le i le N) and if |nrang is a complete orthonormal set then

ntctx nin

i sum= )()(ψ

The density matrix is defined by

nmNi

im

imnm cctctc

Nlowast

=lowast equiv= sum 1 )()(1ρ

It has the following properties

1)( =equiv sumn nnTr ρρ

the ensemble average (denoted by a bar) of the quantum-mechanical expectation value of an operator A is

)( ATrA ρequiv

and the equation of motion of the density operator ρ is given by

][i Ht

ρρ =partpartminus

where the density operator is defined in such a way that langn|ρ|mrang equiv ρnm For a cano-nical ensemble in equilibrium

⎟⎠⎞

⎜⎝⎛ minus=

kTHFexpρ

Thus we can readily link the idea of a density matrix to thermodynamics and hence to measurable quantities For example the internal energy for a system in equilibrium is given by

)][exp()]exp([exp

kTHTrkTHHTr

kTHFHTrHU

minusminus=⎥

⎦

⎤⎢⎣

⎡⎟⎠⎞

⎜⎝⎛ minus==

Alternatively the internal energy can be calculated from the free energy F where for a system in equilibrium

)][exp(ln kTHTrkTF minusminus=

It is fairly common to leave the bar off langA rang so long as the meaning is clear For further properties and references see Patterson [A19] see also Huang [A8]

Time-Dependent Perturbation Theory 663

E Time-Dependent Perturbation Theory

A common problem in solid-state physics (as in other areas of physics) is to find the transition rate between energy levels of a system induced by a small time-dependent perturbation More precisely we want to be able to calculate the time development of a system described by a Hamiltonian that has a small time-dependent part This is a standard problem in quantum mechanics and is solved by the time-dependent perturbation theory However since there are many different aspects of time-dependent perturbation theory it seems appropriate to give a brief review without derivations For further details any good quantum mechanics book such as Merzbacher6 can be consulted

Fig E1 f(t ω) versus ω The area under the curve is 2πt

Let

)()( 0 tVt += HH (E1)

lEl l00 =H (E2)

ltVktV lk )()( = (E3)

00lk

lkEE minus=ω (E4)

In first order in V for V turned on at t = 0 and constant otherwise the probability per unit time of a discrete i rarr f transition for t gt 0 is

)(2 002fififi EEVP minuscongrarr δπ (E5)

6 See Merzbacher [A15 Chap 18]

664 Appendices

In deriving (E5) we have assumed that the f (t ω) in Fig E1 can be replaced by a Dirac delta function via the equation

200

2 2)()(

)(

)cos(1lim ωδπ

ω

ω tfEEttfi

if

if

t=minus=

minusinfinrarr

(E6)

If we have transitions to a group of states with final density of states pf(Ef) a simi-lar calculation gives

)(2 2fffifi EpVP π=rarr (E7)

In the same approximation if we deal with periodic perturbations represented by

tt ggtV ωω idaggeri ee)( minus+= (E8)

which are turned on at t = 0 we obtain for transitions between discrete states

)(2 002ωδπ plusmnminus=rarr fififi EEgP (E9)

In the text we have loosely referred to (E5) (E7) or (E9) as the Golden rule (according to which is appropriate to the physical situation)

F Derivation of The Spin-Orbit Term From Diracrsquos Equation

In this appendix we will indicate how the concepts of spin and spin-orbit interac-tion are introduced by use of Diracrsquos relativistic theory of the electron For further details any good quantum mechanics text such as that of Merzbacher7 or Schiff8 can be consulted We will discuss Diracrsquos equation only for fields described by a potential V For this situation Diracrsquos equation can be written

ψψβ EVcmc =++ ])([ 20pα sdot (F1)

In (F1) c is the speed of light α and β are 4 times 4 matrices defined below p is the momentum operator m0 is the rest mass of the electron ψ is a four-component column matrix (each element of this matrix may be a function of the spatial posi-tion of the electron) and E is the total energy of the electron (including the rest mass energy that is m0c2) The α matrices are defined by

⎟⎟⎠

⎞⎜⎜⎝

⎛=

00σ

σα (F2)

7 See Merzbacher [A15 Chap 23] 8 See Schiff [A23]

Derivation of The Spin-Orbit Term From Diracrsquos Equation 665

where the three components of σ are the 2 times 2 Pauli spin matrices The definition of β is

⎟⎟⎠

⎞⎜⎜⎝

⎛minus

=I

I0

0β (F3)

where I is a 2 times 2 unit matrix For solid-state purposes we are not concerned with the fully relativistic equa-

tion (F1) but rather we are concerned with the relativistic corrections that (F1) predicts should be made to the nonrelativistic Schroumldinger equation That is we want to consider the Dirac equation for the electron in the small velocity limit More precisely we will consider the limit of (F1) when

12

)(2

0

20 ltlt

minusminusequiv

cmVcmEε (F4)

and we want results that are valid to first order in ε ie first-order corrections to the completely nonrelativistic limit To do this it is convenient to make the fol-lowing definitions

20cmEE +prime= (F5)

and

⎟⎟⎠

⎞⎜⎜⎝

⎛=

φχ

ψ (F6)

where both χ and φ are two-component wave functions If we substitute (F5) and (F6) into (F1) we obtain an equation for both χ and

φ We can combine these two equations into a single equation for χ in which φ does not appear We can then use the small velocity limit (F4) together with sev-eral properties of the Pauli spin matrices to obtain the Schroumldinger equation with relativistic corrections

χχ sdottimesnablasdotnablasdotnabla⎥⎥⎦

⎤

⎢⎢⎣

⎡+minus+minus=prime ))((

4482 220

2

220

2

230

4

0

2pσ V

cmV

cmV

cmp

mpE (F7)

This is the form that is appropriate to use in solid-state physics calculations The term

])[(4 22

0

2pσ timesnablasdot V

cm (F8)

is called the spin-orbit term This term is often used by itself as a first-order cor-rection to the nonrelativistic Schroumldinger equation The spin-orbit correction is often applied in band-structure calculations at certain points in the Brillouin zone where bands come together In the case in which the potential is spherically sym-

666 Appendices

metric (which is important for atomic potentials but not crystalline potentials) the spin-orbit term can be cast into the more familiar form

SL middot dd1

2 220

2

rV

rcm (F9)

where L is the orbital angular momentum operator and S is the spin operator (in units of ħ)

It is also interesting to see how Diracrsquos theory works out in the (completely) nonrelativistic limit when an external magnetic field B is present In this case the magnetic moment of the electron is introduced by the term involving S middot B This term automatically appears from the nonrelativistic limit of Diracrsquos equation In addition the correct ratio of magnetic moment to spin angular momentum is ob-tained in this way

G The Second Quantization Notation for Fermions and Bosons

When the second quantization notation is used in a nonrelativistic context it is simply a notation in which we express the wave functions in occupation-number space and the operators as operators on occupation number space It is of course of great utility in considering the many-body problem In this formalism the symmetry or antisymmetry of the wave functions is automatically built into the formalism In relativistic physics annihilation and creation operators (which are the basic operators of the second quantization notation) have physical meaning However we will apply the second quantization notation only in nonrelativistic situations No derivations will be made in this section (The appropriate results will just be concisely written down) There are many good treatments of the sec-ond quantization or occupation number formalism One of the most accessible is by Mattuck9

G1 Bose Particles

For Bose particles we deal with bi and bidagger operators (or other letters where conven-

ient) bidagger creates a Bose particle in the state i bi annihilates a Bose particle in the

state f The bi operators obey the following commutation relations

][

0][

0][

dagger

daggerdagger

ijji

ji

ijjiji

bb

bb

bbbbbb

δ=

=

=minusequiv

9 See Mattuck [A14]

The Second Quantization Notation for Fermions and Bosons 667

The occupation number operator whose eigenvalues are the number of particles in state i is

daggeri i in b b=

and

dagger1 iii bbn =+

The effect of these operators acting on different occupation number kets is

11

1

11dagger

11

helliphelliphelliphellip


++=

minus=

iiii

iiii

nnnnnb

nnnnnb

where |n1hellipnihelliprang means the ket appropriate to the state with n1 particles in state 1 n2 particles in state 2 and so on

The matrix elements of these operators are given by

1

1dagger

iiii

iiii

nnbn

nnbn

=minus

=minus

In this notation any one-particle operator

sum= l lop ff )()1()1( r can be written in the form sum= ki kiop bbkfif

dagger)1()1(

and the |krang are any complete set of one-particle eigenstates In a similar fashion any two-particle operator

sum minus= ml mlop ff )2()2( )( rr

can be written in the form

sum= mlki lmkiop bbbbmlfkif daggerdagger)2()2( )2()1()2()1(

Operators that create or destroy base particles at a given point in space (rather than in a given state) are given by

)()(

)()(daggerdagger sum

sumlowast=

=

α αα

α αα

ψψ

bu

bu

rr

rr

where uα(r) is the single-particle wave function corresponding to state α In gen-eral r would refer to both space and spin variables These operators obey the commutation relation )()]()([ dagger rrrr primeminus= δψψ

668 Appendices

G2 Fermi Particles

For Fermi particles we deal with ai and aidagger operators (or other letters where con-

venient) aidagger creates a fermion in the state i ai annihilates a fermion in the state i

The ai operators obey the following anticommutation relations

0

0

dagger

daggerdagger

ijji

ji

ijjiji

aa

aa

aaaaaa

δ=

=

=+equiv


iii aan dagger= and dagger1 iii aan =minus

Note that (ni)2 = ni so that the only possible eigenvalues of ni are 0 and 1 (the Pauli principle is built in)

The matrix elements of these operators are defined by

sum minusminus=== )11()(10 iiii nan

and sum minusminus=== )11(dagger )(01 i

iii nan

where sum(1i minus 1) equals the sum of the occupation numbers of the states from 1 to i minus 1

In this notation any one-particle operator can be written in the form

sum= ji ji aajfif dagger)1()1(

0

where the | j rang are any complete set of one-particle eigenstates In a similar fashion any two-particle operator can be written in the form

sum= lkji lkijop aaaalkfjif daggerdagger)2()2( )2()1()2()1(

Operators that create or destroy Fermi particles at a given point in space (rather than in a given state) are given by

sum= α ααψ au )()( rr

where uα(r) is the single-particle wave function corresponding to state α and

sum lowast= α ααψ daggerdagger )()( au rr

The Many-Body Problem 669

These operators obey the anticommutation relations

)()()( dagger rrrr primeminus= δψψ

The operators also allow a convenient way of writing Slater determinants eg

)2()1()2()1(

210daggerdagger

ββ

ααβα uu

uuaa harr

|0rang is known as the vacuum ket The easiest way to see that the second quantization notation is consistent is to

show that matrix elements in the second quantization notation have the same val-ues as corresponding matrix elements in the old notation This demonstration will not be done here

H The Many-Body Problem

Richard P Feynman is famous for many things among which is the invention in effect of a new quantum mechanics Or maybe we should say of a new way of looking at quantum mechanics His way involves taking a process going from A to B and looking at all possible paths He then sums the amplitude of the all paths from A to B to find by the square the probability of the process

Related to this is a diagram that defines a process and that contains by implica-tion all the paths as calculated by appropriate integrals Going further one looks at all processes of a certain class and sums up all diagrams (if possible) belonging to this class Ideally (but seldom actually) one eventually treats all classes and hence arrives at an exact description of the interaction

Thus in principle there is not so much to treating interactions by the use of Feynman diagrams The devil is in the details however Certain sums may well be infinitendashalthough hopefully disposable by renormalization Usually doing a non-trivial calculation of this type is a great technical feat

We have found that a common way we use Feynman diagrams is to help us un-derstand what we mean by a given approximation We will note below for exam-ple that the Hartree approximation involves summing a certain class of diagrams while the HartreendashFock approximation involves summing these diagrams along with another class We believe the diagrams give us a very precise idea of what these approximations do

Similarly the diagram expansion can be a useful way to understand why a per-turbation expansion does not work in explaining superconductivity as well as a way to fix it (the Nambu formalism)

The practical use of diagrams and diagram summation may involve great prac-tical skill but it seems that the great utility of the diagram approach is in clearly stating and in keeping track of what we are doing in a given approximation

One should not think that an expertise in the technicalities of Feynman diagrams solves all problems Diagrams have to be summed and integrals still have to be done For some aspects of many-electron physics density functional theory (DFT)

670 Appendices

has become the standard approach Diagrams are usually not used at the beginning of DFT but even here they may often be helpful in discussing some aspects

DFT was discussed in Chap 3 and we briefly review it here because of its great practical importance in the many-electron problem of solid-state physics In the beginning of DFT Hohenberg and Kohn showed that the N-electron Schroumldinger wave equation in three dimensions could be recast They showed that an equation for the electron density in three dimensions would suffice to deter-mine ground-state properties The HohenbergndashKohn formulation may be regarded as a generalization of the ThomasndashFermi approximation Then came the famous KohnndashSham equations that reduced the HohenbergndashKohn formulation to the prob-lem of noninteracting electrons in an effective potential (somewhat analogous to the Hartree equations for example) However part of the potential the exchange correlation part could only be approximately evaluated eg in the local density approximation (LDA) ndash which assumed a locally homogeneous electron gas A problem with DFT-LDA is that it is not necessarily clear what the size of the er-rors are however the DFT is certainly a good way to calculate ab initio certain ground-state properties of finite electronic systems such as the ionization energies of atoms It is also very useful for computing the electronic ground-state proper-ties of periodic solids such as cohesion and stability Excited states as well as approximations for the exchange correlation term in N-electron systems continue to give problems For a nice brief summary of DFT see Mattsson [A13]

For quantum electrodynamics a brief and useful graphical summary can be found at httpwww2slacstanfordeduvvctheoryfeynmanhtml We now present a brief summary of the use of diagrams in many-body physics

In some ways trying to do solid-state physics without Feynman diagrams is a lit-tle like doing electricity and magnetism (EM) without resorting to drawing Fara-dayrsquos lines of electric and magnetic fields However just as field lines have limita-tions in describing EM interactions so do diagrams for discussing the many-body problem [A1] The use of diagrams can certainly augment onersquos understanding

The distinction between quasi- or dressed particles and collective excitations is important and perhaps is made clearer from a diagrammatic point of view Both are lsquoparticlesrsquo and are also elementary energy excitations But after all a polaron (a quasi-particle) is not the same kind of beast as a magnon (a collective excitation) Not everybody makes this distinction Some call all lsquoparticlesrsquo quasiparticles Bo-golons are particles of another type as are excitons (see below for definitions of both) All are elementary excitations and particles but not really collective excita-tions or dressed particles in the usual sense

H1 Propagators

These are the basic quantities Their representation is given in the next section The single-particle propagator is a sum of probability amplitudes for all the ways of going from r1 t1 to r2 t2 (adding a particle at 1 and taking out at 2)

The two-particle propagator is the sum of the probability amplitudes for all the ways two particles can enter a system undergo interactions and emerge again


H2 Green Functions

Propagators are represented by Green functions There are both advanced and retarded propagators Advanced propagators can describe particles traveling back-ward in time ie holes The use of Fourier transforms of time-dependent propaga-tors led to simpler algebraic equations For a retarded propagator the free propaga-tor is

δεω

ωi

1)(0 +minus=+

kG k (H1)

For quasiparticles the real part of the pole of the Fourier transform of the sin-gle-particle propagator gives the energy and the imaginary part gives the width of the energy level For collective excitations one has a similar statement except that two-particle propagators are needed

H3 Feynman Diagrams

Rules for drawing diagrams are found in Economu [A5 pp 251-252] Pines [A22 pp 49-50] and Schrieffer [A24 pp 127-128] Also see Mattuck [A14 p 165] There is a one-to-one correspondence between terms in the perturbation expansion of the Green functions and diagrammatic representation Green functions can also be calculated from a hierarchy of differential equations and an appropriate de-coupling scheme Such approximate decoupling schemes are always equivalent to a partial sum of diagrams

H4 Definitions

Here we remind you of some examples A more complete list is found in Chap 4

Quasiparticle ndash A real particle with a cloud of surrounding disturbed particles with an effective mass and a lifetime In the usual case it is a dressed fermion Examples are listed below

Electrons in a solid ndash These will be dressed electrons They can be dressed by interaction with the static lattice other electrons or interactions with the vibrating lattice It is represented by a straight line with an arrow to the right if time goes that way

Holes in a solid ndash One can view the ground state of a collection of electrons as a vacuum A hole is then what results when an electron is removed from a normally occupied state It is represented by a straight line with an arrow to the left

Polaron ndash An electron moving through a polarizable medium surrounded by its polarization cloud of virtual phonons

672 Appendices

Photon ndash Quanta of electromagnetic radiation (eg light) ndash it is represented by a wavy line

Collective Excitation ndash These are elementary energy excitations that involve wave-like motion of all the particles in the systems Examples are listed below

Phonon ndash Quanta of normal mode vibration of a lattice of ions Also often repre-sented by wavy line

Magnon ndash Quanta of low-energy collective excitations in the spins or quanta of waves in the spins

Plasmon ndash Quanta of energy excitation in the density of electrons in an interacting electron gas (viewing eg the positive ions as a uniform background of charge)

Other Elementary Energy Excitations ndash Excited energy levels of many-particle systems

Bogolon ndash Linear combinations of electrons in a state +k with lsquouprsquo spin and minusk with lsquodownrsquo spin Elementary excitations in a superconductor

Exciton ndash Bound electronndashhole pairs

Some examples of interactions represented by vertices (time going to the right)

Diagrams are built out of vertices with conservation of momentum satisfied at

the vertices For example

represents a coulomb interaction with time going up

H5 Diagrams and the Hartree and HartreendashFock Approximations

In order to make these concepts clearer it is perhaps better to discuss an example that we have already worked out without diagrams Here starting from the Hamiltonian we will discuss briefly how to construct diagrams then explain how to associate single-particle Green functions with the diagrams and how to do the partial sums representing these approximations For details the references must be consulted

In the second quantization notation a Hamiltonian for interacting electrons

sumsum += jii ijViV )(21)(H (H2)


with one- and two-body terms can be written as

sumsum += lijk lkijji ji aaaajkVjiaajVi daggerdagger

dagger )2()1()21()2()1(21)1()1()1(H (H3)

where int lowast= iji dVjVi rrrr 3

111 )()()()1()1()1( φφ (H4) and int lowastlowast= 2

31

32121 )()()21()()()2()1()21()2()1( rrrrrr ddVlkVji lkji φφφφ (H5)

and the annihilation and creation operators have the usual properties

0

daggerdagger

=+=+

ijji

ijijjiaaaaaaaa δ

We now consider the Hartree approximation We assume following Mattuck [A14] that the interactions between electrons is mostly given by the forward scat-tering processes where the interacting electrons have no momentum change in the interaction We want to get an approximation for the single-particle propagator that includes interactions In first order the only possible process is given by a bubble diagram where the hole line joins on itself One thinks of the particle in state k knocking a particle out of and into a state l instantaneously Since this can happen any number of times we get the following partial sum for diagrams repre-senting the single-particle propagator The first diagram on the right-hand side represents the free propagator where nothing happens (Mattuck [A14 p 89]10)

Using the ldquodictionaryrdquo given by Mattuck [A14 p 86] we substitute propagators for diagrams and get

δεω

ωi

1)()occ( +minusminus

=sum

+

l lklkk VG k (H6)

Since the poles give the elementary energy excitations we have

sum+=prime )occ(l lklkkk Vεε (H7)

10 Reproduced with permission from Mattuck RD A Guide to Feynman Diagrams in the

Many-Body Problem 2nd edn (467) p 89 Dover Publications Inc 1992

674 Appendices

which is exactly the same as the Hartree approximation (see (321)) since

sum intintsum lowastlowast= l llkkl lklk VV 13

112223 d)21()()()()(d rrrrrr φφφφ (H8)

It is actually very simple to go from here to the HartreendashFock approximation ndash all we have to do is to include the exchange terms in the interactions These are the ldquoopen-oysterrdquo diagrams

where a particle not only strikes a particle in l and creates an instantaneous hole but is exchanged with it Doing the partial sum of forward scattering and exchange scattering one has (Mattuck [A14 p 91]11)

Associating propagators with the terms in the diagram gives

δεω

ωi)(

1)()occ( +minusminusminus

=sum

+

l lkkllklkk VVG k (H9)

From this we identify the elementary energy excitations as

sum minus+=prime )occ( )(l lkkllklkkk VVεε (H10)

which is just what we got for the HartreendashFock approximation (see (350)) The random-phase approximation [A14] can also be obtained by a partial

summation of diagrams and it is equivalent to the Lindhard theory of screening




H6 The Dyson Equation

This is the starting point for many approximations both diagrammatic and alge-braic Dysonrsquos equation can be regarded as a generalization of the partial sum technique used in the Hartree and HartreendashFock approximations It is exact To state Dysonrsquos equation we need a couple of definitions The self-energy part of a diagram is a diagram that has no incoming or outgoing parts and can be inserted into a particle line The bubbles of the Hartree method are an example An irre-ducible or proper self-energy part is a part that cannot be further reduced into un-connected self-energy parts It is common to define

as the sum over all proper self-energy parts Then one can sum over all repetitions of sigma (sumkω) to get

Dysonrsquos equation yields an exact expression for the propagator

klk

Gδωεω

ωi)(


=sum k

k (H11)

since all diagrams are either proper diagrams or their repetition In the Hartree approximation

and in the HartreendashFock approximation

Although the Dyson equation is in principle exact one still has to evaluate sigma and this is in general not possible except in some approximation

We cannot go into more detail here We have given accurate results for the high and low-density electron gas in Chap 2 In general the ideas of Feynman dia-grams and the many-body problem merit a book of their own We have found the book by Mattuck [A14] to be particularly useful but note the list of references at the end of this section We have used some ideas about diagrams when we dis-cussed superconductivity

676 Appendices

I Brief Summary of Solid-State Physics12

Note the order of review here is not identical to the text as we indicate below

1 Classification and Crystal Structure (See Sects 124 and 125)

fcc (110) (011) (101)

2 2 2

bcc (11 1) ( 111) (1 11)2 2 2

a a a

a a aminus minus minus

Seven Crystal Systems cubic tetragonal orthorhombic monoclinic triclinic hexagonal trigonal 14 Bravais lattices 230 distinct lattices

Bravais Lattice and Reciprocal Lattice Vectors (See Sect 129)

1 2 31 ( )Ω

= timesb a a (See 145)

1 2 3( )Ω = sdot timesa a a (See 146)

Reciprocal Lattice of fcc is bcc Reciprocal Lattice of bcc is fcc

Bragg and von Laue Diffraction (See Sect 129) The two are equivalent

lkhlkhlkh dn θλ sin2= (See 155)

lkh

lkh dG π2

shortest = (See 154)

Δ =k G (See 152)

2 i inπ= sumG b (See 144)

2i imπsdotΔ =a k (See 148 with ai = Rpmn)

Brillouin Zones (See Sect 231 discussed in detail) The first Brillouin zone is the set of all k-space points that enclose the origin and are inside all Bragg planes (planes describing Bragg reflection in k-space) Higher zones are similarly de-fined

Madelungrsquos constant

sumplusmn=ijrR)(α (See 113)

12 A much more extensive survey of solid-state physics is contained in Sybil P Parker

Editor in Chief Solid State Physics Source Book McGraw-Hill Book Co New York 1987

Brief Summary of Solid-State Physics 677

Structure Factor and Atomic Form Factor (See Sect 129)

ie jjjS f minus sdot=sum G r

G (See 141)

i( )ej jf n dVminus sdot= int G ρr (See 142)

Jahn Teller Effect (See Sect 744) Relevant to symmetry considerations A dis-tortion of a symmetric molecule in a degenerate state which reduces symmetry and lowers energy

2 Lattice Vibrations

Lennard-Jones potential (See Sect 111) For van der Waals interactions men-tioned here as illustrative of the potential between atoms in a solid that gives rise to vibrations The 12th power term is used to model the repulsive part of the potential

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎟⎟⎠

⎞⎜⎜⎝

⎛+⎟

⎟⎠

⎞⎜⎜⎝

⎛minus=

126

4)(ijij

ij rrrU σσε

Bose Einstein Statistics for Bosons with zero chemical potential the Planck Distribution (See Sects 223 233)

1e

1 minus

= kTn ω (See 277 2215)

Dispersion Relations (See Sects 222 224) For long wavelength acoustic modes

222 akmks=ω (See 248 288 290)

where ks is the elastic constant k is the wave vector

Thermal Conductivity (See Sect 423 for phonons) Arises from both phonons and electrons A simple kinetic theory argument gives the equation below

lCK v31cong (See 3205)

where C is the specific heat per unit volume v is the average carrier velocity l is the mean free path

Debye Specific Heat (See Sect 233) (assume three modes with same v)

D

DDωωωωωω

gt=leprop

0)( 2

(See 2224)

3126v ⎟

⎟⎠

⎞⎜⎜⎝

⎛==

VN

kk DDD

πθθω (See 2228)

where v is velocity and V is volume

678 Appendices

Heat Capacity (See Sect 233)

intminus

⎟⎠⎞

⎜⎝⎛= T

x

x

VdxxTNkC

0 2

43

)1e(e9 θ

θ (See 2229)

Density of States (See Sect 323 for most general derivation)

3( )

( )( )(2 ) const

dAVD ω

ωω

ωπ= int

k knabla (See 3256)

Van Hove Singularity (See Sect 233) When υg = nablak ω(k) = 0 in Density of States resulting in singularities in D(ω)

Umklapp Relates to phonon interactions but listed here (See Sect 422 with dif-ferent notation) 1 2 3+ = +k k k G (See 412)

Debye-Waller Factor eminus2W (See end of Sect 129) Related to lattice vibrations Scattered x-ray intensity reduced by

)e(e 20

2 WW II minusminus = For low T 2( )W prop Δk and for High T

2( ) TWθ⎛ ⎞prop Δ ⎜ ⎟⎝ ⎠

k

Lindemann Melting Formula relates to lattice vibrations see eg J M Ziman Principles of the Theory of Solids Cambridge 1964 p 63

22

2

9)(

Dm

m mkAT θ=

where Am = amplitude of thermally excited oscillation at melting point

Gruneisen Constant (Debye Model) (See Sect 234)

V

Dln

lnpartpartminus= ωγ (See 2242)

then the coefficient of thermal expansion is

B

C3

vγα = (See 2237 2238 2250)

where B is the bulk modulus (minusV partPpartV the reciprocal of the compressibility Eq 118 2237) and Cv is the specific heat per unit volume


Elastic constants in continuum (See Sect 235) For cubic crystal [100] waves longitudinal wave velocity

ρ11Cv = (See 2270)

transverse wave velocity

ρ44Cv = (See 2270)

where C11 and C44 are elastic stiffness constants

In addition for a summary of crystal mathematics See Sect 231 and Pb 211

3 Electrons

Fermi Function (See Sect 322)

1e

1)( +

= minus kTEf μ (See 3165)

Blochrsquos Theorem for periodic lattices (See Sect 321 and Appendix C)

i( )euψ sdot= k rk k r (See 3128 C24)

( ) ( )u u= +k kr r R (See 3129 C25)

Free Electron Density of States and Fermi Energy (See Sect 322)

EmVEg23

222

2)( ⎟⎟

⎠

⎞⎜⎜⎝

⎛=π

(See 3164)

g(E) = 2N(E) includes spin

32

22

32

)0( ⎟⎠⎞

⎜⎝⎛==

VN

mTEF π (Integrate 3164)

Electrical Conductivity (See Sect 322)

m

ne τσ2

= (See 3214)

where n is the number of electrons per volume

Hall Constant (See Sect 615 1271)

0cgs)in 1 or(1 gtminusminus= enecne

RH (See 6102 1223)

680 Appendices

Wiedemann Franz Law (See Sect 322)

LTK =σ

(See 3215)

22

3⎟⎠⎞

⎜⎝⎛=

ekL π (See 3215)

Nearly Free Electrons (See Sect 323 with different notation used)

0

00 near band edge

E E U

U E E

lowast

+

minus=

minusk G

G k G (See 3230)

Tight Binding Approximation (See Sect 323) ΔU = difference between poten-tial of crystal and potential of isolated atom

i( ) ( )e ja jjψ φ sdot= minussum k R

k r r R (See 3244)

i0 j (nn) e jE E α γ sdot= minus minus sum k R

k (See 3255)

( ) ( )a adV Uα φ φlowast= minus Δint r r (See 3253)

( ) ( )a adV Uγ φ φlowast= minus minus Δint r R r (See 3254)

Wannier Functions (See Sect 324)

i

Bloch Fns

1( ) e ( )jw j Nφ ψminus sdotminus = sum k R

kkr R r (See 3332)

Pseudopotential (See Sect 323) sum minusminus= c cccpseudo EEVV ψψ)( (See 3287 3288)

Cyclotron Frequency (See Sect 322)

cgs)in or (mceB

meB

c =ω (See 3196)

Low Temperature Specific Heat (See Sect 322 and Table 25)

electronsphonons3v +=+= TATC γ

Hartree Equation (See Sect 312)

22

2 2

0 ( )

( )

( ) ( ) ( )2

1( ) ( ) ( )4

el

ioni i

j i i ij i

V

Vm

e d E

ψ ψ

ψ ψ ψπε ne

minus nabla +

⎡ ⎤prime prime+ =⎢ ⎥

primeminus⎢ ⎥⎣ ⎦sum int

r

r r r

r r r rr r

(Equivalent to (315))


Hartree-Fock Equation (See Sect 313)

22

2

0 ( )|| spin

( ) ( ) ( ) ( ) ( )2

1 ( ) ( ) ( ) ( )4

ion eli i i

j i j i ij i

V Vme d E

ψ ψ ψ

ψ ψ ψ ψπε

lowast

ne

minus + +

prime prime primeminus =primeminussum int

r r r r r

r r r r rr r

nabla


Density Functional Theory (See Sect 315) Kohn-Sham Equationsmdashwith ex-change potential derived from jellium

1 32

2 3( ) ( ) ( ) ( ) ( ) ( ) ( )2

ion eli i i i iV V n E

mψ ψ ψ ψ

π⎛ ⎞⎡ ⎤minus + + minus =⎜ ⎟⎣ ⎦ ⎝ ⎠

r r r r r r rnabla

(Implied by (398) with use of (3115) (3116) See especially Marder [334 p 219 (980)]

Lindhard Equation and Screening Length (See Sect 953)

2

2

1 ( ) ( )( ) 1( ) ( )k

e f fV E E iq σ

ε ωω ηε

minus += ++ minus minus +sum

k k qqk q k

(See 9166)

xx

xxxF

kqxxF

qk

qTF

sminus+minus+==+==

11ln

41

21)(

2)(1)0(0

2

2

2ε

For a static screened charge Q

F

srk

F Eemk

rkq s

εεπφ

23e

41 2

02 ==ltlt minus

4 Interactions and Transport

Boltzmann Differential Equation (See Sect 45) 2(2π)3 is the number of states (including spin) in drdk Let f(rkt)dkdr4π3 be the number of electrons in drdk (in equilibrium f = f0 = Fermi function)

collisions

f ff ft t

part part ⎞+ sdot + sdot = ⎟part part ⎠r k

Fv nabla nabla

Relaxation time approximation With simplifying assumptions

0RHS of BDE( )

f fτminus

= minusk

(See 4149 4145)

682 Appendices

Electrical current and Heat flux (one band) (See Sects 452 46)

11 123 v( ) ( ) ( )4

edj e f L E L T

eμ

π⎛ ⎞= minus = + + minus⎜ ⎟⎝ ⎠

intk k k nabla nabla

21 223 ( )v( ) ( ) ( )4

Qdj E f L E L T

eμ

π⎛ ⎞= minus = + + minus⎜ ⎟⎝ ⎠

intk k k k nabla nabla

(See Eqs 4122 4123) See Ashcroft-Mermin p 256 for definitions of Lij

Electrical conductivity σ and thermal conductivity κ (See Sect 46) nablaμe is neg-ligible for metals but not semiconductors We assume metals in the next two Sects

0ej TE

σ = =nabla

0Qe

jj

Tκ = =

nabla

Thermoelectric Power Q and Peltier Effect Π (See Sects 464 462)

0eEQ jT

= =nabla

0Q

e

jT

j

QT

Π = =

Π =

nabla

Mott Transition (metal-insulator transition) (See Sect 44)

constant131 conganc

where nc is the critical electron density and a1 is the Bohr radius When n gt nc electrons are ldquocrowdedrdquo together See Marder [334 p 491] for values

Charge Density Waves (See Sect 561) An electron lattice phenomena At abso-lute zero the deformation amplitude is proportional to

⎟⎟⎠

⎞⎜⎜⎝

⎛minus

VED F )(1exp

where V characterizes the effective electron-electron interaction

Quantum Conductance G (See Sect 1252)

spinfor 2)2()integer(2

heG =

e2h is called the quantum conductance


Bloch Metallic Resistivity (See Sect 453)

D

D

TT

TT

θρ

θρ

ltlt

gtgt5~

~ (See 4146)

May have ρ ~ constant at T ltlt θD due to impurities and there may be other effects

5 Metals (Na Cu Au Mg etc) Alloys and the Fermi Surface

Fermi Surface (See Sect 51) For the nth band with energy En(k) the locus of points such that En(k) = EF (See Table 51)

deHaas van Alphen Effect (See Sect 65) Neglecting spin the number of states per Landau level per area = eBh Interval of susceptibility oscillations

)SI(21

0Ae

Bπ=⎟

⎠⎞

⎜⎝⎛Δ (See 534)

where A0 is extremal area of Fermi surface

Plasma Frequency (See Sect 109) No radiation propagates for frequency below

⎟⎟⎠

⎞⎜⎜⎝

⎛= (cgs) 4 2

0

2

mne

mne

pπ

εω (See 10108)

Hume-Rothery Rules (See Sect 512) (Roughly) when inscribed Fermi sphere makes contact with the Brillouin zone boundary a new phase appears

Kohn Anomalies (See Sects 44 and 953) The Lindhard dielectric constant sin-gularity at q = 2kF introduces (via the screened ion-ion interaction) a kink or infin-ity in partωpartq in the phonon spectrum at values of q corresponding to a diameter of the Fermi surface

6 Semiconductors (Si Ge InSb GaAs etc)

Five Equations for doped semiconductors in equilibrium

1 Charge neutrality 2 Number of electrons in conduction band 3 Number of holes in valence band 4 Number of electrons on donor ions 5 Number of holes on acceptor ions

Law of Mass Action in Equilibrium (See Sect 611)

2innp = (See 613)

⎟⎟⎠

⎞⎜⎜⎝

⎛minus⎟

⎟⎠

⎞⎜⎜⎝

⎛= lowastlowast

kTE

mmkTn ghei 2

exp)(2

2 4323

2π (See 614)

684 Appendices

External Force and k (See Sect 612)

ddt

= kF (See 644)

Group Velocity (See Sect 612)

1 ( )g E= kv knabla (See 629)

Effective Mass (See Sect 612)

2

2

211

kE

m partpart=lowast (See 649)

Einstein Relation (See Sect 614)

eDkT =μ (See 684 685)

Current due to drift and diffusion (See Sect 614)

e e ej ne eD nμ= +E nabla (See 682)

h h hj pe eD pμ= minusE nabla (See 683)

Schottky Barrier (See Sect 635)

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛ ΔΔ== (cgs)

2

222 ne

V

ne

VKZwidth b π

εε (See 6193)

Shockley Diode Equation (See Sect 638)

⎥⎦

⎤⎢⎣

⎡minus⎟

⎠⎞

⎜⎝⎛

⎟⎟⎠

⎞⎜⎜⎝

⎛+= 1exp

00 kTep

LD

nLDej b

np

pp

n

n ϕ (See 6241)

Mobility (See Sect 614)

acoustic phonon scattering

23minusprop Tμ

ionized impurity scattering

23Tpropμ

Depending on the situation there may be several other scattering mechanisms to con-sider With GaAs for example optical phonon scattering may need consideration


7 Magnetism (Fe Ni Co EuS Y3Fe5O12 Gd etc are ferromagnets)

Larmor frequency (See Sect 741)

⎟⎠⎞

⎜⎝⎛ )(

22cgs

mceB

meB (See 7255 7256)

Lande g factor

⎥⎦

⎤⎢⎣

⎡+

+minus++++=)1(

)1()1()1(211

JJLLSSJJg

(Implied by (76) (710))

Pauli Paramagnetism and Landau Diamagnetism (See Sects 712 322 for Pauli and 711 322 for Landau)

)(20 FBp EDμμχ = (See 3181 3201)

pdia χχ31minus= (See 3201)

magnetonBohr =Bμ

Van Vleck Paramagnetism (See footnote 2 of Chap 7)

)groundexcited(22

==minus

= gexcEEgexc

VN

gexc

zμχ

Brillouin Function (See Sect 712)

⎟⎠⎞

⎜⎝⎛minus⎥⎦

⎤⎢⎣⎡ ++=

Jx

JJxJ

JJxBJ 2

coth21

2)12(coth

212)( (See 716)

Heisenberg Hamiltonian (See Sect 721) ij i ji j J= minus sdotsum S SH (See 788)

RKKY Interaction (See Sect 721)

4sincos)()2(

xxxxxFRkFJ ijFij

minus=prop (See 7107-7110)

Weiss Mean Field Theory (See Sect 713) )(xBNgJM JBμ= (See 721)

kT

BgJx eff

Bμ= (See 722)

MBBeff λ+=

686 Appendices

Stoner Criterion for Band ferromagnetism (See Sect 724)

)cgs(

32

0MMH

kEk effB

F

μθθ =primegt

prime (See 7222)

Kondo Temperature (See Sect 752)

⎟⎟⎠

⎞⎜⎜⎝

⎛minusprop

)(1exp

FK EJD

T (See 7300)

Quantum Hall Effect (See Sect 1273) This is an effect of magnetic field on high density electrons in two dimensions The Hall conductance is

h

eGxy

2ν= (See 1247)

where ν is integer or fraction

Bloch Wall Width δ (See Sect 731)

KJpropδ (See 7251)

where K is anisotropy and J is exchange

Spin Wave Theory Ferromagnetic and Antiferromagnetic Dispersion Long waves (See Sect 723)

2 (cubic) F AFk aω ωprop prop k (See 7191)

Low T (Ferro) (See Sect 723)

23

23)()0(

TC

TTMM

M prop

propminus (See 7196)

Critical Exponents (See Sect 725 and Table 73)

)for (1cc

c

c

TTHTTM

TT

TTC

=asympminusasymp

minusasymp

minusasympminus

minus

δβ

γ

α

χ

α β γ α + 2β +γ Mean Field 0 12 1 2 2D Ising 0 18 74 2 3D Ising ~011 ~032 ~124 ~2 Experiment ~0 ~13 ~43 ~2

For a summary of useful group theory results see Sect 121 and p 445ff


8 Superconductivity all cgs (Pb Hg Nb3Ge HTS etc)

London Equation (in London Gauge) (See Sect 82)

2

( ) ( ) (cgs)nemc

= minusJ r A r (See 85)

London Penetration Depth (See Sect 82)

2

2

4 nemc

L πλ = (See 86)

Intrinsic Coherence Length (See Sect 821)

g

FEv

πξ 2~0 (See 849)

Type I and II (See Sect 823) ξ ~ (ξ0l)12 l = mfp of electrons in normal state (See (851))

I ξ gt λ πξλϕ0~cH

II ξ lt λ 20~

1 πλϕ

cH 20~

2 πξϕ

cH 2

~1

2⎟⎟⎠

⎞⎜⎜⎝

⎛ξλ

c

c

HH

Note ξλ defines Type I and II Hc1 and Hc2 are upper and lower critical fields for Type II

Fluxoid Quanta (See Sect 822)

)(20 cgs

ehc=φ (See 845)

BCS Transition Temperature

(a) Weak Coupling (D(EF)V ltlt 1 V = electron-phonon coupling) (See Sect 853)

⎟⎟⎠

⎞⎜⎜⎝

⎛minuscong

)(1exp141

FDc EVD

T θ (See 8217)

(b) Strong Coupling (See Sect 854)

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛

+minus

+minuscong lowast )6201(

1exp

451 epep

epDcT

λμλ

λθ (See first Eq in Sect)

λep is the coupling constant μ is the effective Coulomb repulsion

688 Appendices

GLAG Equation (See Sect 82) |ψ|2 = concentration of superconducting elec-trons = ns

2

20

1 0 ( are related to )2 Lq

m i cα β ψ ψ α β λ ξ

⎡ ⎤⎛ ⎞⎢ minus + + ⎥ =⎜ ⎟⎝ ⎠⎢ ⎥⎣ ⎦

Anabla (See 83)

2

( ) ( 2 2 )2s eiq qj q e m m

m mcψ ψ ψ ψ ψ ψlowast lowast lowast= minus minus minus = minus rarrAnabla nabla (See 84)

Isotope effect (See Sect 853)

αminusprop MTc (for weak coupling α ~ 12)

9 Dielectrics and Ferroelectrics (KH2PO4 BaTiO3)

LST Equation (See Sect 932)

)()0(

2

2

infin=εε

ωω

T

L (See 950)

Lorentz Local Field (See Sect 92)

⎟⎟⎠

⎞⎜⎜⎝

⎛+=+= )SI(

3)guassian(

34

0επ PEPEEloc (See 924)

Clausius-Mossotti Equation (See Sect 92)

⎟⎟⎠

⎞⎜⎜⎝

⎛=

+minus=

+minus

sumsum )SI(31

2)(1)(

34

21

00

0jjjj NN α

εεεεεαπ

εε (See 930)

10 Optical Properties

Direct and Indirect Absorption (See Sect 104)

direct 0eΔ =k

indirect 0e phqΔ + Δ =k

Optical Absorption Coefficient (See Sect 104) Near band edge for allowed direct transitions in parabolic bands

gEhffA minus=α (See 1083)

Frenkel Excitons (tightly bound) (See Sect 107)

)nn(cos2 kaVk += εε (See below 1099)


Mott-Wannier Excitons (weakly bound) (See Sect 107)

)cgs(2 222

4

neEE gn ε

μminus= (See 10100)

)cgs(111

he mm+=

μ (See 10101)

Franck-Condon Effect Roughly speaking absorption of light occurs as if the lattice is frozen

11 Defects

Schottky and Frenkel Defects (See Sect 111) Schottky produces lattice vacancy and surface ion Frenkel produces a lattice vacancy plus interstitial

Mollwo Equation

2bandcenter F ofmax apropλ

a = lattice constant (See Pb 111 ΔE prop andash2 prop λndash1)

Burgerrsquos Vectors (See Sect 116) The jog in a complete (if undistorted) circuit of the dislocation caused by a dislocation

12 Nanophysics Surfaces Interfaces and Amorphous Materials

Surface Reconstruction (See Sect 121) Expressed in 2D for surface with new lattice vectors expressed in terms of undistorted vectors

Bloch Oscillations (See Sect 126)

⎟⎟⎠

⎞⎜⎜⎝

⎛==

pGeEp

Bπω 2 (See 129)

Variable Range Hopping (See Sect 1291)

)(exp 41 MottT

AB ⎟⎠

⎞⎜⎝

⎛minus=σ (See 1255)

for localized states near Fermi energy and at low temperature where σ is electrical conductivity

Richardson-Dushman Equation Thermionic emission a property of surfaces (different notation in text)

function work exp)(2

232surface =⎟

⎠⎞

⎜⎝⎛minusminus= W

kTWkTemj

π (See 1145)

690 Appendices

Fowler Nordheim Equation Field emission a property of surfaces (different nota-tion in text)

3 2

2surface exp BWj AE

E

⎛ ⎞= minus⎜ ⎟⎜ ⎟

⎝ ⎠ (See 1149)

Chapter 12 is substantially related to modern areas of solid-state physics and condensed matter physics There are so many things to mention and review that they obviously cannot all be done here The general area of nanophysics is much in the news now as are the more specific areas of quantum dots wires wells (125) superlattices (113 126) spintronics (which utilizes the spin rather than just the charge of the electron to operate the device (751)) quantum computers (751) entanglement and cryptoanalysis Everyone seems to have a TV with ei-ther LCD (12112) or Plasma displays The latest Nobel prize in physics was for GMR (giant magneto resistance) which is used in iPods The use of GMR is also regarded as the birth of spintronics Nowadays LEDs (where a degenerate pn junc-tion (633 638) under forward bias emits light) are becoming very popular even for Christmas tree lights One could go on and on Solid State Condensed Matter remains a very vibrant area

J Folk Theorems

These are approximate facts which should be easy to remember and are grouped by chapter

Chapter 1

1 X-rays with wavelength comparable to a lattice spacing are diffracted and the diffraction peaks can be used to determine the crystal structure

2 Crystals which have translational symmetry can only have 1 2 3 4 and 6 fold rotational symmetry So why the heck can diffraction patterns show 5-fold symmetry

Chapter 2

3 Phonons carry no momentum

4 Lattice Specific Heat goes as T cubed at low temperaturesmdashin general this just depends on the density of states going as omega squared The Debye model which uses this density of states with a cutoff is correct at both high and low temperatures but not in between

Folk Theorems 691

Chapter 3

5 Band Gaps occur because of Bragg scattering of wave-like electrons

6 Free electron theory often works wellmdashdonrsquot ask why unless you really want to know

7 Using quantum mechanics the electron specific heat is predicted to go line-arly to zero with temperature

8 With quantum mechanics all free electrons contribute to conduction but not to specific heat

9 For extremely narrow bands Bloch functions can be constructed from linear combinations of Atomic functions

10 With band theory we understand the origin of metals insulators and semi-conductors

11 Even though electrons are the only conductors the Hall effect can be posi-tive (see band theory and consider holes)

12 The more a wave function wiggles the higher the kinetic energy

Chapter 4

13 Without Umklapp lattice thermal conductivity would be infinite

14 Anharmonic terms give rise to thermal expansion

15 Quasi particles (dressed particles) and collective energy excitations are a large chunk of solid-state physics

16 Lattice vibrations cause the electrical resistivity to go to zero with tempera-ture but in fact the resistivity usually just goes to a constant value due to imperfections

17 Blochrsquos Theorem then lattice vibrations then electron-phonon and other interactions make resistivity or the lack of it complex to analyze

18 Localization is affected by order and interactions

Chapter 5

19 The free electron Fermi surfaces when mapped into the first Brillouin zone explain a lot

Chapter 6

20 Non-degenerate semiconductors have electrons in the conduction band and holes in the valence band These may be in the Fermi tail and hence behave classically

21 The resistivity of a semiconductor may decrease with increasing temperature

22 For degenerate semiconductors the Fermi energy is in an energy band

692 Appendices

23 Charged impurity scattering is important at lower temperatures and phonon scattering is important at higher temperatures

24 Electrons and holes during their lifetime move a diffusion length before they recombine

25 Recombination centers are most effective when they are near the middle of the band gap

26 An electric field causes energy bands to bend Making this idea rigorous is not trivial

27 Direct band gap semiconductors are much better for LEDs and LASERs than indirect ones Direct band gap materials absorb light better

28 The more abrased the surface is the higher the recombination velocity A large number of surface states can pin the Fermi energy

29 FETs would have been invented long ago if surface states had been under-stood

30 Surface states can have important effects on whether a metal semiconductor contact is ohmic or rectifying

31 BJTs are current controlled

32 FETs are voltage driven

33 The Fermi energy is spatially constant in equilibrium

34 Flat is fatmdashfor effective masses This is true both for energy and momentum effective masses

35 LEDs may someday replace ordinary light bulbs

36 Who would have thought the Hall effect would be so important There is now a spin Hall effect Also (Chap 12) there are two kinds of quantum Hall effect (the integer and the fractional)

Chapter 7

37 Pauli ideas were fundamental for ferromagnetism since his exclusion prin-ciple eventually led to the exchange interaction

38 It takes quantum mechanics to produce magnetism This is the Bohr-Miss JH von Leuwen Theorem Classically paramagnetism and diamagnetism would cancel

39 The mean field theory of magnetism ignores fluctuations and hence does not properly treat critical point phenomena

40 Spin wave theory correctly predicts the magnetization of magnets at low temperatures

41 Demagnetizing fields drive domain formation

Folk Theorems 693

42 For phase transitions you need to know your LCDs and UCDs

43 Broken Symmetry produces Goldstone Bosons eg phonons and magnons

44 Spintronics (controlling spin transport of spin polarized electrons) is now studied in both metals and semiconductors

45 GMR made ipods possible

Chapter 8

46 If it costs too much energy to scatter a carrier it is not scatteredmdashhence su-perconductivity

47 Pauli is still partly rightmdashall theories of superconductivity are wrong (or at least donrsquot completely explain high temperature superconductivity)

48 Cuprates are not the only high temperature superconductors (HTS) HTS has also been found in iron pnictides What is going on

Chapter 9

49 Soft phonon modes are associated with the many ferroelectric transitions

50 Electron screening is important when considering Coulomb interactions

Chapter 10

51 Optical experiments can yield many details about band structure

52 In simple metals total reflection occurs below the plasma edge and this de-termines the electron density

53 ldquoInvisibility cloaksrdquo are being developed so we can ldquocloud menrsquos minds so they cannot seerdquo

54 Optical lattices can be investigated to broaden our understanding of solids

Chapter 11

55 A perfect crystal at any finite temperature is an oxymoron

56 One manrsquos defect is another manrsquos jewel

57 Dislocations cause crystals to be weaker than would be expected They aid plastic deformation

58 Crystals may show ldquowork hardeningrdquo

59 Defects can be shallow or deep

60 The N-V color center may be important for spintronic devices

Chapter 12

61 People in designer jeans are now inventing designer materials

62 Yes Virginia you can see an atom scanning tunneling microscopy (STM)

694 Appendices

63 Graphene has ldquomassless fermionsrdquo

64 Electrons can go ballistic (when there is little scattering)

65 There are ordinary lattices and also superlattices

66 Condensed matter can go ldquosoftrdquo

67 A quantum dot can be thought of as an artificial atom

K Handy Mathematical Results

Gauss law V A A

dV disin

sdot = sdotint intB B Anabla

Stokes law ( ) (for bounding )A L

d d Ltimes sdot = sdotint intV A V l Anabla

2( ) ( ) Ftimes times = sdot minusnablaF Fnabla nabla nabla nabla

10

e +

infinminus =int n

axn

andxx

Normalized Hydrogen ground state

π41Y

e21)(

Y)(

00

23

10

0010001

=

⎟⎟⎠

⎞⎜⎜⎝

⎛=

=Ψ

minus Bar

BarR

rR

Spherical Coordinates

ˆˆ ˆsind dr rd r dθ θ ϕ= + +r r θ ϕ

1 1ˆˆ ˆsinr r rθ θ ϕ

partΦ partΦ partΦΦ = + +part part part

r θnabla ϕ

221 1 1( ) (sin )

sin sinrA

r A Ar r rr

ϕθθ

θ θ θ ϕpartpart partsdot = + +

part part partAnabla

2

2

2222

2 sin1sin

sin11

ϕθθθ

θθ partΦpart+⎟

⎠⎞

⎜⎝⎛

partΦpart

partpart+⎟

⎠⎞

⎜⎝⎛

partΦpart

partpart=Φnabla

rrrr

rr2

Handy Mathematical Results 695

Geometric Progression

111

0 minusminus=sum minus

=primeprime

γγγ

nnn

n

Stirlingrsquos Approximation

n

ennn ⎟⎠⎞

⎜⎝⎛cong π2 (for large n)

Dirac Delta function

at zeroes simpleonly has )( if

)()(

1))((

)()()(0)(

)()( includesregion if1)(

n

n nn

xxxf

xxxf

xf

bfdxbxxfxx

xxbxdxbx

=

minusprime

=

primeminus=minusprime=

minus===minus

sum

int

int

δδ

δδ

δδδ

Complex Variables

sumint = ) curve closed inside )((Res2)( CzfidzzfC

π

f (z) analytic inside C except at singular points where residue (Res) is to be computed

nazfazdzd

naf n

n

n

azorder of at polefor )]()[(lim

)1(1)(Res 1

1

⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧

minusminus

= minus

minus

rarr

Maxwellrsquos Equations

SI Gaussian

0

t

tρ

parttimes = +part

parttimes = minuspart

sdot =sdot =

H J D

BE

DB

nabla

nabla

nablanabla

4 1

1

40

c c t

c t

π

πρ

parttimes = +part


sdot =sdot =

H J D

BE

DB

nabla

nabla

nablanabla

0ε= +D E P

0

1μ

= minusH B M

( )q= + timesF E v B

4π= +D E P 4π= minusH B M

( )qc

= + timesvF E B

696 Appendices

L Condensed Matter Nobel Prize Winners (in Physics or Chemistry)

Name Year Nobel Prize For

W Rontgen 1901 x-rays H A Lorentz P Zeeman

1902 Effect of magnetic fields on radiation

3 Total for Decade J D van der Waals 1910 Equation of State H K Onnes 1913 Liquid Helium Superconductivity Max von Laue 1914 Crystal x-ray diffraction W H Bragg W L Bragg (Son)

1915 Crystal Structurex-rays

5 Total for Decade C E Guillaume 1920 Anomalies in Nickel and Steel Alloys A Einstein 1921 Photoelectric Effect R A Millikan 1923 Charge of ePhotoelectric Effect A H Compton 1927 Compton Effect O W Richardson 1928 Thermionic Phenomena 5 Total for Decade C V Raman 1930 Raman Effect I Langmuir 1932 Surface Chemistry P Debye 1936 Molecular Structure dipole moments x-rays C J Davisson G P Thomson

1937 Diffraction of electrons

5 Total for Decade I I Rabi 1944 Magnetic Resonance of Nuclei P W Bridgman 1946 High Pressure Physics 2 Total for Decade Felix Bloch E M Purcell

1952 Nuclear Magnetic Resonance

W Shockley John Bardeen W H Brattain

1956 Transistor

Condensed Matter Nobel Prize Winners (in Physics or Chemistry) 697


P A Cerenkov I M Frank I E Tamm

1958 Cerenkov Effect

8 Total for Decade R L Mossbauer 1961 Mossbauer Effect L D Landau 1962 Liquid Helium L Onsager 1968 Thermodynamics of Irreversible Processes 3 Total for Decade Louis Neacuteel 1970 FerrimagnetismAntiferromagnetism John Bardeen Leon Cooper J Robert Schrieffer

1972 Theory of Superconductivity

Leo Esaki Ivar Giaver B D Josephson

1973 Tunneling in a Superconductor

P W Anderson N F Mott J H Van Vleck

1977 MagnetismDisorder in Materials

P L Kapitza 1978 Low Temperature Physics 11 Total for Decade Nick Bloembergen 1981 LasersEtc Kai Siegbahn 1981 High Resolution Electron Spectroscopy

(particularly ESCA ndash Electron Spectroscopy for Chemical Analysis)

K G Wilson 1982 Renormalization Group Theory Critical Phenomena (Magnetism)

Klaus von Klitzing 1985 Quantized Hall Resistivity Ernst Ruska Gerd Binnig H Rohrer

1986 Scanning Tunneling Microscopy

Karl A Muller J G Bednorz

1987 High Temperature Superconductivity

9 Total for Decade

698 Appendices


Pierre de Gennes 1991 Liquid Crystals Polymers B Brockhouse Cliff Shull

1994 Diffraction Scattering of Neutrons (Magnetism)

David M Lee D D Osheroff Robert C Richardson

1996 Superfluidity in helium-3

Steve Chu Claude Tannoudji William D Phillips

1997 Methods to cool and trap atoms

John Pople 1998 Quantum Chemical Calculations Walter Kohn 1998 Density Functional Horst Stoumlrmer D Tsui R Laughlin

1998 Half Integer Quantum Hall Effect

14 Total for Decade Z I Alferov 2000 Heterostructures etc Herbert Kroemer 2000 Heterostructures etc Jack Kilby 2000 Integrated Circuits Alan J Heeger Alan G MacDiarmid Hideki Shirakawa

2000 Conducting Plastics

E A Cornell W Ketterle C E Wieman

2001 Bose-Einstein condensation in dilute gases of alkali atoms

A A Abrikosov V L Ginzburg A J Leggett

2003 Theory of superconductors and superfluids

R J Glauber 2005 Quantum theory of optical coherence J L Hall T W Haumlnsch

2005 Laser-based precision spectroscopy

A Fort P Gruumlnberg

2007 Giant Magnetoresistance (GMR)

G Ertl 2007 Chemical processes on solid surfaces C K Kao 2009 Optical fiber communication W S Boyle G E Smith

2009 Charge Coupled Devices

21 (so far) Total for Decade



A Geim K Novoselov

2010 Graphene

The listing of Nobel Laureates suggests even further ways to think about people names and what they did in condensed matter physics One can even play what might be called high school games as in ldquoThe Three Brsquos of Solid State Physicsrdquo as listed below

Every one has heard of the Three Brsquos of Music Bach Brahms and Beethoven But how many have heard of Three Brsquos of Solid State Physics who developed the basis for a large part of our modem electronics industry

Brillouin

Zones ndash These are fundamental for discussing wave like motion in periodic struc-tures and hence for electron motion lattice vibrations and other energy excita-tions in solids

Function ndash This describes paramagnetism as a function of temperature and is used in the mean field theory determines the magnetization below the Curie tempera-ture

Scattering ndash This is scattering of light from acoustic modes in crystals

Bloch

Theorem ndash This is the fundamental theorem in which the effect of lattice periodic-ity is taken into account in writing down a special form for the wave function in solids

Equations ndash These describe the magnetic resonance behavior of the magnetization components in solids

T32 Law ndash This uses spin wave theory to take into account the behavior of mag-netism at low temperatures in ferromagnets

Wall ndash This describes how the magnetism can vary between domains

Bloch also gave the fundamental calculation of the temperature dependence of resistivity in metals due to scattering of electrons by lattice vibrations

Bardeen

He was the only person to win two Nobel prizes in Physics One was with Brattain and Shockley for the development of the transistor ndash the fundamental component of all modern electronic systems

The other was with Schrieffer and Cooper for the development of the theory of superconductivity

700 Appendices

Others

Other Brsquos that could be considered are Bravais (lattice) Bragg WH and WL (equation for x-ray diffraction) and Bridgman (high pressure techniques)

Srsquos that could be considered are Shockley (transistor) Seitz (ldquoMr Solid State Physicsrdquo) Slater (determinant numerical techniques) Schottky (barrier) Stoner (magnetism) and Schrieffer (superconductivity)

Wrsquos that could be considered are Wannier (functions) Weiss (mean field theory) and Wigner (Wigner-Seitz cell group theory) Wilson (Renormalization Group and Critical Phenomena)

Vrsquos include Van der Waals (equation forces) Van Vleck (magnetism) and von Laue (x-ray diffraction)

There is no end to this type of game One can play with any letter of the alphabet and usually find several prominent condensed matter physicists who have the first letter of their last name starting with this letter Such games may be useful when trying to remember ideas on starting a subject Since they have little use for re-search we stop here

M Problem Solutions

M1 Chapter 1 Solutions

Problem (13) Solution

Consider for example NaCl Place the origin on a Na+ cation Suppose the distance to the nearest Clminus anions is d We have 6 Clminus ions at a distance d 12 Na+ ions at a distance 2 d and 8 Clminus ions at 3 d and so on

Evjenrsquos idea was to divide space starting with the Na+ ion into neutral cubic volumes thus making the convergence relatively rapid This requires that we di-vide up the charges Thus for an ion on the face of a cube one half of its charge is inside the cube and one half is outside In a somewhat similar way we divide charges on the edges and corners of the cubes For the first cube we get the fol-lowing series for the Madelung constant M

451388

2412

126

corners)on (ionsedges)on (ionsfaces)on (ions

=+minus=M

If we include the second cube the series gives M = 175 A more precise value of M is M = 174756hellip

Problem Solutions 701


1

A

2

3

456

7

8

g5 g6

g7

g8

The group D4 is of order 8 with 8 distinct elements Let g1 g2 g3 g4 be the clock-wise rotations of 0 90deg 180deg 270deg about an axis through A and perpendicular to the page The other four elements (g5 g6 g7 g8) are rotations of 180deg about the axes as marked by dashed lines in the figure Using the notation of the book we list the group elements

⎟⎠⎞

⎜⎝⎛= 87654321

876543211g

⎟⎠⎞

⎜⎝⎛= 21876543

876543212g

⎟⎠⎞

⎜⎝⎛= 43218765

876543213g

⎟⎠⎞

⎜⎝⎛= 65432187

876543214g

⎟⎠⎞

⎜⎝⎛= 23456781

876543215g

⎟⎠⎞

⎜⎝⎛= 45678123

876543216g

⎟⎠⎞

⎜⎝⎛= 67812345

876543217g

⎟⎠⎞

⎜⎝⎛= 81234567

876543218g

702 Appendices

Then eg

432 6543218787654321 ggg =⎟⎠⎞

⎜⎝⎛=

Note that g1 is the identity so the first line in the table is trivial The rest of the elements can be worked out as we did D3 in the text As an example we worked out g2g3 (above) from the group elements

g1 g2 g3 g4 g5 g6 g7 g8

g1 g1 g2 g3 g4 g5 g6 g7 g8 g2 g2 g2g2=g3 g2g3=g4 g2g4=g1 g2g5=g8 g2g6=g5 g2g7=g6 g2g8=g7 g3 g3 g3g2=g4 g3g3=g1 g3g4=g2 g3g5=g7 g3g6=g8 g3g7=g5 g3g8=g6 g4 g4 g4g2=g1 g4g3=g2 g4g4=g3 g4g5=g6 g4g6=g7 g4g7=g8 g4g8=g5 g5 g5 g5g2=g6 g5g3=g7 g5g4=g8 g5g5=g1 g5g6=g2 g5g7=g3 g5g8=g4 g6 g6 g6g2=g7 g6g3=g8 g6g4=g5 g6g5=g4 g6g6=g1 g6g7=g2 g6g8=g3 g7 g7 g7g2=g8 g7g3=g5 g7g4=g6 g7g5=g3 g7g6=g4 g7g7=g1 g7g8=g2 g8 g8 g8g2=g5 g8g3=g6 g8g4=g7 g8g5=g2 g8g6=g3 g8g7=g4 g8g8=g1


Use primitive unit cell with a1 a2 a3 primitive translation vectors

)(21 kjia minus+= a

)(22 kjia ++minus= a

)(23 kjia +minus= a

x

y

z

a = length of side of conventional unit cell (cubic)

a) Planes could be perpendicular to a1 a2 a3 eg a plane perpendicular to a1 would have intercept (1 infin infin) and reciprocals (1 0 0) and have Miller indices [1 0 0]


b) eg a1 a2 a3 are neither parallel nor perpendicular and [1 0 0] are the Miller indices of a1

The bcc lattice and primitive translation vectors as well as primitive and conven-tional unit cells are clarified in the drawing

y

z

x

a2

a1

a3

a

a) Primitive translation vectors of the bcc lattice

a

b) Conventional and primitive cells of bcc lattice

704 Appendices


If we draw all (hkl) planes they pass through all lattice points If the direct lattice is specified by the primitive vectors a b c then let the intercepts of the planes relative to an origin at a lattice point be denoted by

h k lprime prime primea b c

If (hkl) are the smallest set of numbers with the same ratio (ie Miller indices) then ah bk cl connect adjacent planes and G(hkl) is shortest Orthogonality follows from

2 2 0h kh k h k

π π⎛ ⎞sdot minus = minus =⎜ ⎟⎝ ⎠

a bG

since

1 2 32 ( )h k lπ= + +G b b b

and

1 2 3( )i j ijδ= = = =a b a a a b a csdot

Thus

( ) 2( )( ) ( )h k ld h k lh k l h h k l

π= =G aG G

sdot


A density function that describes the positions of the atoms is given by

ˆ ˆ( ) ( ) ( )n mx nb mbρ δ δ α= minus + minussum sumr i r i

From (139) the scattering amplitude is

i( )eP dVρ minus Δ sdot= int k rr

where

f iΔ = minusk k k

We assume as usual elastic scattering so |kf| = |ki| = k Substituting the first equa-tion in the second and for the geometry shown α = 23


ki

b

b

bα

bα

hellip

x

y

z

kf

0ˆ =sdot ik i

2sin

2sin

e

2sin

2sin

e

e1

e1

e1

e1

ee

2)1(

i2

)1(i

i

i

i

i

10

i10

i

⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜

⎝

⎛

+⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜

⎝

⎛

=

minus

minus+minus

minus=

+=

minusminus

minusminus

minus

minus

minus

minus

minus=

minusminus=

minus sumsum

α

αα

α

α

α

bk

Mbk

bk

bNk

P

fx

fxMbk

fx

fxNbk

bk

Mbk

bk

bNk

Mm

mbkNn

nbk

fxfx

fx

fx

fx

fx

fxfx

The scattering intensity is proportional to |P|2 We assume a limit in which N M rarr very large Following the usual arguments found in optics for interference from many slits we know the principal maxima occur for

)(integer 2

and)(integer 2

)2()1(

pbk

lbk fxfx times=times= π

απ (a)

Since

pl

bk

bk

pl

bk

bk

fx

fx

fx

fx αα

== )2(

)1(

)2(

)1(

or1

)2()1(fxfx kk ne

since α is irrational we need not be concerned with cross terms in the square |P|2 and the maxima of kfx

(1) and kfx(2) are different If we look only in the z = 0 plane

since

22fxfy kkk minus=

then kfy is also determined We thus get sharp Bragg peaks as determined by (a) Similar comments can be made for other observation geometries

706 Appendices


Let

λπ2=K

so θϕ sinnn Kd= We need to evaluate

sum= NnE 0 )iexp( ϕ

Thus we need to evaluate

sum sdot+= N knbbKnbKE 0 ))]2sin(sinsin(iexp[ πεθθ

Define

θsinKB =

)iexp( BbZ =

So

sum= N knbnZZE 0)2sin( πε

But

orderfirst to)2sin(ln1

)]2sin(ln[

1)]2sin()exp[(ln

0

)2sin(

knbZ

knbZm

knbZZ

m

knb

πε

πε

πεπε

+=

=

=

suminfin=

Thus

sumsum minus+= N nN n knbZZZZE 01

0 )2sin()ln( πε

For Z ne 1 we can perform both sums for N rarr infin and obtain13

2)2cos(21)2sin(ln

11

ZkbZkbZZ

ZE

+minus+

minus=

ππε

Notice if ε rarr 0 we retain only the first term that diverges when Z = 1 that is when Bb = 2πm where m is an integer or sinθ = mλb which is like the familiar multi slit 13 IS Gradshteyn and IM Ryzhik Table of Integrals Series and Products Academic

Press San Diego 1980 p 40 (1447)


interference rule The second term must also be considered if ε ne 0 It blows up when (using G = 2πkb) 01cos22 =+minus GZZ

which can be shown to occur when Z = exp(plusmniG) But since Z = exp(iBb)

mGBb π2+plusmn=

where m is an integer This can be rewritten as

b

kbm λθ ][sin plusmn=

a condition for (small subsidiary) maximum in multi-scatterer interference

Problem (113) Solution 1

Aring2255=Ka

Using Braggs law nλ = 2dsinθ

Aring2255100 == ad

)bcc(6125222255

planes lattice === Kd

deg=rArr== 6816225551

2sin θλθ

d

deg=rArr= 04352255

)51(2sin θθ


Using structure factor and conventional unit cell

θλ sin2 100dn =

above as 2)e1( 0435 4510

)51(42

4sin

happent doesn 0)e1( 5125 4510

)51(32

3sin

above as 2)e1( 6816 4510

)51(22

2sin


2sin

i4400

100

i3300

100

i2200

100

i100

100

ffSd

fSd

ffSd

fSd

=+====

=+====

=+====

=+====

minus

minus

minus

minus

π

π

π

π

θλθ

θλθ

θλθ

θλθ

708 Appendices



Lattice vibrations in a three atom monatomic lattice with periodic boundary conditions

mm m

γ γ γx1 x2

1γx3

3 1 2 3

m is the atomic mass xi is the displacement from equilibrium and γ is the spring constant From Newtonrsquos second law and using periodic boundary conditions we have the equations of motion )()( 31121 xxxxxm minusminusminus= γγ (1a)

)()( 12232 xxxxxm minusminusminus= γγ (1b)

)()( 23313 xxxxxm minusminusminus= γγ (1c)

We seek normal mode solutions of the form xn = uneiωt Substituting and canceling eiωt we have

02

22

321

2

2

2

=⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜

⎝

⎛

minusminusminusminusminusminusminusminusminus

uuu

mm

m

ωγγγγωγγγγωγ

(2)

or in matrix form 0=MU (3) We require 0)det( =M (4) and find the eigenfrequencies 0=ω (5a)

twice32mγω = (5b)

Substituting these back into (2) we find the normalized eigenvectors

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛==

111

31

321

0uuu

Uω (6a)

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛minus==12

1

51

3 mU γω (6b)


and

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛minus==

112

51

3 mU γω (6c)

Since (6b) and (6c) are degenerate any linear combination of them works


)(2iexp1

)(2iexp1

)(2iexp1

)1(0

10

10

sum

sum

sum

minusminus= minus

minus=

minus=

⎥⎦⎤

⎢⎣⎡ minus=

⎥⎦⎤

⎢⎣⎡ minusminus=

⎥⎦⎤

⎢⎣⎡ minus=

Ns s

Ns s

Ns snm

nmN

sgN

nmN

sgN

mnN

sgN

G

π

π

π

But by (2104) gminuss = gs Also s rarr s + N leaves gs and ei2πs(m minus n)N invariant since

[ ]

⎥⎦⎤

⎢⎣⎡ minus=

minus⎥⎦⎤

⎢⎣⎡ minus=

⎥⎦⎤

⎢⎣⎡ minus⎥⎦

⎤⎢⎣⎡ minus=⎥⎦

⎤⎢⎣⎡ minus+

)(2iexp

)(2iexp)(2iexp

)(2iexp)(2iexp))((2iexp

nmsN

nmnmsN

nmNN

nmsN

nmNsN

π

ππ

πππ

and (see 2104)

⎟⎠⎞

⎜⎝⎛=⎟

⎠⎞

⎜⎝⎛ +=⎥⎦

⎤⎢⎣⎡ +

Ns

NsNs

Nππππ 222 sinsin)(sin

)(2iexp1 1sum ⎥⎦⎤

⎢⎣⎡ minus= N snm nm

Nsg

NG π

But

⎟⎟⎠

⎞⎜⎜⎝

⎛⎥⎦⎤

⎢⎣⎡ minus=⎥⎦

⎤⎢⎣⎡ minus= )()0(2iexp)(2iexp0 nm

Nnm

NNggN

ππ

so

mnN

snm GnmN

gG =⎥⎦⎤

⎢⎣⎡ minus=sum minus1

0 )(2iexp π

QED

710 Appendices


A detailed solution of this problem has appeared in the literature See K Kesava-samy and N Krishnamurthy ldquoVariations of the One-Dimensional Defect Latticerdquo American Journal of Physics 47(11) 968-973 (1979) We will therefore only outline a few salient features here

The five equations governing the motion assuming Xi denotes displacement from equilibrium are

)2()()(

)()(

)2()()(

)()(

)2()()(

54145515

34454

32423343

12232

15251121

XXXXXXXXm

XXXXXm

XXXXXXXXM

XXXXXm

XXXXXXXXm

minus+=minusminusminus=

minusminusminus=


minusminusminus=


αααβα

βββαβ

ααα

Seeking solutions of the form Xn = uneminusiωt gives us the equations (A = βα Ω2 = mω2α)

0

210011100020001110012

54321

22

212

2

=

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛

minusΩminusminusΩ

minusΩminusminusΩ

minusΩ

minus

uuuuu

AAAAA

AAδ

or MU = 0 and det M = 0 yields the eigenvalues Using δA = (mM)(βα) equiv C we can also write MU = 0 as

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

Ω=

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

minusminusminus+minus


minusminus

54321

2

54321

210011100

020001110012

uuuuu

uuuuu

AACCC

AA

From this it is easy to see that ui = constant is a solution with Ω = 0 (the pure translation mode) What about modes in which

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

minusminus

=

1

0

1

a

aU


This requires

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛

ΩminusΩminus

sdotΩΩΩ

=

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

+minus++minus

minus+minus

=

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

minus+minus+minus+

+minus++minus+minus

22

2

2

2

0

31)1(

01)1(

3

211))(1(

)1(112

a

a

aAa

Aaa

aaAcaca

Aaa

The conditions resulting from this equation (3 minus a = Ω2 a(1 + A) minus 1 = Ω2a ) give two modes These are sketched here

In a similar fashion the remaining two modes can be found Kesavasamy and Krishnamurthy (KampK) find they are of the form

Note the middle (ldquoimpurityrdquo atom) moves in this mode Caution For the two im-purity modes in which the middle atom does not move we obtain

2

842 22 +minusplusmnminus=Ω AAA

while KampK have a different expression Be on the lookout for misprints in KampK


For simplicity we write aq as a We have shown in the text that [a adagger] = 1 so aadagger = adaggera minus 1 and adaggera = aadagger + 1 Defining N = adaggera suppose Nψn = nψn and let φ = aψn We then have

)1()1()1( dagger

dagger

ϕψψ

ψψϕ

minus=minus=minus=

==

nna

aaa

aaaNaN

n

n

nn

712 Appendices

So φ prop ψn-1 or |n minus 1rang changing notation Thus

nKan =minus1 So

112

dagger2

nnnK

naanKnn

=

=minusminus

Assuming langn|nrang = langn minus 1|n minus 1rang = 1 (normalization) then

1or1 minus== nnnan

K

Similarly if η = adaggerψn

)1()1(

)1(dagger

daggerdagger

daggerdaggerdagger

ηψ

ψ

ψψη

+=+=

+=

==

nna

aaa

aaaNaN

n

n

nn

So nKan dagger1 =+ and normalization gives

1

1+

=n

K

so

11dagger ++= nnna

Since we are dealing with one q we are in effect dealing with one harmonic oscil-lator with Hamiltonian

⎟⎠⎞

⎜⎝⎛ +=⎟

⎠⎞

⎜⎝⎛ +=

21

21dagger Naa ωωH

Since itrsquos eigenvalues cannot be negative and since adaggera is Hermitian there must be a smallest n for which |n minus 1rang does not exist This is clearly n = 0 The rest of the n are positive integers with eigenfunction of N generated by adagger


321321

ηηηηηη dddzyxJdxdydz ⎟⎠⎞

⎜⎝⎛=


where J is the Jacobian But

321

321

321

ηηη

ηηη

ηηη

partpart

partpart

partpart

partpart

partpart

partpart

partpart

partpart

partpart

=

zzz

yyy

xxx

J

1 1 2 2 3 3η η η= + +r a a a

so xxx aaax 332211 ηηη ++= so

etc11

xax =partpartη

so we find

1 2 3

1 2 3 1 2 3

1 2 3

( )x x x

y y y

z z z

a a a

J a a a

a a a

= = sdot timesa a a

QED


i j ijδsdot =a b so

1 1

2 1

3 1

100

sdot =sdot =sdot =

a ba ba b

Clearly b1 = Ka2 times a3 satisfies the second two equations and the first equation says

1 2 31 ( ) aK K= sdot times = Ωa a a so

a

KΩ

= 1

and

2 31

a

times=

Ωa a

b

The other two equations are similarly proved

714 Appendices

Problem (211) Solution The relevant quantities are 1 1 2 2 3 3l l l l= + +R a a a

1 1 2 2 3 32 ( )n n n nπ= + +G b b b

where a1 a2 a3 are the primitive lattice translation vectors li and ni are integers aimiddotbj = δij and the Gn are the reciprocal lattice vectors

fpp 1 1 2 2 3 3( )V N N N= sdot timesa a a

with N1N2N3 unit cells in Vfpp

31 21 2 3

1 2 32

nn nN N N

π⎛ ⎞

= + +⎜ ⎟⎝ ⎠

q b b b

are the wave vectors 1 1 2 2 3 3η η η= + +r a a a

is a general vector in real space

a) 31 231 2 31 2

31 2

1 2 3

1 1

11

1

22 2 ii i 11 1i

1 2 3 1 2 30 0 0

i2

2i1

1 1 1 1e e e e

1 1 e etc

1 e0 for multiple of or1 for multiple of

in general for any or

l

l

n

nn

nn n ll l NN NNN N

l l l

n l

n lN

i i

i i

n

N N N N N N

N

n Nn N

ππ π

π

π

δδ

minusminus minussdot

= = ==

minus=

minus= ne= =

=

=

sum sum sum sum

sum

q R

R

Gq

q GG

G

b) 111

1

1

1

11

21

i(BZ)

1 2 3 1 0

2

21

0

1 1e e etc

where the sum is equivalent to summing over a B Z

1 1 e


l

l

ni lNN

l

i n

i lN

i i

i i

N N N N

N

l Nl N

π

π

π

δ

minussdot

==

minus=

minus= ne= =

=

sum sumq Rq

R


c)

1 2 31

1 1

3

33

1 2 31 2 3

31 2 3 03

fpp 33

( ) 2

of s( )Vol of space

1 ( )(2 ) ( )

( )(2 )

( ) (2 )

n n nn

f fN

qf dq

f d

N N NN N N

f d

Vf d

π

π

π

π

⎛ ⎞⎛ ⎞= +⎜ ⎟⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠

=

=sdot times

Ω=

sum sum

int

int

int

int

q q b

q q

q qb b b

q q

q q

d)

i i3fpp3

fpp

i

1 2 30

e e (with )(2 )

1 e

l l

l

l

a ad q VV

N N N

π

δ

Ω Ω= rarrinfin

=

=

sumint

sum

q R q Rq

q Rq

R

sdot sdot

sdot

e)

1 1 1 2 2 2 3 3 3i( ) i2 [( ) ( ) ( ) ]3 31 1e el l l l l l l luc

a ad dπ η η ηprime prime prime primeminus minus + minus + minus=

Ω Ωint intG G r r rsdot

3

1 2 3

x y zd dxdydz J dxdydz

η η η⎛ ⎞

= = ⎜ ⎟⎝ ⎠

r

1 1 2 2 3 3x η η η= sdot + sdot + sdota i a i a i

etc so

1 2 3

1 2 3 1 2 3

1 2 3

( ) aJsdot sdot sdot

= sdot sdot sdot = sdot times = Ωsdot sdot sdot

a i a i a ia j a j a j a a aa k a k a k

So

1 1 1

1 1 111

1i( ) i2 ( )310

1i2 ( )

1 1 0

1 e e

e i2 ( )

l l l l

a

l l

d d

l l

π η

π η

η

δπ

prime primeminus minus

primeminusprime

=Ω

= =prime minus

int intG G r

ll

rsdot

716 Appendices

f)

i ( ) 3all 3

i ( ) i ( ) i ( )

i( )i( ) i( )

1 e(2 )

1 1 1e e e2 2 2( )

1 e 1 e 1 e2 i( ) 2 i( ) 2 i( )

yx z

x y z

yx yx z

zx y

LL Lq x x q y y q z zx y zL L L

x y zLL Ly y Lx x L z z L

LL L

d q

dq dq dq

L L L

x x y y z z

π

π π π

π π π

primeminus

prime prime primeminus minus minusminus minus minus

primeminusprime primeminus minus

minusminus minus

=

rarr infin

⎛ ⎞⎛ ⎞ ⎛ ⎞⎜ ⎟= ⎜ ⎟ ⎜ ⎟⎜ ⎟⎜ ⎟ ⎜ ⎟prime prime primeminus minus minus⎝ ⎠⎝ ⎠ ⎝ ⎠

int

int int int

q r rq

sdot

sin ( )sin ( ) sin ( )Lim Lim Lim

( ) ( ) ( )( ) ( ) ( )( )

z

x y z

yx zL L L

L y yL x x L z zx x y y z z

x x y y z zπ π π

δ δ δδ

rarrinfin rarrinfin rarrinfin

primeminusprime primeminus minus= times times

prime prime primeminus minus minusprime prime prime= minus minus minusprime= minusr r

g) This is the same problem as (f) with notation changed


qm = 2πmNa = wave vector and m is an integer N = Number of atoms separated by a ωm = ωc sin(qma2) ωc = (2V(0)M)12 (see book for derivation) M = Mass of atom V(0) = part2Vpartxi

2 )0 (see book for notation)

We will use the following dimensionless variables

Frequency c

gωω=

Temperature )( k

Ttcω

=

Internal Energy )( kNk

Uucω

=

Entropy NkSs =

Heat Capacity per atom at constant volume

NkC

c vv =


Number of phonons per dω in ω to ω + dω per atom

⎟⎠⎞

⎜⎝⎛ +⎟⎠⎞

⎜⎝⎛==

Nd

ddpP

c

ωωωωω

toin modes of modephonons of

)(

Thus

π

ωω2

)()( NadqdqqNdN ==

1

)2)(2cos(1

2)(

22cc

cN

aqaN

ddqNadqN

ωωπω

ωπωπω

minus=

==

So

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

minus⎟⎟⎠

⎞⎜⎜⎝

⎛

minus==

221

11e

1

ckTc d

dpPωωπω

ω ω

or since

tg

kTkT c==

ωωωω

c

then

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

minus⎟⎟⎠

⎞⎜⎜⎝

⎛

minus=

21

11e

1)(g

gtP tgπ (1)

which is ready for plotting

We now work out the rest of the quantities in dimensionless notation Note we are neglecting zero point energy

int

intsum

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

minus⎟⎟⎠

⎞⎜⎜⎝

⎛

minus=

minus=

minus= =

c

c

m

dN

dNU

cckT

kTNm kT

m

ω

ω

ω

ωω

ωωωω

ωπ

ωωωω

0 22

01

1

11e

2

)(1e

21e

int⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

minus⎟⎟⎠

⎞⎜⎜⎝

⎛

minus==

c

cc

kTc

cd

NUu

ω

ω ωωωω

ωωπω 0 22 )(

1

11e

2

718 Appendices

or

int⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

minus⎟⎟⎠

⎞⎜⎜⎝

⎛

minus=

1

0 21

11e

2 dgg

gu tgπ (2)

ωωωπω

ω

ω

ω

ω

ω

ω

ω

dNkT

kTC

cckT

kT

m kT

kTm

c

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

minus⎟⎟⎠

⎞⎜⎜⎝

⎛

minus=

minus=

int

sum

2202

2

2

2

2

2v

1)1e(e)(2

)1e(e)(1

int⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

minus⎟⎟⎠

⎞⎜⎜⎝

⎛

minus==

c

cc

kT

kT

c

c dTNkN

NkCc

ω

ω

ωωω

ωω

ωπω

ω

0 222

2

22v

v )(1

1)1e(

e)()(

2

int⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

minus⎟⎟⎠

⎞⎜⎜⎝

⎛

minus=

1

0 22

2

v1

1)1e(

e)(2 dgg

tgc tg

tg

π (3)

From basic thermodynamics we can evaluate the entropy (see (274) and (275))

kTFj

kTE jZ ee minusminus ==sum

where j denotes energy levels

VT

FS ⎟⎠⎞

partpartminus=

so

( ) sumsum=

minus=

minus

minus+minusminus=

N

m

mkT

N

m

kT

kTkTkS

mm

12

1

e11e1ln ωω

ω

( )

int

int

minusminus+

minusminusminus=

=

minus

c

c

cc

kT

cc

kT

dkTkTNNk

dNkNk

NkSs

ω

ω

ωω

ωωωω

ωπ

ωωωωπ

0 22

2

0 22

)(1

11e

)2(1

)(1

1e1ln21

( ) intintminusminus

+minus

minusminus= minus1

0 2

1

0 2

1

11e

2

1

1e1ln2 dgg

tgdgg

s tgtg

ππ (4)


The following is a set of calculations and plots from a Mathcad worksheet Number of phonons per atom at temperatures t = 10 100 300 K x = 1999 x is an incremental subscript variable for one-column matrix p = 1000 p is a variable used with x to size the increment of ω bound the

range of ω

ωxxp

= ω is the dimensionless frequency variable

P10x1π

1

e

ωx

10 1minus

sdot1

1 ωx( )2minus

sdot=

P100x1π

1

e

ωx

100 1minus

sdot1

1 ωx( )2minus

sdot=

P300x1π

1

e

ωx

300 1minus

sdot1

1 ωx( )2minus

sdot=

0 05 10

200

400

P300

P100

P10

ω

720 Appendices

Internal energy and Specific heat per atom vs temperature x = 1400 x is an incremental subscript variable for one-column matrix r = 100 r is a variable used with x to size the increment of t and bound

the range of t

txxr

= t is the dimensionless temperature variable

ux2π

0

1

yy

e

y

tx 1minus

1

1 y2minussdot

⌠⎮⎮⎮⎮⌡

dsdot=

spht x2π

0

1

y

ytx

⎛⎜⎝

⎞⎟⎠

2e

y

txsdot

e

y

tx 1minus

⎛⎜⎝

⎞⎟⎠

2

1

1 y2minussdot

⌠⎮⎮⎮⎮⎮⎮⎮⎮⌡

dsdot=

0 1 2 3 40

1

2

3

4

u

spht

t


0 05 1 15 20

05

1

15

2

u

spht

t

Entropy per atom vs temperature t = 1300

Entropy t2π

minus

0

1

ωln 1 e

ωminus

tminus

⎛⎜⎝

⎞⎟⎠

1

1 ω2

minus

sdot

⌠⎮⎮⎮⎮⌡

dsdot2π

0

1

ω

ω

t

e

ω

t 1minus

1

1 ω2

minus

sdot

⌠⎮⎮⎮⎮⎮⌡

dsdot+=

0 100 200 3000

2

4

6

8

Entropy t

t

722 Appendices


For the bcc lattice we have

1 2 3( ) ( ) ( )2 2 2a a a= + minus = minus + + = minus +a i j k a i j k a i j k

By Problem 29

2 31

a

times=

Ωa a

b

where Ωa = a1 sdot (a2 times a3)

2 3 [(1)(1) (1)( 1)] [(1)(1) ( 1)(1)] [( 1)( 1) (1)(1)]

2

(2 2 ) ( )2

a

a a

times = minus minus + minus minus + minus minus minus

= + = +

a a i j k

i j i j

and

2

21 2 3( ) (2)

2a asdot times = =a a a

so Ωa = a2 and

11 ( )a

= +b i j

note a1 sdot b1 = 1 as required Similarly

2

3

1 ( )

1 ( )

a

a

= +

= +

b j k

b i k

So the reciprocal lattice is the fcc lattice Again the bi are stretched by 2π to be-come reciprocal lattice vectors


The reciprocal lattice of the fcc lattice was found in Problem 217 It is the bcc lattice The primitive translation vectors are

1 2 31 1 1( ) ( ) ( )a a a

= minus + + = minus + = + minusb i j k b i j k b i j k

Linear combinations of these stretched by 2π give us the reciprocal lattice vectors The first Brillouin zone is sketched below It is a truncated octahedron


ky

kxkz


a

a

n m + 1

n mn ndash 1 m

n m ndash 1

n + 1 m

xnm ynm

We assume the potential energy of (n m) can be written

])()()()[(2

])()()()[(2

21

21

21

21

21

21

21

21

mnmnmnmnmnmnmnmn

mnmnmnmnmnmnmnmn

yyyyyyyyk

xxxxxxxxkU

minus+minus+minus+minus+

minus+minus+minus+minus=

minus+minus+

minus+minus+

This assumption is more subtle than is perhaps obvious We will discus it at the end Thus

)4( 111

mnmnmnmnmn

mn xxxxkxUxm minus++=

partpartminus= minusminus+

)4( 111

mnmnmnmnmn

mn yyyykyUym minus++=


724 Appendices

Seeking solutions of the form

)i(1 e maknakt

mnyxcx ++minus= ω

)i(2 e maknakt

mnyxcx ++minus= ω

Letting γ = km substituting and canceling

⎟⎠⎞⎜

⎝⎛ minus+++=minus minusminus 4eeee iiii

112 akakakak yyxxcc γω

⎟⎠⎞⎜



For both cases we get )coscos2(22 akak yx minusminus= γω

The subtlety comes in because if we have purely central forces such as furnished by a spring then the terms (xn m plusmn 1 minus xn m)2 and (yn plusmn 1 m minus yn m)2 are missing from U To include these forces we must include angular forces (see Ghatak and Kothari [216 pp 6-8])

We show the absence of terms like this by the following example We calculate the potential energy when atom 1 is at rest and atom 2 is displaced by xi + yj

a x

y

1 2 Using Hookersquos law

⎥⎥⎦

⎤

⎢⎢⎣

⎡+⎟

⎟⎠

⎞⎜⎜⎝

⎛minus+++minus+++=

⎟⎠⎞

⎜⎝⎛ +++minus++=

⎟⎠⎞

⎜⎝⎛ minus++=

22

2

2

222222

22222

222

481

2122

21

)(2)(21

)(21

aax

ax

ayxayxaxak

ayxaayxak

ayxakU

to second order

[ ]

2

22222222

21

2222

kx

axaxyxayxaxakU

cong

++minusminusminusminus+++cong

(no ky22 term to second order)



11 1 p pX m= sdotsum Xq b q b q bb e

(q and b are vectors)

11 11dagger

1 i22

pp p p

pa P X

ωω

= minus qq q q

q

Now

11 1

1

11dagger

p p

p

p

X m

m

X

minus minus minuslowast

minus

= sdot

= sdot

=

sum

sum

X

X

q b q b q bb

b q b q bb

q

e

e

and similarly for 11 11dagger

p pP Pminus=q q since

1dagger 1 andp p p pX Xlowast

minus minus= =q b q b q qe e

We form the sum (α and β denoting vector components) so

1 1

1

1

1

111

1

1 1

p pp p p

pp p

e X m e e X

m e e X

m X m X

β βα αβ

β βαβ

ββ ααβ δ δ

lowast lowast

lowast

=

=

= =

sum sum

sum sum

sum

q b q b q bbqb q b

b q b q bb q b

bb b bq bb q b

or

1 11

1p ppX e X

mα αlowast= sumq b q b q

b (a)

Now

dagger 11dagger 11

1 i22

pp p p

pa P X

ωω

= + qq q q

q

(with ωqp = ωminusqp) Taking the Hermitian conjugate and changing minusq to q

11dagger 11dagger

11 11

1 i22

1 i22

pp p p

p

pp p

p

a P X

P X

ωω

ωω

minus minus minus= minus

= minus

qq q q

q

qq q

q

726 Appendices

so

dagger 11 2i

2p

p p pa a Xω

minusminus = qq q q

11 dagger

i ( )

2p p pp

X a aω minus= minus minusq q q

q

Substituting in (a)

1 dagger

i ( )

2 p p ppp

X e a am ω

lowastminus= minus minussumq b q b q q

b q


The analysis in the book yields three equations for K1 K2 and K3 from which the determinantal (2269) is obtained The first of these equations is

0)(

)(])[(

34412

24412122

442

4411

=++

++minus+minus

Kkkcc

KkkccKkckcc

zx

yxx ρω

but ky = kz = 0 kx = k so this equation becomes

0)( 122

11 =minus Kkc ρω

Similarly the other two equations are

0)( 222

44 =minus Kkc ρω

0)( 322

44 =minus Kkc ρω

If

ρ

ων 1111

ck

==

then K2 = K3 = 0 and K1 is not required to be zero so the wave is longitudinal

If

ρ

ωνων 4433

22

ckk

====

then K1 = 0 and K2 and K3 are not required to be zero so these are transverse waves




Using the notation of the book

)1(

)(

])()([)(

))]()(())()([()(

))]()(())()(()(

2

2

12

1112

11112

11112

1

1

NM

PM

dPM

dxuxuPM

dxuxuQxuxuPM

dxuxPuxuxPuMd

P

qp qq

qp qq

pp NNNNpp

pp NNNNpp

=

=

minus=

minus=

minus=

minus=

sum

sum int

sum int

sum int

sum intint

lowast

primelowastprime+

primelowastprime+lowast

τδ

τ

τ

ττψψ

where Q = Pminus1P


sumNk1 ε counts interactions twice so it is ne E Summing over dummy variables

sumfk = sumfi In the books notation we immediately obtain

)2()1()21()2()1(21

)2()1()21()2()1(21

)2()1()21()2()1(

)2()1()21()2()1(

)1()1()1(

(||)

(||)

sum

sum

sum

sum

sumsum

minus

+=

minus

+

=

ji ijji

ji jiji

ji ijji

ji jiji

i iik

V

VE

V

V

H

ψψψψ

ψψψψ

ψψψψ

ψψψψ

ψψε


In the notation of the book we need to show

22 2

1 21 10 12

| ( ) |( ) d 0

4jN N

II jeV

rψ

τπε= =+ =sum sum int

rr

Note that

221| ( ) |N

jj ψ=sum r

728 Appendices

represents a constant density ρ which equals sumI ρI(r2) where I represents the ions Also then

2

22 11

0 12

( )d ( )

4NI

II Ie V

rρ τ

πε == minussum sumintr

r QED

IonI

Field point

Source point

r1

r2

It would be better for the problem to say |ψj| = constant and is normalized


Let x = kkM

0as2)(21

11ln1limlnlim

2

2

0

22

0

rarr=minus=

minus+minus=

minus+minus

rarrrarr

xxxx

xx

xx

kkkk

kkkk

xM

M

M

Mk

⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢

⎣

⎡

gtrarrrarrminus+rarr

minusminusminus

ltrarrrarrminus+rarr

minus+minusminusminus

rarr

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

gtminusminusminus

ltminusminusminus

rarr

⎥⎦⎤

⎢⎣⎡

gtminusltminusminusminusrarr

⎥⎦⎤

⎢⎣⎡

gtminusltminusminus++minus=

minusminus+minus=minus+minus

rarr

110)1(

)1(1)1(1

or

110)1(

)1(1)1)](1(1[

2

1)1(1)1ln(

1)1(1)1ln(

2

1)1ln(1)1ln(2ln)1(2

1)1ln(1)1ln()1ln()1)(1(

|)1|ln|1|(ln111ln1lim

2

2

22

1

xxx

xx

xxx

xx

xx

x

xxx

xxxxx

xxxxx

xxx

xxxx

xx

xx

x



We determine Jones zones (from the text) by finding all planes such that k middot K = K 22 This determines the Brillouin zone planes

If only those planes are retained so that

i1e 0bm

bS minus sdot== nesum K r

K

where rb determines the positions of the basis vectors then the Jones zones planes are determined If we use these to determine the smallest enclosed volume we have the first Jones zone

The answers are

For the simple cubic lattice the first Jones zone is a simple cube For the bcc lattice it is a rhombic dodecahedron (see Problem 219) and for the fcc lattice it is a truncated octahedron (see Problem 218)

For the simple cubic case there is one atom per unit cell so |SK| = 1 and the first Brillouin zone and first Jones zone are identical If a is the nearest neighbor atom spacing the volume of the first Jones zone is (2πa)3 as it is a cube

Now consider the bcc lattice Using a sc unit cell we must place the basis atoms at (0 0 0) and a(frac12 frac12 frac12)

2 ( )nlm n l maπ= + +K i j k

Thus

i(0) i( )( )e e

1 ( 1)

n l m

n l m

S πminus minus + +

+ +

= +

= + minusK

SK is zero if n + l + m is odd This simply means we construct our reciprocal lattice for the sc case but omit all K vectors such that n + l + m = odd The resulting reci-procal lattice (fcc in reciprocal space) is the same as if we had used primitive trans-lation vectors in the first place Omitting the K vectors for which n + l + m = odd the shortest K vectors are the 12 defined by

2 ( )aπ= plusmn plusmnK i j

2 ( )aπ= plusmn plusmnK i k

2 ( )aπ= plusmn plusmnK j k

and looking at the volume contained by the planes which bisect those K gives the rhombic dodecahedron already mentioned

730 Appendices

For the fcc lattice using a sc unit cell we must place basis atoms at a(0 0 0) a(0 frac12 frac12) a(frac12 0 frac12) a(frac12 frac12 0) The structure factor is

i( )( ) i( )( ) i( )(n )1 e e en l m

l m n m lS π π πminus + minus + minus += + + +K

This vanishes when two of n l m are odd or two are even If all n l m are even or if all n l m are odd the SK ne 0 The shortest K for which SK ne 0 are

2 ( )n l m aπ= plusmn plusmn plusmnK i j k

The next shortest K for which SK ne 0 are

2 2 2( ) or ( ) or ( )n l m a a aπ π π= plusmn plusmn plusmnK i j k

The volume enclosed by planes bisecting these reciprocal lattice vectors is a trun-cated octahedron as we obtain if we had started with primitive lattice vectors in the first place The lattice formed by omitting all K for which SK = 0 is a bcc lat-tice in reciprocal space


For the low lying energy levels 0 lt E lt V0 We define

20

2)(2

2 EVmKmEk

minus==

The potential is symmetric V(x) = +V(minusx) so the wave function can be chosen to be either symmetric or anti-symmetric For the symmetric case

)ee()(22

ee)(2

)(02

If

ii

KxKx

kxkx

Dxbxb

CBxaxb

Vbax

minus

minus

+=ltltminus

+=ltlt

infinrarr=+gt

ψ

ψ

ψ

Requiring the wave function to vanish where the potential is infinite and be con-tinuous and continuous first derivative otherwise we find

)ee()ee(i

)ee(ee

0ee

2i2i2i2i

2i2i2i2i

)2(i)2(i

KbKbkbkb

KbKbkbkb

bakbak

KDCBk

DCB

CB

minusminus

minusminus

+minus+

minus=minus

+=+

=+


Requiring these equations to yield non-trivial solutions gives the eigenvalue relation

⎟⎠⎞

⎜⎝⎛minus=

2coth)tan( Kb

Kkka (a)

The antisymmetric case is solved similarly

)ee()(22

ee)(2

02

ii

KxKx

kxkx

Dxbxb

CBxaxb

bax

minus

minus

+=ltltminus

+=ltlt

=+gt

ψ

ψ

ψ

Going through a similar procedure of requiring the wave function to vanish where the potential is infinite and be continuous and continuous first derivative other-wise we find for non-trivial solutions

⎟⎠⎞

⎜⎝⎛minus=

2tanh)tan( Kb

Kkka (b)

Where b is very large but not infinity

)e21(

2tanh

)e21(2

coth

Kb

Kb

Kb

Kb

minus

minus

minusrarr⎟⎠⎞

⎜⎝⎛

+rarr⎟⎠⎞

⎜⎝⎛

and

00 V

EEV

EKk cong

minuscong

when E ltlt V0 Thus using (a) and (b) we find

⎟⎠⎞

⎜⎝⎛ congrarrminus

rarr+plusmnminus= minus 20

02symmetric-anti

symmetric)e1(tan mVKVEka Kb

For ground state energy

( ) πππ congminuscongminus= kakakaka whentantan

Thus for the ground state energy we find

)e21( 2

0

KbVEka minusplusmnminus=

732 Appendices

and substituting for K and using E cong ħ2π22ma2 (first iteration)

⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛minusplusmnminus=

22exp21

22

20

02

22

2

2 bmVVma

Ema ππ

Thus a weakly interacting double well causes the energy to be split into a doublet The symmetric case is shifted down and the antisymmetric case (with higher cur-vature in the wave function and hence higher kinetic energy) is shifted up If there were many wells the splitting would be into a band as with a solid

As the wells approach b gets smaller and the splitting gets larger As V0 gets smaller the splitting is larger As a gets smaller the ground state energy gets lar-ger as does the splitting Somewhat similar comments can be made about the higher energy levels

A nice numerical treatment of the double well problem without infinite poten-tial boundaries can be found in S Brandt and HD Dahmen The Picture Book of Quantum Mechanics John Wiley and Sons New York 1985 pp 110-114 They also treat the multiple well problem and relate to energy bands


Pieces of energy surfaces are circular arcs (not spherical pieces) per unit area (in real space) There are 2 d 2k(2π)2 states in d 2k (2 includes spin) Looking at energy surfaces d 2k = dldk But

dkm

kdkdkdEdE

2==

so the density of states D(E) is determined by

intint ==ll

dldEk

mdldkdEED 222 21

)2(2)(

ππ

φ

πak

θ

dl

dk

E+dEE = h2k22m

int=kdlmED 222

)(π


For a

k πlt

2222

2)(

ππ

πm

kkmED ==

For a

ka

ππ 2ltlt

)2(42

)( 2222 ϕπ

ϕπ

mkkmED ==

but

⎟⎠⎞

⎜⎝⎛=minus= minus

kaπθθπϕ 1coswhere2

2

So

⎟⎠⎞

⎜⎝⎛ minus= minus

kamED ππ

π1

22 cos4)(

or

⎟⎠⎞


kamED π

ππ1

2 cos41)(



(1) (2) (3) (4) (5)

(6)

(c) (b) (d) (a) (f) (e)


With the notation that the electrons charge is e the Boltzmann equation in first iteration can be written

0 00

z zf fTf f e Eu uE z T

τ τpart partpart= minus minuspart part partk k k

k

734 Appendices

where the electric field E is assumed to be in the z direction Now

( )

0( ) 2e

(e 1)

F

F

E E kTF

E E kTf E EdT dT kT

minus

minuspart minus⎛ ⎞= minus ⎜ ⎟part ⎝ ⎠+

k

k

k

( )

0( ) 2e 1

(e 1)

F

F

E E kT

E E kTfE kT

minus

minuspart

= minuspart +

k

kk

so

0 02

0

F

F

f f EEdkTT E dT kT kT

f EEdTE dT kT T

part part ⎡ ⎤⎛ ⎞= minus +⎢ ⎥⎜ ⎟part part ⎝ ⎠⎣ ⎦part ⎡ ⎤⎛ ⎞= minus +⎢ ⎥⎜ ⎟part ⎝ ⎠⎣ ⎦

k

k

k

k

Thus

00

z F E fEdT df f u eE Tdz dT kT T E

τ⎛ ⎞ part⎡ ⎤⎛ ⎞= + minus + +⎜ ⎟⎢ ⎥⎜ ⎟ part⎝ ⎠⎣ ⎦⎝ ⎠

kk k

k

with the current density flux

33

2(2 )

zzJ e u f d k

π

⎛ ⎞= ⎜ ⎟⎜ ⎟

⎝ ⎠int k k

and the energy or heat flux

33

2(2 )

zzH u f E d k

π

⎛ ⎞= ⎜ ⎟⎜ ⎟

⎝ ⎠int k k k

Thus

2 303 ( )

4z F

zf EEe dT dJ u eE T d kE dz dT kT T

τπ

⎛ ⎞part ⎡ ⎤⎛ ⎞= minus minus +⎜ ⎟⎢ ⎥⎜ ⎟part ⎝ ⎠⎣ ⎦⎝ ⎠int k

kk

and

2 303

1 ( )4

z Fz

f EEdT dH u eEE E T d kE dz dT kT T

τπ

⎛ ⎞part ⎡ ⎤⎛ ⎞= minus + +⎜ ⎟⎢ ⎥⎜ ⎟part ⎝ ⎠⎣ ⎦⎝ ⎠int k

k k kk


where d 3k can be written dsdEkħuk with ds being the area in k space of constant Ek and partf0partEk cong minusδ(Ek minus EF) at low temperature To calculate the electrical con-ductivity we set dTdz = 0 so

23

22

3

23

( ) [ ( )]( )4

( ) ( )12

412

zz F

zF

F

dsdEeJ u E E eEu

dsdEe E u E Eu

e eEu k

τ δπ

τ δπ

τ ππ

= minus minus minus

= minus

=

int

int

kk k

k

kk k

k

k

with ukz evaluated at EF

Assuming as we are a spherical Fermi surface uk = ħkFm with m being effec-tive mass the electrical conductivity becomes (with n = kF

33π2 = number of elec-tronsvolume)

m

neEJ z τσ

2==

For the heat flux we must set Jz = 0 so

230 ( )

12F

z FE dsdEEe dT dJ u E E eE T

dz dT kT T uτ δ

π⎛ ⎞⎡ ⎤⎛ ⎞= = minus minus +⎜ ⎟⎢ ⎥⎜ ⎟

⎝ ⎠⎣ ⎦⎝ ⎠int k k

k kk

or

2 2( ) ( ) FF F

dsdE E dsdEEdT deE u E E u E E Tu dz dT kT T u

τ δ τ δ ⎡ ⎤⎛ ⎞minus = minus +⎢ ⎥⎜ ⎟⎝ ⎠⎣ ⎦

int intk k kk k k k

k k

or

( )2

2 2 2 414F F

F F F FFE E E E

dE E E kdTeE u k uu dz dT T T u

πτ π τ= =

⎞⎛ ⎞ ⎛ ⎞= minus + ⎟⎜ ⎟ ⎜ ⎟ ⎟⎝ ⎠⎝ ⎠ ⎠k k

k kk k

So

dT

dEdzdTeE F=

Thus the thermal conductivity defined by Hz = minusKdTdz is

2

31 [ ( )]

12F F

F

dT Kdz

E dsdEdE EdTu E E eEE Edz dT T T u

τ δπ

=

⎛ ⎞⎡ ⎤minus minus minus minus + minus +⎜ ⎟⎢ ⎥⎣ ⎦⎝ ⎠int k k

k k k kk

so

23

1 412 F

F

F Fk kE E

u dE dEK E E k

dT dTτ π

π ==

⎛ ⎞⎡ ⎤= minus minus minus⎜ ⎟⎢ ⎥⎣ ⎦⎝ ⎠k

kk k

736 Appendices

or since n = kF33π2 and uk = ħkFm

dT

dEEm

nK FF

F 2τminus=

But for quasi free electrons with effective mass m

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡

⎟⎟⎠

⎞⎜⎜⎝

⎛minus=

2

)0(121)0(

FFF E

kTEE π

so

)0(6

22

F

FE

TkdT

dE πminus=

so

)0(6

222

FF

FE

TkEm

nK πτ=

To lowest order in T EF rarr EF(0) and

Tkm

nK F 22

3τπ=

and

22

2

22

33 ⎟

⎠⎞

⎜⎝⎛==

ek

Tm

ne

Tkm

n

TK

F

Fπ

τ

τπ

σ

This is the law of Wiedemann and Franz and applies to metals The number (π23)(ke) is called the Lorenz number and it equals 245 times 10minus8 watt ohmdeg2 This law works very well for many real metals unless we are at very low T Note the derivation assumes τ for σ is the same as τ for K


a) From integral tables if X = a + bx

bXX

dx 12 minus=int

which can be easily verified by differentiation Let

xKKKxKxKX )()1( 21221 minus+=minus+=


so a = K2 b = K1 minus K2

1

)]1([11

)]1([

21

1

02121

10 2

21

KK

xKxKKKxKxKdx

=

⎥⎦

⎤⎢⎣

⎡minus+minus

minus=minus+

int

b) In what follows we will use m for m the effective mass

2

2

22 2

22 2

( )

( ( ))

(2 )2

22

2

L

L

dIq f

dq f

d

q qm

dmq q

m

ω

ω

=

= minusminus

= minus⎡ ⎤

minus sdot minus⎢ ⎥⎢ ⎥⎣ ⎦

= minus⎡ ⎤minus sdot +⎢ ⎥⎣ ⎦

int

int

int

int

qq

qq

q

k q

q

k q

We now apply the result of (a) let K1 = q2 minus 2k middot q + (2mħ) ωL and K2 = q2 Then we can write the integral I as

1

2 20 2 2

2

2(1 ) 2 L

m dxI dmq x q xω

= minus⎡ ⎤⎛ ⎞minus + minus sdot +⎜ ⎟⎢ ⎥⎝ ⎠⎣ ⎦

int intq

k q

which can be written

1

2 20 2 2 2

2

2( ) L

m dI dxmx x x kω

= minus⎡ ⎤⎛ ⎞minus + minus⎜ ⎟⎢ ⎥⎝ ⎠⎣ ⎦

int intq

q k

Since we are integrating over all q this can also be written

1

2 20 2 2 2

2

2L

m dI dxmq x x kω

= minus⎡ ⎤⎛ ⎞+ minus⎜ ⎟⎢ ⎥⎝ ⎠⎣ ⎦

int intq

738 Appendices

Let 222 2 kxxmL minus= ωα so evaluating

2

2 2 2 2 2 20

4[ ] [ ]

d q dqq q

πα α

infin=

+ +int int

q

and looking at the integral we find π2α Substituting α back into the above we find

intminus

minus=1

0 22

22 2

2

kxxmdxmI

Lωπ

But

2

sin2

2

11

0 22

pppxx

dx minus=minus

int

so we find

⎟⎟⎠

⎞⎜⎜⎝

⎛minus= minus

2sin22 12

2k

mk

mm

mIL

L

L ωω

ωπ

Thus

Lm

Kω

=3

c) For small θ sinminus1 θ cong θ + θ36 so we can show

⎟⎟⎠

⎞⎜⎜⎝

⎛+minuscong

2122 2

22

kmm

mILL ω

πω

This evaluates the integral in (4111) So with substitution and chasing down the definitions of CH

2 and αc we get the desired result

2 2

02

16

c L

c

kEm

mm

α ω

α

lowastlowast

lowastlowastlowast

= minus +

=minus

k

where we have used m for m in the above Interaction with the phonons produces a self energy




The displacement of the Fermi surface due to electric and magnetic fields and relaxation process is (gaussian units)

1 1d edt c

δτ

⎛ ⎞ ⎛ ⎞+ = minus +⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

k E v Btimes

where δk = k(t) minus k(0) The effective mass is defined by mv = δk and we assume the direction of the static magnetic field is along the z-axis Thus in the steady state with the cyclotron frequency defined by

mceB

c =ω

v

vv

vv

zz

xcyy

ycxx

Eme

Eme

Eme

τ

τωτ

τωτ

minus=

+minus=

minusminus=

Using jx = minusnevx jy = minusnevy jz = minusnevz with σ0 = ne2τm and solving for vx vy vz we find

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛ minus=

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

z

yx

cc

z

yx

EEE

DDjjj

000101

0 τωτωσ

where D = 1 + (ωcτ)2 Thus

)(10 ycxyxyxxxx EEDEEj τωσσσ minus=+= minus

)(10 yxcyyyxyxy EEDEEj +=+= minus τωσσσ

Dyyxx

0σσσ ==

D

cxyyx

τωσσσ 0=minus=

For the Hall effect

yxyy

xyy

EEj σ

σminus=rArr= 0

740 Appendices

So

xyy

xyxxx Ej ⎟

⎟

⎠

⎞

⎜⎜

⎝

⎛+=σσ

σ2

Substituting for σxx σxy σyy we find

xx Ej 0σ=

independent of B as expected where

m

ne τσ2

0 =

is the effective conductance jxEx The Hall coefficient is

BE

E

BjE

Rx

yxyy

x

x

yH )( 0σ

σσ

minus==

or after substitution for σxy σyy

nec

RH1minus=

Note for high fields after substituting for σ0 and ωc

B

nec

cc

cxyyx ==congminus=

τωσ

τωτωσσσ 02

0

)(

Thus for high fields

B

Ryx

H )(1

σminus=


v

v)()(

)(vvv

32

54

32

540

0

⎥⎦

⎤⎢⎣

⎡partpart

partpartminusminus=

⎥⎥⎦

⎤

⎢⎢⎣

⎡sdot

partpartminus=

partpartminus=

F

F

FF

e

Fe

xxxfxfcm

xfcmEp

π

π


For highly relativistic conditions xF gtgt 1 which leads to

( ) ⎟⎟⎠

⎞⎜⎜⎝

⎛++=+= int hellip2

40

22 11411

FF

xF

xxdxxxxf F

where

312

v3

⎟⎟⎠

⎞⎜⎜⎝

⎛=equiv π

cmcmpx

ee

FF

Differentiating the above two equations

FFF

F xxxxf

21)( 3 +=

partpart

34312

v31)3(

vπ

cmx

e

F minus=partpart

Fe

F xcm

x31

v3

31

vv

312minus=⎟

⎟⎠

⎞⎜⎜⎝

⎛minus=

partpart π

Substituting back into p0

[ ] 12

61

3111

41

2432

54

242

432

54

0

FFe

FFF

Fe

xxcm

xxx

xcm

p

minuscong

⎥⎥⎦

⎤

⎢⎢⎣

⎡++⎟

⎟⎠

⎞⎜⎜⎝

⎛++minuscong hellip

π

π

Using the defined dimensionless quantities

pm

MM δ=

and

⎟⎟⎠

⎞⎜⎜⎝

⎛=

cem

RR

742 Appendices

a quick derivation of xF follows

Mm

R pξπ sdotcong 3

34v

31

31

3

2

491

34

3

⎟⎟⎠

⎞⎜⎜⎝

⎛=

⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜

⎝

⎛

sdot=

ξδπ

ξπ

π

MR

Mm

RRRx

pF

For ξ = 2 (correlates to mp cong mneutron)

32

54

12~and

89

ππδ cm

K e==

⎥⎥⎦

⎤

⎢⎢⎣

⎡minuscong 2

32

4

34

0~

RM

RMKp

Compare to Huang [11 pp 249-250]


As shown in the figure assume we have a well-defined region R for tunneling

EF(2)

EF(1)

E

Δ = eV

R1 2

A voltage V is applied to region 1 relative to region 2 lowering the potential en-ergy of electrons by minuseV Using the same Fermi on both sides and noting

)1()2()2( )( FFF EEEEEE minus=Δminusminus=minusΔ+

We note if

1)2(1exp)(

minus

⎥⎥⎦

⎤

⎢⎢⎣

⎡+

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛ minusminus=

kTEE

Ef F (a)


then

1)1(1exp)(

minus

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡+⎟

⎟

⎠

⎞

⎜⎜

⎝

⎛ minusminus=Δ+

kTEE

Ef F

So f (E) = f2(E) in region 2 f (E + Δ) = f1(E) in region 1 If T denotes the probability of tunneling from region 1 to 2 for each electron

then

[ ] 3

321

)2())(1)((v2

πkdEfEfeTdJ x minusΔ+minus=rarr

The current going the other way is

[ ] 3

312

)2())(1)((v2

πkdEfEfeTdJ x Δ+minusminus=rarr

The total current density is

1221 rarrrarr minus= dJdJdJ

[ ] 3

3

)2()()(v2

πkdTEfEfedJ x minusΔ+minus=

Since

x

x kE

partpart= 1v

We write

dEdkxx1v =

So with d 3k = dkx d 2k||

[ ] ||2

3 )()()2(

2 kddETEfEfedJt minusΔ+minus=π

For region 2 we define

mk

EW2

2||

2minus= (b)

We assume a one-dimensional representation for tunneling so for energy E

)(D1 WTT =

744 Appendices

which we will evaluate by the WKB approximation

⎥⎥⎦

⎤

⎢⎢⎣

⎡minusminus= int 2

1

21))((22exp)( zz dzWzVmWT (c)

where V(z) is the barrier potential between regions 1 and 2 and z1 z2 are the clas-sical turning points We thus obtain

[ ] )( )()()2(

2||

23 WTEfEfkddEeJt intint minusΔ+minus=

π (d)

where use of equations (a) (b) (c) and (d) must be made Some care must be used in setting the limits Often used is

dWmdkkkd 2||||||2 22 ππ minus==

Then note Wmax = E and Wmin will be determined by the smallest W that allows tunneling In simple cases this may be zero



ce EE

mED minus⎟

⎟⎠

⎞⎜⎜⎝

⎛=

lowast 23

222

21)(π

1])exp[(

1)(+minus

=kTE

Efμ

For E minus μ gtgt kT

⎟⎠⎞

⎜⎝⎛ minus=⎟

⎠⎞

⎜⎝⎛ minusminuscong

kTE

kTEEf μμ expexp)(

ce

EmkE += lowast2

22

intinfin

lowast⎟⎠⎞

⎜⎝⎛ minusminus⎟

⎟⎠

⎞⎜⎜⎝

⎛=

cE ce dE

kTEEE

mn μ

πexp

22

123

22


Substitute the integration variable E with k

ce

EmkE += lowast2

22

kdkm

dEe

22

2

lowast=

lowast

=minuse

cm

kEE2

intinfin

lowast

lowast

lowast

lowast

⎟⎟⎠

⎞⎜⎜⎝

⎛ +minus⎟

⎠⎞

⎜⎝⎛

⎟⎟⎠

⎞⎜⎜⎝

⎛=

cEe

ce

e

e kdkmkT

EmkkTm

kmn 2

2)2(

expexp2

22

1 22223

22μ

π

intinfin

lowastlowastlowast

lowast

⎟⎟⎠

⎞⎜⎜⎝

⎛minus⎟

⎠

⎞⎜⎝

⎛ minus⎟⎟⎠

⎞⎜⎜⎝

⎛=

cEee

c

e

e dkkkTm

kmkT

E

m

mn 2

22223

22 2expexp

2

22

1 μπ

intinfin

lowastlowast

lowast

⎟⎟⎠

⎞⎜⎜⎝

⎛minus⎟

⎠

⎞⎜⎝

⎛ minus⎟⎟⎠

⎞⎜⎜⎝

⎛=

cEe

c

e

e dkkkTm

kkT

Em

mn 2

22

23

323

22 2expexp

)2(22

21 μπ

Setting the origin at Ec the integral then has the form14

pp

ndxpxx nn π

)2(2)12()exp(0

22 minus=minusintinfin

where

kTm

pne

2

2 1 ==

So

( )kTmkTm

kTE

m

mn

ee

c

e

e

21

223

323

22

2222

)1(exp2

222

1 πμπ

⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎠

⎞⎜⎝

⎛ minus⎟⎟⎠

⎞⎜⎜⎝

⎛=

lowast

lowast

⎟⎠

⎞⎜⎝

⎛ minus⎟⎟⎠

⎞⎜⎜⎝

⎛=

lowast

kTEkTm

n ce μπ

exp2

223

2

14 IS Gradshteyn and IM Ryzhik Table of Integrals Series and Products Academic


746 Appendices


First let us review the elementary argument to see why something new is needed to get the force from a magnetic field on a charge For an electron with wave vec-tor k its wave velocity is

1 ( ) 1 ( )kdE E

d⎛ ⎞= equiv⎜ ⎟⎝ ⎠

kv kk

nabla

This is the wave packet velocity valid as long as the wave packet is large com-pared to interatomic distances Suppose there is an external force F then

1 ( )

( )

k

k

dE Edt

d Edt

= sdot = sdot

= sdot

F v F k

k k

nabla

nabla

thus

( ) ( )k kdE Edt

sdot = sdotF kk knabla nabla

This is consistent with

ddt

= kF

Notice however that if the above is true 1 q= =F F E would yield

1 ( ) ( )k kdE Edt


but also

T 1ddtperp= = + = kF F F F

where Fperp is perp to v prop nablakE(k) would imply the same equation so

( ) ( )k kdE Edt


does not uniquely argue what F must be So we must find another way to show if we have both E and B fields that for a charge q

( ) dqdt

= + times = kF E v B

For a relatively simple (but sophisticated so we do not use here) way to establish this equation see J Zak ldquoThe kq-Representation in the Dynamics of Electrons in Solidsrdquo in Solid State Physics Advances in Research and Applications edited by H Ehrenreich F Seitz D Turnbull Academic Press New York 1972 Vol 27 See Sect VII Bloch Electrons in a Magnetic Field


The reference we suggest (Smith) also suggests we see AH Wilson Theory of Metals Cambridge University Press 2nd ed (1954 pp 50ndash52) for an appropriate argument

As Smith points out the proof of this relation is ldquonot difficult in principle but is tediousrdquo and since it is readily available in Smith we do not give it here

Combined Problem (66) and (67) Solution We consider only a covalent non-polar semiconductor such as Si The problems together amount to discussing electron mobility at high temperature where lattice scattering is dominant and at low temperature where ionized donors and acceptors (assuming they are present) are dominant

Note the carrier mobility μ is the drift velocity per electric field (μ = | vdriftE |) and it can also be derived from μ = σne where σ is the electrical conductivity and we have assumed for simplicity that we have only electron carriers of density n and magnitude of charge e Thus the mobility measures scattering only and not the variation of carrier concentration with temperature Summary of Results a) Lattice scattering by acoustic phonons ndash F Seitz [635] finds

23constant minustimesprop Tμ

A more detailed discussion of the mobility with similar results has been derived by J Bardeen and W Shockley Phys Rev 80 72 (1950) b) Ionized donor (or acceptor) scattering ndash Conwell and Weisskopf [69] using Rutherford scattering have found 23Tpropμ

Brooks and Herring (H Brooks Phys Rev 83 879 (1951)) have derived a similar result with a quantum mechanical treatment c) Assuming these are the two dominant scattering mechanisms (there may be others eg optic phonon scattering in polar semiconductors (eg GaAs)) we are led to the prediction

ln T

ln μ

low T high T

23Tpropμ 23minusprop Tμ

Ionized impurity scattering

phonon scattering

748 Appendices

Actually if μ1 is for ionized impurity scattering and μ2 is for scattering by lattice vibrations then

if

if

212

121

21

21total

μμμμμμ

μμμμμ

gtgtconggtgtcong

+=

That is whatever is doing the most scattering (producing least mobility) is domi-nant

d) Historically ionized impurity scattering was the first of these two to be well described We sketch its derivation below (a complete description is given eg by Seeger [634 pp 160-162])

1 Use a screened Coulomb potential

rr

rV λminusprop e1)(

where λ(r) prop screening length

2 Calculate the matrix element for the scattering

2 1 2 1( )H V r=k k k k

where k1 and k2 represent the Bloch states (commonly approximated by plane waves) and elastic scattering |k1| = |k2| is assumed

3 Calculate the differential scattering cross section

2 1

2( ) Hσ θ prop k k

4 The momentum relaxation time is then calculated from

int minuspropπ

θθθθστ 0

sin)cos1)((v1 dm

where v is the speed and θ is the scattering angle ([634 pp 154-155] or com-pare (4145))

5 Then the τm must be suitably averaged over the distribution function [634 pp 50 162] or compare (4153) plus manipulation)

Finally we find

23Tm

m prop= lowastτ

μ

Note the calculation of μ for metals and semiconductors differ ndash for semiconduc-tors we are mostly concerned with the Boltzmann tail (non degenerate Fermi gas)


e) The scattering of charge carriers by acoustic phonons follows a somewhat simi-lar line Although the ideas involved in the Seitz and Bardeen-Shockley papers are somewhat similar we will sketch the Bardeen-Shockley result as perhaps more general See Seeger [634 pp 165-168]

1 The scattering potential is assumed to be an acoustic deformation potential Bardeen and Shockley showed that the scattering potential could be assumed as due to the deformation of the band edge which arises from the variation of the lattice constant due to long wavelength acoustic waves One finds that the po-tential for electrons is

cEδ δprop sdotr rnabla

where δEc is the variation of the conduction band edge and δr is the variation of the atoms position If

i( )e tωδ sdot minusprop k rr then

cEδ δprop sdotk r

2 Then one calculates Hk2 k1 But now since k1 minus k2 plusmn kp = 0 (lsquomomentumrsquo correction) with kp being the wave vector of the phonons one has the result that the scattering matrix elements turn out to be proportional to kp and the ampli-tude of the lattice wave

3 Consider both emission and absorption of phonons and assume the energy of the carrier is much greater than the energy of phonons Following a somewhat similar process as in (d) (but with greatly different details which are omitted here as they are difficult to condense) one finds τm and averages it over a Boltzmann distribution (non degenerate electron gas) to find

23minus=prop Tμ


See Fig 618

)( pngVpCn eEEE ϕϕ minusminus+=

Now

dkTEe NkTmn Cn cong⎟

⎟⎠

⎞⎜⎜⎝

⎛= minus )(

23

2 e2

2 μ

π

akTEh NkTmp Vp cong⎟

⎟⎠

⎞⎜⎜⎝

⎛= minus )(

23

2 e2

2 μ

π

750 Appendices

assuming we are beyond depletion regions So

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎟⎟⎠

⎞⎜⎜⎝

⎛=minus

23222

lnkTm

NkTE

e

dCn

πμ

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎟⎟⎠

⎞⎜⎜⎝

⎛=minus

23222

lnkTm

NkTE

h

aVp

πμ

Therefore

⎥⎥⎦

⎤

⎢⎢⎣

⎡=minusminus=minus 323

32

)()()2(

22ln)(

kTmmNN

kTEeEEhe

dagpnCnVp

πϕϕ

Using

kTEg

gkTE

e

1ln minus=

kTEhei

gmmkTn 4323

2 e)(2

2 minus⎟⎟⎠

⎞⎜⎜⎝

⎛=

π

we find

⎟⎟⎠

⎞⎜⎜⎝

⎛=minus 2ln)(

i

dapn

nNN

kTe ϕϕ QED



2 21

12Surface

( )( )

4d

rπΦ = int

M r Sr

sdot

= minus ΦH nabla

12 2 21 2

12

ˆ ( )( )

4d

rπ= int

r M r SH r

sdot

θr

dS2

r 12

z

M

r1 = 0r12 = ndashr2


Using the figure

intminus=π

πθθπθθ

0 24 sin2 cos cos

rrdrMH z

31and

3=minus= DMH z


For a simple moment the Hamiltonian can be written as BH gμ= minus =μ B J Bsdot sdot

(J here is actually Jħ ie dimensionless) where μ is the magnetic moment

)eTr( HZ βminus=

where β = 1kT and Z is the partition function ZkTF lnminus= is the Helmholtz free energy

1 1 1( ) ln

Tr( e ) Tr( e )

zz z z

z z

ZF ZB B Z B

Z Z

β β

μ ββ β β

βμ μβ

part part part= minus = =part part part

= =μ B μ Bsdot sdot

as expected

22211e HHH βββ +minuscongminus

for high temperatures

)Tr(2

)1Tr()eTr( 22

HZ H ββ +== minus

since Tr( ) Tr( ) 0BH gμ= =B Jsdot Note

)Tr(

3)Tr(

3

)Tr(31)Tr()Tr(

2222

222

222222

JBμgJBBμg

JδBBμgJJBBμgH

Bμ μμ

B

μνμν νμBνμμν νμB

==

==

sum

sumsum

)12()1Tr( += j

)1()12()Tr( 2 ++= jjjJ Therefore

⎥⎥⎦

⎤

⎢⎢⎣

⎡+++cong )1(

3211)12( 2

222 jjBgjZ Bμβ

752 Appendices

Using ln(1+ε) cong ε for ε ltlt 1

)1(61)12ln(ln 2222 +++cong jjBgjZ Bμβ

⎥⎦⎤

⎢⎣⎡ +

partpart= )1(

611 2222 jjBg

B Bz

z μββ

μ

BB

BB z

z=

partpart

)1(31 22 += jjBg zBz μβμ

kT

jjgNNB

Bz

zzz

)1(3

22 +=partpart= μμχ QED

(The lack of appearance of μ0 here is because χ = partMpartB rather than χ = partMpartH)


Starting with (7148) and (7149)

ie ii S primeprimeprime=sum k R

kkS sdot

ie ij S primeprimeprimeprimeprimeprime=sum k R

kkS sdot

i( ) ( )eJ J=sum k RRk R sdot

Using (7154) (and Jij equiv J(Rij) = Jji)

i i( ) e ej i j

i jid Jdt

primeprime prime primeprimesdot sdot + sdotprimeprime primeprime primeprimeprime prime primeprime= timessum sumk R k R k R

k k kk k kS S S

Multiply by ie jminus sdotk R and sum j

i( ) i( ) e ej j i j j

i jj i jd Jdt

primeprime prime primeprimesdot minus sdot sdot + sdot minus sdotprimeprime primeprime primeprimeprime prime primeprime= timessum sumk R k R k R k R k R

k k kk k kS S S

The last equation can be rewritten

i( ) i( )i( ) e ( )e ej ij i

d Jdt

primeprimeminus sdot prime primeprimeprimeprime + minus sdotminus minus sdotprimeprime primeprime primeprimeprime prime primeprime= timessum sumk k R k k k Rk k R

k k kk R k kS R S S

Now i 0e Nδ=sum k R

kRsdot


so

( )

( )

( )

kd Jdt

J

J

δ primeminusprimeprime primeprime primeprime primeprime

prime primeminusprime

primeprime primeprimeminusprimeprime

primeprime= minus

prime=primeprime=

sum

sumsum

k kk k kk k

k k kk

k k kk

S k k S S

k S S

k S S

times

times

times

which is (7155) QED


Let φR be the turn angle of spin R Then

)sin( )cos( RRRR SSSSyx

ϕϕ ==

By symmetry the turn angle from spin to spin should be constant φR = qR Substituting

2

2

(cos cos sin sin )

cos( ( ))

x x y yR R R R R R

R R R R

S S S S

S

S q R R

ϕ ϕ ϕ ϕ

prime prime prime

prime prime

= +

= +

prime= minus

S Ssdot

so that

)0)sin()( (since e)(2

)cos()(2

))(cos()(2

i2

2

2

=minus=

minus=

primeminusprimeminusminus=

sumsum

sum

sum prime

RRqR

R

RR

qRRJRJNS

qRRJNS

RRqRRJSH

Let )(e)()( i qJRJqJ R

qR minus== sum Then

)(2

2qJNSH minus=

Thus a stable structure at T = 0 has J(q) rarr max if J(q) gt 0 so H is a min We seek q that does this for our example

)2cos(21)cos(2

21 )2cos(2)cos(2

)ee()ee()(

1

1221

i2i22

ii1

⎥⎦⎤

⎢⎣⎡ minus=

minus=+=

+++= minusminus

qaqaJ

JJqaJqaJ

JJqJ qaqaqaqa

754 Appendices

For a maximum

[ ]0

)cos()sin(2)sin(2

)2sin()2(21)sin(2)(

1

1

=minusminus=

⎥⎦⎤

⎢⎣⎡ minusminusminus=

partpart

qaqaqaaJ

qaaqaaJqqJ

The solution qa = 0 is a minimum Another solution should provide a maximum

angle turn 60

21)cos(

1)cos(2

deg=

=

=

qa

qa

qa

To verify this the second derivative should be nonzero

[ ]

max03

212

212

)2cos(2)cos(2)(

21

21

602

12

2

rArrltminus=

⎥⎦⎤

⎢⎣⎡ +minus=

minusminus=part

partdeg=

aJ

aJ

qaqaaJq

qJqa

Therefore the stable ground state is helical with turn angle 60deg


=minussj mSS

12

112

212

212

21

21

21

+⎟⎠⎞

⎜⎝⎛ minus+=

⎟⎠⎞

⎜⎝⎛ minus=

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛minus=

+

++minus

nSnnS

nSnaS

nSaa

aSnS

j

jjjj

ss mSnmSn minus=equiv and

so

)1(1 minusminus=+ smSn


or

1)1)((

12

2)1(221

minus+minus+=

minus⎟⎠

⎞⎜⎝

⎛ +minusminusminus=minus

sss

ss

ssj

mSmSmS

mSS

mSSmSSmSS

QED


We have

i1 e jjjb a

N= sum k R

ksdot

and

idagger dagger1 e jjjb a

Nminus= sum k R

ksdot

where

0][

0][

][daggerdagger

dagger

=

=

=

ji

ji

ljjj

aa

aa

aa δ

Thus

i( )dagger dagger

i( )

1[ ] e [ ]

1 e

1

j j

i j

j

k jj jk j

j

k k k k

b b a aN

N

NN

δ

δ δ

prime

prime

primeminus minusprimeprime prime

primeminus minus

prime prime

=

=

= =

sum

sum

k R k R

k k R

sdot sdot

sdot

Similarly

i( )

0

1[ ] e [ ] 0j jk k j jj jb b a a

Nprimeprime+

prime primeprime= =sum k R k Rsdot sdot

i( )dagger dagger dagger dagger

0

1[ ] e [ ] 0j jjj jk k jb b a a

Nprimeprimeminus +

primeprime prime= =sum k R k Rsdot sdot

756 Appendices


Ce3+ has configuration 4f 15s2p6 Pm3+ has configuration 4f 45s2p6 and Eu3+ has configuration 4f 65s2p6

We use Hundrsquos rules where maximizing S will maximize L consistent with S

Ce3+ one electron in f shell

ml minus3 minus2 minus1 0 1 2 3 ms spin uarr

Therefore

712

)state F(3max

21221

=+==

=+=

sumL

Lm

SS

l

less than half full

25

213 =minus=minus= SLJ

So stateF 25

2

Pm3+

ml minus3 minus2 minus1 0 1 2 3 ms uarr uarr uarr uarr

426

)state I(123max5122

=minus=minus=++===+=

sumSLJ

mLSS

l

So stateI4

5

Eu3+ six electrons

ml minus3 minus2 minus1 0 1 2 3 ms uarr uarr uarr uarr uarr uarr

033

)state F(3)2()1(123max

7123216

=minus=minus==minus+minus+++==

=+=times=

sumSLJ

mL

SS

l

So stateF0

7



Assume dipole-dipoleUkTc = Magnetic field from a dipole

05 3

3[( ) ] ( )( )4μπ

⎛ ⎞prime primeminus sdot minus⎜ ⎟= minus⎜ ⎟prime primeminus minus⎝ ⎠

r r m r r mB rr r r r

B

r

rprime

m

aEuO = 5143 Aring the same structure as NaCl The distance from Eu2+ to Eu2+ is a 2 and there are 12 of these Of the 12 Eu2+ neighbors to Eu2+ if we assume all moments are parallel and pointing along a cube edge

4 have ( ) 0

28 have ( ) 2 22

a mam

primeminus sdot =

primeminus sdot = =

r r m

r r m

Interaction energy for one moment with another is

[ ]2 2

05 3

3 ( )4

Uμπ

⎛ ⎞primeminus sdot⎜ ⎟= minus sdot = minus minus⎜ ⎟prime primeminus minus⎝ ⎠

r r m mm Br r r r

⎟⎟⎠

⎞⎜⎜⎝

⎛minus= minus

5

252

3

2327

total2)2)(3)(8(2)12(10

ama

amU

( ) 022424)10( 3

27 =minus= minus

amU

no net interaction in this approximation

758 Appendices

Go to the next Eu2+ a distance of a six of them

2 2 2

4 have ( ) 0

2 have [( ) ] a m

primeminus sdot =

primeminus sdot =

r r m

r r m

So the net interaction is

066)10( 3

2

3

27

total =⎟⎟⎠

⎞⎜⎜⎝

⎛minus= minus

am

amU

Even if we assume the interaction does not cancel (if the moments were not all in the direction assumed for example) it would be of order

3

27

total6)10(amU minus=

)JT10279)(8(8

m10524

10

minus

minus

times=cong

timescong

Bm

a

μ

K) 69an (rather thK1

)(10381

101010

1012510)10(

23

2328

51

30

447

total

congtimescong

=congtimes

cong

minus

minusminus

minus

minus

minusminus

c

cT

T

U

Magnetic dipole-dipole interactions are not large enough to account for observed Curie temperatures


In the Bohr-van Leeuwen theorem there is no magnetism with pure classical sta-tistics (the magnetic susceptibility is zero as the para and dia components always cancel)


The free energy is

ZkTF lnminus=

The magnetization is

BFMpartpartminus=

where B is the magnetic induction field The partition function is

int minus= dqdpZ qp hellip)(e Hβ

In the absence of fields

)(0 qpHH rarr

With fields we make the transformation

)()(component potential vector is

momentum cononical is )0(

qpqAp

eeAppqq

i

i

iiii

ii

Π=

gtΠequiv+rarrrarr

0HH

Now

)()(e

e

11)(

11)(

ΠpartparttimesΠΠ=

=

int

intΠminus

minus

qpqdddqdq

dpdpdqdqZ

NNq

NNqp

helliphellip

helliphellip

H

H

β

β

where part(q p)part(q Π) is the Jacobian But since

0 =Πpartpart

=Πpartpart

=partpart

j

iij

j

iij

j

i qpqq δδ

(These follow since Πi = mui = pi0 (ie without field) and qi is the same with or

without a field) and

⎟⎟⎠

⎞⎜⎜⎝

⎛=

Πpartpart

+Π= etc1i

iiii

peAp

760 Appendices

Thus schematically

1

0

100010001

)()( =

Πpartpart

=

Πpartpartpartpart

=

Πpartpart

Πpartpart

partpart

partpart

=Πpart

part

i

i

j

i

i

i

j

i

j

i

j

i

j

ip

pqp

pqqp

qq

qqp 15

int Π= Πminus ddqZ q hellip)(0e Hβ

or the partition function is independent of fields so F is independent of B and

0=partpartminus=

BFM

This problem implies that one cannot have a finite temperature phase transition to a magnetized state Of course there are other parameters beside temperature (eg pressure chemical concentration magnetic field) that can be varied which may come into play into so called quantum phase transitions These are phase transitions at zero temperature See the March 2008 issue of Nature Physics which has a focus on them



Let N1(E)f1(E) be the number of occupied states in 1 so that f (E) is the Fermi function Then N2(E)(1 minus f2(E)) is the number of empty states in 2 After a poten-tial V is applied the current from 1 to 2 is

[ ]int+infininfinminusrarr minusminusminus= dEeVEfeVENEfENCI )(1)()()( 221121

15

1

100

01

100

01

0

=sdot=⎟⎟⎠

⎞⎜⎜⎝

⎛

partpart

⎟⎟⎠

⎞⎜⎜⎝

⎛

Πpartpart

=

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛partpart

partpart

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

Πpartpart=

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛partpart

partpart

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

Πpartpart=

Πpartpartpartpart

partpart

11j

i

j

i

i

i

j

i

j

ii

i

j

i

j

i

j

i

i

i

j

i

qqp

qp

qq

pqp

qq

ppqp

qq

where Block matrices (partqipartqj) eg have been used and also the property of triangular matrices See eg Jon Mathews and R L Walker Mathematical Methods of Physics W A Benjamin New York 1964 p141 and C Ray Wyle and Louis C Bennett Advanced Engineering Mathematics McGraw-Hill Book Co New York Fifth Edn 1982 p167


where the constant C includes the effects of tunneling Similarly

[ ]int+infininfinminusrarr minusminusminus= dEEfENeVEfeVENCI )(1)()()( 112212

Subtracting [ ]int

+infininfinminusrarr minusminusminus= dEeVEfEfeVENENCI Net )()()()( 212121

Note that no current flows until eV ge Δ1 + Δ2 At absolute zero

intΔminus

ΔΔminusminus

minus

Δminus= 2

1 22

221

221)(

)0()0(eV dEEeV

EeV

E

ENNCI

Checks 11

)( Δminus=minus Δ= eVEeV E

222)( Δ=Δ+minus=minus Δminus= eVeVEeV eVE



The problem is for the most part done in the book It is just required here to re-think the following

Eprime0 is the electric field due to charge on the plate after the slab is inserted E0 is the electric field before the slab is inserted d is the distance between plates

Voltage ΔV between the plates is maintained constant by battery Pε0 is the op-posing electric field due to charge on the surface of the slab

0

000

00 εεPEEdPEdEV +=primerArr⎟⎟

⎠

⎞⎜⎜⎝

⎛minusprime==Δ


0

0

0

3

0

0

2

02

3

31)11(

2

3cos

2

)(coscos2

ε

ε

θε

θθε

π

π

P

P

P

dPEE b

=

minusminusminus=

minus=

minus=equiv int

762 Appendices


If the dipole moment is p in an electric field E the energy is U = minusp middot E There will be azimuthal symmetry about the electric field as shown

θ

p

E z

Assuming a classical distribution of directions for p the average p in the z direc-tion is

int

int= π

θ

πθ

θθπ

θθπθ

0

cos

0

cos

sin2e

sin2ecos

d

dpp

kTpE

kTpE

z

Letting c = pE and x = c cos θ we can write

int

int

minus

minus= c

c

x

c

c

x

zdx

dxx

cpp

e

e

After some manipulation this can be written

⎟⎠⎞

⎜⎝⎛ minus=

ccppz

1)coth(

At high T c ltlt 1 so we find

kTp

Ep

kTEpp z

z 3

3

22=== α

This is analogous to paramagnetism


The Hamiltonian is

eEzr

qm

minusminusnablaminus=

0

22

2

2H

H


where q2 = e2(4πε0) and E is the electric field in the z direction For a trial wave function we will use

)1(100 pzN += ψψ

where N is the normalization constant p is the variational parameter ψ100 is the unperturbed (no E field) wave function of the Hydrogen atom in the 1s state

BarBa 23

100 e1 minusminus=π

ψ

where aB is the Bohr radius

2

2

mqaB =

Requiring

12 =int τψ d

a relatively simple calculation yields

221222 1)1( BB apapN minuscong+= minus

We must also evaluate

( )2 3 2 2 3100 0 100 100(1 ) (1 ) (1 ) ( )N pz pz d pz eEz dψ ψ ψ= + + + + minusint intr rH H

Now using

4

3 0100 0 100 2

0

12 (4 )

med Eψ ψπε

= = minusint rH

2 2 3 2100 (1 ) 2 ( )BeE pz zd pa eEψminus + = minusint r

3100 0 100 0pzdψ ψ =int rpzH

(by a tedious but straightforward calculation) and the cross terms

3 3100 0 100 100 0 100d z dψ ψ ψ ψ=int intr rzH H

are also zero so

)(2 202 eEpaE

NBminus=

H

Substituting our result for |N| we have to order p2 (weak electric fields)

02220 )(2 EapeEpaE BB minusminus=H

764 Appendices

Requiring

0=part

partpH

we find

0EeEp minus=

Substituting in the expression for langH rang we find

230

0 8 EaE Bπεminus=H

The polarizability α is given by the second derivative

302

216 Ba

Eπεα =

part

partminus=

H


(a) For G(Psc) = G0 and T = Tc

061

41)(

21 42

0 =++minus scscc PPTT δγβ (1)

For E = 0

0)( 420 =++minus=

partpart= scscc PPTT

PGE δγβ (2)

Solve (2) for 4scPδ and substitute into (1)

20

4 )( sccsc PTTP γβδ minusminusminus=

[ ] 0)(61

41)(

21 2

02

0 =minusminusminus++minus sccscc PTTPTT γβγβ

0121)(

31 2

0 =+minus scc PTT γβ

20 4 scc PTT

βγminus= (3)

(b) From (2) 0)( 42

0 =++minus scscc PPTT δγβ (4) From (3)

20 4

)( scc PTT γβ minus=minus (5)


Substituting (5) into (4)

04

422 =++minus scscsc PPP δγγ

δγ

432 minus=scP

βδγ 2

0 163+= TTc



a) Zeeman effect This is the shift in energy levels of atoms or molecules when they are put in a constant (in space and time) external magnetic field For states with a magnetic moment μ the energy shift is given by the dot product of minusμ and the magnetic field Thus the frequencies of light emitted by an electron making a transition between such levels are shifted This may cause splitting of emission spectral lines on placing the atom for example in a magnetic field which effect is commonly called the Zeeman effect

b) The inverse Zeeman effect is just the Zeeman effect seen in absorption

c) Voigt effect This happens in gases when a magnetic field is perpendicular to the direction of travel of light This can cause birefringence and thus in effect two indices of refraction These correspond to a situation in which the plane of vibration of the light is perpendicular or parallel to the magnetic field

d) CottonndashMouton effect This is the same as the Voigt effect except that it oc-curs in liquids and is generally stronger than the Voigt effect

e) Faraday effect This has already been discussed in the book It is the rotation of the plane of polarization of linearly polarized light when it propagates parallel to the magnetic field

f) Kerr magneto-optic effect This is not to be confused with the Kerr electro-optic effect discussed later Here we are concerned with reflecting polarized light from a magnetized surface If plane polarized light is reflected from a magnetized surface the polarization of the reflected light can be rotated with respect to the incident light Various effects are possible depending on the geometry of the polarized incident light and the direction of magneti-zation at the surface

Electro-optic effects

g) Stark effect Change in energy levels in an atom or molecule due to a uniform (in space or time) external magnetic field

766 Appendices

h) Inverse Stark effect Spectroscopically the Stark effect can be seen in emission or absorption When seen in absorption it is called the inverse Stark effect

i) Electric double refraction This is also called birefringence Electric double refraction is double refraction (two separate indices of refraction depending on whether their plane of vibration is parallel or perpendicular to the electric field)

j) Kerr electro-optic effect See the figure below This relates to ( i) and is dou-ble refraction in an electric field Supposes linearly polarized light is incident on the transparent material (usually a liquid such as benzene) at a 45-degree angle (direction of polarization to the electric field) with the electric field perpendicular to the direction of the light The light as it exits is then in gen-eral elliptically polarized If an analyzer has polarization direction perpen-dicular to the originally polarized light it will not transmit light (in eg ben-zene) when the field is off but it will when the field is on Again the refrac-tive index is one value for light polarized parallel to the electric field and another value for light polarized perpendicular to the electric field

For full details about all of these effects a good optics textbook should be consulted

φ1 (say 45deg)

Polarizer

lightSample

Analyzerz

zφ2

E

z

yx

Light propagates in the minusx direction The electric field is in the z direction The polarizer and analyzer are in the yz plane The electric field of light is polarized at minusφ1 to E


(a) [ ] i =xpx Expanding i=minus xpxp xx


Multiply through by px xxxx pxpxpp i2 =minus

iminus= xx xpxp

xxxx pxppxp i)i( 2 =minusminus

xxx pxpxp i222 minus=minus

If H = px2 m +V(x)

[ ] 0)( =xxV So

[ ] xpm

x i minus=H

With x a component of ˆ sdote r and px a component of ˆ sdote p we can generalize to

[ ] iˆ ˆm

= minuse r e psdot sdotH

(b) Referring to the definition of oscillator strength (see eqn above (1085))

2îj ijj jf b i jω= sdotsum sum e r

2 2ˆ ˆ

2ij ijj ji j i j

f b ωsdot + sdot

=sum sume r e r

( )ˆ ˆ ˆ ˆ2ij ijj jbf i j i j i j i jω lowast lowast= sdot sdot + sdot sdotsum sum e r e r e r e r

ˆ î j j ilowastsdot = sdote r e r for hermitian operator so

( )ˆ ˆ ˆ ˆ2ij ijj jbf i j j i i j j iω= sdot sdot + sdot sdotsum sum e r e r e r e r

jxi

i=minus xpxp xx

xxxppx xx i2 =minus

iminus= xx xpxp

xxpxpx xx i)i(2 =minusminus

768 Appendices

Assume i j are eigenstates of H

jj jω=H

jpm

jxjx iHiHii minus=minus

jpm

jxjx xji iiiiminus=minus ωω

jpm

jx xji iii1)( =minusωω

ωij = ωi minus ωj and generalizing

1ˆ î ij

i j i jmω

sdot = sdote r e p

1 1ˆ ˆ ˆ2 i iij ijj j

ij ji

bf i j j i i j j im m

ωω ω

⎛ ⎞= sdot sdot + sdot sdot⎜ ⎟⎜ ⎟

⎝ ⎠sum sum e p e r e r e p

jiij ωω minus=

( )ˆ ˆ ˆ ˆ2iijj jbf i j j i i j j im

= sdot sdot minus sdot sdotsum sum e p e r e r e p

Summing over j

ˆ ˆ2iijjbf i im

= ⎡ ⎤⎣ ⎦sum e r e psdot sdot

For ˆˆ =e i this leads to [ ]ˆ ˆ i= minuse p e rsdot sdot so that

m

bm

bfj ij 2)i(

i2minus=minus=sum

For b = minus2m 1=sum j ijf


The equation of motion is

eEB

NeG +minus

=+ v13

1vvel

2

0εμ


Assume a solution of the form tωi

0evv minus= Then

eEB

NeG =minus

minus+minus v13

1vvel

2

0

2ε

μω

From the definitions

ion0

ion 31 αε

NB =

and

Ge

Ee 2

locion

v ==α

at ω = 0

GeNB

2

0ion 3

1)0(ε

=

The equation now becomes

eEB

GBG =

minusminus+minus v

1)0(

vvel

ion2μω

eEB

BG =⎟⎟⎠

⎞⎜⎜⎝

⎛minus

minus+minus v1

)0(1v

el

ion2μ

μμω

Define

⎟⎟⎠

⎞⎜⎜⎝

⎛minus

minus=el

ion21

)0(1

BBG

T μω

(ie ω = ωT when E = 0 see (10188))Then

eET =minus v)( 22 ωωμ

For the local field

el00

loc 1v

31

3 BNeEPEEminus

+=+=εε

Substituting

)(

v 22 ωωμ minus=

T

eE

770 Appendices

Then

)(13

122el0

locωωμε minusminus

+=T

eEB

NeEE

)(13

11 22el0

loc

ωωμε minusminus+=

T

eB

NeE

E

)(

11

)0(1 22el

ionloc

ωωμ minusminus+=

TBGB

EE

)()1(

)0()()1(22

el

ion22

loc

ωωμωωμ

minusminus+minusminus

=T

Tel

BGBB

EE

Inverting

)0()()1(

)()1(

ion22

22el

loc GBBB

EE

Tel

T

+minusminusminusminus

=ωωμ

ωωμ

))(1()0(1

1

22el

ion

ωωμ minusminus+

=

T

locB

BGEE

Define

))(1(

)0(22

el

ion

ωωμ minusminus=

TBBGF

Then

FE

E+

=1

1

loc

Using the definition

loc0

ion0

ionv

31

31)(

EeNNB

εα

εω ==

Substitute in v

)(3

1)( 22loc0ion

ωωμεω

minus=

T

eEE

eNB

)(

)0()( 22

ion

locion

ωωμω

minus=

T

GBE

EB

)1(1

1)( elion BFF

B minus+

=ω


Given

[ ] )1()(1)(3)()(

elionel

ion

0 BBBB

minusminusminus=infinminus

ωω

εεωε

Substitute for Bion(ω)

)1()1(

11

)1(1

3)()(

elelel

el

00 BBF

FB

BF

F

minus⎥⎦⎤

⎢⎣⎡ minus

+minusminus

minus++infin=

εε

εωε

)1(

11

13)()(

el00 B

FF

FF

minus⎟⎠⎞

⎜⎝⎛

+minus

++infin=εε

εωε

)1(

3)()(

el00 BF

minus+infin=

εε

εωε

Substitute for F

))(1(

)0()1(

3)()(22

el

ion

el00 ωωμεε

εωε

minusminusminus+infin=

TBBG

B

Define c

2el

ion

)1()0(3

BBGcminus

=μ

Then

)(

)()(2200 ωωε

εεωε

minus+infin=

T

c

Rewriting

202)()(

ωεωε

εω =infinminus

minusc

T

From the Lyddane-Sache-Teller Relation

2

2

)0()(

L

T

ωω

εε =infin

)()0( 2

2infin= ε

ωωε

T

L

772 Appendices

Substituting ε(0) and letting ω = 0

0)()0(

02 =infinminus

minusεε

εω cT

0)()(2

202 =

infinminusinfinminus

εεωω

εω

T

LT

c

0)()( 22

202 =

infinminusminus

εωωωεωTL

TT

c

0)(

)(2

0222 =infin

minusminusεωεωωω T

TLTc

202)( LT

c ωεεω =infin

+



ψψ Em

=nablaminus 22

2

For each coordinate

0222

2=+minus i

i

i

mEdxd ψψ

where E1 + E2 + E3 = ETotal

⎟⎟⎠

⎞⎜⎜⎝

⎛+⎟

⎟⎠

⎞⎜⎜⎝

⎛= i

ii

ii xmEBxmEA 22

2cos2sinψ

assuming the walls of the rigid box are at xi = 0 and xi = a ψi(0) = 0 ψi(a) = 0 Thus B =0 and

2

222

2 2or2

manEnamE i

iii ππ ==

)(2

23

22

212

22nnn

maET ++= π


For a non-vanishing wave function we have for example the lowest and next lowest energy tabulated as follows

n1 n2 n3 lowest energy 1 1 1 next lowest energy 2 1 1

Therefore the excitation energy from ground to first excited state is

2

22

2

22

23

2)111114(

mamaE ππ =minusminusminus++=Δ

Then ΔE = ħω = hf so

2

22

23

mahf π=

for a frequency of transition from the first excited state to the ground state There-fore

constant432 ==

mfa π


For the Hydrogen atom in 2D (where we assume for simplicity εε0 = K = 1) we can write in polar coordinates (r φ)

0

22

2

2

2

2

22

4where1

2 πεφeq

rq

rrrm=minus⎟

⎟⎠

⎞⎜⎜⎝

⎛

partpart+

partpart+

partpartminus=H

and m is the (reduced) mass of the electron The Schroumldinger equation for station-ary states is then

)()( φρφρ Eψψ =H

It is convenient to use the coordinates and energy

εεε RRYEmqEmq

r ==== )(2

2

4

2

2ρ

where RY is a Rydberg

ψψρ

ψφρρρρ

ε2

4

2

22

2

2

22

2

4

422

2)(11

22mqmqqqm

m=minus⎟

⎟⎠

⎞⎜⎜⎝

⎛

partpart+

partpart+

partpartminus

774 Appendices

Canceling we obtain

ψρψψ

φρρρρε

2111

21

2

2

22

2=minus⎟

⎟⎠

⎞⎜⎜⎝

⎛

partpart+

partpart+

partpartminus

We seek a solution of the form

)()()( φρφρψ ΦΨ=

where by separation of variables we can choose

φφ mie)( =Φ

The differential equation becomes

Ψ=ΨminusΨ⎟⎟⎠

⎞⎜⎜⎝

⎛minus+minus

21

21 2

2

2 ερρρρρ

mdd

dd

We let

2

and1 2

2xαρ

α=minus=ε

The differential equation can be written

Ψ=Ψ+⎟⎟⎠

⎞⎜⎜⎝

⎛ΨminusΨ+Ψminus α2

241

21 22

2

2 xxm

dxd

xdxd

Comparing to Park [19 (2nd edn) p 215 prob 78] we see

21122 +=+= nn αα

(where n = zero or positive integer + |2m|) and 2m = 0 plusmn1 plusmn2 etc so the n are positive integers and

2212

21 )(

or)(

1+

minus=+

minus=n

REn

ε

This is the desired answer since

21)1(

21 minus+=+ nn

and then n + 1 is a nonzero integer



This problem is almost the same result as problem 114 Note the quantum well represents the donor atom which is assumed to be a hydrogen atom with unusual boundary conditions We have to separate out the z dependence to have

)()(2 2

22zEz

dzd

mψψ =minus

where ψ(0) = 0 ψ(w) = 0 The x y dependent equation is the same as in 114 and gives the same eigenvalues For the z dependence

0)(2)(22

2=+ zmE

dzzd ψψ

⎟⎟⎠

⎞⎜⎜⎝

⎛+⎟

⎟⎠

⎞⎜⎜⎝

⎛= zmEBzmEAz 22

2cos2sin)(ψ

πψψ pwmEwB =rArr==rArr= 220)(and00)0(

where p is a nonzero integer so

2

222

2mwpE p

π=

Adding together the two eigenvalues obtained by separation of variables we have the desired result


Suppose the metallic surface is at x = 0 (with the metal for x le 0) For a charge of minuse at x the image charge is +e at minusx Thus the two charges are separated by 2x The electric force between the charges is

)attractivefor ()2(4

12

2

0minusminus=

xeF

πε

Since F = minusdVdx with V the potential energy

x

ex

edxx

eFdxdVxxxx

0

2

0

2

20

2

161

1616 πεπεπεminus=minus=+=minus=

infininfininfininfinintintint

Let V(infin) = E0 the barrier height in a vacuum (see Fig 1111) Then

x

eExV0

2

0 16)(

πεminus=

776 Appendices

An external electric field will add minuseE1x to the potential energy (field in minusx direc-tion assuming metal is grounded so V = 0 at x = 0 if no image charge) Then

x

exeEExV0

2

10 16)(

πεminusminus=

V(x) has a maximum at x1 where

01

=⎟⎠⎞

⎜⎝⎛

=xxdxdV

which gives

10

1 16 Eex

πε=

or

0

13

01 21)(

πεEeExV minus=

Thus

0

13

21

πεEe=Δ



a

V(r)

ndashV(0)

E (bound state)

r

The l = 0 (presumably including ground state case) is worked out for a three di-mensional square well potential by Schiff [A23 pp 83-84]


We assume a spherically symmetric square well as shown Defining for E lt 0

20 )(2 EVm +

=α

22mEminus=β

The eigenvalue condition for l = 0 is

βαα minus=)cot( a

This implies

2

so0)tan( παβαα gtltminus= aa

Also

E

EV

aa

minus=

+=

+= 0

22

2

22

)(cot11)(sin

βαα

αα

We estimate V0 by assuming the quantum dot is formed by AlAsGaAs boundaries with V0 cong 125 eV [1247 p 133] so for a binding energy of 1 eV V0 minus |E| = 025 eV Using

2πα gta

to estimate a

eV)1(22)(22

22

0 mEVma ππ

απ gt

+=cong

nm931)1061)(1019(2

)10636)(143(1931

34

min congtimestimes

timescongminusminus

minusa

See Schiff for a more precise development


A superlattice can be created by using alternating layers of two kinds of semicon-ductor with different bandgaps This can make a set of quantum wells which when sufficiently close together allow tunneling between the wells In the direction per-pendicular to the layers one is immediately reminded of the Kronig-Penny model We illustrate this below

778 Appendices

Energy Gap

Ec(z)

Eg1 Eg2

Valence Band

Conduction Band

d

z

Ev(z)

Consider a Type I compositional superlattice By (1123) we can write

EFFzEzmyxm

c =⎥⎥⎦

⎤

⎢⎢⎣

⎡+

partpartminus⎟

⎟⎠

⎞⎜⎜⎝

⎛

partpart+

partpartminus lowastlowast )(

22 2

22

2

2

2

22

where we have assumed a constant effective mass everywhere If we seek a solu-tion of the form

)(e )(i zF ykxk yx ψ+=

where exp[i(kxx+kyy] is the wave function for the in-plane motion then

32||

2

322

2)(

2)(

2Ek

mEkk

mE yx +equiv++= lowastlowast

where for the transverse motion

ψψ 32

22)(

2EzE

dzd

mc =⎟

⎟⎠

⎞⎜⎜⎝

⎛+minus lowast

This is just a square well periodic potential as considered by Kronig and Penny We could also solve this in a tight binding model by combining wave functions

χi(z) for single well problems If we assume for N wells

sum=

minus=N

ni

ndiq ndz

Nz

1

i )(e1)( χψ

then following the usual tight binding approximation for one dimension and using the ground state wave functions χ0 and energies E0

0 we find for E3

qdEE q cos003 γα minusminus=


The total energy would be

2

2 2 0 0( ) cos

2x yk k q x yE k k E qdm

α γlowast= + + minus minus

The bands introduced by the superlattice E3q are called superlattice minibands or just minibands

Note there are several assumptions involved in this model These include

1 The assumption of rectangular boundaries for the walls

2 The effective masses are the same in the wells and the barriers

3 Note the assumption of effective masses is not even valid if the thickness of the barriers and the walls are only a few atomic layers

4 The assumption that the (xy) motion is free particle like

5 For the tight binding model we are limited to narrow minibands


a) Under the assumptions of the problem the Schroumldinger wave equation sepa-rates The motion in the (xy) direction is free electron like and the motion in the z direction is like a particle in a box with infinite sides (clearly an approximation for realistic cases) Thus the allowed energies are

)(22

||

222

2

222

EE

kkmmw

nE

n

yx

+equiv

++= π

)(22

222

2||

2

|| yx kkm

km

E +==

b) Including a factor of 2 for spin the density of states can be represented by summing a delta function over all eigenvalues so

sum minusminus=yx kkn n EEEE || )(2)( δρ

Now in (kx ky) space

ndEm

dkdkkkd

2

2||||||

2

2

2

π

ππ

=

==

780 Appendices

so sumkxky can be replaced by an integral Suppose D(E) = ρ(E) per unit area Since there are d 2k(2π)2 states in d 2k per unit area

)(

)(2)2(

2)(

2

0||||22

sum

sum int

minus=

minusminus=infin

n n

n n

EEm

dEEEEmED

θπ

δππ

D(E) is called the 2D density of states for this situation

c) Assume E gtsim E3 then

23)(π

mED = (1)

In 3D

E

dEmdkmkE 2

22 221

2==

2

212)4(

)2(2

)unit volper (4)2(

2)(

223

23D3

EdEmmE

dkkdEED

ππ

ππ

=

=

or we get the usual result

32

22

1)(223

22D3

E

wmmED π

π ⎟⎟⎠

⎞⎜⎜⎝

⎛=

where

2

222

32

3mw

EE π==

Per unit area

222D3332

21)(

ππ

πmw

wmwED == (2)

which checks (1)


d)

D3D(E)w

2hπm

23hπm

E1 E2 E3

The stair case pattern is the two dimensional density of states per unit area

M13 Appendix B Solutions

Problem (B1) Solution

1001

512

512

51

2

51

2~

21

21

22

11

I

NN

NN

NN

NN

AA

=⎟⎠⎞

⎜⎝⎛=

⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜

⎝

⎛minus+

⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜

⎝

⎛

minus

+=

by using matrix multiplication and the given values of N1 N2 (see (B14a) and (B14b))

Bibliography

Chapter 1

11 Anderson PW Science 177 393-396 (1972) 12 Bacon GE Neutron Diffraction Clarendon Press Oxford 2nd edn (1962) 13 Bradley CJ and Cracknell AP The Mathematical Theory of Symmetry in Solids

Representation Theory for Point Groups and Space Groups Clarendon Press Oxford (1972)

14 Brown PJ and Forsyth JB The Crystal Structure of Solids Edward Arnold London 1975 (Chap 3)

15 Buerger MJ Elementary Crystallography John Wiley and Sons New York 1956 16 Daw MS ldquoModel of metallic cohesion The embedded-atom methodrdquo Phys Rev B

39 7441-7452 (1989) 17 de Gennes PG and Prost J The Physics of Liquid Crystals Clarendon Press

Oxford 2nd edn (1993) 18 Evjan HM Physical Review 39 675 (1932) 19 Ghatak AK and Kothari LS An Introduction to Lattice Dynamics Addison Wesley

Publishing Co Reading Mass 1972 p165ff 110 Herzfield CM and Meijer PHE ldquoGroup Theory and Crystal Field Theoryrdquo Solid

State Physics Advances in Research and Applications 12 1-91 (1961) 111 Horton GK ldquoIdeal Rare-Gas Crystalsrdquo American Journal of Physics 36(2) 93

(1968) 112 Janot C Quasicrystals A Primer Clarendon Press Oxford (1992) 113 Kittel C Introduction to Solid State Physics John Wiley and Sons New York

Seventh Edition 1996 Chaps 1-3 19 114 Koster GF ldquoSpace Groups and Their Representationsrdquo Solid State Physics

Advances in Research and Applications 5 174-256 (1957) 115 Levine D and Steinhardt PJ Phys Rev Lett 53 2477-83 (1984) 116 Maradudin AA Montrol EW and Weiss GW Theory of Lattice Dynamics in the

Harmonic Approximation Academic Press NY 1963 p 245 117 Moffatt WG Pearsall GW and Wulff J The Structure and Properties of

Materials Vol 1 John Wiley and Sons Inc New York 1964 118 Pauling L Nature of the Chemical Bond 2nd edn Cornell University Press

Ithaca 1945 119 Phillips FC An Introduction to Crystallography John Wiley and Sons Inc New

York 4th edn 1971 120 Pollock Daniel D ldquoPhysical Properties of Materials for Engineersrdquo CRC Press

Boca Raton 1993 121 Shechtman D Blech I Gratias D and Cahn JW Phys Rev Lett 53 1951-3 (1984)

784 Bibliography

122 Steinhardt PJ and Ostlund S The Physics of Quasicrystals World Scientific Publishing Singapore 1987

123 Streetman BG Solid State Electronic Devices Prentice Hall Englewood Cliffs NJ 4th edn (1995)

124 Tinkham M Group Theory and Quantum Mechanics McGraw-Hill Book Company New York 1964

125 Tosi MP ldquoCohesion of Ionic Solids in the Born Modelrdquo Solid State Physics Advances in Research and Applications 16 1-120 (1964)

126 Tran HT and Perdew JP ldquoHow Metals Bind The deformable jellium model with correlated electronsrdquo Am J Phys 71 1048-1061 (2003)

127 Webb MB and Lagally MG ldquoElastic Scattering of Low Energy Electrons from Surfacesrdquo Solid State Physics Advances in Research and Applications 28 301-405 (1973)

128 West Anthony R Solid State Chemistry and its Properties John Wiley and Sons New York 1984

129 Wigner EP and Seitz F ldquoQualitative Analysis of the Cohesion in Metalsrdquo Solid State Physics Advances in Research and Applications 1 97-126 (1955)

130 Wyckoff RWG ldquoCrystal Structuresrdquo Vols 1-5 John Wiley and Sons New York 1963-1968

Chapter 2

21 Anderson PW Science 177 393-396 (1972) 22 Bak TA (ed) Phonons and Phonon Interactions W A Benjamin New York 1964 23 Bilz H and Kress W Phonon Dispersion Relations in Insulators Springer 1979 24 Blackman M ldquoThe Specific Heat of Solidsrdquo Encyclopedia of Physics Vol VII

Part 1 Crystal Physics 1 Springer Verlag Berlin 1955 p 325 25 Born M and Huang K Dynamical Theory of Crystal Lattices Oxford University

Press New York 1954 26 Brockhouse BN and Stewart AT Rev Mod Phys 30 236 (1958) 27 Brown FC The Physics of Solids W A Benjamin Inc New York 1967 Chap 5 28 Choquard P The Anharmonic Crystal W A Benjamin Inc New York (1967) 29 Cochran W ldquoInterpretation of Phonon Dispersion Curvesrdquo in Proceedings of the

International Conference on Lattice Dynamics Copenhagen 1963 Pergamon Press New York 1965

210 Cochran W ldquoLattice Vibrationsrdquo Reports on Progress in Physics Vol XXVI The Institute of Physics and the Physical Society London 1963 p 1 See also Cochran W The Dynamics of Atoms in Crystals Edward Arnold London 1973

211 deLauney J ldquoThe Theory of Specific Heats and Lattice Vibrationsrdquo Solid State Physics Advances in Research and Applications 2 220-303 (1956)

212 Dick BG Jr and Overhauser AW Physical Review 112 90 (1958) 213 Dorner B Burkel E Illini T and Peisl J Z fuumlr Physik 69 179-183 (1989) 214 Dove MT Structure and Dynamics Oxford University Press 2003 215 Elliott RJ and Dawber DG Proc Roy Soc A223 222 (1963) 216 Ghatak K and Kothari LS An Introduction to Lattice Dynamics Addison-Wesley

Publ Co Reading MA 1972 Chap 4

Chapter 2 785

217 Grosso G and Paravicini GP Solid State Physics Academic Press New York 2000 Chaps VIII and IX

218 Huntington HB ldquoThe Elastic Constants of Crystalsrdquo Solid State Physics Advances in Research and Applications 7 214-351 (1958)

219 Jensen HH ldquoIntroductory Lectures on the Free Phonon Fieldrdquo in Phonons and Phonon Interactions Bak TA (ed) W A Benjamin New York 1964

220 Jones W and March NA Theoretical Solid State Physics John Wiley and Sons (1973) Vol I Chap 3

221 Joshi SK and Rajagopal AK ldquoLattice Dynamics of Metalsrdquo Solid State Physics Advances in Research and Applications 22 159-312 (1968)

222 Kunc K Balkanski M and Nusimovici MA Phys Stat Sol B 71 341 72 229 249 (1975)

223 Lehman GW Wolfram T and DeWames RE Physical Review 128(4) 1593 (1962)

224 Leibfried G and Ludwig W ldquoTheory of Anharmonic Effects in Crystalsrdquo Solid State Physics Advances in Research and Applications 12 276-444 (1961)

225 Lifshitz M and Kosevich AM ldquoThe Dynamics of a Crystal Lattice with Defectsrdquo Reports on Progress in Physics Vol XXIX Part 1 The Institute of Physics and the Physical Society London 1966 p 217

226 Maradudin A Montroll EW and Weiss GH ldquoTheory of Lattice Dynamics in the Harmonic Approximationrdquo Solid State Physics Advances in Research and Applications Supplement 3 (1963)

227 Messiah A Quantum Mechanics North Holland Publishing Company Amsterdam 1961 Vol 1 p 69

228 Montroll EW J Chem Phys 10 218 (1942) 11 481 (1943) 229 Schaefer G Journal of Physics and Chemistry of Solids 12 233 (1960) 230 Scottish Universities Summer School Phonon in Perfect Lattices and in Lattices

with Point Imperfections 1965 Plenum Press New York 1960 231 Shull CG and Wollan EO ldquoApplication of Neutron Diffraction to Solid State

Problemsrdquo Solid State Physics Advances in Research and Applications 2 137 (1956)

232 Srivastava GP The Physics of Phonons Adam Hilger Bristol 1990 233 Strauch D Pavone P Meyer AP Karch K Sterner H Schmid A Pleti Th Bauer R

Schmitt M ldquoFestkoumlrperproblemerdquo Advances in Solid State Physics 37 99-124 (1998) Helbig R (ed) BraunschweigWeisbaden Vieweg

234 Toya T ldquoLattice Dynamics of Leadrdquo in Proceedings of the International Conference on Lattice Dynamics Copenhagen 1963 Pergamon Press New York 1965

235 Van Hove L Phys Rev 89 1189 (1953) 236 Vogelgesang R et al Phys Rev B54 3989 (1996) 237 Wallis RF (ed) Proceedings of the International Conference on Lattice Dynamics

Copenhagen 1963 Pergamon Press New York 1965 238 Ziman JM Electrons and Phonons Oxford Clarendon Press (1962) 239 1962 Brandeis University Summer Institute Lectures in Theoretical Physics Vol 2

W A Benjamin New York 1963

786 Bibliography

Chapter 3

31 Altman SL Band Theory of Solids Clarendon Press Oxford 1994 See also Singleton J Band Theory and Electronic Structure Oxford University Press 2001

32 Aryasetiawan F and Gunnarson D Rep Prog Phys 61 237 (1998) 33 Austin BJ et al Phys Rev 127 276 (1962) 34 Berman R Thermal Conduction in Solid Clarendon Press Oxford 1976 p 125 35 Blount EI ldquoFormalisms of Band Theoryrdquo Solid State Physics Advances in

Research and Applications 13 305-373 (1962) 36 Blount EI Lectures in Theoretical Physics Vol V Interscience Publishers New

York 1963 p422ff 37 Bouckaert LP Smoluchowski R and Wigner E Physical Review 50 58 (1936) 38 Callaway J and March NH ldquoDensity Functional Methods Theory and

Applicationsrdquo Solid State Physics Advances in Research and Applications 38 135-221 (1984)

39 Ceperley DM and Alder BJ Phys Rev Lett 45 566 (1980) 310 Chelikowsky JR and Louie SG (eds) Quantum Theory of Real Materials Kluwer

Academic Publishers Dordrecht 1996 311 Cohen ML Physics Today 33 40-44 (1979) 312 Cohen ML and Chelikowsky JR Electronic Structure and Optical Properties of

Semiconductors Springer-Verlag Berlin 2nd edn (1989) 313 Cohen ML and Heine V ldquoThe Fitting of Pseudopotentials to Experimental Data

and their Subsequent Applicationrdquo Solid State Physics Advances in Research and Applications 24 37-248 (1970)

314 Cohen M and Heine V Phys Rev 122 1821 (1961) 315 Cusack NE The Physics of Disordered Matter Adam Hilger Bristol 1987 see

especially Chaps 7 and 9 316 Dimmock JO ldquoThe Calculation of Electronic Energy Bands by the Augmented

Plane Wave Methodrdquo Solid State Physics Advances in Research and Applications 26 103-274 (1971)

317 Fermi E Nuovo Cimento 2 157 (1934) 318 Friedman B Principles and Techniques of Applied Mathematics John Wiley and

Sons New York 1956 319 Harrison WA Pseudopotentials in the theory of Metals W A Benjamin Inc New

York 1966 320 Heine V ldquoThe Pseudopotential Conceptrdquo Solid State Physics Advances in

Research and Applications 24 1-36 (1970) 321 Herring C Phys Rev 57 1169 (1940) 322 Herring C Phys Rev 58 132 (1940) 323 Hohenberg PC and Kohn W ldquoInhomogeneous Electron Gasrdquo Phys Rev 136

B804-871 (1964) 324 lzynmov YA Advances in Physics 14(56) 569 (1965) 325 Jones RO and Gunnaisson O ldquoThe Density Functional Formalism its Applications

and Prospectsrdquo Rev Modern Phys 61 689-746 (1989) 326 Jones W and March NH Theoretical Solid State Physics Vol 1 and 2 John Wiley

and Sons 1973 327 Kohn W ldquoElectronic Structure of MatterndashWave Functions and Density

Functionalsrdquo Rev Modern Phys 71 1253-1266 (1999)

Chapter 3 787

328 Kohn W and Sham LJ ldquoSelf Consistent Equations Including Exchange and Correlation Effectsrdquo Phys Rev 140 A1133-1138 (1965)

329 Kohn W and Sham LJ Phys Rev 145 561 (1966) 330 Kronig and Penny Proceedings of the Royal Society (London) A130 499 (1931) 331 Landau L Soviet Physics JETP 3 920 (1956) 332 Loucks TL Phys Rev Lett 14 693 (1965) 333 Loumlwdin PO Advances in Physics 5 1 (1956) 334 Marder MP Condensed Matter Physics John Wiley and Sons Inc New York

2000 335 Mattuck RD A Guide to Feynman Diagrams in the Many-Body Problem McGraw-

Hill Book Company New York 2nd Ed 1976 See particularly Chap 4 336 Negele JW and Orland H Quantum Many Particle Systems Addison-Wesley

Publishing Company Redwood City California (1988) 337 Nemoshkalenko VV and Antonov VN Computational Methods in Solid State

Physics Gordon and Breach Science Publishers The Netherlands 1998 338 Parr RG and Yang W Oxford Univ Press New York 1989 339 Pewdew JP and Zunger A Phys Rev B23 5048 (1981) 340 Phillips JC and Kleinman L Phys Rev 116 287-294 (1959) 341 Pines D The Many-Body Problem W A Benjamin New York 1961 342 Raimes S The Wave Mechanics of Electrons in Metals North-Holland Publishing

Company Amsterdam (1961) 343 Reitz JR ldquoMethods of the One-Electron Theory of Solidsrdquo Solid State Physics

Advances in Research and Applications 1 1-95 (1955) 344 Schluumlter M and Sham LJ ldquoDensity Functional Techniquesrdquo Physics Today Feb

1982 pp 36-43 345 Singh DJ Plane Waves Pseudopotentials and the APW Method Kluwer

Academic Publishers Boston (1994) 346 Singleton J Band Theory and Electronic Properties of Solids Oxford University

Press (2001) 347 Slater JC [88 89 90] 348 Slater JC ldquoThe Current State of Solid-State and Molecular Theoryrdquo International

Journal of Quantum Chemistry I 37-102 (1967) 349 Slater JC and Koster GF Phys Rev 95 1167 (1954) 350 Slater JC and Koster GF Phys Rev 96 1208 (1954) 351 Smith N ldquoScience with soft x-raysrdquo Physics Today 54(1) 29-54 (2001) 352 Spicer WE Phys Rev 112 p114ff (1958) 353 Stern EA ldquoRigid-Band Model of Alloysrdquo Phys Rev 157(3) 544 (1967) 354 Thouless DJ The Quantum Mechanics of Many-Body Systems Academic Press

New York 1961 355 Tran HT and Pewdew JP ldquoHow metals bind The deformable-jellium model with

correlated electronsrdquo Am J Phys 71(10) 1048-1061 (2003) 356 Wannier GH ldquoThe Structure of Electronic Excitation Levels in Insulating

Crystalsrdquo Phys Rev 52 191-197 (1937) 357 Wigner EP and Seitz F ldquoQualitative Analysis of the Cohesion in Metalsrdquo Solid

State Physics Advances in Research and Applications 1 97-126 (1955) 358 Woodruff TO ldquoThe Orthogonalized Plane-Wave Methodrdquo Solid State Physics

Advances in Research and Applications 4 367-411 (1957) 359 Ziman JM ldquoThe Calculation of Bloch Functionsrdquo Solid State Physics Advances in

Research and Applications 26 1-101 (1971)

788 Bibliography

Chapter 4

41 Anderson HL (ed) A Physicists Desk Reference 2nd edn Article 20 Frederikse HPR p 310 AIP Press New York 1989

42 Appel J ldquoPolaronsrdquo Solid State Physics Advances in Research and Applications 21 193-391 (1968) A comprehensive treatment

43 Arajs S American Journal of Physics 37 (7) 752 (1969) 44 Bergmann DJ Physics Reports 43 377 (1978) 45 Brockhouse BN Rev Modern Physics 67 735-751 (1995) 46 Callaway J ldquoModel for Lattice Thermal Conductivity at Low Temperaturesrdquo

Physical Review 113 1046 (1959) 47 Feynman RP Statistical Mechanics Addison-Wesley Publ Co Reading MA

1972 Chap 8 48 Fisher ME and Langer JS ldquoResistive Anomalies at Magnetic Critical Pointsrdquo

Physical Review Letters 20(13) 665 (1968) 49 Garnett M Philos Trans R Soc (London) 203 385 (1904) 410 Geiger Jr FE and Cunningham FG ldquoAmbipolar Diffusion in Semiconductorsrdquo

American Journal of Physics 32 336 (1964) 411 Halperin BI and Hohenberg PC ldquoScaling Laws for Dynamical Critical

Phenomenardquo Physical Review 177(2) 952 (1969) 412 Holland MG ldquoPhonon Scattering in Semiconductors from Thermal Conductivity

Studiesrdquo Physical Review 134 A471 (1964) 413 Howarth DJ and Sondheimer EH Proc Roy Soc A219 53 (1953) 414 Jan JP ldquoGalvanomagnetic and Thermomagnetic Effects in Metalsrdquo Solid State

Physics Advances in Research and Applications 5 1-96 (1957) 415 Kadanoff LP ldquoTransport Coefficients Near Critical Pointsrdquo Comments on Solid

State Physics 1(1) 5 (1968) 416 Katsnelson AA Stepanyuk VS Szaacutesz AI and Farberovich DV Computational

Methods in Condensed Matter Electronic Structure American Institute of Physics 1992

417 Kawasaki K ldquoOn the Behavior of Thermal Conductivity Near the Magnetic Transition Pointrdquo Progress in Theoretical Physics (Kyoto) 29(6) 801 (1963)

418 Klemens PG ldquoThermal Conductivity and Lattice Vibration Modesrdquo Solid State Physics Advances in Research and Applications 7 1-98 (1958)

419 Kohn W Physical Review 126 1693 (1962) 420 Kohn W ldquoNobel Lecture Electronic Structure of MatterndashWave Functions and

density Functionalsrdquo Rev Modern Phys 71 1253-1266 (1998) 421 Kondo J ldquoResistance Minimum in Dilute Magnetic Alloysrdquo Progress in

Theoretical Physics (Kyoto) 32 37 (1964) 422 Kothari LS and Singwi KS ldquoInteraction of Thermal Neutrons with Solidsrdquo Solid

State Physics Advances in Research and Applications 8 109-190 (1959) 423 Kuper CG and Whitfield GD POLARONS AND EXCITONS Plenum Press New

York 1962 There are lucid articles by Froumlhlich Pines and others here as well as a chapter by F C Brown on experimental aspects of the polaron

424 Langer JS and Vosko SH Journal of Physics and Chemistry of Solids 12 196 (1960)

Chapter 5 789

425 MacDonald DKC ldquoElectrical Conductivity of Metals and Alloys at Low Temperaturesrdquo Encyclopedia of Physics Vol XIV Low Temperature Physics I Springer-Verlag Berlin 1956 p 137

426 Madelung O Introduction to Solid State Theory Springer-Verlag 1978 pp 153-155 183-187 370-373 A relatively simple and clear exposition of both the large and small polaron

427 Mahan GD Many Particle Physics Plenum Press New York 1981 Chaps 1 and 6 Greenrsquos functions and diagrams will be found here

428 Mattuck RD A Guide to Feynman Diagrams in the Many-Body Problem McGraw-Hill Book Company New York 1967

429 McMillan WL and Rowell JM Physical Review Letters 14 (4) 108 (1965) 430 Mendelssohn K and Rosenberg HM ldquoThe Thermal Conductivity of Metals a Low

Temperaturesrdquo Solid State Physics Advances in Research and Applications 12 223-274 (1961)

431 Mott NF Metal-Insulator Transitions Taylor and Francis London 1990 (2nd edn)

432 Olsen JL Electron Transport in Metals Interscience New York 1962 433 Patterson JD ldquoModern Study of Solidsrdquo Am J Phys 32 269-278 (1964) 434 Patterson JD ldquoError Analysis and Equations for the Thermal Conductivity of

Compositesrdquo Thermal Conductivity 18 Ashworth T and Smith DR (eds) Plenum Press New York 1985 pp 733-742

435 Pines D ldquoElectron Interactions in Metalsrdquo Solid State Physics Advances in Research and Applications 1 373-450 (1955)

436 Reynolds JA and Hough JM Proc Roy Soc (London) B70 769-775 (1957) 437 Sham LJ and Ziman JM ldquoThe Electron-Phonon Interactionrdquo Solid State Physics

Advances in Research and Applications 15 223-298 (1963) 438 Stratton JA Electromagnetic Theory McGraw Hill 1941 p 211ff 439 Ziman JM Electrons and Phonons Oxford London 1962 Chap 5 and later

chapters (esp p 497)

Chapter 5

51 Alexander W and Street A Metals in the Service of Man 7th edn Middlesex England Penguin 1979

52 Blatt FJ Physics of Electronic Conduction in Solids McGraw-Hill (1968) 53 Borg RJ and Dienes GJ An Introduction to Solid State Diffusion Academic Press

San Diego 1988 p 148-151 54 Cottrell A Introduction to the Modern Theory of Metals the Institute of Metals

London 1988 55 Cracknell AP and Wong KC The Fermi Surface Its Concept Determination and

Use in the Physics of Metals Clarendon Press Oxford 1973 56 Duke CB ldquoTunneling in Solidsrdquo in Supplement 10 Solid State Physics Advances

in Research and Applications (1969) 57 Fiks VB Sov Phys Solid State 1 14 (1959) 58 Fisk Z et al ldquoThe Physics and Chemistry of Heavy Fermionsrdquo Proc Natl Acad Sci

USA 92 6663-6667 (1995)

790 Bibliography

59 Gantmakher VF ldquoRadio Frequency Size Effect in Metalsrdquo Progress in Low Temperature Physics Vol V Gorter CJ (ed) North-Holland Publishing Company Amsterdam 1967 p 181

510 Harrison WA Applied Quantum Mechanics World Scientific Singapore 2000 Chap 21

511 Harrison WA and Webb MB (eds) The Fermi Surface John Wiley and Sons New York 1960

512 Huang K Statistical Mechanics John Wiley and Sons 2nd edn 1987 pp 247-255 513 Huntington HB and Grove AR J Phys Chem Solids 20 76 (1961) 514 Kahn AH and Frederikse HPR ldquoOscillatory Behavior of Magnetic Susceptibility

and Electronic Conductivityrdquo Solid State Physics Advances in Research and Applications 9 259-291 (1959)

515 Kittel C and Kroemer H Thermal Physics W H Freeman and Company San Francisco 2nd edn 1980 Chap 11

516 Langenberg DN ldquoResource Letter OEPM-1 on the Ordinary Electronic Properties of Metalsrdquo American Journal of Physics 36 (9) 777 (1968)

517 Lax B and Mavroides JG ldquoCyclotron Resonancerdquo Solid State Physics Advances in Research and Applications 11 261-400 (1960)

518 Lloyd JR ldquoElectromigration in integrated circuit conductorsrdquo J Phys D Appl Phys 32 R109-R118 (1999)

519 Mackintosh AR Sci Am 209 110 (1963) 520 Onsager L Phil Mag 93 1006-1008 (1952) 521 Overhauser AW ldquoCharge Density Waverdquo Solid State Physics Source Book Parker

SP (ed) McGraw-Hill Book Co 1987 pp 142-143 522 Overhauser AW ldquoSpin-Density Waverdquo Solid State Physics Source Book Parker

SP (ed) McGraw-Hill Book Co 1987 pp 143-145 523 Peierls R More Surprises in Theoretical Physics Princeton University Press NJ

1991 p29 524 Pippard AB ldquoThe Dynamics of Conduction Electronsrdquo Low Temperature Physics

deWitt C Dreyfus B and deGennes PG (eds) Gordon and Breach New York 1962 Also Pippard AB Magnetoresistance in Metals Cambridge University Press 1988

525 Radousky HB Magnetism in Heavy Fermion Systems World Scientific Singapore 2000

526 Shapiro SL and Teukolsky SA Black Holes White Dwarfs and Neutron Stars The Physics of Compact Objects John Wiley and Sons Inc New York 1983

527 Shoenberg D Magnetic Oscillations in Metals Cambridge University Press 1984 528 Sorbello RS ldquoTheory of Electromigrationrdquo Solid State Physics Advances in

Research and Applications 51 159-231 (1997) 529 Stark RW and Falicov LM ldquoMagnetic Breakdown in Metalsrdquo Progress in Low

Temperature Physics Vol V Gorter CJ (ed) North-Holland Publishing Company Amsterdam 1967 p 235

530 Stewart GR ldquoHeavy-Fermion Systemsrdquo Rev Modern Physics 56 755-787 (1984) 531 Thorne RE ldquoCharge Density Wave Conductorsrdquo Physics Today pp 42-47 (May

1996) 532 Wigner E and Huntington HB J Chem Phys 3 764-770 (1935) 533 Wilson AH The Theory of Metals Cambridge 1954

Chapter 6 791

534 Zak J ldquoDynamics of Electrons in Solids in External Fieldsrdquo I Physical Review 168(3) 686 (1968) II Physical Review 177(3) 1151 (1969) Also ldquoThe kq-Representation in the Dynamics of Metalsrdquo Solid State Physics Advances in Research and Applications 27 1-62 (1972)

535 Ziman JM Principles of Theory of Solids 2nd edn Cambridge 1972 536 Ziman JM Electrons in Metals A Short Guide to the Fermi Surface Taylor and

Francis London 1963

Chapter 6

61 Alferov ZI ldquoNobel Lecture The Double Heterostructure Concept and its application in Physics Electronics and Technologyrdquo Rev Modern Phys 73(3) 767-782 (2001)

62 Ashcroft NW and Mermin ND Solid State Physics Holt Rinehart and Winston New York 1976 Chapters 28 and 29

63 Bardeen J ldquoSurface States and Rectification at a Metal-Semiconductor Contactrdquo Physical Review 71 717-727 (1947)

64 Blakemore JS Solid State Physics Second Edition W B Saunders Co Philadelphia 1974

65 Boer KW Survey of Semiconductor Physics Electrons and Other Particles in Bulk Semiconductors Van Nostrand Reinhold New York 1990

66 Bube R Electronics in Solids Academic Press Inc New York 1992 3rd edn 67 Chen A and Sher A Semiconductor Alloys Plenum Press New York 1995 68 Cohen ML and Chelikowsky JR Electronic Structure and Optical Properties of

Semiconductors Springer-Verlag Berlin 2nd edn 1989 69 Conwell E and Weisskopf VF Physical Review 77 388 (1950) 610 Dalven R Introduction to Applied Solid State Physics Plenum Press New York

1980 See also second edition 1990 611 Dresselhaus G Kip AF and Kittel C Phys Rev 98 368 (1955) 612 Einspruch NG ldquoUltrasonic Effects in Semiconductorsrdquo Solid State Physics

Advances in Research and Applications 17 217-268 (1965) 613 Fan HY ldquoValence Semiconductors Ge and Sirdquo Solid State Physics Advances in

Research and Applications 1 283-265 (1955) 614 Fraser DA The Physics of Semiconductor Devices Clarendon Press Oxford 4th

edition 1986 615 Handler P ldquoResource Letter Scr-1 on Semiconductorsrdquo American Journal of

Physics 32 (5) 329 (1964) 616 Kane EO J Phys Chem Solids 1 249 (1957) 617 Kittel C Introduction to Solid State Physics Seventh Edition John Wiley and

Sons New York 1996 Chap 8 618 Kohn W ldquoShallow Impurity States in Si and Gerdquo Solid State Physics Advances in

Research and Applications 5 257-320 (1957) 619 Kroemer H ldquoNobel Lecture Quasielectronic Fields and Band Offsets Teaching

Electrons New Tricksrdquo Rev Modern Phys 73(3) 783-793 (2001) 620 Li M-F ldquoModern Semiconductor Quantum Physicsrdquo World Scientific Singapore

1994

792 Bibliography

621 Long D Energy Bands in Semiconductors Interscience Publishers New York 1968

622 Ludwig GW and Woodbury HH ldquoElectron Spin Resonance in Semiconductorsrdquo Solid State Physics Advances in Research and Applications 13 223-304 (1962)

623 McKelvey JP Solid State and Semiconductor Physics Harper and Row Publishers New York 1966

624 Merzbacher E Quantum Mechanics 2nd edn John Wiley amp Sons Inc New York 1970 Chap 2

625 Moss TS (ed) Handbook on Semiconductors Vol 1 Landberg PT (ed) North HollandElsevier (1992) Amsterdam (There are additional volumes)

626 Nakamura S Pearton S Fasol G The Blue Laser Diode The Complete Story Springer-Verlag New York 2000

627 Ovshinsky SR ldquoReversible Electrical Switching Phenomena in Disordered Structuresrdquo Physical Review Letters 21 1450 (1968)

628 Pankove JI and Moustaka TD (eds) ldquoGallium Nitride Irdquo Semiconductors and Semimetals Vol 50 Academic Press New York 1997

629 Pantiledes ST (editor) Deep Centers in Semiconductors Gordon and Breach Publishers Yverdon Switzerland 1992

630 Patterson JD ldquoNarrow Gap Semiconductorsrdquo Condensed Matter News 3 (1) 4-11 (1994)

631 Perkowitz S Optical Characterization of Semiconductors Academic Press San Diego 1993

632 Ridley BK Quantum Processes in Semiconductors Clarendon Press Oxford 1988

633 Sapoval B and Hermann C Physics of Semiconductors Springer-Verlag New York 1995

634 Seeger K Semiconductor Physics Springer-Verlag Berlin 4th edn 1989 635 Seitz F Physical Review 73 549 (1948) 636 Shockley W Electrons and Holes in Semiconductors D Van Nostrand New York

1950 637 Slater JC Quantum Theory of Molecules and Solids Vol III Insulators

Semiconductors and Metals McGraw-Hill Book Company New York 1967 638 Smith RA Wave Mechanics of Crystalline Solids John Wiley and Sons 1961

section 88 and appendix 1 639 Smith RA (ed) Semiconductors Proceedings of the International School of

Physics ldquoEnrico Fermirdquo Course XXII Academic Press New York 1963 640 Streetman BG Solid State Electronic Devices 2nd ed Prentice Hall Englewood

Cliffs NJ 1980 Also see the third edition (1990) 641 Sze SM Physics of Semiconductor Devices 2nd edn Wiley New York 1981 642 Sze SM (ed) Modern Semiconductor Device Physics John Wiley and Sons Inc

New York 1998 643 Willardson RK and Weber ER ldquoGallium Nitride IIrdquo Semiconductors and

Semimetals Vol 57 Academic Press 1998 644 Yu PY and Cardona M Fundamentals of Semiconductors Springer Verlag Berlin

1996

Chapter 7 793

Chapter 7

71 Anderson PW ldquoTheory of Magnetic Exchange Interactions Exchange in Insulators and Semiconductorsrdquo Solid State Physics Advances in Research and Applications 14 99-214 (1963)

72 Ashcroft NW and Mermin ND Solid State Physics Holt Rinehart and Winston New York 1976 Chaps 31 32 and 33

73 Auld BA ldquoMagnetostatic and Magnetoelastic Wave Propagation in Solidsrdquo Applied Solid State Science Vol 2 Wolfe R and Kriessman CJ (eds) Academic Press 1971

74 Baibich MN Broto JM Fert A Nguyen Van Dau F Petroff F Eitenne P Creuzet G Friederich A and Chazelas J Phys Rev Lett 61 2472 (1988)

75 Bennett C ldquoQuantum Information and Computationrdquo Physics Today October 1995 pp 24-30

76 Bertram HN Theory of Magnetic Recording Cambridge University Press 1994 Chap 2

77 Bitko D et al J Research of NIST 102(2) 207-211 (1997)) 78 Blackman JA and Taguumlentildea J Disorder in Condensed Matter Physics A Volume in

Honour of Roger Elliott Clarendon Press Oxford 1991 79 Blundell S Magnetism in Condensed Matter Oxford University Press 2001 710 Charap SH and Boyd EL Physical Review 133 A811 (1964) 711 Chikazumi S Physics of Ferromagnetism (Translation editor Graham CD)

Oxford at Clarendon Press 1977 712 Chowdhury D Spin Glasses and Other Frustrated Systems Princeton University

Press 1986 713 Cooper B ldquoMagnetic Properties of Rare Earth Metalsrdquo Solid State Physics

Advances in Research and Applications 21 393-490 (1968) 714 Cracknell AP and Vaughn RA Magnetism in Solids Some Current Topics Scottish

Universities Summer School 1981 715 Craik D Magnetism Principles and Applications John Wiley and Sons 1995 716 Cullity BD Introduction to Magnetic Materials Addison-Wesley Reading Mass

1972 717 Damon R and Eshbach J J Phys Chem Solids 19 308 (1961) 718 Dyson FJ Physical Review 102 1217 (1956) 719 Elliott RJ Magnetic Properties of Rare Earth Metals Plenum Press London 1972 720 Fetter AL and Walecka JD Theoretical Mechanics of Particles and Continua

McGraw-Hill pp 399-402 1980 721 Fisher ME ldquoThe Theory of Equilibrium Critical Phenomenardquo Reports on Progress

in Physics XXX(II) 615 (1967) 722 Fischer KH and Hertz JA Spin Glasses Cambridge University Press 1991 723 Fontcuberta J ldquoColossal Magnetoresistancerdquo Physics World February 1999 pp

33-38 724 Gibbs MRJ (ed) Modern Trends in Magnetostriction Study and Application

Kluwer Academic Publishers Dordrecht 2000 725 Gilbert W De Magnete (originally published in 1600) Translated by P Fleury

Mottelay Dover New York (1958) 726 Griffiths RB Physical Review 136(2) 437 (1964)

794 Bibliography

727 Heitler W Elementary Wave Mechanics Oxford 1956 2nd edn Chap IX 728 Heller P ldquoExperimental Investigations of Critical Phenomenardquo Reports on

Progress in Physics XXX(II) 731 (1967) 729 Herbst JF Rev Modern Physics 63(4) 819-898 (1991) 730 Herring C Exchange Interactions among Itinerant Electrons in Magnetism Rado

GT and Suhl H (eds) Academic Press New York 1966 731 Herzfield CM and Meijer HE ldquoGroup Theory and Crystal Field Theoryrdquo Solid

State Physics Advances in Research and Applications 12 1-91 (1961) 732 Huang K Statistical Mechanics 2nd edn John Wiley and Sons New York 1987 733 Ibach H and Luth H Solid State Physics Springer-Verlag Berlin 1991 p 152 734 Julliere M Phys Lett 54A 225 (1975) 735 Kadanoff LP et al Reviews of Modern Physics 39 (2) 395 (1967) 736 Kasuya T Progress in Theoretical Physics (Kyoto) 16 45 and 58 (1956) 737 Keffer F ldquoSpin Wavesrdquo Encyclopedia of Physics Vol XVIII Part 2

Ferromagnetism Springer-Verlag Berlin 1966 738 Kittel C ldquoMagnonsrdquo Low Temperature Physics DeWitt C Dreyfus B and

deGennes PG (eds) Gordon and Breach New York 1962 739 Kittel C Introduction to Solid State Physics 7th edn John Wiley and Sons New

York 1996 Chapters 14 15 and 16 740 Kosterlitz JM and Thouless DJ J Phys C 6 1181 (1973) 741 Kouwenhoven L and Glazman L Physics World pp 33-38 Jan 2001 742 Langer JS and S H Vosko J Phys Chem Solids 12 196 (1960) 743 Levy RA and Hasegawa R Amorphous Magnetism II Plenum Press New York

1977 744 Malozemoff AP and Slonczewski JC Magnetic Domain Walls in Bubble Materials

Academic Press New York 1979 745 Manenkov AA and Orbach R (eds) Spin-Lattice Relaxation in Ionic Solids Harper

and Row Publishers New York 1966 746 Marshall W (ed) Theory of Magnetism in Transition Metals Proceedings of the

International School of Physics ldquoEnrico Fermirdquo Course XXXVII Academic Press New York 1967

747 Mathews J and Walker RL Mathematical Methods of Physics W A Benjamin New York 1967

748 Mattis DC The Theory of Magnetism I Statics and Dynamics Springer-Verlag 1988 and II Thermodynamics and Statistical Mechanics Springer-Verlag 1985

749 Mermin ND and Wagner H Physical Review Letters 17(22) 1133 (1966) 750 Muller B and Reinhardt J Neural Networks An Introduction Springer-Verlag

Berlin 1990 751 Pake GE ldquoNuclear Magnetic Resonancerdquo Solid State Physics Advances in

Research and Applications 2 1-91 (1956) 752 Parkin S J App Phys 85 5828 (1999) 753 Patterson JD Introduction to the Theory of Solid State Physics Addison-Wesley

Publishing Co Reading MA 1971 p176ff 754 Patterson JD et al Journal of Applied Physics 39 (3) 1629 (1968) and references

cited therein 755 Prinz GA Science Vol 282 27 Nov 1998 p 1660

Chapter 8 795

756 Rado GT and Suhl H (eds) Vol II Part A Statistical Models Magnetic Symmetry Hyperfine Interactions and Metals Academic Press New York 1965 Vol IV Exchange Interactions among Itinerant Electrons by Conyers Herring Academic Press New York 1966

757 Ruderman MA and Kittel C Physical Review 96 99 (1954)1 758 Salamon MB and Jaime M ldquoThe Physics of Manganites Structure and Transportrdquo

Rev Modern Physics 73 583-628 (2001) 759 Schrieffer JR ldquoThe Kondo EffectminusThe Link Between Magnetic and NonMagnetic

Impurities in Metalsrdquo Journal of Applied Physics 38(3) 1143 (1967) 760 Slichter CP Principles of Magnetic Resonance Harper and Row Evanston1963 761 Slonczewski JC Phys Rev B39 6995 (1989) 762 Tyalblikov SV Methods in the Quantum Theory of Magnetism Plenum Press New

York 1967 763 Van Vleck JH The Theory of Electric and Magnetic Susceptibilities Oxford

University Press 1932 764 Von der Lage FC and Bethe HA Phys Rev 71 612 (1947) 765 Walker LR Phys Rev 105 309 (1957) 766 Waller I Z Physik 79 370 (1932) 767 Weinberg S The Quantum Theory of Fields Vol I Modern Applications

Cambridge University Press 1996 pp 332-352 768 White RM Quantum Theory of Magnetism McGraw Hill New York 1970 769 Wohlfarth EP Magnetism Vol III Rado GT and Suhl H (eds) Academic Press

New York 1963 Wohlfarth EP Rev Mod Phys 25 211 (1953) Wohlfarth EP Handbook of Magnetic Materials Elsevier several volumes (~1993)

770 Wojtowicz PJ Journal of Applied Physics 35 991 (1964) 771 Yosida K Physical Review 106 893 (1957)1 772 Yosida K Theory of Magnetism Springer Berlin 1998 773 Zutic I et al ldquoSpintronics Fundamentals and applicationsrdquo Rev Mod Phys 76 325

(2004)

Chapter 8

81 Allen PB and Mitrovic B ldquoTheory of Superconductivity Tcrdquo Solid State Physics Advances in Research and Applications 37 2-92 (1982)

82 Anderson PW ldquoThe Josephson Effect and Quantum Coherence Measurements in Superconductors and Superfluidsrdquo Progress in Low Temperature Physics Vol V Gorter CJ (ed) North-Holland Publishing Company Amsterdam 1967 p 1 See also Annett JF Superconductivity Superfluids and Condensates Oxford University Press 2004

83 Annett JF Superconductivity Superfluids and Condensates Oxford University Press 2004

1 These papers deal with the indirect interaction of nuclei by their interaction with the

conduction electrons and of the related indirect interaction of ions with atomic magnetic moments by their interaction with the conduction electrons In the first case the hyperfine interaction is important and in the second the exchange interaction is important

796 Bibliography

84 Bardeen J ldquoSuperconductivityrdquo 1962 Cargese Lectures in Theoretical Physics Levy M (ed) W A Benjamin New York 1963

85 Bardeen J and Schrieffer JR ldquoRecent Developments in Superconductivityrdquo Progress in Low Temperature Physics Vol III Gorter CJ (ed) North-Holland Publishing Company Amsterdam 1961 p 170

86 Bardeen J Cooper LN and Schreiffer JR ldquoTheory of Superconductivityrdquo Phys Rev 108 1175-1204 (1957)

87 Beyers R and Shaw TM ldquoThe Structure of Y1Ba2Cu3O7-δ and its Derivativesrdquo Solid State Physics Advances in Research and Applications 42 135-212 (1989)

88 Bogoliubov NN (ed) The Theory of Superconductivity Gordon and Breoch New York 1962

89 Burns G High Temperature Superconductivity an Introduction Academic Press Inc Boston 1992

810 Cooper L ldquoBound Electron Pairs in a Degenerate Fermi Gasrdquo Phys Rev 104 1187-1190 (1956)

811 Dalven R Introduction to Applied Solid State Physics Plenum Press New York 1990 2nd edn Ch 8

812 deGennes PG Superconductivity of Metals and Alloys W A Benjamin New York 1966

813 Feynman RP Leighton RB and Sands M The Feynman Lectures on Physics Vol III Addison-Wesley Publishing Co 1965

814 Giaever I ldquoElectron Tunneling and Superconductivityrdquo Rev Modern Phys 46(2) 245-250 (1974)

815 Goodman BB ldquoType II Superconductorsrdquo Reports on Progress in Physics Vol XXIX Part 11 The Institute of Physics and The Physical Society London 1966 p 445

816 Hass KC ldquoElectronics Structure of Copper-Oxide Superconductorsrdquo Solid State Physics Advances in Research and Applications 42 213-270 (1989)

817 Jones W and March NH Theoretical Solid State Physics Vol 2 Dover 1985 p895ff and p1151ff

818 Josephson B ldquoThe Discovery of Tunneling Supercurrentsrdquo Rev Modern Phys 46(2) 251-254 (1974)

819 Kittel C [60] Chap 8 820 Kuper CG An Introduction to the Theory of Superconductivity Clarendon Press

Oxford 1968 821 Mahan G Many-Particle Physics Plenum Press New York 1981 Ch 9 822 Marder Condensed Matter Physics John Wiley and Sons 2000 p 819 823 Mattuck RD A Guide to Feynman Diagrams in the Many-Body Problem 2nd edn

Dover New York 1992 Chap 15 824 Parker WH Taylor BN and Langenberg DN Physical Review Letters 18 (8) 287

(1967) 825 Parks RD (ed) Superconductivity Vols 1 and 2 Marcel Dekker New York 1969 826 Rickaysen G Theory of Superconductivity Interscience New York 1965 827 Saint-James D Thomas EJ and Sarma G Type II Superconductivity Pergamon

Oxford 1969 828 Scalapino DJ ldquoThe Theory of Josephson Tunnelingrdquo Tunneling Phenomena in

Solids Burstein E and Lundquist S (eds) Plenum Press New York 1969

Chapter 9 797

829 Schafroth MR ldquoTheoretical Aspects of Superconductivityrdquo Solid State Physics Advances in Research and Applications 10 293-498 (1960)

830 Schrieffer JR Theory of Superconductivity W A Benjamin New York 1964 831 Silver AH and Zimmerman JE Phys Rev 157 317 (1967) 832 Tinkham M Introduction to Superconductivity McGraw-Hill New York 2nd edn

1996 833 Tinkham M and Lobb CJ ldquoPhysical Properties of the New Superconductorsrdquo Solid

State Physics Advances in Research and Applications 42 91-134 (1989)

Chapter 9

91 Bauer S Gerhard-Multhaupt R and Sessler GM ldquoFerroelectrets Soft Electroactive Foams for Transducersrdquo Physics Today 57 39-43 (Feb 2004)

92 Boumlttcher CJF Theory of Electric Polarization Elsevier Publishing Company New York 1952

93 Brown WF Jr ldquoDielectricsrdquo Encyclopedia of Physics Vol XVII Fluumlgge S (ed) Springer-Verlag Berlin 1956

94 Devonshire AF ldquoSome Recent Work on Ferroelectricsrdquo Reports on Progress in Physics Vol XXVII The Institute of Physics and The Physical Society London 1964 p 1

95 Elliot RJ and Gibson AF An Introduction to Solid State Physics and its Applications Harper and Row 1974 p277ff

96 Fatuzzo E and Merz WJ Ferroelectricity John Wiley and Sons New York 1967 97 Forsbergh PW Jr ldquoPiezoelectricity Electrostriction and Ferroelectricityrdquo

Encyclopedia of Physics Vol XVII Fluumlgge S (ed) Springer Verlag Berlin 1956 98 Froumlhlich H Theory of DielectricsmdashDielectric Constant and Dielectric Loss

Oxford University Press New York 1949 99 Gutmann F Rev Modern Phys 70 457 (1948) 910 Jona F and Shirane G Ferroelectric Crystals Pergamon Press New York 1962 911 Kanzig W ldquoFerroelectrics and Antiferroelectricsrdquo Solid State Physics Advances in

Research and Applications 4 1-97 (1957) 912 Lines ME and Glass AM Principles and Applications of Ferroelectrics and Related

Materials Oxford 1977 913 Moss TS Optical Properties of Semi-Conductors Butterworth and Company

Pubs London 1959 914 Pines D ldquoElectron Interaction in Metalsrdquo Solid State Physics Advances in

Research and Applications 1 373-450 (1955) 915 Platzman PM and Wolff PA Waves and Interactions in Solid State Plasmas

Academic Press New York 1973 Chaps VI and VII 916 Smyth CP Dielectric Behavior and Structure McGraw-Hill Book Company New

York 1955 917 Samara GA and Peercy PS ldquoThe Study of Soft-Mode Transitions at High

Pressurerdquo Solid State Physics Advances in Research and Applications 36 1-118 (1981)

918 Steele MC and Vural B Wave Interactions in Solid State Plasmas McGraw-Hill Book Company New York 1969

798 Bibliography

919 Tonks L and Langmuir I Phys Rev 33 195-210 (1929) 920 Uehling EA ldquoTheories of Ferroelectricity in KH2P04rdquo in Lectures in Theoretical

Physics Vol V Briton WE Downs BW and Downs J (eds) Interscience Publishers New York 1963

921 Zheludev IS ldquoFerroelectricity and Symmetryrdquo Solid State Physics Advances in Research and Applications 26 429-464 (1971)

Chapter 10

101 Born M and Wolf E Principles of Optics 2nd (Revised) edn MacMillan 1964 especially Optics of Metal (Chap XIII) and Optics of Crystals (Chap XIV)

102 Born M and Huang K Dynamical Theory of Crystal Lattices Oxford at the Clarenden Press 1954 see especially the optical effects (Chap VII)

103 Brown FC ldquoUltraviolet Spectroscopy of Solids with the Use of Synchrotron Radiationrdquo Solid State Physics Advances in Research and Applications 29 1-73 (1974)

104 Bube RH Photoconductivity of Solids John Wiley and Sons New York 1960 105 Caldwell DJ ldquoSome Observations of the Faraday Effectrdquo Proc Natl Acad Sci 56

1391-1398 (1966) 106 Callaway J ldquoOptical Absorption in an Electric-Fieldrdquo Physical Review 130(2)

549 (1963) 107 Cochran W The Dynamics of Atoms in Crystals Edward Arnold London 1973 p

90 108 Cohen MH Phil Mag 3 762 (1958) 109 Cohen ML and Chelikowsky JR Electronic Structure and Optical Properties of

Semiconductors 2nd edn Springer-Verlag Berlin 1989 1010 Dexter DL ldquoTheory of the Optical Properties of Imperfections in Nonmetalsrdquo

Solid State Physics Advances in Research and Applications 6 353-411 (1958) 1011 Elliott RJ and Gibson AF An Introduction to Solid State Physics Macmillan 1974

Chap 6 7 1012 Fox M Optical Properties of Solids Oxford University Press 2002 1013 Frova A Handler P Germano FA and Aspnes DE ldquoElectro-Absorption Effects at

the Band Edges of Silicon and Germaniumrdquo Physical Review 145(2) 575 (1966) 1014 Givens MP ldquoOptical Properties of Metalsrdquo Solid State Physics Advances in

Research and Applications 6 313-352 (1958) 1015 Gobeli GW and Fan HY Phys Rev 119(2) 613-620 (1960) 1016 Greenaway DL and Harbeke G Optical Properties and Band Structures of

Semiconductors Pergamon Press Oxford 1968 1017 Hagen E and Rubens H Ann d Physik (4) 11 873 (1903) 1018 Kane EO J Phys Chem Solids 12 181 (1959) 1019 Knox RS ldquoTheory of Excitonsrdquo Solid State Physics Advances in Research and

Applications Supplement 5 1963 1020 Lyddane RH Sachs RG and Teller E Phys Rev 59 673 (1941) 1021 Moss TS Optical Properties of Semiconductors Butterworth London 1961 1022 Pankove JL Optical Processes in Semiconductors Dover New York 1975

Chapter 11 799

1023 Phillips JC ldquoThe Fundamental Optical Spectra of Solidsrdquo Solid State Physics Advances in Research and Applications 18 55-164 (1966)

1024 Stern F ldquoElementary Theory of the Optical Properties of Solidsrdquo Solid State Physics Advances in Research and Applications 15 299-408 (1963)

1025 Tauc J (ed) The Optical Properties of Solids Proceedings of the International School of Physics ldquoEnrico Fermirdquo Course XXXIV Academic Press New York 1966

1026 Tauc J The Optical Properties of Semiconductors Academic Press New York 1966

1027 Yu PY and Cordona M Fundamentals of Semiconductors Springer-Verlag Berlin 1996 chapters 678

Chapter 11

111 Bastard G Wave Mechanics Applied to Semiconductor Heterostructures Halsted (1988)

112 Borg RJ and Dienes GJ An Introduction to Solid State Diffusion Academic Press San Diego 1988

113 Bube RH ldquoImperfection Ionization Energies in CdS-Type Materials by Photo-electronic Techniquesrdquo Solid State Physics Advances in Research and Applications 11 223-260 (1960)

114 Chelikowsky JR and Louie SG Quantum Theory of Real Materials Kluwer Academic Publishers Dordrecht 1996

115 Compton WD and Rabin H ldquoF-Aggregate Centers in Alkali Halide Crystalsrdquo Solid State Physics Advances in Research and Applications 16 121-226 (1964)

116 Cottrell AH Dislocations and Plastic Flow in Crystals Oxford University Press New York 1953

117 Crawford JH Jr and Slifkin LM Point Defects in Solids Vol 1 General and Ionic Crystals Vol 2 Defects in Semiconductors Plenum Press New York Vol 1 (1972) Vol 2 (1975)

118 Davison SG and Steslicka M Basic Theory of Surface States Clarendon Press Oxford 1992 p 155

119 deWit R ldquoThe Continuum Theory of Stationary Dislocationsrdquo Solid State Physics Advances in Research and Applications 10 249-292 (1960)

1110 Dexter DL ldquoTheory of Optical Properties of Imperfections in Nonmetalsrdquo Solid State Physics Advances in Research and Applications 6 353-411 (1958)

1111 Eshelby JD ldquoThe Continuum Theory of Lattice Defectsrdquo Solid State Physics Advances in Research and Applications 3 79-144 (1956)

1112 Fowler WB (ed) Physics of Color Centers Academic Press New York 1968 1113 Gilman JJ and Johnston WG ldquoDislocations in Lithium Fluoride Crystalsrdquo Solid

State Physics Advances in Research and Applications 13 147-222 (1962) 1114 Gundry PM and Tompkins FC ldquoSurface Potentialsrdquo in Experimental Methods of

Catalysis Research Anderson RB (ed) Academic Press New York 1968 pp 100-168

800 Bibliography

1115 Gourary BS and Adrian FJ ldquoWave Functions for Electron-Excess Color Centers in Alkali Halide Crystalsrdquo Solid State Physics Advances in Research and Applications 10 127-247 (1960)

1116 Henderson B Defects in Crystalline Solids Crane Russak and Company Inc New York 1972

1117 Kohn W ldquoShallow Impurity States in Silicon and Germaniumrdquo Solid State Physics Advances in Research and Applications 5 257-320 (1957)

1118 Kroumlger FA and Vink HJ ldquoRelations between the Concentrations of Imperfections in Crystalline Solidsrdquo Solid State Physics Advances in Research and Applications 3 307-435 (1956)

1119 Lehoczky SL et al NASA CR-101598 ldquoAdvanced Methods for Preparation and Characterization of Infrared Detector Materials Part Irdquo July 5 1981

1120 Li W and Patterson JD ldquoDeep Defects in Narrow-Gap Semiconductorsrdquo Phys Rev 50 14903-14910 (1994)

1121 Li W and Patterson JD ldquoElectronic and Formation Energies for Deep Defects in Narrow-Gap Semiconductorsrdquo Phys Rev 53 15622-15630 (1996) and references cited therein

1122 Luttinger JM and Kohn W ldquoMotion of Electrons and Holes in Perturbed Periodic Fieldsrdquo Phys Rev 99 869-883 (1955)

1123 Madelung O Introduction to Solid-State Theory Springer-Verlag Berlin (1978) Chaps 2 and 9

1124 Markham JJ ldquoF-Centers in the Alkali Halidesrdquo Solid State Physics Advances in Research and Applications Supplement 8 (1966)

1125 Mitin V Kochelap VA and Stroscio MA Quantum Heterostructures Cambridge University Press 1999

1126 Pantelides ST Deep Centers in Semiconductors 2nd edn Gordon and Breach Yverdon Switzerland (1992)

1127 Sarid D Exploring Scanning Probe Microscopy with Mathematica John Wiley and Sons Inc 1997 Chap 11

1128 Schulman JH and Compton WD Color Centers in Solids The Macmillan Company New York 1962

1129 Seitz F and Koehler JS ldquoDisplacement of Atoms During Irradiationrdquo Solid State Physics Advances in Research and Applications 2 305-448 (1956)

1130 Stoneham HM Theory of Defects in Solids Oxford 1973 1131 Wallis RF (ed) Localized Excitations in Solids Plenum Press New York 1968 1132 West AR Solid State Chemistry and its Applications John Wiley and Sons New

York 1984 1133 Zanquill A Physics at Surfaces Cambridge University Press 1988 p 293

Chapter 12

121 Barnham K and Vvendensky D Low Dimensional Semiconductor Structures Cambridge University Press Cambridge 2001

122 Bastard G Wave Mechanics Applied to Semiconductor Heterostructures Halsted Press New York 1988

Chapter 12 801

123 Blakemore JS Solid State Physics 2nd edn W B Saunders Company Philadelphia 1974 p 168

124 Brown TL LeMay HE Jr and Bursten BE Chemistry The Central Science 6th edn Prentice Hall Englewood Cliff NJ 07632 1994

125 Bullis WM Seiler DG and Diebold AC (eds) Semiconductor CharacterizationmdashPresent Status and Future Needs AIP Press Woodbury New York 1996

126 Butcher P March NH and Tosi MP Physics of Low-Dimensional Semiconductor Structures Plenum Press New York 1993

127 Callen HB Thermodynamics and an introduction to Thermostatistics John Wiley and Sons New York 1985 p339ff

128 Capasso F and Datta S ldquoQuantum Electron Devicesrdquo Physics Today 43 74-82 (1990)

129 Capasso F Gmachl C Siveo D and Cho A ldquoQuantum Cascade Lasersrdquo Physics Today 55 34-40 (May 2002)

1210 Cargill GS ldquoStructure of Metallic Alloy Glassesrdquo Solid State Physics Advances in Research and Applications 30 227-320 (1975)

1211 Chaikin PM and Lubensky TC Principles of Condensed Matter Physics Cambridge University Press 1995

1212 Chen CT and Ho KM Chap 20 ldquoMetal Surface Reconstructionsrdquo in Chelikowsky JR and Louie SG Quantum Theory of Real Materials Kluwer Academic Publishers Dordrecht 1996

1213 Davies JH and Long AR (eds) Nanostructures Scottish Universities Summer School and Institute of Physics Bristol and Philadelphia 1992

1214 Davison SG and Steslika M Basic Theory of Surface States Clarendon Press Oxford 1992

1215 deGennes PG and Prost J The Physics of Liquid Crystals Clarendon Press Oxford 2nd edn (1993)

1216 Doi M and Edwards SF The Theory of Polymer Dynamics Oxford University Oxford 1986

1217 Dresselhaus MS Dresselhaus G and Avouris P Carbon Nanotubes Springer-Verlag 2000

1218 Esaki L and Tsu R IBM J Res Devel 14 61 (1970) 1219 Fergason JL The Scientific American 74 (Aug 1964) 1220 Fisher KH and Hertz JA Spin Glasses Cambridge University Press 1991 see

especially p 55 pp 346-353 1221 Gaponenko SV Optical Properties of Semiconductor Nanocrystals Cambridge

University Press 1998 1222 Girvin S ldquoSpin and Isospin Exotic Order in Quantum Hall Ferromagnetsrdquo Physics

Today 39-45 (June 2000) 1223 Grahn HT (ed) Semiconductor SuperlatticesndashGrowth and Electronic Properties

World Scientific Singapore 1998 1224 Halperin BI ldquoThe Quantized Hall Effectrdquo Scientific American April 1986 52-60 1225 Hebard A ldquoSuperconductivity in Doped Fullerenesrdquo Physics Today 26-32

(November 1992) 1226 Herbst JF ldquoR2Fe14B Materials Intrinsic Properties and Technological Aspectsrdquo

Rev Modern Physics 63 819-898 (1991) 1227 Isihara A Condensed Matter Physics Oxford University Press New York 1991 1228 Jacak L Hawrylak P Wojs A Quantum Dots Springer Berlin 1998

802 Bibliography

1229 Jain JK ldquoThe Composite Fermion A Quantum Particle and its Quantum Fluidrdquo Physics Today April 2000 39-45

1230 Jones RAL Soft Condensed Matter Oxford University Press 2002 1231 Joyce BA Rep Prog Physics 48 1637 (1985) 1232 Kastner M ldquoArtificial Atomsrdquo Physics Today 46(1) 24-31 (Jan 1993) 1233 Kelly MJ Low Dimensional SemiconductorsndashMaterials Physics Technology

Devices Clarendon Press Oxford 1995 1234 Kivelson S Lee D-H and Zhang S-C ldquoGlobal phase diagram in the quantum Hall

effectrdquo Phys Rev B 46 2223-2238 (1992) 1235 Kivelson S Lee D-H and Zhang S-C ldquoElectrons in Flatlandrdquo Scientific American

March 1996 86-91 1236 Levy RA and Hasegana R (eds) Amorphous Magnetism II Plenum Press New

York 1977 1237 Lockwood DJ and Pinzuk A (eds) Optical Phenomena in Semiconductor

Structures of Reduced Dimensions Kluwer Academic Publishers Dordrecht 1993 1238 Laughlin RB ldquoQuantized Hall conductivity in two dimensionsrdquo Phys Rev B 23

5632-5633 (1981) 1239 Laughlin RB ldquoAnomalous quantum Hall effect An incompressible quantum fluid

with fractionally charged excitationsrdquo Phys Rev Lett 50 1395-1398 (1983) 1240 Laughlin RB Phys Rev Lett 80 2677 (1988) 1241 Laughlin RB Rev Mod Phys 71 863-874 (1999) 1242 Lee DH Phys Rev Lett 80 2677 (1988) 1243 Lee DH ldquoAnyon superconductivity and the fractional quantum Hall effectrdquo

International Journal of Modern Physics B 5 1695 (1991) 1244 Lu ZP et al Phys Rev Lett 92 245503 (2004) 1245 Lyssenko VG et al ldquoDirect Measurement of the Spatial Displacement of Bloch-

Oscillating Electrons in Semiconductor Superlatticesrdquo Phys Rev Lett 79 301 (1997)

1246 Mendez EE and Bastard G ldquoWannier-Stark Ladders and Bloch Oscillations in Superlatticesrdquo Physics Today 46(6) 39-42 (June 1993)

1247 Mitin VV Kochelap VA and Stroscio MA Quantum Heterostructures Cambridge University Press 1999

1248 Mott NF Metal-Insulator Transitions Taylor and Francis London 1990 2nd edn See especially pp 50-54

1249 Perkowitz S Optical Characterization of Semiconductors Academic Press New York 1993

1250 Poon W McLeish T and Donald A ldquoSoft Condensed Matter Where Physics meets Biologyrdquo Physics Education 37(1) 25-33 (2002)

1251 Prange RE and Girvin SM (eds) The Quantum Hall Effect 2nd edn Springer New York 1990

1252 Prutton M Introduction to Surface Physics Clarendon Press Oxford 1994 1253 Schab K et al Nature 404 974 (2000) 1254 Shik A Quantum Wells Physics and Electronics of Two-Dimensional Systems

World Scientific Singapore 1997 1255 Shklovskii BI and Efros AL Electronic Properties of Doped Semiconductors

Springer-Verlag Berlin 1984 1256 Stormer HL Rev Mod Phys 71 875-889 (1999) 1257 Strobl G The Physics of Polymers Springer-Verlag Berlin 2nd edn 1997

Chapter 12 803

1258 Tarton R The Quantum Dot Oxford Press New York 1995 1259 Tsui DC Rev Modern Phys 71 891-895 (1999) 1260 Tsui DC Stormer HL and Gossard AC ldquoTwo-dimensional magnetotransport in the

extreme quantum limitrdquo Phys Rev Lett 48 1559-1562 (1982) 1261 Vasko FT and Kuznetsov AM Electronic States and Optical Transitions in

Semiconductor Heterostructures Springer Berlin 1993 1262 von Klitzing K Dorda G and Pepper M ldquoNew Method for high-accuracy

determination of the fine-structure constant based on quantum Hall resistancerdquo Phys Rev Lett 45 1545-1547 (1980)

1263 von Klitzing K ldquoThe quantized Hall effectrdquo Rev Modern Phys 58 519-531 (1986) 1264 Wannier GH Phys Rev 117 432 (1969) 1265 Weisbuch C and Vinter B Quantum Semiconductor Structures Academic Press

Inc Boston 1991 1266 Wilczek F ldquoAnyonsrdquo Scientific American May 1991 58-65 1267 Zallen R The Physics of Amorphous Solids John Wiley New York 1983 1268 Zargwill A Physics at Surfaces Cambridge University Press New York 1988 1269 Zhang SC ldquoThe Chern-Simons-Landau-Ginzburg theory of the fractional quantum

Hall effectrdquo International Journal of Modern Physics B 6 25 1992 Additional References on nanophysics especially nanomagnetism (some of this material also relates to Chap 7 see section 751 on spintronics) Thanks to D J Sellmyer Univ of Nebraska-Lincoln for this list 1270 Hadjipanayis GC and Prinz GA (eds) ldquoScience and Technology of Nanostructured

Magnetic Materialsrdquo NATO Proceedings Kluwer Dordrecht (1991) 1271 Hadjipanayis GC and Siegel RW (eds) Nanophase Materials Synthesis -

Properties - Applications Kluwer Dortrecht (1994) 1272 Hernando A (Ed) ldquoNanomagnetismrdquo NATO Proceedings Kluwer Dordrecht

(1992) 1273 Jena P Khanna SN and Rao BK (eds) ldquoCluster and Nanostructure Interfacesrdquo

Proceedings of International Symposium World Scientific Singapore (2000) 1274 Maekawa S and Shinjo T Spin Dependent Transport in Magnetic Nanostructures

Taylor amp Francis London (2002) 1275 Nalwa HS (ed) Magnetic Nanostructures American Scientific Publishers Los

Angeles (2001) 1276 Nedkov I and Ausloos M (eds) Nano-Crystalline and Thin Film Magnetic Oxides

Kluwer Dordrecht (1999) 1277 Shi D Aktas B Pust L and Mikallov F (eds) Nanostructured Magnetic Materials

and Their Applications Springer Berlin (2003) 1278 Wang ZL Liu Y and Zhang Z (eds) Handbook of Nanophase and Nanostructured

Materials Kluwer Dortrecht (2002) 1279 Zhang J et al (eds) Self-Assembled Nanostructures Kluwer Dordrecht (2002)

804 Bibliography

Appendices

A1 Anderson PW ldquoBrainwashed by Feynmanrdquo Physics Today 53(2) 11-12 (Feb 2000)

A2 Anderson PW Concepts in Solids W A Benjamin New York 1963 A3 Ashcroft NW and Mermin ND Solid State Physics Holt Rhinhart and Wilson

New York 1976 pp 133-141 A4 Dekker AJ Solid State Physics Prentice-Hall Inc Englewood Cliffs NJ 1957

pp 240-242 A5 Economou EN Greenrsquos Functions in Quantum Physics Springer Berlin 1990 A6 Enz CP ldquoA Course on Many-Body Theory Applied to Solid-State Physicsrdquo World

Scientific Singapore 1992 A7 Fradkin E Field Theories of Condensed Matter Systems Addison-Wesley

Publishing Co Redwood City CA 1991 A8 Huang K Statistical Mechanics 2nd edn John Wiley and Sons New York 1987

pp 174-178 A9 Huang K Quantum Field Theory From Operators to Path Integrals John Wiley

and Sons Inc New York 1998 A10 Jones H The Theory of Brillouin Zones and Electronic States in Crystals North-

Holland Pub Co Amsterdam 1960 Chap 1 A11 Levy M (ed) 1962 Cargese Lectures in Theoretical Physics W A Benjamin Inc

New York 1963 A12 Mahan GD Many-Particle Physics Plenum New York 1981 A13 Mattsson AE ldquoIn Pursuit of the ldquoDivinerdquo Functionalrdquo Science 298 759-760 (25

October 2002) A14 Mattuck RD A Guide to Feynman Diagrams in the Many-Body Problem 2nd edn

Dover edition New York 1992 A15 Merzbacher E Quantum Mechanics 2nd edn John Wiley and Sons Inc New

York 1970 A16 Mills R Propagators for Many-particle Systems Gordon and Breach Science

Publishers New York 1969 A17 Negele JW and Henri Orland Quantum Many-Particle Systems Addison-Wesley

Publishing Co Redwood City CA 1988 A18 Nozieres P Theory of Interacting Fermi Systems W A Benjamin Inc New York

1964 see especially pp 155-167 for rules about Feynman diagrams A19 Patterson JD American Journal of Physics 30 894 (1962) A20 Phillips P Advanced Solid State Physics Westview Press Boulder CO 2003 A21 Pines D The Many-Body Problem W A Benjamin New York 1961 A22 Pines D Elementary Excitation in Solids W A Benjamin New York 1963 A23 Schiff LI Quantum Mechanics 3rd edn McGraw-Hill Book Company New York

1968 A24 Schrieffer JR Theory of Superconductivity W A Benjamin Inc New York 1964 A25 Starzak ME Mathematical Methods in Chemistry and Physics Plenum Press New

York 1989 Chap 5 A26 Van Hove L Hugenholtz NM and Howland LP Quantum Theory of Many-

Particle Systems W A Benjamin Inc New York 1961 A27 Zagoskin AM Quantum Theory of Many-Body Systems Springer Berlin 1998

Subject References 805

Subject References

Solid state of necessity draws on many other disciplines Suggested background reading is listed in this bibliography

Mechanics

1 Fetter AL and Walecka JD Theoretical Mechanics of Particles and Continua McGraw-Hill Book Co New York 1980 Advanced

2 Goldstein H Classical Mechanics 2nd edn Addison-Wesley Publishing Co Reading MA 1980 Advanced

3 Marion JB and Thornton ST Classical Dynamics of Particles and Systems Saunders College Publ Co Fort Worth 1995 Intermediate

Electricity

4 Jackson JD Classical Electrodynamics John Wiley and Sons 2nd edn New York 1975 Advanced

5 Reitz JD Milford FJ and Christy RW Foundations of Electromagnetic Theory Addison-Wesley Publishing Co Reading MA 1993 Intermediate

Optics

6 Guenther RD Modern Optics John Wiley and Sons New York 1990 Intermediate

7 Klein MV and Furtak TE Optics 2nd edn John Wiley and Sons New York 1986 Intermediate

Thermodynamics

8 Espinosa TP Introduction to Thermophysics W C Brown Dubuque IA 1994 Intermediate

9 Callen HB Thermodynamics and an Introduction to Thermostatics John Wiley and Sons New York 1985 Intermediate to Advanced

Statistical Mechanics

10 Kittel C and Kroemer H Thermal Physics 2nd edn W H Freeman and Co San Francisco 1980 Intermediate

11 Huang K Statistical Physics 2nd edn John Wiley and Sons New York 1987 Advanced

Critical Phenomena

12 Binney JJ Dowrick NJ Fisher AJ and Newman MEJ The Theory of Critical Phenomena Clarendon Press Oxford 1992 Advanced

806 Bibliography

Crystal Growth

13 Tiller WA The Science of Crystallization-Macroscopic Phenomena and Defect Generation Cambridge U Press 1991 and The Science of Crystallization-Microscopic Interfacial Phenomena Cambridge University Press Cambridge 1991 Advanced

Modern Physics

14 Born M Atomic Physics 7th edn Hafner Publishing Company New York 1962 Intermediate

15 Eisberg R and Resnick R Quantum Physics of Atoms Molecules Solids Nuclei and Particles 2nd edn John Wiley and Sons 1985 Intermediate

Quantum Mechanics

16 Bjorken JD and Drell SD Relativistic Quantum Mechanics McGraw-Hill New York 1964 and Relativistic Quantum Fields McGraw-Hill New York 1965 Advanced

17 Mattuck RD A Guide to Feynman Diagrams in the Many-body Problem 2nd edn McGraw-Hill Book Company New York 1976 Intermediate to Advanced

18 Merzbacher E Quantum Mechanics 2nd edn John Wiley New York 1970 Intermediate to Advanced

19 Park D Introduction to the Quantum Mechanics 3rd edn McGraw-Hill Inc New York 1992 Intermediate and very readable

Math Physics

20 Arfken G Mathematical Methods for Physicists 3rd edn Academic Press Orlando 1980 Intermediate

Solid State

21 Ashcroft NW and Mermin ND Solid State Physics Holt Reiehart and Winston New York 1976 Intermediate to Advanced

22 Jones W and March NH Theoretical Solid State Physics Vol 1 Perfect Lattices in Equilibrium Vol 2 Non-equilibrium and Disorder John Wiley and Sons London 1973 (also available in a Dover edition) Advanced

23 Kittel C Introduction to Solid State Physics 7th edn John Wiley and Sons Inc New York 1996 Intermediate

24 Parker SP Editor in Chief Solid State Physics Source Book McGraw-Hill Book Co New York 1987 Intermediate

25 Ziman JM Principles of the Theory of Solids Second Edition Cambridge University Press Cambridge 1972 Advanced


Condensed Matter

26 Chaikin PM and Lubensky TC Condensed Matter Physics Cambridge University Press Cambridge 1995 Advanced

27 Isihara A Condensed Matter Physics Oxford University Press Oxford 1991 Advanced

Computational Physics

28 Koonin SE Computational Physics BenjaminCummings Menlo Park CA 1986 Intermediate to Advanced

29 Press WH Flannery BP Teukolsky SA and Vetterling WT Numerical Recipes-The Art of Scientific Computing Cambridge University Press Cambridge 1986 Advanced

Problems

30 Goldsmid HJ (ed) Problems in Solid State Physics Academic Press New York 1968 Intermediate

General Comprehensive Reference

31 Seitz F Turnbull D Ehrenreich H (and others depending upon volume) Solid State Physics Advances in Research and Applications Academic Press New York a continuing series at research level

Applied Physics

32 Dalven R Introduction to Applied Solid State Physics 2nd edn Plenum New York 1990 Intermediate

33 Fraser DA The Physics of Semiconductor Devices 4th edn Oxford University Press Oxford 1986 Intermediate

34 Kroemer H Quantum Mechanics for Engineering Materials Science and Applied Physics Prentice-Hall Englewood Cliffs NJ 1994 Intermediate

35 Sze SM Semiconductor Devices Physics and Technology John Wiley and Sons 2nd edn New York 1985 Advanced

Rocks

36 Gueguen Y and Palciauskas V Introduction to the Physics of Rocks Princeton University Press Princeton 1994 Intermediate

History of Solid State Physics

37 Seitz F On the Frontier-My Life in Science AIP Press New York 1994 Descriptive

808 Bibliography

38 Hoddeson L Braun E Teichmann J and Weart S (eds) Out of the Crystal MazendashChapters from the History of Solid State Physics Oxford University Press Oxford 1992 Descriptive plus technical

The Internet

39 (httpxxxlanlgovlanl) This gets to arXiv which is an e-print source in several fields including physics It is presently owned by Cornell University

40 (httponlineitpucsbonline) Institute of Theoretical Physics at the University of California Santa Barbara programs and conferences available on line

Periodic Table

When thinking about solids it is often useful to have a good tabulation of atomic properties handy The Welch periodic chart of the atoms by Hubbard and Meggers is often useful as a reference tool

Further Reading

The following mostly older books have also been useful in the preparation of this book and hence the student may wish to consult some of them from time to time

41 Anderson PW Concepts in Solids W A Benjamin New York 1963 Emphasizes modern and quantum ideas of solids

42 Bates LF Modern Magnetism Cambridge University Press New York 1961 An experimental point of view

43 Billington DS and Crawford JH Jr Radiation Damage in Solids Princeton University Press Princeton New Jersey 1961 Describes a means for introducing defects in solids

44 Bloembergen N Nuclear Magnetic Relaxation W A Benjamin New York 1961 A reprint volume with a pleasant mixture of theory and experiment

45 Bloembergen N Nonlinear Optics W A Benjamin New York 1965 Describes the types of optics one needs with high intensity laser beams

46 Born M and Huang K Dynamical Theory of Crystal Lattices Oxford University Press New York 1954 Useful for the study of lattice vibrations

47 Brillouin L Wave Propagation in Periodic Structures McGraw-Hill Book Company New York 1946 Gives a unifying treatment of the properties of different kinds of waves in periodic media

48 Brout R Phase Transitions W A Benjamin New York 1965 A very advanced treatment of freezing ferromagnetism and superconductivity

49 Brown FC The Physics of SolidsmdashIonic Crystals Lattice Vibrations and Imperfections W A Benjamin New York 1967 A textbook with an unusual emphasis on ionic crystals The book has a particularly complete chapter on color centers

50 Buerger MJ Elementary Crystallography John Wiley and Sons New York 1956 A very complete and elementary account of the symmetry properties of solids

Further Reading 809

51 Choquard P The Anharmonic Crystal W A Benjamin New York 1967 This book is intended mainly for theoreticians except for a chapter on thermal properties The book should convince you that there are still many things to do in the field of lattice dynamics

52 Debye P Polar Molecules The Chemical Catalog Company 1929 reprinted by Dover Publications New York Among other things this book should aid the student in understanding the concept of the dielectric constant

53 Dekker AJ Solid State Physics Prentice-Hall Engelwood Cliffs New Jersey 1957 Has many elementary topics and treats them well

54 Frauenfelder H The Mossbauer Effect W A Benjamin New York 1962 A good example of relationships between solid state and nuclear physics

55 Grosso G and Paravicini GP Solid State Physics Academic Press 2000 modern 56 Harrison WA Pseudopotentials in the Theory of Metals W A Benjamin New

York 1966 The first book-length review of pseudopotentials 57 Holden A The Nature of Solids Columbia University Press New York 1965 A

greatly simplified view of solids May be quite useful for beginners 58 Jones H The Theory of Brillouin Zones and Electronic States in Crystals North-

Holland Publishing Company Amsterdam 1960 Uses group theory to indicate how the symmetry of crystals determines in large measure the electronic band structure

59 Kittel C Introduction to Solid State Physics John Wiley and Sons New York All editions have some differences and can be useful The latest is listed in [23] The standard introductory text in the field

60 Kittel C Quantum Theory of Solids John Wiley and Sons New York 1963 Gives a good picture of how the techniques of field theory have been applied to solids Most of the material is on a high level

61 Knox RS and Gold A Symmetry in the Solid State W A Benjamin New York 1964 Group theory is vital for solid state physics and this is a good review and reprint volume

62 Lieb EH and Mattis DC Mathematical Physics in One Dimension Academic Press New York 1966 This book is a collection of reprints with an introductory text Because of mathematical simplicity many topics in solid state physics can best be introduced in one dimension This book offers many examples of one dimensional calculations which are of interest to solid state physics

63 Loucks T Augmented Plane Wave Method W A Benjamin New York 1967 With the use of the digital computer the APW method developed by J C Slater in 1937 has been found to be a practical and useful technique for doing electronic band structure calculations This lecture note and reprint volume is by the man who developed a relativistic generalization of the APW method

64 Magnetism and Magnetic Materials Digest A Survey of Technical Literature of the Preceding Year Academic Press New York This is a useful continuing series put out by different editors in different years Mention should also be made of the survey volumes of Bell Telephone Laboratories called Index to the Literature of Magnetism

65 Materials A Scientific American Book W H Freeman and Company San Francisco 1967 A good elementary and modern view of many of the properties of solids Written in the typical Scientific American style

810 Bibliography

66 Mattis DC The Theory of MagnetismmdashAn Introduction to the Study of Cooperative Phenomena Harper and Row Publishers New York 1965 A modern authoritative account of magnetism advanced See also The Theory of Magnetism I and II Springer-Verlag Berlin 1988 (I) 1985 (II)

67 Mihaly L and Martin MC Solid State PhysicsmdashProblems and Solutions John Wiley 1996

68 Morrish AH The Physical Principles of Magnetism John Wiley and Sons New York 1965 A rather complete and modern exposition of intermediate level topics in magnetism

69 Moss TS Optical Properties of Semi-Conductors Butterworth and Company Publishers London 1959 A rather special treatise but it gives a good picture of the power of optical measurements in determining the properties of solids

70 Mott NF and Gurney RW Electronic Processes in Ionic Crystals Oxford University Press New York 1948 A good introduction to the properties of the alkali halides

71 Mott NF and Jones H Theory of the Properties of Metals and Alloys Oxford University Press New York 1936 A classic presentation of the free-electron properties of metals and alloys

72 Nozieres P Theory of Interacting Fermi Systems W A Benjamin New York 1964 A good account of Landaursquos ideas of quasi-particles Very advanced but helps to explain why ldquofree-electron theoryrdquo seems to work for many metals In general it discusses the many-body problem which is a central problem of solid state physics

73 Olsen JL Electron Transport in Metals Interscience Publishers New York 1962 A simple outline of theory and experiment

74 Pake GE Paramagnetic Resonance W A Benjamin New York 1962 This book is particularly useful for the discussion of crystal field theory

75 Peierls RE Quantum Theory of Solids Oxford University Press New York 1955 Very useful for physical insight into the basic nature of a wide variety of topics

76 Pines D Elementary Excitations in Solids W A Benjamin New York 1963 The preface states that the course on which the book is based concerns itself with the ldquoview of a solid as a system of interacting particles which under suitable circumstances behaves like a collection of nearly independent elementary excitationsrdquo

77 Rado GT and Suhl H (eds) Magnetism Vols I IIA IIB III and IV Academic Press New York Good summaries in various fields of magnetism which take one up to the level of current research

78 Raimes S The Wave Mechanics of Electrons in Metals North-Holland Publishing Company Amsterdam 1961 Gives a fairly simple approach to the applications of quantum mechanics in atoms and metals

79 Rice FO and Teller E The Structure of Matter John Wiley and Sons New York 1949 A very simply written book mostly words and no equations

80 Schrieffer JR Theory of Superconductivity W A Benjamin New York 1964 An account of the Bardeen Cooper and Schrieffer theory of superconductivity by one of the originators of the theory

81 Schulman JH and Compton WD Color Centers in Solids The Macmillan Company New York 1962 A nonmathematical account of color center research

Further Reading 811

82 Seitz F The Modern Theory of Solids McGraw-Hill Book Company New York 1940 This is still probably the most complete book on the properties of solids but it may be out of date in certain sections

83 Seitz F and Turnbull D (eds) (these are the original editors later volumes have other editors) Solid State Physics Advances in Research and Applications Academic Press New York Several volumes a continuing series This series provides excellent detailed reviews of many topics

84 Shive JN Physics of Solid State Electronics Charles E Merrill Books Columbus Ohio 1966 An undergraduate level presentation of some of the solid state topics of interest to electrical engineers

85 Shockley W Electrons and Holes in Semiconductors D van Nostrand Company Princeton New Jersey 1950 An applied point of view

86 Slater JC Quantum Theory of Matter McGraw-Hill Book Company New York 1951 also 2nd edn 1968 Good for physical insight

87 Slater JC Atomic Structure Vols I II McGraw-Hill Book Company New York 1960

88 Slater JC Quantum Theory of Molecules and Solids Vol I Electronic Structure of Molecules McGraw-Hill Book Company New York 1963

89 Slater JC Quantum Theory of Molecules and Solids Vol II Symmetry and Energy Bands in Crystals McGraw-Hill Book Company New York 1965

90 Slater JC Quantum Theory of Molecules and Solids Vol 111 Insulators Semiconductors and Metals McGraw-Hill Book Company New York 1967 The titles of these books [87 through 90] are self-descriptive They are all good books With the advent of computers Slaterrsquos ideas have gained in prominence

91 Slichter CP Principles of Magnetic Resonance with Examples from Solid State Physics Harper and Row Publishers New York 1963 This is a special topic but the book is very good and it has many transparent applications of quantum mechanics Also see 3rd edn Springer-Verlag Berlin 1980

92 Smart JS Effective Field Theories of Magnetism W B Saunders Company Philadelphia 1966 A good summary of Weiss field theory and its generalizations

93 Smith RA Wave Mechanics of Crystalline Solids John Wiley and Sons New York 1961 Among other things this book has some good sections on one-dimensional lattice vibrations

94 Van Vleck JH Theory of Electric and Magnetic Susceptibilities Oxford University Press New York 1932 Old but still very useful

95 Wannier GH Elements of Solid State Theory Cambridge University Press New York 1959 Has novel points of view on many topics

96 Weinreich G Solids Elementary Theory for Advanced Students John Wiley and Sons New York 1965 The title is descriptive of the book The preface states that the bookrsquos ldquopurpose is to give the reader some feeling for what solid state physics is all about rather than to cover any appreciable fractionrdquo of the theory of solids

97 Wilson AH The Theory of Metals Cambridge University Press New York 1954 2nd edn This book gives an excellent detailed account of the quasi-free electron picture of metals and its application to transport properties

98 Wood EA Crystals and Light D van Nostrand Company Princeton New Jersey 1964 An elementary viewpoint of this subject

812 Bibliography

99 Ziman JM Electrons and Phonons Oxford University Press New York 1960 Has interesting treatments of electrons phonons their interactions and applications to transport processes

100 Ziman JM Electrons in MetalsmdashA Short Guide to the Fermi Surface Taylor and Francis London 1963 Short qualitative and excellent

101 Ziman JM Elements of Advanced Quantum Theory Cambridge University Press New York 1969 Excellent for gaining an understanding of the many-body techniques now in vogue in solid state physics

Index

A

Abelian group 16 661 Absolute zero temperature 133 162 Absorption by excitons 545 Absorption coefficient 350 467

546 553 Absorptivity 549 AC Josephson Effect 482 486 Acceptors 295 563 594 Accidental degeneracy 199 451 Acoustic mode 64 92 223 528 Actinides 267 Adiabatic approximation 44 46 47 Ag 293 Al 96 286 292 295 454 508 661 Alfveacuten Waves 533 Alkali Halides 96 563 Alkali metals 184 216 267 273 Alkaline earth metals 273 Allowed and forbidden regions of

energy 178 Alloys 280 787 Amorphous chalcogenide

semiconductors 642 Amorphous magnet 433 644 Amorphous semiconductors 642 Amorphous solids 1 Anderson localization transition

642 Angle-resolved Photoemission 172 Angular momentum operators

82 446 Anharmonic terms 44 113 Anisotropy energy 424 426 444 Annihilation operator 80 396

Anomalous skin effect 275 276 Anticommutation relations 125

499 668 Antiferromagnetic resonance 442 Antiferromagnetism 367 372

385 420 Anti-site defects 592 Anti-Stokes line 582 583 Antisymmetric spatial wave

function 378 Ar 2 Asperomagnetic 433 Associative Law 14 Atomic Force Microscopy 615 Atomic form factor 35 Atomic number of the nucleus

42 116 Atomic polyhedra 184 Atomic wave functions 179 190

207 409 Attenuation 157 275 466 531 558 Au 273 275 293 448 458 459 Au1-xSix 281 Auger Electron Spectroscopy 615 Augmented plane wave 173 185 Avalanche mechanism 346 Average drift velocity 171 Axially symmetric bond bending 48 Axis of symmetry 18 21 426 Azbel-Kaner 276

B

BCS theory 465 Band bending 336 Band ferromagnetism 408 453

814 Index

Band filling 556 Band structure 172 Base-centered orthorhombic cell 24 Basis vectors 17 BaTi03 29 30 BaxLa2-xCuO4-y 506 Bcc lattice 38 271 Becquerel 600 Bell Labs 347 353 Beryllium 191 Binding forces 11 41 Bipolar junction transistor 352 Bitter Patterns 432 Black hole 291 Bloch Ansatz 251 Bloch condition 174 179 186

191 207 Bloch frequency 627 Bloch T32 law 215 Bloch Walls 431 Body-centered cubic cell 25 Body-centered cubic lattice 111

181 210 271 Body-centered orthorhombic cell

24 Bogoliubov-Valatin transformation

499 Bogolons 214 218 499 Bohr magneton 358 652 Boltzmann equation 244 309 633 Boltzmann gas constant 111 Boltzmann statistics 313 607 Bond stretching 48 Born approximation 457 Born-Haber cycle 9 Born-Mayer Theory 6 9 Born-Oppenheimer approximation

41 44 191 225 Born-von Kaacuterman or cyclic

boundary conditions 51 Bose statistics 224 Bose-Einstein condensation 62

510 639 Boson annihilation operators 90 Boson creation operators 90

Bosons 62 403 480 639 Bragg and von Laue Diffraction 31 Bragg peaks 21 22 39 97 285 Bragg reflection 175 176 268 Brass 29 292 Bravais lattice 13 23 177 Brillouin function 360 Brillouin scattering 97 582 Brillouin zone 54 86 Brillouin zone surface 177 Brownian motion 646 Buckyball 640 Built-in potential 331 345 348 Bulk negative conductivity 329 Burgers Vector 604

C

Ca 191 273 275 Canonical ensemble 61 167

358 662 Canonical equations 59 Carbon 11 292 614 640 641 Carbon nanotubes 640 641 Carrier drift currents 344 CaTiO3 29 Cauchy principal value 550 CCD 353 Cd 295 320 CdS 242 295 CdSe 295 297 CdTe 242 322 CeAl2 285 CeAl3 285 CeCu2Si2 285 506 508 Center of mass 50 68 480 Central forces 3 6 11 107 Ceramic oxide 506 Chandrasekhar Limit 291 292 Character 449 451 Charge density oscillations 382 Charge density waves 284 Chemical potential 249 497

501 623 Chemical vapor deposition 617

Index 815

Chemically saturated units 2 6 Classical Diatomic Lattices 64 Classical elastic isotropic continuum

waves 94 Classical equipartition theorem

43 94 Classical Heisenberg Ferromagnet

391 Classical specific heat 222 Clausius-Mossotti Equation

519 524 Close packing 6 Close-packed crystal structures 3 Closure 14 Co 427 Coercive Force 423 430 Coercivity 431 644 Coherence length 472 477 Cohesive energy 8 9 Cold-field emission 605 608 Color center 600 601 Commutation relations 77 80 396

492 560 666 Complex conjugate 117 206 470 Complex dielectric constant

515 546 Complex index of refraction

514 547 Complex refractive index 515 Composite 216 217 257 258 259

263 639 Compressibility 9 Concentration gradients 309 310 Conductance 293 620 622 624

625 633 636 638 641 Conducting polymers 648 Conduction band 133 216 552

562 594 618 623 631 Conservation law 84 218 225 571 Constant energy surfaces 182

268 277 Constraint of normalization 117 Continuity equation 330 342 Continuum frequencies 94 Controlled doping 295 327

Cooper pairs 494 Core electrons 178 192 356 Core repulsion 6 Correlation length 418 Correlations 2 115 129 135 163

193 217 242 267 643 Coulomb blockade model 620 Coulomb gauge 552 Coulomb potential energy 7 42

116 244 Covalent bonds 11 446 594 595 Covalent crystals 47 Cr 293 Creation operator 80 409 498

666 673 Critical exponents 417 Critical magnetic field 464 506 Critical point 98 418 562 580 Critical temperature 355 369 395

464 503 506 513 Crystal classes 13 Crystal field 358 378 386 445 Crystal growth 328 594 604

611 614 Crystal Hamiltonian 42 179 Crystal lattice with defects 70 206 Crystal mathematics 86 110 Crystal structure determination 30 Crystal symmetry operations 13 22 Crystal systems 13 23 26 Crystalline anisotropy 377 426 Crystalline potential 186 192 666 Crystalline solid 1 2 19 22

23 644 Crystalline state 1 Crystalline symmetry 2 13 Crystallography 1 13 21 23 CsCl structure 29 Cu 9 267 272 273 292 347 456

457 458 506 Cubic point group 18 19 Cubic symmetry 205 429 Cuprates 506 508 Curie constants 368 Curie law 359

816 Index

Curie temperature 360 362 416 644

Curie-Weiss behavior 522 525 Cutoff frequency 38 94 172 493 Cyclic group 16 661 Cyclotron frequency 167 215 276

311 317 531 588 632 Cyclotron resonance 275 304 313

316 317

D

Damon-Eshbach wave solutions 406

DC Josephson Effect 481 485 de Broglie wavelength 157 175

288 565 de Haas-Schubnikov Effect 275 de Haas-van Alphen effect 165 Debye approximation 38 92 251 Debye density of states 98 Debye function 96 Debye temperature 95 251

504 509 Debye-Huckel Theory 535 538 Debye-Waller factor 37 38 Deep defects 205 563 599 Defect 70 73 205 303 564 591

593 599 Degenerate semiconductors

349 557 deGennes factor 383 Degree of polymerization 646 Demagnetization field 390 442 Density functional theory 137 147

192 669 Density matrix 662 Density of states 63 94 95 98

159 182 321 415 479 538 Density of states for magnons 405 Depletion width 333 Destructive interference 157 Diamagnetic 164 356 372 Diamond 11 28 29 191 297

594 640

Diatomic linear lattice 64 65 67 Dielectric constant 233 259 513

516 518 595 Dielectric function 282 535 580 Dielectric screening 513 Differential conductivity 328 Diffusion 222 286 310 334 340

344 602 Diodes 619 Dipole moment 234 238 386 518 Dipole-dipole interactions 406 Dirac delta function 86 110 253

536 664 Dirac delta function potential 149 Dirac Hamiltonian 193 Direct band gap 238 556 557 Direct optical transitions 553 Direct product group 16 Dodecahedron 22 271 Domain wall 422 430 460 Donors 295 300 563 593 Drift current density 310 Drift velocity 171 296 309 319

328 531 635 Drude theory 565 d-wave pairing 506 Dynamical matrix 65 92 Dyson equation 675

E

Edge dislocation 603 Effective magneton number 359 Effective mass 156 276 285

296 306 Effective mass theory 595 626 Effusion cell 617 Einstein A and B coefficients

581 582 Einstein relation 311 Einstein theory of specific heat

of a crystal 112 Elastic continuum 41 93 96 Elastic restoring force 5 Elastically scattered 31

Index 817

Electric current density 171 246 257 286

Electric dipole interactions 47 Electric polarization 526 Electrical and thermal conductors

10 Electrical conductivity 171 251

275 295 298 304 309 Electrical current density 258

310 329 Electrical mobility 309 Electrical neutrality condition

302 333 Electrical resistivity 41 46 247

251 267 295 463 487 489 Electrochemical deposition 617 Electromagnetic wave 32 68 69

215 514 531 533 545 571 Electromechanical transducers

520 Electromigration 286 Electron correlations 137 243

252 267 Electron Energy Loss Spectroscopy

615 Electron paramagnetic resonance

436 Electron spin resonance 436 Electron tunneling 467 Electron volt 6 39 136 295

326 616 Electron-electron interaction 10

136 193 203 242 493 620 Electron-hole pairs 347 353 Electronic conduction spin density

381 Electronic conductivity 11 Electronic configuration 2 Electronic flux of heat energy 246 Electronic mass 42 Electronic surface states 591 Electron-lattice interaction 282

478 509 Electron-phonon coupling 233 480

503 566

Electron-phonon interaction 41 158 216 225 232 238 281 463 465 486

Ellipsometry 615 Empty lattice 268 Energy Dispersive X-ray

Spectroscopy 615 Energy tubes 279 Entropy 8 62 111 423 592 646 Envelope Functions 595 626 Epitaxial layers 626 Equilibrium properties 417 Equivalent one-electron problem

113 Euler-Lagrange 141 428 EuO 432 462 Eutectic 280 Exchange charge density 129 Exchange correlation energy 142

143 144 148 Exchange coupled spin system 390 Exchange energy 144 374 412

424 426 Exchange enhancement factor 415 Exchange integral 365 367 375

392 424 Exchange operator 128 132 Excitation and ionization of

impurities 545 Excitation of lattice vibrations 546 Exciton absorption 308 Excitons 215 218 Exhaustion region 302 Extended X-ray Absorption

Spectroscopy 615 Eyjen counting scheme 39

F

Face-centered cubic cell 25 Face-centered cubic lattice 112

181 210 Factor group 16 23 Faraday and Kerr effects 432 Faraday effect 587 589

818 Index

Fast Fourier Transform 617 f-band superconductivity 285 F-centers 600 Fe 292 428 447 Fe3O4 433 Fermi energy 160 253 303

493 538 Fermi function 160 248 298 303

501 537 623 Fermi gas 289 Fermi hole 129 133 134 Fermi liquid 135 217 Fermi momentum 136 290 Fermi polaron 233 Fermi surface 244 267 270 274 Fermi temperature 163 538 Fermi wave vector 285 Fermi-Dirac distribution 136 161

245 250 252 Fermion pairing 509 Fermions 119 124 138 639 Fermi-Thomas potential 543 Ferrimagnetism 367 369 420 Ferroelectric 12 29 513 520 Ferroelectric crystals 520 Ferromagnetic 190 355 360 362

373 385 399 407 422 426 456 486 513

Ferromagnetic resonance 441 Feynman diagrams 671 f-fold axis of rotational symmetry

26 Field effect transistor 327 352 Field Ion Microscopy 615 Finite phonon lifetimes 222 Five-fold symmetry 21 Fluorescence 30 581 Flux penetration 465 466 475 Fluxoid 477 506 Fourier analysis 71 88 174 Fourier coefficient 178 Fourier transform infrared

spectroscopy 617 Fowler-Nordheim equation 609 Fractional Quantum Hall Effect 639

Free carrier absorption 546 565 Free electrons 134 161 268 565 Free energy 1 100 662 Free ions 8 Free-electron gas 130 134 165 Frenkel excitons 215 563 Friction drag 309 Friedel oscillation 134 381

530 543 Fullerides 640 Fundamental absorption edge

556 557 Fundamental symmetries 225

G

GaAs 242 295 297 324 326 328 329 337 347 563 564 579 618 628 629 637

GaAs-AlAs 337 347 Gamma functions 403 GaN 326 Gap parameter 498 504 511 Gas crystals 2 Gas discharge plasmas 529 Gauge symmetry 422 Ge 29 191 354 Generator 16 Gibbs free energy 522 544 Ginzburg-Landau equation

470 471 Glide plane symmetry 20 Golden rule of perturbation theory

219 220 221 226 228 664 Goldstone excitations 422 Graded junction 330 334 Grain boundary 594 611 Grand canonical ensemble 300 Graphene 640 641 Gravitational energy 291 Green functions 671 672 Group element 15 448 449 Group multiplication 15 39 Group theory 448 661

Index 819

Group velocity 55 306 532 562 627

Gruneisen Parameter 99 102 Gunn effect 328 329 Gyromagnetic ratio 362 435 436

H

Hall coefficient 293 312 531 632 Hall effect 311 631 633 635 Halogen 6 Harmonic approximation 44 51 Harmonic oscillator 5 50 Hartree approximation 115 158

174 669 673 Hartree atomic units 651 Hartree-Fock analysis 413 Hartree-Fock approximation 119

137 243 669 674 He 1 209 421 463 506 Heat transport by phonons 223 Heavy electron superconductors

463 506 Heavy holes 296 321 326 Heisenberg Hamiltonian 355 361

373 377 379 398 Heitler-London approximation

11 374 Helicons 215 531 Helimagnetism 420 Helmholtz free energy 61 168 358 Hermann-Mauguin 26 Heterostructures 618 Hexagonal symmetry 25 Hg 295 320 504 508 511 Hg1-xCdxTe 320 322 HgTe 322 Higgs mode 422 Highly correlated electrons 244 Hilbert space 422 Hohenberg-Kohn Theorem

137 147 Hole Conduction 307 Hole effective mass 308 318 Holes 215 671

Holstein-Primakoff transformation 396 398

Homopolar bonds 11 Homostructures 613 Hopping conductivity 643 Hubbard Hamiltonian 408 Hume-Rothery 272 Hund-Mulliken method 379 Hydrogen atom 12 81 373 544

595 611 612 Hydrogen bond 12 520 Hydrogen Metal 273 Hydrogenic wave functions 598 Hyperfine interaction 445 468 795 Hyperfine splitting 483 Hysteresis loop 430 433

I

Ice 12 Icosahedron 22 Ideal crystals 17 Imperfections 430 563 Impurity mode 75 109 Impurity states 205 244 643 Index of refraction 513 547

561 584 Induced transition 581 Inelastic neutron diffraction 229 Infinite crystal 17 55 Infinite one-dimensional periodic

potential 148 Infrared Absorption 467 Infrared detector 295 322 Inhomogeneous Semiconductors

340 Injected minority carrier densities

345 Injection current 341 Inner product 16 199 InP 328 Interatomic potential 3 Interatomic spacing 3 223 244 Inter-band frequencies 513 Internal energy 8 62 102 367 662

820 Index

Interpolation methods 173 194 Interstitial atoms 593 Interstitial defects 603 Intra-band absorption 565 Ion core potential energies 130 Ionic conductivity 6 11 Ionic crystals 6 12 514 592 601 Iron oxypnictides 506 508 Irreducible representation 13 661 Isomorphic 16 23 39 80 322 Isoprene group 646 Isothermal compressibility 8 101 Itinerant electron magnetism 137

J

Jahn-Teller effect 447 Jellium 137 242 267 539 Joint density of states 555 561 Jones zones 175 Josephson effects 467 479 483 Junction capacitance 334 336

K

K 40 191 520 k space 159 275 KH2PO4 520 525 Kinematic correlations 129 Kohn anomalies 543 Kohn Effect 530 Kohn-Sham Equations 141 148 Kondo effect 284 355 457 Korteweg-de Vries equation 460 Kronecker delta 36 108 199

221 450 Kronig-Kramers equations 550 Kronig-Penney model 148 657

L

Lagrange equations 53 428 Lagrange multiplier 117 140 Lagrangian 53 59 77 146 Lagrangian mechanics 53 146

Landau diamagnetism 165 167 168 356 605 634

Landau levels 636 637 Landau quasi-particles 158 217 Landau theory 135 418 463 Landauer equation 624 Landau-Lifshitz equations 444 Lande g-factors 368 Lanthanides 267 Larmor frequency 441 Lasers 630 Latent heat 416 521 Lattice constant 9 298 530 626 Lattice of point ions 37 Lattice thermal conductivity 223 Lattice vibrations 41 42 47 55

62 65 Laughlin 637 638 Law of constancy of angle 13 Law of Dulong and Petit 64 Law of geometric progression 36 Law of Mass Action 300 332 Law of Wiedemann and Franz

171 264 LiF 6 Light Emitting Diode (LED)

327 346 Light holes 320 321 Lindhard theory 535 538 Line defects 593 Linear combination of atomic

orbitals 178 Linear lattice 61 64 67 76 109 Linear metal 267 281 Liquid crystals 1 614 644 645 Liquid nitrogen 322 506 Liquidus branches 280 Local density approximation 137

143 192 203 670 Local density of states 616 London penetration depth 470 471 Longitudinal mode 64 95 Longitudinal optic modes 323 Longitudinal plasma oscillations

529 570

Index 821

Lorentz-Lorenz Equation 519 Lorenz number 172 Low Energy Electron Diffraction

616 Low temperature magnon specific

heat 404 Lower critical field 466 476 509 Luminescence 581 Lyddane-Sachs-Teller Relation 576

M

Madelung constant 7 38 39 Magnetic anisotropy 385 Magnetic charge 429 Magnetic domains 422 423 Magnetic flux 258 464 509 Magnetic hysteresis 423 Magnetic induction 32 Magnetic interactions 116 Magnetic moment 164 355

425 795 Magnetic phase transition 416 Magnetic potential 389 Magnetic resonance 355 435 617 Magnetic specific heat 367

417 458 Magnetic structure 355 360

395 420 Magnetic susceptibility 164 275

279 356 372 381 458 461 Magnetization 164 168 356 Magnetoacoustic 276 Magnetoelectronics 453 Magnetoresistance 275 453

455 457 Magnetostatic energy 388 424 Magnetostatic self energy 389 Magnetostatic Spin Waves 406 Magnetostriction 428 Magnetostrictive energy 424 Magnon-magnon interactions 396 Magnons 215 394 399 Mass defect in a linear chain 73 Mass of the electron 46 158 651

Mass of the nucleus 42 Maxwell equations 32 263 389 Mean field theory 355 361

368 418 Medium crystal field 446 Meissner effect 422 471 509 Membranes 644 Mesoscopic 624 Metal Oxide Semiconductor Field

Effect Transistor 352 Metal Semiconductor Junctions

336 Metal-Barrier-Metal Tunneling 623 Metallic binding 10 Metallic densities 135 193 243 Mg 274 505 MgB2 463 505 Microgravity 324 610 Miller Indices 31 Minibands 626 Minority carrier concentrations

344 350 Mobility gap 642 Models of Band Structure 320 Molecular Beam Epitaxy 617 Molecular crystals 2 3 215 Molecular field constant 362

363 407 Monatomic case 66 67 Monatomic Lattice 51 Monoclinic Symmetry 24 Monomer 646 647 Monovalent metal 130 158 MOS transistors 352 MOSFET 339 352 637 Mott transition 173 643 Mott-Wannier excitons 215

N

N 2 506 N interacting atoms 59 Na 9 56 77 155 184 273

531 600 NaCl 6 9 28 579 600

822 Index

NaKC4H4O64H2O (Rochelle salt)

520 Nanomagnetism 435 Nanostructure 435 586 Narrow gap insulator 295 Narrow gap semiconductor 295

619 629 630 N-body problem 59 Nd2Fe14B 434 Nearest neighbor repulsive

interactions 8 Nearly free-electron approximation

173 186 201 Neacuteel temperature 367 370 Neacuteel walls 431 Negative Index of Refraction 584 Neutron diffraction 91 96 285 371 Neutron star 291 Ni 188 292 361 364 407 427 Noble metals 267 273 Nonequilibrium statistical properties

218 Non-radiative (Auger) transitions

564 Normal coordinate transformation

4 57 78 88 653 654 Normal coordinates 50 57 77 653 Normal mode 41 49 65 654 672 Normal or N-process 221 Normal subgroup 16 n-type semiconductor 336 354 Nuclear coordinates 42 Nuclear magnetic resonance 436 Nuclear spin relaxation time 468 N-V center 600

O

O 337 645 Occupation number space 80 666 Octahedron 22 One-dimensional crystal 53 207 One-dimensional harmonic

oscillators 3 One-dimensional lattices 47

One-dimensional potential well 210 211

One-electron Hamiltonian 356 One-electron models 148 One-particle operator 116 126 667 Optic mode 64 92 234 524 528 Optical absorption 557 601 Optical fibers 584 Optical lattice 587 Optical magnons 401 Optical phenomena 545 Optical phonons 216 546 571 582 Orbital angular momentum operator

358 666 Order parameter 418 458 Order-disorder transition 525 Orthogonality constraints 142 Orthogonalized plane wave 173

190 203 Orthorhombic symmetry 24 Oscillating polarization 382 Oscillator strength 516 555

560 590 Overlap catastrophe 377

P

Padeacute approximant 417 Pair tunneling 511 Parabolic bands 298 323 555 Paraelectric phase 521 522 Parallelepiped 17 22 85 159 Paramagnetic Curie temperature

363 Paramagnetic effects 357 Paramagnetic ions 360 445 Paramagnetic resonance 601 Paramagnetic susceptibility

357 415 Paramagnetism 166 357 367 415 Particle tunneling 511 512 Particle-in-a-box 154 Partition function 61 100 167

417 461 Passivation 337

Index 823

Pauli paramagnetism 135 357 605 Pauli principle 120 129 168 289 Pauli spin paramagnetism 164 Pauli susceptibility 415 Peierls transitions 281 284 Peltier coefficient 254 257 Penetration depth 471 475 477

506 580 Perfect diamagnetism 511 Periodic boundary conditions

48 51 Permalloy 431 Permittivity of free space 4 234 Permutation operator 120 Perovskite 29 521 Perpendicular twofold axis 26 Perturbation expansion 44 671 Phase space 182 245 Phase transition 203 281 361 408

416 458 521 525 Phonon 41 61 62 82 558 Phonon absorption 229 Phonon current density 223 Phonon density of states 99 229 Phonon emission 229 Phonon frequencies 222 Phonon radiation 222 Phononics 225 Phonon-phonon interaction 41 113

219 221 Phosphorescence 581 Photoconductivity 589 601 Photoelectric effect 172 349 Photoemission 172 545 580 Photoluminescent 581 Photon absorption 553 Photonics 584 Photons 172 565 581 Photovoltaic effect 346 348 Physical observables 78 Piezoelectric crystals 520 Pinned 208 338 442 Planar defects 593 Planck distribution 581 Plane polarized light 587 615

Plane wave solution 132 210 Plasma frequency 216 267 519

530 566 588 Plasmonics 533 Plasmons 136 216 530 615 Platinum 293 Pnictides 506 508 pn-junction 333 340 346 Point defects 591 600 Point group 13 17 22 23 26 27

39 445 Point scatterers 32 36 Point transformations 17 Poisson bracket relations 77 Polar crystals 216 520 Polar solids 546 Polaritons 571 Polarization 234 513 518 526

565 574 588 Polarization catastrophe 524 Polarization vectors 90 92 112

230 238 Polarons 216 233 241 Polyhedron 22 184 Polymers 1 614 644 645 Polyvalent metals 267 Population inversion 630 Positive definite Hermitian operator

114 Positrons 213 Potential barrier 338 340

558 609 Potential gradients 309 Primitive cells 17 Primitive translation 17 23 85

271 614 660 Principal threefold axis 26 Projection operators 179 Propagators 670 671 Proper subgroup 15 16 Pseudo binary alloys 320 Pseudo-Hamiltonian 200 Pseudopotential 178 190 194 267

320 322 323 504 615 p-type semiconductor 337 338 339

824 Index

p-wave pairing 506 Pyroelectric crystals 520

Q

Quantum computing 454 Quantum conductance 613 624 Quantum dot 613 619 620 Quantum electrodynamics 217

233 483 670 Quantum Hall Effect 165 636 Quantum mechanical inter-band

tunneling 346 Quantum wells 327 613 630 Quantum wires 613 619 641 Quasi Periodic 21 Quasi-classical approximation 346 Quasicrystals 21 Quasi-electrons 135 136 217 488 Quasi-free electron 113 158 513 Quasi-particles 135 217 670 Qubit 454

R

Radiation damage 592 Radiative transitions 564 630 Raman scattering 97 582 617 Rare earths 267 380 385 407

433 446 Rayleigh-Ritz variational principle

114 Real orthogonal transformation 89 Real solids 6 173 243 390 Reciprocal lattice 31 36 37 85

174 221 Reciprocal lattice vectors 35

270 614 Reciprocal space 36 176 Reducible representation 448 Reflection coefficient 547 Reflection High Energy Electron

Diffraction 616 Reflection symmetry 18 19 Reflectivity 203 549 566 578

Refractive index 513 Registry 617 Regular polyhedron 22 Relativistic corrections 116 192

193 204 243 665 Relativistic dynamics 289 Relativistic effects 43 Relativistic pressure 290 Relaxation region 567 Relaxation time 171 172 250 309 Relaxation time approximation 251

254 264 Remanence 423 430 Renormalization 417 419 488 669 Reptation 646 Repulsive force 3 6 Resonance frequencies 315 513 Resonant tunneling 619 629 Rest mass 193 289 664 Restrahl frequency 69 578 Restrahlen effect 514 Reverse bias breakdown 346 Reversible processes 8 Richardson-Dushmann equation

607 Riemann zeta functions 403 Rigid ion approximation 230 RKKY interaction 380 Rochelle salt 520 Rotary reflection 18 Rotation inversion axis 26 Rotational operators 82 Rotational symmetry 18 20 81 91

188 422 Rubber 645 Rushbrooke inequality 419

S

Saturation magnetization 407 411 423 431

s-band 179 184 409 sc lattice 38 210 Scaling laws 417 Scanning Auger microscope 616

Index 825

Scanning Electron Microscopy 432 616

Scanning Tunneling Microscopy 616

Scattered amplitude 40 Schoenflies 26 27 Schottky and Frenkel defects 592 Schottky Barrier 336 Schottky emission 609 Screening 539 Screening parameter 537 Screw axis symmetry 19 20 Screw dislocation 603 604 605 Second classical turning point 609 Second quantization 124 666 Secondary Ion Mass Spectrometry

616 Second-order phase transitions 416 Secular equation 50 Selection rules 225 229 571 Self-consistent one-particle

Hamiltonian 126 Semiconductor 295 557 570 591

594 642 Seven crystal systems 23 Shallow defects 298 593 595 Shell structure 115 Shockley diode 341 346 Shockley state 591 Short range forces 3 Si 29 191 317 322 Similarity transformation 448 Simple cubic cell 25 Simple cubic lattice 181 210

394 425 Simple monoclinic cell 24 Simple orthorhombic cell 24 Simple tetragonal cell 24 Single crystal 275 327 427 594 Single domain 422 423 Single electron transistors 619 Single particle wave functions 141 Single-ion anisotropy 387 388 Singlet state 375 506 SiO2 337 637

Skin-depth 276 Slater determinant 120 669 Slater-Koster model 205 599 Slow neutron diffraction 229 SmCo5 434 Sn 267 292 Soft condensed matter 1 613 644 Soft mode theory 520 525 Soft x-ray emission 210 Soft x-ray emission spectra 168 Solar cell 296 347 Solid state symmetry 17 Solitons 355 460 Space degrees of freedom 43 Space groups 13 23 28 Specific heat of an electron gas

135 161 Specific heat of an insulator 52 98 Specific heat of linear lattice 61 Specific heat of spin waves 402 Specific heat of the one-dimensional

crystal 61 Speed of light 32 639 664 Speromagnetic 433 Spherical harmonics 187 Spin 12 particle 43 Spin coordinate 119 375 Spin degeneracy 606 Spin density waves 285 Spin deviation quantum number

399 404 Spin diffusion length 454 Spin glass 355 433 458 Spin Hall Effect 313 Spin Hamiltonian 375 Spin polarization 453 Spin wave theory 366 391

415 420 Spin-lattice interaction 222 Spin-orbit interaction 204 446 664 Spin-polarized transport 453 Spin-spin relaxation time 436 440 Spintronics 453 454 456 601 Split-off band 320 Spontaneous emission 581

826 Index

Spontaneous magnetism 361 Spontaneous polarization 521 Spontaneously broken symmetry

422 510 Stark-Wannier Ladder 628 Steel 292 644 Stereograms 26 39 Stokes line 582 Stoner criterion 414 Stoner model 408 453 Strained layer 626 Strong crystal field 446 Structure factor 35 38

177 201 Subgroup 16 23 39 Substitutional atoms 70 593 Superconducting metals 464 Superconducting wave function

472 476 Superconductive state 463 Superconductivity 469 Superconductors 463 Superlattice 626 628 630 Surface defects 591 Surface reconstruction 616 Surface states 337 347 352 641 s-wave pairing 506 Symmetry operations 17 18 23

26 192 Symmorphic 23

T

Tamm states 591 Tensor effective mass 158 Tetragonal Symmetry 24 Tetrahedron 22 Thermal conductivity 100 169

223 246 255 Thermal energy 172 563 Thermal neutrons 30 229 Thermal resistance 52 222 Thermionic emission 135 605 Thermodynamic fluctuations

361 418

Thermodynamics of irreversible process 256

Thermoelectric power 255 256 Thomas-Fermi approximation 535 Thomas-Fermi-Dirac method 137 Three-dimensional lattice vibration

47 86 Three-dimensional periodic

potential 148 Threefold axis 20 26 Ti 293 Tight binding approximation 113

173 178 184 190 207 Total cohesive energy 8 Total exchange charge 129 Total reflection 570 Transistors 296 330 340 352

619 641 Transition metals 267 275 644 Translation operator 81 660 Translational symmetry 17 Transmission coefficient 609 623 Transmission Electron Microscopy

432 616 Transport coefficients 218 247 Transverse and longitudinal acoustic

modes 230 Trial wave function 115 190 243

494 497 544 Triclinic Symmetry 23 Triglycine selenate 520 Triglycine sulfate 520 Trigonal Symmetry 25 Triplet state 375 Triplet superconductivity 285 Two-atom crystal 50 Two-body forces 3 42 Two-dimensional defect 594 Two-fold axis of symmetry 26 Two-fold degeneracy 4 150 447 Two-particle operator 116 121 667 Type I superconductors 465

475 510 Type II superconductors 463

475 509

Index 827

U

UAl2 285 UBe13 285 506 508 Ultrasonic absorption 276 Ultrasonic attenuation 467 Ultrasonic wave 276 Ultraviolet photoemission 580 Umklapp process 221 232

251 487 Uncertainty principle 5 Unit cells 17 Unitary transformation 89 207 454 Unrestricted force constants

approach 47 Upper critical field 466 474 506

V

Vacancies 287 592 599 600 602 Valence band 594 Valence crystals 11 12 373 van der Waals forces 4 6 Van Hove singularities 561 Varactor 334 Variational principle 114 Variational procedure 140 Vector potential 165 166 470 552 Velocity operator 195 Verdet constant 589 Vertical transitions 556 Vibrating dipoles 4 Virgin curve 430 Virtual crystal approximation 320 Virtual excited states 2 Virtual magnons 506 Virtual phonons 233 237 506

Volume coefficient of thermal expansion α 101

Vortex region 466

W

W 293 Wall energy 427 428 Wannier excitons 563 Wannier function 207 409 Wave vector 55 Weak crystal fields 446 Weak superconductors 509 Weiss theory 361 363 408 417 Whiskers 603 White dwarf 288 WiedemannndashFranz Law 169 Wigner-Seitz cell 184 186 231 Wigner-Seitz method 9 184 187 WKB approximation 559 609 624

X

X-ray photoemission 580 X-rays 30 97 168

Y

YBa2Cu3O7 506 508

Z

Zeeman energy 390 Zener Breakdown 346 558 Zero point energy 5 62 Zincblende 297 324 325 Zn 292

Appendices

A Units


C Derivations of Blochtrades Theorem

C1 Simple One-Dimensional Derivation-


C3 Derivation of Blochtrades Theorem by Group Theory



F Derivation of The Spin-Orbit Term From Diractrades Equation


G1 Bose Particles

G2 Fermi Particles


H1 Propagators

H2 Green Functions

H3 Feynman Diagrams

H4 Definitions

H5 Diagrams and the Hartree and Hartree-Fock Approximations


I Brief Summary of Solid-State Physics

J Folk Theorems



M Problem Solutions











































Combined Problem (66) and (67) Solution



































Bibliography

Mechanics

Electricity

Optics

Thermodynamics


Critical Phenomena

Crystal Growth

Modern Physics

Quantum Mechanics

Math Physics

Solid State

Condensed Matter


Problems


Applied Physics

Rocks


The Internet

Periodic Table

Index

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 ESP 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 FRA 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 ITA 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 JPN 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 KOR ltFEFFc7740020c124c815c7440020c0acc6a9d558c5ec0020d654ba740020d45cc2dc002c0020c804c7900020ba54c77c002c0020c778d130b137c5d00020ac00c7a50020c801d569d55c002000410064006f0062006500200050004400460020bb38c11cb97c0020c791c131d569b2c8b2e4002e0020c774b807ac8c0020c791c131b41c00200050004400460020bb38c11cb2940020004100630072006f0062006100740020bc0f002000410064006f00620065002000520065006100640065007200200035002e00300020c774c0c1c5d0c11c0020c5f40020c2180020c788c2b5b2c8b2e4002egt NLD (Gebruik deze instellingen om Adobe PDF-documenten te maken die zijn geoptimaliseerd voor weergave op een beeldscherm e-mail en internet De gemaakte PDF-documenten kunnen worden geopend met Acrobat en Adobe Reader 50 en hoger) NOR ltFEFF004200720075006b00200064006900730073006500200069006e006e007300740069006c006c0069006e00670065006e0065002000740069006c002000e50020006f0070007000720065007400740065002000410064006f006200650020005000440046002d0064006f006b0075006d0065006e00740065007200200073006f006d00200065007200200062006500730074002000650067006e0065007400200066006f007200200073006b006a00650072006d007600690073006e0069006e0067002c00200065002d0070006f007300740020006f006700200049006e007400650072006e006500740074002e0020005000440046002d0064006f006b0075006d0065006e00740065006e00650020006b0061006e002000e50070006e00650073002000690020004100630072006f00620061007400200065006c006c00650072002000410064006f00620065002000520065006100640065007200200035002e003000200065006c006c00650072002000730065006e006500720065002egt PTB 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 SUO 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 SVE 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 ENU (Use these settings to create Adobe PDF documents best suited for on-screen display e-mail and the Internet Created PDF documents can be opened with Acrobat and Adobe Reader 50 and later) DEU ltFEFF004a006f0062006f007000740069006f006e007300200066006f00720020004100630072006f006200610074002000440069007300740069006c006c0065007200200037000d00500072006f006400750063006500730020005000440046002000660069006c0065007300200077006800690063006800200061007200650020007500730065006400200066006f00720020006f006e006c0069006e0065002e000d0028006300290020003200300031003000200053007000720069006e006700650072002d005600650072006c0061006700200047006d006200480020gt gtgt Namespace [ (Adobe) (Common) (10) ] OtherNamespaces [ ltlt AsReaderSpreads false CropImagesToFrames true ErrorControl WarnAndContinue FlattenerIgnoreSpreadOverrides false IncludeGuidesGrids false IncludeNonPrinting false IncludeSlug false Namespace [ (Adobe) (InDesign) (40) ] OmitPlacedBitmaps false OmitPlacedEPS false OmitPlacedPDF false SimulateOverprint Legacy gtgt ltlt AddBleedMarks false AddColorBars false AddCropMarks false AddPageInfo false AddRegMarks false ConvertColors ConvertToRGB DestinationProfileName (sRGB IEC61966-21) DestinationProfileSelector UseName Downsample16BitImages true FlattenerPreset ltlt PresetSelector MediumResolution gtgt FormElements false GenerateStructure false IncludeBookmarks false IncludeHyperlinks false IncludeInteractive false IncludeLayers false IncludeProfiles true MultimediaHandling UseObjectSettings Namespace [ (Adobe) (CreativeSuite) (20) ] PDFXOutputIntentProfileSelector NA PreserveEditing false UntaggedCMYKHandling UseDocumentProfile UntaggedRGBHandling UseDocumentProfile UseDocumentBleed false gtgt ]gtgt setdistillerparamsltlt HWResolution [2400 2400] PageSize [595276 841890]gtgt setpagedevice

652 Appendices

Table A1 Fundamental physical quantities

Quantity Symbol Expression value in mksa units

Expression value in Gaussian units

Value in Hartree units

Charge on electron

e 16 times 10minus19 coulomb 480 times 10minus10 esu 1

Mass of elec-tron

m 091 times 10minus30 kg 091 times 10minus27 g 1

Planckrsquos con-stant

1054 times 10minus34 joule s 1054 times 10minus27 erg s 1

Compton wavelength of electron

λc 2π( mc) 243 times 10minus12 m

2π( mc) 243 times 10minus10 cm

(2π) 1371

Bohr radius of hydrogen

a0 4πε02me2

053 times 10minus10 m

2me2 053 times 10minus8 cm

1

Fine structure constant

α e24πε0 c

1371 (approx)

e2 c

1371

1371

Speed of light c 3 times 108 m sminus1 3 times 1010 cm sminus1 137

Classical electron ra-dius

r0 e24πε0mc2 282 times 10minus15 m

e2mc2 282 times 10minus13 cm

2137

1 )(

Energy of ground state of hydrogen (1 Rydberg)

E0 e4m32(πε0 )2

1361dagger eV me42 2

1361dagger eV 21

Bohr magne-ton (calcu-lated from above)

μB e 2m 0927 times 10minus23 amp meter2

e 2mc 0927 times 10minus20 erg gaussminus1

2741

Cyclotron frequency

ωc or ωh

(μ0e2m)(2H) (e2mc)(2H) 2741 (2H)

The values given are greatly rounded off from the standard values The list of fundamen-tal constants has been updated and published yearly in part B of the August issue of Phys-ics Today See eg Peter J Mohr and Barry N Taylor ldquoThe Fundamental Physical Con-stantsrdquo Physics Today pp BG6-BG13 August 2003 Now see httpwwwphysicstodayorgguidefundconhtml dagger 1 eV = 16 times 10minus12 erg = 16 times 10minus19 joule



Quantity Conversion



1 calorie = 419 J

1 T = 1 Wbm2 = 104 gauss

me c2 = 051 MeV

mp c2 = 938 MeV















654 Appendices
















ΩAVA = (B6)


IAAAA == ~~ (B7)


Ω=VAA~ (B8)



aAE ~= (B9a)

or

AEa = (B9b)


)~()~(

)~()~(

~~~~

2

sumsum

sumsumsum

==

==

===

kj jkiijj jjj

kj kjkkjjk kjkj

jiij

EEEE

EEEEEEVAEAEVaaaa

δωΩ

δΩΩΩυ



032

21det 2

2=⎟

⎟⎠

⎞⎜⎜⎝

⎛

minusminus

ωω (B10)


5221 +=ω (B11a)


656 Appendices


51

2 21

21 +

== aaωω (B12a)

and for

51

2 21

22 minus

== aaωω (B12b)


⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜

⎝

⎛

minus

+=2

2

11

51

2

51

2~

NN

NN

A (B13)

where

21

21

1 1)15(

4)(⎥⎥⎦

⎤

⎢⎢⎣

⎡+

+=minusN (B14a)

and

21

21

2 1)15(

4)(⎥⎥⎦

⎤

⎢⎢⎣

⎡+

minus=minusN (B14b)


⎟⎟⎠

⎞⎜⎜⎝

⎛

⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜

⎝

⎛

minus

+=⎟⎟⎠

⎞⎜⎜⎝

⎛=

2

1

22

11

2

1

51

2

51

2

aa

NN

NN

EE

E (B15)

Problems



⎟⎟⎠

⎞⎜⎜⎝

⎛

minus+=⎟⎟

⎠

⎞⎜⎜⎝

⎛

520052

3221~ AA







0)()(d

)(d2

2=+ xxU

xx ψψ (C1)


)()( xUnaxU =+ (C2)








0d

d =x

W (C6)

where

21


=W (C7)


658 Appendices



0)()(d

)(d2

2=++++ axaxU

xax ψψ


0)()(d

)(d2

2=+++ axxU

xax ψψ (C8)


)()()( 211 xBxAax ψψψ +=+ (C9)

)()()( 212 xDxCax ψψψ +=+ (C10)


21

21

21

21

2121

2121ψψψψ

ψψψψ

ψψψψψψψψ

primeprime=

primeprime=


DBCA

DCBADCBA

or

1=DBCA

or

1=minus BCAD (C11)


)()( xΔax ψψ =+ (C12)


)()()( 21 xxx βψαψψ += (C13a)

or

)()()( 21 axaxax +++=+ βψαψψ (C13b)


)()()()()()()(

2121

xxxDBxCAax


+=+++=+ (C14)



αΔβα =+ CA

and

βΔβα =+ DB


0=minus

minusΔ

ΔDB

CA (C15)






DA +=+ minus++

1)(ΔΔ


1)(ΔΔ (C17)

and

DA +=+ minus+ ΔΔ



If we set

be=+Δ (C19)

and



)()( 11


and

)()( 12


660 Appendices





where

)()( xuaxu =+ (C25)



Let

)()( 11 NN xxExx ψψ =H (C26)



0][ =lTH (C27)

and






lkmiddotl

ie=t (C31)

Further




so that

12 =lt (C32)










)(02)(0

22

21 Gnnn G =+++


5 In general sum= j

kj

klij

kil AT ψψ

where the Ail




kklkl AT ψψ =


662 Appendices




ntctx nin

i sum= )()(ψ


nmNi

im

imnm cctctc

Nlowast





)( ATrA ρequiv


][i Ht

ρρ =partpartminus


⎟⎠⎞

⎜⎝⎛ minus=

kTHFexpρ


)][exp()]exp([exp

kTHTrkTHHTr

kTHFHTrHU

minusminus=⎥

⎦

⎤⎢⎣

⎡⎟⎠⎞

⎜⎝⎛ minus==








Let

)()( 0 tVt += HH (E1)

lEl l00 =H (E2)


00lk

lkEE minus=ω (E4)




664 Appendices


200

2 2)()(

)(

)cos(1lim ωδπ

ω

ω tfEEttfi

if

if

t=minus=

minusinfinrarr

(E6)












⎟⎟⎠

⎞⎜⎜⎝

⎛=

00σ

σα (F2)




⎟⎟⎠

⎞⎜⎜⎝

⎛minus

=I

I0

0β (F3)



12

)(2

0

20 ltlt

minusminusequiv

cmVcmEε (F4)


20cmEE +prime= (F5)

and

⎟⎟⎠

⎞⎜⎜⎝

⎛=

φχ

ψ (F6)




⎤

⎢⎢⎣


4482 220

2

220

2

230

4

0

2pσ V

cmV

cmV

cmp

mpE (F7)


])[(4 22

0


cm (F8)


666 Appendices


SL middot dd1

2 220

2

rV

rcm (F9)





G1 Bose Particles




][

0][

0][

dagger

daggerdagger

ijji

ji

ijjiji

bb

bb

bbbbbb

δ=

=

=minusequiv

9 See Mattuck [A14]



daggeri i in b b=

and

dagger1 iii bbn =+


11

1

11dagger

11



++=

minus=

iiii

iiii

nnnnnb

nnnnnb



1

1dagger

iiii

iiii

nnbn

nnbn

=minus

=minus



dagger)1()1(






)()(


sumlowast=

=

α αα

α αα

ψψ

bu

bu

rr

rr


668 Appendices

G2 Fermi Particles




0

0

dagger

daggerdagger

ijji

ji

ijjiji

aa

aa

aaaaaa

δ=

=

=+equiv







iii nan




0











)2()1()2()1(

210daggerdagger

ββ

ααβα uu

uuaa harr











670 Appendices






H1 Propagators




H2 Green Functions


δεω

ωi

1)(0 +minus=+

kG k (H1)


H3 Feynman Diagrams


H4 Definitions






672 Appendices




















dagger )2()1()21()2()1(21)1()1()1(H (H3)



31



0

daggerdagger

=+=+

ijji

ijijjiaaaaaaaa δ



δεω

ωi


=sum

+

l lklkk VG k (H6)





674 Appendices



112223 d)21()()()()(d rrrrrr φφφφ (H8)




δεω

ωi)(


=sum

+













klk

Gδωεω

ωi)(


=sum k

k (H11)





676 Appendices




fcc (110) (011) (101)

2 2 2

bcc (11 1) ( 111) (1 11)2 2 2

a a a




1 2 31 ( )Ω






lkh

lkh dG π2


Δ =k G (See 152)











G (See 141)





⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎟⎟⎠

⎞⎜⎜⎝

⎛+⎟

⎟⎠

⎞⎜⎜⎝

⎛minus=

126

4)(ijij

ij rrrU σσε


1e

1 minus

= kTn ω (See 277 2215)


222 akmks=ω (See 248 288 290)






D

DDωωωωωω

gt=leprop

0)( 2

(See 2224)

3126v ⎟

⎟⎠

⎞⎜⎜⎝

⎛==

VN

kk DDD

πθθω (See 2228)


678 Appendices


intminus

⎟⎠⎞

⎜⎝⎛= T

x

x

VdxxTNkC

0 2

43

)1e(e9 θ

θ (See 2229)


3( )

( )( )(2 ) const

dAVD ω

ωω

ωπ= int

k knabla (See 3256)




)e(e 20


2( ) TWθ⎛ ⎞prop Δ ⎜ ⎟⎝ ⎠

k


22

2

9)(

Dm

m mkAT θ=



V

Dln



B

C3

vγα = (See 2237 2238 2250)




ρ11Cv = (See 2270)


ρ44Cv = (See 2270)



3 Electrons


1e

1)( +




( ) ( )u u= +k kr r R (See 3129 C25)


EmVEg23

222

2)( ⎟⎟

⎠

⎞⎜⎜⎝

⎛=π

(See 3164)


32

22

32

)0( ⎟⎠⎞

⎜⎝⎛==

VN



m

ne τσ2

= (See 3214)




RH (See 6102 1223)

680 Appendices


LTK =σ

(See 3215)

22

3⎟⎠⎞

⎜⎝⎛=

ekL π (See 3215)


0

00 near band edge

E E U

U E E

lowast

+

minus=

minusk G

G k G (See 3230)



k r r R (See 3244)


k (See 3255)




i

Bloch Fns


kkr R r (See 3332)



cgs)in or (mceB

meB

c =ω (See 3196)




22

2 2

0 ( )

( )

( ) ( ) ( )2

1( ) ( ) ( )4

el

ioni i

j i i ij i

V

Vm

e d E

ψ ψ

ψ ψ ψπε ne

minus nabla +



r

r r r

r r r rr r




22

2

0 ( )|| spin

( ) ( ) ( ) ( ) ( )2

1 ( ) ( ) ( ) ( )4

ion eli i i

j i j i ij i

V Vme d E

ψ ψ ψ

ψ ψ ψ ψπε

lowast

ne

minus + +


r r r r r

r r r r rr r

nabla



1 32

2 3( ) ( ) ( ) ( ) ( ) ( ) ( )2


mψ ψ ψ ψ

π⎛ ⎞⎡ ⎤minus + + minus =⎜ ⎟⎣ ⎦ ⎝ ⎠

r r r r r r rnabla



2

2

1 ( ) ( )( ) 1( ) ( )k

e f fV E E iq σ

ε ωω ηε


k k qqk q k

(See 9166)

xx

xxxF

kqxxF

qk

qTF

sminus+minus+==+==

11ln

41

21)(

2)(1)0(0

2

2

2ε


F

srk

F Eemk

rkq s

εεπφ

23e

41 2

02 ==ltlt minus



collisions

f ff ft t


Fv nabla nabla


0RHS of BDE( )

f fτminus

= minusk

(See 4149 4145)

682 Appendices


11 123 v( ) ( ) ( )4

edj e f L E L T

eμ

π⎛ ⎞= minus = + + minus⎜ ⎟⎝ ⎠


21 223 ( )v( ) ( ) ( )4

Qdj E f L E L T

eμ

π⎛ ⎞= minus = + + minus⎜ ⎟⎝ ⎠




0ej TE

σ = =nabla

0Qe

jj

Tκ = =

nabla


0eEQ jT

= =nabla

0Q

e

jT

j

QT

Π = =

Π =

nabla


constant131 conganc



⎟⎟⎠

⎞⎜⎜⎝

⎛minus

VED F )(1exp




heG =




D

D

TT

TT

θρ

θρ

ltlt

gtgt5~

~ (See 4146)





)SI(21

0Ae

Bπ=⎟

⎠⎞

⎜⎝⎛Δ (See 534)



⎟⎟⎠

⎞⎜⎜⎝

⎛= (cgs) 4 2

0

2

mne

mne

pπ

εω (See 10108)







2innp = (See 613)

⎟⎟⎠

⎞⎜⎜⎝

⎛minus⎟

⎟⎠

⎞⎜⎜⎝

⎛= lowastlowast

kTE

mmkTn ghei 2

exp)(2

2 4323

2π (See 614)

684 Appendices


ddt

= kF (See 644)




2

2

211

kE



eDkT =μ (See 684 685)





⎟⎟

⎠

⎞

⎜⎜

⎝

⎛ ΔΔ== (cgs)

2

222 ne

V

ne

VKZwidth b π

εε (See 6193)


⎥⎦

⎤⎢⎣

⎡minus⎟

⎠⎞

⎜⎝⎛

⎟⎟⎠

⎞⎜⎜⎝

⎛+= 1exp

00 kTep

LD

nLDej b

np

pp

n

n ϕ (See 6241)



23minusprop Tμ


23Tpropμ





⎟⎠⎞

⎜⎝⎛ )(

22cgs

mceB

meB (See 7255 7256)

Lande g factor

⎥⎦

⎤⎢⎣

⎡+

+minus++++=)1(

)1()1()1(211

JJLLSSJJg

(Implied by (76) (710))


)(20 FBp EDμμχ = (See 3181 3201)


magnetonBohr =Bμ


)groundexcited(22

==minus

= gexcEEgexc

VN

gexc

zμχ


⎟⎠⎞


⎤⎢⎣⎡ ++=

Jx

JJxJ

JJxBJ 2

coth21

2)12(coth

212)( (See 716)



4sincos)()2(

xxxxxFRkFJ ijFij



kT

BgJx eff

Bμ= (See 722)

MBBeff λ+=

686 Appendices


)cgs(

32

0MMH

kEk effB

F

μθθ =primegt

prime (See 7222)


⎟⎟⎠

⎞⎜⎜⎝

⎛minusprop

)(1exp

FK EJD

T (See 7300)


h

eGxy

2ν= (See 1247)



KJpropδ (See 7251)





23

23)()0(

TC

TTMM

M prop



)for (1cc

c

c

TTHTTM

TT

TTC

=asympminusasymp

minusasymp

minusasympminus

minus

δβ

γ

α

χ






2

( ) ( ) (cgs)nemc



2

2

4 nemc

L πλ = (See 86)


g

FEv

πξ 2~0 (See 849)



II ξ lt λ 20~

1 πλϕ

cH 20~

2 πξϕ

cH 2

~1

2⎟⎟⎠

⎞⎜⎜⎝

⎛ξλ

c

c

HH



)(20 cgs

ehc=φ (See 845)



⎟⎟⎠

⎞⎜⎜⎝

⎛minuscong

)(1exp141

FDc EVD

T θ (See 8217)


⎟⎟

⎠

⎞

⎜⎜

⎝

⎛

+minus


1exp

451 epep

epDcT

λμλ



688 Appendices


2

20



⎡ ⎤⎛ ⎞⎢ minus + + ⎥ =⎜ ⎟⎝ ⎠⎢ ⎥⎣ ⎦

Anabla (See 83)

2

( ) ( 2 2 )2s eiq qj q e m m






)()0(

2

2

infin=εε

ωω

T

L (See 950)


⎟⎟⎠

⎞⎜⎜⎝

⎛+=+= )SI(

3)guassian(

34



⎟⎟⎠

⎞⎜⎜⎝

⎛=

+minus=

+minus

sumsum )SI(31

2)(1)(

34

21

00

0jjjj NN α

εεεεεαπ

εε (See 930)



direct 0eΔ =k








)cgs(2 222

4

neEE gn ε


)cgs(111

he mm+=

μ (See 10101)


11 Defects


Mollwo Equation







⎟⎟⎠

⎞⎜⎜⎝

⎛==

pGeEp

Bπω 2 (See 129)


)(exp 41 MottT

AB ⎟⎠

⎞⎜⎝





232surface =⎟

⎠⎞


kTWkTemj

π (See 1145)

690 Appendices


3 2

2surface exp BWj AE

E

⎛ ⎞= minus⎜ ⎟⎜ ⎟

⎝ ⎠ (See 1149)


J Folk Theorems


Chapter 1



Chapter 2



Folk Theorems 691

Chapter 3









Chapter 4







Chapter 5


Chapter 6




692 Appendices















Chapter 7






Folk Theorems 693





Chapter 8




Chapter 9



Chapter 10





Chapter 11







Chapter 12



694 Appendices







Gauss law V A A

dV disin





10

e +

infinminus =int n

axn

andxx


π41Y

e21)(

Y)(

00

23

10

0010001

=

⎟⎟⎠

⎞⎜⎜⎝

⎛=

=Ψ

minus Bar

BarR

rR





r θnabla ϕ

221 1 1( ) (sin )

sin sinrA

r A Ar r rr

ϕθθ



2

2

2222

2 sin1sin

sin11

ϕθθθ

θθ partΦpart+⎟

⎠⎞

⎜⎝⎛

partΦpart

partpart+⎟

⎠⎞

⎜⎝⎛

partΦpart

partpart=Φnabla

rrrr

rr2



111


=primeprime

γγγ

nnn

n


n

ennn ⎟⎠⎞




)()(

1))((

)()()(0)(


n

n nn

xxxf

xxxf

xf

bfdxbxxfxx

xxbxdxbx

=

minusprime

=


minus===minus

sum

int

int

δδ

δδ

δδδ

Complex Variables


π


nazfazdzd

naf n

n

n


)1(1)(Res 1

1

⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧

minusminus

= minus

minus

rarr


SI Gaussian

0

t

tρ

parttimes = +part


sdot =sdot =

H J D

BE

DB

nabla

nabla

nablanabla

4 1

1

40

c c t

c t

π

πρ

parttimes = +part


sdot =sdot =

H J D

BE

DB

nabla

nabla

nablanabla

0ε= +D E P

0

1μ

= minusH B M



( )qc

= + timesvF E B

696 Appendices












1956 Transistor


















9 Total for Decade

698 Appendices

























A Geim K Novoselov

2010 Graphene



Brillouin




Bloch






Bardeen



700 Appendices

Others






M Problem Solutions





451388

2412

126


=+minus=M




1

A

2

3

456

7

8

g5 g6

g7

g8


⎟⎠⎞

⎜⎝⎛= 87654321

876543211g

⎟⎠⎞

⎜⎝⎛= 21876543

876543212g

⎟⎠⎞

⎜⎝⎛= 43218765

876543213g

⎟⎠⎞

⎜⎝⎛= 65432187

876543214g

⎟⎠⎞

⎜⎝⎛= 23456781

876543215g

⎟⎠⎞

⎜⎝⎛= 45678123

876543216g

⎟⎠⎞

⎜⎝⎛= 67812345

876543217g

⎟⎠⎞

⎜⎝⎛= 81234567

876543218g

702 Appendices

Then eg

432 6543218787654321 ggg =⎟⎠⎞

⎜⎝⎛=


g1 g2 g3 g4 g5 g6 g7 g8




)(21 kjia minus+= a


)(23 kjia +minus= a

x

y

z






y

z

x

a2

a1

a3

a


a


704 Appendices





2 2 0h kh k h k


a bG

since

1 2 32 ( )h k lπ= + +G b b b

and


Thus

( ) 2( )( ) ( )h k ld h k lh k l h h k l

π= =G aG G

sdot






where

f iΔ = minusk k k



ki

b

b

bα

bα

hellip

x

y

z

kf

0ˆ =sdot ik i

2sin

2sin

e

2sin

2sin

e

e1

e1

e1

e1

ee

2)1(

i2

)1(i

i

i

i

i

10

i10

i

⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜

⎝

⎛

+⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜

⎝

⎛

=

minus

minus+minus

minus=

+=

minusminus

minusminus

minus

minus

minus

minus

minus=

minusminus=

minus sumsum

α

αα

α

α

α

bk

Mbk

bk

bNk

P

fx

fxMbk

fx

fxNbk

bk

Mbk

bk

bNk

Mm

mbkNn

nbk

fxfx

fx

fx

fx

fx

fxfx


)(integer 2

and)(integer 2

)2()1(

pbk


απ (a)

Since

pl

bk

bk

pl

bk

bk

fx

fx

fx

fx αα

== )2(

)1(

)2(

)1(

or1

)2()1(fxfx kk ne



since

22fxfy kkk minus=


706 Appendices


Let

λπ2=K





Define

θsinKB =

)iexp( BbZ =

So


But


)]2sin(ln[

1)]2sin()exp[(ln

0

)2sin(

knbZ

knbZm

knbZZ

m

knb

πε

πε

πεπε

+=

=

=

suminfin=

Thus


0 )2sin()ln( πε


2)2cos(21)2sin(ln

11

ZkbZkbZZ

ZE

+minus+

minus=

ππε






mGBb π2+plusmn=


b




Aring2255=Ka


Aring2255100 == ad

)bcc(6125222255


deg=rArr== 6816225551

2sin θλθ

d

deg=rArr= 04352255

)51(2sin θθ



θλ sin2 100dn =

above as 2)e1( 0435 4510

)51(42

4sin


)51(32

3sin

above as 2)e1( 6816 4510

)51(22

2sin


2sin

i4400

100

i3300

100

i2200

100

i100

100

ffSd

fSd

ffSd

fSd

=+====

=+====

=+====

=+====

minus

minus

minus

minus

π

π

π

π

θλθ

θλθ

θλθ

θλθ

708 Appendices




mm m

γ γ γx1 x2

1γx3

3 1 2 3





02

22

321

2

2

2

=⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜

⎝

⎛


uuu

mm

m


(2)


twice32mγω = (5b)


⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛==

111

31

321

0uuu

Uω (6a)

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛minus==12

1

51

3 mU γω (6b)


and

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛minus==

112

51

3 mU γω (6c)



)(2iexp1

)(2iexp1

)(2iexp1

)1(0

10

10

sum

sum

sum

minusminus= minus

minus=

minus=

⎥⎦⎤

⎢⎣⎡ minus=

⎥⎦⎤


⎥⎦⎤

⎢⎣⎡ minus=

Ns s

Ns s

Ns snm

nmN

sgN

nmN

sgN

mnN

sgN

G

π

π

π


[ ]

⎥⎦⎤

⎢⎣⎡ minus=

minus⎥⎦⎤

⎢⎣⎡ minus=

⎥⎦⎤


⎤⎢⎣⎡ minus=⎥⎦

⎤⎢⎣⎡ minus+

)(2iexp

)(2iexp)(2iexp


nmsN

nmnmsN

nmNN

nmsN

nmNsN

π

ππ

πππ

and (see 2104)

⎟⎠⎞

⎜⎝⎛=⎟

⎠⎞

⎜⎝⎛ +=⎥⎦

⎤⎢⎣⎡ +

Ns

NsNs




Nsg

NG π

But

⎟⎟⎠

⎞⎜⎜⎝

⎛⎥⎦⎤



Nnm

NNggN

ππ

so

mnN

snm GnmN

gG =⎥⎦⎤


0 )(2iexp π

QED

710 Appendices




)2()()(

)()(

)2()()(

)()(

)2()()(

54145515

34454

32423343

12232

15251121

XXXXXXXXm

XXXXXm

XXXXXXXXM

XXXXXm

XXXXXXXXm


minusminusminus=


minusminusminus=


αααβα

βββαβ

ααα


0

210011100020001110012

54321

22

212

2

=

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛

minusΩminusminusΩ

minusΩminusminusΩ

minusΩ

minus

uuuuu

AAAAA

AAδ


⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

Ω=

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛



minusminus

54321

2

54321

210011100

020001110012

uuuuu

uuuuu

AACCC

AA


⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

minusminus

=

1

0

1

a

aU


This requires

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛

ΩminusΩminus

sdotΩΩΩ

=

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

+minus++minus

minus+minus

=

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

minus+minus+minus+

+minus++minus+minus

22

2

2

2

0

31)1(

01)1(

3

211))(1(

)1(112

a

a

aAa

Aaa

aaAcaca

Aaa




2





)1()1()1( dagger

dagger

ϕψψ

ψψϕ

minus=minus=minus=

==

nna

aaa

aaaNaN

n

n

nn

712 Appendices


nKan =minus1 So

112

dagger2

nnnK

naanKnn

=

=minusminus


1or1 minus== nnnan

K


)1()1(

)1(dagger

daggerdagger

daggerdaggerdagger

ηψ

ψ

ψψη

+=+=

+=

==

nna

aaa

aaaNaN

n

n

nn


1

1+

=n

K

so

11dagger ++= nnna


⎟⎠⎞

⎜⎝⎛ +=⎟

⎠⎞

⎜⎝⎛ +=

21

21dagger Naa ωωH



321321


⎜⎝⎛=



321

321

321

ηηη

ηηη

ηηη

partpart

partpart

partpart

partpart

partpart

partpart

partpart

partpart

partpart

=

zzz

yyy

xxx

J

1 1 2 2 3 3η η η= + +r a a a


etc11

xax =partpartη

so we find

1 2 3

1 2 3 1 2 3

1 2 3

( )x x x

y y y

z z z

a a a

J a a a

a a a

= = sdot timesa a a

QED



1 1

2 1

3 1

100

sdot =sdot =sdot =

a ba ba b



a

KΩ

= 1

and

2 31

a

times=

Ωa a

b


714 Appendices


1 1 2 2 3 32 ( )n n n nπ= + +G b b b




31 21 2 3

1 2 32

nn nN N N

π⎛ ⎞

= + +⎜ ⎟⎝ ⎠

q b b b



a) 31 231 2 31 2

31 2

1 2 3

1 1

11

1

22 2 ii i 11 1i

1 2 3 1 2 30 0 0

i2

2i1

1 1 1 1e e e e

1 1 e etc



l

l

n

nn

nn n ll l NN NNN N

l l l

n l

n lN

i i

i i

n

N N N N N N

N

n Nn N

ππ π

π

π

δδ


= = ==

minus=

minus= ne= =

=

=

sum sum sum sum

sum

q R

R

Gq

q GG

G

b) 111

1

1

1

11

21

i(BZ)

1 2 3 1 0

2

21

0

1 1e e etc


1 1 e


l

l

ni lNN

l

i n

i lN

i i

i i

N N N N

N

l Nl N

π

π

π

δ

minussdot

==

minus=

minus= ne= =

=

sum sumq Rq

R


c)

1 2 31

1 1

3

33

1 2 31 2 3

31 2 3 03

fpp 33

( ) 2

of s( )Vol of space

1 ( )(2 ) ( )

( )(2 )

( ) (2 )

n n nn

f fN

qf dq

f d

N N NN N N

f d

Vf d

π

π

π

π

⎛ ⎞⎛ ⎞= +⎜ ⎟⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠

=

=sdot times

Ω=

sum sum

int

int

int

int

q q b

q q

q qb b b

q q

q q

d)

i i3fpp3

fpp

i

1 2 30

e e (with )(2 )

1 e

l l

l

l

a ad q VV

N N N

π

δ

Ω Ω= rarrinfin

=

=

sumint

sum

q R q Rq

q Rq

R

sdot sdot

sdot

e)

1 1 1 2 2 2 3 3 3i( ) i2 [( ) ( ) ( ) ]3 31 1e el l l l l l l luc



3

1 2 3


η η η⎛ ⎞

= = ⎜ ⎟⎝ ⎠

r


etc so

1 2 3

1 2 3 1 2 3

1 2 3




So

1 1 1

1 1 111

1i( ) i2 ( )310

1i2 ( )

1 1 0

1 e e

e i2 ( )

l l l l

a

l l

d d

l l

π η

π η

η

δπ


primeminusprime

=Ω

= =prime minus

int intG G r

ll

rsdot

716 Appendices

f)

i ( ) 3all 3

i ( ) i ( ) i ( )

i( )i( ) i( )

1 e(2 )

1 1 1e e e2 2 2( )

1 e 1 e 1 e2 i( ) 2 i( ) 2 i( )

yx z

x y z

yx yx z

zx y



LL L

d q

dq dq dq

L L L

x x y y z z

π

π π π

π π π

primeminus



minusminus minus

=

rarr infin


int

int int int

q r rq

sdot


( ) ( ) ( )( ) ( ) ( )( )

z

x y z

yx zL L L


x x y y z zπ π π

δ δ δδ









Frequency c

gωω=

Temperature )( k

Ttcω

=


Uucω

=

Entropy NkSs =


NkC

c vv =



⎟⎠⎞

⎜⎝⎛ +⎟⎠⎞

⎜⎝⎛==

Nd

ddpP

c

ωωωωω


)(

Thus

π

ωω2

)()( NadqdqqNdN ==

1

)2)(2cos(1

2)(

22cc

cN

aqaN

ddqNadqN

ωωπω

ωπωπω

minus=

==

So

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

minus⎟⎟⎠

⎞⎜⎜⎝

⎛

minus==

221

11e

1

ckTc d

dpPωωπω

ω ω

or since

tg

kTkT c==

ωωωω

c

then

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

minus⎟⎟⎠

⎞⎜⎜⎝

⎛

minus=

21

11e

1)(g

gtP tgπ (1)



int

intsum

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

minus⎟⎟⎠

⎞⎜⎜⎝

⎛

minus=

minus=

minus= =

c

c

m

dN

dNU

cckT

kTNm kT

m

ω

ω

ω

ωω

ωωωω

ωπ

ωωωω

0 22

01

1

11e

2

)(1e

21e

int⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

minus⎟⎟⎠

⎞⎜⎜⎝

⎛

minus==

c

cc

kTc

cd

NUu

ω

ω ωωωω

ωωπω 0 22 )(

1

11e

2

718 Appendices

or

int⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

minus⎟⎟⎠

⎞⎜⎜⎝

⎛

minus=

1

0 21

11e

2 dgg

gu tgπ (2)

ωωωπω

ω

ω

ω

ω

ω

ω

ω

dNkT

kTC

cckT

kT

m kT

kTm

c

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

minus⎟⎟⎠

⎞⎜⎜⎝

⎛

minus=

minus=

int

sum

2202

2

2

2

2

2v

1)1e(e)(2

)1e(e)(1

int⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

minus⎟⎟⎠

⎞⎜⎜⎝

⎛

minus==

c

cc

kT

kT

c

c dTNkN

NkCc

ω

ω

ωωω

ωω

ωπω

ω

0 222

2

22v

v )(1

1)1e(

e)()(

2

int⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

minus⎟⎟⎠

⎞⎜⎜⎝

⎛

minus=

1

0 22

2

v1

1)1e(

e)(2 dgg

tgc tg

tg

π (3)


kTFj



VT

FS ⎟⎠⎞

partpartminus=

so

( ) sumsum=

minus=

minus

minus+minusminus=

N

m

mkT

N

m

kT

kTkTkS

mm

12

1

e11e1ln ωω

ω

( )

int

int

minusminus+

minusminusminus=

=

minus

c

c

cc

kT

cc

kT

dkTkTNNk

dNkNk

NkSs

ω

ω

ωω

ωωωω

ωπ

ωωωωπ

0 22

2

0 22

)(1

11e

)2(1

)(1

1e1ln21


+minus

minusminus= minus1

0 2

1

0 2

1

11e

2

1

1e1ln2 dgg

tgdgg

s tgtg

ππ (4)



range of ω

ωxxp


P10x1π

1

e

ωx

10 1minus

sdot1

1 ωx( )2minus

sdot=

P100x1π

1

e

ωx

100 1minus

sdot1

1 ωx( )2minus

sdot=

P300x1π

1

e

ωx

300 1minus

sdot1

1 ωx( )2minus

sdot=

0 05 10

200

400

P300

P100

P10

ω

720 Appendices


the range of t

txxr


ux2π

0

1

yy

e

y

tx 1minus

1

1 y2minussdot

⌠⎮⎮⎮⎮⌡

dsdot=

spht x2π

0

1

y

ytx

⎛⎜⎝

⎞⎟⎠

2e

y

txsdot

e

y

tx 1minus

⎛⎜⎝

⎞⎟⎠

2

1

1 y2minussdot

⌠⎮⎮⎮⎮⎮⎮⎮⎮⌡

dsdot=

0 1 2 3 40

1

2

3

4

u

spht

t


0 05 1 15 20

05

1

15

2

u

spht

t


Entropy t2π

minus

0

1

ωln 1 e

ωminus

tminus

⎛⎜⎝

⎞⎟⎠

1

1 ω2

minus

sdot

⌠⎮⎮⎮⎮⌡

dsdot2π

0

1

ω

ω

t

e

ω

t 1minus

1

1 ω2

minus

sdot

⌠⎮⎮⎮⎮⎮⌡

dsdot+=

0 100 200 3000

2

4

6

8

Entropy t

t

722 Appendices




By Problem 29

2 31

a

times=

Ωa a

b


2 3 [(1)(1) (1)( 1)] [(1)(1) ( 1)(1)] [( 1)( 1) (1)(1)]

2

(2 2 ) ( )2

a

a a


= + = +

a a i j k

i j i j

and

2

21 2 3( ) (2)


so Ωa = a2 and

11 ( )a

= +b i j


2

3

1 ( )

1 ( )

a

a

= +

= +

b j k

b i k




1 2 31 1 1( ) ( ) ( )a a a




ky

kxkz


a

a

n m + 1

n mn ndash 1 m

n m ndash 1

n + 1 m

xnm ynm


])()()()[(2

])()()()[(2

21

21

21

21

21

21

21

21

mnmnmnmnmnmnmnmn

mnmnmnmnmnmnmnmn

yyyyyyyyk

xxxxxxxxkU



minus+minus+

minus+minus+


)4( 111

mnmnmnmnmn



)4( 111

mnmnmnmnmn



724 Appendices


)i(1 e maknakt

mnyxcx ++minus= ω

)i(2 e maknakt

mnyxcx ++minus= ω


⎟⎠⎞⎜



⎟⎠⎞⎜






a x

y


⎥⎥⎦

⎤

⎢⎢⎣

⎡+⎟

⎟⎠

⎞⎜⎜⎝


⎟⎠⎞

⎜⎝⎛ +++minus++=

⎟⎠⎞

⎜⎝⎛ minus++=

22

2

2

222222

22222

222

481

2122

21

)(2)(21

)(21

aax

ax

ayxayxaxak

ayxaayxak

ayxakU

to second order

[ ]

2

22222222

21

2222

kx

axaxyxayxaxakU

cong







11 11dagger

1 i22

pp p p

pa P X

ωω

= minus qq q q

q

Now

11 1

1

11dagger

p p

p

p

X m

m

X


minus

= sdot

= sdot

=

sum

sum

X

X

q b q b q bb

b q b q bb

q

e

e






1 1

1

1

1

111

1

1 1

p pp p p

pp p

e X m e e X

m e e X

m X m X

β βα αβ

β βαβ

ββ ααβ δ δ

lowast lowast

lowast

=

=

= =

sum sum

sum sum

sum

q b q b q bbqb q b

b q b q bb q b

bb b bq bb q b

or

1 11

1p ppX e X


b (a)

Now

dagger 11dagger 11

1 i22

pp p p

pa P X

ωω

= + qq q q

q


11dagger 11dagger

11 11

1 i22

1 i22

pp p p

p

pp p

p

a P X

P X

ωω

ωω


= minus

qq q q

q

qq q

q

726 Appendices

so

dagger 11 2i

2p

p p pa a Xω

minusminus = qq q q

11 dagger

i ( )

2p p pp


q

Substituting in (a)

1 dagger

i ( )

2 p p ppp

X e a am ω


b q



0)(

)(])[(

34412

24412122

442

4411

=++

++minus+minus

Kkkcc

KkkccKkckcc

zx

yxx ρω


0)( 122

11 =minus Kkc ρω


0)( 222

44 =minus Kkc ρω

0)( 322

44 =minus Kkc ρω

If

ρ

ων 1111

ck

==


If

ρ

ωνων 4433

22

ckk

====






)1(

)(

])()([)(

))]()(())()([()(

))]()(())()(()(

2

2

12

1112

11112

11112

1

1

NM

PM

dPM

dxuxuPM

dxuxuQxuxuPM

dxuxPuxuxPuMd

P

qp qq

qp qq

pp NNNNpp

pp NNNNpp

=

=

minus=

minus=

minus=

minus=

sum

sum int

sum int

sum int

sum intint

lowast

primelowastprime+


τδ

τ

τ

ττψψ

where Q = Pminus1P




)2()1()21()2()1(21

)2()1()21()2()1(21

)2()1()21()2()1(

)2()1()21()2()1(

)1()1()1(

(||)

(||)

sum

sum

sum

sum

sumsum

minus

+=

minus

+

=

ji ijji

ji jiji

ji ijji

ji jiji

i iik

V

VE

V

V

H

ψψψψ

ψψψψ

ψψψψ

ψψψψ

ψψε



22 2

1 21 10 12

| ( ) |( ) d 0

4jN N

II jeV

rψ


rr

Note that

221| ( ) |N

jj ψ=sum r

728 Appendices


2

22 11

0 12

( )d ( )

4NI

II Ie V

rρ τ


r QED

IonI

Field point

Source point

r1

r2



Let x = kkM

0as2)(21

11ln1limlnlim

2

2

0

22

0

rarr=minus=

minus+minus=

minus+minus

rarrrarr

xxxx

xx

xx

kkkk

kkkk

xM

M

M

Mk

⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢

⎣

⎡


minusminusminus



rarr

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

gtminusminusminus

ltminusminusminus

rarr

⎥⎦⎤

⎢⎣⎡


⎥⎦⎤

⎢⎣⎡



rarr

110)1(

)1(1)1(1

or

110)1(

)1(1)1)](1(1[

2

1)1(1)1ln(

1)1(1)1ln(

2

1)1ln(1)1ln(2ln)1(2

1)1ln(1)1ln()1ln()1)(1(


2

2

22

1

xxx

xx

xxx

xx

xx

x

xxx

xxxxx

xxxxx

xxx

xxxx

xx

xx

x





i1e 0bm


K


The answers are





Thus

i(0) i( )( )e e

1 ( 1)

n l m

n l m

S πminus minus + +

+ +

= +

= + minusK






730 Appendices


i( )( ) i( )( ) i( )(n )1 e e en l m









20

2)(2

2 EVmKmEk

minus==


)ee()(22

ee)(2

)(02

If

ii

KxKx

kxkx

Dxbxb

CBxaxb

Vbax

minus

minus

+=ltltminus

+=ltlt

infinrarr=+gt

ψ

ψ

ψ


)ee()ee(i

)ee(ee

0ee

2i2i2i2i

2i2i2i2i

)2(i)2(i

KbKbkbkb

KbKbkbkb

bakbak

KDCBk

DCB

CB

minusminus

minusminus

+minus+

minus=minus

+=+

=+



⎟⎠⎞

⎜⎝⎛minus=

2coth)tan( Kb

Kkka (a)


)ee()(22

ee)(2

02

ii

KxKx

kxkx

Dxbxb

CBxaxb

bax

minus

minus

+=ltltminus

+=ltlt

=+gt

ψ

ψ

ψ


⎟⎠⎞

⎜⎝⎛minus=

2tanh)tan( Kb

Kkka (b)


)e21(

2tanh

)e21(2

coth

Kb

Kb

Kb

Kb

minus

minus

minusrarr⎟⎠⎞

⎜⎝⎛

+rarr⎟⎠⎞

⎜⎝⎛

and

00 V

EEV

EKk cong

minuscong


⎟⎠⎞



02symmetric-anti





)e21( 2

0


732 Appendices


⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝


22exp21

22

20

02

22

2

2 bmVVma

Ema ππ






dkm

kdkdkdEdE

2==


intint ==ll

dldEk

mdldkdEED 222 21

)2(2)(

ππ

φ

πak

θ

dl

dk

E+dEE = h2k22m

int=kdlmED 222

)(π


For a

k πlt

2222

2)(

ππ

πm

kkmED ==

For a

ka

ππ 2ltlt

)2(42

)( 2222 ϕπ

ϕπ

mkkmED ==

but

⎟⎠⎞



2

So

⎟⎠⎞


kamED ππ

π1

22 cos4)(

or

⎟⎠⎞


kamED π

ππ1

2 cos41)(



(1) (2) (3) (4) (5)

(6)

(c) (b) (d) (a) (f) (e)



0 00



k

734 Appendices


( )

0( ) 2e

(e 1)

F

F

E E kTF

E E kTf E EdT dT kT

minus


k

k

k

( )

0( ) 2e 1

(e 1)

F

F

E E kT

E E kTfE kT

minus

minuspart

= minuspart +

k

kk

so

0 02

0

F

F


f EEdTE dT kT T


k

k

k

k

Thus

00


τ⎛ ⎞ part⎡ ⎤⎛ ⎞= + minus + +⎜ ⎟⎢ ⎥⎜ ⎟ part⎝ ⎠⎣ ⎦⎝ ⎠

kk k

k


33

2(2 )

zzJ e u f d k

π

⎛ ⎞= ⎜ ⎟⎜ ⎟

⎝ ⎠int k k


33

2(2 )

zzH u f E d k

π

⎛ ⎞= ⎜ ⎟⎜ ⎟

⎝ ⎠int k k k

Thus

2 303 ( )

4z F


τπ


kk

and

2 303

1 ( )4

z Fz


τπ


k k kk



23

22

3

23

( ) [ ( )]( )4

( ) ( )12

412

zz F

zF

F

dsdEeJ u E E eEu

dsdEe E u E Eu

e eEu k

τ δπ

τ δπ

τ ππ

= minus minus minus

= minus

=

int

int

kk k

k

kk k

k

k




m

neEJ z τσ

2==


230 ( )

12F


dz dT kT T uτ δ

π⎛ ⎞⎡ ⎤⎛ ⎞= = minus minus +⎜ ⎟⎢ ⎥⎜ ⎟

⎝ ⎠⎣ ⎦⎝ ⎠int k k

k kk

or

2 2( ) ( ) FF F


τ δ τ δ ⎡ ⎤⎛ ⎞minus = minus +⎢ ⎥⎜ ⎟⎝ ⎠⎣ ⎦

int intk k kk k k k

k k

or

( )2

2 2 2 414F F

F F F FFE E E E


πτ π τ= =

⎞⎛ ⎞ ⎛ ⎞= minus + ⎟⎜ ⎟ ⎜ ⎟ ⎟⎝ ⎠⎝ ⎠ ⎠k k

k kk k

So

dT

dEdzdTeE F=


2

31 [ ( )]

12F F

F

dT Kdz


τ δπ

=


k k k kk

so

23

1 412 F

F

F Fk kE E

u dE dEK E E k

dT dTτ π

π ==


kk k

736 Appendices


dT

dEEm

nK FF

F 2τminus=


⎥⎥

⎦

⎤

⎢⎢

⎣

⎡

⎟⎟⎠

⎞⎜⎜⎝

⎛minus=

2

)0(121)0(

FFF E

kTEE π

so

)0(6

22

F

FE

TkdT

dE πminus=

so

)0(6

222

FF

FE

TkEm

nK πτ=


Tkm

nK F 22

3τπ=

and

22

2

22

33 ⎟

⎠⎞

⎜⎝⎛==

ek

Tm

ne

Tkm

n

TK

F

Fπ

τ

τπ

σ




bXX

dx 12 minus=int





1

)]1([11

)]1([

21

1

02121

10 2

21

KK

xKxKKKxKxKdx

=

⎥⎦

⎤⎢⎣

⎡minus+minus

minus=minus+

int


2

2

22 2

22 2

( )

( ( ))

(2 )2

22

2

L

L

dIq f

dq f

d

q qm

dmq q

m

ω

ω

=

= minusminus

= minus⎡ ⎤



int

int

int

int

qq

qq

q

k q

q

k q


1

2 20 2 2

2

2(1 ) 2 L

m dxI dmq x q xω


int intq

k q


1

2 20 2 2 2

2

2( ) L

m dI dxmx x x kω


int intq

q k


1

2 20 2 2 2

2

2L

m dI dxmq x x kω

= minus⎡ ⎤⎛ ⎞+ minus⎜ ⎟⎢ ⎥⎝ ⎠⎣ ⎦

int intq

738 Appendices


2

2 2 2 2 2 20

4[ ] [ ]

d q dqq q

πα α

infin=

+ +int int

q


intminus

minus=1

0 22

22 2

2

kxxmdxmI

Lωπ

But

2

sin2

2

11

0 22

pppxx

dx minus=minus

int

so we find

⎟⎟⎠

⎞⎜⎜⎝

⎛minus= minus

2sin22 12

2k

mk

mm

mIL

L

L ωω

ωπ

Thus

Lm

Kω

=3


⎟⎟⎠

⎞⎜⎜⎝

⎛+minuscong

2122 2

22

kmm

mILL ω

πω



2 2

02

16

c L

c

kEm

mm

α ω

α

lowastlowast

lowastlowastlowast

= minus +

=minus

k






1 1d edt c

δτ

⎛ ⎞ ⎛ ⎞+ = minus +⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

k E v Btimes


mceB

c =ω

v

vv

vv

zz

xcyy

ycxx

Eme

Eme

Eme

τ

τωτ

τωτ

minus=

+minus=

minusminus=


⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛ minus=

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

z

yx

cc

z

yx

EEE

DDjjj

000101

0 τωτωσ




Dyyxx

0σσσ ==

D

cxyyx

τωσσσ 0=minus=

For the Hall effect

yxyy

xyy

EEj σ

σminus=rArr= 0

740 Appendices

So

xyy

xyxxx Ej ⎟

⎟

⎠

⎞

⎜⎜

⎝

⎛+=σσ

σ2


xx Ej 0σ=


m

ne τσ2

0 =


BE

E

BjE

Rx

yxyy

x

x

yH )( 0σ

σσ

minus==


nec

RH1minus=


B

nec

cc

cxyyx ==congminus=

τωσ

τωτωσσσ 02

0

)(


B

Ryx

H )(1

σminus=


v

v)()(

)(vvv

32

54

32

540

0

⎥⎦

⎤⎢⎣

⎡partpart

partpartminusminus=

⎥⎥⎦

⎤

⎢⎢⎣

⎡sdot

partpartminus=

partpartminus=

F

F

FF

e

Fe

xxxfxfcm

xfcmEp

π

π



( ) ⎟⎟⎠

⎞⎜⎜⎝


40

22 11411

FF

xF

xxdxxxxf F

where

312

v3

⎟⎟⎠

⎞⎜⎜⎝

⎛=equiv π

cmcmpx

ee

FF


FFF

F xxxxf

21)( 3 +=

partpart

34312

v31)3(

vπ

cmx

e

F minus=partpart

Fe

F xcm

x31

v3

31

vv

312minus=⎟

⎟⎠

⎞⎜⎜⎝

⎛minus=

partpart π


[ ] 12

61

3111

41

2432

54

242

432

54

0

FFe

FFF

Fe

xxcm

xxx

xcm

p

minuscong

⎥⎥⎦

⎤

⎢⎢⎣

⎡++⎟

⎟⎠

⎞⎜⎜⎝


π

π


pm

MM δ=

and

⎟⎟⎠

⎞⎜⎜⎝

⎛=

cem

RR

742 Appendices


Mm

R pξπ sdotcong 3

34v

31

31

3

2

491

34

3

⎟⎟⎠

⎞⎜⎜⎝

⎛=

⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜

⎝

⎛

sdot=

ξδπ

ξπ

π

MR

Mm

RRRx

pF


32

54

12~and

89

ππδ cm

K e==

⎥⎥⎦

⎤

⎢⎢⎣

⎡minuscong 2

32

4

34

0~

RM

RMKp




EF(2)

EF(1)

E

Δ = eV

R1 2



We note if

1)2(1exp)(

minus

⎥⎥⎦

⎤

⎢⎢⎣

⎡+

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛ minusminus=

kTEE

Ef F (a)


then

1)1(1exp)(

minus

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡+⎟

⎟

⎠

⎞

⎜⎜

⎝

⎛ minusminus=Δ+

kTEE

Ef F


then

[ ] 3

321

)2())(1)((v2



[ ] 3

312

)2())(1)((v2




[ ] 3

3

)2()()(v2


Since

x

x kE

partpart= 1v

We write

dEdkxx1v =


[ ] ||2

3 )()()2(



mk

EW2

2||

2minus= (b)


)(D1 WTT =

744 Appendices


⎥⎥⎦

⎤

⎢⎢⎣


1



[ ] )( )()()2(

2||


π (d)






ce EE

mED minus⎟

⎟⎠

⎞⎜⎜⎝

⎛=

lowast 23

222

21)(π

1])exp[(

1)(+minus

=kTE

Efμ


⎟⎠⎞

⎜⎝⎛ minus=⎟

⎠⎞


kTE

kTEEf μμ expexp)(

ce

EmkE += lowast2

22

intinfin

lowast⎟⎠⎞


⎟⎠

⎞⎜⎜⎝

⎛=

cE ce dE

kTEEE

mn μ

πexp

22

123

22



ce

EmkE += lowast2

22

kdkm

dEe

22

2

lowast=

lowast

=minuse

cm

kEE2

intinfin

lowast

lowast

lowast

lowast

⎟⎟⎠

⎞⎜⎜⎝

⎛ +minus⎟

⎠⎞

⎜⎝⎛

⎟⎟⎠

⎞⎜⎜⎝

⎛=

cEe

ce

e

e kdkmkT

EmkkTm

kmn 2

2)2(

expexp2

22

1 22223

22μ

π

intinfin

lowastlowastlowast

lowast

⎟⎟⎠

⎞⎜⎜⎝

⎛minus⎟

⎠

⎞⎜⎝

⎛ minus⎟⎟⎠

⎞⎜⎜⎝

⎛=

cEee

c

e

e dkkkTm

kmkT

E

m

mn 2

22223

22 2expexp

2

22

1 μπ

intinfin

lowastlowast

lowast

⎟⎟⎠

⎞⎜⎜⎝

⎛minus⎟

⎠

⎞⎜⎝

⎛ minus⎟⎟⎠

⎞⎜⎜⎝

⎛=

cEe

c

e

e dkkkTm

kkT

Em

mn 2

22

23

323

22 2expexp

)2(22

21 μπ


pp

ndxpxx nn π

)2(2)12()exp(0


where

kTm

pne

2

2 1 ==

So

( )kTmkTm

kTE

m

mn

ee

c

e

e

21

223

323

22

2222

)1(exp2

222

1 πμπ

⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎠

⎞⎜⎝

⎛ minus⎟⎟⎠

⎞⎜⎜⎝

⎛=

lowast

lowast

⎟⎠

⎞⎜⎝

⎛ minus⎟⎟⎠

⎞⎜⎜⎝

⎛=

lowast

kTEkTm

n ce μπ

exp2

223

2



746 Appendices



1 ( ) 1 ( )kdE E

d⎛ ⎞= equiv⎜ ⎟⎝ ⎠

kv kk

nabla


1 ( )

( )

k

k

dE Edt

d Edt

= sdot = sdot

= sdot

F v F k

k k

nabla

nabla

thus

( ) ( )k kdE Edt



ddt

= kF


1 ( ) ( )k kdE Edt


but also



( ) ( )k kdE Edt



( ) dqdt











ln T

ln μ

low T high T



phonon scattering

748 Appendices


if

if

212

121

21

21total

μμμμμμ

μμμμμ

gtgtconggtgtcong

+=




rr

rV λminusprop e1)(



2 1 2 1( )H V r=k k k k



2 1



int minuspropπ

θθθθστ 0

sin)cos1)((v1 dm



Finally we find

23Tm

m prop= lowastτ

μ








cEδ δprop sdotk r



23minus=prop Tμ


See Fig 618


Now


⎟⎠

⎞⎜⎜⎝

⎛= minus )(

23

2 e2

2 μ

π


⎟⎠

⎞⎜⎜⎝

⎛= minus )(

23

2 e2

2 μ

π

750 Appendices


⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎟⎟⎠

⎞⎜⎜⎝

⎛=minus

23222

lnkTm

NkTE

e

dCn

πμ

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎟⎟⎠

⎞⎜⎜⎝

⎛=minus

23222

lnkTm

NkTE

h

aVp

πμ

Therefore

⎥⎥⎦

⎤

⎢⎢⎣


32

)()()2(

22ln)(

kTmmNN

kTEeEEhe

dagpnCnVp

πϕϕ

Using

kTEg

gkTE

e

1ln minus=

kTEhei

gmmkTn 4323

2 e)(2

2 minus⎟⎟⎠

⎞⎜⎜⎝

⎛=

π

we find

⎟⎟⎠

⎞⎜⎜⎝

⎛=minus 2ln)(

i

dapn

nNN

kTe ϕϕ QED



2 21

12Surface

( )( )

4d

rπΦ = int

M r Sr

sdot

= minus ΦH nabla

12 2 21 2

12

ˆ ( )( )

4d

rπ= int

r M r SH r

sdot

θr

dS2

r 12

z

M

r1 = 0r12 = ndashr2


Using the figure

intminus=π

πθθπθθ

0 24 sin2 cos cos

rrdrMH z

31and

3=minus= DMH z




)eTr( HZ βminus=


1 1 1( ) ln

Tr( e ) Tr( e )

zz z z

z z

ZF ZB B Z B

Z Z

β β

μ ββ β β

βμ μβ



as expected



)Tr(2

)1Tr()eTr( 22

HZ H ββ +== minus


)Tr(

3)Tr(

3

)Tr(31)Tr()Tr(

2222

222

222222

JBμgJBBμg

JδBBμgJJBBμgH

Bμ μμ

B


==

==

sum

sumsum

)12()1Tr( += j


⎥⎥⎦

⎤

⎢⎢⎣

⎡+++cong )1(

3211)12( 2

222 jjBgjZ Bμβ

752 Appendices



⎥⎦⎤

⎢⎣⎡ +

partpart= )1(

611 2222 jjBg

B Bz

z μββ

μ

BB

BB z

z=

partpart


kT

jjgNNB

Bz

zzz

)1(3






kkS sdot


kkS sdot



i i( ) e ej i j

i jid Jdt


k k kk k kS S S



i jj i jd Jdt


k k kk k kS S S


i( ) i( )i( ) e ( )e ej ij i

d Jdt


k k kk R k kS R S S


kRsdot


so

( )

( )

( )

kd Jdt

J

J




primeprime= minus

prime=primeprime=

sum

sumsum

k kk k kk k

k k kk

k k kk

S k k S S

k S S

k S S

times

times

times

which is (7155) QED




ϕϕ ==


2

2

(cos cos sin sin )

cos( ( ))

x x y yR R R R R R

R R R R

S S S S

S

S q R R

ϕ ϕ ϕ ϕ

prime prime prime

prime prime

= +

= +

prime= minus

S Ssdot

so that


)cos()(2

))(cos()(2

i2

2

2

=minus=

minus=


sumsum

sum

sum prime

RRqR

R

RR

qRRJRJNS

qRRJNS

RRqRRJSH


qR minus== sum Then

)(2

2qJNSH minus=


)2cos(21)cos(2

21 )2cos(2)cos(2

)ee()ee()(

1

1221

i2i22

ii1

⎥⎦⎤

⎢⎣⎡ minus=

minus=+=

+++= minusminus

qaqaJ

JJqaJqaJ

JJqJ qaqaqaqa

754 Appendices

For a maximum

[ ]0

)cos()sin(2)sin(2

)2sin()2(21)sin(2)(

1

1

=minusminus=

⎥⎦⎤


partpart

qaqaqaaJ

qaaqaaJqqJ


angle turn 60

21)cos(

1)cos(2

deg=

=

=

qa

qa

qa


[ ]

max03

212

212

)2cos(2)cos(2)(

21

21

602

12

2

rArrltminus=

⎥⎦⎤

⎢⎣⎡ +minus=

minusminus=part

partdeg=

aJ

aJ

qaqaaJq

qJqa



=minussj mSS

12

112

212

212

21

21

21

+⎟⎠⎞

⎜⎝⎛ minus+=

⎟⎠⎞

⎜⎝⎛ minus=

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛minus=

+

++minus

nSnnS

nSnaS

nSaa

aSnS

j

jjjj


so



or

1)1)((

12

2)1(221

minus+minus+=

minus⎟⎠

⎞⎜⎝


sss

ss

ssj

mSmSmS

mSS

mSSmSSmSS

QED


We have

i1 e jjjb a

N= sum k R

ksdot

and


Nminus= sum k R

ksdot

where

0][

0][

][daggerdagger

dagger

=

=

=

ji

ji

ljjj

aa

aa

aa δ

Thus

i( )dagger dagger

i( )

1[ ] e [ ]

1 e

1

j j

i j

j

k jj jk j

j

k k k k

b b a aN

N

NN

δ

δ δ

prime

prime


primeminus minus

prime prime

=

=

= =

sum

sum

k R k R

k k R

sdot sdot

sdot

Similarly

i( )

0


Nprimeprime+



0


Nprimeprimeminus +


756 Appendices






Therefore

712

)state F(3max

21221

=+==

=+=

sumL

Lm

SS

l

less than half full

25


So stateF 25

2

Pm3+


426

)state I(123max5122


sumSLJ

mLSS

l

So stateI4

5

Eu3+ six electrons


033


7123216


=+=times=

sumSLJ

mL

SS

l

So stateF0

7




05 3

3[( ) ] ( )( )4μπ



B

r

rprime

m


4 have ( ) 0

28 have ( ) 2 22

a mam

primeminus sdot =

primeminus sdot = =

r r m

r r m


[ ]2 2

05 3

3 ( )4

Uμπ


r r m mm Br r r r

⎟⎟⎠

⎞⎜⎜⎝

⎛minus= minus

5

252

3

2327

total2)2)(3)(8(2)12(10

ama

amU

( ) 022424)10( 3

27 =minus= minus

amU


758 Appendices


2 2 2

4 have ( ) 0

2 have [( ) ] a m

primeminus sdot =

primeminus sdot =

r r m

r r m


066)10( 3

2

3

27

total =⎟⎟⎠

⎞⎜⎜⎝

⎛minus= minus

am

amU


3

27


)JT10279)(8(8

m10524

10

minus

minus

times=cong

timescong

Bm

a

μ


)(10381

101010

1012510)10(

23

2328

51

30

447

total

congtimescong

=congtimes

cong

minus

minusminus

minus

minus

minusminus

c

cT

T

U





The free energy is

ZkTF lnminus=


BFMpartpartminus=




)(0 qpHH rarr




qpqAp

eeAppqq

i

i

iiii

ii

Π=

gtΠequiv+rarrrarr

0HH

Now

)()(e

e

11)(

11)(


=

int

intΠminus

minus

qpqdddqdq

dpdpdqdqZ

NNq

NNqp

helliphellip

helliphellip

H

H

β

β


0 =Πpartpart

=Πpartpart

=partpart

j

iij

j

iij

j

i qpqq δδ



⎟⎟⎠

⎞⎜⎜⎝

⎛=

Πpartpart

+Π= etc1i

iiii

peAp

760 Appendices

Thus schematically

1

0

100010001

)()( =

Πpartpart

=

Πpartpartpartpart

=

Πpartpart

Πpartpart

partpart

partpart

=Πpart

part

i

i

j

i

i

i

j

i

j

i

j

i

j

ip

pqp

pqqp

qq

qqp 15



0=partpartminus=

BFM






15

1

100

01

100

01

0

=sdot=⎟⎟⎠

⎞⎜⎜⎝

⎛

partpart

⎟⎟⎠

⎞⎜⎜⎝

⎛

Πpartpart

=

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛partpart

partpart

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

Πpartpart=

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛partpart

partpart

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

Πpartpart=

Πpartpartpartpart

partpart

11j

i

j

i

i

i

j

i

j

ii

i

j

i

j

i

j

i

i

i

j

i

qqp

qp

qq

pqp

qq

ppqp

qq





Subtracting [ ]int



intΔminus

ΔΔminusminus

minus

Δminus= 2

1 22

221

221)(

)0()0(eV dEEeV

EeV

E

ENNCI

Checks 11








0

000


⎠

⎞⎜⎜⎝

⎛minusprime==Δ


0

0

0

3

0

0

2

02

3

31)11(

2

3cos

2

)(coscos2

ε

ε

θε

θθε

π

π

P

P

P

dPEE b

=

minusminusminus=

minus=

minus=equiv int

762 Appendices



θ

p

E z


int

int= π

θ

πθ

θθπ

θθπθ

0

cos

0

cos

sin2e

sin2ecos

d

dpp

kTpE

kTpE

z


int

int

minus

minus= c

c

x

c

c

x

zdx

dxx

cpp

e

e


⎟⎠⎞

⎜⎝⎛ minus=

ccppz

1)coth(


kTp

Ep

kTEpp z

z 3

3

22=== α



The Hamiltonian is

eEzr

qm


0

22

2

2H

H



)1(100 pzN += ψψ


BarBa 23


ψ


2

2

mqaB =

Requiring

12 =int τψ d





Now using

4

3 0100 0 100 2

0

12 (4 )

med Eψ ψπε

= = minusint rH





are also zero so

)(2 202 eEpaE

NBminus=

H



764 Appendices

Requiring

0=part

partpH

we find

0EeEp minus=


230



302

216 Ba

Eπεα =

part

partminus=

H



061

41)(

21 42


For E = 0

0)( 420 =++minus=


PGE δγβ (2)


20


[ ] 0)(61

41)(

21 2

02


0121)(

31 2


20 4 scc PTT

βγminus= (3)

(b) From (2) 0)( 42


20 4




04


δγ

432 minus=scP

βδγ 2

0 163+= TTc











766 Appendices





φ1 (say 45deg)

Polarizer

lightSample

Analyzerz

zφ2

E

z

yx






iminus= xx xpxp



If H = px2 m +V(x)

[ ] 0)( =xxV So

[ ] xpm

x i minus=H


[ ] iˆ ˆm




2 2ˆ ˆ

2ij ijj ji j i j

f b ωsdot + sdot

=sum sume r e r




jxi

i=minus xpxp xx

xxxppx xx i2 =minus

iminus= xx xpxp


768 Appendices


jj jω=H

jpm


jpm


jpm



1ˆ ˆi ij

i j i jmω

sdot = sdote r e p


ij ji


ωω ω



jiij ωω minus=



Summing over j

ˆ ˆ2iijjbf i im



m

bm

bfj ij 2)i(

i2minus=minus=sum




eEB

NeG +minus

=+ v13

1vvel

2

0εμ



0evv minus= Then

eEB

NeG =minus

minus+minus v13

1vvel

2

0

2ε

μω


ion0

ion 31 αε

NB =

and

Ge

Ee 2

locion

v ==α

at ω = 0

GeNB

2

0ion 3

1)0(ε

=


eEB

GBG =

minusminus+minus v

1)0(

vvel

ion2μω

eEB

BG =⎟⎟⎠

⎞⎜⎜⎝

⎛minus

minus+minus v1

)0(1v

el

ion2μ

μμω

Define

⎟⎟⎠

⎞⎜⎜⎝

⎛minus

minus=el

ion21

)0(1

BBG

T μω



For the local field

el00

loc 1v

31

3 BNeEPEEminus

+=+=εε

Substituting

)(

v 22 ωωμ minus=

T

eE

770 Appendices

Then

)(13

122el0


+=T

eEB

NeEE

)(13

11 22el0

loc


T

eB

NeE

E

)(

11

)0(1 22el

ionloc

ωωμ minusminus+=

TBGB

EE

)()1(

)0()()1(22

el

ion22

loc

ωωμωωμ


=T

Tel

BGBB

EE

Inverting

)0()()1(

)()1(

ion22

22el

loc GBBB

EE

Tel

T


=ωωμ

ωωμ

))(1()0(1

1

22el

ion

ωωμ minusminus+

=

T

locB

BGEE

Define

))(1(

)0(22

el

ion

ωωμ minusminus=

TBBGF

Then

FE

E+

=1

1

loc


loc0

ion0

ionv

31

31)(

EeNNB

εα

εω ==

Substitute in v

)(3

1)( 22loc0ion

ωωμεω

minus=

T

eEE

eNB

)(

)0()( 22

ion

locion

ωωμω

minus=

T

GBE

EB

)1(1

1)( elion BFF

B minus+

=ω


Given

[ ] )1()(1)(3)()(

elionel

ion

0 BBBB


ωω

εεωε


)1()1(

11

)1(1

3)()(

elelel

el

00 BBF

FB

BF

F

minus⎥⎦⎤

⎢⎣⎡ minus

+minusminus

minus++infin=

εε

εωε

)1(

11

13)()(

el00 B

FF

FF

minus⎟⎠⎞

⎜⎝⎛

+minus

++infin=εε

εωε

)1(

3)()(

el00 BF

minus+infin=

εε

εωε

Substitute for F

))(1(

)0()1(

3)()(22

el

ion

el00 ωωμεε

εωε


TBBG

B

Define c

2el

ion

)1()0(3

BBGcminus

=μ

Then

)(

)()(2200 ωωε

εεωε

minus+infin=

T

c

Rewriting

202)()(

ωεωε

εω =infinminus

minusc

T


2

2

)0()(

L

T

ωω

εε =infin

)()0( 2

2infin= ε

ωωε

T

L

772 Appendices


0)()0(

02 =infinminus

minusεε

εω cT

0)()(2

202 =


εεωω

εω

T

LT

c

0)()( 22

202 =

infinminusminus

εωωωεωTL

TT

c

0)(

)(2

0222 =infin


TLTc

202)( LT

c ωεεω =infin

+



ψψ Em

=nablaminus 22

2

For each coordinate

0222

2=+minus i

i

i

mEdxd ψψ


⎟⎟⎠

⎞⎜⎜⎝

⎛+⎟

⎟⎠

⎞⎜⎜⎝

⎛= i

ii

ii xmEBxmEA 22

2cos2sinψ


2

222

2 2or2

manEnamE i

iii ππ ==

)(2

23

22

212

22nnn

maET ++= π





2

22

2

22

23

2)111114(



2

22

23

mahf π=


constant432 ==

mfa π



0

22

2

2

2

2

22

4where1

2 πεφeq

rq

rrrm=minus⎟

⎟⎠

⎞⎜⎜⎝

⎛

partpart+

partpart+

partpartminus=H




εεε RRYEmqEmq

r ==== )(2

2

4

2

2ρ


ψψρ

ψφρρρρ

ε2

4

2

22

2

2

22

2

4

422

2)(11

22mqmqqqm

m=minus⎟

⎟⎠

⎞⎜⎜⎝

⎛

partpart+

partpart+

partpartminus

774 Appendices

Canceling we obtain

ψρψψ

φρρρρε

2111

21

2

2

22

2=minus⎟

⎟⎠

⎞⎜⎜⎝

⎛

partpart+

partpart+

partpartminus




φφ mie)( =Φ



⎞⎜⎜⎝

⎛minus+minus

21

21 2

2

2 ερρρρρ

mdd

dd

We let

2

and1 2

2xαρ

α=minus=ε


Ψ=Ψ+⎟⎟⎠

⎞⎜⎜⎝


241

21 22

2

2 xxm

dxd

xdxd


21122 +=+= nn αα


2212

21 )(

or)(

1+

minus=+

minus=n

REn

ε


21)1(

21 minus+=+ nn





)()(2 2

22zEz

dzd

mψψ =minus


0)(2)(22

2=+ zmE

dzzd ψψ

⎟⎟⎠

⎞⎜⎜⎝

⎛+⎟

⎟⎠

⎞⎜⎜⎝

⎛= zmEBzmEAz 22

2cos2sin)(ψ



2

222

2mwpE p

π=





12

2

0minusminus=

xeF

πε


x

ex

edxx

eFdxdVxxxx

0

2

0

2

20

2

161




x

eExV0

2

0 16)(

πεminus=

776 Appendices


x

exeEExV0

2

10 16)(

πεminusminus=


01

=⎟⎠⎞

⎜⎝⎛

=xxdxdV

which gives

10

1 16 Eex

πε=

or

0

13

01 21)(

πεEeExV minus=

Thus

0

13

21

πεEe=Δ



a

V(r)

ndashV(0)

E (bound state)

r




20 )(2 EVm +

=α

22mEminus=β



This implies

2


Also

E

EV

aa

minus=

+=

+= 0

22

2

22

)(cot11)(sin

βαα

αα


2πα gta

to estimate a

eV)1(22)(22

22

0 mEVma ππ

απ gt

+=cong

nm931)1061)(1019(2

)10636)(143(1931

34

min congtimestimes

timescongminusminus

minusa




778 Appendices

Energy Gap

Ec(z)

Eg1 Eg2

Valence Band

Conduction Band

d

z

Ev(z)


EFFzEzmyxm

c =⎥⎥⎦

⎤

⎢⎢⎣

⎡+

partpartminus⎟

⎟⎠

⎞⎜⎜⎝

⎛

partpart+


22 2

22

2

2

2

22




32||

2

322

2)(

2)(

2Ek

mEkk



ψψ 32

22)(

2EzE

dzd

mc =⎟

⎟⎠

⎞⎜⎜⎝

⎛+minus lowast



sum=

minus=N

ni

ndiq ndz

Nz

1

i )(e1)( χψ


0 we find for E3




2

2 2 0 0( ) cos












)(22

||

222

2

222

EE

kkmmw

nE

n

yx

+equiv

++= π

)(22

222

2||

2

|| yx kkm

km

E +==




ndEm

dkdkkkd

2

2||||||

2

2

2

π

ππ

=

==

780 Appendices


)(

)(2)2(

2)(

2

0||||22

sum

sum int

minus=

minusminus=infin

n n

n n

EEm

dEEEEmED

θπ

δππ



23)(π

mED = (1)

In 3D

E

dEmdkmkE 2

22 221

2==

2

212)4(

)2(2

)unit volper (4)2(

2)(

223

23D3

EdEmmE

dkkdEED

ππ

ππ

=

=


32

22

1)(223

22D3

E

wmmED π

π ⎟⎟⎠

⎞⎜⎜⎝

⎛=

where

2

222

32

3mw

EE π==

Per unit area

222D3332

21)(

ππ

πmw

wmwED == (2)

which checks (1)


d)

D3D(E)w

2hπm

23hπm

E1 E2 E3




1001

512

512

51

2

51

2~

21

21

22

11

I

NN

NN

NN

NN

AA

=⎟⎠⎞

⎜⎝⎛=

⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜

⎝

⎛minus+

⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜

⎝

⎛

minus

+=


Bibliography

Chapter 1

















784 Bibliography










Chapter 2









Chapter 2 785





















786 Bibliography

Chapter 3





















Chapter 3 787



















788 Bibliography

Chapter 4





















Chapter 5 789














Chapter 5







USA 92 6663-6667 (1995)

790 Bibliography






















Chapter 6 791



Francis London 1963

Chapter 6
















1994

792 Bibliography






















1996

Chapter 7 793

Chapter 7



















794 Bibliography



















Chapter 8 795










(2004)

Chapter 8






796 Bibliography






















Chapter 9 797





Chapter 9

















798 Bibliography




Chapter 10















Chapter 11 799






Chapter 11















800 Bibliography


















Chapter 12



Chapter 12 801























802 Bibliography





















Chapter 12 803

















804 Bibliography

Appendices





















Subject References


Mechanics




Electricity



Optics



Thermodynamics






Critical Phenomena


806 Bibliography

Crystal Growth


Modern Physics



Quantum Mechanics





Math Physics


Solid State







Condensed Matter






Problems




Applied Physics





Rocks




808 Bibliography


The Internet



Periodic Table


Further Reading












Further Reading 809
















810 Bibliography

















Further Reading 811


















812 Bibliography




Index

A













B


814 Index










C




Index 815





















816 Index




316 317

D









594 640






E





Index 817

















503 566










F



818 Index


493 538 Fermi function 160 248 298 303



373 385 399 407 422 426 456 486 513







G

GaAs 242 295 297 324 326 328 329 337 347 563 564 579 618 628 629 637




Index 819

Group velocity 55 306 532 562 627


H











I






820 Index


J


K



L









529 570

Index 821




M













N


531 600 NaCl 6 9 28 579 600

822 Index









O








P





Index 823















824 Index


Q




R












S



Index 825













826 Index






T



361 418











475 509

Index 827

U




V



Vortex region 466

W


X


Y

YBa2Cu3O7 506 508

Z


Appendices

A Units










G1 Bose Particles

G2 Fermi Particles


H1 Propagators

H2 Green Functions

H3 Feynman Diagrams

H4 Definitions




J Folk Theorems



M Problem Solutions














































































Bibliography

Mechanics

Electricity

Optics

Thermodynamics


Critical Phenomena

Crystal Growth

Modern Physics

Quantum Mechanics

Math Physics

Solid State

Condensed Matter


Problems


Applied Physics

Rocks


The Internet

Periodic Table

Index

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Quantity Conversion



1 calorie = 419 J

1 T = 1 Wbm2 = 104 gauss

me c2 = 051 MeV

mp c2 = 938 MeV















654 Appendices
















ΩAVA = (B6)


IAAAA == ~~ (B7)


Ω=VAA~ (B8)



aAE ~= (B9a)

or

AEa = (B9b)


)~()~(

)~()~(

~~~~

2

sumsum

sumsumsum

==

==

===

kj jkiijj jjj

kj kjkkjjk kjkj

jiij

EEEE

EEEEEEVAEAEVaaaa

δωΩ

δΩΩΩυ



032

21det 2

2=⎟

⎟⎠

⎞⎜⎜⎝

⎛

minusminus

ωω (B10)


5221 +=ω (B11a)


656 Appendices


51

2 21

21 +

== aaωω (B12a)

and for

51

2 21

22 minus

== aaωω (B12b)


⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜

⎝

⎛

minus

+=2

2

11

51

2

51

2~

NN

NN

A (B13)

where

21

21

1 1)15(

4)(⎥⎥⎦

⎤

⎢⎢⎣

⎡+

+=minusN (B14a)

and

21

21

2 1)15(

4)(⎥⎥⎦

⎤

⎢⎢⎣

⎡+

minus=minusN (B14b)


⎟⎟⎠

⎞⎜⎜⎝

⎛

⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜

⎝

⎛

minus

+=⎟⎟⎠

⎞⎜⎜⎝

⎛=

2

1

22

11

2

1

51

2

51

2

aa

NN

NN

EE

E (B15)

Problems



⎟⎟⎠

⎞⎜⎜⎝

⎛

minus+=⎟⎟

⎠

⎞⎜⎜⎝

⎛

520052

3221~ AA







0)()(d

)(d2

2=+ xxU

xx ψψ (C1)


)()( xUnaxU =+ (C2)








0d

d =x

W (C6)

where

21


=W (C7)


658 Appendices



0)()(d

)(d2

2=++++ axaxU

xax ψψ


0)()(d

)(d2

2=+++ axxU

xax ψψ (C8)


)()()( 211 xBxAax ψψψ +=+ (C9)

)()()( 212 xDxCax ψψψ +=+ (C10)


21

21

21

21

2121

2121ψψψψ

ψψψψ

ψψψψψψψψ

primeprime=

primeprime=


DBCA

DCBADCBA

or

1=DBCA

or

1=minus BCAD (C11)


)()( xΔax ψψ =+ (C12)


)()()( 21 xxx βψαψψ += (C13a)

or

)()()( 21 axaxax +++=+ βψαψψ (C13b)


)()()()()()()(

2121

xxxDBxCAax


+=+++=+ (C14)



αΔβα =+ CA

and

βΔβα =+ DB


0=minus

minusΔ

ΔDB

CA (C15)






DA +=+ minus++

1)(ΔΔ


1)(ΔΔ (C17)

and

DA +=+ minus+ ΔΔ



If we set

be=+Δ (C19)

and



)()( 11


and

)()( 12


660 Appendices





where

)()( xuaxu =+ (C25)



Let

)()( 11 NN xxExx ψψ =H (C26)



0][ =lTH (C27)

and






lkmiddotl

ie=t (C31)

Further




so that

12 =lt (C32)










)(02)(0

22

21 Gnnn G =+++


5 In general sum= j

kj

klij

kil AT ψψ

where the Ail




kklkl AT ψψ =


662 Appendices




ntctx nin

i sum= )()(ψ


nmNi

im

imnm cctctc

Nlowast





)( ATrA ρequiv


][i Ht

ρρ =partpartminus


⎟⎠⎞

⎜⎝⎛ minus=

kTHFexpρ


)][exp()]exp([exp

kTHTrkTHHTr

kTHFHTrHU

minusminus=⎥

⎦

⎤⎢⎣

⎡⎟⎠⎞

⎜⎝⎛ minus==








Let

)()( 0 tVt += HH (E1)

lEl l00 =H (E2)


00lk

lkEE minus=ω (E4)




664 Appendices


200

2 2)()(

)(

)cos(1lim ωδπ

ω

ω tfEEttfi

if

if

t=minus=

minusinfinrarr

(E6)












⎟⎟⎠

⎞⎜⎜⎝

⎛=

00σ

σα (F2)




⎟⎟⎠

⎞⎜⎜⎝

⎛minus

=I

I0

0β (F3)



12

)(2

0

20 ltlt

minusminusequiv

cmVcmEε (F4)


20cmEE +prime= (F5)

and

⎟⎟⎠

⎞⎜⎜⎝

⎛=

φχ

ψ (F6)




⎤

⎢⎢⎣


4482 220

2

220

2

230

4

0

2pσ V

cmV

cmV

cmp

mpE (F7)


])[(4 22

0


cm (F8)


666 Appendices


SL middot dd1

2 220

2

rV

rcm (F9)





G1 Bose Particles




][

0][

0][

dagger

daggerdagger

ijji

ji

ijjiji

bb

bb

bbbbbb

δ=

=

=minusequiv

9 See Mattuck [A14]



daggeri i in b b=

and

dagger1 iii bbn =+


11

1

11dagger

11



++=

minus=

iiii

iiii

nnnnnb

nnnnnb



1

1dagger

iiii

iiii

nnbn

nnbn

=minus

=minus



dagger)1()1(






)()(


sumlowast=

=

α αα

α αα

ψψ

bu

bu

rr

rr


668 Appendices

G2 Fermi Particles




0

0

dagger

daggerdagger

ijji

ji

ijjiji

aa

aa

aaaaaa

δ=

=

=+equiv







iii nan




0











)2()1()2()1(

210daggerdagger

ββ

ααβα uu

uuaa harr











670 Appendices






H1 Propagators




H2 Green Functions


δεω

ωi

1)(0 +minus=+

kG k (H1)


H3 Feynman Diagrams


H4 Definitions






672 Appendices




















dagger )2()1()21()2()1(21)1()1()1(H (H3)



31



0

daggerdagger

=+=+

ijji

ijijjiaaaaaaaa δ



δεω

ωi


=sum

+

l lklkk VG k (H6)





674 Appendices



112223 d)21()()()()(d rrrrrr φφφφ (H8)




δεω

ωi)(


=sum

+













klk

Gδωεω

ωi)(


=sum k

k (H11)





676 Appendices




fcc (110) (011) (101)

2 2 2

bcc (11 1) ( 111) (1 11)2 2 2

a a a




1 2 31 ( )Ω






lkh

lkh dG π2


Δ =k G (See 152)











G (See 141)





⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎟⎟⎠

⎞⎜⎜⎝

⎛+⎟

⎟⎠

⎞⎜⎜⎝

⎛minus=

126

4)(ijij

ij rrrU σσε


1e

1 minus

= kTn ω (See 277 2215)


222 akmks=ω (See 248 288 290)






D

DDωωωωωω

gt=leprop

0)( 2

(See 2224)

3126v ⎟

⎟⎠

⎞⎜⎜⎝

⎛==

VN

kk DDD

πθθω (See 2228)


678 Appendices


intminus

⎟⎠⎞

⎜⎝⎛= T

x

x

VdxxTNkC

0 2

43

)1e(e9 θ

θ (See 2229)


3( )

( )( )(2 ) const

dAVD ω

ωω

ωπ= int

k knabla (See 3256)




)e(e 20


2( ) TWθ⎛ ⎞prop Δ ⎜ ⎟⎝ ⎠

k


22

2

9)(

Dm

m mkAT θ=



V

Dln



B

C3

vγα = (See 2237 2238 2250)




ρ11Cv = (See 2270)


ρ44Cv = (See 2270)



3 Electrons


1e

1)( +




( ) ( )u u= +k kr r R (See 3129 C25)


EmVEg23

222

2)( ⎟⎟

⎠

⎞⎜⎜⎝

⎛=π

(See 3164)


32

22

32

)0( ⎟⎠⎞

⎜⎝⎛==

VN



m

ne τσ2

= (See 3214)




RH (See 6102 1223)

680 Appendices


LTK =σ

(See 3215)

22

3⎟⎠⎞

⎜⎝⎛=

ekL π (See 3215)


0

00 near band edge

E E U

U E E

lowast

+

minus=

minusk G

G k G (See 3230)



k r r R (See 3244)


k (See 3255)




i

Bloch Fns


kkr R r (See 3332)



cgs)in or (mceB

meB

c =ω (See 3196)




22

2 2

0 ( )

( )

( ) ( ) ( )2

1( ) ( ) ( )4

el

ioni i

j i i ij i

V

Vm

e d E

ψ ψ

ψ ψ ψπε ne

minus nabla +



r

r r r

r r r rr r




22

2

0 ( )|| spin

( ) ( ) ( ) ( ) ( )2

1 ( ) ( ) ( ) ( )4

ion eli i i

j i j i ij i

V Vme d E

ψ ψ ψ

ψ ψ ψ ψπε

lowast

ne

minus + +


r r r r r

r r r r rr r

nabla



1 32

2 3( ) ( ) ( ) ( ) ( ) ( ) ( )2


mψ ψ ψ ψ

π⎛ ⎞⎡ ⎤minus + + minus =⎜ ⎟⎣ ⎦ ⎝ ⎠

r r r r r r rnabla



2

2

1 ( ) ( )( ) 1( ) ( )k

e f fV E E iq σ

ε ωω ηε


k k qqk q k

(See 9166)

xx

xxxF

kqxxF

qk

qTF

sminus+minus+==+==

11ln

41

21)(

2)(1)0(0

2

2

2ε


F

srk

F Eemk

rkq s

εεπφ

23e

41 2

02 ==ltlt minus



collisions

f ff ft t


Fv nabla nabla


0RHS of BDE( )

f fτminus

= minusk

(See 4149 4145)

682 Appendices


11 123 v( ) ( ) ( )4

edj e f L E L T

eμ

π⎛ ⎞= minus = + + minus⎜ ⎟⎝ ⎠


21 223 ( )v( ) ( ) ( )4

Qdj E f L E L T

eμ

π⎛ ⎞= minus = + + minus⎜ ⎟⎝ ⎠




0ej TE

σ = =nabla

0Qe

jj

Tκ = =

nabla


0eEQ jT

= =nabla

0Q

e

jT

j

QT

Π = =

Π =

nabla


constant131 conganc



⎟⎟⎠

⎞⎜⎜⎝

⎛minus

VED F )(1exp




heG =




D

D

TT

TT

θρ

θρ

ltlt

gtgt5~

~ (See 4146)





)SI(21

0Ae

Bπ=⎟

⎠⎞

⎜⎝⎛Δ (See 534)



⎟⎟⎠

⎞⎜⎜⎝

⎛= (cgs) 4 2

0

2

mne

mne

pπ

εω (See 10108)







2innp = (See 613)

⎟⎟⎠

⎞⎜⎜⎝

⎛minus⎟

⎟⎠

⎞⎜⎜⎝

⎛= lowastlowast

kTE

mmkTn ghei 2

exp)(2

2 4323

2π (See 614)

684 Appendices


ddt

= kF (See 644)




2

2

211

kE



eDkT =μ (See 684 685)





⎟⎟

⎠

⎞

⎜⎜

⎝

⎛ ΔΔ== (cgs)

2

222 ne

V

ne

VKZwidth b π

εε (See 6193)


⎥⎦

⎤⎢⎣

⎡minus⎟

⎠⎞

⎜⎝⎛

⎟⎟⎠

⎞⎜⎜⎝

⎛+= 1exp

00 kTep

LD

nLDej b

np

pp

n

n ϕ (See 6241)



23minusprop Tμ


23Tpropμ





⎟⎠⎞

⎜⎝⎛ )(

22cgs

mceB

meB (See 7255 7256)

Lande g factor

⎥⎦

⎤⎢⎣

⎡+

+minus++++=)1(

)1()1()1(211

JJLLSSJJg

(Implied by (76) (710))


)(20 FBp EDμμχ = (See 3181 3201)


magnetonBohr =Bμ


)groundexcited(22

==minus

= gexcEEgexc

VN

gexc

zμχ


⎟⎠⎞


⎤⎢⎣⎡ ++=

Jx

JJxJ

JJxBJ 2

coth21

2)12(coth

212)( (See 716)



4sincos)()2(

xxxxxFRkFJ ijFij



kT

BgJx eff

Bμ= (See 722)

MBBeff λ+=

686 Appendices


)cgs(

32

0MMH

kEk effB

F

μθθ =primegt

prime (See 7222)


⎟⎟⎠

⎞⎜⎜⎝

⎛minusprop

)(1exp

FK EJD

T (See 7300)


h

eGxy

2ν= (See 1247)



KJpropδ (See 7251)





23

23)()0(

TC

TTMM

M prop



)for (1cc

c

c

TTHTTM

TT

TTC

=asympminusasymp

minusasymp

minusasympminus

minus

δβ

γ

α

χ






2

( ) ( ) (cgs)nemc



2

2

4 nemc

L πλ = (See 86)


g

FEv

πξ 2~0 (See 849)



II ξ lt λ 20~

1 πλϕ

cH 20~

2 πξϕ

cH 2

~1

2⎟⎟⎠

⎞⎜⎜⎝

⎛ξλ

c

c

HH



)(20 cgs

ehc=φ (See 845)



⎟⎟⎠

⎞⎜⎜⎝

⎛minuscong

)(1exp141

FDc EVD

T θ (See 8217)


⎟⎟

⎠

⎞

⎜⎜

⎝

⎛

+minus


1exp

451 epep

epDcT

λμλ



688 Appendices


2

20



⎡ ⎤⎛ ⎞⎢ minus + + ⎥ =⎜ ⎟⎝ ⎠⎢ ⎥⎣ ⎦

Anabla (See 83)

2

( ) ( 2 2 )2s eiq qj q e m m






)()0(

2

2

infin=εε

ωω

T

L (See 950)


⎟⎟⎠

⎞⎜⎜⎝

⎛+=+= )SI(

3)guassian(

34



⎟⎟⎠

⎞⎜⎜⎝

⎛=

+minus=

+minus

sumsum )SI(31

2)(1)(

34

21

00

0jjjj NN α

εεεεεαπ

εε (See 930)



direct 0eΔ =k








)cgs(2 222

4

neEE gn ε


)cgs(111

he mm+=

μ (See 10101)


11 Defects


Mollwo Equation







⎟⎟⎠

⎞⎜⎜⎝

⎛==

pGeEp

Bπω 2 (See 129)


)(exp 41 MottT

AB ⎟⎠

⎞⎜⎝





232surface =⎟

⎠⎞


kTWkTemj

π (See 1145)

690 Appendices


3 2

2surface exp BWj AE

E

⎛ ⎞= minus⎜ ⎟⎜ ⎟

⎝ ⎠ (See 1149)


J Folk Theorems


Chapter 1



Chapter 2



Folk Theorems 691

Chapter 3









Chapter 4







Chapter 5


Chapter 6




692 Appendices















Chapter 7






Folk Theorems 693





Chapter 8




Chapter 9



Chapter 10





Chapter 11







Chapter 12



694 Appendices







Gauss law V A A

dV disin





10

e +

infinminus =int n

axn

andxx


π41Y

e21)(

Y)(

00

23

10

0010001

=

⎟⎟⎠

⎞⎜⎜⎝

⎛=

=Ψ

minus Bar

BarR

rR





r θnabla ϕ

221 1 1( ) (sin )

sin sinrA

r A Ar r rr

ϕθθ



2

2

2222

2 sin1sin

sin11

ϕθθθ

θθ partΦpart+⎟

⎠⎞

⎜⎝⎛

partΦpart

partpart+⎟

⎠⎞

⎜⎝⎛

partΦpart

partpart=Φnabla

rrrr

rr2



111


=primeprime

γγγ

nnn

n


n

ennn ⎟⎠⎞




)()(

1))((

)()()(0)(


n

n nn

xxxf

xxxf

xf

bfdxbxxfxx

xxbxdxbx

=

minusprime

=


minus===minus

sum

int

int

δδ

δδ

δδδ

Complex Variables


π


nazfazdzd

naf n

n

n


)1(1)(Res 1

1

⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧

minusminus

= minus

minus

rarr


SI Gaussian

0

t

tρ

parttimes = +part


sdot =sdot =

H J D

BE

DB

nabla

nabla

nablanabla

4 1

1

40

c c t

c t

π

πρ

parttimes = +part


sdot =sdot =

H J D

BE

DB

nabla

nabla

nablanabla

0ε= +D E P

0

1μ

= minusH B M



( )qc

= + timesvF E B

696 Appendices












1956 Transistor


















9 Total for Decade

698 Appendices

























A Geim K Novoselov

2010 Graphene



Brillouin




Bloch






Bardeen



700 Appendices

Others






M Problem Solutions





451388

2412

126


=+minus=M




1

A

2

3

456

7

8

g5 g6

g7

g8


⎟⎠⎞

⎜⎝⎛= 87654321

876543211g

⎟⎠⎞

⎜⎝⎛= 21876543

876543212g

⎟⎠⎞

⎜⎝⎛= 43218765

876543213g

⎟⎠⎞

⎜⎝⎛= 65432187

876543214g

⎟⎠⎞

⎜⎝⎛= 23456781

876543215g

⎟⎠⎞

⎜⎝⎛= 45678123

876543216g

⎟⎠⎞

⎜⎝⎛= 67812345

876543217g

⎟⎠⎞

⎜⎝⎛= 81234567

876543218g

702 Appendices

Then eg

432 6543218787654321 ggg =⎟⎠⎞

⎜⎝⎛=


g1 g2 g3 g4 g5 g6 g7 g8




)(21 kjia minus+= a


)(23 kjia +minus= a

x

y

z






y

z

x

a2

a1

a3

a


a


704 Appendices





2 2 0h kh k h k


a bG

since

1 2 32 ( )h k lπ= + +G b b b

and


Thus

( ) 2( )( ) ( )h k ld h k lh k l h h k l

π= =G aG G

sdot






where

f iΔ = minusk k k



ki

b

b

bα

bα

hellip

x

y

z

kf

0ˆ =sdot ik i

2sin

2sin

e

2sin

2sin

e

e1

e1

e1

e1

ee

2)1(

i2

)1(i

i

i

i

i

10

i10

i

⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜

⎝

⎛

+⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜

⎝

⎛

=

minus

minus+minus

minus=

+=

minusminus

minusminus

minus

minus

minus

minus

minus=

minusminus=

minus sumsum

α

αα

α

α

α

bk

Mbk

bk

bNk

P

fx

fxMbk

fx

fxNbk

bk

Mbk

bk

bNk

Mm

mbkNn

nbk

fxfx

fx

fx

fx

fx

fxfx


)(integer 2

and)(integer 2

)2()1(

pbk


απ (a)

Since

pl

bk

bk

pl

bk

bk

fx

fx

fx

fx αα

== )2(

)1(

)2(

)1(

or1

)2()1(fxfx kk ne



since

22fxfy kkk minus=


706 Appendices


Let

λπ2=K





Define

θsinKB =

)iexp( BbZ =

So


But


)]2sin(ln[

1)]2sin()exp[(ln

0

)2sin(

knbZ

knbZm

knbZZ

m

knb

πε

πε

πεπε

+=

=

=

suminfin=

Thus


0 )2sin()ln( πε


2)2cos(21)2sin(ln

11

ZkbZkbZZ

ZE

+minus+

minus=

ππε






mGBb π2+plusmn=


b




Aring2255=Ka


Aring2255100 == ad

)bcc(6125222255


deg=rArr== 6816225551

2sin θλθ

d

deg=rArr= 04352255

)51(2sin θθ



θλ sin2 100dn =

above as 2)e1( 0435 4510

)51(42

4sin


)51(32

3sin

above as 2)e1( 6816 4510

)51(22

2sin


2sin

i4400

100

i3300

100

i2200

100

i100

100

ffSd

fSd

ffSd

fSd

=+====

=+====

=+====

=+====

minus

minus

minus

minus

π

π

π

π

θλθ

θλθ

θλθ

θλθ

708 Appendices




mm m

γ γ γx1 x2

1γx3

3 1 2 3





02

22

321

2

2

2

=⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜

⎝

⎛


uuu

mm

m


(2)


twice32mγω = (5b)


⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛==

111

31

321

0uuu

Uω (6a)

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛minus==12

1

51

3 mU γω (6b)


and

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛minus==

112

51

3 mU γω (6c)



)(2iexp1

)(2iexp1

)(2iexp1

)1(0

10

10

sum

sum

sum

minusminus= minus

minus=

minus=

⎥⎦⎤

⎢⎣⎡ minus=

⎥⎦⎤


⎥⎦⎤

⎢⎣⎡ minus=

Ns s

Ns s

Ns snm

nmN

sgN

nmN

sgN

mnN

sgN

G

π

π

π


[ ]

⎥⎦⎤

⎢⎣⎡ minus=

minus⎥⎦⎤

⎢⎣⎡ minus=

⎥⎦⎤


⎤⎢⎣⎡ minus=⎥⎦

⎤⎢⎣⎡ minus+

)(2iexp

)(2iexp)(2iexp


nmsN

nmnmsN

nmNN

nmsN

nmNsN

π

ππ

πππ

and (see 2104)

⎟⎠⎞

⎜⎝⎛=⎟

⎠⎞

⎜⎝⎛ +=⎥⎦

⎤⎢⎣⎡ +

Ns

NsNs




Nsg

NG π

But

⎟⎟⎠

⎞⎜⎜⎝

⎛⎥⎦⎤



Nnm

NNggN

ππ

so

mnN

snm GnmN

gG =⎥⎦⎤


0 )(2iexp π

QED

710 Appendices




)2()()(

)()(

)2()()(

)()(

)2()()(

54145515

34454

32423343

12232

15251121

XXXXXXXXm

XXXXXm

XXXXXXXXM

XXXXXm

XXXXXXXXm


minusminusminus=


minusminusminus=


αααβα

βββαβ

ααα


0

210011100020001110012

54321

22

212

2

=

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛

minusΩminusminusΩ

minusΩminusminusΩ

minusΩ

minus

uuuuu

AAAAA

AAδ


⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

Ω=

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛



minusminus

54321

2

54321

210011100

020001110012

uuuuu

uuuuu

AACCC

AA


⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

minusminus

=

1

0

1

a

aU


This requires

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛

ΩminusΩminus

sdotΩΩΩ

=

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

+minus++minus

minus+minus

=

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

minus+minus+minus+

+minus++minus+minus

22

2

2

2

0

31)1(

01)1(

3

211))(1(

)1(112

a

a

aAa

Aaa

aaAcaca

Aaa




2





)1()1()1( dagger

dagger

ϕψψ

ψψϕ

minus=minus=minus=

==

nna

aaa

aaaNaN

n

n

nn

712 Appendices


nKan =minus1 So

112

dagger2

nnnK

naanKnn

=

=minusminus


1or1 minus== nnnan

K


)1()1(

)1(dagger

daggerdagger

daggerdaggerdagger

ηψ

ψ

ψψη

+=+=

+=

==

nna

aaa

aaaNaN

n

n

nn


1

1+

=n

K

so

11dagger ++= nnna


⎟⎠⎞

⎜⎝⎛ +=⎟

⎠⎞

⎜⎝⎛ +=

21

21dagger Naa ωωH



321321


⎜⎝⎛=



321

321

321

ηηη

ηηη

ηηη

partpart

partpart

partpart

partpart

partpart

partpart

partpart

partpart

partpart

=

zzz

yyy

xxx

J

1 1 2 2 3 3η η η= + +r a a a


etc11

xax =partpartη

so we find

1 2 3

1 2 3 1 2 3

1 2 3

( )x x x

y y y

z z z

a a a

J a a a

a a a

= = sdot timesa a a

QED



1 1

2 1

3 1

100

sdot =sdot =sdot =

a ba ba b



a

KΩ

= 1

and

2 31

a

times=

Ωa a

b


714 Appendices


1 1 2 2 3 32 ( )n n n nπ= + +G b b b




31 21 2 3

1 2 32

nn nN N N

π⎛ ⎞

= + +⎜ ⎟⎝ ⎠

q b b b



a) 31 231 2 31 2

31 2

1 2 3

1 1

11

1

22 2 ii i 11 1i

1 2 3 1 2 30 0 0

i2

2i1

1 1 1 1e e e e

1 1 e etc



l

l

n

nn

nn n ll l NN NNN N

l l l

n l

n lN

i i

i i

n

N N N N N N

N

n Nn N

ππ π

π

π

δδ


= = ==

minus=

minus= ne= =

=

=

sum sum sum sum

sum

q R

R

Gq

q GG

G

b) 111

1

1

1

11

21

i(BZ)

1 2 3 1 0

2

21

0

1 1e e etc


1 1 e


l

l

ni lNN

l

i n

i lN

i i

i i

N N N N

N

l Nl N

π

π

π

δ

minussdot

==

minus=

minus= ne= =

=

sum sumq Rq

R


c)

1 2 31

1 1

3

33

1 2 31 2 3

31 2 3 03

fpp 33

( ) 2

of s( )Vol of space

1 ( )(2 ) ( )

( )(2 )

( ) (2 )

n n nn

f fN

qf dq

f d

N N NN N N

f d

Vf d

π

π

π

π

⎛ ⎞⎛ ⎞= +⎜ ⎟⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠

=

=sdot times

Ω=

sum sum

int

int

int

int

q q b

q q

q qb b b

q q

q q

d)

i i3fpp3

fpp

i

1 2 30

e e (with )(2 )

1 e

l l

l

l

a ad q VV

N N N

π

δ

Ω Ω= rarrinfin

=

=

sumint

sum

q R q Rq

q Rq

R

sdot sdot

sdot

e)

1 1 1 2 2 2 3 3 3i( ) i2 [( ) ( ) ( ) ]3 31 1e el l l l l l l luc



3

1 2 3


η η η⎛ ⎞

= = ⎜ ⎟⎝ ⎠

r


etc so

1 2 3

1 2 3 1 2 3

1 2 3




So

1 1 1

1 1 111

1i( ) i2 ( )310

1i2 ( )

1 1 0

1 e e

e i2 ( )

l l l l

a

l l

d d

l l

π η

π η

η

δπ


primeminusprime

=Ω

= =prime minus

int intG G r

ll

rsdot

716 Appendices

f)

i ( ) 3all 3

i ( ) i ( ) i ( )

i( )i( ) i( )

1 e(2 )

1 1 1e e e2 2 2( )

1 e 1 e 1 e2 i( ) 2 i( ) 2 i( )

yx z

x y z

yx yx z

zx y



LL L

d q

dq dq dq

L L L

x x y y z z

π

π π π

π π π

primeminus



minusminus minus

=

rarr infin


int

int int int

q r rq

sdot


( ) ( ) ( )( ) ( ) ( )( )

z

x y z

yx zL L L


x x y y z zπ π π

δ δ δδ









Frequency c

gωω=

Temperature )( k

Ttcω

=


Uucω

=

Entropy NkSs =


NkC

c vv =



⎟⎠⎞

⎜⎝⎛ +⎟⎠⎞

⎜⎝⎛==

Nd

ddpP

c

ωωωωω


)(

Thus

π

ωω2

)()( NadqdqqNdN ==

1

)2)(2cos(1

2)(

22cc

cN

aqaN

ddqNadqN

ωωπω

ωπωπω

minus=

==

So

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

minus⎟⎟⎠

⎞⎜⎜⎝

⎛

minus==

221

11e

1

ckTc d

dpPωωπω

ω ω

or since

tg

kTkT c==

ωωωω

c

then

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

minus⎟⎟⎠

⎞⎜⎜⎝

⎛

minus=

21

11e

1)(g

gtP tgπ (1)



int

intsum

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

minus⎟⎟⎠

⎞⎜⎜⎝

⎛

minus=

minus=

minus= =

c

c

m

dN

dNU

cckT

kTNm kT

m

ω

ω

ω

ωω

ωωωω

ωπ

ωωωω

0 22

01

1

11e

2

)(1e

21e

int⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

minus⎟⎟⎠

⎞⎜⎜⎝

⎛

minus==

c

cc

kTc

cd

NUu

ω

ω ωωωω

ωωπω 0 22 )(

1

11e

2

718 Appendices

or

int⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

minus⎟⎟⎠

⎞⎜⎜⎝

⎛

minus=

1

0 21

11e

2 dgg

gu tgπ (2)

ωωωπω

ω

ω

ω

ω

ω

ω

ω

dNkT

kTC

cckT

kT

m kT

kTm

c

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

minus⎟⎟⎠

⎞⎜⎜⎝

⎛

minus=

minus=

int

sum

2202

2

2

2

2

2v

1)1e(e)(2

)1e(e)(1

int⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

minus⎟⎟⎠

⎞⎜⎜⎝

⎛

minus==

c

cc

kT

kT

c

c dTNkN

NkCc

ω

ω

ωωω

ωω

ωπω

ω

0 222

2

22v

v )(1

1)1e(

e)()(

2

int⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

minus⎟⎟⎠

⎞⎜⎜⎝

⎛

minus=

1

0 22

2

v1

1)1e(

e)(2 dgg

tgc tg

tg

π (3)


kTFj



VT

FS ⎟⎠⎞

partpartminus=

so

( ) sumsum=

minus=

minus

minus+minusminus=

N

m

mkT

N

m

kT

kTkTkS

mm

12

1

e11e1ln ωω

ω

( )

int

int

minusminus+

minusminusminus=

=

minus

c

c

cc

kT

cc

kT

dkTkTNNk

dNkNk

NkSs

ω

ω

ωω

ωωωω

ωπ

ωωωωπ

0 22

2

0 22

)(1

11e

)2(1

)(1

1e1ln21


+minus

minusminus= minus1

0 2

1

0 2

1

11e

2

1

1e1ln2 dgg

tgdgg

s tgtg

ππ (4)



range of ω

ωxxp


P10x1π

1

e

ωx

10 1minus

sdot1

1 ωx( )2minus

sdot=

P100x1π

1

e

ωx

100 1minus

sdot1

1 ωx( )2minus

sdot=

P300x1π

1

e

ωx

300 1minus

sdot1

1 ωx( )2minus

sdot=

0 05 10

200

400

P300

P100

P10

ω

720 Appendices


the range of t

txxr


ux2π

0

1

yy

e

y

tx 1minus

1

1 y2minussdot

⌠⎮⎮⎮⎮⌡

dsdot=

spht x2π

0

1

y

ytx

⎛⎜⎝

⎞⎟⎠

2e

y

txsdot

e

y

tx 1minus

⎛⎜⎝

⎞⎟⎠

2

1

1 y2minussdot

⌠⎮⎮⎮⎮⎮⎮⎮⎮⌡

dsdot=

0 1 2 3 40

1

2

3

4

u

spht

t


0 05 1 15 20

05

1

15

2

u

spht

t


Entropy t2π

minus

0

1

ωln 1 e

ωminus

tminus

⎛⎜⎝

⎞⎟⎠

1

1 ω2

minus

sdot

⌠⎮⎮⎮⎮⌡

dsdot2π

0

1

ω

ω

t

e

ω

t 1minus

1

1 ω2

minus

sdot

⌠⎮⎮⎮⎮⎮⌡

dsdot+=

0 100 200 3000

2

4

6

8

Entropy t

t

722 Appendices




By Problem 29

2 31

a

times=

Ωa a

b


2 3 [(1)(1) (1)( 1)] [(1)(1) ( 1)(1)] [( 1)( 1) (1)(1)]

2

(2 2 ) ( )2

a

a a


= + = +

a a i j k

i j i j

and

2

21 2 3( ) (2)


so Ωa = a2 and

11 ( )a

= +b i j


2

3

1 ( )

1 ( )

a

a

= +

= +

b j k

b i k




1 2 31 1 1( ) ( ) ( )a a a




ky

kxkz


a

a

n m + 1

n mn ndash 1 m

n m ndash 1

n + 1 m

xnm ynm


])()()()[(2

])()()()[(2

21

21

21

21

21

21

21

21

mnmnmnmnmnmnmnmn

mnmnmnmnmnmnmnmn

yyyyyyyyk

xxxxxxxxkU



minus+minus+

minus+minus+


)4( 111

mnmnmnmnmn



)4( 111

mnmnmnmnmn



724 Appendices


)i(1 e maknakt

mnyxcx ++minus= ω

)i(2 e maknakt

mnyxcx ++minus= ω


⎟⎠⎞⎜



⎟⎠⎞⎜






a x

y


⎥⎥⎦

⎤

⎢⎢⎣

⎡+⎟

⎟⎠

⎞⎜⎜⎝


⎟⎠⎞

⎜⎝⎛ +++minus++=

⎟⎠⎞

⎜⎝⎛ minus++=

22

2

2

222222

22222

222

481

2122

21

)(2)(21

)(21

aax

ax

ayxayxaxak

ayxaayxak

ayxakU

to second order

[ ]

2

22222222

21

2222

kx

axaxyxayxaxakU

cong







11 11dagger

1 i22

pp p p

pa P X

ωω

= minus qq q q

q

Now

11 1

1

11dagger

p p

p

p

X m

m

X


minus

= sdot

= sdot

=

sum

sum

X

X

q b q b q bb

b q b q bb

q

e

e






1 1

1

1

1

111

1

1 1

p pp p p

pp p

e X m e e X

m e e X

m X m X

β βα αβ

β βαβ

ββ ααβ δ δ

lowast lowast

lowast

=

=

= =

sum sum

sum sum

sum

q b q b q bbqb q b

b q b q bb q b

bb b bq bb q b

or

1 11

1p ppX e X


b (a)

Now

dagger 11dagger 11

1 i22

pp p p

pa P X

ωω

= + qq q q

q


11dagger 11dagger

11 11

1 i22

1 i22

pp p p

p

pp p

p

a P X

P X

ωω

ωω


= minus

qq q q

q

qq q

q

726 Appendices

so

dagger 11 2i

2p

p p pa a Xω

minusminus = qq q q

11 dagger

i ( )

2p p pp


q

Substituting in (a)

1 dagger

i ( )

2 p p ppp

X e a am ω


b q



0)(

)(])[(

34412

24412122

442

4411

=++

++minus+minus

Kkkcc

KkkccKkckcc

zx

yxx ρω


0)( 122

11 =minus Kkc ρω


0)( 222

44 =minus Kkc ρω

0)( 322

44 =minus Kkc ρω

If

ρ

ων 1111

ck

==


If

ρ

ωνων 4433

22

ckk

====






)1(

)(

])()([)(

))]()(())()([()(

))]()(())()(()(

2

2

12

1112

11112

11112

1

1

NM

PM

dPM

dxuxuPM

dxuxuQxuxuPM

dxuxPuxuxPuMd

P

qp qq

qp qq

pp NNNNpp

pp NNNNpp

=

=

minus=

minus=

minus=

minus=

sum

sum int

sum int

sum int

sum intint

lowast

primelowastprime+


τδ

τ

τ

ττψψ

where Q = Pminus1P




)2()1()21()2()1(21

)2()1()21()2()1(21

)2()1()21()2()1(

)2()1()21()2()1(

)1()1()1(

(||)

(||)

sum

sum

sum

sum

sumsum

minus

+=

minus

+

=

ji ijji

ji jiji

ji ijji

ji jiji

i iik

V

VE

V

V

H

ψψψψ

ψψψψ

ψψψψ

ψψψψ

ψψε



22 2

1 21 10 12

| ( ) |( ) d 0

4jN N

II jeV

rψ


rr

Note that

221| ( ) |N

jj ψ=sum r

728 Appendices


2

22 11

0 12

( )d ( )

4NI

II Ie V

rρ τ


r QED

IonI

Field point

Source point

r1

r2



Let x = kkM

0as2)(21

11ln1limlnlim

2

2

0

22

0

rarr=minus=

minus+minus=

minus+minus

rarrrarr

xxxx

xx

xx

kkkk

kkkk

xM

M

M

Mk

⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢

⎣

⎡


minusminusminus



rarr

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

gtminusminusminus

ltminusminusminus

rarr

⎥⎦⎤

⎢⎣⎡


⎥⎦⎤

⎢⎣⎡



rarr

110)1(

)1(1)1(1

or

110)1(

)1(1)1)](1(1[

2

1)1(1)1ln(

1)1(1)1ln(

2

1)1ln(1)1ln(2ln)1(2

1)1ln(1)1ln()1ln()1)(1(


2

2

22

1

xxx

xx

xxx

xx

xx

x

xxx

xxxxx

xxxxx

xxx

xxxx

xx

xx

x





i1e 0bm


K


The answers are





Thus

i(0) i( )( )e e

1 ( 1)

n l m

n l m

S πminus minus + +

+ +

= +

= + minusK






730 Appendices


i( )( ) i( )( ) i( )(n )1 e e en l m









20

2)(2

2 EVmKmEk

minus==


)ee()(22

ee)(2

)(02

If

ii

KxKx

kxkx

Dxbxb

CBxaxb

Vbax

minus

minus

+=ltltminus

+=ltlt

infinrarr=+gt

ψ

ψ

ψ


)ee()ee(i

)ee(ee

0ee

2i2i2i2i

2i2i2i2i

)2(i)2(i

KbKbkbkb

KbKbkbkb

bakbak

KDCBk

DCB

CB

minusminus

minusminus

+minus+

minus=minus

+=+

=+



⎟⎠⎞

⎜⎝⎛minus=

2coth)tan( Kb

Kkka (a)


)ee()(22

ee)(2

02

ii

KxKx

kxkx

Dxbxb

CBxaxb

bax

minus

minus

+=ltltminus

+=ltlt

=+gt

ψ

ψ

ψ


⎟⎠⎞

⎜⎝⎛minus=

2tanh)tan( Kb

Kkka (b)


)e21(

2tanh

)e21(2

coth

Kb

Kb

Kb

Kb

minus

minus

minusrarr⎟⎠⎞

⎜⎝⎛

+rarr⎟⎠⎞

⎜⎝⎛

and

00 V

EEV

EKk cong

minuscong


⎟⎠⎞



02symmetric-anti





)e21( 2

0


732 Appendices


⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝


22exp21

22

20

02

22

2

2 bmVVma

Ema ππ






dkm

kdkdkdEdE

2==


intint ==ll

dldEk

mdldkdEED 222 21

)2(2)(

ππ

φ

πak

θ

dl

dk

E+dEE = h2k22m

int=kdlmED 222

)(π


For a

k πlt

2222

2)(

ππ

πm

kkmED ==

For a

ka

ππ 2ltlt

)2(42

)( 2222 ϕπ

ϕπ

mkkmED ==

but

⎟⎠⎞



2

So

⎟⎠⎞


kamED ππ

π1

22 cos4)(

or

⎟⎠⎞


kamED π

ππ1

2 cos41)(



(1) (2) (3) (4) (5)

(6)

(c) (b) (d) (a) (f) (e)



0 00



k

734 Appendices


( )

0( ) 2e

(e 1)

F

F

E E kTF

E E kTf E EdT dT kT

minus


k

k

k

( )

0( ) 2e 1

(e 1)

F

F

E E kT

E E kTfE kT

minus

minuspart

= minuspart +

k

kk

so

0 02

0

F

F


f EEdTE dT kT T


k

k

k

k

Thus

00


τ⎛ ⎞ part⎡ ⎤⎛ ⎞= + minus + +⎜ ⎟⎢ ⎥⎜ ⎟ part⎝ ⎠⎣ ⎦⎝ ⎠

kk k

k


33

2(2 )

zzJ e u f d k

π

⎛ ⎞= ⎜ ⎟⎜ ⎟

⎝ ⎠int k k


33

2(2 )

zzH u f E d k

π

⎛ ⎞= ⎜ ⎟⎜ ⎟

⎝ ⎠int k k k

Thus

2 303 ( )

4z F


τπ


kk

and

2 303

1 ( )4

z Fz


τπ


k k kk



23

22

3

23

( ) [ ( )]( )4

( ) ( )12

412

zz F

zF

F

dsdEeJ u E E eEu

dsdEe E u E Eu

e eEu k

τ δπ

τ δπ

τ ππ

= minus minus minus

= minus

=

int

int

kk k

k

kk k

k

k




m

neEJ z τσ

2==


230 ( )

12F


dz dT kT T uτ δ

π⎛ ⎞⎡ ⎤⎛ ⎞= = minus minus +⎜ ⎟⎢ ⎥⎜ ⎟

⎝ ⎠⎣ ⎦⎝ ⎠int k k

k kk

or

2 2( ) ( ) FF F


τ δ τ δ ⎡ ⎤⎛ ⎞minus = minus +⎢ ⎥⎜ ⎟⎝ ⎠⎣ ⎦

int intk k kk k k k

k k

or

( )2

2 2 2 414F F

F F F FFE E E E


πτ π τ= =

⎞⎛ ⎞ ⎛ ⎞= minus + ⎟⎜ ⎟ ⎜ ⎟ ⎟⎝ ⎠⎝ ⎠ ⎠k k

k kk k

So

dT

dEdzdTeE F=


2

31 [ ( )]

12F F

F

dT Kdz


τ δπ

=


k k k kk

so

23

1 412 F

F

F Fk kE E

u dE dEK E E k

dT dTτ π

π ==


kk k

736 Appendices


dT

dEEm

nK FF

F 2τminus=


⎥⎥

⎦

⎤

⎢⎢

⎣

⎡

⎟⎟⎠

⎞⎜⎜⎝

⎛minus=

2

)0(121)0(

FFF E

kTEE π

so

)0(6

22

F

FE

TkdT

dE πminus=

so

)0(6

222

FF

FE

TkEm

nK πτ=


Tkm

nK F 22

3τπ=

and

22

2

22

33 ⎟

⎠⎞

⎜⎝⎛==

ek

Tm

ne

Tkm

n

TK

F

Fπ

τ

τπ

σ




bXX

dx 12 minus=int





1

)]1([11

)]1([

21

1

02121

10 2

21

KK

xKxKKKxKxKdx

=

⎥⎦

⎤⎢⎣

⎡minus+minus

minus=minus+

int


2

2

22 2

22 2

( )

( ( ))

(2 )2

22

2

L

L

dIq f

dq f

d

q qm

dmq q

m

ω

ω

=

= minusminus

= minus⎡ ⎤



int

int

int

int

qq

qq

q

k q

q

k q


1

2 20 2 2

2

2(1 ) 2 L

m dxI dmq x q xω


int intq

k q


1

2 20 2 2 2

2

2( ) L

m dI dxmx x x kω


int intq

q k


1

2 20 2 2 2

2

2L

m dI dxmq x x kω

= minus⎡ ⎤⎛ ⎞+ minus⎜ ⎟⎢ ⎥⎝ ⎠⎣ ⎦

int intq

738 Appendices


2

2 2 2 2 2 20

4[ ] [ ]

d q dqq q

πα α

infin=

+ +int int

q


intminus

minus=1

0 22

22 2

2

kxxmdxmI

Lωπ

But

2

sin2

2

11

0 22

pppxx

dx minus=minus

int

so we find

⎟⎟⎠

⎞⎜⎜⎝

⎛minus= minus

2sin22 12

2k

mk

mm

mIL

L

L ωω

ωπ

Thus

Lm

Kω

=3


⎟⎟⎠

⎞⎜⎜⎝

⎛+minuscong

2122 2

22

kmm

mILL ω

πω



2 2

02

16

c L

c

kEm

mm

α ω

α

lowastlowast

lowastlowastlowast

= minus +

=minus

k






1 1d edt c

δτ

⎛ ⎞ ⎛ ⎞+ = minus +⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

k E v Btimes


mceB

c =ω

v

vv

vv

zz

xcyy

ycxx

Eme

Eme

Eme

τ

τωτ

τωτ

minus=

+minus=

minusminus=


⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛ minus=

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

z

yx

cc

z

yx

EEE

DDjjj

000101

0 τωτωσ




Dyyxx

0σσσ ==

D

cxyyx

τωσσσ 0=minus=

For the Hall effect

yxyy

xyy

EEj σ

σminus=rArr= 0

740 Appendices

So

xyy

xyxxx Ej ⎟

⎟

⎠

⎞

⎜⎜

⎝

⎛+=σσ

σ2


xx Ej 0σ=


m

ne τσ2

0 =


BE

E

BjE

Rx

yxyy

x

x

yH )( 0σ

σσ

minus==


nec

RH1minus=


B

nec

cc

cxyyx ==congminus=

τωσ

τωτωσσσ 02

0

)(


B

Ryx

H )(1

σminus=


v

v)()(

)(vvv

32

54

32

540

0

⎥⎦

⎤⎢⎣

⎡partpart

partpartminusminus=

⎥⎥⎦

⎤

⎢⎢⎣

⎡sdot

partpartminus=

partpartminus=

F

F

FF

e

Fe

xxxfxfcm

xfcmEp

π

π



( ) ⎟⎟⎠

⎞⎜⎜⎝


40

22 11411

FF

xF

xxdxxxxf F

where

312

v3

⎟⎟⎠

⎞⎜⎜⎝

⎛=equiv π

cmcmpx

ee

FF


FFF

F xxxxf

21)( 3 +=

partpart

34312

v31)3(

vπ

cmx

e

F minus=partpart

Fe

F xcm

x31

v3

31

vv

312minus=⎟

⎟⎠

⎞⎜⎜⎝

⎛minus=

partpart π


[ ] 12

61

3111

41

2432

54

242

432

54

0

FFe

FFF

Fe

xxcm

xxx

xcm

p

minuscong

⎥⎥⎦

⎤

⎢⎢⎣

⎡++⎟

⎟⎠

⎞⎜⎜⎝


π

π


pm

MM δ=

and

⎟⎟⎠

⎞⎜⎜⎝

⎛=

cem

RR

742 Appendices


Mm

R pξπ sdotcong 3

34v

31

31

3

2

491

34

3

⎟⎟⎠

⎞⎜⎜⎝

⎛=

⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜

⎝

⎛

sdot=

ξδπ

ξπ

π

MR

Mm

RRRx

pF


32

54

12~and

89

ππδ cm

K e==

⎥⎥⎦

⎤

⎢⎢⎣

⎡minuscong 2

32

4

34

0~

RM

RMKp




EF(2)

EF(1)

E

Δ = eV

R1 2



We note if

1)2(1exp)(

minus

⎥⎥⎦

⎤

⎢⎢⎣

⎡+

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛ minusminus=

kTEE

Ef F (a)


then

1)1(1exp)(

minus

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡+⎟

⎟

⎠

⎞

⎜⎜

⎝

⎛ minusminus=Δ+

kTEE

Ef F


then

[ ] 3

321

)2())(1)((v2



[ ] 3

312

)2())(1)((v2




[ ] 3

3

)2()()(v2


Since

x

x kE

partpart= 1v

We write

dEdkxx1v =


[ ] ||2

3 )()()2(



mk

EW2

2||

2minus= (b)


)(D1 WTT =

744 Appendices


⎥⎥⎦

⎤

⎢⎢⎣


1



[ ] )( )()()2(

2||


π (d)






ce EE

mED minus⎟

⎟⎠

⎞⎜⎜⎝

⎛=

lowast 23

222

21)(π

1])exp[(

1)(+minus

=kTE

Efμ


⎟⎠⎞

⎜⎝⎛ minus=⎟

⎠⎞


kTE

kTEEf μμ expexp)(

ce

EmkE += lowast2

22

intinfin

lowast⎟⎠⎞


⎟⎠

⎞⎜⎜⎝

⎛=

cE ce dE

kTEEE

mn μ

πexp

22

123

22



ce

EmkE += lowast2

22

kdkm

dEe

22

2

lowast=

lowast

=minuse

cm

kEE2

intinfin

lowast

lowast

lowast

lowast

⎟⎟⎠

⎞⎜⎜⎝

⎛ +minus⎟

⎠⎞

⎜⎝⎛

⎟⎟⎠

⎞⎜⎜⎝

⎛=

cEe

ce

e

e kdkmkT

EmkkTm

kmn 2

2)2(

expexp2

22

1 22223

22μ

π

intinfin

lowastlowastlowast

lowast

⎟⎟⎠

⎞⎜⎜⎝

⎛minus⎟

⎠

⎞⎜⎝

⎛ minus⎟⎟⎠

⎞⎜⎜⎝

⎛=

cEee

c

e

e dkkkTm

kmkT

E

m

mn 2

22223

22 2expexp

2

22

1 μπ

intinfin

lowastlowast

lowast

⎟⎟⎠

⎞⎜⎜⎝

⎛minus⎟

⎠

⎞⎜⎝

⎛ minus⎟⎟⎠

⎞⎜⎜⎝

⎛=

cEe

c

e

e dkkkTm

kkT

Em

mn 2

22

23

323

22 2expexp

)2(22

21 μπ


pp

ndxpxx nn π

)2(2)12()exp(0


where

kTm

pne

2

2 1 ==

So

( )kTmkTm

kTE

m

mn

ee

c

e

e

21

223

323

22

2222

)1(exp2

222

1 πμπ

⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎠

⎞⎜⎝

⎛ minus⎟⎟⎠

⎞⎜⎜⎝

⎛=

lowast

lowast

⎟⎠

⎞⎜⎝

⎛ minus⎟⎟⎠

⎞⎜⎜⎝

⎛=

lowast

kTEkTm

n ce μπ

exp2

223

2



746 Appendices



1 ( ) 1 ( )kdE E

d⎛ ⎞= equiv⎜ ⎟⎝ ⎠

kv kk

nabla


1 ( )

( )

k

k

dE Edt

d Edt

= sdot = sdot

= sdot

F v F k

k k

nabla

nabla

thus

( ) ( )k kdE Edt



ddt

= kF


1 ( ) ( )k kdE Edt


but also



( ) ( )k kdE Edt



( ) dqdt











ln T

ln μ

low T high T



phonon scattering

748 Appendices


if

if

212

121

21

21total

μμμμμμ

μμμμμ

gtgtconggtgtcong

+=




rr

rV λminusprop e1)(



2 1 2 1( )H V r=k k k k



2 1



int minuspropπ

θθθθστ 0

sin)cos1)((v1 dm



Finally we find

23Tm

m prop= lowastτ

μ








cEδ δprop sdotk r



23minus=prop Tμ


See Fig 618


Now


⎟⎠

⎞⎜⎜⎝

⎛= minus )(

23

2 e2

2 μ

π


⎟⎠

⎞⎜⎜⎝

⎛= minus )(

23

2 e2

2 μ

π

750 Appendices


⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎟⎟⎠

⎞⎜⎜⎝

⎛=minus

23222

lnkTm

NkTE

e

dCn

πμ

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎟⎟⎠

⎞⎜⎜⎝

⎛=minus

23222

lnkTm

NkTE

h

aVp

πμ

Therefore

⎥⎥⎦

⎤

⎢⎢⎣


32

)()()2(

22ln)(

kTmmNN

kTEeEEhe

dagpnCnVp

πϕϕ

Using

kTEg

gkTE

e

1ln minus=

kTEhei

gmmkTn 4323

2 e)(2

2 minus⎟⎟⎠

⎞⎜⎜⎝

⎛=

π

we find

⎟⎟⎠

⎞⎜⎜⎝

⎛=minus 2ln)(

i

dapn

nNN

kTe ϕϕ QED



2 21

12Surface

( )( )

4d

rπΦ = int

M r Sr

sdot

= minus ΦH nabla

12 2 21 2

12

ˆ ( )( )

4d

rπ= int

r M r SH r

sdot

θr

dS2

r 12

z

M

r1 = 0r12 = ndashr2


Using the figure

intminus=π

πθθπθθ

0 24 sin2 cos cos

rrdrMH z

31and

3=minus= DMH z




)eTr( HZ βminus=


1 1 1( ) ln

Tr( e ) Tr( e )

zz z z

z z

ZF ZB B Z B

Z Z

β β

μ ββ β β

βμ μβ



as expected



)Tr(2

)1Tr()eTr( 22

HZ H ββ +== minus


)Tr(

3)Tr(

3

)Tr(31)Tr()Tr(

2222

222

222222

JBμgJBBμg

JδBBμgJJBBμgH

Bμ μμ

B


==

==

sum

sumsum

)12()1Tr( += j


⎥⎥⎦

⎤

⎢⎢⎣

⎡+++cong )1(

3211)12( 2

222 jjBgjZ Bμβ

752 Appendices



⎥⎦⎤

⎢⎣⎡ +

partpart= )1(

611 2222 jjBg

B Bz

z μββ

μ

BB

BB z

z=

partpart


kT

jjgNNB

Bz

zzz

)1(3






kkS sdot


kkS sdot



i i( ) e ej i j

i jid Jdt


k k kk k kS S S



i jj i jd Jdt


k k kk k kS S S


i( ) i( )i( ) e ( )e ej ij i

d Jdt


k k kk R k kS R S S


kRsdot


so

( )

( )

( )

kd Jdt

J

J




primeprime= minus

prime=primeprime=

sum

sumsum

k kk k kk k

k k kk

k k kk

S k k S S

k S S

k S S

times

times

times

which is (7155) QED




ϕϕ ==


2

2

(cos cos sin sin )

cos( ( ))

x x y yR R R R R R

R R R R

S S S S

S

S q R R

ϕ ϕ ϕ ϕ

prime prime prime

prime prime

= +

= +

prime= minus

S Ssdot

so that


)cos()(2

))(cos()(2

i2

2

2

=minus=

minus=


sumsum

sum

sum prime

RRqR

R

RR

qRRJRJNS

qRRJNS

RRqRRJSH


qR minus== sum Then

)(2

2qJNSH minus=


)2cos(21)cos(2

21 )2cos(2)cos(2

)ee()ee()(

1

1221

i2i22

ii1

⎥⎦⎤

⎢⎣⎡ minus=

minus=+=

+++= minusminus

qaqaJ

JJqaJqaJ

JJqJ qaqaqaqa

754 Appendices

For a maximum

[ ]0

)cos()sin(2)sin(2

)2sin()2(21)sin(2)(

1

1

=minusminus=

⎥⎦⎤


partpart

qaqaqaaJ

qaaqaaJqqJ


angle turn 60

21)cos(

1)cos(2

deg=

=

=

qa

qa

qa


[ ]

max03

212

212

)2cos(2)cos(2)(

21

21

602

12

2

rArrltminus=

⎥⎦⎤

⎢⎣⎡ +minus=

minusminus=part

partdeg=

aJ

aJ

qaqaaJq

qJqa



=minussj mSS

12

112

212

212

21

21

21

+⎟⎠⎞

⎜⎝⎛ minus+=

⎟⎠⎞

⎜⎝⎛ minus=

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛minus=

+

++minus

nSnnS

nSnaS

nSaa

aSnS

j

jjjj


so



or

1)1)((

12

2)1(221

minus+minus+=

minus⎟⎠

⎞⎜⎝


sss

ss

ssj

mSmSmS

mSS

mSSmSSmSS

QED


We have

i1 e jjjb a

N= sum k R

ksdot

and


Nminus= sum k R

ksdot

where

0][

0][

][daggerdagger

dagger

=

=

=

ji

ji

ljjj

aa

aa

aa δ

Thus

i( )dagger dagger

i( )

1[ ] e [ ]

1 e

1

j j

i j

j

k jj jk j

j

k k k k

b b a aN

N

NN

δ

δ δ

prime

prime


primeminus minus

prime prime

=

=

= =

sum

sum

k R k R

k k R

sdot sdot

sdot

Similarly

i( )

0


Nprimeprime+



0


Nprimeprimeminus +


756 Appendices






Therefore

712

)state F(3max

21221

=+==

=+=

sumL

Lm

SS

l

less than half full

25


So stateF 25

2

Pm3+


426

)state I(123max5122


sumSLJ

mLSS

l

So stateI4

5

Eu3+ six electrons


033


7123216


=+=times=

sumSLJ

mL

SS

l

So stateF0

7




05 3

3[( ) ] ( )( )4μπ



B

r

rprime

m


4 have ( ) 0

28 have ( ) 2 22

a mam

primeminus sdot =

primeminus sdot = =

r r m

r r m


[ ]2 2

05 3

3 ( )4

Uμπ


r r m mm Br r r r

⎟⎟⎠

⎞⎜⎜⎝

⎛minus= minus

5

252

3

2327

total2)2)(3)(8(2)12(10

ama

amU

( ) 022424)10( 3

27 =minus= minus

amU


758 Appendices


2 2 2

4 have ( ) 0

2 have [( ) ] a m

primeminus sdot =

primeminus sdot =

r r m

r r m


066)10( 3

2

3

27

total =⎟⎟⎠

⎞⎜⎜⎝

⎛minus= minus

am

amU


3

27


)JT10279)(8(8

m10524

10

minus

minus

times=cong

timescong

Bm

a

μ


)(10381

101010

1012510)10(

23

2328

51

30

447

total

congtimescong

=congtimes

cong

minus

minusminus

minus

minus

minusminus

c

cT

T

U





The free energy is

ZkTF lnminus=


BFMpartpartminus=




)(0 qpHH rarr




qpqAp

eeAppqq

i

i

iiii

ii

Π=

gtΠequiv+rarrrarr

0HH

Now

)()(e

e

11)(

11)(


=

int

intΠminus

minus

qpqdddqdq

dpdpdqdqZ

NNq

NNqp

helliphellip

helliphellip

H

H

β

β


0 =Πpartpart

=Πpartpart

=partpart

j

iij

j

iij

j

i qpqq δδ



⎟⎟⎠

⎞⎜⎜⎝

⎛=

Πpartpart

+Π= etc1i

iiii

peAp

760 Appendices

Thus schematically

1

0

100010001

)()( =

Πpartpart

=

Πpartpartpartpart

=

Πpartpart

Πpartpart

partpart

partpart

=Πpart

part

i

i

j

i

i

i

j

i

j

i

j

i

j

ip

pqp

pqqp

qq

qqp 15



0=partpartminus=

BFM






15

1

100

01

100

01

0

=sdot=⎟⎟⎠

⎞⎜⎜⎝

⎛

partpart

⎟⎟⎠

⎞⎜⎜⎝

⎛

Πpartpart

=

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛partpart

partpart

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

Πpartpart=

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛partpart

partpart

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

Πpartpart=

Πpartpartpartpart

partpart

11j

i

j

i

i

i

j

i

j

ii

i

j

i

j

i

j

i

i

i

j

i

qqp

qp

qq

pqp

qq

ppqp

qq





Subtracting [ ]int



intΔminus

ΔΔminusminus

minus

Δminus= 2

1 22

221

221)(

)0()0(eV dEEeV

EeV

E

ENNCI

Checks 11








0

000


⎠

⎞⎜⎜⎝

⎛minusprime==Δ


0

0

0

3

0

0

2

02

3

31)11(

2

3cos

2

)(coscos2

ε

ε

θε

θθε

π

π

P

P

P

dPEE b

=

minusminusminus=

minus=

minus=equiv int

762 Appendices



θ

p

E z


int

int= π

θ

πθ

θθπ

θθπθ

0

cos

0

cos

sin2e

sin2ecos

d

dpp

kTpE

kTpE

z


int

int

minus

minus= c

c

x

c

c

x

zdx

dxx

cpp

e

e


⎟⎠⎞

⎜⎝⎛ minus=

ccppz

1)coth(


kTp

Ep

kTEpp z

z 3

3

22=== α



The Hamiltonian is

eEzr

qm


0

22

2

2H

H



)1(100 pzN += ψψ


BarBa 23


ψ


2

2

mqaB =

Requiring

12 =int τψ d





Now using

4

3 0100 0 100 2

0

12 (4 )

med Eψ ψπε

= = minusint rH





are also zero so

)(2 202 eEpaE

NBminus=

H



764 Appendices

Requiring

0=part

partpH

we find

0EeEp minus=


230



302

216 Ba

Eπεα =

part

partminus=

H



061

41)(

21 42


For E = 0

0)( 420 =++minus=


PGE δγβ (2)


20


[ ] 0)(61

41)(

21 2

02


0121)(

31 2


20 4 scc PTT

βγminus= (3)

(b) From (2) 0)( 42


20 4




04


δγ

432 minus=scP

βδγ 2

0 163+= TTc











766 Appendices





φ1 (say 45deg)

Polarizer

lightSample

Analyzerz

zφ2

E

z

yx






iminus= xx xpxp



If H = px2 m +V(x)

[ ] 0)( =xxV So

[ ] xpm

x i minus=H


[ ] iˆ ˆm




2 2ˆ ˆ

2ij ijj ji j i j

f b ωsdot + sdot

=sum sume r e r




jxi

i=minus xpxp xx

xxxppx xx i2 =minus

iminus= xx xpxp


768 Appendices


jj jω=H

jpm


jpm


jpm



1ˆ ˆi ij

i j i jmω

sdot = sdote r e p


ij ji


ωω ω



jiij ωω minus=



Summing over j

ˆ ˆ2iijjbf i im



m

bm

bfj ij 2)i(

i2minus=minus=sum




eEB

NeG +minus

=+ v13

1vvel

2

0εμ



0evv minus= Then

eEB

NeG =minus

minus+minus v13

1vvel

2

0

2ε

μω


ion0

ion 31 αε

NB =

and

Ge

Ee 2

locion

v ==α

at ω = 0

GeNB

2

0ion 3

1)0(ε

=


eEB

GBG =

minusminus+minus v

1)0(

vvel

ion2μω

eEB

BG =⎟⎟⎠

⎞⎜⎜⎝

⎛minus

minus+minus v1

)0(1v

el

ion2μ

μμω

Define

⎟⎟⎠

⎞⎜⎜⎝

⎛minus

minus=el

ion21

)0(1

BBG

T μω



For the local field

el00

loc 1v

31

3 BNeEPEEminus

+=+=εε

Substituting

)(

v 22 ωωμ minus=

T

eE

770 Appendices

Then

)(13

122el0


+=T

eEB

NeEE

)(13

11 22el0

loc


T

eB

NeE

E

)(

11

)0(1 22el

ionloc

ωωμ minusminus+=

TBGB

EE

)()1(

)0()()1(22

el

ion22

loc

ωωμωωμ


=T

Tel

BGBB

EE

Inverting

)0()()1(

)()1(

ion22

22el

loc GBBB

EE

Tel

T


=ωωμ

ωωμ

))(1()0(1

1

22el

ion

ωωμ minusminus+

=

T

locB

BGEE

Define

))(1(

)0(22

el

ion

ωωμ minusminus=

TBBGF

Then

FE

E+

=1

1

loc


loc0

ion0

ionv

31

31)(

EeNNB

εα

εω ==

Substitute in v

)(3

1)( 22loc0ion

ωωμεω

minus=

T

eEE

eNB

)(

)0()( 22

ion

locion

ωωμω

minus=

T

GBE

EB

)1(1

1)( elion BFF

B minus+

=ω


Given

[ ] )1()(1)(3)()(

elionel

ion

0 BBBB


ωω

εεωε


)1()1(

11

)1(1

3)()(

elelel

el

00 BBF

FB

BF

F

minus⎥⎦⎤

⎢⎣⎡ minus

+minusminus

minus++infin=

εε

εωε

)1(

11

13)()(

el00 B

FF

FF

minus⎟⎠⎞

⎜⎝⎛

+minus

++infin=εε

εωε

)1(

3)()(

el00 BF

minus+infin=

εε

εωε

Substitute for F

))(1(

)0()1(

3)()(22

el

ion

el00 ωωμεε

εωε


TBBG

B

Define c

2el

ion

)1()0(3

BBGcminus

=μ

Then

)(

)()(2200 ωωε

εεωε

minus+infin=

T

c

Rewriting

202)()(

ωεωε

εω =infinminus

minusc

T


2

2

)0()(

L

T

ωω

εε =infin

)()0( 2

2infin= ε

ωωε

T

L

772 Appendices


0)()0(

02 =infinminus

minusεε

εω cT

0)()(2

202 =


εεωω

εω

T

LT

c

0)()( 22

202 =

infinminusminus

εωωωεωTL

TT

c

0)(

)(2

0222 =infin


TLTc

202)( LT

c ωεεω =infin

+



ψψ Em

=nablaminus 22

2

For each coordinate

0222

2=+minus i

i

i

mEdxd ψψ


⎟⎟⎠

⎞⎜⎜⎝

⎛+⎟

⎟⎠

⎞⎜⎜⎝

⎛= i

ii

ii xmEBxmEA 22

2cos2sinψ


2

222

2 2or2

manEnamE i

iii ππ ==

)(2

23

22

212

22nnn

maET ++= π





2

22

2

22

23

2)111114(



2

22

23

mahf π=


constant432 ==

mfa π



0

22

2

2

2

2

22

4where1

2 πεφeq

rq

rrrm=minus⎟

⎟⎠

⎞⎜⎜⎝

⎛

partpart+

partpart+

partpartminus=H




εεε RRYEmqEmq

r ==== )(2

2

4

2

2ρ


ψψρ

ψφρρρρ

ε2

4

2

22

2

2

22

2

4

422

2)(11

22mqmqqqm

m=minus⎟

⎟⎠

⎞⎜⎜⎝

⎛

partpart+

partpart+

partpartminus

774 Appendices

Canceling we obtain

ψρψψ

φρρρρε

2111

21

2

2

22

2=minus⎟

⎟⎠

⎞⎜⎜⎝

⎛

partpart+

partpart+

partpartminus




φφ mie)( =Φ



⎞⎜⎜⎝

⎛minus+minus

21

21 2

2

2 ερρρρρ

mdd

dd

We let

2

and1 2

2xαρ

α=minus=ε


Ψ=Ψ+⎟⎟⎠

⎞⎜⎜⎝


241

21 22

2

2 xxm

dxd

xdxd


21122 +=+= nn αα


2212

21 )(

or)(

1+

minus=+

minus=n

REn

ε


21)1(

21 minus+=+ nn





)()(2 2

22zEz

dzd

mψψ =minus


0)(2)(22

2=+ zmE

dzzd ψψ

⎟⎟⎠

⎞⎜⎜⎝

⎛+⎟

⎟⎠

⎞⎜⎜⎝

⎛= zmEBzmEAz 22

2cos2sin)(ψ



2

222

2mwpE p

π=





12

2

0minusminus=

xeF

πε


x

ex

edxx

eFdxdVxxxx

0

2

0

2

20

2

161




x

eExV0

2

0 16)(

πεminus=

776 Appendices


x

exeEExV0

2

10 16)(

πεminusminus=


01

=⎟⎠⎞

⎜⎝⎛

=xxdxdV

which gives

10

1 16 Eex

πε=

or

0

13

01 21)(

πεEeExV minus=

Thus

0

13

21

πεEe=Δ



a

V(r)

ndashV(0)

E (bound state)

r




20 )(2 EVm +

=α

22mEminus=β



This implies

2


Also

E

EV

aa

minus=

+=

+= 0

22

2

22

)(cot11)(sin

βαα

αα


2πα gta

to estimate a

eV)1(22)(22

22

0 mEVma ππ

απ gt

+=cong

nm931)1061)(1019(2

)10636)(143(1931

34

min congtimestimes

timescongminusminus

minusa




778 Appendices

Energy Gap

Ec(z)

Eg1 Eg2

Valence Band

Conduction Band

d

z

Ev(z)


EFFzEzmyxm

c =⎥⎥⎦

⎤

⎢⎢⎣

⎡+

partpartminus⎟

⎟⎠

⎞⎜⎜⎝

⎛

partpart+


22 2

22

2

2

2

22




32||

2

322

2)(

2)(

2Ek

mEkk



ψψ 32

22)(

2EzE

dzd

mc =⎟

⎟⎠

⎞⎜⎜⎝

⎛+minus lowast



sum=

minus=N

ni

ndiq ndz

Nz

1

i )(e1)( χψ


0 we find for E3




2

2 2 0 0( ) cos












)(22

||

222

2

222

EE

kkmmw

nE

n

yx

+equiv

++= π

)(22

222

2||

2

|| yx kkm

km

E +==




ndEm

dkdkkkd

2

2||||||

2

2

2

π

ππ

=

==

780 Appendices


)(

)(2)2(

2)(

2

0||||22

sum

sum int

minus=

minusminus=infin

n n

n n

EEm

dEEEEmED

θπ

δππ



23)(π

mED = (1)

In 3D

E

dEmdkmkE 2

22 221

2==

2

212)4(

)2(2

)unit volper (4)2(

2)(

223

23D3

EdEmmE

dkkdEED

ππ

ππ

=

=


32

22

1)(223

22D3

E

wmmED π

π ⎟⎟⎠

⎞⎜⎜⎝

⎛=

where

2

222

32

3mw

EE π==

Per unit area

222D3332

21)(

ππ

πmw

wmwED == (2)

which checks (1)


d)

D3D(E)w

2hπm

23hπm

E1 E2 E3




1001

512

512

51

2

51

2~

21

21

22

11

I

NN

NN

NN

NN

AA

=⎟⎠⎞

⎜⎝⎛=

⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜

⎝

⎛minus+

⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜

⎝

⎛

minus

+=


Bibliography

Chapter 1

















784 Bibliography










Chapter 2









Chapter 2 785





















786 Bibliography

Chapter 3





















Chapter 3 787



















788 Bibliography

Chapter 4





















Chapter 5 789














Chapter 5







USA 92 6663-6667 (1995)

790 Bibliography






















Chapter 6 791



Francis London 1963

Chapter 6
















1994

792 Bibliography






















1996

Chapter 7 793

Chapter 7



















794 Bibliography



















Chapter 8 795










(2004)

Chapter 8






796 Bibliography






















Chapter 9 797





Chapter 9

















798 Bibliography




Chapter 10















Chapter 11 799






Chapter 11















800 Bibliography


















Chapter 12



Chapter 12 801























802 Bibliography





















Chapter 12 803

















804 Bibliography

Appendices





















Subject References


Mechanics




Electricity



Optics



Thermodynamics






Critical Phenomena


806 Bibliography

Crystal Growth


Modern Physics



Quantum Mechanics





Math Physics


Solid State







Condensed Matter






Problems




Applied Physics





Rocks




808 Bibliography


The Internet



Periodic Table


Further Reading












Further Reading 809
















810 Bibliography

















Further Reading 811


















812 Bibliography




Index

A













B


814 Index










C




Index 815





















816 Index




316 317

D









594 640






E





Index 817

















503 566










F



818 Index


493 538 Fermi function 160 248 298 303



373 385 399 407 422 426 456 486 513







G

GaAs 242 295 297 324 326 328 329 337 347 563 564 579 618 628 629 637




Index 819

Group velocity 55 306 532 562 627


H











I






820 Index


J


K



L









529 570

Index 821




M













N


531 600 NaCl 6 9 28 579 600

822 Index









O








P





Index 823















824 Index


Q




R












S



Index 825













826 Index






T



361 418











475 509

Index 827

U




V



Vortex region 466

W


X


Y

YBa2Cu3O7 506 508

Z


Appendices

A Units










G1 Bose Particles

G2 Fermi Particles


H1 Propagators

H2 Green Functions

H3 Feynman Diagrams

H4 Definitions




J Folk Theorems



M Problem Solutions














































































Bibliography

Mechanics

Electricity

Optics

Thermodynamics


Critical Phenomena

Crystal Growth

Modern Physics

Quantum Mechanics

Math Physics

Solid State

Condensed Matter


Problems


Applied Physics

Rocks


The Internet

Periodic Table

Index


654 Appendices
















ΩAVA = (B6)


IAAAA == ~~ (B7)


Ω=VAA~ (B8)



aAE ~= (B9a)

or

AEa = (B9b)


)~()~(

)~()~(

~~~~

2

sumsum

sumsumsum

==

==

===

kj jkiijj jjj

kj kjkkjjk kjkj

jiij

EEEE

EEEEEEVAEAEVaaaa

δωΩ

δΩΩΩυ



032

21det 2

2=⎟

⎟⎠

⎞⎜⎜⎝

⎛

minusminus

ωω (B10)


5221 +=ω (B11a)


656 Appendices


51

2 21

21 +

== aaωω (B12a)

and for

51

2 21

22 minus

== aaωω (B12b)


⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜

⎝

⎛

minus

+=2

2

11

51

2

51

2~

NN

NN

A (B13)

where

21

21

1 1)15(

4)(⎥⎥⎦

⎤

⎢⎢⎣

⎡+

+=minusN (B14a)

and

21

21

2 1)15(

4)(⎥⎥⎦

⎤

⎢⎢⎣

⎡+

minus=minusN (B14b)


⎟⎟⎠

⎞⎜⎜⎝

⎛

⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜

⎝

⎛

minus

+=⎟⎟⎠

⎞⎜⎜⎝

⎛=

2

1

22

11

2

1

51

2

51

2

aa

NN

NN

EE

E (B15)

Problems



⎟⎟⎠

⎞⎜⎜⎝

⎛

minus+=⎟⎟

⎠

⎞⎜⎜⎝

⎛

520052

3221~ AA







0)()(d

)(d2

2=+ xxU

xx ψψ (C1)


)()( xUnaxU =+ (C2)








0d

d =x

W (C6)

where

21


=W (C7)


658 Appendices



0)()(d

)(d2

2=++++ axaxU

xax ψψ


0)()(d

)(d2

2=+++ axxU

xax ψψ (C8)


)()()( 211 xBxAax ψψψ +=+ (C9)

)()()( 212 xDxCax ψψψ +=+ (C10)


21

21

21

21

2121

2121ψψψψ

ψψψψ

ψψψψψψψψ

primeprime=

primeprime=


DBCA

DCBADCBA

or

1=DBCA

or

1=minus BCAD (C11)


)()( xΔax ψψ =+ (C12)


)()()( 21 xxx βψαψψ += (C13a)

or

)()()( 21 axaxax +++=+ βψαψψ (C13b)


)()()()()()()(

2121

xxxDBxCAax


+=+++=+ (C14)



αΔβα =+ CA

and

βΔβα =+ DB


0=minus

minusΔ

ΔDB

CA (C15)






DA +=+ minus++

1)(ΔΔ


1)(ΔΔ (C17)

and

DA +=+ minus+ ΔΔ



If we set

be=+Δ (C19)

and



)()( 11


and

)()( 12


660 Appendices





where

)()( xuaxu =+ (C25)



Let

)()( 11 NN xxExx ψψ =H (C26)



0][ =lTH (C27)

and






lkmiddotl

ie=t (C31)

Further




so that

12 =lt (C32)










)(02)(0

22

21 Gnnn G =+++


5 In general sum= j

kj

klij

kil AT ψψ

where the Ail




kklkl AT ψψ =


662 Appendices




ntctx nin

i sum= )()(ψ


nmNi

im

imnm cctctc

Nlowast





)( ATrA ρequiv


][i Ht

ρρ =partpartminus


⎟⎠⎞

⎜⎝⎛ minus=

kTHFexpρ


)][exp()]exp([exp

kTHTrkTHHTr

kTHFHTrHU

minusminus=⎥

⎦

⎤⎢⎣

⎡⎟⎠⎞

⎜⎝⎛ minus==








Let

)()( 0 tVt += HH (E1)

lEl l00 =H (E2)


00lk

lkEE minus=ω (E4)




664 Appendices


200

2 2)()(

)(

)cos(1lim ωδπ

ω

ω tfEEttfi

if

if

t=minus=

minusinfinrarr

(E6)












⎟⎟⎠

⎞⎜⎜⎝

⎛=

00σ

σα (F2)




⎟⎟⎠

⎞⎜⎜⎝

⎛minus

=I

I0

0β (F3)



12

)(2

0

20 ltlt

minusminusequiv

cmVcmEε (F4)


20cmEE +prime= (F5)

and

⎟⎟⎠

⎞⎜⎜⎝

⎛=

φχ

ψ (F6)




⎤

⎢⎢⎣


4482 220

2

220

2

230

4

0

2pσ V

cmV

cmV

cmp

mpE (F7)


])[(4 22

0


cm (F8)


666 Appendices


SL middot dd1

2 220

2

rV

rcm (F9)





G1 Bose Particles




][

0][

0][

dagger

daggerdagger

ijji

ji

ijjiji

bb

bb

bbbbbb

δ=

=

=minusequiv

9 See Mattuck [A14]



daggeri i in b b=

and

dagger1 iii bbn =+


11

1

11dagger

11



++=

minus=

iiii

iiii

nnnnnb

nnnnnb



1

1dagger

iiii

iiii

nnbn

nnbn

=minus

=minus



dagger)1()1(






)()(


sumlowast=

=

α αα

α αα

ψψ

bu

bu

rr

rr


668 Appendices

G2 Fermi Particles




0

0

dagger

daggerdagger

ijji

ji

ijjiji

aa

aa

aaaaaa

δ=

=

=+equiv







iii nan




0











)2()1()2()1(

210daggerdagger

ββ

ααβα uu

uuaa harr











670 Appendices






H1 Propagators




H2 Green Functions


δεω

ωi

1)(0 +minus=+

kG k (H1)


H3 Feynman Diagrams


H4 Definitions






672 Appendices




















dagger )2()1()21()2()1(21)1()1()1(H (H3)



31



0

daggerdagger

=+=+

ijji

ijijjiaaaaaaaa δ



δεω

ωi


=sum

+

l lklkk VG k (H6)





674 Appendices



112223 d)21()()()()(d rrrrrr φφφφ (H8)




δεω

ωi)(


=sum

+













klk

Gδωεω

ωi)(


=sum k

k (H11)





676 Appendices




fcc (110) (011) (101)

2 2 2

bcc (11 1) ( 111) (1 11)2 2 2

a a a




1 2 31 ( )Ω






lkh

lkh dG π2


Δ =k G (See 152)











G (See 141)





⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎟⎟⎠

⎞⎜⎜⎝

⎛+⎟

⎟⎠

⎞⎜⎜⎝

⎛minus=

126

4)(ijij

ij rrrU σσε


1e

1 minus

= kTn ω (See 277 2215)


222 akmks=ω (See 248 288 290)






D

DDωωωωωω

gt=leprop

0)( 2

(See 2224)

3126v ⎟

⎟⎠

⎞⎜⎜⎝

⎛==

VN

kk DDD

πθθω (See 2228)


678 Appendices


intminus

⎟⎠⎞

⎜⎝⎛= T

x

x

VdxxTNkC

0 2

43

)1e(e9 θ

θ (See 2229)


3( )

( )( )(2 ) const

dAVD ω

ωω

ωπ= int

k knabla (See 3256)




)e(e 20


2( ) TWθ⎛ ⎞prop Δ ⎜ ⎟⎝ ⎠

k


22

2

9)(

Dm

m mkAT θ=



V

Dln



B

C3

vγα = (See 2237 2238 2250)




ρ11Cv = (See 2270)


ρ44Cv = (See 2270)



3 Electrons


1e

1)( +




( ) ( )u u= +k kr r R (See 3129 C25)


EmVEg23

222

2)( ⎟⎟

⎠

⎞⎜⎜⎝

⎛=π

(See 3164)


32

22

32

)0( ⎟⎠⎞

⎜⎝⎛==

VN



m

ne τσ2

= (See 3214)




RH (See 6102 1223)

680 Appendices


LTK =σ

(See 3215)

22

3⎟⎠⎞

⎜⎝⎛=

ekL π (See 3215)


0

00 near band edge

E E U

U E E

lowast

+

minus=

minusk G

G k G (See 3230)



k r r R (See 3244)


k (See 3255)




i

Bloch Fns


kkr R r (See 3332)



cgs)in or (mceB

meB

c =ω (See 3196)




22

2 2

0 ( )

( )

( ) ( ) ( )2

1( ) ( ) ( )4

el

ioni i

j i i ij i

V

Vm

e d E

ψ ψ

ψ ψ ψπε ne

minus nabla +



r

r r r

r r r rr r




22

2

0 ( )|| spin

( ) ( ) ( ) ( ) ( )2

1 ( ) ( ) ( ) ( )4

ion eli i i

j i j i ij i

V Vme d E

ψ ψ ψ

ψ ψ ψ ψπε

lowast

ne

minus + +


r r r r r

r r r r rr r

nabla



1 32

2 3( ) ( ) ( ) ( ) ( ) ( ) ( )2


mψ ψ ψ ψ

π⎛ ⎞⎡ ⎤minus + + minus =⎜ ⎟⎣ ⎦ ⎝ ⎠

r r r r r r rnabla



2

2

1 ( ) ( )( ) 1( ) ( )k

e f fV E E iq σ

ε ωω ηε


k k qqk q k

(See 9166)

xx

xxxF

kqxxF

qk

qTF

sminus+minus+==+==

11ln

41

21)(

2)(1)0(0

2

2

2ε


F

srk

F Eemk

rkq s

εεπφ

23e

41 2

02 ==ltlt minus



collisions

f ff ft t


Fv nabla nabla


0RHS of BDE( )

f fτminus

= minusk

(See 4149 4145)

682 Appendices


11 123 v( ) ( ) ( )4

edj e f L E L T

eμ

π⎛ ⎞= minus = + + minus⎜ ⎟⎝ ⎠


21 223 ( )v( ) ( ) ( )4

Qdj E f L E L T

eμ

π⎛ ⎞= minus = + + minus⎜ ⎟⎝ ⎠




0ej TE

σ = =nabla

0Qe

jj

Tκ = =

nabla


0eEQ jT

= =nabla

0Q

e

jT

j

QT

Π = =

Π =

nabla


constant131 conganc



⎟⎟⎠

⎞⎜⎜⎝

⎛minus

VED F )(1exp




heG =




D

D

TT

TT

θρ

θρ

ltlt

gtgt5~

~ (See 4146)





)SI(21

0Ae

Bπ=⎟

⎠⎞

⎜⎝⎛Δ (See 534)



⎟⎟⎠

⎞⎜⎜⎝

⎛= (cgs) 4 2

0

2

mne

mne

pπ

εω (See 10108)







2innp = (See 613)

⎟⎟⎠

⎞⎜⎜⎝

⎛minus⎟

⎟⎠

⎞⎜⎜⎝

⎛= lowastlowast

kTE

mmkTn ghei 2

exp)(2

2 4323

2π (See 614)

684 Appendices


ddt

= kF (See 644)




2

2

211

kE



eDkT =μ (See 684 685)





⎟⎟

⎠

⎞

⎜⎜

⎝

⎛ ΔΔ== (cgs)

2

222 ne

V

ne

VKZwidth b π

εε (See 6193)


⎥⎦

⎤⎢⎣

⎡minus⎟

⎠⎞

⎜⎝⎛

⎟⎟⎠

⎞⎜⎜⎝

⎛+= 1exp

00 kTep

LD

nLDej b

np

pp

n

n ϕ (See 6241)



23minusprop Tμ


23Tpropμ





⎟⎠⎞

⎜⎝⎛ )(

22cgs

mceB

meB (See 7255 7256)

Lande g factor

⎥⎦

⎤⎢⎣

⎡+

+minus++++=)1(

)1()1()1(211

JJLLSSJJg

(Implied by (76) (710))


)(20 FBp EDμμχ = (See 3181 3201)


magnetonBohr =Bμ


)groundexcited(22

==minus

= gexcEEgexc

VN

gexc

zμχ


⎟⎠⎞


⎤⎢⎣⎡ ++=

Jx

JJxJ

JJxBJ 2

coth21

2)12(coth

212)( (See 716)



4sincos)()2(

xxxxxFRkFJ ijFij



kT

BgJx eff

Bμ= (See 722)

MBBeff λ+=

686 Appendices


)cgs(

32

0MMH

kEk effB

F

μθθ =primegt

prime (See 7222)


⎟⎟⎠

⎞⎜⎜⎝

⎛minusprop

)(1exp

FK EJD

T (See 7300)


h

eGxy

2ν= (See 1247)



KJpropδ (See 7251)





23

23)()0(

TC

TTMM

M prop



)for (1cc

c

c

TTHTTM

TT

TTC

=asympminusasymp

minusasymp

minusasympminus

minus

δβ

γ

α

χ






2

( ) ( ) (cgs)nemc



2

2

4 nemc

L πλ = (See 86)


g

FEv

πξ 2~0 (See 849)



II ξ lt λ 20~

1 πλϕ

cH 20~

2 πξϕ

cH 2

~1

2⎟⎟⎠

⎞⎜⎜⎝

⎛ξλ

c

c

HH



)(20 cgs

ehc=φ (See 845)



⎟⎟⎠

⎞⎜⎜⎝

⎛minuscong

)(1exp141

FDc EVD

T θ (See 8217)


⎟⎟

⎠

⎞

⎜⎜

⎝

⎛

+minus


1exp

451 epep

epDcT

λμλ



688 Appendices


2

20



⎡ ⎤⎛ ⎞⎢ minus + + ⎥ =⎜ ⎟⎝ ⎠⎢ ⎥⎣ ⎦

Anabla (See 83)

2

( ) ( 2 2 )2s eiq qj q e m m






)()0(

2

2

infin=εε

ωω

T

L (See 950)


⎟⎟⎠

⎞⎜⎜⎝

⎛+=+= )SI(

3)guassian(

34



⎟⎟⎠

⎞⎜⎜⎝

⎛=

+minus=

+minus

sumsum )SI(31

2)(1)(

34

21

00

0jjjj NN α

εεεεεαπ

εε (See 930)



direct 0eΔ =k








)cgs(2 222

4

neEE gn ε


)cgs(111

he mm+=

μ (See 10101)


11 Defects


Mollwo Equation







⎟⎟⎠

⎞⎜⎜⎝

⎛==

pGeEp

Bπω 2 (See 129)


)(exp 41 MottT

AB ⎟⎠

⎞⎜⎝





232surface =⎟

⎠⎞


kTWkTemj

π (See 1145)

690 Appendices


3 2

2surface exp BWj AE

E

⎛ ⎞= minus⎜ ⎟⎜ ⎟

⎝ ⎠ (See 1149)


J Folk Theorems


Chapter 1



Chapter 2



Folk Theorems 691

Chapter 3









Chapter 4







Chapter 5


Chapter 6




692 Appendices















Chapter 7






Folk Theorems 693





Chapter 8




Chapter 9



Chapter 10





Chapter 11







Chapter 12



694 Appendices







Gauss law V A A

dV disin





10

e +

infinminus =int n

axn

andxx


π41Y

e21)(

Y)(

00

23

10

0010001

=

⎟⎟⎠

⎞⎜⎜⎝

⎛=

=Ψ

minus Bar

BarR

rR





r θnabla ϕ

221 1 1( ) (sin )

sin sinrA

r A Ar r rr

ϕθθ



2

2

2222

2 sin1sin

sin11

ϕθθθ

θθ partΦpart+⎟

⎠⎞

⎜⎝⎛

partΦpart

partpart+⎟

⎠⎞

⎜⎝⎛

partΦpart

partpart=Φnabla

rrrr

rr2



111


=primeprime

γγγ

nnn

n


n

ennn ⎟⎠⎞




)()(

1))((

)()()(0)(


n

n nn

xxxf

xxxf

xf

bfdxbxxfxx

xxbxdxbx

=

minusprime

=


minus===minus

sum

int

int

δδ

δδ

δδδ

Complex Variables


π


nazfazdzd

naf n

n

n


)1(1)(Res 1

1

⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧

minusminus

= minus

minus

rarr


SI Gaussian

0

t

tρ

parttimes = +part


sdot =sdot =

H J D

BE

DB

nabla

nabla

nablanabla

4 1

1

40

c c t

c t

π

πρ

parttimes = +part


sdot =sdot =

H J D

BE

DB

nabla

nabla

nablanabla

0ε= +D E P

0

1μ

= minusH B M



( )qc

= + timesvF E B

696 Appendices












1956 Transistor


















9 Total for Decade

698 Appendices

























A Geim K Novoselov

2010 Graphene



Brillouin




Bloch






Bardeen



700 Appendices

Others






M Problem Solutions





451388

2412

126


=+minus=M




1

A

2

3

456

7

8

g5 g6

g7

g8


⎟⎠⎞

⎜⎝⎛= 87654321

876543211g

⎟⎠⎞

⎜⎝⎛= 21876543

876543212g

⎟⎠⎞

⎜⎝⎛= 43218765

876543213g

⎟⎠⎞

⎜⎝⎛= 65432187

876543214g

⎟⎠⎞

⎜⎝⎛= 23456781

876543215g

⎟⎠⎞

⎜⎝⎛= 45678123

876543216g

⎟⎠⎞

⎜⎝⎛= 67812345

876543217g

⎟⎠⎞

⎜⎝⎛= 81234567

876543218g

702 Appendices

Then eg

432 6543218787654321 ggg =⎟⎠⎞

⎜⎝⎛=


g1 g2 g3 g4 g5 g6 g7 g8




)(21 kjia minus+= a


)(23 kjia +minus= a

x

y

z






y

z

x

a2

a1

a3

a


a


704 Appendices





2 2 0h kh k h k


a bG

since

1 2 32 ( )h k lπ= + +G b b b

and


Thus

( ) 2( )( ) ( )h k ld h k lh k l h h k l

π= =G aG G

sdot






where

f iΔ = minusk k k



ki

b

b

bα

bα

hellip

x

y

z

kf

0ˆ =sdot ik i

2sin

2sin

e

2sin

2sin

e

e1

e1

e1

e1

ee

2)1(

i2

)1(i

i

i

i

i

10

i10

i

⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜

⎝

⎛

+⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜

⎝

⎛

=

minus

minus+minus

minus=

+=

minusminus

minusminus

minus

minus

minus

minus

minus=

minusminus=

minus sumsum

α

αα

α

α

α

bk

Mbk

bk

bNk

P

fx

fxMbk

fx

fxNbk

bk

Mbk

bk

bNk

Mm

mbkNn

nbk

fxfx

fx

fx

fx

fx

fxfx


)(integer 2

and)(integer 2

)2()1(

pbk


απ (a)

Since

pl

bk

bk

pl

bk

bk

fx

fx

fx

fx αα

== )2(

)1(

)2(

)1(

or1

)2()1(fxfx kk ne



since

22fxfy kkk minus=


706 Appendices


Let

λπ2=K





Define

θsinKB =

)iexp( BbZ =

So


But


)]2sin(ln[

1)]2sin()exp[(ln

0

)2sin(

knbZ

knbZm

knbZZ

m

knb

πε

πε

πεπε

+=

=

=

suminfin=

Thus


0 )2sin()ln( πε


2)2cos(21)2sin(ln

11

ZkbZkbZZ

ZE

+minus+

minus=

ππε






mGBb π2+plusmn=


b




Aring2255=Ka


Aring2255100 == ad

)bcc(6125222255


deg=rArr== 6816225551

2sin θλθ

d

deg=rArr= 04352255

)51(2sin θθ



θλ sin2 100dn =

above as 2)e1( 0435 4510

)51(42

4sin


)51(32

3sin

above as 2)e1( 6816 4510

)51(22

2sin


2sin

i4400

100

i3300

100

i2200

100

i100

100

ffSd

fSd

ffSd

fSd

=+====

=+====

=+====

=+====

minus

minus

minus

minus

π

π

π

π

θλθ

θλθ

θλθ

θλθ

708 Appendices




mm m

γ γ γx1 x2

1γx3

3 1 2 3





02

22

321

2

2

2

=⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜

⎝

⎛


uuu

mm

m


(2)


twice32mγω = (5b)


⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛==

111

31

321

0uuu

Uω (6a)

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛minus==12

1

51

3 mU γω (6b)


and

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛minus==

112

51

3 mU γω (6c)



)(2iexp1

)(2iexp1

)(2iexp1

)1(0

10

10

sum

sum

sum

minusminus= minus

minus=

minus=

⎥⎦⎤

⎢⎣⎡ minus=

⎥⎦⎤


⎥⎦⎤

⎢⎣⎡ minus=

Ns s

Ns s

Ns snm

nmN

sgN

nmN

sgN

mnN

sgN

G

π

π

π


[ ]

⎥⎦⎤

⎢⎣⎡ minus=

minus⎥⎦⎤

⎢⎣⎡ minus=

⎥⎦⎤


⎤⎢⎣⎡ minus=⎥⎦

⎤⎢⎣⎡ minus+

)(2iexp

)(2iexp)(2iexp


nmsN

nmnmsN

nmNN

nmsN

nmNsN

π

ππ

πππ

and (see 2104)

⎟⎠⎞

⎜⎝⎛=⎟

⎠⎞

⎜⎝⎛ +=⎥⎦

⎤⎢⎣⎡ +

Ns

NsNs




Nsg

NG π

But

⎟⎟⎠

⎞⎜⎜⎝

⎛⎥⎦⎤



Nnm

NNggN

ππ

so

mnN

snm GnmN

gG =⎥⎦⎤


0 )(2iexp π

QED

710 Appendices




)2()()(

)()(

)2()()(

)()(

)2()()(

54145515

34454

32423343

12232

15251121

XXXXXXXXm

XXXXXm

XXXXXXXXM

XXXXXm

XXXXXXXXm


minusminusminus=


minusminusminus=


αααβα

βββαβ

ααα


0

210011100020001110012

54321

22

212

2

=

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛

minusΩminusminusΩ

minusΩminusminusΩ

minusΩ

minus

uuuuu

AAAAA

AAδ


⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

Ω=

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛



minusminus

54321

2

54321

210011100

020001110012

uuuuu

uuuuu

AACCC

AA


⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

minusminus

=

1

0

1

a

aU


This requires

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛

ΩminusΩminus

sdotΩΩΩ

=

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

+minus++minus

minus+minus

=

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

minus+minus+minus+

+minus++minus+minus

22

2

2

2

0

31)1(

01)1(

3

211))(1(

)1(112

a

a

aAa

Aaa

aaAcaca

Aaa




2





)1()1()1( dagger

dagger

ϕψψ

ψψϕ

minus=minus=minus=

==

nna

aaa

aaaNaN

n

n

nn

712 Appendices


nKan =minus1 So

112

dagger2

nnnK

naanKnn

=

=minusminus


1or1 minus== nnnan

K


)1()1(

)1(dagger

daggerdagger

daggerdaggerdagger

ηψ

ψ

ψψη

+=+=

+=

==

nna

aaa

aaaNaN

n

n

nn


1

1+

=n

K

so

11dagger ++= nnna


⎟⎠⎞

⎜⎝⎛ +=⎟

⎠⎞

⎜⎝⎛ +=

21

21dagger Naa ωωH



321321


⎜⎝⎛=



321

321

321

ηηη

ηηη

ηηη

partpart

partpart

partpart

partpart

partpart

partpart

partpart

partpart

partpart

=

zzz

yyy

xxx

J

1 1 2 2 3 3η η η= + +r a a a


etc11

xax =partpartη

so we find

1 2 3

1 2 3 1 2 3

1 2 3

( )x x x

y y y

z z z

a a a

J a a a

a a a

= = sdot timesa a a

QED



1 1

2 1

3 1

100

sdot =sdot =sdot =

a ba ba b



a

KΩ

= 1

and

2 31

a

times=

Ωa a

b


714 Appendices


1 1 2 2 3 32 ( )n n n nπ= + +G b b b




31 21 2 3

1 2 32

nn nN N N

π⎛ ⎞

= + +⎜ ⎟⎝ ⎠

q b b b



a) 31 231 2 31 2

31 2

1 2 3

1 1

11

1

22 2 ii i 11 1i

1 2 3 1 2 30 0 0

i2

2i1

1 1 1 1e e e e

1 1 e etc



l

l

n

nn

nn n ll l NN NNN N

l l l

n l

n lN

i i

i i

n

N N N N N N

N

n Nn N

ππ π

π

π

δδ


= = ==

minus=

minus= ne= =

=

=

sum sum sum sum

sum

q R

R

Gq

q GG

G

b) 111

1

1

1

11

21

i(BZ)

1 2 3 1 0

2

21

0

1 1e e etc


1 1 e


l

l

ni lNN

l

i n

i lN

i i

i i

N N N N

N

l Nl N

π

π

π

δ

minussdot

==

minus=

minus= ne= =

=

sum sumq Rq

R


c)

1 2 31

1 1

3

33

1 2 31 2 3

31 2 3 03

fpp 33

( ) 2

of s( )Vol of space

1 ( )(2 ) ( )

( )(2 )

( ) (2 )

n n nn

f fN

qf dq

f d

N N NN N N

f d

Vf d

π

π

π

π

⎛ ⎞⎛ ⎞= +⎜ ⎟⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠

=

=sdot times

Ω=

sum sum

int

int

int

int

q q b

q q

q qb b b

q q

q q

d)

i i3fpp3

fpp

i

1 2 30

e e (with )(2 )

1 e

l l

l

l

a ad q VV

N N N

π

δ

Ω Ω= rarrinfin

=

=

sumint

sum

q R q Rq

q Rq

R

sdot sdot

sdot

e)

1 1 1 2 2 2 3 3 3i( ) i2 [( ) ( ) ( ) ]3 31 1e el l l l l l l luc



3

1 2 3


η η η⎛ ⎞

= = ⎜ ⎟⎝ ⎠

r


etc so

1 2 3

1 2 3 1 2 3

1 2 3




So

1 1 1

1 1 111

1i( ) i2 ( )310

1i2 ( )

1 1 0

1 e e

e i2 ( )

l l l l

a

l l

d d

l l

π η

π η

η

δπ


primeminusprime

=Ω

= =prime minus

int intG G r

ll

rsdot

716 Appendices

f)

i ( ) 3all 3

i ( ) i ( ) i ( )

i( )i( ) i( )

1 e(2 )

1 1 1e e e2 2 2( )

1 e 1 e 1 e2 i( ) 2 i( ) 2 i( )

yx z

x y z

yx yx z

zx y



LL L

d q

dq dq dq

L L L

x x y y z z

π

π π π

π π π

primeminus



minusminus minus

=

rarr infin


int

int int int

q r rq

sdot


( ) ( ) ( )( ) ( ) ( )( )

z

x y z

yx zL L L


x x y y z zπ π π

δ δ δδ









Frequency c

gωω=

Temperature )( k

Ttcω

=


Uucω

=

Entropy NkSs =


NkC

c vv =



⎟⎠⎞

⎜⎝⎛ +⎟⎠⎞

⎜⎝⎛==

Nd

ddpP

c

ωωωωω


)(

Thus

π

ωω2

)()( NadqdqqNdN ==

1

)2)(2cos(1

2)(

22cc

cN

aqaN

ddqNadqN

ωωπω

ωπωπω

minus=

==

So

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

minus⎟⎟⎠

⎞⎜⎜⎝

⎛

minus==

221

11e

1

ckTc d

dpPωωπω

ω ω

or since

tg

kTkT c==

ωωωω

c

then

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

minus⎟⎟⎠

⎞⎜⎜⎝

⎛

minus=

21

11e

1)(g

gtP tgπ (1)



int

intsum

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

minus⎟⎟⎠

⎞⎜⎜⎝

⎛

minus=

minus=

minus= =

c

c

m

dN

dNU

cckT

kTNm kT

m

ω

ω

ω

ωω

ωωωω

ωπ

ωωωω

0 22

01

1

11e

2

)(1e

21e

int⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

minus⎟⎟⎠

⎞⎜⎜⎝

⎛

minus==

c

cc

kTc

cd

NUu

ω

ω ωωωω

ωωπω 0 22 )(

1

11e

2

718 Appendices

or

int⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

minus⎟⎟⎠

⎞⎜⎜⎝

⎛

minus=

1

0 21

11e

2 dgg

gu tgπ (2)

ωωωπω

ω

ω

ω

ω

ω

ω

ω

dNkT

kTC

cckT

kT

m kT

kTm

c

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

minus⎟⎟⎠

⎞⎜⎜⎝

⎛

minus=

minus=

int

sum

2202

2

2

2

2

2v

1)1e(e)(2

)1e(e)(1

int⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

minus⎟⎟⎠

⎞⎜⎜⎝

⎛

minus==

c

cc

kT

kT

c

c dTNkN

NkCc

ω

ω

ωωω

ωω

ωπω

ω

0 222

2

22v

v )(1

1)1e(

e)()(

2

int⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

minus⎟⎟⎠

⎞⎜⎜⎝

⎛

minus=

1

0 22

2

v1

1)1e(

e)(2 dgg

tgc tg

tg

π (3)


kTFj



VT

FS ⎟⎠⎞

partpartminus=

so

( ) sumsum=

minus=

minus

minus+minusminus=

N

m

mkT

N

m

kT

kTkTkS

mm

12

1

e11e1ln ωω

ω

( )

int

int

minusminus+

minusminusminus=

=

minus

c

c

cc

kT

cc

kT

dkTkTNNk

dNkNk

NkSs

ω

ω

ωω

ωωωω

ωπ

ωωωωπ

0 22

2

0 22

)(1

11e

)2(1

)(1

1e1ln21


+minus

minusminus= minus1

0 2

1

0 2

1

11e

2

1

1e1ln2 dgg

tgdgg

s tgtg

ππ (4)



range of ω

ωxxp


P10x1π

1

e

ωx

10 1minus

sdot1

1 ωx( )2minus

sdot=

P100x1π

1

e

ωx

100 1minus

sdot1

1 ωx( )2minus

sdot=

P300x1π

1

e

ωx

300 1minus

sdot1

1 ωx( )2minus

sdot=

0 05 10

200

400

P300

P100

P10

ω

720 Appendices


the range of t

txxr


ux2π

0

1

yy

e

y

tx 1minus

1

1 y2minussdot

⌠⎮⎮⎮⎮⌡

dsdot=

spht x2π

0

1

y

ytx

⎛⎜⎝

⎞⎟⎠

2e

y

txsdot

e

y

tx 1minus

⎛⎜⎝

⎞⎟⎠

2

1

1 y2minussdot

⌠⎮⎮⎮⎮⎮⎮⎮⎮⌡

dsdot=

0 1 2 3 40

1

2

3

4

u

spht

t


0 05 1 15 20

05

1

15

2

u

spht

t


Entropy t2π

minus

0

1

ωln 1 e

ωminus

tminus

⎛⎜⎝

⎞⎟⎠

1

1 ω2

minus

sdot

⌠⎮⎮⎮⎮⌡

dsdot2π

0

1

ω

ω

t

e

ω

t 1minus

1

1 ω2

minus

sdot

⌠⎮⎮⎮⎮⎮⌡

dsdot+=

0 100 200 3000

2

4

6

8

Entropy t

t

722 Appendices




By Problem 29

2 31

a

times=

Ωa a

b


2 3 [(1)(1) (1)( 1)] [(1)(1) ( 1)(1)] [( 1)( 1) (1)(1)]

2

(2 2 ) ( )2

a

a a


= + = +

a a i j k

i j i j

and

2

21 2 3( ) (2)


so Ωa = a2 and

11 ( )a

= +b i j


2

3

1 ( )

1 ( )

a

a

= +

= +

b j k

b i k




1 2 31 1 1( ) ( ) ( )a a a




ky

kxkz


a

a

n m + 1

n mn ndash 1 m

n m ndash 1

n + 1 m

xnm ynm


])()()()[(2

])()()()[(2

21

21

21

21

21

21

21

21

mnmnmnmnmnmnmnmn

mnmnmnmnmnmnmnmn

yyyyyyyyk

xxxxxxxxkU



minus+minus+

minus+minus+


)4( 111

mnmnmnmnmn



)4( 111

mnmnmnmnmn



724 Appendices


)i(1 e maknakt

mnyxcx ++minus= ω

)i(2 e maknakt

mnyxcx ++minus= ω


⎟⎠⎞⎜



⎟⎠⎞⎜






a x

y


⎥⎥⎦

⎤

⎢⎢⎣

⎡+⎟

⎟⎠

⎞⎜⎜⎝


⎟⎠⎞

⎜⎝⎛ +++minus++=

⎟⎠⎞

⎜⎝⎛ minus++=

22

2

2

222222

22222

222

481

2122

21

)(2)(21

)(21

aax

ax

ayxayxaxak

ayxaayxak

ayxakU

to second order

[ ]

2

22222222

21

2222

kx

axaxyxayxaxakU

cong







11 11dagger

1 i22

pp p p

pa P X

ωω

= minus qq q q

q

Now

11 1

1

11dagger

p p

p

p

X m

m

X


minus

= sdot

= sdot

=

sum

sum

X

X

q b q b q bb

b q b q bb

q

e

e






1 1

1

1

1

111

1

1 1

p pp p p

pp p

e X m e e X

m e e X

m X m X

β βα αβ

β βαβ

ββ ααβ δ δ

lowast lowast

lowast

=

=

= =

sum sum

sum sum

sum

q b q b q bbqb q b

b q b q bb q b

bb b bq bb q b

or

1 11

1p ppX e X


b (a)

Now

dagger 11dagger 11

1 i22

pp p p

pa P X

ωω

= + qq q q

q


11dagger 11dagger

11 11

1 i22

1 i22

pp p p

p

pp p

p

a P X

P X

ωω

ωω


= minus

qq q q

q

qq q

q

726 Appendices

so

dagger 11 2i

2p

p p pa a Xω

minusminus = qq q q

11 dagger

i ( )

2p p pp


q

Substituting in (a)

1 dagger

i ( )

2 p p ppp

X e a am ω


b q



0)(

)(])[(

34412

24412122

442

4411

=++

++minus+minus

Kkkcc

KkkccKkckcc

zx

yxx ρω


0)( 122

11 =minus Kkc ρω


0)( 222

44 =minus Kkc ρω

0)( 322

44 =minus Kkc ρω

If

ρ

ων 1111

ck

==


If

ρ

ωνων 4433

22

ckk

====






)1(

)(

])()([)(

))]()(())()([()(

))]()(())()(()(

2

2

12

1112

11112

11112

1

1

NM

PM

dPM

dxuxuPM

dxuxuQxuxuPM

dxuxPuxuxPuMd

P

qp qq

qp qq

pp NNNNpp

pp NNNNpp

=

=

minus=

minus=

minus=

minus=

sum

sum int

sum int

sum int

sum intint

lowast

primelowastprime+


τδ

τ

τ

ττψψ

where Q = Pminus1P




)2()1()21()2()1(21

)2()1()21()2()1(21

)2()1()21()2()1(

)2()1()21()2()1(

)1()1()1(

(||)

(||)

sum

sum

sum

sum

sumsum

minus

+=

minus

+

=

ji ijji

ji jiji

ji ijji

ji jiji

i iik

V

VE

V

V

H

ψψψψ

ψψψψ

ψψψψ

ψψψψ

ψψε



22 2

1 21 10 12

| ( ) |( ) d 0

4jN N

II jeV

rψ


rr

Note that

221| ( ) |N

jj ψ=sum r

728 Appendices


2

22 11

0 12

( )d ( )

4NI

II Ie V

rρ τ


r QED

IonI

Field point

Source point

r1

r2



Let x = kkM

0as2)(21

11ln1limlnlim

2

2

0

22

0

rarr=minus=

minus+minus=

minus+minus

rarrrarr

xxxx

xx

xx

kkkk

kkkk

xM

M

M

Mk

⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢

⎣

⎡


minusminusminus



rarr

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

gtminusminusminus

ltminusminusminus

rarr

⎥⎦⎤

⎢⎣⎡


⎥⎦⎤

⎢⎣⎡



rarr

110)1(

)1(1)1(1

or

110)1(

)1(1)1)](1(1[

2

1)1(1)1ln(

1)1(1)1ln(

2

1)1ln(1)1ln(2ln)1(2

1)1ln(1)1ln()1ln()1)(1(


2

2

22

1

xxx

xx

xxx

xx

xx

x

xxx

xxxxx

xxxxx

xxx

xxxx

xx

xx

x





i1e 0bm


K


The answers are





Thus

i(0) i( )( )e e

1 ( 1)

n l m

n l m

S πminus minus + +

+ +

= +

= + minusK






730 Appendices


i( )( ) i( )( ) i( )(n )1 e e en l m









20

2)(2

2 EVmKmEk

minus==


)ee()(22

ee)(2

)(02

If

ii

KxKx

kxkx

Dxbxb

CBxaxb

Vbax

minus

minus

+=ltltminus

+=ltlt

infinrarr=+gt

ψ

ψ

ψ


)ee()ee(i

)ee(ee

0ee

2i2i2i2i

2i2i2i2i

)2(i)2(i

KbKbkbkb

KbKbkbkb

bakbak

KDCBk

DCB

CB

minusminus

minusminus

+minus+

minus=minus

+=+

=+



⎟⎠⎞

⎜⎝⎛minus=

2coth)tan( Kb

Kkka (a)


)ee()(22

ee)(2

02

ii

KxKx

kxkx

Dxbxb

CBxaxb

bax

minus

minus

+=ltltminus

+=ltlt

=+gt

ψ

ψ

ψ


⎟⎠⎞

⎜⎝⎛minus=

2tanh)tan( Kb

Kkka (b)


)e21(

2tanh

)e21(2

coth

Kb

Kb

Kb

Kb

minus

minus

minusrarr⎟⎠⎞

⎜⎝⎛

+rarr⎟⎠⎞

⎜⎝⎛

and

00 V

EEV

EKk cong

minuscong


⎟⎠⎞



02symmetric-anti





)e21( 2

0


732 Appendices


⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝


22exp21

22

20

02

22

2

2 bmVVma

Ema ππ






dkm

kdkdkdEdE

2==


intint ==ll

dldEk

mdldkdEED 222 21

)2(2)(

ππ

φ

πak

θ

dl

dk

E+dEE = h2k22m

int=kdlmED 222

)(π


For a

k πlt

2222

2)(

ππ

πm

kkmED ==

For a

ka

ππ 2ltlt

)2(42

)( 2222 ϕπ

ϕπ

mkkmED ==

but

⎟⎠⎞



2

So

⎟⎠⎞


kamED ππ

π1

22 cos4)(

or

⎟⎠⎞


kamED π

ππ1

2 cos41)(



(1) (2) (3) (4) (5)

(6)

(c) (b) (d) (a) (f) (e)



0 00



k

734 Appendices


( )

0( ) 2e

(e 1)

F

F

E E kTF

E E kTf E EdT dT kT

minus


k

k

k

( )

0( ) 2e 1

(e 1)

F

F

E E kT

E E kTfE kT

minus

minuspart

= minuspart +

k

kk

so

0 02

0

F

F


f EEdTE dT kT T


k

k

k

k

Thus

00


τ⎛ ⎞ part⎡ ⎤⎛ ⎞= + minus + +⎜ ⎟⎢ ⎥⎜ ⎟ part⎝ ⎠⎣ ⎦⎝ ⎠

kk k

k


33

2(2 )

zzJ e u f d k

π

⎛ ⎞= ⎜ ⎟⎜ ⎟

⎝ ⎠int k k


33

2(2 )

zzH u f E d k

π

⎛ ⎞= ⎜ ⎟⎜ ⎟

⎝ ⎠int k k k

Thus

2 303 ( )

4z F


τπ


kk

and

2 303

1 ( )4

z Fz


τπ


k k kk



23

22

3

23

( ) [ ( )]( )4

( ) ( )12

412

zz F

zF

F

dsdEeJ u E E eEu

dsdEe E u E Eu

e eEu k

τ δπ

τ δπ

τ ππ

= minus minus minus

= minus

=

int

int

kk k

k

kk k

k

k




m

neEJ z τσ

2==


230 ( )

12F


dz dT kT T uτ δ

π⎛ ⎞⎡ ⎤⎛ ⎞= = minus minus +⎜ ⎟⎢ ⎥⎜ ⎟

⎝ ⎠⎣ ⎦⎝ ⎠int k k

k kk

or

2 2( ) ( ) FF F


τ δ τ δ ⎡ ⎤⎛ ⎞minus = minus +⎢ ⎥⎜ ⎟⎝ ⎠⎣ ⎦

int intk k kk k k k

k k

or

( )2

2 2 2 414F F

F F F FFE E E E


πτ π τ= =

⎞⎛ ⎞ ⎛ ⎞= minus + ⎟⎜ ⎟ ⎜ ⎟ ⎟⎝ ⎠⎝ ⎠ ⎠k k

k kk k

So

dT

dEdzdTeE F=


2

31 [ ( )]

12F F

F

dT Kdz


τ δπ

=


k k k kk

so

23

1 412 F

F

F Fk kE E

u dE dEK E E k

dT dTτ π

π ==


kk k

736 Appendices


dT

dEEm

nK FF

F 2τminus=


⎥⎥

⎦

⎤

⎢⎢

⎣

⎡

⎟⎟⎠

⎞⎜⎜⎝

⎛minus=

2

)0(121)0(

FFF E

kTEE π

so

)0(6

22

F

FE

TkdT

dE πminus=

so

)0(6

222

FF

FE

TkEm

nK πτ=


Tkm

nK F 22

3τπ=

and

22

2

22

33 ⎟

⎠⎞

⎜⎝⎛==

ek

Tm

ne

Tkm

n

TK

F

Fπ

τ

τπ

σ




bXX

dx 12 minus=int





1

)]1([11

)]1([

21

1

02121

10 2

21

KK

xKxKKKxKxKdx

=

⎥⎦

⎤⎢⎣

⎡minus+minus

minus=minus+

int


2

2

22 2

22 2

( )

( ( ))

(2 )2

22

2

L

L

dIq f

dq f

d

q qm

dmq q

m

ω

ω

=

= minusminus

= minus⎡ ⎤



int

int

int

int

qq

qq

q

k q

q

k q


1

2 20 2 2

2

2(1 ) 2 L

m dxI dmq x q xω


int intq

k q


1

2 20 2 2 2

2

2( ) L

m dI dxmx x x kω


int intq

q k


1

2 20 2 2 2

2

2L

m dI dxmq x x kω

= minus⎡ ⎤⎛ ⎞+ minus⎜ ⎟⎢ ⎥⎝ ⎠⎣ ⎦

int intq

738 Appendices


2

2 2 2 2 2 20

4[ ] [ ]

d q dqq q

πα α

infin=

+ +int int

q


intminus

minus=1

0 22

22 2

2

kxxmdxmI

Lωπ

But

2

sin2

2

11

0 22

pppxx

dx minus=minus

int

so we find

⎟⎟⎠

⎞⎜⎜⎝

⎛minus= minus

2sin22 12

2k

mk

mm

mIL

L

L ωω

ωπ

Thus

Lm

Kω

=3


⎟⎟⎠

⎞⎜⎜⎝

⎛+minuscong

2122 2

22

kmm

mILL ω

πω



2 2

02

16

c L

c

kEm

mm

α ω

α

lowastlowast

lowastlowastlowast

= minus +

=minus

k






1 1d edt c

δτ

⎛ ⎞ ⎛ ⎞+ = minus +⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

k E v Btimes


mceB

c =ω

v

vv

vv

zz

xcyy

ycxx

Eme

Eme

Eme

τ

τωτ

τωτ

minus=

+minus=

minusminus=


⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛ minus=

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

z

yx

cc

z

yx

EEE

DDjjj

000101

0 τωτωσ




Dyyxx

0σσσ ==

D

cxyyx

τωσσσ 0=minus=

For the Hall effect

yxyy

xyy

EEj σ

σminus=rArr= 0

740 Appendices

So

xyy

xyxxx Ej ⎟

⎟

⎠

⎞

⎜⎜

⎝

⎛+=σσ

σ2


xx Ej 0σ=


m

ne τσ2

0 =


BE

E

BjE

Rx

yxyy

x

x

yH )( 0σ

σσ

minus==


nec

RH1minus=


B

nec

cc

cxyyx ==congminus=

τωσ

τωτωσσσ 02

0

)(


B

Ryx

H )(1

σminus=


v

v)()(

)(vvv

32

54

32

540

0

⎥⎦

⎤⎢⎣

⎡partpart

partpartminusminus=

⎥⎥⎦

⎤

⎢⎢⎣

⎡sdot

partpartminus=

partpartminus=

F

F

FF

e

Fe

xxxfxfcm

xfcmEp

π

π



( ) ⎟⎟⎠

⎞⎜⎜⎝


40

22 11411

FF

xF

xxdxxxxf F

where

312

v3

⎟⎟⎠

⎞⎜⎜⎝

⎛=equiv π

cmcmpx

ee

FF


FFF

F xxxxf

21)( 3 +=

partpart

34312

v31)3(

vπ

cmx

e

F minus=partpart

Fe

F xcm

x31

v3

31

vv

312minus=⎟

⎟⎠

⎞⎜⎜⎝

⎛minus=

partpart π


[ ] 12

61

3111

41

2432

54

242

432

54

0

FFe

FFF

Fe

xxcm

xxx

xcm

p

minuscong

⎥⎥⎦

⎤

⎢⎢⎣

⎡++⎟

⎟⎠

⎞⎜⎜⎝


π

π


pm

MM δ=

and

⎟⎟⎠

⎞⎜⎜⎝

⎛=

cem

RR

742 Appendices


Mm

R pξπ sdotcong 3

34v

31

31

3

2

491

34

3

⎟⎟⎠

⎞⎜⎜⎝

⎛=

⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜

⎝

⎛

sdot=

ξδπ

ξπ

π

MR

Mm

RRRx

pF


32

54

12~and

89

ππδ cm

K e==

⎥⎥⎦

⎤

⎢⎢⎣

⎡minuscong 2

32

4

34

0~

RM

RMKp




EF(2)

EF(1)

E

Δ = eV

R1 2



We note if

1)2(1exp)(

minus

⎥⎥⎦

⎤

⎢⎢⎣

⎡+

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛ minusminus=

kTEE

Ef F (a)


then

1)1(1exp)(

minus

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡+⎟

⎟

⎠

⎞

⎜⎜

⎝

⎛ minusminus=Δ+

kTEE

Ef F


then

[ ] 3

321

)2())(1)((v2



[ ] 3

312

)2())(1)((v2




[ ] 3

3

)2()()(v2


Since

x

x kE

partpart= 1v

We write

dEdkxx1v =


[ ] ||2

3 )()()2(



mk

EW2

2||

2minus= (b)


)(D1 WTT =

744 Appendices


⎥⎥⎦

⎤

⎢⎢⎣


1



[ ] )( )()()2(

2||


π (d)






ce EE

mED minus⎟

⎟⎠

⎞⎜⎜⎝

⎛=

lowast 23

222

21)(π

1])exp[(

1)(+minus

=kTE

Efμ


⎟⎠⎞

⎜⎝⎛ minus=⎟

⎠⎞


kTE

kTEEf μμ expexp)(

ce

EmkE += lowast2

22

intinfin

lowast⎟⎠⎞


⎟⎠

⎞⎜⎜⎝

⎛=

cE ce dE

kTEEE

mn μ

πexp

22

123

22



ce

EmkE += lowast2

22

kdkm

dEe

22

2

lowast=

lowast

=minuse

cm

kEE2

intinfin

lowast

lowast

lowast

lowast

⎟⎟⎠

⎞⎜⎜⎝

⎛ +minus⎟

⎠⎞

⎜⎝⎛

⎟⎟⎠

⎞⎜⎜⎝

⎛=

cEe

ce

e

e kdkmkT

EmkkTm

kmn 2

2)2(

expexp2

22

1 22223

22μ

π

intinfin

lowastlowastlowast

lowast

⎟⎟⎠

⎞⎜⎜⎝

⎛minus⎟

⎠

⎞⎜⎝

⎛ minus⎟⎟⎠

⎞⎜⎜⎝

⎛=

cEee

c

e

e dkkkTm

kmkT

E

m

mn 2

22223

22 2expexp

2

22

1 μπ

intinfin

lowastlowast

lowast

⎟⎟⎠

⎞⎜⎜⎝

⎛minus⎟

⎠

⎞⎜⎝

⎛ minus⎟⎟⎠

⎞⎜⎜⎝

⎛=

cEe

c

e

e dkkkTm

kkT

Em

mn 2

22

23

323

22 2expexp

)2(22

21 μπ


pp

ndxpxx nn π

)2(2)12()exp(0


where

kTm

pne

2

2 1 ==

So

( )kTmkTm

kTE

m

mn

ee

c

e

e

21

223

323

22

2222

)1(exp2

222

1 πμπ

⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎠

⎞⎜⎝

⎛ minus⎟⎟⎠

⎞⎜⎜⎝

⎛=

lowast

lowast

⎟⎠

⎞⎜⎝

⎛ minus⎟⎟⎠

⎞⎜⎜⎝

⎛=

lowast

kTEkTm

n ce μπ

exp2

223

2



746 Appendices



1 ( ) 1 ( )kdE E

d⎛ ⎞= equiv⎜ ⎟⎝ ⎠

kv kk

nabla


1 ( )

( )

k

k

dE Edt

d Edt

= sdot = sdot

= sdot

F v F k

k k

nabla

nabla

thus

( ) ( )k kdE Edt



ddt

= kF


1 ( ) ( )k kdE Edt


but also



( ) ( )k kdE Edt



( ) dqdt











ln T

ln μ

low T high T



phonon scattering

748 Appendices


if

if

212

121

21

21total

μμμμμμ

μμμμμ

gtgtconggtgtcong

+=




rr

rV λminusprop e1)(



2 1 2 1( )H V r=k k k k



2 1



int minuspropπ

θθθθστ 0

sin)cos1)((v1 dm



Finally we find

23Tm

m prop= lowastτ

μ








cEδ δprop sdotk r



23minus=prop Tμ


See Fig 618


Now


⎟⎠

⎞⎜⎜⎝

⎛= minus )(

23

2 e2

2 μ

π


⎟⎠

⎞⎜⎜⎝

⎛= minus )(

23

2 e2

2 μ

π

750 Appendices


⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎟⎟⎠

⎞⎜⎜⎝

⎛=minus

23222

lnkTm

NkTE

e

dCn

πμ

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎟⎟⎠

⎞⎜⎜⎝

⎛=minus

23222

lnkTm

NkTE

h

aVp

πμ

Therefore

⎥⎥⎦

⎤

⎢⎢⎣


32

)()()2(

22ln)(

kTmmNN

kTEeEEhe

dagpnCnVp

πϕϕ

Using

kTEg

gkTE

e

1ln minus=

kTEhei

gmmkTn 4323

2 e)(2

2 minus⎟⎟⎠

⎞⎜⎜⎝

⎛=

π

we find

⎟⎟⎠

⎞⎜⎜⎝

⎛=minus 2ln)(

i

dapn

nNN

kTe ϕϕ QED



2 21

12Surface

( )( )

4d

rπΦ = int

M r Sr

sdot

= minus ΦH nabla

12 2 21 2

12

ˆ ( )( )

4d

rπ= int

r M r SH r

sdot

θr

dS2

r 12

z

M

r1 = 0r12 = ndashr2


Using the figure

intminus=π

πθθπθθ

0 24 sin2 cos cos

rrdrMH z

31and

3=minus= DMH z




)eTr( HZ βminus=


1 1 1( ) ln

Tr( e ) Tr( e )

zz z z

z z

ZF ZB B Z B

Z Z

β β

μ ββ β β

βμ μβ



as expected



)Tr(2

)1Tr()eTr( 22

HZ H ββ +== minus


)Tr(

3)Tr(

3

)Tr(31)Tr()Tr(

2222

222

222222

JBμgJBBμg

JδBBμgJJBBμgH

Bμ μμ

B


==

==

sum

sumsum

)12()1Tr( += j


⎥⎥⎦

⎤

⎢⎢⎣

⎡+++cong )1(

3211)12( 2

222 jjBgjZ Bμβ

752 Appendices



⎥⎦⎤

⎢⎣⎡ +

partpart= )1(

611 2222 jjBg

B Bz

z μββ

μ

BB

BB z

z=

partpart


kT

jjgNNB

Bz

zzz

)1(3






kkS sdot


kkS sdot



i i( ) e ej i j

i jid Jdt


k k kk k kS S S



i jj i jd Jdt


k k kk k kS S S


i( ) i( )i( ) e ( )e ej ij i

d Jdt


k k kk R k kS R S S


kRsdot


so

( )

( )

( )

kd Jdt

J

J




primeprime= minus

prime=primeprime=

sum

sumsum

k kk k kk k

k k kk

k k kk

S k k S S

k S S

k S S

times

times

times

which is (7155) QED




ϕϕ ==


2

2

(cos cos sin sin )

cos( ( ))

x x y yR R R R R R

R R R R

S S S S

S

S q R R

ϕ ϕ ϕ ϕ

prime prime prime

prime prime

= +

= +

prime= minus

S Ssdot

so that


)cos()(2

))(cos()(2

i2

2

2

=minus=

minus=


sumsum

sum

sum prime

RRqR

R

RR

qRRJRJNS

qRRJNS

RRqRRJSH


qR minus== sum Then

)(2

2qJNSH minus=


)2cos(21)cos(2

21 )2cos(2)cos(2

)ee()ee()(

1

1221

i2i22

ii1

⎥⎦⎤

⎢⎣⎡ minus=

minus=+=

+++= minusminus

qaqaJ

JJqaJqaJ

JJqJ qaqaqaqa

754 Appendices

For a maximum

[ ]0

)cos()sin(2)sin(2

)2sin()2(21)sin(2)(

1

1

=minusminus=

⎥⎦⎤


partpart

qaqaqaaJ

qaaqaaJqqJ


angle turn 60

21)cos(

1)cos(2

deg=

=

=

qa

qa

qa


[ ]

max03

212

212

)2cos(2)cos(2)(

21

21

602

12

2

rArrltminus=

⎥⎦⎤

⎢⎣⎡ +minus=

minusminus=part

partdeg=

aJ

aJ

qaqaaJq

qJqa



=minussj mSS

12

112

212

212

21

21

21

+⎟⎠⎞

⎜⎝⎛ minus+=

⎟⎠⎞

⎜⎝⎛ minus=

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛minus=

+

++minus

nSnnS

nSnaS

nSaa

aSnS

j

jjjj


so



or

1)1)((

12

2)1(221

minus+minus+=

minus⎟⎠

⎞⎜⎝


sss

ss

ssj

mSmSmS

mSS

mSSmSSmSS

QED


We have

i1 e jjjb a

N= sum k R

ksdot

and


Nminus= sum k R

ksdot

where

0][

0][

][daggerdagger

dagger

=

=

=

ji

ji

ljjj

aa

aa

aa δ

Thus

i( )dagger dagger

i( )

1[ ] e [ ]

1 e

1

j j

i j

j

k jj jk j

j

k k k k

b b a aN

N

NN

δ

δ δ

prime

prime


primeminus minus

prime prime

=

=

= =

sum

sum

k R k R

k k R

sdot sdot

sdot

Similarly

i( )

0


Nprimeprime+



0


Nprimeprimeminus +


756 Appendices






Therefore

712

)state F(3max

21221

=+==

=+=

sumL

Lm

SS

l

less than half full

25


So stateF 25

2

Pm3+


426

)state I(123max5122


sumSLJ

mLSS

l

So stateI4

5

Eu3+ six electrons


033


7123216


=+=times=

sumSLJ

mL

SS

l

So stateF0

7




05 3

3[( ) ] ( )( )4μπ



B

r

rprime

m


4 have ( ) 0

28 have ( ) 2 22

a mam

primeminus sdot =

primeminus sdot = =

r r m

r r m


[ ]2 2

05 3

3 ( )4

Uμπ


r r m mm Br r r r

⎟⎟⎠

⎞⎜⎜⎝

⎛minus= minus

5

252

3

2327

total2)2)(3)(8(2)12(10

ama

amU

( ) 022424)10( 3

27 =minus= minus

amU


758 Appendices


2 2 2

4 have ( ) 0

2 have [( ) ] a m

primeminus sdot =

primeminus sdot =

r r m

r r m


066)10( 3

2

3

27

total =⎟⎟⎠

⎞⎜⎜⎝

⎛minus= minus

am

amU


3

27


)JT10279)(8(8

m10524

10

minus

minus

times=cong

timescong

Bm

a

μ


)(10381

101010

1012510)10(

23

2328

51

30

447

total

congtimescong

=congtimes

cong

minus

minusminus

minus

minus

minusminus

c

cT

T

U





The free energy is

ZkTF lnminus=


BFMpartpartminus=




)(0 qpHH rarr




qpqAp

eeAppqq

i

i

iiii

ii

Π=

gtΠequiv+rarrrarr

0HH

Now

)()(e

e

11)(

11)(


=

int

intΠminus

minus

qpqdddqdq

dpdpdqdqZ

NNq

NNqp

helliphellip

helliphellip

H

H

β

β


0 =Πpartpart

=Πpartpart

=partpart

j

iij

j

iij

j

i qpqq δδ



⎟⎟⎠

⎞⎜⎜⎝

⎛=

Πpartpart

+Π= etc1i

iiii

peAp

760 Appendices

Thus schematically

1

0

100010001

)()( =

Πpartpart

=

Πpartpartpartpart

=

Πpartpart

Πpartpart

partpart

partpart

=Πpart

part

i

i

j

i

i

i

j

i

j

i

j

i

j

ip

pqp

pqqp

qq

qqp 15



0=partpartminus=

BFM






15

1

100

01

100

01

0

=sdot=⎟⎟⎠

⎞⎜⎜⎝

⎛

partpart

⎟⎟⎠

⎞⎜⎜⎝

⎛

Πpartpart

=

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛partpart

partpart

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

Πpartpart=

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛partpart

partpart

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

Πpartpart=

Πpartpartpartpart

partpart

11j

i

j

i

i

i

j

i

j

ii

i

j

i

j

i

j

i

i

i

j

i

qqp

qp

qq

pqp

qq

ppqp

qq





Subtracting [ ]int



intΔminus

ΔΔminusminus

minus

Δminus= 2

1 22

221

221)(

)0()0(eV dEEeV

EeV

E

ENNCI

Checks 11








0

000


⎠

⎞⎜⎜⎝

⎛minusprime==Δ


0

0

0

3

0

0

2

02

3

31)11(

2

3cos

2

)(coscos2

ε

ε

θε

θθε

π

π

P

P

P

dPEE b

=

minusminusminus=

minus=

minus=equiv int

762 Appendices



θ

p

E z


int

int= π

θ

πθ

θθπ

θθπθ

0

cos

0

cos

sin2e

sin2ecos

d

dpp

kTpE

kTpE

z


int

int

minus

minus= c

c

x

c

c

x

zdx

dxx

cpp

e

e


⎟⎠⎞

⎜⎝⎛ minus=

ccppz

1)coth(


kTp

Ep

kTEpp z

z 3

3

22=== α



The Hamiltonian is

eEzr

qm


0

22

2

2H

H



)1(100 pzN += ψψ


BarBa 23


ψ


2

2

mqaB =

Requiring

12 =int τψ d





Now using

4

3 0100 0 100 2

0

12 (4 )

med Eψ ψπε

= = minusint rH





are also zero so

)(2 202 eEpaE

NBminus=

H



764 Appendices

Requiring

0=part

partpH

we find

0EeEp minus=


230



302

216 Ba

Eπεα =

part

partminus=

H



061

41)(

21 42


For E = 0

0)( 420 =++minus=


PGE δγβ (2)


20


[ ] 0)(61

41)(

21 2

02


0121)(

31 2


20 4 scc PTT

βγminus= (3)

(b) From (2) 0)( 42


20 4




04


δγ

432 minus=scP

βδγ 2

0 163+= TTc











766 Appendices





φ1 (say 45deg)

Polarizer

lightSample

Analyzerz

zφ2

E

z

yx






iminus= xx xpxp



If H = px2 m +V(x)

[ ] 0)( =xxV So

[ ] xpm

x i minus=H


[ ] iˆ ˆm




2 2ˆ ˆ

2ij ijj ji j i j

f b ωsdot + sdot

=sum sume r e r




jxi

i=minus xpxp xx

xxxppx xx i2 =minus

iminus= xx xpxp


768 Appendices


jj jω=H

jpm


jpm


jpm



1ˆ ˆi ij

i j i jmω

sdot = sdote r e p


ij ji


ωω ω



jiij ωω minus=



Summing over j

ˆ ˆ2iijjbf i im



m

bm

bfj ij 2)i(

i2minus=minus=sum




eEB

NeG +minus

=+ v13

1vvel

2

0εμ



0evv minus= Then

eEB

NeG =minus

minus+minus v13

1vvel

2

0

2ε

μω


ion0

ion 31 αε

NB =

and

Ge

Ee 2

locion

v ==α

at ω = 0

GeNB

2

0ion 3

1)0(ε

=


eEB

GBG =

minusminus+minus v

1)0(

vvel

ion2μω

eEB

BG =⎟⎟⎠

⎞⎜⎜⎝

⎛minus

minus+minus v1

)0(1v

el

ion2μ

μμω

Define

⎟⎟⎠

⎞⎜⎜⎝

⎛minus

minus=el

ion21

)0(1

BBG

T μω



For the local field

el00

loc 1v

31

3 BNeEPEEminus

+=+=εε

Substituting

)(

v 22 ωωμ minus=

T

eE

770 Appendices

Then

)(13

122el0


+=T

eEB

NeEE

)(13

11 22el0

loc


T

eB

NeE

E

)(

11

)0(1 22el

ionloc

ωωμ minusminus+=

TBGB

EE

)()1(

)0()()1(22

el

ion22

loc

ωωμωωμ


=T

Tel

BGBB

EE

Inverting

)0()()1(

)()1(

ion22

22el

loc GBBB

EE

Tel

T


=ωωμ

ωωμ

))(1()0(1

1

22el

ion

ωωμ minusminus+

=

T

locB

BGEE

Define

))(1(

)0(22

el

ion

ωωμ minusminus=

TBBGF

Then

FE

E+

=1

1

loc


loc0

ion0

ionv

31

31)(

EeNNB

εα

εω ==

Substitute in v

)(3

1)( 22loc0ion

ωωμεω

minus=

T

eEE

eNB

)(

)0()( 22

ion

locion

ωωμω

minus=

T

GBE

EB

)1(1

1)( elion BFF

B minus+

=ω


Given

[ ] )1()(1)(3)()(

elionel

ion

0 BBBB


ωω

εεωε


)1()1(

11

)1(1

3)()(

elelel

el

00 BBF

FB

BF

F

minus⎥⎦⎤

⎢⎣⎡ minus

+minusminus

minus++infin=

εε

εωε

)1(

11

13)()(

el00 B

FF

FF

minus⎟⎠⎞

⎜⎝⎛

+minus

++infin=εε

εωε

)1(

3)()(

el00 BF

minus+infin=

εε

εωε

Substitute for F

))(1(

)0()1(

3)()(22

el

ion

el00 ωωμεε

εωε


TBBG

B

Define c

2el

ion

)1()0(3

BBGcminus

=μ

Then

)(

)()(2200 ωωε

εεωε

minus+infin=

T

c

Rewriting

202)()(

ωεωε

εω =infinminus

minusc

T


2

2

)0()(

L

T

ωω

εε =infin

)()0( 2

2infin= ε

ωωε

T

L

772 Appendices


0)()0(

02 =infinminus

minusεε

εω cT

0)()(2

202 =


εεωω

εω

T

LT

c

0)()( 22

202 =

infinminusminus

εωωωεωTL

TT

c

0)(

)(2

0222 =infin


TLTc

202)( LT

c ωεεω =infin

+



ψψ Em

=nablaminus 22

2

For each coordinate

0222

2=+minus i

i

i

mEdxd ψψ


⎟⎟⎠

⎞⎜⎜⎝

⎛+⎟

⎟⎠

⎞⎜⎜⎝

⎛= i

ii

ii xmEBxmEA 22

2cos2sinψ


2

222

2 2or2

manEnamE i

iii ππ ==

)(2

23

22

212

22nnn

maET ++= π





2

22

2

22

23

2)111114(



2

22

23

mahf π=


constant432 ==

mfa π



0

22

2

2

2

2

22

4where1

2 πεφeq

rq

rrrm=minus⎟

⎟⎠

⎞⎜⎜⎝

⎛

partpart+

partpart+

partpartminus=H




εεε RRYEmqEmq

r ==== )(2

2

4

2

2ρ


ψψρ

ψφρρρρ

ε2

4

2

22

2

2

22

2

4

422

2)(11

22mqmqqqm

m=minus⎟

⎟⎠

⎞⎜⎜⎝

⎛

partpart+

partpart+

partpartminus

774 Appendices

Canceling we obtain

ψρψψ

φρρρρε

2111

21

2

2

22

2=minus⎟

⎟⎠

⎞⎜⎜⎝

⎛

partpart+

partpart+

partpartminus




φφ mie)( =Φ



⎞⎜⎜⎝

⎛minus+minus

21

21 2

2

2 ερρρρρ

mdd

dd

We let

2

and1 2

2xαρ

α=minus=ε


Ψ=Ψ+⎟⎟⎠

⎞⎜⎜⎝


241

21 22

2

2 xxm

dxd

xdxd


21122 +=+= nn αα


2212

21 )(

or)(

1+

minus=+

minus=n

REn

ε


21)1(

21 minus+=+ nn





)()(2 2

22zEz

dzd

mψψ =minus


0)(2)(22

2=+ zmE

dzzd ψψ

⎟⎟⎠

⎞⎜⎜⎝

⎛+⎟

⎟⎠

⎞⎜⎜⎝

⎛= zmEBzmEAz 22

2cos2sin)(ψ



2

222

2mwpE p

π=





12

2

0minusminus=

xeF

πε


x

ex

edxx

eFdxdVxxxx

0

2

0

2

20

2

161




x

eExV0

2

0 16)(

πεminus=

776 Appendices


x

exeEExV0

2

10 16)(

πεminusminus=


01

=⎟⎠⎞

⎜⎝⎛

=xxdxdV

which gives

10

1 16 Eex

πε=

or

0

13

01 21)(

πεEeExV minus=

Thus

0

13

21

πεEe=Δ



a

V(r)

ndashV(0)

E (bound state)

r




20 )(2 EVm +

=α

22mEminus=β



This implies

2


Also

E

EV

aa

minus=

+=

+= 0

22

2

22

)(cot11)(sin

βαα

αα


2πα gta

to estimate a

eV)1(22)(22

22

0 mEVma ππ

απ gt

+=cong

nm931)1061)(1019(2

)10636)(143(1931

34

min congtimestimes

timescongminusminus

minusa




778 Appendices

Energy Gap

Ec(z)

Eg1 Eg2

Valence Band

Conduction Band

d

z

Ev(z)


EFFzEzmyxm

c =⎥⎥⎦

⎤

⎢⎢⎣

⎡+

partpartminus⎟

⎟⎠

⎞⎜⎜⎝

⎛

partpart+


22 2

22

2

2

2

22




32||

2

322

2)(

2)(

2Ek

mEkk



ψψ 32

22)(

2EzE

dzd

mc =⎟

⎟⎠

⎞⎜⎜⎝

⎛+minus lowast



sum=

minus=N

ni

ndiq ndz

Nz

1

i )(e1)( χψ


0 we find for E3




2

2 2 0 0( ) cos












)(22

||

222

2

222

EE

kkmmw

nE

n

yx

+equiv

++= π

)(22

222

2||

2

|| yx kkm

km

E +==




ndEm

dkdkkkd

2

2||||||

2

2

2

π

ππ

=

==

780 Appendices


)(

)(2)2(

2)(

2

0||||22

sum

sum int

minus=

minusminus=infin

n n

n n

EEm

dEEEEmED

θπ

δππ



23)(π

mED = (1)

In 3D

E

dEmdkmkE 2

22 221

2==

2

212)4(

)2(2

)unit volper (4)2(

2)(

223

23D3

EdEmmE

dkkdEED

ππ

ππ

=

=


32

22

1)(223

22D3

E

wmmED π

π ⎟⎟⎠

⎞⎜⎜⎝

⎛=

where

2

222

32

3mw

EE π==

Per unit area

222D3332

21)(

ππ

πmw

wmwED == (2)

which checks (1)


d)

D3D(E)w

2hπm

23hπm

E1 E2 E3




1001

512

512

51

2

51

2~

21

21

22

11

I

NN

NN

NN

NN

AA

=⎟⎠⎞

⎜⎝⎛=

⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜

⎝

⎛minus+

⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜

⎝

⎛

minus

+=


Bibliography

Chapter 1

















784 Bibliography










Chapter 2









Chapter 2 785





















786 Bibliography

Chapter 3





















Chapter 3 787



















788 Bibliography

Chapter 4





















Chapter 5 789














Chapter 5







USA 92 6663-6667 (1995)

790 Bibliography






















Chapter 6 791



Francis London 1963

Chapter 6
















1994

792 Bibliography






















1996

Chapter 7 793

Chapter 7



















794 Bibliography



















Chapter 8 795










(2004)

Chapter 8






796 Bibliography






















Chapter 9 797





Chapter 9

















798 Bibliography




Chapter 10















Chapter 11 799






Chapter 11















800 Bibliography


















Chapter 12



Chapter 12 801























802 Bibliography





















Chapter 12 803

















804 Bibliography

Appendices





















Subject References


Mechanics




Electricity



Optics



Thermodynamics






Critical Phenomena


806 Bibliography

Crystal Growth


Modern Physics



Quantum Mechanics





Math Physics


Solid State







Condensed Matter






Problems




Applied Physics





Rocks




808 Bibliography


The Internet



Periodic Table


Further Reading












Further Reading 809
















810 Bibliography

















Further Reading 811


















812 Bibliography




Index

A













B


814 Index










C




Index 815





















816 Index




316 317

D









594 640






E





Index 817

















503 566










F



818 Index


493 538 Fermi function 160 248 298 303



373 385 399 407 422 426 456 486 513







G

GaAs 242 295 297 324 326 328 329 337 347 563 564 579 618 628 629 637




Index 819

Group velocity 55 306 532 562 627


H











I






820 Index


J


K



L









529 570

Index 821




M













N


531 600 NaCl 6 9 28 579 600

822 Index









O








P





Index 823















824 Index


Q




R












S



Index 825













826 Index






T



361 418











475 509

Index 827

U




V



Vortex region 466

W


X


Y

YBa2Cu3O7 506 508

Z


Appendices

A Units










G1 Bose Particles

G2 Fermi Particles


H1 Propagators

H2 Green Functions

H3 Feynman Diagrams

H4 Definitions




J Folk Theorems



M Problem Solutions














































































Bibliography

Mechanics

Electricity

Optics

Thermodynamics


Critical Phenomena

Crystal Growth

Modern Physics

Quantum Mechanics

Math Physics

Solid State

Condensed Matter


Problems


Applied Physics

Rocks


The Internet

Periodic Table

Index




ΩAVA = (B6)


IAAAA == ~~ (B7)


Ω=VAA~ (B8)



aAE ~= (B9a)

or

AEa = (B9b)


)~()~(

)~()~(

~~~~

2

sumsum

sumsumsum

==

==

===

kj jkiijj jjj

kj kjkkjjk kjkj

jiij

EEEE

EEEEEEVAEAEVaaaa

δωΩ

δΩΩΩυ



032

21det 2

2=⎟

⎟⎠

⎞⎜⎜⎝

⎛

minusminus

ωω (B10)


5221 +=ω (B11a)


656 Appendices


51

2 21

21 +

== aaωω (B12a)

and for

51

2 21

22 minus

== aaωω (B12b)


⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜

⎝

⎛

minus

+=2

2

11

51

2

51

2~

NN

NN

A (B13)

where

21

21

1 1)15(

4)(⎥⎥⎦

⎤

⎢⎢⎣

⎡+

+=minusN (B14a)

and

21

21

2 1)15(

4)(⎥⎥⎦

⎤

⎢⎢⎣

⎡+

minus=minusN (B14b)


⎟⎟⎠

⎞⎜⎜⎝

⎛

⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜

⎝

⎛

minus

+=⎟⎟⎠

⎞⎜⎜⎝

⎛=

2

1

22

11

2

1

51

2

51

2

aa

NN

NN

EE

E (B15)

Problems



⎟⎟⎠

⎞⎜⎜⎝

⎛

minus+=⎟⎟

⎠

⎞⎜⎜⎝

⎛

520052

3221~ AA







0)()(d

)(d2

2=+ xxU

xx ψψ (C1)


)()( xUnaxU =+ (C2)








0d

d =x

W (C6)

where

21


=W (C7)


658 Appendices



0)()(d

)(d2

2=++++ axaxU

xax ψψ


0)()(d

)(d2

2=+++ axxU

xax ψψ (C8)


)()()( 211 xBxAax ψψψ +=+ (C9)

)()()( 212 xDxCax ψψψ +=+ (C10)


21

21

21

21

2121

2121ψψψψ

ψψψψ

ψψψψψψψψ

primeprime=

primeprime=


DBCA

DCBADCBA

or

1=DBCA

or

1=minus BCAD (C11)


)()( xΔax ψψ =+ (C12)


)()()( 21 xxx βψαψψ += (C13a)

or

)()()( 21 axaxax +++=+ βψαψψ (C13b)


)()()()()()()(

2121

xxxDBxCAax


+=+++=+ (C14)



αΔβα =+ CA

and

βΔβα =+ DB


0=minus

minusΔ

ΔDB

CA (C15)






DA +=+ minus++

1)(ΔΔ


1)(ΔΔ (C17)

and

DA +=+ minus+ ΔΔ



If we set

be=+Δ (C19)

and



)()( 11


and

)()( 12


660 Appendices





where

)()( xuaxu =+ (C25)



Let

)()( 11 NN xxExx ψψ =H (C26)



0][ =lTH (C27)

and






lkmiddotl

ie=t (C31)

Further




so that

12 =lt (C32)










)(02)(0

22

21 Gnnn G =+++


5 In general sum= j

kj

klij

kil AT ψψ

where the Ail




kklkl AT ψψ =


662 Appendices




ntctx nin

i sum= )()(ψ


nmNi

im

imnm cctctc

Nlowast





)( ATrA ρequiv


][i Ht

ρρ =partpartminus


⎟⎠⎞

⎜⎝⎛ minus=

kTHFexpρ


)][exp()]exp([exp

kTHTrkTHHTr

kTHFHTrHU

minusminus=⎥

⎦

⎤⎢⎣

⎡⎟⎠⎞

⎜⎝⎛ minus==








Let

)()( 0 tVt += HH (E1)

lEl l00 =H (E2)


00lk

lkEE minus=ω (E4)




664 Appendices


200

2 2)()(

)(

)cos(1lim ωδπ

ω

ω tfEEttfi

if

if

t=minus=

minusinfinrarr

(E6)












⎟⎟⎠

⎞⎜⎜⎝

⎛=

00σ

σα (F2)




⎟⎟⎠

⎞⎜⎜⎝

⎛minus

=I

I0

0β (F3)



12

)(2

0

20 ltlt

minusminusequiv

cmVcmEε (F4)


20cmEE +prime= (F5)

and

⎟⎟⎠

⎞⎜⎜⎝

⎛=

φχ

ψ (F6)




⎤

⎢⎢⎣


4482 220

2

220

2

230

4

0

2pσ V

cmV

cmV

cmp

mpE (F7)


])[(4 22

0


cm (F8)


666 Appendices


SL middot dd1

2 220

2

rV

rcm (F9)





G1 Bose Particles




][

0][

0][

dagger

daggerdagger

ijji

ji

ijjiji

bb

bb

bbbbbb

δ=

=

=minusequiv

9 See Mattuck [A14]



daggeri i in b b=

and

dagger1 iii bbn =+


11

1

11dagger

11



++=

minus=

iiii

iiii

nnnnnb

nnnnnb



1

1dagger

iiii

iiii

nnbn

nnbn

=minus

=minus



dagger)1()1(






)()(


sumlowast=

=

α αα

α αα

ψψ

bu

bu

rr

rr


668 Appendices

G2 Fermi Particles




0

0

dagger

daggerdagger

ijji

ji

ijjiji

aa

aa

aaaaaa

δ=

=

=+equiv







iii nan




0











)2()1()2()1(

210daggerdagger

ββ

ααβα uu

uuaa harr











670 Appendices






H1 Propagators




H2 Green Functions


δεω

ωi

1)(0 +minus=+

kG k (H1)


H3 Feynman Diagrams


H4 Definitions






672 Appendices




















dagger )2()1()21()2()1(21)1()1()1(H (H3)



31



0

daggerdagger

=+=+

ijji

ijijjiaaaaaaaa δ



δεω

ωi


=sum

+

l lklkk VG k (H6)





674 Appendices



112223 d)21()()()()(d rrrrrr φφφφ (H8)




δεω

ωi)(


=sum

+













klk

Gδωεω

ωi)(


=sum k

k (H11)





676 Appendices




fcc (110) (011) (101)

2 2 2

bcc (11 1) ( 111) (1 11)2 2 2

a a a




1 2 31 ( )Ω






lkh

lkh dG π2


Δ =k G (See 152)











G (See 141)





⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎟⎟⎠

⎞⎜⎜⎝

⎛+⎟

⎟⎠

⎞⎜⎜⎝

⎛minus=

126

4)(ijij

ij rrrU σσε


1e

1 minus

= kTn ω (See 277 2215)


222 akmks=ω (See 248 288 290)






D

DDωωωωωω

gt=leprop

0)( 2

(See 2224)

3126v ⎟

⎟⎠

⎞⎜⎜⎝

⎛==

VN

kk DDD

πθθω (See 2228)


678 Appendices


intminus

⎟⎠⎞

⎜⎝⎛= T

x

x

VdxxTNkC

0 2

43

)1e(e9 θ

θ (See 2229)


3( )

( )( )(2 ) const

dAVD ω

ωω

ωπ= int

k knabla (See 3256)




)e(e 20


2( ) TWθ⎛ ⎞prop Δ ⎜ ⎟⎝ ⎠

k


22

2

9)(

Dm

m mkAT θ=



V

Dln



B

C3

vγα = (See 2237 2238 2250)




ρ11Cv = (See 2270)


ρ44Cv = (See 2270)



3 Electrons


1e

1)( +




( ) ( )u u= +k kr r R (See 3129 C25)


EmVEg23

222

2)( ⎟⎟

⎠

⎞⎜⎜⎝

⎛=π

(See 3164)


32

22

32

)0( ⎟⎠⎞

⎜⎝⎛==

VN



m

ne τσ2

= (See 3214)




RH (See 6102 1223)

680 Appendices


LTK =σ

(See 3215)

22

3⎟⎠⎞

⎜⎝⎛=

ekL π (See 3215)


0

00 near band edge

E E U

U E E

lowast

+

minus=

minusk G

G k G (See 3230)



k r r R (See 3244)


k (See 3255)




i

Bloch Fns


kkr R r (See 3332)



cgs)in or (mceB

meB

c =ω (See 3196)




22

2 2

0 ( )

( )

( ) ( ) ( )2

1( ) ( ) ( )4

el

ioni i

j i i ij i

V

Vm

e d E

ψ ψ

ψ ψ ψπε ne

minus nabla +



r

r r r

r r r rr r




22

2

0 ( )|| spin

( ) ( ) ( ) ( ) ( )2

1 ( ) ( ) ( ) ( )4

ion eli i i

j i j i ij i

V Vme d E

ψ ψ ψ

ψ ψ ψ ψπε

lowast

ne

minus + +


r r r r r

r r r r rr r

nabla



1 32

2 3( ) ( ) ( ) ( ) ( ) ( ) ( )2


mψ ψ ψ ψ

π⎛ ⎞⎡ ⎤minus + + minus =⎜ ⎟⎣ ⎦ ⎝ ⎠

r r r r r r rnabla



2

2

1 ( ) ( )( ) 1( ) ( )k

e f fV E E iq σ

ε ωω ηε


k k qqk q k

(See 9166)

xx

xxxF

kqxxF

qk

qTF

sminus+minus+==+==

11ln

41

21)(

2)(1)0(0

2

2

2ε


F

srk

F Eemk

rkq s

εεπφ

23e

41 2

02 ==ltlt minus



collisions

f ff ft t


Fv nabla nabla


0RHS of BDE( )

f fτminus

= minusk

(See 4149 4145)

682 Appendices


11 123 v( ) ( ) ( )4

edj e f L E L T

eμ

π⎛ ⎞= minus = + + minus⎜ ⎟⎝ ⎠


21 223 ( )v( ) ( ) ( )4

Qdj E f L E L T

eμ

π⎛ ⎞= minus = + + minus⎜ ⎟⎝ ⎠




0ej TE

σ = =nabla

0Qe

jj

Tκ = =

nabla


0eEQ jT

= =nabla

0Q

e

jT

j

QT

Π = =

Π =

nabla


constant131 conganc



⎟⎟⎠

⎞⎜⎜⎝

⎛minus

VED F )(1exp




heG =




D

D

TT

TT

θρ

θρ

ltlt

gtgt5~

~ (See 4146)





)SI(21

0Ae

Bπ=⎟

⎠⎞

⎜⎝⎛Δ (See 534)



⎟⎟⎠

⎞⎜⎜⎝

⎛= (cgs) 4 2

0

2

mne

mne

pπ

εω (See 10108)







2innp = (See 613)

⎟⎟⎠

⎞⎜⎜⎝

⎛minus⎟

⎟⎠

⎞⎜⎜⎝

⎛= lowastlowast

kTE

mmkTn ghei 2

exp)(2

2 4323

2π (See 614)

684 Appendices


ddt

= kF (See 644)




2

2

211

kE



eDkT =μ (See 684 685)





⎟⎟

⎠

⎞

⎜⎜

⎝

⎛ ΔΔ== (cgs)

2

222 ne

V

ne

VKZwidth b π

εε (See 6193)


⎥⎦

⎤⎢⎣

⎡minus⎟

⎠⎞

⎜⎝⎛

⎟⎟⎠

⎞⎜⎜⎝

⎛+= 1exp

00 kTep

LD

nLDej b

np

pp

n

n ϕ (See 6241)



23minusprop Tμ


23Tpropμ





⎟⎠⎞

⎜⎝⎛ )(

22cgs

mceB

meB (See 7255 7256)

Lande g factor

⎥⎦

⎤⎢⎣

⎡+

+minus++++=)1(

)1()1()1(211

JJLLSSJJg

(Implied by (76) (710))


)(20 FBp EDμμχ = (See 3181 3201)


magnetonBohr =Bμ


)groundexcited(22

==minus

= gexcEEgexc

VN

gexc

zμχ


⎟⎠⎞


⎤⎢⎣⎡ ++=

Jx

JJxJ

JJxBJ 2

coth21

2)12(coth

212)( (See 716)



4sincos)()2(

xxxxxFRkFJ ijFij



kT

BgJx eff

Bμ= (See 722)

MBBeff λ+=

686 Appendices


)cgs(

32

0MMH

kEk effB

F

μθθ =primegt

prime (See 7222)


⎟⎟⎠

⎞⎜⎜⎝

⎛minusprop

)(1exp

FK EJD

T (See 7300)


h

eGxy

2ν= (See 1247)



KJpropδ (See 7251)





23

23)()0(

TC

TTMM

M prop



)for (1cc

c

c

TTHTTM

TT

TTC

=asympminusasymp

minusasymp

minusasympminus

minus

δβ

γ

α

χ






2

( ) ( ) (cgs)nemc



2

2

4 nemc

L πλ = (See 86)


g

FEv

πξ 2~0 (See 849)



II ξ lt λ 20~

1 πλϕ

cH 20~

2 πξϕ

cH 2

~1

2⎟⎟⎠

⎞⎜⎜⎝

⎛ξλ

c

c

HH



)(20 cgs

ehc=φ (See 845)



⎟⎟⎠

⎞⎜⎜⎝

⎛minuscong

)(1exp141

FDc EVD

T θ (See 8217)


⎟⎟

⎠

⎞

⎜⎜

⎝

⎛

+minus


1exp

451 epep

epDcT

λμλ



688 Appendices


2

20



⎡ ⎤⎛ ⎞⎢ minus + + ⎥ =⎜ ⎟⎝ ⎠⎢ ⎥⎣ ⎦

Anabla (See 83)

2

( ) ( 2 2 )2s eiq qj q e m m






)()0(

2

2

infin=εε

ωω

T

L (See 950)


⎟⎟⎠

⎞⎜⎜⎝

⎛+=+= )SI(

3)guassian(

34



⎟⎟⎠

⎞⎜⎜⎝

⎛=

+minus=

+minus

sumsum )SI(31

2)(1)(

34

21

00

0jjjj NN α

εεεεεαπ

εε (See 930)



direct 0eΔ =k








)cgs(2 222

4

neEE gn ε


)cgs(111

he mm+=

μ (See 10101)


11 Defects


Mollwo Equation







⎟⎟⎠

⎞⎜⎜⎝

⎛==

pGeEp

Bπω 2 (See 129)


)(exp 41 MottT

AB ⎟⎠

⎞⎜⎝





232surface =⎟

⎠⎞


kTWkTemj

π (See 1145)

690 Appendices


3 2

2surface exp BWj AE

E

⎛ ⎞= minus⎜ ⎟⎜ ⎟

⎝ ⎠ (See 1149)


J Folk Theorems


Chapter 1



Chapter 2



Folk Theorems 691

Chapter 3









Chapter 4







Chapter 5


Chapter 6




692 Appendices















Chapter 7






Folk Theorems 693





Chapter 8




Chapter 9



Chapter 10





Chapter 11







Chapter 12



694 Appendices







Gauss law V A A

dV disin





10

e +

infinminus =int n

axn

andxx


π41Y

e21)(

Y)(

00

23

10

0010001

=

⎟⎟⎠

⎞⎜⎜⎝

⎛=

=Ψ

minus Bar

BarR

rR





r θnabla ϕ

221 1 1( ) (sin )

sin sinrA

r A Ar r rr

ϕθθ



2

2

2222

2 sin1sin

sin11

ϕθθθ

θθ partΦpart+⎟

⎠⎞

⎜⎝⎛

partΦpart

partpart+⎟

⎠⎞

⎜⎝⎛

partΦpart

partpart=Φnabla

rrrr

rr2



111


=primeprime

γγγ

nnn

n


n

ennn ⎟⎠⎞




)()(

1))((

)()()(0)(


n

n nn

xxxf

xxxf

xf

bfdxbxxfxx

xxbxdxbx

=

minusprime

=


minus===minus

sum

int

int

δδ

δδ

δδδ

Complex Variables


π


nazfazdzd

naf n

n

n


)1(1)(Res 1

1

⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧

minusminus

= minus

minus

rarr


SI Gaussian

0

t

tρ

parttimes = +part


sdot =sdot =

H J D

BE

DB

nabla

nabla

nablanabla

4 1

1

40

c c t

c t

π

πρ

parttimes = +part


sdot =sdot =

H J D

BE

DB

nabla

nabla

nablanabla

0ε= +D E P

0

1μ

= minusH B M



( )qc

= + timesvF E B

696 Appendices












1956 Transistor


















9 Total for Decade

698 Appendices

























A Geim K Novoselov

2010 Graphene



Brillouin




Bloch






Bardeen



700 Appendices

Others






M Problem Solutions





451388

2412

126


=+minus=M




1

A

2

3

456

7

8

g5 g6

g7

g8


⎟⎠⎞

⎜⎝⎛= 87654321

876543211g

⎟⎠⎞

⎜⎝⎛= 21876543

876543212g

⎟⎠⎞

⎜⎝⎛= 43218765

876543213g

⎟⎠⎞

⎜⎝⎛= 65432187

876543214g

⎟⎠⎞

⎜⎝⎛= 23456781

876543215g

⎟⎠⎞

⎜⎝⎛= 45678123

876543216g

⎟⎠⎞

⎜⎝⎛= 67812345

876543217g

⎟⎠⎞

⎜⎝⎛= 81234567

876543218g

702 Appendices

Then eg

432 6543218787654321 ggg =⎟⎠⎞

⎜⎝⎛=


g1 g2 g3 g4 g5 g6 g7 g8




)(21 kjia minus+= a


)(23 kjia +minus= a

x

y

z






y

z

x

a2

a1

a3

a


a


704 Appendices





2 2 0h kh k h k


a bG

since

1 2 32 ( )h k lπ= + +G b b b

and


Thus

( ) 2( )( ) ( )h k ld h k lh k l h h k l

π= =G aG G

sdot






where

f iΔ = minusk k k



ki

b

b

bα

bα

hellip

x

y

z

kf

0ˆ =sdot ik i

2sin

2sin

e

2sin

2sin

e

e1

e1

e1

e1

ee

2)1(

i2

)1(i

i

i

i

i

10

i10

i

⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜

⎝

⎛

+⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜

⎝

⎛

=

minus

minus+minus

minus=

+=

minusminus

minusminus

minus

minus

minus

minus

minus=

minusminus=

minus sumsum

α

αα

α

α

α

bk

Mbk

bk

bNk

P

fx

fxMbk

fx

fxNbk

bk

Mbk

bk

bNk

Mm

mbkNn

nbk

fxfx

fx

fx

fx

fx

fxfx


)(integer 2

and)(integer 2

)2()1(

pbk


απ (a)

Since

pl

bk

bk

pl

bk

bk

fx

fx

fx

fx αα

== )2(

)1(

)2(

)1(

or1

)2()1(fxfx kk ne



since

22fxfy kkk minus=


706 Appendices


Let

λπ2=K





Define

θsinKB =

)iexp( BbZ =

So


But


)]2sin(ln[

1)]2sin()exp[(ln

0

)2sin(

knbZ

knbZm

knbZZ

m

knb

πε

πε

πεπε

+=

=

=

suminfin=

Thus


0 )2sin()ln( πε


2)2cos(21)2sin(ln

11

ZkbZkbZZ

ZE

+minus+

minus=

ππε






mGBb π2+plusmn=


b




Aring2255=Ka


Aring2255100 == ad

)bcc(6125222255


deg=rArr== 6816225551

2sin θλθ

d

deg=rArr= 04352255

)51(2sin θθ



θλ sin2 100dn =

above as 2)e1( 0435 4510

)51(42

4sin


)51(32

3sin

above as 2)e1( 6816 4510

)51(22

2sin


2sin

i4400

100

i3300

100

i2200

100

i100

100

ffSd

fSd

ffSd

fSd

=+====

=+====

=+====

=+====

minus

minus

minus

minus

π

π

π

π

θλθ

θλθ

θλθ

θλθ

708 Appendices




mm m

γ γ γx1 x2

1γx3

3 1 2 3





02

22

321

2

2

2

=⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜

⎝

⎛


uuu

mm

m


(2)


twice32mγω = (5b)


⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛==

111

31

321

0uuu

Uω (6a)

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛minus==12

1

51

3 mU γω (6b)


and

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛minus==

112

51

3 mU γω (6c)



)(2iexp1

)(2iexp1

)(2iexp1

)1(0

10

10

sum

sum

sum

minusminus= minus

minus=

minus=

⎥⎦⎤

⎢⎣⎡ minus=

⎥⎦⎤


⎥⎦⎤

⎢⎣⎡ minus=

Ns s

Ns s

Ns snm

nmN

sgN

nmN

sgN

mnN

sgN

G

π

π

π


[ ]

⎥⎦⎤

⎢⎣⎡ minus=

minus⎥⎦⎤

⎢⎣⎡ minus=

⎥⎦⎤


⎤⎢⎣⎡ minus=⎥⎦

⎤⎢⎣⎡ minus+

)(2iexp

)(2iexp)(2iexp


nmsN

nmnmsN

nmNN

nmsN

nmNsN

π

ππ

πππ

and (see 2104)

⎟⎠⎞

⎜⎝⎛=⎟

⎠⎞

⎜⎝⎛ +=⎥⎦

⎤⎢⎣⎡ +

Ns

NsNs




Nsg

NG π

But

⎟⎟⎠

⎞⎜⎜⎝

⎛⎥⎦⎤



Nnm

NNggN

ππ

so

mnN

snm GnmN

gG =⎥⎦⎤


0 )(2iexp π

QED

710 Appendices




)2()()(

)()(

)2()()(

)()(

)2()()(

54145515

34454

32423343

12232

15251121

XXXXXXXXm

XXXXXm

XXXXXXXXM

XXXXXm

XXXXXXXXm


minusminusminus=


minusminusminus=


αααβα

βββαβ

ααα


0

210011100020001110012

54321

22

212

2

=

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛

minusΩminusminusΩ

minusΩminusminusΩ

minusΩ

minus

uuuuu

AAAAA

AAδ


⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

Ω=

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛



minusminus

54321

2

54321

210011100

020001110012

uuuuu

uuuuu

AACCC

AA


⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

minusminus

=

1

0

1

a

aU


This requires

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛

ΩminusΩminus

sdotΩΩΩ

=

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

+minus++minus

minus+minus

=

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

minus+minus+minus+

+minus++minus+minus

22

2

2

2

0

31)1(

01)1(

3

211))(1(

)1(112

a

a

aAa

Aaa

aaAcaca

Aaa




2





)1()1()1( dagger

dagger

ϕψψ

ψψϕ

minus=minus=minus=

==

nna

aaa

aaaNaN

n

n

nn

712 Appendices


nKan =minus1 So

112

dagger2

nnnK

naanKnn

=

=minusminus


1or1 minus== nnnan

K


)1()1(

)1(dagger

daggerdagger

daggerdaggerdagger

ηψ

ψ

ψψη

+=+=

+=

==

nna

aaa

aaaNaN

n

n

nn


1

1+

=n

K

so

11dagger ++= nnna


⎟⎠⎞

⎜⎝⎛ +=⎟

⎠⎞

⎜⎝⎛ +=

21

21dagger Naa ωωH



321321


⎜⎝⎛=



321

321

321

ηηη

ηηη

ηηη

partpart

partpart

partpart

partpart

partpart

partpart

partpart

partpart

partpart

=

zzz

yyy

xxx

J

1 1 2 2 3 3η η η= + +r a a a


etc11

xax =partpartη

so we find

1 2 3

1 2 3 1 2 3

1 2 3

( )x x x

y y y

z z z

a a a

J a a a

a a a

= = sdot timesa a a

QED



1 1

2 1

3 1

100

sdot =sdot =sdot =

a ba ba b



a

KΩ

= 1

and

2 31

a

times=

Ωa a

b


714 Appendices


1 1 2 2 3 32 ( )n n n nπ= + +G b b b




31 21 2 3

1 2 32

nn nN N N

π⎛ ⎞

= + +⎜ ⎟⎝ ⎠

q b b b



a) 31 231 2 31 2

31 2

1 2 3

1 1

11

1

22 2 ii i 11 1i

1 2 3 1 2 30 0 0

i2

2i1

1 1 1 1e e e e

1 1 e etc



l

l

n

nn

nn n ll l NN NNN N

l l l

n l

n lN

i i

i i

n

N N N N N N

N

n Nn N

ππ π

π

π

δδ


= = ==

minus=

minus= ne= =

=

=

sum sum sum sum

sum

q R

R

Gq

q GG

G

b) 111

1

1

1

11

21

i(BZ)

1 2 3 1 0

2

21

0

1 1e e etc


1 1 e


l

l

ni lNN

l

i n

i lN

i i

i i

N N N N

N

l Nl N

π

π

π

δ

minussdot

==

minus=

minus= ne= =

=

sum sumq Rq

R


c)

1 2 31

1 1

3

33

1 2 31 2 3

31 2 3 03

fpp 33

( ) 2

of s( )Vol of space

1 ( )(2 ) ( )

( )(2 )

( ) (2 )

n n nn

f fN

qf dq

f d

N N NN N N

f d

Vf d

π

π

π

π

⎛ ⎞⎛ ⎞= +⎜ ⎟⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠

=

=sdot times

Ω=

sum sum

int

int

int

int

q q b

q q

q qb b b

q q

q q

d)

i i3fpp3

fpp

i

1 2 30

e e (with )(2 )

1 e

l l

l

l

a ad q VV

N N N

π

δ

Ω Ω= rarrinfin

=

=

sumint

sum

q R q Rq

q Rq

R

sdot sdot

sdot

e)

1 1 1 2 2 2 3 3 3i( ) i2 [( ) ( ) ( ) ]3 31 1e el l l l l l l luc



3

1 2 3


η η η⎛ ⎞

= = ⎜ ⎟⎝ ⎠

r


etc so

1 2 3

1 2 3 1 2 3

1 2 3




So

1 1 1

1 1 111

1i( ) i2 ( )310

1i2 ( )

1 1 0

1 e e

e i2 ( )

l l l l

a

l l

d d

l l

π η

π η

η

δπ


primeminusprime

=Ω

= =prime minus

int intG G r

ll

rsdot

716 Appendices

f)

i ( ) 3all 3

i ( ) i ( ) i ( )

i( )i( ) i( )

1 e(2 )

1 1 1e e e2 2 2( )

1 e 1 e 1 e2 i( ) 2 i( ) 2 i( )

yx z

x y z

yx yx z

zx y



LL L

d q

dq dq dq

L L L

x x y y z z

π

π π π

π π π

primeminus



minusminus minus

=

rarr infin


int

int int int

q r rq

sdot


( ) ( ) ( )( ) ( ) ( )( )

z

x y z

yx zL L L


x x y y z zπ π π

δ δ δδ









Frequency c

gωω=

Temperature )( k

Ttcω

=


Uucω

=

Entropy NkSs =


NkC

c vv =



⎟⎠⎞

⎜⎝⎛ +⎟⎠⎞

⎜⎝⎛==

Nd

ddpP

c

ωωωωω


)(

Thus

π

ωω2

)()( NadqdqqNdN ==

1

)2)(2cos(1

2)(

22cc

cN

aqaN

ddqNadqN

ωωπω

ωπωπω

minus=

==

So

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

minus⎟⎟⎠

⎞⎜⎜⎝

⎛

minus==

221

11e

1

ckTc d

dpPωωπω

ω ω

or since

tg

kTkT c==

ωωωω

c

then

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

minus⎟⎟⎠

⎞⎜⎜⎝

⎛

minus=

21

11e

1)(g

gtP tgπ (1)



int

intsum

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

minus⎟⎟⎠

⎞⎜⎜⎝

⎛

minus=

minus=

minus= =

c

c

m

dN

dNU

cckT

kTNm kT

m

ω

ω

ω

ωω

ωωωω

ωπ

ωωωω

0 22

01

1

11e

2

)(1e

21e

int⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

minus⎟⎟⎠

⎞⎜⎜⎝

⎛

minus==

c

cc

kTc

cd

NUu

ω

ω ωωωω

ωωπω 0 22 )(

1

11e

2

718 Appendices

or

int⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

minus⎟⎟⎠

⎞⎜⎜⎝

⎛

minus=

1

0 21

11e

2 dgg

gu tgπ (2)

ωωωπω

ω

ω

ω

ω

ω

ω

ω

dNkT

kTC

cckT

kT

m kT

kTm

c

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

minus⎟⎟⎠

⎞⎜⎜⎝

⎛

minus=

minus=

int

sum

2202

2

2

2

2

2v

1)1e(e)(2

)1e(e)(1

int⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

minus⎟⎟⎠

⎞⎜⎜⎝

⎛

minus==

c

cc

kT

kT

c

c dTNkN

NkCc

ω

ω

ωωω

ωω

ωπω

ω

0 222

2

22v

v )(1

1)1e(

e)()(

2

int⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

minus⎟⎟⎠

⎞⎜⎜⎝

⎛

minus=

1

0 22

2

v1

1)1e(

e)(2 dgg

tgc tg

tg

π (3)


kTFj



VT

FS ⎟⎠⎞

partpartminus=

so

( ) sumsum=

minus=

minus

minus+minusminus=

N

m

mkT

N

m

kT

kTkTkS

mm

12

1

e11e1ln ωω

ω

( )

int

int

minusminus+

minusminusminus=

=

minus

c

c

cc

kT

cc

kT

dkTkTNNk

dNkNk

NkSs

ω

ω

ωω

ωωωω

ωπ

ωωωωπ

0 22

2

0 22

)(1

11e

)2(1

)(1

1e1ln21


+minus

minusminus= minus1

0 2

1

0 2

1

11e

2

1

1e1ln2 dgg

tgdgg

s tgtg

ππ (4)



range of ω

ωxxp


P10x1π

1

e

ωx

10 1minus

sdot1

1 ωx( )2minus

sdot=

P100x1π

1

e

ωx

100 1minus

sdot1

1 ωx( )2minus

sdot=

P300x1π

1

e

ωx

300 1minus

sdot1

1 ωx( )2minus

sdot=

0 05 10

200

400

P300

P100

P10

ω

720 Appendices


the range of t

txxr


ux2π

0

1

yy

e

y

tx 1minus

1

1 y2minussdot

⌠⎮⎮⎮⎮⌡

dsdot=

spht x2π

0

1

y

ytx

⎛⎜⎝

⎞⎟⎠

2e

y

txsdot

e

y

tx 1minus

⎛⎜⎝

⎞⎟⎠

2

1

1 y2minussdot

⌠⎮⎮⎮⎮⎮⎮⎮⎮⌡

dsdot=

0 1 2 3 40

1

2

3

4

u

spht

t


0 05 1 15 20

05

1

15

2

u

spht

t


Entropy t2π

minus

0

1

ωln 1 e

ωminus

tminus

⎛⎜⎝

⎞⎟⎠

1

1 ω2

minus

sdot

⌠⎮⎮⎮⎮⌡

dsdot2π

0

1

ω

ω

t

e

ω

t 1minus

1

1 ω2

minus

sdot

⌠⎮⎮⎮⎮⎮⌡

dsdot+=

0 100 200 3000

2

4

6

8

Entropy t

t

722 Appendices




By Problem 29

2 31

a

times=

Ωa a

b


2 3 [(1)(1) (1)( 1)] [(1)(1) ( 1)(1)] [( 1)( 1) (1)(1)]

2

(2 2 ) ( )2

a

a a


= + = +

a a i j k

i j i j

and

2

21 2 3( ) (2)


so Ωa = a2 and

11 ( )a

= +b i j


2

3

1 ( )

1 ( )

a

a

= +

= +

b j k

b i k




1 2 31 1 1( ) ( ) ( )a a a




ky

kxkz


a

a

n m + 1

n mn ndash 1 m

n m ndash 1

n + 1 m

xnm ynm


])()()()[(2

])()()()[(2

21

21

21

21

21

21

21

21

mnmnmnmnmnmnmnmn

mnmnmnmnmnmnmnmn

yyyyyyyyk

xxxxxxxxkU



minus+minus+

minus+minus+


)4( 111

mnmnmnmnmn



)4( 111

mnmnmnmnmn



724 Appendices


)i(1 e maknakt

mnyxcx ++minus= ω

)i(2 e maknakt

mnyxcx ++minus= ω


⎟⎠⎞⎜



⎟⎠⎞⎜






a x

y


⎥⎥⎦

⎤

⎢⎢⎣

⎡+⎟

⎟⎠

⎞⎜⎜⎝


⎟⎠⎞

⎜⎝⎛ +++minus++=

⎟⎠⎞

⎜⎝⎛ minus++=

22

2

2

222222

22222

222

481

2122

21

)(2)(21

)(21

aax

ax

ayxayxaxak

ayxaayxak

ayxakU

to second order

[ ]

2

22222222

21

2222

kx

axaxyxayxaxakU

cong







11 11dagger

1 i22

pp p p

pa P X

ωω

= minus qq q q

q

Now

11 1

1

11dagger

p p

p

p

X m

m

X


minus

= sdot

= sdot

=

sum

sum

X

X

q b q b q bb

b q b q bb

q

e

e






1 1

1

1

1

111

1

1 1

p pp p p

pp p

e X m e e X

m e e X

m X m X

β βα αβ

β βαβ

ββ ααβ δ δ

lowast lowast

lowast

=

=

= =

sum sum

sum sum

sum

q b q b q bbqb q b

b q b q bb q b

bb b bq bb q b

or

1 11

1p ppX e X


b (a)

Now

dagger 11dagger 11

1 i22

pp p p

pa P X

ωω

= + qq q q

q


11dagger 11dagger

11 11

1 i22

1 i22

pp p p

p

pp p

p

a P X

P X

ωω

ωω


= minus

qq q q

q

qq q

q

726 Appendices

so

dagger 11 2i

2p

p p pa a Xω

minusminus = qq q q

11 dagger

i ( )

2p p pp


q

Substituting in (a)

1 dagger

i ( )

2 p p ppp

X e a am ω


b q



0)(

)(])[(

34412

24412122

442

4411

=++

++minus+minus

Kkkcc

KkkccKkckcc

zx

yxx ρω


0)( 122

11 =minus Kkc ρω


0)( 222

44 =minus Kkc ρω

0)( 322

44 =minus Kkc ρω

If

ρ

ων 1111

ck

==


If

ρ

ωνων 4433

22

ckk

====






)1(

)(

])()([)(

))]()(())()([()(

))]()(())()(()(

2

2

12

1112

11112

11112

1

1

NM

PM

dPM

dxuxuPM

dxuxuQxuxuPM

dxuxPuxuxPuMd

P

qp qq

qp qq

pp NNNNpp

pp NNNNpp

=

=

minus=

minus=

minus=

minus=

sum

sum int

sum int

sum int

sum intint

lowast

primelowastprime+


τδ

τ

τ

ττψψ

where Q = Pminus1P




)2()1()21()2()1(21

)2()1()21()2()1(21

)2()1()21()2()1(

)2()1()21()2()1(

)1()1()1(

(||)

(||)

sum

sum

sum

sum

sumsum

minus

+=

minus

+

=

ji ijji

ji jiji

ji ijji

ji jiji

i iik

V

VE

V

V

H

ψψψψ

ψψψψ

ψψψψ

ψψψψ

ψψε



22 2

1 21 10 12

| ( ) |( ) d 0

4jN N

II jeV

rψ


rr

Note that

221| ( ) |N

jj ψ=sum r

728 Appendices


2

22 11

0 12

( )d ( )

4NI

II Ie V

rρ τ


r QED

IonI

Field point

Source point

r1

r2



Let x = kkM

0as2)(21

11ln1limlnlim

2

2

0

22

0

rarr=minus=

minus+minus=

minus+minus

rarrrarr

xxxx

xx

xx

kkkk

kkkk

xM

M

M

Mk

⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢

⎣

⎡


minusminusminus



rarr

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

gtminusminusminus

ltminusminusminus

rarr

⎥⎦⎤

⎢⎣⎡


⎥⎦⎤

⎢⎣⎡



rarr

110)1(

)1(1)1(1

or

110)1(

)1(1)1)](1(1[

2

1)1(1)1ln(

1)1(1)1ln(

2

1)1ln(1)1ln(2ln)1(2

1)1ln(1)1ln()1ln()1)(1(


2

2

22

1

xxx

xx

xxx

xx

xx

x

xxx

xxxxx

xxxxx

xxx

xxxx

xx

xx

x





i1e 0bm


K


The answers are





Thus

i(0) i( )( )e e

1 ( 1)

n l m

n l m

S πminus minus + +

+ +

= +

= + minusK






730 Appendices


i( )( ) i( )( ) i( )(n )1 e e en l m









20

2)(2

2 EVmKmEk

minus==


)ee()(22

ee)(2

)(02

If

ii

KxKx

kxkx

Dxbxb

CBxaxb

Vbax

minus

minus

+=ltltminus

+=ltlt

infinrarr=+gt

ψ

ψ

ψ


)ee()ee(i

)ee(ee

0ee

2i2i2i2i

2i2i2i2i

)2(i)2(i

KbKbkbkb

KbKbkbkb

bakbak

KDCBk

DCB

CB

minusminus

minusminus

+minus+

minus=minus

+=+

=+



⎟⎠⎞

⎜⎝⎛minus=

2coth)tan( Kb

Kkka (a)


)ee()(22

ee)(2

02

ii

KxKx

kxkx

Dxbxb

CBxaxb

bax

minus

minus

+=ltltminus

+=ltlt

=+gt

ψ

ψ

ψ


⎟⎠⎞

⎜⎝⎛minus=

2tanh)tan( Kb

Kkka (b)


)e21(

2tanh

)e21(2

coth

Kb

Kb

Kb

Kb

minus

minus

minusrarr⎟⎠⎞

⎜⎝⎛

+rarr⎟⎠⎞

⎜⎝⎛

and

00 V

EEV

EKk cong

minuscong


⎟⎠⎞



02symmetric-anti





)e21( 2

0


732 Appendices


⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝


22exp21

22

20

02

22

2

2 bmVVma

Ema ππ






dkm

kdkdkdEdE

2==


intint ==ll

dldEk

mdldkdEED 222 21

)2(2)(

ππ

φ

πak

θ

dl

dk

E+dEE = h2k22m

int=kdlmED 222

)(π


For a

k πlt

2222

2)(

ππ

πm

kkmED ==

For a

ka

ππ 2ltlt

)2(42

)( 2222 ϕπ

ϕπ

mkkmED ==

but

⎟⎠⎞



2

So

⎟⎠⎞


kamED ππ

π1

22 cos4)(

or

⎟⎠⎞


kamED π

ππ1

2 cos41)(



(1) (2) (3) (4) (5)

(6)

(c) (b) (d) (a) (f) (e)



0 00



k

734 Appendices


( )

0( ) 2e

(e 1)

F

F

E E kTF

E E kTf E EdT dT kT

minus


k

k

k

( )

0( ) 2e 1

(e 1)

F

F

E E kT

E E kTfE kT

minus

minuspart

= minuspart +

k

kk

so

0 02

0

F

F


f EEdTE dT kT T


k

k

k

k

Thus

00


τ⎛ ⎞ part⎡ ⎤⎛ ⎞= + minus + +⎜ ⎟⎢ ⎥⎜ ⎟ part⎝ ⎠⎣ ⎦⎝ ⎠

kk k

k


33

2(2 )

zzJ e u f d k

π

⎛ ⎞= ⎜ ⎟⎜ ⎟

⎝ ⎠int k k


33

2(2 )

zzH u f E d k

π

⎛ ⎞= ⎜ ⎟⎜ ⎟

⎝ ⎠int k k k

Thus

2 303 ( )

4z F


τπ


kk

and

2 303

1 ( )4

z Fz


τπ


k k kk



23

22

3

23

( ) [ ( )]( )4

( ) ( )12

412

zz F

zF

F

dsdEeJ u E E eEu

dsdEe E u E Eu

e eEu k

τ δπ

τ δπ

τ ππ

= minus minus minus

= minus

=

int

int

kk k

k

kk k

k

k




m

neEJ z τσ

2==


230 ( )

12F


dz dT kT T uτ δ

π⎛ ⎞⎡ ⎤⎛ ⎞= = minus minus +⎜ ⎟⎢ ⎥⎜ ⎟

⎝ ⎠⎣ ⎦⎝ ⎠int k k

k kk

or

2 2( ) ( ) FF F


τ δ τ δ ⎡ ⎤⎛ ⎞minus = minus +⎢ ⎥⎜ ⎟⎝ ⎠⎣ ⎦

int intk k kk k k k

k k

or

( )2

2 2 2 414F F

F F F FFE E E E


πτ π τ= =

⎞⎛ ⎞ ⎛ ⎞= minus + ⎟⎜ ⎟ ⎜ ⎟ ⎟⎝ ⎠⎝ ⎠ ⎠k k

k kk k

So

dT

dEdzdTeE F=


2

31 [ ( )]

12F F

F

dT Kdz


τ δπ

=


k k k kk

so

23

1 412 F

F

F Fk kE E

u dE dEK E E k

dT dTτ π

π ==


kk k

736 Appendices


dT

dEEm

nK FF

F 2τminus=


⎥⎥

⎦

⎤

⎢⎢

⎣

⎡

⎟⎟⎠

⎞⎜⎜⎝

⎛minus=

2

)0(121)0(

FFF E

kTEE π

so

)0(6

22

F

FE

TkdT

dE πminus=

so

)0(6

222

FF

FE

TkEm

nK πτ=


Tkm

nK F 22

3τπ=

and

22

2

22

33 ⎟

⎠⎞

⎜⎝⎛==

ek

Tm

ne

Tkm

n

TK

F

Fπ

τ

τπ

σ




bXX

dx 12 minus=int





1

)]1([11

)]1([

21

1

02121

10 2

21

KK

xKxKKKxKxKdx

=

⎥⎦

⎤⎢⎣

⎡minus+minus

minus=minus+

int


2

2

22 2

22 2

( )

( ( ))

(2 )2

22

2

L

L

dIq f

dq f

d

q qm

dmq q

m

ω

ω

=

= minusminus

= minus⎡ ⎤



int

int

int

int

qq

qq

q

k q

q

k q


1

2 20 2 2

2

2(1 ) 2 L

m dxI dmq x q xω


int intq

k q


1

2 20 2 2 2

2

2( ) L

m dI dxmx x x kω


int intq

q k


1

2 20 2 2 2

2

2L

m dI dxmq x x kω

= minus⎡ ⎤⎛ ⎞+ minus⎜ ⎟⎢ ⎥⎝ ⎠⎣ ⎦

int intq

738 Appendices


2

2 2 2 2 2 20

4[ ] [ ]

d q dqq q

πα α

infin=

+ +int int

q


intminus

minus=1

0 22

22 2

2

kxxmdxmI

Lωπ

But

2

sin2

2

11

0 22

pppxx

dx minus=minus

int

so we find

⎟⎟⎠

⎞⎜⎜⎝

⎛minus= minus

2sin22 12

2k

mk

mm

mIL

L

L ωω

ωπ

Thus

Lm

Kω

=3


⎟⎟⎠

⎞⎜⎜⎝

⎛+minuscong

2122 2

22

kmm

mILL ω

πω



2 2

02

16

c L

c

kEm

mm

α ω

α

lowastlowast

lowastlowastlowast

= minus +

=minus

k






1 1d edt c

δτ

⎛ ⎞ ⎛ ⎞+ = minus +⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

k E v Btimes


mceB

c =ω

v

vv

vv

zz

xcyy

ycxx

Eme

Eme

Eme

τ

τωτ

τωτ

minus=

+minus=

minusminus=


⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛ minus=

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

z

yx

cc

z

yx

EEE

DDjjj

000101

0 τωτωσ




Dyyxx

0σσσ ==

D

cxyyx

τωσσσ 0=minus=

For the Hall effect

yxyy

xyy

EEj σ

σminus=rArr= 0

740 Appendices

So

xyy

xyxxx Ej ⎟

⎟

⎠

⎞

⎜⎜

⎝

⎛+=σσ

σ2


xx Ej 0σ=


m

ne τσ2

0 =


BE

E

BjE

Rx

yxyy

x

x

yH )( 0σ

σσ

minus==


nec

RH1minus=


B

nec

cc

cxyyx ==congminus=

τωσ

τωτωσσσ 02

0

)(


B

Ryx

H )(1

σminus=


v

v)()(

)(vvv

32

54

32

540

0

⎥⎦

⎤⎢⎣

⎡partpart

partpartminusminus=

⎥⎥⎦

⎤

⎢⎢⎣

⎡sdot

partpartminus=

partpartminus=

F

F

FF

e

Fe

xxxfxfcm

xfcmEp

π

π



( ) ⎟⎟⎠

⎞⎜⎜⎝


40

22 11411

FF

xF

xxdxxxxf F

where

312

v3

⎟⎟⎠

⎞⎜⎜⎝

⎛=equiv π

cmcmpx

ee

FF


FFF

F xxxxf

21)( 3 +=

partpart

34312

v31)3(

vπ

cmx

e

F minus=partpart

Fe

F xcm

x31

v3

31

vv

312minus=⎟

⎟⎠

⎞⎜⎜⎝

⎛minus=

partpart π


[ ] 12

61

3111

41

2432

54

242

432

54

0

FFe

FFF

Fe

xxcm

xxx

xcm

p

minuscong

⎥⎥⎦

⎤

⎢⎢⎣

⎡++⎟

⎟⎠

⎞⎜⎜⎝


π

π


pm

MM δ=

and

⎟⎟⎠

⎞⎜⎜⎝

⎛=

cem

RR

742 Appendices


Mm

R pξπ sdotcong 3

34v

31

31

3

2

491

34

3

⎟⎟⎠

⎞⎜⎜⎝

⎛=

⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜

⎝

⎛

sdot=

ξδπ

ξπ

π

MR

Mm

RRRx

pF


32

54

12~and

89

ππδ cm

K e==

⎥⎥⎦

⎤

⎢⎢⎣

⎡minuscong 2

32

4

34

0~

RM

RMKp




EF(2)

EF(1)

E

Δ = eV

R1 2



We note if

1)2(1exp)(

minus

⎥⎥⎦

⎤

⎢⎢⎣

⎡+

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛ minusminus=

kTEE

Ef F (a)


then

1)1(1exp)(

minus

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡+⎟

⎟

⎠

⎞

⎜⎜

⎝

⎛ minusminus=Δ+

kTEE

Ef F


then

[ ] 3

321

)2())(1)((v2



[ ] 3

312

)2())(1)((v2




[ ] 3

3

)2()()(v2


Since

x

x kE

partpart= 1v

We write

dEdkxx1v =


[ ] ||2

3 )()()2(



mk

EW2

2||

2minus= (b)


)(D1 WTT =

744 Appendices


⎥⎥⎦

⎤

⎢⎢⎣


1



[ ] )( )()()2(

2||


π (d)






ce EE

mED minus⎟

⎟⎠

⎞⎜⎜⎝

⎛=

lowast 23

222

21)(π

1])exp[(

1)(+minus

=kTE

Efμ


⎟⎠⎞

⎜⎝⎛ minus=⎟

⎠⎞


kTE

kTEEf μμ expexp)(

ce

EmkE += lowast2

22

intinfin

lowast⎟⎠⎞


⎟⎠

⎞⎜⎜⎝

⎛=

cE ce dE

kTEEE

mn μ

πexp

22

123

22



ce

EmkE += lowast2

22

kdkm

dEe

22

2

lowast=

lowast

=minuse

cm

kEE2

intinfin

lowast

lowast

lowast

lowast

⎟⎟⎠

⎞⎜⎜⎝

⎛ +minus⎟

⎠⎞

⎜⎝⎛

⎟⎟⎠

⎞⎜⎜⎝

⎛=

cEe

ce

e

e kdkmkT

EmkkTm

kmn 2

2)2(

expexp2

22

1 22223

22μ

π

intinfin

lowastlowastlowast

lowast

⎟⎟⎠

⎞⎜⎜⎝

⎛minus⎟

⎠

⎞⎜⎝

⎛ minus⎟⎟⎠

⎞⎜⎜⎝

⎛=

cEee

c

e

e dkkkTm

kmkT

E

m

mn 2

22223

22 2expexp

2

22

1 μπ

intinfin

lowastlowast

lowast

⎟⎟⎠

⎞⎜⎜⎝

⎛minus⎟

⎠

⎞⎜⎝

⎛ minus⎟⎟⎠

⎞⎜⎜⎝

⎛=

cEe

c

e

e dkkkTm

kkT

Em

mn 2

22

23

323

22 2expexp

)2(22

21 μπ


pp

ndxpxx nn π

)2(2)12()exp(0


where

kTm

pne

2

2 1 ==

So

( )kTmkTm

kTE

m

mn

ee

c

e

e

21

223

323

22

2222

)1(exp2

222

1 πμπ

⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎠

⎞⎜⎝

⎛ minus⎟⎟⎠

⎞⎜⎜⎝

⎛=

lowast

lowast

⎟⎠

⎞⎜⎝

⎛ minus⎟⎟⎠

⎞⎜⎜⎝

⎛=

lowast

kTEkTm

n ce μπ

exp2

223

2



746 Appendices



1 ( ) 1 ( )kdE E

d⎛ ⎞= equiv⎜ ⎟⎝ ⎠

kv kk

nabla


1 ( )

( )

k

k

dE Edt

d Edt

= sdot = sdot

= sdot

F v F k

k k

nabla

nabla

thus

( ) ( )k kdE Edt



ddt

= kF


1 ( ) ( )k kdE Edt


but also



( ) ( )k kdE Edt



( ) dqdt











ln T

ln μ

low T high T



phonon scattering

748 Appendices


if

if

212

121

21

21total

μμμμμμ

μμμμμ

gtgtconggtgtcong

+=




rr

rV λminusprop e1)(



2 1 2 1( )H V r=k k k k



2 1



int minuspropπ

θθθθστ 0

sin)cos1)((v1 dm



Finally we find

23Tm

m prop= lowastτ

μ








cEδ δprop sdotk r



23minus=prop Tμ


See Fig 618


Now


⎟⎠

⎞⎜⎜⎝

⎛= minus )(

23

2 e2

2 μ

π


⎟⎠

⎞⎜⎜⎝

⎛= minus )(

23

2 e2

2 μ

π

750 Appendices


⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎟⎟⎠

⎞⎜⎜⎝

⎛=minus

23222

lnkTm

NkTE

e

dCn

πμ

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎟⎟⎠

⎞⎜⎜⎝

⎛=minus

23222

lnkTm

NkTE

h

aVp

πμ

Therefore

⎥⎥⎦

⎤

⎢⎢⎣


32

)()()2(

22ln)(

kTmmNN

kTEeEEhe

dagpnCnVp

πϕϕ

Using

kTEg

gkTE

e

1ln minus=

kTEhei

gmmkTn 4323

2 e)(2

2 minus⎟⎟⎠

⎞⎜⎜⎝

⎛=

π

we find

⎟⎟⎠

⎞⎜⎜⎝

⎛=minus 2ln)(

i

dapn

nNN

kTe ϕϕ QED



2 21

12Surface

( )( )

4d

rπΦ = int

M r Sr

sdot

= minus ΦH nabla

12 2 21 2

12

ˆ ( )( )

4d

rπ= int

r M r SH r

sdot

θr

dS2

r 12

z

M

r1 = 0r12 = ndashr2


Using the figure

intminus=π

πθθπθθ

0 24 sin2 cos cos

rrdrMH z

31and

3=minus= DMH z




)eTr( HZ βminus=


1 1 1( ) ln

Tr( e ) Tr( e )

zz z z

z z

ZF ZB B Z B

Z Z

β β

μ ββ β β

βμ μβ



as expected



)Tr(2

)1Tr()eTr( 22

HZ H ββ +== minus


)Tr(

3)Tr(

3

)Tr(31)Tr()Tr(

2222

222

222222

JBμgJBBμg

JδBBμgJJBBμgH

Bμ μμ

B


==

==

sum

sumsum

)12()1Tr( += j


⎥⎥⎦

⎤

⎢⎢⎣

⎡+++cong )1(

3211)12( 2

222 jjBgjZ Bμβ

752 Appendices



⎥⎦⎤

⎢⎣⎡ +

partpart= )1(

611 2222 jjBg

B Bz

z μββ

μ

BB

BB z

z=

partpart


kT

jjgNNB

Bz

zzz

)1(3






kkS sdot


kkS sdot



i i( ) e ej i j

i jid Jdt


k k kk k kS S S



i jj i jd Jdt


k k kk k kS S S


i( ) i( )i( ) e ( )e ej ij i

d Jdt


k k kk R k kS R S S


kRsdot


so

( )

( )

( )

kd Jdt

J

J




primeprime= minus

prime=primeprime=

sum

sumsum

k kk k kk k

k k kk

k k kk

S k k S S

k S S

k S S

times

times

times

which is (7155) QED




ϕϕ ==


2

2

(cos cos sin sin )

cos( ( ))

x x y yR R R R R R

R R R R

S S S S

S

S q R R

ϕ ϕ ϕ ϕ

prime prime prime

prime prime

= +

= +

prime= minus

S Ssdot

so that


)cos()(2

))(cos()(2

i2

2

2

=minus=

minus=


sumsum

sum

sum prime

RRqR

R

RR

qRRJRJNS

qRRJNS

RRqRRJSH


qR minus== sum Then

)(2

2qJNSH minus=


)2cos(21)cos(2

21 )2cos(2)cos(2

)ee()ee()(

1

1221

i2i22

ii1

⎥⎦⎤

⎢⎣⎡ minus=

minus=+=

+++= minusminus

qaqaJ

JJqaJqaJ

JJqJ qaqaqaqa

754 Appendices

For a maximum

[ ]0

)cos()sin(2)sin(2

)2sin()2(21)sin(2)(

1

1

=minusminus=

⎥⎦⎤


partpart

qaqaqaaJ

qaaqaaJqqJ


angle turn 60

21)cos(

1)cos(2

deg=

=

=

qa

qa

qa


[ ]

max03

212

212

)2cos(2)cos(2)(

21

21

602

12

2

rArrltminus=

⎥⎦⎤

⎢⎣⎡ +minus=

minusminus=part

partdeg=

aJ

aJ

qaqaaJq

qJqa



=minussj mSS

12

112

212

212

21

21

21

+⎟⎠⎞

⎜⎝⎛ minus+=

⎟⎠⎞

⎜⎝⎛ minus=

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛minus=

+

++minus

nSnnS

nSnaS

nSaa

aSnS

j

jjjj


so



or

1)1)((

12

2)1(221

minus+minus+=

minus⎟⎠

⎞⎜⎝


sss

ss

ssj

mSmSmS

mSS

mSSmSSmSS

QED


We have

i1 e jjjb a

N= sum k R

ksdot

and


Nminus= sum k R

ksdot

where

0][

0][

][daggerdagger

dagger

=

=

=

ji

ji

ljjj

aa

aa

aa δ

Thus

i( )dagger dagger

i( )

1[ ] e [ ]

1 e

1

j j

i j

j

k jj jk j

j

k k k k

b b a aN

N

NN

δ

δ δ

prime

prime


primeminus minus

prime prime

=

=

= =

sum

sum

k R k R

k k R

sdot sdot

sdot

Similarly

i( )

0


Nprimeprime+



0


Nprimeprimeminus +


756 Appendices






Therefore

712

)state F(3max

21221

=+==

=+=

sumL

Lm

SS

l

less than half full

25


So stateF 25

2

Pm3+


426

)state I(123max5122


sumSLJ

mLSS

l

So stateI4

5

Eu3+ six electrons


033


7123216


=+=times=

sumSLJ

mL

SS

l

So stateF0

7




05 3

3[( ) ] ( )( )4μπ



B

r

rprime

m


4 have ( ) 0

28 have ( ) 2 22

a mam

primeminus sdot =

primeminus sdot = =

r r m

r r m


[ ]2 2

05 3

3 ( )4

Uμπ


r r m mm Br r r r

⎟⎟⎠

⎞⎜⎜⎝

⎛minus= minus

5

252

3

2327

total2)2)(3)(8(2)12(10

ama

amU

( ) 022424)10( 3

27 =minus= minus

amU


758 Appendices


2 2 2

4 have ( ) 0

2 have [( ) ] a m

primeminus sdot =

primeminus sdot =

r r m

r r m


066)10( 3

2

3

27

total =⎟⎟⎠

⎞⎜⎜⎝

⎛minus= minus

am

amU


3

27


)JT10279)(8(8

m10524

10

minus

minus

times=cong

timescong

Bm

a

μ


)(10381

101010

1012510)10(

23

2328

51

30

447

total

congtimescong

=congtimes

cong

minus

minusminus

minus

minus

minusminus

c

cT

T

U





The free energy is

ZkTF lnminus=


BFMpartpartminus=




)(0 qpHH rarr




qpqAp

eeAppqq

i

i

iiii

ii

Π=

gtΠequiv+rarrrarr

0HH

Now

)()(e

e

11)(

11)(


=

int

intΠminus

minus

qpqdddqdq

dpdpdqdqZ

NNq

NNqp

helliphellip

helliphellip

H

H

β

β


0 =Πpartpart

=Πpartpart

=partpart

j

iij

j

iij

j

i qpqq δδ



⎟⎟⎠

⎞⎜⎜⎝

⎛=

Πpartpart

+Π= etc1i

iiii

peAp

760 Appendices

Thus schematically

1

0

100010001

)()( =

Πpartpart

=

Πpartpartpartpart

=

Πpartpart

Πpartpart

partpart

partpart

=Πpart

part

i

i

j

i

i

i

j

i

j

i

j

i

j

ip

pqp

pqqp

qq

qqp 15



0=partpartminus=

BFM






15

1

100

01

100

01

0

=sdot=⎟⎟⎠

⎞⎜⎜⎝

⎛

partpart

⎟⎟⎠

⎞⎜⎜⎝

⎛

Πpartpart

=

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛partpart

partpart

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

Πpartpart=

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛partpart

partpart

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

Πpartpart=

Πpartpartpartpart

partpart

11j

i

j

i

i

i

j

i

j

ii

i

j

i

j

i

j

i

i

i

j

i

qqp

qp

qq

pqp

qq

ppqp

qq





Subtracting [ ]int



intΔminus

ΔΔminusminus

minus

Δminus= 2

1 22

221

221)(

)0()0(eV dEEeV

EeV

E

ENNCI

Checks 11








0

000


⎠

⎞⎜⎜⎝

⎛minusprime==Δ


0

0

0

3

0

0

2

02

3

31)11(

2

3cos

2

)(coscos2

ε

ε

θε

θθε

π

π

P

P

P

dPEE b

=

minusminusminus=

minus=

minus=equiv int

762 Appendices



θ

p

E z


int

int= π

θ

πθ

θθπ

θθπθ

0

cos

0

cos

sin2e

sin2ecos

d

dpp

kTpE

kTpE

z


int

int

minus

minus= c

c

x

c

c

x

zdx

dxx

cpp

e

e


⎟⎠⎞

⎜⎝⎛ minus=

ccppz

1)coth(


kTp

Ep

kTEpp z

z 3

3

22=== α



The Hamiltonian is

eEzr

qm


0

22

2

2H

H



)1(100 pzN += ψψ


BarBa 23


ψ


2

2

mqaB =

Requiring

12 =int τψ d





Now using

4

3 0100 0 100 2

0

12 (4 )

med Eψ ψπε

= = minusint rH





are also zero so

)(2 202 eEpaE

NBminus=

H



764 Appendices

Requiring

0=part

partpH

we find

0EeEp minus=


230



302

216 Ba

Eπεα =

part

partminus=

H



061

41)(

21 42


For E = 0

0)( 420 =++minus=


PGE δγβ (2)


20


[ ] 0)(61

41)(

21 2

02


0121)(

31 2


20 4 scc PTT

βγminus= (3)

(b) From (2) 0)( 42


20 4




04


δγ

432 minus=scP

βδγ 2

0 163+= TTc











766 Appendices





φ1 (say 45deg)

Polarizer

lightSample

Analyzerz

zφ2

E

z

yx






iminus= xx xpxp



If H = px2 m +V(x)

[ ] 0)( =xxV So

[ ] xpm

x i minus=H


[ ] iˆ ˆm




2 2ˆ ˆ

2ij ijj ji j i j

f b ωsdot + sdot

=sum sume r e r




jxi

i=minus xpxp xx

xxxppx xx i2 =minus

iminus= xx xpxp


768 Appendices


jj jω=H

jpm


jpm


jpm



1ˆ ˆi ij

i j i jmω

sdot = sdote r e p


ij ji


ωω ω



jiij ωω minus=



Summing over j

ˆ ˆ2iijjbf i im



m

bm

bfj ij 2)i(

i2minus=minus=sum




eEB

NeG +minus

=+ v13

1vvel

2

0εμ



0evv minus= Then

eEB

NeG =minus

minus+minus v13

1vvel

2

0

2ε

μω


ion0

ion 31 αε

NB =

and

Ge

Ee 2

locion

v ==α

at ω = 0

GeNB

2

0ion 3

1)0(ε

=


eEB

GBG =

minusminus+minus v

1)0(

vvel

ion2μω

eEB

BG =⎟⎟⎠

⎞⎜⎜⎝

⎛minus

minus+minus v1

)0(1v

el

ion2μ

μμω

Define

⎟⎟⎠

⎞⎜⎜⎝

⎛minus

minus=el

ion21

)0(1

BBG

T μω



For the local field

el00

loc 1v

31

3 BNeEPEEminus

+=+=εε

Substituting

)(

v 22 ωωμ minus=

T

eE

770 Appendices

Then

)(13

122el0


+=T

eEB

NeEE

)(13

11 22el0

loc


T

eB

NeE

E

)(

11

)0(1 22el

ionloc

ωωμ minusminus+=

TBGB

EE

)()1(

)0()()1(22

el

ion22

loc

ωωμωωμ


=T

Tel

BGBB

EE

Inverting

)0()()1(

)()1(

ion22

22el

loc GBBB

EE

Tel

T


=ωωμ

ωωμ

))(1()0(1

1

22el

ion

ωωμ minusminus+

=

T

locB

BGEE

Define

))(1(

)0(22

el

ion

ωωμ minusminus=

TBBGF

Then

FE

E+

=1

1

loc


loc0

ion0

ionv

31

31)(

EeNNB

εα

εω ==

Substitute in v

)(3

1)( 22loc0ion

ωωμεω

minus=

T

eEE

eNB

)(

)0()( 22

ion

locion

ωωμω

minus=

T

GBE

EB

)1(1

1)( elion BFF

B minus+

=ω


Given

[ ] )1()(1)(3)()(

elionel

ion

0 BBBB


ωω

εεωε


)1()1(

11

)1(1

3)()(

elelel

el

00 BBF

FB

BF

F

minus⎥⎦⎤

⎢⎣⎡ minus

+minusminus

minus++infin=

εε

εωε

)1(

11

13)()(

el00 B

FF

FF

minus⎟⎠⎞

⎜⎝⎛

+minus

++infin=εε

εωε

)1(

3)()(

el00 BF

minus+infin=

εε

εωε

Substitute for F

))(1(

)0()1(

3)()(22

el

ion

el00 ωωμεε

εωε


TBBG

B

Define c

2el

ion

)1()0(3

BBGcminus

=μ

Then

)(

)()(2200 ωωε

εεωε

minus+infin=

T

c

Rewriting

202)()(

ωεωε

εω =infinminus

minusc

T


2

2

)0()(

L

T

ωω

εε =infin

)()0( 2

2infin= ε

ωωε

T

L

772 Appendices


0)()0(

02 =infinminus

minusεε

εω cT

0)()(2

202 =


εεωω

εω

T

LT

c

0)()( 22

202 =

infinminusminus

εωωωεωTL

TT

c

0)(

)(2

0222 =infin


TLTc

202)( LT

c ωεεω =infin

+



ψψ Em

=nablaminus 22

2

For each coordinate

0222

2=+minus i

i

i

mEdxd ψψ


⎟⎟⎠

⎞⎜⎜⎝

⎛+⎟

⎟⎠

⎞⎜⎜⎝

⎛= i

ii

ii xmEBxmEA 22

2cos2sinψ


2

222

2 2or2

manEnamE i

iii ππ ==

)(2

23

22

212

22nnn

maET ++= π





2

22

2

22

23

2)111114(



2

22

23

mahf π=


constant432 ==

mfa π



0

22

2

2

2

2

22

4where1

2 πεφeq

rq

rrrm=minus⎟

⎟⎠

⎞⎜⎜⎝

⎛

partpart+

partpart+

partpartminus=H




εεε RRYEmqEmq

r ==== )(2

2

4

2

2ρ


ψψρ

ψφρρρρ

ε2

4

2

22

2

2

22

2

4

422

2)(11

22mqmqqqm

m=minus⎟

⎟⎠

⎞⎜⎜⎝

⎛

partpart+

partpart+

partpartminus

774 Appendices

Canceling we obtain

ψρψψ

φρρρρε

2111

21

2

2

22

2=minus⎟

⎟⎠

⎞⎜⎜⎝

⎛

partpart+

partpart+

partpartminus




φφ mie)( =Φ



⎞⎜⎜⎝

⎛minus+minus

21

21 2

2

2 ερρρρρ

mdd

dd

We let

2

and1 2

2xαρ

α=minus=ε


Ψ=Ψ+⎟⎟⎠

⎞⎜⎜⎝


241

21 22

2

2 xxm

dxd

xdxd


21122 +=+= nn αα


2212

21 )(

or)(

1+

minus=+

minus=n

REn

ε


21)1(

21 minus+=+ nn





)()(2 2

22zEz

dzd

mψψ =minus


0)(2)(22

2=+ zmE

dzzd ψψ

⎟⎟⎠

⎞⎜⎜⎝

⎛+⎟

⎟⎠

⎞⎜⎜⎝

⎛= zmEBzmEAz 22

2cos2sin)(ψ



2

222

2mwpE p

π=





12

2

0minusminus=

xeF

πε


x

ex

edxx

eFdxdVxxxx

0

2

0

2

20

2

161




x

eExV0

2

0 16)(

πεminus=

776 Appendices


x

exeEExV0

2

10 16)(

πεminusminus=


01

=⎟⎠⎞

⎜⎝⎛

=xxdxdV

which gives

10

1 16 Eex

πε=

or

0

13

01 21)(

πεEeExV minus=

Thus

0

13

21

πεEe=Δ



a

V(r)

ndashV(0)

E (bound state)

r




20 )(2 EVm +

=α

22mEminus=β



This implies

2


Also

E

EV

aa

minus=

+=

+= 0

22

2

22

)(cot11)(sin

βαα

αα


2πα gta

to estimate a

eV)1(22)(22

22

0 mEVma ππ

απ gt

+=cong

nm931)1061)(1019(2

)10636)(143(1931

34

min congtimestimes

timescongminusminus

minusa




778 Appendices

Energy Gap

Ec(z)

Eg1 Eg2

Valence Band

Conduction Band

d

z

Ev(z)


EFFzEzmyxm

c =⎥⎥⎦

⎤

⎢⎢⎣

⎡+

partpartminus⎟

⎟⎠

⎞⎜⎜⎝

⎛

partpart+


22 2

22

2

2

2

22




32||

2

322

2)(

2)(

2Ek

mEkk



ψψ 32

22)(

2EzE

dzd

mc =⎟

⎟⎠

⎞⎜⎜⎝

⎛+minus lowast



sum=

minus=N

ni

ndiq ndz

Nz

1

i )(e1)( χψ


0 we find for E3




2

2 2 0 0( ) cos












)(22

||

222

2

222

EE

kkmmw

nE

n

yx

+equiv

++= π

)(22

222

2||

2

|| yx kkm

km

E +==




ndEm

dkdkkkd

2

2||||||

2

2

2

π

ππ

=

==

780 Appendices


)(

)(2)2(

2)(

2

0||||22

sum

sum int

minus=

minusminus=infin

n n

n n

EEm

dEEEEmED

θπ

δππ



23)(π

mED = (1)

In 3D

E

dEmdkmkE 2

22 221

2==

2

212)4(

)2(2

)unit volper (4)2(

2)(

223

23D3

EdEmmE

dkkdEED

ππ

ππ

=

=


32

22

1)(223

22D3

E

wmmED π

π ⎟⎟⎠

⎞⎜⎜⎝

⎛=

where

2

222

32

3mw

EE π==

Per unit area

222D3332

21)(

ππ

πmw

wmwED == (2)

which checks (1)


d)

D3D(E)w

2hπm

23hπm

E1 E2 E3




1001

512

512

51

2

51

2~

21

21

22

11

I

NN

NN

NN

NN

AA

=⎟⎠⎞

⎜⎝⎛=

⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜

⎝

⎛minus+

⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜

⎝

⎛

minus

+=


Bibliography

Chapter 1

















784 Bibliography










Chapter 2









Chapter 2 785





















786 Bibliography

Chapter 3





















Chapter 3 787



















788 Bibliography

Chapter 4





















Chapter 5 789














Chapter 5







USA 92 6663-6667 (1995)

790 Bibliography






















Chapter 6 791



Francis London 1963

Chapter 6
















1994

792 Bibliography






















1996

Chapter 7 793

Chapter 7



















794 Bibliography



















Chapter 8 795










(2004)

Chapter 8






796 Bibliography






















Chapter 9 797





Chapter 9

















798 Bibliography




Chapter 10















Chapter 11 799






Chapter 11















800 Bibliography


















Chapter 12



Chapter 12 801























802 Bibliography





















Chapter 12 803

















804 Bibliography

Appendices





















Subject References


Mechanics




Electricity



Optics



Thermodynamics






Critical Phenomena


806 Bibliography

Crystal Growth


Modern Physics



Quantum Mechanics





Math Physics


Solid State







Condensed Matter






Problems




Applied Physics





Rocks




808 Bibliography


The Internet



Periodic Table


Further Reading












Further Reading 809
















810 Bibliography

















Further Reading 811


















812 Bibliography




Index

A













B


814 Index










C




Index 815





















816 Index




316 317

D









594 640






E





Index 817

















503 566










F



818 Index


493 538 Fermi function 160 248 298 303



373 385 399 407 422 426 456 486 513







G

GaAs 242 295 297 324 326 328 329 337 347 563 564 579 618 628 629 637




Index 819

Group velocity 55 306 532 562 627


H











I






820 Index


J


K



L









529 570

Index 821




M













N


531 600 NaCl 6 9 28 579 600

822 Index









O








P





Index 823















824 Index


Q




R












S



Index 825













826 Index






T



361 418











475 509

Index 827

U




V



Vortex region 466

W


X


Y

YBa2Cu3O7 506 508

Z


Appendices

A Units










G1 Bose Particles

G2 Fermi Particles


H1 Propagators

H2 Green Functions

H3 Feynman Diagrams

H4 Definitions




J Folk Theorems



M Problem Solutions














































































Bibliography

Mechanics

Electricity

Optics

Thermodynamics


Critical Phenomena

Crystal Growth

Modern Physics

Quantum Mechanics

Math Physics

Solid State

Condensed Matter


Problems


Applied Physics

Rocks


The Internet

Periodic Table

Index


656 Appendices


51

2 21

21 +

== aaωω (B12a)

and for

51

2 21

22 minus

== aaωω (B12b)


⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜

⎝

⎛

minus

+=2

2

11

51

2

51

2~

NN

NN

A (B13)

where

21

21

1 1)15(

4)(⎥⎥⎦

⎤

⎢⎢⎣

⎡+

+=minusN (B14a)

and

21

21

2 1)15(

4)(⎥⎥⎦

⎤

⎢⎢⎣

⎡+

minus=minusN (B14b)


⎟⎟⎠

⎞⎜⎜⎝

⎛

⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜

⎝

⎛

minus

+=⎟⎟⎠

⎞⎜⎜⎝

⎛=

2

1

22

11

2

1

51

2

51

2

aa

NN

NN

EE

E (B15)

Problems



⎟⎟⎠

⎞⎜⎜⎝

⎛

minus+=⎟⎟

⎠

⎞⎜⎜⎝

⎛

520052

3221~ AA







0)()(d

)(d2

2=+ xxU

xx ψψ (C1)


)()( xUnaxU =+ (C2)








0d

d =x

W (C6)

where

21


=W (C7)


658 Appendices



0)()(d

)(d2

2=++++ axaxU

xax ψψ


0)()(d

)(d2

2=+++ axxU

xax ψψ (C8)


)()()( 211 xBxAax ψψψ +=+ (C9)

)()()( 212 xDxCax ψψψ +=+ (C10)


21

21

21

21

2121

2121ψψψψ

ψψψψ

ψψψψψψψψ

primeprime=

primeprime=


DBCA

DCBADCBA

or

1=DBCA

or

1=minus BCAD (C11)


)()( xΔax ψψ =+ (C12)


)()()( 21 xxx βψαψψ += (C13a)

or

)()()( 21 axaxax +++=+ βψαψψ (C13b)


)()()()()()()(

2121

xxxDBxCAax


+=+++=+ (C14)



αΔβα =+ CA

and

βΔβα =+ DB


0=minus

minusΔ

ΔDB

CA (C15)






DA +=+ minus++

1)(ΔΔ


1)(ΔΔ (C17)

and

DA +=+ minus+ ΔΔ



If we set

be=+Δ (C19)

and



)()( 11


and

)()( 12


660 Appendices





where

)()( xuaxu =+ (C25)



Let

)()( 11 NN xxExx ψψ =H (C26)



0][ =lTH (C27)

and






lkmiddotl

ie=t (C31)

Further




so that

12 =lt (C32)










)(02)(0

22

21 Gnnn G =+++


5 In general sum= j

kj

klij

kil AT ψψ

where the Ail




kklkl AT ψψ =


662 Appendices




ntctx nin

i sum= )()(ψ


nmNi

im

imnm cctctc

Nlowast





)( ATrA ρequiv


][i Ht

ρρ =partpartminus


⎟⎠⎞

⎜⎝⎛ minus=

kTHFexpρ


)][exp()]exp([exp

kTHTrkTHHTr

kTHFHTrHU

minusminus=⎥

⎦

⎤⎢⎣

⎡⎟⎠⎞

⎜⎝⎛ minus==








Let

)()( 0 tVt += HH (E1)

lEl l00 =H (E2)


00lk

lkEE minus=ω (E4)




664 Appendices


200

2 2)()(

)(

)cos(1lim ωδπ

ω

ω tfEEttfi

if

if

t=minus=

minusinfinrarr

(E6)












⎟⎟⎠

⎞⎜⎜⎝

⎛=

00σ

σα (F2)




⎟⎟⎠

⎞⎜⎜⎝

⎛minus

=I

I0

0β (F3)



12

)(2

0

20 ltlt

minusminusequiv

cmVcmEε (F4)


20cmEE +prime= (F5)

and

⎟⎟⎠

⎞⎜⎜⎝

⎛=

φχ

ψ (F6)




⎤

⎢⎢⎣


4482 220

2

220

2

230

4

0

2pσ V

cmV

cmV

cmp

mpE (F7)


])[(4 22

0


cm (F8)


666 Appendices


SL middot dd1

2 220

2

rV

rcm (F9)





G1 Bose Particles




][

0][

0][

dagger

daggerdagger

ijji

ji

ijjiji

bb

bb

bbbbbb

δ=

=

=minusequiv

9 See Mattuck [A14]



daggeri i in b b=

and

dagger1 iii bbn =+


11

1

11dagger

11



++=

minus=

iiii

iiii

nnnnnb

nnnnnb



1

1dagger

iiii

iiii

nnbn

nnbn

=minus

=minus



dagger)1()1(






)()(


sumlowast=

=

α αα

α αα

ψψ

bu

bu

rr

rr


668 Appendices

G2 Fermi Particles




0

0

dagger

daggerdagger

ijji

ji

ijjiji

aa

aa

aaaaaa

δ=

=

=+equiv







iii nan




0











)2()1()2()1(

210daggerdagger

ββ

ααβα uu

uuaa harr











670 Appendices






H1 Propagators




H2 Green Functions


δεω

ωi

1)(0 +minus=+

kG k (H1)


H3 Feynman Diagrams


H4 Definitions






672 Appendices




















dagger )2()1()21()2()1(21)1()1()1(H (H3)



31



0

daggerdagger

=+=+

ijji

ijijjiaaaaaaaa δ



δεω

ωi


=sum

+

l lklkk VG k (H6)





674 Appendices



112223 d)21()()()()(d rrrrrr φφφφ (H8)




δεω

ωi)(


=sum

+













klk

Gδωεω

ωi)(


=sum k

k (H11)





676 Appendices




fcc (110) (011) (101)

2 2 2

bcc (11 1) ( 111) (1 11)2 2 2

a a a




1 2 31 ( )Ω






lkh

lkh dG π2


Δ =k G (See 152)











G (See 141)





⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎟⎟⎠

⎞⎜⎜⎝

⎛+⎟

⎟⎠

⎞⎜⎜⎝

⎛minus=

126

4)(ijij

ij rrrU σσε


1e

1 minus

= kTn ω (See 277 2215)


222 akmks=ω (See 248 288 290)






D

DDωωωωωω

gt=leprop

0)( 2

(See 2224)

3126v ⎟

⎟⎠

⎞⎜⎜⎝

⎛==

VN

kk DDD

πθθω (See 2228)


678 Appendices


intminus

⎟⎠⎞

⎜⎝⎛= T

x

x

VdxxTNkC

0 2

43

)1e(e9 θ

θ (See 2229)


3( )

( )( )(2 ) const

dAVD ω

ωω

ωπ= int

k knabla (See 3256)




)e(e 20


2( ) TWθ⎛ ⎞prop Δ ⎜ ⎟⎝ ⎠

k


22

2

9)(

Dm

m mkAT θ=



V

Dln



B

C3

vγα = (See 2237 2238 2250)




ρ11Cv = (See 2270)


ρ44Cv = (See 2270)



3 Electrons


1e

1)( +




( ) ( )u u= +k kr r R (See 3129 C25)


EmVEg23

222

2)( ⎟⎟

⎠

⎞⎜⎜⎝

⎛=π

(See 3164)


32

22

32

)0( ⎟⎠⎞

⎜⎝⎛==

VN



m

ne τσ2

= (See 3214)




RH (See 6102 1223)

680 Appendices


LTK =σ

(See 3215)

22

3⎟⎠⎞

⎜⎝⎛=

ekL π (See 3215)


0

00 near band edge

E E U

U E E

lowast

+

minus=

minusk G

G k G (See 3230)



k r r R (See 3244)


k (See 3255)




i

Bloch Fns


kkr R r (See 3332)



cgs)in or (mceB

meB

c =ω (See 3196)




22

2 2

0 ( )

( )

( ) ( ) ( )2

1( ) ( ) ( )4

el

ioni i

j i i ij i

V

Vm

e d E

ψ ψ

ψ ψ ψπε ne

minus nabla +



r

r r r

r r r rr r




22

2

0 ( )|| spin

( ) ( ) ( ) ( ) ( )2

1 ( ) ( ) ( ) ( )4

ion eli i i

j i j i ij i

V Vme d E

ψ ψ ψ

ψ ψ ψ ψπε

lowast

ne

minus + +


r r r r r

r r r r rr r

nabla



1 32

2 3( ) ( ) ( ) ( ) ( ) ( ) ( )2


mψ ψ ψ ψ

π⎛ ⎞⎡ ⎤minus + + minus =⎜ ⎟⎣ ⎦ ⎝ ⎠

r r r r r r rnabla



2

2

1 ( ) ( )( ) 1( ) ( )k

e f fV E E iq σ

ε ωω ηε


k k qqk q k

(See 9166)

xx

xxxF

kqxxF

qk

qTF

sminus+minus+==+==

11ln

41

21)(

2)(1)0(0

2

2

2ε


F

srk

F Eemk

rkq s

εεπφ

23e

41 2

02 ==ltlt minus



collisions

f ff ft t


Fv nabla nabla


0RHS of BDE( )

f fτminus

= minusk

(See 4149 4145)

682 Appendices


11 123 v( ) ( ) ( )4

edj e f L E L T

eμ

π⎛ ⎞= minus = + + minus⎜ ⎟⎝ ⎠


21 223 ( )v( ) ( ) ( )4

Qdj E f L E L T

eμ

π⎛ ⎞= minus = + + minus⎜ ⎟⎝ ⎠




0ej TE

σ = =nabla

0Qe

jj

Tκ = =

nabla


0eEQ jT

= =nabla

0Q

e

jT

j

QT

Π = =

Π =

nabla


constant131 conganc



⎟⎟⎠

⎞⎜⎜⎝

⎛minus

VED F )(1exp




heG =




D

D

TT

TT

θρ

θρ

ltlt

gtgt5~

~ (See 4146)





)SI(21

0Ae

Bπ=⎟

⎠⎞

⎜⎝⎛Δ (See 534)



⎟⎟⎠

⎞⎜⎜⎝

⎛= (cgs) 4 2

0

2

mne

mne

pπ

εω (See 10108)







2innp = (See 613)

⎟⎟⎠

⎞⎜⎜⎝

⎛minus⎟

⎟⎠

⎞⎜⎜⎝

⎛= lowastlowast

kTE

mmkTn ghei 2

exp)(2

2 4323

2π (See 614)

684 Appendices


ddt

= kF (See 644)




2

2

211

kE



eDkT =μ (See 684 685)





⎟⎟

⎠

⎞

⎜⎜

⎝

⎛ ΔΔ== (cgs)

2

222 ne

V

ne

VKZwidth b π

εε (See 6193)


⎥⎦

⎤⎢⎣

⎡minus⎟

⎠⎞

⎜⎝⎛

⎟⎟⎠

⎞⎜⎜⎝

⎛+= 1exp

00 kTep

LD

nLDej b

np

pp

n

n ϕ (See 6241)



23minusprop Tμ


23Tpropμ





⎟⎠⎞

⎜⎝⎛ )(

22cgs

mceB

meB (See 7255 7256)

Lande g factor

⎥⎦

⎤⎢⎣

⎡+

+minus++++=)1(

)1()1()1(211

JJLLSSJJg

(Implied by (76) (710))


)(20 FBp EDμμχ = (See 3181 3201)


magnetonBohr =Bμ


)groundexcited(22

==minus

= gexcEEgexc

VN

gexc

zμχ


⎟⎠⎞


⎤⎢⎣⎡ ++=

Jx

JJxJ

JJxBJ 2

coth21

2)12(coth

212)( (See 716)



4sincos)()2(

xxxxxFRkFJ ijFij



kT

BgJx eff

Bμ= (See 722)

MBBeff λ+=

686 Appendices


)cgs(

32

0MMH

kEk effB

F

μθθ =primegt

prime (See 7222)


⎟⎟⎠

⎞⎜⎜⎝

⎛minusprop

)(1exp

FK EJD

T (See 7300)


h

eGxy

2ν= (See 1247)



KJpropδ (See 7251)





23

23)()0(

TC

TTMM

M prop



)for (1cc

c

c

TTHTTM

TT

TTC

=asympminusasymp

minusasymp

minusasympminus

minus

δβ

γ

α

χ






2

( ) ( ) (cgs)nemc



2

2

4 nemc

L πλ = (See 86)


g

FEv

πξ 2~0 (See 849)



II ξ lt λ 20~

1 πλϕ

cH 20~

2 πξϕ

cH 2

~1

2⎟⎟⎠

⎞⎜⎜⎝

⎛ξλ

c

c

HH



)(20 cgs

ehc=φ (See 845)



⎟⎟⎠

⎞⎜⎜⎝

⎛minuscong

)(1exp141

FDc EVD

T θ (See 8217)


⎟⎟

⎠

⎞

⎜⎜

⎝

⎛

+minus


1exp

451 epep

epDcT

λμλ



688 Appendices


2

20



⎡ ⎤⎛ ⎞⎢ minus + + ⎥ =⎜ ⎟⎝ ⎠⎢ ⎥⎣ ⎦

Anabla (See 83)

2

( ) ( 2 2 )2s eiq qj q e m m






)()0(

2

2

infin=εε

ωω

T

L (See 950)


⎟⎟⎠

⎞⎜⎜⎝

⎛+=+= )SI(

3)guassian(

34



⎟⎟⎠

⎞⎜⎜⎝

⎛=

+minus=

+minus

sumsum )SI(31

2)(1)(

34

21

00

0jjjj NN α

εεεεεαπ

εε (See 930)



direct 0eΔ =k








)cgs(2 222

4

neEE gn ε


)cgs(111

he mm+=

μ (See 10101)


11 Defects


Mollwo Equation







⎟⎟⎠

⎞⎜⎜⎝

⎛==

pGeEp

Bπω 2 (See 129)


)(exp 41 MottT

AB ⎟⎠

⎞⎜⎝





232surface =⎟

⎠⎞


kTWkTemj

π (See 1145)

690 Appendices


3 2

2surface exp BWj AE

E

⎛ ⎞= minus⎜ ⎟⎜ ⎟

⎝ ⎠ (See 1149)


J Folk Theorems


Chapter 1



Chapter 2



Folk Theorems 691

Chapter 3









Chapter 4







Chapter 5


Chapter 6




692 Appendices















Chapter 7






Folk Theorems 693





Chapter 8




Chapter 9



Chapter 10





Chapter 11







Chapter 12



694 Appendices







Gauss law V A A

dV disin





10

e +

infinminus =int n

axn

andxx


π41Y

e21)(

Y)(

00

23

10

0010001

=

⎟⎟⎠

⎞⎜⎜⎝

⎛=

=Ψ

minus Bar

BarR

rR





r θnabla ϕ

221 1 1( ) (sin )

sin sinrA

r A Ar r rr

ϕθθ



2

2

2222

2 sin1sin

sin11

ϕθθθ

θθ partΦpart+⎟

⎠⎞

⎜⎝⎛

partΦpart

partpart+⎟

⎠⎞

⎜⎝⎛

partΦpart

partpart=Φnabla

rrrr

rr2



111


=primeprime

γγγ

nnn

n


n

ennn ⎟⎠⎞




)()(

1))((

)()()(0)(


n

n nn

xxxf

xxxf

xf

bfdxbxxfxx

xxbxdxbx

=

minusprime

=


minus===minus

sum

int

int

δδ

δδ

δδδ

Complex Variables


π


nazfazdzd

naf n

n

n


)1(1)(Res 1

1

⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧

minusminus

= minus

minus

rarr


SI Gaussian

0

t

tρ

parttimes = +part


sdot =sdot =

H J D

BE

DB

nabla

nabla

nablanabla

4 1

1

40

c c t

c t

π

πρ

parttimes = +part


sdot =sdot =

H J D

BE

DB

nabla

nabla

nablanabla

0ε= +D E P

0

1μ

= minusH B M



( )qc

= + timesvF E B

696 Appendices












1956 Transistor


















9 Total for Decade

698 Appendices

























A Geim K Novoselov

2010 Graphene



Brillouin




Bloch






Bardeen



700 Appendices

Others






M Problem Solutions





451388

2412

126


=+minus=M




1

A

2

3

456

7

8

g5 g6

g7

g8


⎟⎠⎞

⎜⎝⎛= 87654321

876543211g

⎟⎠⎞

⎜⎝⎛= 21876543

876543212g

⎟⎠⎞

⎜⎝⎛= 43218765

876543213g

⎟⎠⎞

⎜⎝⎛= 65432187

876543214g

⎟⎠⎞

⎜⎝⎛= 23456781

876543215g

⎟⎠⎞

⎜⎝⎛= 45678123

876543216g

⎟⎠⎞

⎜⎝⎛= 67812345

876543217g

⎟⎠⎞

⎜⎝⎛= 81234567

876543218g

702 Appendices

Then eg

432 6543218787654321 ggg =⎟⎠⎞

⎜⎝⎛=


g1 g2 g3 g4 g5 g6 g7 g8




)(21 kjia minus+= a


)(23 kjia +minus= a

x

y

z






y

z

x

a2

a1

a3

a


a


704 Appendices





2 2 0h kh k h k


a bG

since

1 2 32 ( )h k lπ= + +G b b b

and


Thus

( ) 2( )( ) ( )h k ld h k lh k l h h k l

π= =G aG G

sdot






where

f iΔ = minusk k k



ki

b

b

bα

bα

hellip

x

y

z

kf

0ˆ =sdot ik i

2sin

2sin

e

2sin

2sin

e

e1

e1

e1

e1

ee

2)1(

i2

)1(i

i

i

i

i

10

i10

i

⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜

⎝

⎛

+⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜

⎝

⎛

=

minus

minus+minus

minus=

+=

minusminus

minusminus

minus

minus

minus

minus

minus=

minusminus=

minus sumsum

α

αα

α

α

α

bk

Mbk

bk

bNk

P

fx

fxMbk

fx

fxNbk

bk

Mbk

bk

bNk

Mm

mbkNn

nbk

fxfx

fx

fx

fx

fx

fxfx


)(integer 2

and)(integer 2

)2()1(

pbk


απ (a)

Since

pl

bk

bk

pl

bk

bk

fx

fx

fx

fx αα

== )2(

)1(

)2(

)1(

or1

)2()1(fxfx kk ne



since

22fxfy kkk minus=


706 Appendices


Let

λπ2=K





Define

θsinKB =

)iexp( BbZ =

So


But


)]2sin(ln[

1)]2sin()exp[(ln

0

)2sin(

knbZ

knbZm

knbZZ

m

knb

πε

πε

πεπε

+=

=

=

suminfin=

Thus


0 )2sin()ln( πε


2)2cos(21)2sin(ln

11

ZkbZkbZZ

ZE

+minus+

minus=

ππε






mGBb π2+plusmn=


b




Aring2255=Ka


Aring2255100 == ad

)bcc(6125222255


deg=rArr== 6816225551

2sin θλθ

d

deg=rArr= 04352255

)51(2sin θθ



θλ sin2 100dn =

above as 2)e1( 0435 4510

)51(42

4sin


)51(32

3sin

above as 2)e1( 6816 4510

)51(22

2sin


2sin

i4400

100

i3300

100

i2200

100

i100

100

ffSd

fSd

ffSd

fSd

=+====

=+====

=+====

=+====

minus

minus

minus

minus

π

π

π

π

θλθ

θλθ

θλθ

θλθ

708 Appendices




mm m

γ γ γx1 x2

1γx3

3 1 2 3





02

22

321

2

2

2

=⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜

⎝

⎛


uuu

mm

m


(2)


twice32mγω = (5b)


⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛==

111

31

321

0uuu

Uω (6a)

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛minus==12

1

51

3 mU γω (6b)


and

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛minus==

112

51

3 mU γω (6c)



)(2iexp1

)(2iexp1

)(2iexp1

)1(0

10

10

sum

sum

sum

minusminus= minus

minus=

minus=

⎥⎦⎤

⎢⎣⎡ minus=

⎥⎦⎤


⎥⎦⎤

⎢⎣⎡ minus=

Ns s

Ns s

Ns snm

nmN

sgN

nmN

sgN

mnN

sgN

G

π

π

π


[ ]

⎥⎦⎤

⎢⎣⎡ minus=

minus⎥⎦⎤

⎢⎣⎡ minus=

⎥⎦⎤


⎤⎢⎣⎡ minus=⎥⎦

⎤⎢⎣⎡ minus+

)(2iexp

)(2iexp)(2iexp


nmsN

nmnmsN

nmNN

nmsN

nmNsN

π

ππ

πππ

and (see 2104)

⎟⎠⎞

⎜⎝⎛=⎟

⎠⎞

⎜⎝⎛ +=⎥⎦

⎤⎢⎣⎡ +

Ns

NsNs




Nsg

NG π

But

⎟⎟⎠

⎞⎜⎜⎝

⎛⎥⎦⎤



Nnm

NNggN

ππ

so

mnN

snm GnmN

gG =⎥⎦⎤


0 )(2iexp π

QED

710 Appendices




)2()()(

)()(

)2()()(

)()(

)2()()(

54145515

34454

32423343

12232

15251121

XXXXXXXXm

XXXXXm

XXXXXXXXM

XXXXXm

XXXXXXXXm


minusminusminus=


minusminusminus=


αααβα

βββαβ

ααα


0

210011100020001110012

54321

22

212

2

=

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛

minusΩminusminusΩ

minusΩminusminusΩ

minusΩ

minus

uuuuu

AAAAA

AAδ


⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

Ω=

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛



minusminus

54321

2

54321

210011100

020001110012

uuuuu

uuuuu

AACCC

AA


⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

minusminus

=

1

0

1

a

aU


This requires

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛

ΩminusΩminus

sdotΩΩΩ

=

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

+minus++minus

minus+minus

=

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

minus+minus+minus+

+minus++minus+minus

22

2

2

2

0

31)1(

01)1(

3

211))(1(

)1(112

a

a

aAa

Aaa

aaAcaca

Aaa




2





)1()1()1( dagger

dagger

ϕψψ

ψψϕ

minus=minus=minus=

==

nna

aaa

aaaNaN

n

n

nn

712 Appendices


nKan =minus1 So

112

dagger2

nnnK

naanKnn

=

=minusminus


1or1 minus== nnnan

K


)1()1(

)1(dagger

daggerdagger

daggerdaggerdagger

ηψ

ψ

ψψη

+=+=

+=

==

nna

aaa

aaaNaN

n

n

nn


1

1+

=n

K

so

11dagger ++= nnna


⎟⎠⎞

⎜⎝⎛ +=⎟

⎠⎞

⎜⎝⎛ +=

21

21dagger Naa ωωH



321321


⎜⎝⎛=



321

321

321

ηηη

ηηη

ηηη

partpart

partpart

partpart

partpart

partpart

partpart

partpart

partpart

partpart

=

zzz

yyy

xxx

J

1 1 2 2 3 3η η η= + +r a a a


etc11

xax =partpartη

so we find

1 2 3

1 2 3 1 2 3

1 2 3

( )x x x

y y y

z z z

a a a

J a a a

a a a

= = sdot timesa a a

QED



1 1

2 1

3 1

100

sdot =sdot =sdot =

a ba ba b



a

KΩ

= 1

and

2 31

a

times=

Ωa a

b


714 Appendices


1 1 2 2 3 32 ( )n n n nπ= + +G b b b




31 21 2 3

1 2 32

nn nN N N

π⎛ ⎞

= + +⎜ ⎟⎝ ⎠

q b b b



a) 31 231 2 31 2

31 2

1 2 3

1 1

11

1

22 2 ii i 11 1i

1 2 3 1 2 30 0 0

i2

2i1

1 1 1 1e e e e

1 1 e etc



l

l

n

nn

nn n ll l NN NNN N

l l l

n l

n lN

i i

i i

n

N N N N N N

N

n Nn N

ππ π

π

π

δδ


= = ==

minus=

minus= ne= =

=

=

sum sum sum sum

sum

q R

R

Gq

q GG

G

b) 111

1

1

1

11

21

i(BZ)

1 2 3 1 0

2

21

0

1 1e e etc


1 1 e


l

l

ni lNN

l

i n

i lN

i i

i i

N N N N

N

l Nl N

π

π

π

δ

minussdot

==

minus=

minus= ne= =

=

sum sumq Rq

R


c)

1 2 31

1 1

3

33

1 2 31 2 3

31 2 3 03

fpp 33

( ) 2

of s( )Vol of space

1 ( )(2 ) ( )

( )(2 )

( ) (2 )

n n nn

f fN

qf dq

f d

N N NN N N

f d

Vf d

π

π

π

π

⎛ ⎞⎛ ⎞= +⎜ ⎟⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠

=

=sdot times

Ω=

sum sum

int

int

int

int

q q b

q q

q qb b b

q q

q q

d)

i i3fpp3

fpp

i

1 2 30

e e (with )(2 )

1 e

l l

l

l

a ad q VV

N N N

π

δ

Ω Ω= rarrinfin

=

=

sumint

sum

q R q Rq

q Rq

R

sdot sdot

sdot

e)

1 1 1 2 2 2 3 3 3i( ) i2 [( ) ( ) ( ) ]3 31 1e el l l l l l l luc



3

1 2 3


η η η⎛ ⎞

= = ⎜ ⎟⎝ ⎠

r


etc so

1 2 3

1 2 3 1 2 3

1 2 3




So

1 1 1

1 1 111

1i( ) i2 ( )310

1i2 ( )

1 1 0

1 e e

e i2 ( )

l l l l

a

l l

d d

l l

π η

π η

η

δπ


primeminusprime

=Ω

= =prime minus

int intG G r

ll

rsdot

716 Appendices

f)

i ( ) 3all 3

i ( ) i ( ) i ( )

i( )i( ) i( )

1 e(2 )

1 1 1e e e2 2 2( )

1 e 1 e 1 e2 i( ) 2 i( ) 2 i( )

yx z

x y z

yx yx z

zx y



LL L

d q

dq dq dq

L L L

x x y y z z

π

π π π

π π π

primeminus



minusminus minus

=

rarr infin


int

int int int

q r rq

sdot


( ) ( ) ( )( ) ( ) ( )( )

z

x y z

yx zL L L


x x y y z zπ π π

δ δ δδ









Frequency c

gωω=

Temperature )( k

Ttcω

=


Uucω

=

Entropy NkSs =


NkC

c vv =



⎟⎠⎞

⎜⎝⎛ +⎟⎠⎞

⎜⎝⎛==

Nd

ddpP

c

ωωωωω


)(

Thus

π

ωω2

)()( NadqdqqNdN ==

1

)2)(2cos(1

2)(

22cc

cN

aqaN

ddqNadqN

ωωπω

ωπωπω

minus=

==

So

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

minus⎟⎟⎠

⎞⎜⎜⎝

⎛

minus==

221

11e

1

ckTc d

dpPωωπω

ω ω

or since

tg

kTkT c==

ωωωω

c

then

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

minus⎟⎟⎠

⎞⎜⎜⎝

⎛

minus=

21

11e

1)(g

gtP tgπ (1)



int

intsum

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

minus⎟⎟⎠

⎞⎜⎜⎝

⎛

minus=

minus=

minus= =

c

c

m

dN

dNU

cckT

kTNm kT

m

ω

ω

ω

ωω

ωωωω

ωπ

ωωωω

0 22

01

1

11e

2

)(1e

21e

int⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

minus⎟⎟⎠

⎞⎜⎜⎝

⎛

minus==

c

cc

kTc

cd

NUu

ω

ω ωωωω

ωωπω 0 22 )(

1

11e

2

718 Appendices

or

int⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

minus⎟⎟⎠

⎞⎜⎜⎝

⎛

minus=

1

0 21

11e

2 dgg

gu tgπ (2)

ωωωπω

ω

ω

ω

ω

ω

ω

ω

dNkT

kTC

cckT

kT

m kT

kTm

c

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

minus⎟⎟⎠

⎞⎜⎜⎝

⎛

minus=

minus=

int

sum

2202

2

2

2

2

2v

1)1e(e)(2

)1e(e)(1

int⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

minus⎟⎟⎠

⎞⎜⎜⎝

⎛

minus==

c

cc

kT

kT

c

c dTNkN

NkCc

ω

ω

ωωω

ωω

ωπω

ω

0 222

2

22v

v )(1

1)1e(

e)()(

2

int⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

minus⎟⎟⎠

⎞⎜⎜⎝

⎛

minus=

1

0 22

2

v1

1)1e(

e)(2 dgg

tgc tg

tg

π (3)


kTFj



VT

FS ⎟⎠⎞

partpartminus=

so

( ) sumsum=

minus=

minus

minus+minusminus=

N

m

mkT

N

m

kT

kTkTkS

mm

12

1

e11e1ln ωω

ω

( )

int

int

minusminus+

minusminusminus=

=

minus

c

c

cc

kT

cc

kT

dkTkTNNk

dNkNk

NkSs

ω

ω

ωω

ωωωω

ωπ

ωωωωπ

0 22

2

0 22

)(1

11e

)2(1

)(1

1e1ln21


+minus

minusminus= minus1

0 2

1

0 2

1

11e

2

1

1e1ln2 dgg

tgdgg

s tgtg

ππ (4)



range of ω

ωxxp


P10x1π

1

e

ωx

10 1minus

sdot1

1 ωx( )2minus

sdot=

P100x1π

1

e

ωx

100 1minus

sdot1

1 ωx( )2minus

sdot=

P300x1π

1

e

ωx

300 1minus

sdot1

1 ωx( )2minus

sdot=

0 05 10

200

400

P300

P100

P10

ω

720 Appendices


the range of t

txxr


ux2π

0

1

yy

e

y

tx 1minus

1

1 y2minussdot

⌠⎮⎮⎮⎮⌡

dsdot=

spht x2π

0

1

y

ytx

⎛⎜⎝

⎞⎟⎠

2e

y

txsdot

e

y

tx 1minus

⎛⎜⎝

⎞⎟⎠

2

1

1 y2minussdot

⌠⎮⎮⎮⎮⎮⎮⎮⎮⌡

dsdot=

0 1 2 3 40

1

2

3

4

u

spht

t


0 05 1 15 20

05

1

15

2

u

spht

t


Entropy t2π

minus

0

1

ωln 1 e

ωminus

tminus

⎛⎜⎝

⎞⎟⎠

1

1 ω2

minus

sdot

⌠⎮⎮⎮⎮⌡

dsdot2π

0

1

ω

ω

t

e

ω

t 1minus

1

1 ω2

minus

sdot

⌠⎮⎮⎮⎮⎮⌡

dsdot+=

0 100 200 3000

2

4

6

8

Entropy t

t

722 Appendices




By Problem 29

2 31

a

times=

Ωa a

b


2 3 [(1)(1) (1)( 1)] [(1)(1) ( 1)(1)] [( 1)( 1) (1)(1)]

2

(2 2 ) ( )2

a

a a


= + = +

a a i j k

i j i j

and

2

21 2 3( ) (2)


so Ωa = a2 and

11 ( )a

= +b i j


2

3

1 ( )

1 ( )

a

a

= +

= +

b j k

b i k




1 2 31 1 1( ) ( ) ( )a a a




ky

kxkz


a

a

n m + 1

n mn ndash 1 m

n m ndash 1

n + 1 m

xnm ynm


])()()()[(2

])()()()[(2

21

21

21

21

21

21

21

21

mnmnmnmnmnmnmnmn

mnmnmnmnmnmnmnmn

yyyyyyyyk

xxxxxxxxkU



minus+minus+

minus+minus+


)4( 111

mnmnmnmnmn



)4( 111

mnmnmnmnmn



724 Appendices


)i(1 e maknakt

mnyxcx ++minus= ω

)i(2 e maknakt

mnyxcx ++minus= ω


⎟⎠⎞⎜



⎟⎠⎞⎜






a x

y


⎥⎥⎦

⎤

⎢⎢⎣

⎡+⎟

⎟⎠

⎞⎜⎜⎝


⎟⎠⎞

⎜⎝⎛ +++minus++=

⎟⎠⎞

⎜⎝⎛ minus++=

22

2

2

222222

22222

222

481

2122

21

)(2)(21

)(21

aax

ax

ayxayxaxak

ayxaayxak

ayxakU

to second order

[ ]

2

22222222

21

2222

kx

axaxyxayxaxakU

cong







11 11dagger

1 i22

pp p p

pa P X

ωω

= minus qq q q

q

Now

11 1

1

11dagger

p p

p

p

X m

m

X


minus

= sdot

= sdot

=

sum

sum

X

X

q b q b q bb

b q b q bb

q

e

e






1 1

1

1

1

111

1

1 1

p pp p p

pp p

e X m e e X

m e e X

m X m X

β βα αβ

β βαβ

ββ ααβ δ δ

lowast lowast

lowast

=

=

= =

sum sum

sum sum

sum

q b q b q bbqb q b

b q b q bb q b

bb b bq bb q b

or

1 11

1p ppX e X


b (a)

Now

dagger 11dagger 11

1 i22

pp p p

pa P X

ωω

= + qq q q

q


11dagger 11dagger

11 11

1 i22

1 i22

pp p p

p

pp p

p

a P X

P X

ωω

ωω


= minus

qq q q

q

qq q

q

726 Appendices

so

dagger 11 2i

2p

p p pa a Xω

minusminus = qq q q

11 dagger

i ( )

2p p pp


q

Substituting in (a)

1 dagger

i ( )

2 p p ppp

X e a am ω


b q



0)(

)(])[(

34412

24412122

442

4411

=++

++minus+minus

Kkkcc

KkkccKkckcc

zx

yxx ρω


0)( 122

11 =minus Kkc ρω


0)( 222

44 =minus Kkc ρω

0)( 322

44 =minus Kkc ρω

If

ρ

ων 1111

ck

==


If

ρ

ωνων 4433

22

ckk

====






)1(

)(

])()([)(

))]()(())()([()(

))]()(())()(()(

2

2

12

1112

11112

11112

1

1

NM

PM

dPM

dxuxuPM

dxuxuQxuxuPM

dxuxPuxuxPuMd

P

qp qq

qp qq

pp NNNNpp

pp NNNNpp

=

=

minus=

minus=

minus=

minus=

sum

sum int

sum int

sum int

sum intint

lowast

primelowastprime+


τδ

τ

τ

ττψψ

where Q = Pminus1P




)2()1()21()2()1(21

)2()1()21()2()1(21

)2()1()21()2()1(

)2()1()21()2()1(

)1()1()1(

(||)

(||)

sum

sum

sum

sum

sumsum

minus

+=

minus

+

=

ji ijji

ji jiji

ji ijji

ji jiji

i iik

V

VE

V

V

H

ψψψψ

ψψψψ

ψψψψ

ψψψψ

ψψε



22 2

1 21 10 12

| ( ) |( ) d 0

4jN N

II jeV

rψ


rr

Note that

221| ( ) |N

jj ψ=sum r

728 Appendices


2

22 11

0 12

( )d ( )

4NI

II Ie V

rρ τ


r QED

IonI

Field point

Source point

r1

r2



Let x = kkM

0as2)(21

11ln1limlnlim

2

2

0

22

0

rarr=minus=

minus+minus=

minus+minus

rarrrarr

xxxx

xx

xx

kkkk

kkkk

xM

M

M

Mk

⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢

⎣

⎡


minusminusminus



rarr

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

gtminusminusminus

ltminusminusminus

rarr

⎥⎦⎤

⎢⎣⎡


⎥⎦⎤

⎢⎣⎡



rarr

110)1(

)1(1)1(1

or

110)1(

)1(1)1)](1(1[

2

1)1(1)1ln(

1)1(1)1ln(

2

1)1ln(1)1ln(2ln)1(2

1)1ln(1)1ln()1ln()1)(1(


2

2

22

1

xxx

xx

xxx

xx

xx

x

xxx

xxxxx

xxxxx

xxx

xxxx

xx

xx

x





i1e 0bm


K


The answers are





Thus

i(0) i( )( )e e

1 ( 1)

n l m

n l m

S πminus minus + +

+ +

= +

= + minusK






730 Appendices


i( )( ) i( )( ) i( )(n )1 e e en l m









20

2)(2

2 EVmKmEk

minus==


)ee()(22

ee)(2

)(02

If

ii

KxKx

kxkx

Dxbxb

CBxaxb

Vbax

minus

minus

+=ltltminus

+=ltlt

infinrarr=+gt

ψ

ψ

ψ


)ee()ee(i

)ee(ee

0ee

2i2i2i2i

2i2i2i2i

)2(i)2(i

KbKbkbkb

KbKbkbkb

bakbak

KDCBk

DCB

CB

minusminus

minusminus

+minus+

minus=minus

+=+

=+



⎟⎠⎞

⎜⎝⎛minus=

2coth)tan( Kb

Kkka (a)


)ee()(22

ee)(2

02

ii

KxKx

kxkx

Dxbxb

CBxaxb

bax

minus

minus

+=ltltminus

+=ltlt

=+gt

ψ

ψ

ψ


⎟⎠⎞

⎜⎝⎛minus=

2tanh)tan( Kb

Kkka (b)


)e21(

2tanh

)e21(2

coth

Kb

Kb

Kb

Kb

minus

minus

minusrarr⎟⎠⎞

⎜⎝⎛

+rarr⎟⎠⎞

⎜⎝⎛

and

00 V

EEV

EKk cong

minuscong


⎟⎠⎞



02symmetric-anti





)e21( 2

0


732 Appendices


⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝


22exp21

22

20

02

22

2

2 bmVVma

Ema ππ






dkm

kdkdkdEdE

2==


intint ==ll

dldEk

mdldkdEED 222 21

)2(2)(

ππ

φ

πak

θ

dl

dk

E+dEE = h2k22m

int=kdlmED 222

)(π


For a

k πlt

2222

2)(

ππ

πm

kkmED ==

For a

ka

ππ 2ltlt

)2(42

)( 2222 ϕπ

ϕπ

mkkmED ==

but

⎟⎠⎞



2

So

⎟⎠⎞


kamED ππ

π1

22 cos4)(

or

⎟⎠⎞


kamED π

ππ1

2 cos41)(



(1) (2) (3) (4) (5)

(6)

(c) (b) (d) (a) (f) (e)



0 00



k

734 Appendices


( )

0( ) 2e

(e 1)

F

F

E E kTF

E E kTf E EdT dT kT

minus


k

k

k

( )

0( ) 2e 1

(e 1)

F

F

E E kT

E E kTfE kT

minus

minuspart

= minuspart +

k

kk

so

0 02

0

F

F


f EEdTE dT kT T


k

k

k

k

Thus

00


τ⎛ ⎞ part⎡ ⎤⎛ ⎞= + minus + +⎜ ⎟⎢ ⎥⎜ ⎟ part⎝ ⎠⎣ ⎦⎝ ⎠

kk k

k


33

2(2 )

zzJ e u f d k

π

⎛ ⎞= ⎜ ⎟⎜ ⎟

⎝ ⎠int k k


33

2(2 )

zzH u f E d k

π

⎛ ⎞= ⎜ ⎟⎜ ⎟

⎝ ⎠int k k k

Thus

2 303 ( )

4z F


τπ


kk

and

2 303

1 ( )4

z Fz


τπ


k k kk



23

22

3

23

( ) [ ( )]( )4

( ) ( )12

412

zz F

zF

F

dsdEeJ u E E eEu

dsdEe E u E Eu

e eEu k

τ δπ

τ δπ

τ ππ

= minus minus minus

= minus

=

int

int

kk k

k

kk k

k

k




m

neEJ z τσ

2==


230 ( )

12F


dz dT kT T uτ δ

π⎛ ⎞⎡ ⎤⎛ ⎞= = minus minus +⎜ ⎟⎢ ⎥⎜ ⎟

⎝ ⎠⎣ ⎦⎝ ⎠int k k

k kk

or

2 2( ) ( ) FF F


τ δ τ δ ⎡ ⎤⎛ ⎞minus = minus +⎢ ⎥⎜ ⎟⎝ ⎠⎣ ⎦

int intk k kk k k k

k k

or

( )2

2 2 2 414F F

F F F FFE E E E


πτ π τ= =

⎞⎛ ⎞ ⎛ ⎞= minus + ⎟⎜ ⎟ ⎜ ⎟ ⎟⎝ ⎠⎝ ⎠ ⎠k k

k kk k

So

dT

dEdzdTeE F=


2

31 [ ( )]

12F F

F

dT Kdz


τ δπ

=


k k k kk

so

23

1 412 F

F

F Fk kE E

u dE dEK E E k

dT dTτ π

π ==


kk k

736 Appendices


dT

dEEm

nK FF

F 2τminus=


⎥⎥

⎦

⎤

⎢⎢

⎣

⎡

⎟⎟⎠

⎞⎜⎜⎝

⎛minus=

2

)0(121)0(

FFF E

kTEE π

so

)0(6

22

F

FE

TkdT

dE πminus=

so

)0(6

222

FF

FE

TkEm

nK πτ=


Tkm

nK F 22

3τπ=

and

22

2

22

33 ⎟

⎠⎞

⎜⎝⎛==

ek

Tm

ne

Tkm

n

TK

F

Fπ

τ

τπ

σ




bXX

dx 12 minus=int





1

)]1([11

)]1([

21

1

02121

10 2

21

KK

xKxKKKxKxKdx

=

⎥⎦

⎤⎢⎣

⎡minus+minus

minus=minus+

int


2

2

22 2

22 2

( )

( ( ))

(2 )2

22

2

L

L

dIq f

dq f

d

q qm

dmq q

m

ω

ω

=

= minusminus

= minus⎡ ⎤



int

int

int

int

qq

qq

q

k q

q

k q


1

2 20 2 2

2

2(1 ) 2 L

m dxI dmq x q xω


int intq

k q


1

2 20 2 2 2

2

2( ) L

m dI dxmx x x kω


int intq

q k


1

2 20 2 2 2

2

2L

m dI dxmq x x kω

= minus⎡ ⎤⎛ ⎞+ minus⎜ ⎟⎢ ⎥⎝ ⎠⎣ ⎦

int intq

738 Appendices


2

2 2 2 2 2 20

4[ ] [ ]

d q dqq q

πα α

infin=

+ +int int

q


intminus

minus=1

0 22

22 2

2

kxxmdxmI

Lωπ

But

2

sin2

2

11

0 22

pppxx

dx minus=minus

int

so we find

⎟⎟⎠

⎞⎜⎜⎝

⎛minus= minus

2sin22 12

2k

mk

mm

mIL

L

L ωω

ωπ

Thus

Lm

Kω

=3


⎟⎟⎠

⎞⎜⎜⎝

⎛+minuscong

2122 2

22

kmm

mILL ω

πω



2 2

02

16

c L

c

kEm

mm

α ω

α

lowastlowast

lowastlowastlowast

= minus +

=minus

k






1 1d edt c

δτ

⎛ ⎞ ⎛ ⎞+ = minus +⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

k E v Btimes


mceB

c =ω

v

vv

vv

zz

xcyy

ycxx

Eme

Eme

Eme

τ

τωτ

τωτ

minus=

+minus=

minusminus=


⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛ minus=

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

z

yx

cc

z

yx

EEE

DDjjj

000101

0 τωτωσ




Dyyxx

0σσσ ==

D

cxyyx

τωσσσ 0=minus=

For the Hall effect

yxyy

xyy

EEj σ

σminus=rArr= 0

740 Appendices

So

xyy

xyxxx Ej ⎟

⎟

⎠

⎞

⎜⎜

⎝

⎛+=σσ

σ2


xx Ej 0σ=


m

ne τσ2

0 =


BE

E

BjE

Rx

yxyy

x

x

yH )( 0σ

σσ

minus==


nec

RH1minus=


B

nec

cc

cxyyx ==congminus=

τωσ

τωτωσσσ 02

0

)(


B

Ryx

H )(1

σminus=


v

v)()(

)(vvv

32

54

32

540

0

⎥⎦

⎤⎢⎣

⎡partpart

partpartminusminus=

⎥⎥⎦

⎤

⎢⎢⎣

⎡sdot

partpartminus=

partpartminus=

F

F

FF

e

Fe

xxxfxfcm

xfcmEp

π

π



( ) ⎟⎟⎠

⎞⎜⎜⎝


40

22 11411

FF

xF

xxdxxxxf F

where

312

v3

⎟⎟⎠

⎞⎜⎜⎝

⎛=equiv π

cmcmpx

ee

FF


FFF

F xxxxf

21)( 3 +=

partpart

34312

v31)3(

vπ

cmx

e

F minus=partpart

Fe

F xcm

x31

v3

31

vv

312minus=⎟

⎟⎠

⎞⎜⎜⎝

⎛minus=

partpart π


[ ] 12

61

3111

41

2432

54

242

432

54

0

FFe

FFF

Fe

xxcm

xxx

xcm

p

minuscong

⎥⎥⎦

⎤

⎢⎢⎣

⎡++⎟

⎟⎠

⎞⎜⎜⎝


π

π


pm

MM δ=

and

⎟⎟⎠

⎞⎜⎜⎝

⎛=

cem

RR

742 Appendices


Mm

R pξπ sdotcong 3

34v

31

31

3

2

491

34

3

⎟⎟⎠

⎞⎜⎜⎝

⎛=

⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜

⎝

⎛

sdot=

ξδπ

ξπ

π

MR

Mm

RRRx

pF


32

54

12~and

89

ππδ cm

K e==

⎥⎥⎦

⎤

⎢⎢⎣

⎡minuscong 2

32

4

34

0~

RM

RMKp




EF(2)

EF(1)

E

Δ = eV

R1 2



We note if

1)2(1exp)(

minus

⎥⎥⎦

⎤

⎢⎢⎣

⎡+

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛ minusminus=

kTEE

Ef F (a)


then

1)1(1exp)(

minus

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡+⎟

⎟

⎠

⎞

⎜⎜

⎝

⎛ minusminus=Δ+

kTEE

Ef F


then

[ ] 3

321

)2())(1)((v2



[ ] 3

312

)2())(1)((v2




[ ] 3

3

)2()()(v2


Since

x

x kE

partpart= 1v

We write

dEdkxx1v =


[ ] ||2

3 )()()2(



mk

EW2

2||

2minus= (b)


)(D1 WTT =

744 Appendices


⎥⎥⎦

⎤

⎢⎢⎣


1



[ ] )( )()()2(

2||


π (d)






ce EE

mED minus⎟

⎟⎠

⎞⎜⎜⎝

⎛=

lowast 23

222

21)(π

1])exp[(

1)(+minus

=kTE

Efμ


⎟⎠⎞

⎜⎝⎛ minus=⎟

⎠⎞


kTE

kTEEf μμ expexp)(

ce

EmkE += lowast2

22

intinfin

lowast⎟⎠⎞


⎟⎠

⎞⎜⎜⎝

⎛=

cE ce dE

kTEEE

mn μ

πexp

22

123

22



ce

EmkE += lowast2

22

kdkm

dEe

22

2

lowast=

lowast

=minuse

cm

kEE2

intinfin

lowast

lowast

lowast

lowast

⎟⎟⎠

⎞⎜⎜⎝

⎛ +minus⎟

⎠⎞

⎜⎝⎛

⎟⎟⎠

⎞⎜⎜⎝

⎛=

cEe

ce

e

e kdkmkT

EmkkTm

kmn 2

2)2(

expexp2

22

1 22223

22μ

π

intinfin

lowastlowastlowast

lowast

⎟⎟⎠

⎞⎜⎜⎝

⎛minus⎟

⎠

⎞⎜⎝

⎛ minus⎟⎟⎠

⎞⎜⎜⎝

⎛=

cEee

c

e

e dkkkTm

kmkT

E

m

mn 2

22223

22 2expexp

2

22

1 μπ

intinfin

lowastlowast

lowast

⎟⎟⎠

⎞⎜⎜⎝

⎛minus⎟

⎠

⎞⎜⎝

⎛ minus⎟⎟⎠

⎞⎜⎜⎝

⎛=

cEe

c

e

e dkkkTm

kkT

Em

mn 2

22

23

323

22 2expexp

)2(22

21 μπ


pp

ndxpxx nn π

)2(2)12()exp(0


where

kTm

pne

2

2 1 ==

So

( )kTmkTm

kTE

m

mn

ee

c

e

e

21

223

323

22

2222

)1(exp2

222

1 πμπ

⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎠

⎞⎜⎝

⎛ minus⎟⎟⎠

⎞⎜⎜⎝

⎛=

lowast

lowast

⎟⎠

⎞⎜⎝

⎛ minus⎟⎟⎠

⎞⎜⎜⎝

⎛=

lowast

kTEkTm

n ce μπ

exp2

223

2



746 Appendices



1 ( ) 1 ( )kdE E

d⎛ ⎞= equiv⎜ ⎟⎝ ⎠

kv kk

nabla


1 ( )

( )

k

k

dE Edt

d Edt

= sdot = sdot

= sdot

F v F k

k k

nabla

nabla

thus

( ) ( )k kdE Edt



ddt

= kF


1 ( ) ( )k kdE Edt


but also



( ) ( )k kdE Edt



( ) dqdt











ln T

ln μ

low T high T



phonon scattering

748 Appendices


if

if

212

121

21

21total

μμμμμμ

μμμμμ

gtgtconggtgtcong

+=




rr

rV λminusprop e1)(



2 1 2 1( )H V r=k k k k



2 1



int minuspropπ

θθθθστ 0

sin)cos1)((v1 dm



Finally we find

23Tm

m prop= lowastτ

μ








cEδ δprop sdotk r



23minus=prop Tμ


See Fig 618


Now


⎟⎠

⎞⎜⎜⎝

⎛= minus )(

23

2 e2

2 μ

π


⎟⎠

⎞⎜⎜⎝

⎛= minus )(

23

2 e2

2 μ

π

750 Appendices


⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎟⎟⎠

⎞⎜⎜⎝

⎛=minus

23222

lnkTm

NkTE

e

dCn

πμ

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎟⎟⎠

⎞⎜⎜⎝

⎛=minus

23222

lnkTm

NkTE

h

aVp

πμ

Therefore

⎥⎥⎦

⎤

⎢⎢⎣


32

)()()2(

22ln)(

kTmmNN

kTEeEEhe

dagpnCnVp

πϕϕ

Using

kTEg

gkTE

e

1ln minus=

kTEhei

gmmkTn 4323

2 e)(2

2 minus⎟⎟⎠

⎞⎜⎜⎝

⎛=

π

we find

⎟⎟⎠

⎞⎜⎜⎝

⎛=minus 2ln)(

i

dapn

nNN

kTe ϕϕ QED



2 21

12Surface

( )( )

4d

rπΦ = int

M r Sr

sdot

= minus ΦH nabla

12 2 21 2

12

ˆ ( )( )

4d

rπ= int

r M r SH r

sdot

θr

dS2

r 12

z

M

r1 = 0r12 = ndashr2


Using the figure

intminus=π

πθθπθθ

0 24 sin2 cos cos

rrdrMH z

31and

3=minus= DMH z




)eTr( HZ βminus=


1 1 1( ) ln

Tr( e ) Tr( e )

zz z z

z z

ZF ZB B Z B

Z Z

β β

μ ββ β β

βμ μβ



as expected



)Tr(2

)1Tr()eTr( 22

HZ H ββ +== minus


)Tr(

3)Tr(

3

)Tr(31)Tr()Tr(

2222

222

222222

JBμgJBBμg

JδBBμgJJBBμgH

Bμ μμ

B


==

==

sum

sumsum

)12()1Tr( += j


⎥⎥⎦

⎤

⎢⎢⎣

⎡+++cong )1(

3211)12( 2

222 jjBgjZ Bμβ

752 Appendices



⎥⎦⎤

⎢⎣⎡ +

partpart= )1(

611 2222 jjBg

B Bz

z μββ

μ

BB

BB z

z=

partpart


kT

jjgNNB

Bz

zzz

)1(3






kkS sdot


kkS sdot



i i( ) e ej i j

i jid Jdt


k k kk k kS S S



i jj i jd Jdt


k k kk k kS S S


i( ) i( )i( ) e ( )e ej ij i

d Jdt


k k kk R k kS R S S


kRsdot


so

( )

( )

( )

kd Jdt

J

J




primeprime= minus

prime=primeprime=

sum

sumsum

k kk k kk k

k k kk

k k kk

S k k S S

k S S

k S S

times

times

times

which is (7155) QED




ϕϕ ==


2

2

(cos cos sin sin )

cos( ( ))

x x y yR R R R R R

R R R R

S S S S

S

S q R R

ϕ ϕ ϕ ϕ

prime prime prime

prime prime

= +

= +

prime= minus

S Ssdot

so that


)cos()(2

))(cos()(2

i2

2

2

=minus=

minus=


sumsum

sum

sum prime

RRqR

R

RR

qRRJRJNS

qRRJNS

RRqRRJSH


qR minus== sum Then

)(2

2qJNSH minus=


)2cos(21)cos(2

21 )2cos(2)cos(2

)ee()ee()(

1

1221

i2i22

ii1

⎥⎦⎤

⎢⎣⎡ minus=

minus=+=

+++= minusminus

qaqaJ

JJqaJqaJ

JJqJ qaqaqaqa

754 Appendices

For a maximum

[ ]0

)cos()sin(2)sin(2

)2sin()2(21)sin(2)(

1

1

=minusminus=

⎥⎦⎤


partpart

qaqaqaaJ

qaaqaaJqqJ


angle turn 60

21)cos(

1)cos(2

deg=

=

=

qa

qa

qa


[ ]

max03

212

212

)2cos(2)cos(2)(

21

21

602

12

2

rArrltminus=

⎥⎦⎤

⎢⎣⎡ +minus=

minusminus=part

partdeg=

aJ

aJ

qaqaaJq

qJqa



=minussj mSS

12

112

212

212

21

21

21

+⎟⎠⎞

⎜⎝⎛ minus+=

⎟⎠⎞

⎜⎝⎛ minus=

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛minus=

+

++minus

nSnnS

nSnaS

nSaa

aSnS

j

jjjj


so



or

1)1)((

12

2)1(221

minus+minus+=

minus⎟⎠

⎞⎜⎝


sss

ss

ssj

mSmSmS

mSS

mSSmSSmSS

QED


We have

i1 e jjjb a

N= sum k R

ksdot

and


Nminus= sum k R

ksdot

where

0][

0][

][daggerdagger

dagger

=

=

=

ji

ji

ljjj

aa

aa

aa δ

Thus

i( )dagger dagger

i( )

1[ ] e [ ]

1 e

1

j j

i j

j

k jj jk j

j

k k k k

b b a aN

N

NN

δ

δ δ

prime

prime


primeminus minus

prime prime

=

=

= =

sum

sum

k R k R

k k R

sdot sdot

sdot

Similarly

i( )

0


Nprimeprime+



0


Nprimeprimeminus +


756 Appendices






Therefore

712

)state F(3max

21221

=+==

=+=

sumL

Lm

SS

l

less than half full

25


So stateF 25

2

Pm3+


426

)state I(123max5122


sumSLJ

mLSS

l

So stateI4

5

Eu3+ six electrons


033


7123216


=+=times=

sumSLJ

mL

SS

l

So stateF0

7




05 3

3[( ) ] ( )( )4μπ



B

r

rprime

m


4 have ( ) 0

28 have ( ) 2 22

a mam

primeminus sdot =

primeminus sdot = =

r r m

r r m


[ ]2 2

05 3

3 ( )4

Uμπ


r r m mm Br r r r

⎟⎟⎠

⎞⎜⎜⎝

⎛minus= minus

5

252

3

2327

total2)2)(3)(8(2)12(10

ama

amU

( ) 022424)10( 3

27 =minus= minus

amU


758 Appendices


2 2 2

4 have ( ) 0

2 have [( ) ] a m

primeminus sdot =

primeminus sdot =

r r m

r r m


066)10( 3

2

3

27

total =⎟⎟⎠

⎞⎜⎜⎝

⎛minus= minus

am

amU


3

27


)JT10279)(8(8

m10524

10

minus

minus

times=cong

timescong

Bm

a

μ


)(10381

101010

1012510)10(

23

2328

51

30

447

total

congtimescong

=congtimes

cong

minus

minusminus

minus

minus

minusminus

c

cT

T

U





The free energy is

ZkTF lnminus=


BFMpartpartminus=




)(0 qpHH rarr




qpqAp

eeAppqq

i

i

iiii

ii

Π=

gtΠequiv+rarrrarr

0HH

Now

)()(e

e

11)(

11)(


=

int

intΠminus

minus

qpqdddqdq

dpdpdqdqZ

NNq

NNqp

helliphellip

helliphellip

H

H

β

β


0 =Πpartpart

=Πpartpart

=partpart

j

iij

j

iij

j

i qpqq δδ



⎟⎟⎠

⎞⎜⎜⎝

⎛=

Πpartpart

+Π= etc1i

iiii

peAp

760 Appendices

Thus schematically

1

0

100010001

)()( =

Πpartpart

=

Πpartpartpartpart

=

Πpartpart

Πpartpart

partpart

partpart

=Πpart

part

i

i

j

i

i

i

j

i

j

i

j

i

j

ip

pqp

pqqp

qq

qqp 15



0=partpartminus=

BFM






15

1

100

01

100

01

0

=sdot=⎟⎟⎠

⎞⎜⎜⎝

⎛

partpart

⎟⎟⎠

⎞⎜⎜⎝

⎛

Πpartpart

=

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛partpart

partpart

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

Πpartpart=

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛partpart

partpart

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

Πpartpart=

Πpartpartpartpart

partpart

11j

i

j

i

i

i

j

i

j

ii

i

j

i

j

i

j

i

i

i

j

i

qqp

qp

qq

pqp

qq

ppqp

qq





Subtracting [ ]int



intΔminus

ΔΔminusminus

minus

Δminus= 2

1 22

221

221)(

)0()0(eV dEEeV

EeV

E

ENNCI

Checks 11








0

000


⎠

⎞⎜⎜⎝

⎛minusprime==Δ


0

0

0

3

0

0

2

02

3

31)11(

2

3cos

2

)(coscos2

ε

ε

θε

θθε

π

π

P

P

P

dPEE b

=

minusminusminus=

minus=

minus=equiv int

762 Appendices



θ

p

E z


int

int= π

θ

πθ

θθπ

θθπθ

0

cos

0

cos

sin2e

sin2ecos

d

dpp

kTpE

kTpE

z


int

int

minus

minus= c

c

x

c

c

x

zdx

dxx

cpp

e

e


⎟⎠⎞

⎜⎝⎛ minus=

ccppz

1)coth(


kTp

Ep

kTEpp z

z 3

3

22=== α



The Hamiltonian is

eEzr

qm


0

22

2

2H

H



)1(100 pzN += ψψ


BarBa 23


ψ


2

2

mqaB =

Requiring

12 =int τψ d





Now using

4

3 0100 0 100 2

0

12 (4 )

med Eψ ψπε

= = minusint rH





are also zero so

)(2 202 eEpaE

NBminus=

H



764 Appendices

Requiring

0=part

partpH

we find

0EeEp minus=


230



302

216 Ba

Eπεα =

part

partminus=

H



061

41)(

21 42


For E = 0

0)( 420 =++minus=


PGE δγβ (2)


20


[ ] 0)(61

41)(

21 2

02


0121)(

31 2


20 4 scc PTT

βγminus= (3)

(b) From (2) 0)( 42


20 4




04


δγ

432 minus=scP

βδγ 2

0 163+= TTc











766 Appendices





φ1 (say 45deg)

Polarizer

lightSample

Analyzerz

zφ2

E

z

yx






iminus= xx xpxp



If H = px2 m +V(x)

[ ] 0)( =xxV So

[ ] xpm

x i minus=H


[ ] iˆ ˆm




2 2ˆ ˆ

2ij ijj ji j i j

f b ωsdot + sdot

=sum sume r e r




jxi

i=minus xpxp xx

xxxppx xx i2 =minus

iminus= xx xpxp


768 Appendices


jj jω=H

jpm


jpm


jpm



1ˆ ˆi ij

i j i jmω

sdot = sdote r e p


ij ji


ωω ω



jiij ωω minus=



Summing over j

ˆ ˆ2iijjbf i im



m

bm

bfj ij 2)i(

i2minus=minus=sum




eEB

NeG +minus

=+ v13

1vvel

2

0εμ



0evv minus= Then

eEB

NeG =minus

minus+minus v13

1vvel

2

0

2ε

μω


ion0

ion 31 αε

NB =

and

Ge

Ee 2

locion

v ==α

at ω = 0

GeNB

2

0ion 3

1)0(ε

=


eEB

GBG =

minusminus+minus v

1)0(

vvel

ion2μω

eEB

BG =⎟⎟⎠

⎞⎜⎜⎝

⎛minus

minus+minus v1

)0(1v

el

ion2μ

μμω

Define

⎟⎟⎠

⎞⎜⎜⎝

⎛minus

minus=el

ion21

)0(1

BBG

T μω



For the local field

el00

loc 1v

31

3 BNeEPEEminus

+=+=εε

Substituting

)(

v 22 ωωμ minus=

T

eE

770 Appendices

Then

)(13

122el0


+=T

eEB

NeEE

)(13

11 22el0

loc


T

eB

NeE

E

)(

11

)0(1 22el

ionloc

ωωμ minusminus+=

TBGB

EE

)()1(

)0()()1(22

el

ion22

loc

ωωμωωμ


=T

Tel

BGBB

EE

Inverting

)0()()1(

)()1(

ion22

22el

loc GBBB

EE

Tel

T


=ωωμ

ωωμ

))(1()0(1

1

22el

ion

ωωμ minusminus+

=

T

locB

BGEE

Define

))(1(

)0(22

el

ion

ωωμ minusminus=

TBBGF

Then

FE

E+

=1

1

loc


loc0

ion0

ionv

31

31)(

EeNNB

εα

εω ==

Substitute in v

)(3

1)( 22loc0ion

ωωμεω

minus=

T

eEE

eNB

)(

)0()( 22

ion

locion

ωωμω

minus=

T

GBE

EB

)1(1

1)( elion BFF

B minus+

=ω


Given

[ ] )1()(1)(3)()(

elionel

ion

0 BBBB


ωω

εεωε


)1()1(

11

)1(1

3)()(

elelel

el

00 BBF

FB

BF

F

minus⎥⎦⎤

⎢⎣⎡ minus

+minusminus

minus++infin=

εε

εωε

)1(

11

13)()(

el00 B

FF

FF

minus⎟⎠⎞

⎜⎝⎛

+minus

++infin=εε

εωε

)1(

3)()(

el00 BF

minus+infin=

εε

εωε

Substitute for F

))(1(

)0()1(

3)()(22

el

ion

el00 ωωμεε

εωε


TBBG

B

Define c

2el

ion

)1()0(3

BBGcminus

=μ

Then

)(

)()(2200 ωωε

εεωε

minus+infin=

T

c

Rewriting

202)()(

ωεωε

εω =infinminus

minusc

T


2

2

)0()(

L

T

ωω

εε =infin

)()0( 2

2infin= ε

ωωε

T

L

772 Appendices


0)()0(

02 =infinminus

minusεε

εω cT

0)()(2

202 =


εεωω

εω

T

LT

c

0)()( 22

202 =

infinminusminus

εωωωεωTL

TT

c

0)(

)(2

0222 =infin


TLTc

202)( LT

c ωεεω =infin

+



ψψ Em

=nablaminus 22

2

For each coordinate

0222

2=+minus i

i

i

mEdxd ψψ


⎟⎟⎠

⎞⎜⎜⎝

⎛+⎟

⎟⎠

⎞⎜⎜⎝

⎛= i

ii

ii xmEBxmEA 22

2cos2sinψ


2

222

2 2or2

manEnamE i

iii ππ ==

)(2

23

22

212

22nnn

maET ++= π





2

22

2

22

23

2)111114(



2

22

23

mahf π=


constant432 ==

mfa π



0

22

2

2

2

2

22

4where1

2 πεφeq

rq

rrrm=minus⎟

⎟⎠

⎞⎜⎜⎝

⎛

partpart+

partpart+

partpartminus=H




εεε RRYEmqEmq

r ==== )(2

2

4

2

2ρ


ψψρ

ψφρρρρ

ε2

4

2

22

2

2

22

2

4

422

2)(11

22mqmqqqm

m=minus⎟

⎟⎠

⎞⎜⎜⎝

⎛

partpart+

partpart+

partpartminus

774 Appendices

Canceling we obtain

ψρψψ

φρρρρε

2111

21

2

2

22

2=minus⎟

⎟⎠

⎞⎜⎜⎝

⎛

partpart+

partpart+

partpartminus




φφ mie)( =Φ



⎞⎜⎜⎝

⎛minus+minus

21

21 2

2

2 ερρρρρ

mdd

dd

We let

2

and1 2

2xαρ

α=minus=ε


Ψ=Ψ+⎟⎟⎠

⎞⎜⎜⎝


241

21 22

2

2 xxm

dxd

xdxd


21122 +=+= nn αα


2212

21 )(

or)(

1+

minus=+

minus=n

REn

ε


21)1(

21 minus+=+ nn





)()(2 2

22zEz

dzd

mψψ =minus


0)(2)(22

2=+ zmE

dzzd ψψ

⎟⎟⎠

⎞⎜⎜⎝

⎛+⎟

⎟⎠

⎞⎜⎜⎝

⎛= zmEBzmEAz 22

2cos2sin)(ψ



2

222

2mwpE p

π=





12

2

0minusminus=

xeF

πε


x

ex

edxx

eFdxdVxxxx

0

2

0

2

20

2

161




x

eExV0

2

0 16)(

πεminus=

776 Appendices


x

exeEExV0

2

10 16)(

πεminusminus=


01

=⎟⎠⎞

⎜⎝⎛

=xxdxdV

which gives

10

1 16 Eex

πε=

or

0

13

01 21)(

πεEeExV minus=

Thus

0

13

21

πεEe=Δ



a

V(r)

ndashV(0)

E (bound state)

r




20 )(2 EVm +

=α

22mEminus=β



This implies

2


Also

E

EV

aa

minus=

+=

+= 0

22

2

22

)(cot11)(sin

βαα

αα


2πα gta

to estimate a

eV)1(22)(22

22

0 mEVma ππ

απ gt

+=cong

nm931)1061)(1019(2

)10636)(143(1931

34

min congtimestimes

timescongminusminus

minusa




778 Appendices

Energy Gap

Ec(z)

Eg1 Eg2

Valence Band

Conduction Band

d

z

Ev(z)


EFFzEzmyxm

c =⎥⎥⎦

⎤

⎢⎢⎣

⎡+

partpartminus⎟

⎟⎠

⎞⎜⎜⎝

⎛

partpart+


22 2

22

2

2

2

22




32||

2

322

2)(

2)(

2Ek

mEkk



ψψ 32

22)(

2EzE

dzd

mc =⎟

⎟⎠

⎞⎜⎜⎝

⎛+minus lowast



sum=

minus=N

ni

ndiq ndz

Nz

1

i )(e1)( χψ


0 we find for E3




2

2 2 0 0( ) cos












)(22

||

222

2

222

EE

kkmmw

nE

n

yx

+equiv

++= π

)(22

222

2||

2

|| yx kkm

km

E +==




ndEm

dkdkkkd

2

2||||||

2

2

2

π

ππ

=

==

780 Appendices


)(

)(2)2(

2)(

2

0||||22

sum

sum int

minus=

minusminus=infin

n n

n n

EEm

dEEEEmED

θπ

δππ



23)(π

mED = (1)

In 3D

E

dEmdkmkE 2

22 221

2==

2

212)4(

)2(2

)unit volper (4)2(

2)(

223

23D3

EdEmmE

dkkdEED

ππ

ππ

=

=


32

22

1)(223

22D3

E

wmmED π

π ⎟⎟⎠

⎞⎜⎜⎝

⎛=

where

2

222

32

3mw

EE π==

Per unit area

222D3332

21)(

ππ

πmw

wmwED == (2)

which checks (1)


d)

D3D(E)w

2hπm

23hπm

E1 E2 E3




1001

512

512

51

2

51

2~

21

21

22

11

I

NN

NN

NN

NN

AA

=⎟⎠⎞

⎜⎝⎛=

⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜

⎝

⎛minus+

⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜

⎝

⎛

minus

+=


Bibliography

Chapter 1

















784 Bibliography










Chapter 2









Chapter 2 785





















786 Bibliography

Chapter 3





















Chapter 3 787



















788 Bibliography

Chapter 4





















Chapter 5 789














Chapter 5







USA 92 6663-6667 (1995)

790 Bibliography






















Chapter 6 791



Francis London 1963

Chapter 6
















1994

792 Bibliography






















1996

Chapter 7 793

Chapter 7



















794 Bibliography



















Chapter 8 795










(2004)

Chapter 8






796 Bibliography






















Chapter 9 797





Chapter 9

















798 Bibliography




Chapter 10















Chapter 11 799






Chapter 11















800 Bibliography


















Chapter 12



Chapter 12 801























802 Bibliography





















Chapter 12 803

















804 Bibliography

Appendices





















Subject References


Mechanics




Electricity



Optics



Thermodynamics






Critical Phenomena


806 Bibliography

Crystal Growth


Modern Physics



Quantum Mechanics





Math Physics


Solid State







Condensed Matter






Problems




Applied Physics





Rocks




808 Bibliography


The Internet



Periodic Table


Further Reading












Further Reading 809
















810 Bibliography

















Further Reading 811


















812 Bibliography




Index

A













B


814 Index










C




Index 815





















816 Index




316 317

D









594 640






E





Index 817

















503 566










F



818 Index


493 538 Fermi function 160 248 298 303



373 385 399 407 422 426 456 486 513







G

GaAs 242 295 297 324 326 328 329 337 347 563 564 579 618 628 629 637




Index 819

Group velocity 55 306 532 562 627


H











I






820 Index


J


K



L









529 570

Index 821




M













N


531 600 NaCl 6 9 28 579 600

822 Index









O








P





Index 823















824 Index


Q




R












S



Index 825













826 Index






T



361 418











475 509

Index 827

U




V



Vortex region 466

W


X


Y

YBa2Cu3O7 506 508

Z


Appendices

A Units










G1 Bose Particles

G2 Fermi Particles


H1 Propagators

H2 Green Functions

H3 Feynman Diagrams

H4 Definitions




J Folk Theorems



M Problem Solutions














































































Bibliography

Mechanics

Electricity

Optics

Thermodynamics


Critical Phenomena

Crystal Growth

Modern Physics

Quantum Mechanics

Math Physics

Solid State

Condensed Matter


Problems


Applied Physics

Rocks


The Internet

Periodic Table

Index







0)()(d

)(d2

2=+ xxU

xx ψψ (C1)


)()( xUnaxU =+ (C2)








0d

d =x

W (C6)

where

21


=W (C7)


658 Appendices



0)()(d

)(d2

2=++++ axaxU

xax ψψ


0)()(d

)(d2

2=+++ axxU

xax ψψ (C8)


)()()( 211 xBxAax ψψψ +=+ (C9)

)()()( 212 xDxCax ψψψ +=+ (C10)


21

21

21

21

2121

2121ψψψψ

ψψψψ

ψψψψψψψψ

primeprime=

primeprime=


DBCA

DCBADCBA

or

1=DBCA

or

1=minus BCAD (C11)


)()( xΔax ψψ =+ (C12)


)()()( 21 xxx βψαψψ += (C13a)

or

)()()( 21 axaxax +++=+ βψαψψ (C13b)


)()()()()()()(

2121

xxxDBxCAax


+=+++=+ (C14)



αΔβα =+ CA

and

βΔβα =+ DB


0=minus

minusΔ

ΔDB

CA (C15)






DA +=+ minus++

1)(ΔΔ


1)(ΔΔ (C17)

and

DA +=+ minus+ ΔΔ



If we set

be=+Δ (C19)

and



)()( 11


and

)()( 12


660 Appendices





where

)()( xuaxu =+ (C25)



Let

)()( 11 NN xxExx ψψ =H (C26)



0][ =lTH (C27)

and






lkmiddotl

ie=t (C31)

Further




so that

12 =lt (C32)










)(02)(0

22

21 Gnnn G =+++


5 In general sum= j

kj

klij

kil AT ψψ

where the Ail




kklkl AT ψψ =


662 Appendices




ntctx nin

i sum= )()(ψ


nmNi

im

imnm cctctc

Nlowast





)( ATrA ρequiv


][i Ht

ρρ =partpartminus


⎟⎠⎞

⎜⎝⎛ minus=

kTHFexpρ


)][exp()]exp([exp

kTHTrkTHHTr

kTHFHTrHU

minusminus=⎥

⎦

⎤⎢⎣

⎡⎟⎠⎞

⎜⎝⎛ minus==








Let

)()( 0 tVt += HH (E1)

lEl l00 =H (E2)


00lk

lkEE minus=ω (E4)




664 Appendices


200

2 2)()(

)(

)cos(1lim ωδπ

ω

ω tfEEttfi

if

if

t=minus=

minusinfinrarr

(E6)












⎟⎟⎠

⎞⎜⎜⎝

⎛=

00σ

σα (F2)




⎟⎟⎠

⎞⎜⎜⎝

⎛minus

=I

I0

0β (F3)



12

)(2

0

20 ltlt

minusminusequiv

cmVcmEε (F4)


20cmEE +prime= (F5)

and

⎟⎟⎠

⎞⎜⎜⎝

⎛=

φχ

ψ (F6)




⎤

⎢⎢⎣


4482 220

2

220

2

230

4

0

2pσ V

cmV

cmV

cmp

mpE (F7)


])[(4 22

0


cm (F8)


666 Appendices


SL middot dd1

2 220

2

rV

rcm (F9)





G1 Bose Particles




][

0][

0][

dagger

daggerdagger

ijji

ji

ijjiji

bb

bb

bbbbbb

δ=

=

=minusequiv

9 See Mattuck [A14]



daggeri i in b b=

and

dagger1 iii bbn =+


11

1

11dagger

11



++=

minus=

iiii

iiii

nnnnnb

nnnnnb



1

1dagger

iiii

iiii

nnbn

nnbn

=minus

=minus



dagger)1()1(






)()(


sumlowast=

=

α αα

α αα

ψψ

bu

bu

rr

rr


668 Appendices

G2 Fermi Particles




0

0

dagger

daggerdagger

ijji

ji

ijjiji

aa

aa

aaaaaa

δ=

=

=+equiv







iii nan




0











)2()1()2()1(

210daggerdagger

ββ

ααβα uu

uuaa harr











670 Appendices






H1 Propagators




H2 Green Functions


δεω

ωi

1)(0 +minus=+

kG k (H1)


H3 Feynman Diagrams


H4 Definitions






672 Appendices




















dagger )2()1()21()2()1(21)1()1()1(H (H3)



31



0

daggerdagger

=+=+

ijji

ijijjiaaaaaaaa δ



δεω

ωi


=sum

+

l lklkk VG k (H6)





674 Appendices



112223 d)21()()()()(d rrrrrr φφφφ (H8)




δεω

ωi)(


=sum

+













klk

Gδωεω

ωi)(


=sum k

k (H11)





676 Appendices




fcc (110) (011) (101)

2 2 2

bcc (11 1) ( 111) (1 11)2 2 2

a a a




1 2 31 ( )Ω






lkh

lkh dG π2


Δ =k G (See 152)











G (See 141)





⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎟⎟⎠

⎞⎜⎜⎝

⎛+⎟

⎟⎠

⎞⎜⎜⎝

⎛minus=

126

4)(ijij

ij rrrU σσε


1e

1 minus

= kTn ω (See 277 2215)


222 akmks=ω (See 248 288 290)






D

DDωωωωωω

gt=leprop

0)( 2

(See 2224)

3126v ⎟

⎟⎠

⎞⎜⎜⎝

⎛==

VN

kk DDD

πθθω (See 2228)


678 Appendices


intminus

⎟⎠⎞

⎜⎝⎛= T

x

x

VdxxTNkC

0 2

43

)1e(e9 θ

θ (See 2229)


3( )

( )( )(2 ) const

dAVD ω

ωω

ωπ= int

k knabla (See 3256)




)e(e 20


2( ) TWθ⎛ ⎞prop Δ ⎜ ⎟⎝ ⎠

k


22

2

9)(

Dm

m mkAT θ=



V

Dln



B

C3

vγα = (See 2237 2238 2250)




ρ11Cv = (See 2270)


ρ44Cv = (See 2270)



3 Electrons


1e

1)( +




( ) ( )u u= +k kr r R (See 3129 C25)


EmVEg23

222

2)( ⎟⎟

⎠

⎞⎜⎜⎝

⎛=π

(See 3164)


32

22

32

)0( ⎟⎠⎞

⎜⎝⎛==

VN



m

ne τσ2

= (See 3214)




RH (See 6102 1223)

680 Appendices


LTK =σ

(See 3215)

22

3⎟⎠⎞

⎜⎝⎛=

ekL π (See 3215)


0

00 near band edge

E E U

U E E

lowast

+

minus=

minusk G

G k G (See 3230)



k r r R (See 3244)


k (See 3255)




i

Bloch Fns


kkr R r (See 3332)



cgs)in or (mceB

meB

c =ω (See 3196)




22

2 2

0 ( )

( )

( ) ( ) ( )2

1( ) ( ) ( )4

el

ioni i

j i i ij i

V

Vm

e d E

ψ ψ

ψ ψ ψπε ne

minus nabla +



r

r r r

r r r rr r




22

2

0 ( )|| spin

( ) ( ) ( ) ( ) ( )2

1 ( ) ( ) ( ) ( )4

ion eli i i

j i j i ij i

V Vme d E

ψ ψ ψ

ψ ψ ψ ψπε

lowast

ne

minus + +


r r r r r

r r r r rr r

nabla



1 32

2 3( ) ( ) ( ) ( ) ( ) ( ) ( )2


mψ ψ ψ ψ

π⎛ ⎞⎡ ⎤minus + + minus =⎜ ⎟⎣ ⎦ ⎝ ⎠

r r r r r r rnabla



2

2

1 ( ) ( )( ) 1( ) ( )k

e f fV E E iq σ

ε ωω ηε


k k qqk q k

(See 9166)

xx

xxxF

kqxxF

qk

qTF

sminus+minus+==+==

11ln

41

21)(

2)(1)0(0

2

2

2ε


F

srk

F Eemk

rkq s

εεπφ

23e

41 2

02 ==ltlt minus



collisions

f ff ft t


Fv nabla nabla


0RHS of BDE( )

f fτminus

= minusk

(See 4149 4145)

682 Appendices


11 123 v( ) ( ) ( )4

edj e f L E L T

eμ

π⎛ ⎞= minus = + + minus⎜ ⎟⎝ ⎠


21 223 ( )v( ) ( ) ( )4

Qdj E f L E L T

eμ

π⎛ ⎞= minus = + + minus⎜ ⎟⎝ ⎠




0ej TE

σ = =nabla

0Qe

jj

Tκ = =

nabla


0eEQ jT

= =nabla

0Q

e

jT

j

QT

Π = =

Π =

nabla


constant131 conganc



⎟⎟⎠

⎞⎜⎜⎝

⎛minus

VED F )(1exp




heG =




D

D

TT

TT

θρ

θρ

ltlt

gtgt5~

~ (See 4146)





)SI(21

0Ae

Bπ=⎟

⎠⎞

⎜⎝⎛Δ (See 534)



⎟⎟⎠

⎞⎜⎜⎝

⎛= (cgs) 4 2

0

2

mne

mne

pπ

εω (See 10108)







2innp = (See 613)

⎟⎟⎠

⎞⎜⎜⎝

⎛minus⎟

⎟⎠

⎞⎜⎜⎝

⎛= lowastlowast

kTE

mmkTn ghei 2

exp)(2

2 4323

2π (See 614)

684 Appendices


ddt

= kF (See 644)




2

2

211

kE



eDkT =μ (See 684 685)





⎟⎟

⎠

⎞

⎜⎜

⎝

⎛ ΔΔ== (cgs)

2

222 ne

V

ne

VKZwidth b π

εε (See 6193)


⎥⎦

⎤⎢⎣

⎡minus⎟

⎠⎞

⎜⎝⎛

⎟⎟⎠

⎞⎜⎜⎝

⎛+= 1exp

00 kTep

LD

nLDej b

np

pp

n

n ϕ (See 6241)



23minusprop Tμ


23Tpropμ





⎟⎠⎞

⎜⎝⎛ )(

22cgs

mceB

meB (See 7255 7256)

Lande g factor

⎥⎦

⎤⎢⎣

⎡+

+minus++++=)1(

)1()1()1(211

JJLLSSJJg

(Implied by (76) (710))


)(20 FBp EDμμχ = (See 3181 3201)


magnetonBohr =Bμ


)groundexcited(22

==minus

= gexcEEgexc

VN

gexc

zμχ


⎟⎠⎞


⎤⎢⎣⎡ ++=

Jx

JJxJ

JJxBJ 2

coth21

2)12(coth

212)( (See 716)



4sincos)()2(

xxxxxFRkFJ ijFij



kT

BgJx eff

Bμ= (See 722)

MBBeff λ+=

686 Appendices


)cgs(

32

0MMH

kEk effB

F

μθθ =primegt

prime (See 7222)


⎟⎟⎠

⎞⎜⎜⎝

⎛minusprop

)(1exp

FK EJD

T (See 7300)


h

eGxy

2ν= (See 1247)



KJpropδ (See 7251)





23

23)()0(

TC

TTMM

M prop



)for (1cc

c

c

TTHTTM

TT

TTC

=asympminusasymp

minusasymp

minusasympminus

minus

δβ

γ

α

χ






2

( ) ( ) (cgs)nemc



2

2

4 nemc

L πλ = (See 86)


g

FEv

πξ 2~0 (See 849)



II ξ lt λ 20~

1 πλϕ

cH 20~

2 πξϕ

cH 2

~1

2⎟⎟⎠

⎞⎜⎜⎝

⎛ξλ

c

c

HH



)(20 cgs

ehc=φ (See 845)



⎟⎟⎠

⎞⎜⎜⎝

⎛minuscong

)(1exp141

FDc EVD

T θ (See 8217)


⎟⎟

⎠

⎞

⎜⎜

⎝

⎛

+minus


1exp

451 epep

epDcT

λμλ



688 Appendices


2

20



⎡ ⎤⎛ ⎞⎢ minus + + ⎥ =⎜ ⎟⎝ ⎠⎢ ⎥⎣ ⎦

Anabla (See 83)

2

( ) ( 2 2 )2s eiq qj q e m m






)()0(

2

2

infin=εε

ωω

T

L (See 950)


⎟⎟⎠

⎞⎜⎜⎝

⎛+=+= )SI(

3)guassian(

34



⎟⎟⎠

⎞⎜⎜⎝

⎛=

+minus=

+minus

sumsum )SI(31

2)(1)(

34

21

00

0jjjj NN α

εεεεεαπ

εε (See 930)



direct 0eΔ =k








)cgs(2 222

4

neEE gn ε


)cgs(111

he mm+=

μ (See 10101)


11 Defects


Mollwo Equation







⎟⎟⎠

⎞⎜⎜⎝

⎛==

pGeEp

Bπω 2 (See 129)


)(exp 41 MottT

AB ⎟⎠

⎞⎜⎝





232surface =⎟

⎠⎞


kTWkTemj

π (See 1145)

690 Appendices


3 2

2surface exp BWj AE

E

⎛ ⎞= minus⎜ ⎟⎜ ⎟

⎝ ⎠ (See 1149)


J Folk Theorems


Chapter 1



Chapter 2



Folk Theorems 691

Chapter 3









Chapter 4







Chapter 5


Chapter 6




692 Appendices















Chapter 7






Folk Theorems 693





Chapter 8




Chapter 9



Chapter 10





Chapter 11







Chapter 12



694 Appendices







Gauss law V A A

dV disin





10

e +

infinminus =int n

axn

andxx


π41Y

e21)(

Y)(

00

23

10

0010001

=

⎟⎟⎠

⎞⎜⎜⎝

⎛=

=Ψ

minus Bar

BarR

rR





r θnabla ϕ

221 1 1( ) (sin )

sin sinrA

r A Ar r rr

ϕθθ



2

2

2222

2 sin1sin

sin11

ϕθθθ

θθ partΦpart+⎟

⎠⎞

⎜⎝⎛

partΦpart

partpart+⎟

⎠⎞

⎜⎝⎛

partΦpart

partpart=Φnabla

rrrr

rr2



111


=primeprime

γγγ

nnn

n


n

ennn ⎟⎠⎞




)()(

1))((

)()()(0)(


n

n nn

xxxf

xxxf

xf

bfdxbxxfxx

xxbxdxbx

=

minusprime

=


minus===minus

sum

int

int

δδ

δδ

δδδ

Complex Variables


π


nazfazdzd

naf n

n

n


)1(1)(Res 1

1

⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧

minusminus

= minus

minus

rarr


SI Gaussian

0

t

tρ

parttimes = +part


sdot =sdot =

H J D

BE

DB

nabla

nabla

nablanabla

4 1

1

40

c c t

c t

π

πρ

parttimes = +part


sdot =sdot =

H J D

BE

DB

nabla

nabla

nablanabla

0ε= +D E P

0

1μ

= minusH B M



( )qc

= + timesvF E B

696 Appendices












1956 Transistor


















9 Total for Decade

698 Appendices

























A Geim K Novoselov

2010 Graphene



Brillouin




Bloch






Bardeen



700 Appendices

Others






M Problem Solutions





451388

2412

126


=+minus=M




1

A

2

3

456

7

8

g5 g6

g7

g8


⎟⎠⎞

⎜⎝⎛= 87654321

876543211g

⎟⎠⎞

⎜⎝⎛= 21876543

876543212g

⎟⎠⎞

⎜⎝⎛= 43218765

876543213g

⎟⎠⎞

⎜⎝⎛= 65432187

876543214g

⎟⎠⎞

⎜⎝⎛= 23456781

876543215g

⎟⎠⎞

⎜⎝⎛= 45678123

876543216g

⎟⎠⎞

⎜⎝⎛= 67812345

876543217g

⎟⎠⎞

⎜⎝⎛= 81234567

876543218g

702 Appendices

Then eg

432 6543218787654321 ggg =⎟⎠⎞

⎜⎝⎛=


g1 g2 g3 g4 g5 g6 g7 g8




)(21 kjia minus+= a


)(23 kjia +minus= a

x

y

z






y

z

x

a2

a1

a3

a


a


704 Appendices





2 2 0h kh k h k


a bG

since

1 2 32 ( )h k lπ= + +G b b b

and


Thus

( ) 2( )( ) ( )h k ld h k lh k l h h k l

π= =G aG G

sdot






where

f iΔ = minusk k k



ki

b

b

bα

bα

hellip

x

y

z

kf

0ˆ =sdot ik i

2sin

2sin

e

2sin

2sin

e

e1

e1

e1

e1

ee

2)1(

i2

)1(i

i

i

i

i

10

i10

i

⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜

⎝

⎛

+⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜

⎝

⎛

=

minus

minus+minus

minus=

+=

minusminus

minusminus

minus

minus

minus

minus

minus=

minusminus=

minus sumsum

α

αα

α

α

α

bk

Mbk

bk

bNk

P

fx

fxMbk

fx

fxNbk

bk

Mbk

bk

bNk

Mm

mbkNn

nbk

fxfx

fx

fx

fx

fx

fxfx


)(integer 2

and)(integer 2

)2()1(

pbk


απ (a)

Since

pl

bk

bk

pl

bk

bk

fx

fx

fx

fx αα

== )2(

)1(

)2(

)1(

or1

)2()1(fxfx kk ne



since

22fxfy kkk minus=


706 Appendices


Let

λπ2=K





Define

θsinKB =

)iexp( BbZ =

So


But


)]2sin(ln[

1)]2sin()exp[(ln

0

)2sin(

knbZ

knbZm

knbZZ

m

knb

πε

πε

πεπε

+=

=

=

suminfin=

Thus


0 )2sin()ln( πε


2)2cos(21)2sin(ln

11

ZkbZkbZZ

ZE

+minus+

minus=

ππε






mGBb π2+plusmn=


b




Aring2255=Ka


Aring2255100 == ad

)bcc(6125222255


deg=rArr== 6816225551

2sin θλθ

d

deg=rArr= 04352255

)51(2sin θθ



θλ sin2 100dn =

above as 2)e1( 0435 4510

)51(42

4sin


)51(32

3sin

above as 2)e1( 6816 4510

)51(22

2sin


2sin

i4400

100

i3300

100

i2200

100

i100

100

ffSd

fSd

ffSd

fSd

=+====

=+====

=+====

=+====

minus

minus

minus

minus

π

π

π

π

θλθ

θλθ

θλθ

θλθ

708 Appendices




mm m

γ γ γx1 x2

1γx3

3 1 2 3





02

22

321

2

2

2

=⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜

⎝

⎛


uuu

mm

m


(2)


twice32mγω = (5b)


⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛==

111

31

321

0uuu

Uω (6a)

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛minus==12

1

51

3 mU γω (6b)


and

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛minus==

112

51

3 mU γω (6c)



)(2iexp1

)(2iexp1

)(2iexp1

)1(0

10

10

sum

sum

sum

minusminus= minus

minus=

minus=

⎥⎦⎤

⎢⎣⎡ minus=

⎥⎦⎤


⎥⎦⎤

⎢⎣⎡ minus=

Ns s

Ns s

Ns snm

nmN

sgN

nmN

sgN

mnN

sgN

G

π

π

π


[ ]

⎥⎦⎤

⎢⎣⎡ minus=

minus⎥⎦⎤

⎢⎣⎡ minus=

⎥⎦⎤


⎤⎢⎣⎡ minus=⎥⎦

⎤⎢⎣⎡ minus+

)(2iexp

)(2iexp)(2iexp


nmsN

nmnmsN

nmNN

nmsN

nmNsN

π

ππ

πππ

and (see 2104)

⎟⎠⎞

⎜⎝⎛=⎟

⎠⎞

⎜⎝⎛ +=⎥⎦

⎤⎢⎣⎡ +

Ns

NsNs




Nsg

NG π

But

⎟⎟⎠

⎞⎜⎜⎝

⎛⎥⎦⎤



Nnm

NNggN

ππ

so

mnN

snm GnmN

gG =⎥⎦⎤


0 )(2iexp π

QED

710 Appendices




)2()()(

)()(

)2()()(

)()(

)2()()(

54145515

34454

32423343

12232

15251121

XXXXXXXXm

XXXXXm

XXXXXXXXM

XXXXXm

XXXXXXXXm


minusminusminus=


minusminusminus=


αααβα

βββαβ

ααα


0

210011100020001110012

54321

22

212

2

=

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛

minusΩminusminusΩ

minusΩminusminusΩ

minusΩ

minus

uuuuu

AAAAA

AAδ


⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

Ω=

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛



minusminus

54321

2

54321

210011100

020001110012

uuuuu

uuuuu

AACCC

AA


⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

minusminus

=

1

0

1

a

aU


This requires

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛

ΩminusΩminus

sdotΩΩΩ

=

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

+minus++minus

minus+minus

=

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

minus+minus+minus+

+minus++minus+minus

22

2

2

2

0

31)1(

01)1(

3

211))(1(

)1(112

a

a

aAa

Aaa

aaAcaca

Aaa




2





)1()1()1( dagger

dagger

ϕψψ

ψψϕ

minus=minus=minus=

==

nna

aaa

aaaNaN

n

n

nn

712 Appendices


nKan =minus1 So

112

dagger2

nnnK

naanKnn

=

=minusminus


1or1 minus== nnnan

K


)1()1(

)1(dagger

daggerdagger

daggerdaggerdagger

ηψ

ψ

ψψη

+=+=

+=

==

nna

aaa

aaaNaN

n

n

nn


1

1+

=n

K

so

11dagger ++= nnna


⎟⎠⎞

⎜⎝⎛ +=⎟

⎠⎞

⎜⎝⎛ +=

21

21dagger Naa ωωH



321321


⎜⎝⎛=



321

321

321

ηηη

ηηη

ηηη

partpart

partpart

partpart

partpart

partpart

partpart

partpart

partpart

partpart

=

zzz

yyy

xxx

J

1 1 2 2 3 3η η η= + +r a a a


etc11

xax =partpartη

so we find

1 2 3

1 2 3 1 2 3

1 2 3

( )x x x

y y y

z z z

a a a

J a a a

a a a

= = sdot timesa a a

QED



1 1

2 1

3 1

100

sdot =sdot =sdot =

a ba ba b



a

KΩ

= 1

and

2 31

a

times=

Ωa a

b


714 Appendices


1 1 2 2 3 32 ( )n n n nπ= + +G b b b




31 21 2 3

1 2 32

nn nN N N

π⎛ ⎞

= + +⎜ ⎟⎝ ⎠

q b b b



a) 31 231 2 31 2

31 2

1 2 3

1 1

11

1

22 2 ii i 11 1i

1 2 3 1 2 30 0 0

i2

2i1

1 1 1 1e e e e

1 1 e etc



l

l

n

nn

nn n ll l NN NNN N

l l l

n l

n lN

i i

i i

n

N N N N N N

N

n Nn N

ππ π

π

π

δδ


= = ==

minus=

minus= ne= =

=

=

sum sum sum sum

sum

q R

R

Gq

q GG

G

b) 111

1

1

1

11

21

i(BZ)

1 2 3 1 0

2

21

0

1 1e e etc


1 1 e


l

l

ni lNN

l

i n

i lN

i i

i i

N N N N

N

l Nl N

π

π

π

δ

minussdot

==

minus=

minus= ne= =

=

sum sumq Rq

R


c)

1 2 31

1 1

3

33

1 2 31 2 3

31 2 3 03

fpp 33

( ) 2

of s( )Vol of space

1 ( )(2 ) ( )

( )(2 )

( ) (2 )

n n nn

f fN

qf dq

f d

N N NN N N

f d

Vf d

π

π

π

π

⎛ ⎞⎛ ⎞= +⎜ ⎟⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠

=

=sdot times

Ω=

sum sum

int

int

int

int

q q b

q q

q qb b b

q q

q q

d)

i i3fpp3

fpp

i

1 2 30

e e (with )(2 )

1 e

l l

l

l

a ad q VV

N N N

π

δ

Ω Ω= rarrinfin

=

=

sumint

sum

q R q Rq

q Rq

R

sdot sdot

sdot

e)

1 1 1 2 2 2 3 3 3i( ) i2 [( ) ( ) ( ) ]3 31 1e el l l l l l l luc



3

1 2 3


η η η⎛ ⎞

= = ⎜ ⎟⎝ ⎠

r


etc so

1 2 3

1 2 3 1 2 3

1 2 3




So

1 1 1

1 1 111

1i( ) i2 ( )310

1i2 ( )

1 1 0

1 e e

e i2 ( )

l l l l

a

l l

d d

l l

π η

π η

η

δπ


primeminusprime

=Ω

= =prime minus

int intG G r

ll

rsdot

716 Appendices

f)

i ( ) 3all 3

i ( ) i ( ) i ( )

i( )i( ) i( )

1 e(2 )

1 1 1e e e2 2 2( )

1 e 1 e 1 e2 i( ) 2 i( ) 2 i( )

yx z

x y z

yx yx z

zx y



LL L

d q

dq dq dq

L L L

x x y y z z

π

π π π

π π π

primeminus



minusminus minus

=

rarr infin


int

int int int

q r rq

sdot


( ) ( ) ( )( ) ( ) ( )( )

z

x y z

yx zL L L


x x y y z zπ π π

δ δ δδ









Frequency c

gωω=

Temperature )( k

Ttcω

=


Uucω

=

Entropy NkSs =


NkC

c vv =



⎟⎠⎞

⎜⎝⎛ +⎟⎠⎞

⎜⎝⎛==

Nd

ddpP

c

ωωωωω


)(

Thus

π

ωω2

)()( NadqdqqNdN ==

1

)2)(2cos(1

2)(

22cc

cN

aqaN

ddqNadqN

ωωπω

ωπωπω

minus=

==

So

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

minus⎟⎟⎠

⎞⎜⎜⎝

⎛

minus==

221

11e

1

ckTc d

dpPωωπω

ω ω

or since

tg

kTkT c==

ωωωω

c

then

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

minus⎟⎟⎠

⎞⎜⎜⎝

⎛

minus=

21

11e

1)(g

gtP tgπ (1)



int

intsum

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

minus⎟⎟⎠

⎞⎜⎜⎝

⎛

minus=

minus=

minus= =

c

c

m

dN

dNU

cckT

kTNm kT

m

ω

ω

ω

ωω

ωωωω

ωπ

ωωωω

0 22

01

1

11e

2

)(1e

21e

int⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

minus⎟⎟⎠

⎞⎜⎜⎝

⎛

minus==

c

cc

kTc

cd

NUu

ω

ω ωωωω

ωωπω 0 22 )(

1

11e

2

718 Appendices

or

int⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

minus⎟⎟⎠

⎞⎜⎜⎝

⎛

minus=

1

0 21

11e

2 dgg

gu tgπ (2)

ωωωπω

ω

ω

ω

ω

ω

ω

ω

dNkT

kTC

cckT

kT

m kT

kTm

c

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

minus⎟⎟⎠

⎞⎜⎜⎝

⎛

minus=

minus=

int

sum

2202

2

2

2

2

2v

1)1e(e)(2

)1e(e)(1

int⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

minus⎟⎟⎠

⎞⎜⎜⎝

⎛

minus==

c

cc

kT

kT

c

c dTNkN

NkCc

ω

ω

ωωω

ωω

ωπω

ω

0 222

2

22v

v )(1

1)1e(

e)()(

2

int⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

minus⎟⎟⎠

⎞⎜⎜⎝

⎛

minus=

1

0 22

2

v1

1)1e(

e)(2 dgg

tgc tg

tg

π (3)


kTFj



VT

FS ⎟⎠⎞

partpartminus=

so

( ) sumsum=

minus=

minus

minus+minusminus=

N

m

mkT

N

m

kT

kTkTkS

mm

12

1

e11e1ln ωω

ω

( )

int

int

minusminus+

minusminusminus=

=

minus

c

c

cc

kT

cc

kT

dkTkTNNk

dNkNk

NkSs

ω

ω

ωω

ωωωω

ωπ

ωωωωπ

0 22

2

0 22

)(1

11e

)2(1

)(1

1e1ln21


+minus

minusminus= minus1

0 2

1

0 2

1

11e

2

1

1e1ln2 dgg

tgdgg

s tgtg

ππ (4)



range of ω

ωxxp


P10x1π

1

e

ωx

10 1minus

sdot1

1 ωx( )2minus

sdot=

P100x1π

1

e

ωx

100 1minus

sdot1

1 ωx( )2minus

sdot=

P300x1π

1

e

ωx

300 1minus

sdot1

1 ωx( )2minus

sdot=

0 05 10

200

400

P300

P100

P10

ω

720 Appendices


the range of t

txxr


ux2π

0

1

yy

e

y

tx 1minus

1

1 y2minussdot

⌠⎮⎮⎮⎮⌡

dsdot=

spht x2π

0

1

y

ytx

⎛⎜⎝

⎞⎟⎠

2e

y

txsdot

e

y

tx 1minus

⎛⎜⎝

⎞⎟⎠

2

1

1 y2minussdot

⌠⎮⎮⎮⎮⎮⎮⎮⎮⌡

dsdot=

0 1 2 3 40

1

2

3

4

u

spht

t


0 05 1 15 20

05

1

15

2

u

spht

t


Entropy t2π

minus

0

1

ωln 1 e

ωminus

tminus

⎛⎜⎝

⎞⎟⎠

1

1 ω2

minus

sdot

⌠⎮⎮⎮⎮⌡

dsdot2π

0

1

ω

ω

t

e

ω

t 1minus

1

1 ω2

minus

sdot

⌠⎮⎮⎮⎮⎮⌡

dsdot+=

0 100 200 3000

2

4

6

8

Entropy t

t

722 Appendices




By Problem 29

2 31

a

times=

Ωa a

b


2 3 [(1)(1) (1)( 1)] [(1)(1) ( 1)(1)] [( 1)( 1) (1)(1)]

2

(2 2 ) ( )2

a

a a


= + = +

a a i j k

i j i j

and

2

21 2 3( ) (2)


so Ωa = a2 and

11 ( )a

= +b i j


2

3

1 ( )

1 ( )

a

a

= +

= +

b j k

b i k




1 2 31 1 1( ) ( ) ( )a a a




ky

kxkz


a

a

n m + 1

n mn ndash 1 m

n m ndash 1

n + 1 m

xnm ynm


])()()()[(2

])()()()[(2

21

21

21

21

21

21

21

21

mnmnmnmnmnmnmnmn

mnmnmnmnmnmnmnmn

yyyyyyyyk

xxxxxxxxkU



minus+minus+

minus+minus+


)4( 111

mnmnmnmnmn



)4( 111

mnmnmnmnmn



724 Appendices


)i(1 e maknakt

mnyxcx ++minus= ω

)i(2 e maknakt

mnyxcx ++minus= ω


⎟⎠⎞⎜



⎟⎠⎞⎜






a x

y


⎥⎥⎦

⎤

⎢⎢⎣

⎡+⎟

⎟⎠

⎞⎜⎜⎝


⎟⎠⎞

⎜⎝⎛ +++minus++=

⎟⎠⎞

⎜⎝⎛ minus++=

22

2

2

222222

22222

222

481

2122

21

)(2)(21

)(21

aax

ax

ayxayxaxak

ayxaayxak

ayxakU

to second order

[ ]

2

22222222

21

2222

kx

axaxyxayxaxakU

cong







11 11dagger

1 i22

pp p p

pa P X

ωω

= minus qq q q

q

Now

11 1

1

11dagger

p p

p

p

X m

m

X


minus

= sdot

= sdot

=

sum

sum

X

X

q b q b q bb

b q b q bb

q

e

e






1 1

1

1

1

111

1

1 1

p pp p p

pp p

e X m e e X

m e e X

m X m X

β βα αβ

β βαβ

ββ ααβ δ δ

lowast lowast

lowast

=

=

= =

sum sum

sum sum

sum

q b q b q bbqb q b

b q b q bb q b

bb b bq bb q b

or

1 11

1p ppX e X


b (a)

Now

dagger 11dagger 11

1 i22

pp p p

pa P X

ωω

= + qq q q

q


11dagger 11dagger

11 11

1 i22

1 i22

pp p p

p

pp p

p

a P X

P X

ωω

ωω


= minus

qq q q

q

qq q

q

726 Appendices

so

dagger 11 2i

2p

p p pa a Xω

minusminus = qq q q

11 dagger

i ( )

2p p pp


q

Substituting in (a)

1 dagger

i ( )

2 p p ppp

X e a am ω


b q



0)(

)(])[(

34412

24412122

442

4411

=++

++minus+minus

Kkkcc

KkkccKkckcc

zx

yxx ρω


0)( 122

11 =minus Kkc ρω


0)( 222

44 =minus Kkc ρω

0)( 322

44 =minus Kkc ρω

If

ρ

ων 1111

ck

==


If

ρ

ωνων 4433

22

ckk

====






)1(

)(

])()([)(

))]()(())()([()(

))]()(())()(()(

2

2

12

1112

11112

11112

1

1

NM

PM

dPM

dxuxuPM

dxuxuQxuxuPM

dxuxPuxuxPuMd

P

qp qq

qp qq

pp NNNNpp

pp NNNNpp

=

=

minus=

minus=

minus=

minus=

sum

sum int

sum int

sum int

sum intint

lowast

primelowastprime+


τδ

τ

τ

ττψψ

where Q = Pminus1P




)2()1()21()2()1(21

)2()1()21()2()1(21

)2()1()21()2()1(

)2()1()21()2()1(

)1()1()1(

(||)

(||)

sum

sum

sum

sum

sumsum

minus

+=

minus

+

=

ji ijji

ji jiji

ji ijji

ji jiji

i iik

V

VE

V

V

H

ψψψψ

ψψψψ

ψψψψ

ψψψψ

ψψε



22 2

1 21 10 12

| ( ) |( ) d 0

4jN N

II jeV

rψ


rr

Note that

221| ( ) |N

jj ψ=sum r

728 Appendices


2

22 11

0 12

( )d ( )

4NI

II Ie V

rρ τ


r QED

IonI

Field point

Source point

r1

r2



Let x = kkM

0as2)(21

11ln1limlnlim

2

2

0

22

0

rarr=minus=

minus+minus=

minus+minus

rarrrarr

xxxx

xx

xx

kkkk

kkkk

xM

M

M

Mk

⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢

⎣

⎡


minusminusminus



rarr

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

gtminusminusminus

ltminusminusminus

rarr

⎥⎦⎤

⎢⎣⎡


⎥⎦⎤

⎢⎣⎡



rarr

110)1(

)1(1)1(1

or

110)1(

)1(1)1)](1(1[

2

1)1(1)1ln(

1)1(1)1ln(

2

1)1ln(1)1ln(2ln)1(2

1)1ln(1)1ln()1ln()1)(1(


2

2

22

1

xxx

xx

xxx

xx

xx

x

xxx

xxxxx

xxxxx

xxx

xxxx

xx

xx

x





i1e 0bm


K


The answers are





Thus

i(0) i( )( )e e

1 ( 1)

n l m

n l m

S πminus minus + +

+ +

= +

= + minusK






730 Appendices


i( )( ) i( )( ) i( )(n )1 e e en l m









20

2)(2

2 EVmKmEk

minus==


)ee()(22

ee)(2

)(02

If

ii

KxKx

kxkx

Dxbxb

CBxaxb

Vbax

minus

minus

+=ltltminus

+=ltlt

infinrarr=+gt

ψ

ψ

ψ


)ee()ee(i

)ee(ee

0ee

2i2i2i2i

2i2i2i2i

)2(i)2(i

KbKbkbkb

KbKbkbkb

bakbak

KDCBk

DCB

CB

minusminus

minusminus

+minus+

minus=minus

+=+

=+



⎟⎠⎞

⎜⎝⎛minus=

2coth)tan( Kb

Kkka (a)


)ee()(22

ee)(2

02

ii

KxKx

kxkx

Dxbxb

CBxaxb

bax

minus

minus

+=ltltminus

+=ltlt

=+gt

ψ

ψ

ψ


⎟⎠⎞

⎜⎝⎛minus=

2tanh)tan( Kb

Kkka (b)


)e21(

2tanh

)e21(2

coth

Kb

Kb

Kb

Kb

minus

minus

minusrarr⎟⎠⎞

⎜⎝⎛

+rarr⎟⎠⎞

⎜⎝⎛

and

00 V

EEV

EKk cong

minuscong


⎟⎠⎞



02symmetric-anti





)e21( 2

0


732 Appendices


⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝


22exp21

22

20

02

22

2

2 bmVVma

Ema ππ






dkm

kdkdkdEdE

2==


intint ==ll

dldEk

mdldkdEED 222 21

)2(2)(

ππ

φ

πak

θ

dl

dk

E+dEE = h2k22m

int=kdlmED 222

)(π


For a

k πlt

2222

2)(

ππ

πm

kkmED ==

For a

ka

ππ 2ltlt

)2(42

)( 2222 ϕπ

ϕπ

mkkmED ==

but

⎟⎠⎞



2

So

⎟⎠⎞


kamED ππ

π1

22 cos4)(

or

⎟⎠⎞


kamED π

ππ1

2 cos41)(



(1) (2) (3) (4) (5)

(6)

(c) (b) (d) (a) (f) (e)



0 00



k

734 Appendices


( )

0( ) 2e

(e 1)

F

F

E E kTF

E E kTf E EdT dT kT

minus


k

k

k

( )

0( ) 2e 1

(e 1)

F

F

E E kT

E E kTfE kT

minus

minuspart

= minuspart +

k

kk

so

0 02

0

F

F


f EEdTE dT kT T


k

k

k

k

Thus

00


τ⎛ ⎞ part⎡ ⎤⎛ ⎞= + minus + +⎜ ⎟⎢ ⎥⎜ ⎟ part⎝ ⎠⎣ ⎦⎝ ⎠

kk k

k


33

2(2 )

zzJ e u f d k

π

⎛ ⎞= ⎜ ⎟⎜ ⎟

⎝ ⎠int k k


33

2(2 )

zzH u f E d k

π

⎛ ⎞= ⎜ ⎟⎜ ⎟

⎝ ⎠int k k k

Thus

2 303 ( )

4z F


τπ


kk

and

2 303

1 ( )4

z Fz


τπ


k k kk



23

22

3

23

( ) [ ( )]( )4

( ) ( )12

412

zz F

zF

F

dsdEeJ u E E eEu

dsdEe E u E Eu

e eEu k

τ δπ

τ δπ

τ ππ

= minus minus minus

= minus

=

int

int

kk k

k

kk k

k

k




m

neEJ z τσ

2==


230 ( )

12F


dz dT kT T uτ δ

π⎛ ⎞⎡ ⎤⎛ ⎞= = minus minus +⎜ ⎟⎢ ⎥⎜ ⎟

⎝ ⎠⎣ ⎦⎝ ⎠int k k

k kk

or

2 2( ) ( ) FF F


τ δ τ δ ⎡ ⎤⎛ ⎞minus = minus +⎢ ⎥⎜ ⎟⎝ ⎠⎣ ⎦

int intk k kk k k k

k k

or

( )2

2 2 2 414F F

F F F FFE E E E


πτ π τ= =

⎞⎛ ⎞ ⎛ ⎞= minus + ⎟⎜ ⎟ ⎜ ⎟ ⎟⎝ ⎠⎝ ⎠ ⎠k k

k kk k

So

dT

dEdzdTeE F=


2

31 [ ( )]

12F F

F

dT Kdz


τ δπ

=


k k k kk

so

23

1 412 F

F

F Fk kE E

u dE dEK E E k

dT dTτ π

π ==


kk k

736 Appendices


dT

dEEm

nK FF

F 2τminus=


⎥⎥

⎦

⎤

⎢⎢

⎣

⎡

⎟⎟⎠

⎞⎜⎜⎝

⎛minus=

2

)0(121)0(

FFF E

kTEE π

so

)0(6

22

F

FE

TkdT

dE πminus=

so

)0(6

222

FF

FE

TkEm

nK πτ=


Tkm

nK F 22

3τπ=

and

22

2

22

33 ⎟

⎠⎞

⎜⎝⎛==

ek

Tm

ne

Tkm

n

TK

F

Fπ

τ

τπ

σ




bXX

dx 12 minus=int





1

)]1([11

)]1([

21

1

02121

10 2

21

KK

xKxKKKxKxKdx

=

⎥⎦

⎤⎢⎣

⎡minus+minus

minus=minus+

int


2

2

22 2

22 2

( )

( ( ))

(2 )2

22

2

L

L

dIq f

dq f

d

q qm

dmq q

m

ω

ω

=

= minusminus

= minus⎡ ⎤



int

int

int

int

qq

qq

q

k q

q

k q


1

2 20 2 2

2

2(1 ) 2 L

m dxI dmq x q xω


int intq

k q


1

2 20 2 2 2

2

2( ) L

m dI dxmx x x kω


int intq

q k


1

2 20 2 2 2

2

2L

m dI dxmq x x kω

= minus⎡ ⎤⎛ ⎞+ minus⎜ ⎟⎢ ⎥⎝ ⎠⎣ ⎦

int intq

738 Appendices


2

2 2 2 2 2 20

4[ ] [ ]

d q dqq q

πα α

infin=

+ +int int

q


intminus

minus=1

0 22

22 2

2

kxxmdxmI

Lωπ

But

2

sin2

2

11

0 22

pppxx

dx minus=minus

int

so we find

⎟⎟⎠

⎞⎜⎜⎝

⎛minus= minus

2sin22 12

2k

mk

mm

mIL

L

L ωω

ωπ

Thus

Lm

Kω

=3


⎟⎟⎠

⎞⎜⎜⎝

⎛+minuscong

2122 2

22

kmm

mILL ω

πω



2 2

02

16

c L

c

kEm

mm

α ω

α

lowastlowast

lowastlowastlowast

= minus +

=minus

k






1 1d edt c

δτ

⎛ ⎞ ⎛ ⎞+ = minus +⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

k E v Btimes


mceB

c =ω

v

vv

vv

zz

xcyy

ycxx

Eme

Eme

Eme

τ

τωτ

τωτ

minus=

+minus=

minusminus=


⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛ minus=

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

z

yx

cc

z

yx

EEE

DDjjj

000101

0 τωτωσ




Dyyxx

0σσσ ==

D

cxyyx

τωσσσ 0=minus=

For the Hall effect

yxyy

xyy

EEj σ

σminus=rArr= 0

740 Appendices

So

xyy

xyxxx Ej ⎟

⎟

⎠

⎞

⎜⎜

⎝

⎛+=σσ

σ2


xx Ej 0σ=


m

ne τσ2

0 =


BE

E

BjE

Rx

yxyy

x

x

yH )( 0σ

σσ

minus==


nec

RH1minus=


B

nec

cc

cxyyx ==congminus=

τωσ

τωτωσσσ 02

0

)(


B

Ryx

H )(1

σminus=


v

v)()(

)(vvv

32

54

32

540

0

⎥⎦

⎤⎢⎣

⎡partpart

partpartminusminus=

⎥⎥⎦

⎤

⎢⎢⎣

⎡sdot

partpartminus=

partpartminus=

F

F

FF

e

Fe

xxxfxfcm

xfcmEp

π

π



( ) ⎟⎟⎠

⎞⎜⎜⎝


40

22 11411

FF

xF

xxdxxxxf F

where

312

v3

⎟⎟⎠

⎞⎜⎜⎝

⎛=equiv π

cmcmpx

ee

FF


FFF

F xxxxf

21)( 3 +=

partpart

34312

v31)3(

vπ

cmx

e

F minus=partpart

Fe

F xcm

x31

v3

31

vv

312minus=⎟

⎟⎠

⎞⎜⎜⎝

⎛minus=

partpart π


[ ] 12

61

3111

41

2432

54

242

432

54

0

FFe

FFF

Fe

xxcm

xxx

xcm

p

minuscong

⎥⎥⎦

⎤

⎢⎢⎣

⎡++⎟

⎟⎠

⎞⎜⎜⎝


π

π


pm

MM δ=

and

⎟⎟⎠

⎞⎜⎜⎝

⎛=

cem

RR

742 Appendices


Mm

R pξπ sdotcong 3

34v

31

31

3

2

491

34

3

⎟⎟⎠

⎞⎜⎜⎝

⎛=

⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜

⎝

⎛

sdot=

ξδπ

ξπ

π

MR

Mm

RRRx

pF


32

54

12~and

89

ππδ cm

K e==

⎥⎥⎦

⎤

⎢⎢⎣

⎡minuscong 2

32

4

34

0~

RM

RMKp




EF(2)

EF(1)

E

Δ = eV

R1 2



We note if

1)2(1exp)(

minus

⎥⎥⎦

⎤

⎢⎢⎣

⎡+

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛ minusminus=

kTEE

Ef F (a)


then

1)1(1exp)(

minus

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡+⎟

⎟

⎠

⎞

⎜⎜

⎝

⎛ minusminus=Δ+

kTEE

Ef F


then

[ ] 3

321

)2())(1)((v2



[ ] 3

312

)2())(1)((v2




[ ] 3

3

)2()()(v2


Since

x

x kE

partpart= 1v

We write

dEdkxx1v =


[ ] ||2

3 )()()2(



mk

EW2

2||

2minus= (b)


)(D1 WTT =

744 Appendices


⎥⎥⎦

⎤

⎢⎢⎣


1



[ ] )( )()()2(

2||


π (d)






ce EE

mED minus⎟

⎟⎠

⎞⎜⎜⎝

⎛=

lowast 23

222

21)(π

1])exp[(

1)(+minus

=kTE

Efμ


⎟⎠⎞

⎜⎝⎛ minus=⎟

⎠⎞


kTE

kTEEf μμ expexp)(

ce

EmkE += lowast2

22

intinfin

lowast⎟⎠⎞


⎟⎠

⎞⎜⎜⎝

⎛=

cE ce dE

kTEEE

mn μ

πexp

22

123

22



ce

EmkE += lowast2

22

kdkm

dEe

22

2

lowast=

lowast

=minuse

cm

kEE2

intinfin

lowast

lowast

lowast

lowast

⎟⎟⎠

⎞⎜⎜⎝

⎛ +minus⎟

⎠⎞

⎜⎝⎛

⎟⎟⎠

⎞⎜⎜⎝

⎛=

cEe

ce

e

e kdkmkT

EmkkTm

kmn 2

2)2(

expexp2

22

1 22223

22μ

π

intinfin

lowastlowastlowast

lowast

⎟⎟⎠

⎞⎜⎜⎝

⎛minus⎟

⎠

⎞⎜⎝

⎛ minus⎟⎟⎠

⎞⎜⎜⎝

⎛=

cEee

c

e

e dkkkTm

kmkT

E

m

mn 2

22223

22 2expexp

2

22

1 μπ

intinfin

lowastlowast

lowast

⎟⎟⎠

⎞⎜⎜⎝

⎛minus⎟

⎠

⎞⎜⎝

⎛ minus⎟⎟⎠

⎞⎜⎜⎝

⎛=

cEe

c

e

e dkkkTm

kkT

Em

mn 2

22

23

323

22 2expexp

)2(22

21 μπ


pp

ndxpxx nn π

)2(2)12()exp(0


where

kTm

pne

2

2 1 ==

So

( )kTmkTm

kTE

m

mn

ee

c

e

e

21

223

323

22

2222

)1(exp2

222

1 πμπ

⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎠

⎞⎜⎝

⎛ minus⎟⎟⎠

⎞⎜⎜⎝

⎛=

lowast

lowast

⎟⎠

⎞⎜⎝

⎛ minus⎟⎟⎠

⎞⎜⎜⎝

⎛=

lowast

kTEkTm

n ce μπ

exp2

223

2



746 Appendices



1 ( ) 1 ( )kdE E

d⎛ ⎞= equiv⎜ ⎟⎝ ⎠

kv kk

nabla


1 ( )

( )

k

k

dE Edt

d Edt

= sdot = sdot

= sdot

F v F k

k k

nabla

nabla

thus

( ) ( )k kdE Edt



ddt

= kF


1 ( ) ( )k kdE Edt


but also



( ) ( )k kdE Edt



( ) dqdt











ln T

ln μ

low T high T



phonon scattering

748 Appendices


if

if

212

121

21

21total

μμμμμμ

μμμμμ

gtgtconggtgtcong

+=




rr

rV λminusprop e1)(



2 1 2 1( )H V r=k k k k



2 1



int minuspropπ

θθθθστ 0

sin)cos1)((v1 dm



Finally we find

23Tm

m prop= lowastτ

μ








cEδ δprop sdotk r



23minus=prop Tμ


See Fig 618


Now


⎟⎠

⎞⎜⎜⎝

⎛= minus )(

23

2 e2

2 μ

π


⎟⎠

⎞⎜⎜⎝

⎛= minus )(

23

2 e2

2 μ

π

750 Appendices


⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎟⎟⎠

⎞⎜⎜⎝

⎛=minus

23222

lnkTm

NkTE

e

dCn

πμ

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎟⎟⎠

⎞⎜⎜⎝

⎛=minus

23222

lnkTm

NkTE

h

aVp

πμ

Therefore

⎥⎥⎦

⎤

⎢⎢⎣


32

)()()2(

22ln)(

kTmmNN

kTEeEEhe

dagpnCnVp

πϕϕ

Using

kTEg

gkTE

e

1ln minus=

kTEhei

gmmkTn 4323

2 e)(2

2 minus⎟⎟⎠

⎞⎜⎜⎝

⎛=

π

we find

⎟⎟⎠

⎞⎜⎜⎝

⎛=minus 2ln)(

i

dapn

nNN

kTe ϕϕ QED



2 21

12Surface

( )( )

4d

rπΦ = int

M r Sr

sdot

= minus ΦH nabla

12 2 21 2

12

ˆ ( )( )

4d

rπ= int

r M r SH r

sdot

θr

dS2

r 12

z

M

r1 = 0r12 = ndashr2


Using the figure

intminus=π

πθθπθθ

0 24 sin2 cos cos

rrdrMH z

31and

3=minus= DMH z




)eTr( HZ βminus=


1 1 1( ) ln

Tr( e ) Tr( e )

zz z z

z z

ZF ZB B Z B

Z Z

β β

μ ββ β β

βμ μβ



as expected



)Tr(2

)1Tr()eTr( 22

HZ H ββ +== minus


)Tr(

3)Tr(

3

)Tr(31)Tr()Tr(

2222

222

222222

JBμgJBBμg

JδBBμgJJBBμgH

Bμ μμ

B


==

==

sum

sumsum

)12()1Tr( += j


⎥⎥⎦

⎤

⎢⎢⎣

⎡+++cong )1(

3211)12( 2

222 jjBgjZ Bμβ

752 Appendices



⎥⎦⎤

⎢⎣⎡ +

partpart= )1(

611 2222 jjBg

B Bz

z μββ

μ

BB

BB z

z=

partpart


kT

jjgNNB

Bz

zzz

)1(3






kkS sdot


kkS sdot



i i( ) e ej i j

i jid Jdt


k k kk k kS S S



i jj i jd Jdt


k k kk k kS S S


i( ) i( )i( ) e ( )e ej ij i

d Jdt


k k kk R k kS R S S


kRsdot


so

( )

( )

( )

kd Jdt

J

J




primeprime= minus

prime=primeprime=

sum

sumsum

k kk k kk k

k k kk

k k kk

S k k S S

k S S

k S S

times

times

times

which is (7155) QED




ϕϕ ==


2

2

(cos cos sin sin )

cos( ( ))

x x y yR R R R R R

R R R R

S S S S

S

S q R R

ϕ ϕ ϕ ϕ

prime prime prime

prime prime

= +

= +

prime= minus

S Ssdot

so that


)cos()(2

))(cos()(2

i2

2

2

=minus=

minus=


sumsum

sum

sum prime

RRqR

R

RR

qRRJRJNS

qRRJNS

RRqRRJSH


qR minus== sum Then

)(2

2qJNSH minus=


)2cos(21)cos(2

21 )2cos(2)cos(2

)ee()ee()(

1

1221

i2i22

ii1

⎥⎦⎤

⎢⎣⎡ minus=

minus=+=

+++= minusminus

qaqaJ

JJqaJqaJ

JJqJ qaqaqaqa

754 Appendices

For a maximum

[ ]0

)cos()sin(2)sin(2

)2sin()2(21)sin(2)(

1

1

=minusminus=

⎥⎦⎤


partpart

qaqaqaaJ

qaaqaaJqqJ


angle turn 60

21)cos(

1)cos(2

deg=

=

=

qa

qa

qa


[ ]

max03

212

212

)2cos(2)cos(2)(

21

21

602

12

2

rArrltminus=

⎥⎦⎤

⎢⎣⎡ +minus=

minusminus=part

partdeg=

aJ

aJ

qaqaaJq

qJqa



=minussj mSS

12

112

212

212

21

21

21

+⎟⎠⎞

⎜⎝⎛ minus+=

⎟⎠⎞

⎜⎝⎛ minus=

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛minus=

+

++minus

nSnnS

nSnaS

nSaa

aSnS

j

jjjj


so



or

1)1)((

12

2)1(221

minus+minus+=

minus⎟⎠

⎞⎜⎝


sss

ss

ssj

mSmSmS

mSS

mSSmSSmSS

QED


We have

i1 e jjjb a

N= sum k R

ksdot

and


Nminus= sum k R

ksdot

where

0][

0][

][daggerdagger

dagger

=

=

=

ji

ji

ljjj

aa

aa

aa δ

Thus

i( )dagger dagger

i( )

1[ ] e [ ]

1 e

1

j j

i j

j

k jj jk j

j

k k k k

b b a aN

N

NN

δ

δ δ

prime

prime


primeminus minus

prime prime

=

=

= =

sum

sum

k R k R

k k R

sdot sdot

sdot

Similarly

i( )

0


Nprimeprime+



0


Nprimeprimeminus +


756 Appendices






Therefore

712

)state F(3max

21221

=+==

=+=

sumL

Lm

SS

l

less than half full

25


So stateF 25

2

Pm3+


426

)state I(123max5122


sumSLJ

mLSS

l

So stateI4

5

Eu3+ six electrons


033


7123216


=+=times=

sumSLJ

mL

SS

l

So stateF0

7




05 3

3[( ) ] ( )( )4μπ



B

r

rprime

m


4 have ( ) 0

28 have ( ) 2 22

a mam

primeminus sdot =

primeminus sdot = =

r r m

r r m


[ ]2 2

05 3

3 ( )4

Uμπ


r r m mm Br r r r

⎟⎟⎠

⎞⎜⎜⎝

⎛minus= minus

5

252

3

2327

total2)2)(3)(8(2)12(10

ama

amU

( ) 022424)10( 3

27 =minus= minus

amU


758 Appendices


2 2 2

4 have ( ) 0

2 have [( ) ] a m

primeminus sdot =

primeminus sdot =

r r m

r r m


066)10( 3

2

3

27

total =⎟⎟⎠

⎞⎜⎜⎝

⎛minus= minus

am

amU


3

27


)JT10279)(8(8

m10524

10

minus

minus

times=cong

timescong

Bm

a

μ


)(10381

101010

1012510)10(

23

2328

51

30

447

total

congtimescong

=congtimes

cong

minus

minusminus

minus

minus

minusminus

c

cT

T

U





The free energy is

ZkTF lnminus=


BFMpartpartminus=




)(0 qpHH rarr




qpqAp

eeAppqq

i

i

iiii

ii

Π=

gtΠequiv+rarrrarr

0HH

Now

)()(e

e

11)(

11)(


=

int

intΠminus

minus

qpqdddqdq

dpdpdqdqZ

NNq

NNqp

helliphellip

helliphellip

H

H

β

β


0 =Πpartpart

=Πpartpart

=partpart

j

iij

j

iij

j

i qpqq δδ



⎟⎟⎠

⎞⎜⎜⎝

⎛=

Πpartpart

+Π= etc1i

iiii

peAp

760 Appendices

Thus schematically

1

0

100010001

)()( =

Πpartpart

=

Πpartpartpartpart

=

Πpartpart

Πpartpart

partpart

partpart

=Πpart

part

i

i

j

i

i

i

j

i

j

i

j

i

j

ip

pqp

pqqp

qq

qqp 15



0=partpartminus=

BFM






15

1

100

01

100

01

0

=sdot=⎟⎟⎠

⎞⎜⎜⎝

⎛

partpart

⎟⎟⎠

⎞⎜⎜⎝

⎛

Πpartpart

=

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛partpart

partpart

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

Πpartpart=

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛partpart

partpart

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

Πpartpart=

Πpartpartpartpart

partpart

11j

i

j

i

i

i

j

i

j

ii

i

j

i

j

i

j

i

i

i

j

i

qqp

qp

qq

pqp

qq

ppqp

qq





Subtracting [ ]int



intΔminus

ΔΔminusminus

minus

Δminus= 2

1 22

221

221)(

)0()0(eV dEEeV

EeV

E

ENNCI

Checks 11








0

000


⎠

⎞⎜⎜⎝

⎛minusprime==Δ


0

0

0

3

0

0

2

02

3

31)11(

2

3cos

2

)(coscos2

ε

ε

θε

θθε

π

π

P

P

P

dPEE b

=

minusminusminus=

minus=

minus=equiv int

762 Appendices



θ

p

E z


int

int= π

θ

πθ

θθπ

θθπθ

0

cos

0

cos

sin2e

sin2ecos

d

dpp

kTpE

kTpE

z


int

int

minus

minus= c

c

x

c

c

x

zdx

dxx

cpp

e

e


⎟⎠⎞

⎜⎝⎛ minus=

ccppz

1)coth(


kTp

Ep

kTEpp z

z 3

3

22=== α



The Hamiltonian is

eEzr

qm


0

22

2

2H

H



)1(100 pzN += ψψ


BarBa 23


ψ


2

2

mqaB =

Requiring

12 =int τψ d





Now using

4

3 0100 0 100 2

0

12 (4 )

med Eψ ψπε

= = minusint rH





are also zero so

)(2 202 eEpaE

NBminus=

H



764 Appendices

Requiring

0=part

partpH

we find

0EeEp minus=


230



302

216 Ba

Eπεα =

part

partminus=

H



061

41)(

21 42


For E = 0

0)( 420 =++minus=


PGE δγβ (2)


20


[ ] 0)(61

41)(

21 2

02


0121)(

31 2


20 4 scc PTT

βγminus= (3)

(b) From (2) 0)( 42


20 4




04


δγ

432 minus=scP

βδγ 2

0 163+= TTc











766 Appendices





φ1 (say 45deg)

Polarizer

lightSample

Analyzerz

zφ2

E

z

yx






iminus= xx xpxp



If H = px2 m +V(x)

[ ] 0)( =xxV So

[ ] xpm

x i minus=H


[ ] iˆ ˆm




2 2ˆ ˆ

2ij ijj ji j i j

f b ωsdot + sdot

=sum sume r e r




jxi

i=minus xpxp xx

xxxppx xx i2 =minus

iminus= xx xpxp


768 Appendices


jj jω=H

jpm


jpm


jpm



1ˆ ˆi ij

i j i jmω

sdot = sdote r e p


ij ji


ωω ω



jiij ωω minus=



Summing over j

ˆ ˆ2iijjbf i im



m

bm

bfj ij 2)i(

i2minus=minus=sum




eEB

NeG +minus

=+ v13

1vvel

2

0εμ



0evv minus= Then

eEB

NeG =minus

minus+minus v13

1vvel

2

0

2ε

μω


ion0

ion 31 αε

NB =

and

Ge

Ee 2

locion

v ==α

at ω = 0

GeNB

2

0ion 3

1)0(ε

=


eEB

GBG =

minusminus+minus v

1)0(

vvel

ion2μω

eEB

BG =⎟⎟⎠

⎞⎜⎜⎝

⎛minus

minus+minus v1

)0(1v

el

ion2μ

μμω

Define

⎟⎟⎠

⎞⎜⎜⎝

⎛minus

minus=el

ion21

)0(1

BBG

T μω



For the local field

el00

loc 1v

31

3 BNeEPEEminus

+=+=εε

Substituting

)(

v 22 ωωμ minus=

T

eE

770 Appendices

Then

)(13

122el0


+=T

eEB

NeEE

)(13

11 22el0

loc


T

eB

NeE

E

)(

11

)0(1 22el

ionloc

ωωμ minusminus+=

TBGB

EE

)()1(

)0()()1(22

el

ion22

loc

ωωμωωμ


=T

Tel

BGBB

EE

Inverting

)0()()1(

)()1(

ion22

22el

loc GBBB

EE

Tel

T


=ωωμ

ωωμ

))(1()0(1

1

22el

ion

ωωμ minusminus+

=

T

locB

BGEE

Define

))(1(

)0(22

el

ion

ωωμ minusminus=

TBBGF

Then

FE

E+

=1

1

loc


loc0

ion0

ionv

31

31)(

EeNNB

εα

εω ==

Substitute in v

)(3

1)( 22loc0ion

ωωμεω

minus=

T

eEE

eNB

)(

)0()( 22

ion

locion

ωωμω

minus=

T

GBE

EB

)1(1

1)( elion BFF

B minus+

=ω


Given

[ ] )1()(1)(3)()(

elionel

ion

0 BBBB


ωω

εεωε


)1()1(

11

)1(1

3)()(

elelel

el

00 BBF

FB

BF

F

minus⎥⎦⎤

⎢⎣⎡ minus

+minusminus

minus++infin=

εε

εωε

)1(

11

13)()(

el00 B

FF

FF

minus⎟⎠⎞

⎜⎝⎛

+minus

++infin=εε

εωε

)1(

3)()(

el00 BF

minus+infin=

εε

εωε

Substitute for F

))(1(

)0()1(

3)()(22

el

ion

el00 ωωμεε

εωε


TBBG

B

Define c

2el

ion

)1()0(3

BBGcminus

=μ

Then

)(

)()(2200 ωωε

εεωε

minus+infin=

T

c

Rewriting

202)()(

ωεωε

εω =infinminus

minusc

T


2

2

)0()(

L

T

ωω

εε =infin

)()0( 2

2infin= ε

ωωε

T

L

772 Appendices


0)()0(

02 =infinminus

minusεε

εω cT

0)()(2

202 =


εεωω

εω

T

LT

c

0)()( 22

202 =

infinminusminus

εωωωεωTL

TT

c

0)(

)(2

0222 =infin


TLTc

202)( LT

c ωεεω =infin

+



ψψ Em

=nablaminus 22

2

For each coordinate

0222

2=+minus i

i

i

mEdxd ψψ


⎟⎟⎠

⎞⎜⎜⎝

⎛+⎟

⎟⎠

⎞⎜⎜⎝

⎛= i

ii

ii xmEBxmEA 22

2cos2sinψ


2

222

2 2or2

manEnamE i

iii ππ ==

)(2

23

22

212

22nnn

maET ++= π





2

22

2

22

23

2)111114(



2

22

23

mahf π=


constant432 ==

mfa π



0

22

2

2

2

2

22

4where1

2 πεφeq

rq

rrrm=minus⎟

⎟⎠

⎞⎜⎜⎝

⎛

partpart+

partpart+

partpartminus=H




εεε RRYEmqEmq

r ==== )(2

2

4

2

2ρ


ψψρ

ψφρρρρ

ε2

4

2

22

2

2

22

2

4

422

2)(11

22mqmqqqm

m=minus⎟

⎟⎠

⎞⎜⎜⎝

⎛

partpart+

partpart+

partpartminus

774 Appendices

Canceling we obtain

ψρψψ

φρρρρε

2111

21

2

2

22

2=minus⎟

⎟⎠

⎞⎜⎜⎝

⎛

partpart+

partpart+

partpartminus




φφ mie)( =Φ



⎞⎜⎜⎝

⎛minus+minus

21

21 2

2

2 ερρρρρ

mdd

dd

We let

2

and1 2

2xαρ

α=minus=ε


Ψ=Ψ+⎟⎟⎠

⎞⎜⎜⎝


241

21 22

2

2 xxm

dxd

xdxd


21122 +=+= nn αα


2212

21 )(

or)(

1+

minus=+

minus=n

REn

ε


21)1(

21 minus+=+ nn





)()(2 2

22zEz

dzd

mψψ =minus


0)(2)(22

2=+ zmE

dzzd ψψ

⎟⎟⎠

⎞⎜⎜⎝

⎛+⎟

⎟⎠

⎞⎜⎜⎝

⎛= zmEBzmEAz 22

2cos2sin)(ψ



2

222

2mwpE p

π=





12

2

0minusminus=

xeF

πε


x

ex

edxx

eFdxdVxxxx

0

2

0

2

20

2

161




x

eExV0

2

0 16)(

πεminus=

776 Appendices


x

exeEExV0

2

10 16)(

πεminusminus=


01

=⎟⎠⎞

⎜⎝⎛

=xxdxdV

which gives

10

1 16 Eex

πε=

or

0

13

01 21)(

πεEeExV minus=

Thus

0

13

21

πεEe=Δ



a

V(r)

ndashV(0)

E (bound state)

r




20 )(2 EVm +

=α

22mEminus=β



This implies

2


Also

E

EV

aa

minus=

+=

+= 0

22

2

22

)(cot11)(sin

βαα

αα


2πα gta

to estimate a

eV)1(22)(22

22

0 mEVma ππ

απ gt

+=cong

nm931)1061)(1019(2

)10636)(143(1931

34

min congtimestimes

timescongminusminus

minusa




778 Appendices

Energy Gap

Ec(z)

Eg1 Eg2

Valence Band

Conduction Band

d

z

Ev(z)


EFFzEzmyxm

c =⎥⎥⎦

⎤

⎢⎢⎣

⎡+

partpartminus⎟

⎟⎠

⎞⎜⎜⎝

⎛

partpart+


22 2

22

2

2

2

22




32||

2

322

2)(

2)(

2Ek

mEkk



ψψ 32

22)(

2EzE

dzd

mc =⎟

⎟⎠

⎞⎜⎜⎝

⎛+minus lowast



sum=

minus=N

ni

ndiq ndz

Nz

1

i )(e1)( χψ


0 we find for E3




2

2 2 0 0( ) cos












)(22

||

222

2

222

EE

kkmmw

nE

n

yx

+equiv

++= π

)(22

222

2||

2

|| yx kkm

km

E +==




ndEm

dkdkkkd

2

2||||||

2

2

2

π

ππ

=

==

780 Appendices


)(

)(2)2(

2)(

2

0||||22

sum

sum int

minus=

minusminus=infin

n n

n n

EEm

dEEEEmED

θπ

δππ



23)(π

mED = (1)

In 3D

E

dEmdkmkE 2

22 221

2==

2

212)4(

)2(2

)unit volper (4)2(

2)(

223

23D3

EdEmmE

dkkdEED

ππ

ππ

=

=


32

22

1)(223

22D3

E

wmmED π

π ⎟⎟⎠

⎞⎜⎜⎝

⎛=

where

2

222

32

3mw

EE π==

Per unit area

222D3332

21)(

ππ

πmw

wmwED == (2)

which checks (1)


d)

D3D(E)w

2hπm

23hπm

E1 E2 E3




1001

512

512

51

2

51

2~

21

21

22

11

I

NN

NN

NN

NN

AA

=⎟⎠⎞

⎜⎝⎛=

⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜

⎝

⎛minus+

⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜

⎝

⎛

minus

+=


Bibliography

Chapter 1

















784 Bibliography










Chapter 2









Chapter 2 785





















786 Bibliography

Chapter 3





















Chapter 3 787



















788 Bibliography

Chapter 4





















Chapter 5 789














Chapter 5







USA 92 6663-6667 (1995)

790 Bibliography






















Chapter 6 791



Francis London 1963

Chapter 6
















1994

792 Bibliography






















1996

Chapter 7 793

Chapter 7



















794 Bibliography



















Chapter 8 795










(2004)

Chapter 8






796 Bibliography






















Chapter 9 797





Chapter 9

















798 Bibliography




Chapter 10















Chapter 11 799






Chapter 11















800 Bibliography


















Chapter 12



Chapter 12 801























802 Bibliography





















Chapter 12 803

















804 Bibliography

Appendices





















Subject References


Mechanics




Electricity



Optics



Thermodynamics






Critical Phenomena


806 Bibliography

Crystal Growth


Modern Physics



Quantum Mechanics





Math Physics


Solid State







Condensed Matter






Problems




Applied Physics





Rocks




808 Bibliography


The Internet



Periodic Table


Further Reading












Further Reading 809
















810 Bibliography

















Further Reading 811


















812 Bibliography




Index

A













B


814 Index










C




Index 815





















816 Index




316 317

D









594 640






E





Index 817

















503 566










F



818 Index


493 538 Fermi function 160 248 298 303



373 385 399 407 422 426 456 486 513







G

GaAs 242 295 297 324 326 328 329 337 347 563 564 579 618 628 629 637




Index 819

Group velocity 55 306 532 562 627


H











I






820 Index


J


K



L









529 570

Index 821




M













N


531 600 NaCl 6 9 28 579 600

822 Index









O








P





Index 823















824 Index


Q




R












S



Index 825













826 Index






T



361 418











475 509

Index 827

U




V



Vortex region 466

W


X


Y

YBa2Cu3O7 506 508

Z


Appendices

A Units










G1 Bose Particles

G2 Fermi Particles


H1 Propagators

H2 Green Functions

H3 Feynman Diagrams

H4 Definitions




J Folk Theorems



M Problem Solutions














































































Bibliography

Mechanics

Electricity

Optics

Thermodynamics


Critical Phenomena

Crystal Growth

Modern Physics

Quantum Mechanics

Math Physics

Solid State

Condensed Matter


Problems


Applied Physics

Rocks


The Internet

Periodic Table

Index


appendices - cern · appendices a units the choice of a system of units to use is sometimes...

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