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A CLASSICAL THEORY OF THE DIELECTRIC SUSCEPTIBILITY
OF ANHARMONIC CRYSTALS
DISSERTATION
Presented to the Graduate Council of the
North Texas State University in Partial
Fulfillment of the Requirements
For the Degree of
DOCTOR OF PHILOSOPHY
By
Howard V. Kennedy, B. A., M. S.
Denton, Texas
- May, 1976
Kennedy, Howard V., A Classical Theory of the Dielectric
Constant of Anharmonic Crystals, Doctor of Philosophy
(Physics), May, 1976 161 pp., 2 tables, bibliography,
35 titles.
An expression for the dielectric susceptibility tensor
of a cubic ionic crystal has been derived using the classical
Liouville operator. The effect of cubic anharmonic forces
is included as a perturbation on the harmonic crystal
solution, and a series expansion for the dielectric sus-
ceptibility is developed. The most important terms in the
series are identified and summed, yielding an expression for
the complex susceptibility with an anharmonic contribution
which is linearly dependent on temperature. A numerical
example shows that both the real and imaginary parts of the
susceptibility are continuous, finite functions of frequency.
TABLE OF CONTENTS
PageLIST OF TABLES .... ....... . .. . . . . . . ivLIST OF ILLUSTRATIONSv.............. y
Chapter
I. INTRODUCTION . . . ........ .. 1
Survey of Related WorkMethod of Calculation
II. DEVELOPMENT OF THE PERTURBATION SERIESEXPANSION FOR THE DIELECTRICSUSCEPTIBILITY . . . . . . . . . . . . . . 10
III. NUMERICAL EXAMPLE . . . . . . . . . . . . . . 36
IV. DISCUSSION OF RESULTS . . . . . . . . . . . . 49
APPENDIX . - - . . - * ................53
BIBLIOGRAPHY . - - - . . . . . . . . . . . . . . . . . 159
iii
LIST OF TABLES
Table Page
I. 2nd Order Contributing Frequencies,Contributions, and FactorialMultipliers . . . . . . . . ...... 28
II. 4th Order Contributing Frequency Sets,Vertex Contributions and FactorialMultipliers - - -. . . ........... . 31
iv
LIST OF ILLUSTRATIONS
Figure Page
1. Diagrams Contributing to the Sus-ceptibility .......... . . . . 26
2. Plot of Squared Frequency Spectrum 4,l j(W&A)vs. Squared Frequency Ratiofor a Diatomic Cubic Lattice... .... 40
3. Plot of Frequency Spectra g-(4))and g(4) (w ) vs. Frequency Ratio
e/u. for a Diatomic Cubic Lattice . . . 42
4. Plot of Frequency Spectra g(e),_ 4)..)and g(4+..) vs. Frequency Ratioev4 . - - - . . -. -. -. -. -. -. -. . . . . . . 43
5. Plot of Squared Frequency Spectrum.)?G I'(.fx) vs. Squared FrequencyRatio ... . . . . . . . . . . . . . . . . . 44
6. Plot of the Normalized Squared FrequencyDisplacement Jd ,/O 4 ? vs.Squared Frequency Ratio (z/o. 0 . . . . . 46
7. Plot of the Normalized Real Part of theSusceptibilityOCj)vs. Squared Fre-quency Ratio (z/w. , )' * . . ......... **47
8. Plot of the Normalized Imaginary Part ofthe SusceptibilityQ(&)vs. SquaredFrequency Ratio (z/w.)' - . . . . . . . . 48
9. The Contributions to the Real andImaginary Parts of the DielectricSusceptibility of KBr at 3000 K . . . . . . 50
10. The Infrared E (4) Spectrum (IR) andthe Raman spectrum (R) of NaCl . . . . . . 50
11. Flow Chart of Diagram Evaluation Program . . . 124
12. Flow Chart of Program to Calculate FrequencySpectra and Susceptibility . . . . . . . . 151
V
CHAPTER I
INTRODUCTION
Survey of Related Work
The theory of absorption and refraction in solids has
been treated in a variety of ways. The earliest theories
(16, 7) assumed that the bound electrons in a solid were
set into oscillatory motion by the electric field of a
light wave as it passed through the solid. With this
approach, the dielectric constant was shown to be dependent
on the density of the oscillating electrons, and absorption
was shown to occur at the natural frequency of the oscil-
lators. In the simplest case, the oscillation was assumed
to be undamped, giving rise to an infinitely sharp ab-
sorption line. In other calculations, phenomenological
arguments were used to postulate a damping force propor-
tional to the velocity of the oscillators, thereby giving
a continuous absorption with still a single absorption
maximum, now finite, slightly shifted from the natural fre-
quency of the undamped oscillators.
In a pair of pioneering papers (7, 5), Born and
Blackman considered the coupling of the oscillation of
nearest-neighbors in an ionic crystal lattice due to
1
2
anharmonic forces. They used the normal coordinate ex-
pansion developed by Pauli (18) and Peierls (19) to find
the normal frequencies in the absence of damping or coupling
terms and applied the anharmonic interaction as a per-
turbation. They were able to show that secondary absorption
maximum occurred, in agreement with experimental work of
the period (3). The first paper by Born and Blackman (7)
dealt with a one-dimensional chain; Blackman's later
paper (5) extended the theory to three dimensions.
All of the work described above made use of purely
classical ideas. With the advent of quantum mechanics,
it was natural to make similar calculations using the
postulates of quantum mechanics. One of the earlier
papers (13) went back to the idea of a continuous medium
of randomly distributed, independent oscillators and
duplicated the classical results. Barnes, Brattain, and
Seitz (4) used ordinary perturbation theory to include
anharmonic coupling and obtained results in the quantum
case very similar to those of the classical work of Born
and Blackman.
The earlier theories were satisfactory only in a
qualitative sense. None predicted a continuous absorption;
damping was entered by a phenomenological argument if at
all. Polarizability of ions was either ignored, or if
assumed, gave answers in poor agreement with experiment.
3
In 1949-1950, Szigeti (21, 22) introduced the concept
of an "effective" ionic charge in an attempt to improve
the agreement between theory and experiment, with moderate
success. Quantum field theory (23) was applied to the
calculation of dielectric constants, with no new results.
It was argued by some researchers (8, 15) that the second-
order electric moment must be the primary mode for intrinsic
lattice absorption, with anharmonicity accounting for the
broadening of the main absorption line, especially for
homopolar crystals (diamond, Germanium, Silicon) in which
the linear electric moment vanishes.
More sophisticated models of an ionic crystal were
considered (10, 29, 30, 31) in which short range forces
between nearest or next nearest neighbors were assumed
to cause electron shell deformation, giving rise to addi-
tional terms in the Hamiltonian because of the resulting
polarization. Elaborate calculations using an electronic
computer were made for the purpose of fitting the arbitrary
parameters of the model to observed data. Such calculations
were made possible by the development of the experimental
technique of neutron scattering for measuring the phonon
dispersion relation of a crystal. Quite detailed calcu-
lations of the absorption and the dielectric constant were
then possible, using even a simple assumed form for the
anharmonic potential.
4
Most recent calculations have used a quantum-mechanical
approach. Born and Huang (6, pp. 328-381) used a method
attributed to Weisskopf and Wigner (28) to derive ex-
pressions for the shift of the normal frequencies from
those of the harmonic crystal and for the absorption widths.
Vinogradov (24) took a slightly different approach which
he claimed to be more exact because of his choice of
different initial conditions than those of Born and Huang.
R. A. Cowley (9) used thermodynamic Green's function to
make similar calculations, while S. C. Adler (1) extended
the band theory treatment of Nozieres and Pines (17) and
Ehrenreich and Cohen (11) to estimate the dielectric
constant.
In a number of papers, A. A. Maradudin and R. F. Wallis
(25, 26, 27) investigated many aspects of the anharmonic
crystal, including the dielectric constant and the ab-
sorption coefficient. They used both ordinary second-order
perturbation theory and the method of Born and Huang to
examine the properties of the linear chain (25). The
linear approximation to a three-dimensional crystal (26)
was approached by a method similar to that used by Kubo
(14) in treating the magnetic susceptibility. Finally,
a modification of the Kubo method was used to make a
quantum-mechanical calculation (27) of the linear approxi-
mation to the dielectric constant and absorption coefficient.
5
Method of Calculation
This dissertation takes a classical approach to the
calculation of the dielectric susceptibility and absorption
coefficient of an ionic crystal using a mathematical
approach developed by Prigogine and Balescu and co-workers
(2, pp. 26-39; 12; 20, pp. 36-42). The three-dimensional
calculation is restricted to non-polarizable ions and
central forces, for a simple cubic lattice of the NaCl
type. Ions of alternating sign and different mass are
assumed to be located at alternating lattice sites.
The potential energy at each lattice site is expanded
in a Taylor series, then truncated after the cubic terms.
The calculation proceeds via an initial value perturbation
technique using the Green's function of the classical
Liouville operator. The induced polarization is calculated
as the linear electric moment of the ions, averaged over
the thermal distribution function of a set of normal co-
ordinates. The Fourier transformation of the expression
is taken and the susceptibility tensor extracted from the
result.
Following a change of variables, a series expansion
for the Fourier transformation of the Green's function
is substituted which introduces anharmonic interactions.
Finally, the terms in this series are grouped in similar
classes which can be summed. The lowest order summations
6
represent the harmonic crystal susceptibility and the first
order correction due to anharmonic forces between ions.
CHAPTER BIBLIOGRAPHY
1. Adler, S. L., "Quantum Theory of the DielectricConstant in Real Solids," Physical Review 126,413 (1962).
2. Balescu, R., Statistical Mechanics of Charged Particles,Vol. IV of Monographs intatisTicTl~FPhysics, 4 vols,eidted by I. Prigogine~(Interscience, New-York, 1963).
3. Barnes, B., Zeitschrift fur Physik 75, 723 (1932).
4. Barnes, R. B., R. R. Brattain and F. Seitz, "On theStructure and Interpretation of the InfraredAbsorption Spectra of Crystals," Physical Review 48,582 (1935).
5. Blackman, M., "Die Feinstruktur der Reststrahlen,"Zeitschrift fur Physik 86, 421 (1933).
6. Born, M. and K. Huang, Dynamical Theory of CrystalLattices (Oxford University-Press, London, 1954).
7. and M. Blackman, "Uber die Feinstruktur derRessTrahlen," Zeitschrift fur Physik 82, 551 (1933).
8. Burnstein, E., "The Intrinsic Infrared and the RamanLattice Vibration Spectra of Cubic Diatomic Crystals,"Lattice Dynamics, Supplement I to Journal of Physicsand CHiemistry of Solids (Permagon PressW York,MT5)T-
9. Cowley, R. A., "The Lattice Dynamics of an AnharmonicCrystal," Advances in Physics 12, 421 (1963).
10. Dick, B. G., Jr. and A. W. Overhauser, "Theory of theDielectric Constants of Alkali Halide Crystals,"Physical Review 112, 90 (1958).
11. Ehrenreich, H. and M. H. Cohen, "Self-Consistent FieldApproach to the Many-Electron Problem," PhysicalReview 115, 786 (1959).
12. Henin, F., I. Prigogine, C. C. L. George and F. Mayne,"Kinetic Equations of Quasiparticle Description,"Physica 32, 1828 (1966).
7
8
13. Korff, S. A. and G. Breit, "Optical Dispersion," Reviewof Modern Physics 4, 471 (1932).
14. Kubo, R., "Statistical-Mechanical Theory of IrreversibleProcesses," Journal of the Physical Society of Japan12, 570 (1957).
15. Lax, M. and E. Burnstein, "Infrared Lattice Absorptionin Ionic and Homopolar Crystals," Physical Review 9739 (1955).
16. Lorentz, H. A., The Theory of Electrons (Reprinted byDover Publications , New Y6r-k, 1952)-.
17. Nozieres, P. and D. Pines, "Electron Interaction inSolids, General Formulation and Collective Approachto Dielectric Constants," Physical Review 109, 741(1958).
18. Pauli, W., Verhandlung der Deutsche PhysikalischeGesellschaft 6, 10 (1925).
19. Peierls, R., Annalen der Physik 3, 1055 (1929).
20. Prigogine, I., Non-Equilibrium Statistical Mechanics,Vol. I of Monographs in Statistical Physics, 4 vois.,edited by I. Prigogine (Interscience, New York, 1962).
21. Szigeti, B., "Compressibility and Absorption Frequencyof Ionic Crystals," Royal Society (London) ProceedingsA204, 51 (1950).
22. , "Polarisability and Dielectric Constant ofIonic Crystals," Transactions of the Faraday Society45, 155 (1949).
23. Tidman, D. A., "A Quantum Theory of Refractive Index,Cerenkov Radiation, and the Energy Loss of a FastCharged Particle," Nuclear Physics 2, 289 (1956).
24. Vinogradov, V. S., "The Shape of the Infrared AbsorptionBands and the Dielectric Losses in Ionic Crystals atUltrahigh Frequencies," Soviet Physics Solid State 31249 (1961).
25. Wallis, R. F. and A. A. Maradudin, "Lattice Anharmonicityand Optical Absorption in Polar Crystals I. The LinearChain," Physical Review 120, 442 (1960).
9
26 _ _ _ _ _ _ _ _, "Lattice Anharmonicityand Optical Absorption in Polar Crystals II. ClassicalTreatment in the Linear Approximation," PhysicalReview 123, 777 (1961).
27., "Lattice Anharmonicityand Optical Absorption in Polar Crystals III. QuantumMechanical Treatment in the Linear Approximation,"Physical Review 125, 1277 (1962).
28. Weisskopf, V. and E. Wigner, "Uber die NaturlichLinienbreite in der Strahlung des HarmonischenOszillators," Zeitschrift fur Physik 63, 18 (1930).
29. Woods, A. D. B., W. Cochran and B. N. Brockhouse,"Lattice Dynamics of Alkali Halide Crystals," PhysicalReview 119, 980 (1960).
30. andR. A. Cowley.k "Lattice Dynamics ot ATKali~Hal~deCrystals II. Experimental Studies of KBr and NaI "Physical Review 131, 1030 (1963).
31."Lattice Dynamics ot Alkali Halide
Crystais111. Theoretical," Physical Review 1311025 (1963). -ci
CHAPTER II
DEVELOPMENT OF THE PERTURBATION SERIES EXPANSION
FOR THE DIELECTRIC SUSCEPTIBILITY
The time dependent polarization P(t) per unit volume
of the crystal can be expressed as the sum over the distri-
bution function F(0,t) of the linear electric moment M($)
of the crystal:
where 0 is a set of canonical coordinates giving the
position and momenta of the ions making up the crystal.
The distribution function F(0,t) is obtained by an initial
value perturbation technique using the Green's function G
of the Liouville operator defined by
4Z ? d - 4 [ H go( Jo fim- JH)
The brackets in Equation 2. are the classical Poisson
brackets in which is any function of the generalized
coordinates and their conjugate momenta {J,, and
H is the Hamiltonian. The symbol 0 will be used to
represent the set f J . The angle is a generalized
Is ^
11
position coordinate, and the action Jk is a generalized
momentum. The development of the mathematics of an
anharmonic crystal in these coordinates is found in
references 1-3 and in Appendix C of this dissertation.
The Hamiltonian is broken up into three parts for
the present calculation. The major part is the "harmonic"
part HH, which is the Hamiltonian of an unperturbed system
of harmonic oscillators. Two perturbing parts are added
to this: the anharmonic correction HA to the Hamiltonian
of the unperturbed crystal and the Hamiltonian HE,resulting from the interaction of the crystal with an
external uniform electric field.
Inserting H = HH + HA + H1E into Equation 2 gives
Defining the operators XZ t, and X by
PE 54)and substituting into Equation 3 gives
XA ( E . (s)
12
The A are inserted to indicate the perturbation order
and will be set to unity at the appropriate point in the
calculation.
The desired initial value solution follows from setting
Equation 5 equal to zero. The formal solution to Equation 5
will then be found by successively applying perturbations
to the equilibrium, anharmonic distribution function FA(0 )
to obtain the time dependent distribution function FA(0,t)
as a perturbation series involving the operators and
. Only terms of first order in c will be kept for
present interest.
The first step in the calculation is to find that
solution to the equation
which reduces to the function FA(0) at t = 0. Following
Balescu (1), the Green's function for the operator is
defined as the solution to the equation
which also satisfies the causality condition G(0t; 0't') = 0
for t < t'. The adjoint Green's function G* is similarly
defined by the equation
13
with G* (0t; 0't') = 0 for t > t'. Applying the generalized
Green's formula for the Liouville operator X allows the
solution to be found. The Green's formula is given in
Appendix A as
where and are any functions of 0 = (e, J) which vanish
for J = 00 and which are periodic in G.
To find the function FA(0,t) which satisfies the
equation ''FA(0,t) = 0 and the boundary condition
FA(0,0) = FA(0), the generalized Green's formula is used
with
The resulting equation is
fd f,"Od IfF^A ,i'),'r6 " )*LiI 07 ZF tj
The right hand side of Equation 10 becomes
-fdF(&)6( ; o) =- -fdr )674S710 10) (C)
because the condition t = 00> t" makes G vanish at the
upper limit. The first term on the left hand side of
Equation 10 becomes
14
and the second term on the left hand side of Equation 10
is zero from Equation 6. Rearranging terms and inter-
changing superscripts gives the initial-value solution
F' + ) = fd'C) t; 01o)F '.(3)
The next step adds the electric field part of the
Liouville operator as a perturbation on the anharmonic
solution:
It is assumed that the system is at equilibrium with the
distribution function FA(V) for t(0, and that the electric
field is turned on at t = 0. The problem is to solve
for the distribution function F(0, t) with the initial
condition given. The canonical form of the distribution
function will be assumed, namely
FA(O)= Z (5e
where H' = HH + HA, ( = l/kT, and Zis the partitionfunction for this distribution. A new Green's function
is now defined as the solution to the equation
which obeys the same boundary conditions as those defined
previously for G and has the property that G'(0t; 0"t") = 0
15
for t <t". The solution to the initial value problem
follows as before and is
F (0t) f d 01, '(ot - "o) F"(o" (17)
For a solution in which the perturbation is linear in
the electric field, the function G' can be written as a
truncated infinite series of terms (see Appendix B ):
Substituting Equation 18 into Equation 17 gives
F(A) F'(v4)f fd'f ' '
Since FA(0) is the equilibrium distribution, it will
be unaffected by the direct Green's function operation.
The integral over 0" in the first term of Equation 19
therefore simplifies to
f]j 0" (Ot 0#) F,4(04) = f(O) )
and in the second term the integral over 0" becomes
Making('e' bituon F give) te re). e
Making these substitutions gives the reduced expression
16
Equation 22 expresses the distribution function at
time t in terms of the initial, equilibrium distribution
function FA(0), the linear electric field perturbation
operator XC, and the time development Green's function
G(0t; O't').
Inserting Equation 22 into Equation 1 and dropping
the term corresponding to a permanent electric moment gives
the first order perturbation value for the induced polari-
zation EfrIft.(14%.;h') fEf4
PM= -) do . (~tr~ f ( .0as
The time integral is a convolution integral because the
Green's function G is a function only of the time difference
t-t', while the electric field operator is a function
only of t'. Taking the Fourier transform of Equation 23
and applying the Fourier integral theorem gives the trans-
form of the induced polarization
p()=- Eff ?') Qd'o) A4i ), (
where R is the Fourier transform of G and ; is a frequency.
Appendix E develops more explicit expressions for iand 4 for the particular case of a simple cubic crystal
having oppositely charged ions at alternating lattice
sites. Making these substitutions into Equation 24
gives
17
P ) A t(lm)'\c, Co, 1) 04' R (0tse
Xf (-/)' EC E4) (o,))0F'( '], ( s)
where q(0,1) = ... , q(O,l) . E(-) is the transform of
the applied electric field. Vtis a unit vector in the
direction of one of the harmonic normal mode eigenvectors.
(- is the electronic charge on one of the ions. rh = Ml1h2/
(ml + M 2) is the reduced mass of the two ions in a unit
cell. The amplitude of the normal mode "1" is q(0,1).
Rearranging as
IV~~ ~ ~~ W=O f O 7,'7/ I I)fd 4 R o,)F ')E ) (.
shows the similarity in form of the equation for P with
the macroscopic expression for induced polarization
where )1(f) is the dielectric susceptibility tensor.
Making this identification, dropping the multiplier )
and using Y = , the identity tensor, gives
(A//m) 6, iJd'J%1)fd6'R jY,) F' ) (.s)
The set of canonical coordinates 0 is now identified
explicitly as the set of action-angle variables fJ(fs),
e (f,s) defined in Appendix C, where f = (f1 , f2 f3)
is the wave-number and s is the mode label of the normal
18
coordinate sets. In most of what follows, however, the
single subscript k will be used to stand for both f and s.
Only where specific differentiation is required will the
more cumbersome (f,s) be used. Also, any quantity written
without a subscript will mean the entire set of that
quantity. Details of the development of the Hamiltonian
and the various operators in terms of these variables are
contained in the appendices.
For convenience in calculations, the transformation
is made to the basis
where rik = 0,+1 .... Rules for the transformation are
given in Appendix F. The transformation changes the
integrals over 9 to summations over M. Functions of J
are unchanged. The result forX can be written
Xc L If di'df'J,, <ao/Y1) 1Ih>
(30)The Dirac brackets are used for convenience in notation;
the calculation is purely classical.
The resolvent R, the Fourier transform of the Green's
function G, can be expressed as a series in the anharmonic
perturbation operator Z and the harmonic resolvent RH.
This series for a cubic anharmonic potential is developed
19
in Appendices B and F. Substituting from those appendices
gives
XZ2. I-Afdf r j)s f~t ~M~ArI>J
X4'r< I F AI>. (3A)
Appendix G shows that the matrix element of the product
RH XAcan be simplified as
<hi gR'XIh>: ' EK lwX'1[) r )(h<)'/I
Since the operator RH is paired with the operator each
time the latter appears in Equation 31, the expression for
the product RHXA obtained from Equation 32 can be used
to eliminate multiple sets of J in the expression for the
susceptibility. Substituting and integrating successively
over each set of J's simply replaces each set in turn by
the next until only a single integration over the initial
set remains. The result of the substitution of Equation 32
into Equation 31 and integrating over all J's except one is
I fdJ2Z0 IqvO)1) Ii >
)( r( A)rD(r)4 2: <Kn(V4)1XIht)> r
20
where D
Explicit forms for the matrix elements have been determined
in Appendix F. Substituting from that appendix gives
(r # ) (1-4 )
>(7T~ foa(2Z V{re) V K) / 0I 71))< 7T(6 ( )r+if I) -1)/fL....- t 1Z 1 (;D
X f(h h/)(J/2 2i2I fC kCC)'hX, /
X 17[F4v's 3Z eKI V~eij IUT/QJ/ '7TJ (t) to)- C,(3Ywhere
1\JV AX .br) t,4, and has been set to unity.
The significance of the |[ ]l brackets is given in
Appendix F.
Equation 34 contains a series of delta functions
relating each of the n's to the value in the summation to
its right. It is reasonable to think of each summation as
an "interaction," and to refer to the change in value of
a particular set of n's as the result of that interaction.
The delta functions associated with the cubic anharmonic
21
interactions restrict three and only three n's to change
in an interaction (see Appendix F). The sum over all
n's then reduces to the sum over just the three n's which
change; that is, those n's taking part in the interaction.
The remaining n's have an effect only on the frequency
sum, the denominator, since all non-zero n's are included.
Only the three n's which change in the interaction affect
the anharmonic coefficient V {e,ki and then only by virtue
of the changes they undergo, since the e's in the subscript
3kS = fe es, e"1 are the values +1 by which the
n's change.
The delta functions involving the n's cause a
"chaining" of the n's so that a particular set of e's,
which are the changes in an interaction, determine the
n's which follow on the left in the expression for sus-
ceptibility. However, the leftmost set of delta functions
forces all n's to be zero after this final interaction.
Consequently, for a given interaction order (determined
by the value of r on the anharmonic term), there are only
certain sequences of the n's which are allowed. Since
only no has a special part in the susceptibility expression,
most of these sequences for the n's are identical except
for the particular set of lattice frequencies with which
they are associated. All allowed combinations are
eventually summed and integrated over, so that sequence
22
types are significant rather than the particular lattice
frequencies they represent. There is still a great
"bookkeeping" problem, however, since there are three
n's changing for each anharmonic interaction, and the
permutations and combinations go up at an astronomical
rate for relatively low interaction orders.
A computer program has been written to aid in the
bookkeeping problem. A listing of the Fortran statements
comprising the program is given in Appendix I along with
a detailed description of all its operations. The program
tries all possible sequencies of n's, retaining those which
are allowed by the selection rules. Restrictions on starting
and ending sets are included, and only interactions through
the eighth order may be examined. Because of these re-
strictions, the number of allowed sequences is reduced to
a manageable level.
A further simplification is due to the special form
of the functions of the Jk's. Only multiplication and
differentiation of terms of the form JPe-"' appear and
all such terms are then integrated over the entire range
of the J's, resulting in a value for each term of the
general form (p!) ((3 )-(P + 1). Appendix H shows that the
coefficient of each such term and the power p is uniquely
determined by the sequence of the n's and that the result
of a given sequence can be written down from an inspection
23
of the sequence of n's and the changes of the n's. Conse-
quently, each product of r anharmonic terms can be written
in the form3 C [K3(-1)(1
where fk (r) = the set of three lattice
frequencies associated with the
rth interaction;
k = one of the set of three lattice
frequencies;
V (e,k}(") = the anharmonic coefficient
associated with the anharmonic
coefficient of order r;
CU) (k(r) .... k(.)) = the computer-determined value
of the interaction coefficient
associated with a particular
set of lattice frequencies,
chosen one from each set of
three associated with each inter-
action, down to and including
the jth. Each CO) depends on
the choice of lattice frequencies
to its left in the diagram, so
24
that there are 3r sets of C's,
each set giving a term to the
susceptibility, for an anharmonic
interaction of order r. C(E)
is the contribution from the
electric field interaction;
k (r)... k (lM] = the factorial multiplier
associated with the particular
lattice frequencies chosen;
[6(r)J 3/2 ( o, l1/2 for the set of
k's at the rth interaction.
The computer program of Appendix I also evaluates the
factorial multiplier and the interaction coefficients
CO) for each sequence. Because one coefficient from each
interaction is included in the product of coefficients
multiplying the entire term, any null coefficient will
eliminate the term completely. This fact makes it pos-
sible to eliminate many sequences from consideration which
would otherwise qualify for consideration. A further
restriction is contained in the definition of anharmonic
constants V fefsj (Appendix C) which includes the
generalized delta function A (f + f' + f") defined by
A(f + f' + f") = 1 if f + f' + f" = 0 or
a reciprocal lattice vector,
= 0 otherwise.
25
This selection rule severely limits the sequencing of the
n's. Even so, the number and type of terms to be considered
multiplies very rapidly with only a few interactions.
After using the computer program to sort out the terms
of lowest order, it becomes evident that certain sequences
of lattice frequencies and their corresponding n's occur
repeatedly. Fig. 1 shows the lowest order repeating sequence
in diagrammatic form. The first diagram is that of the
harmonic, non-interacting crystal. The second represents
the coupling of the square of the cubic anharmonic potential
to the infinite wavelength mode (k = 0), through a pair of
modes of oscillation having wavenumber +k. The third
represents two such couplings, to different modes, etc.
Each diagram represents several similar terms in the series
expansion of the susceptibility given by Equation 35. For
the unperturbed harmonic crystal represented by the first
diagram in Fig. 1, the contribution to the susceptibility
is
where the summation is over all sign permutations con-
sistent with the diagram. The results of computer evalu-
ation of this diagram gives
0s that* )
so that
L . I '' ZhT. )IV.
where n0 = n(0,1) and J0 W
no
No n c
A- 4
TwoAht4rmo"all
Four Ahkdi"o6c.
f% vi 0
0r~ d~mni
Fig. 1--Diagrams contributing to the susceptibility
26
(38)
7 T (?.)4 )
27
Summing the two terms gives the total harmonic contribution
to the susceptibility
()N EJIA/l,. ), yo)where use has been made of the definition of JC in
Equation 34 and the harmonic partition function Z /O
and Y/is the volume. (aW
(4/)The next diagram is the lowest order interaction in-
volving anharmonic interactions. It represents the term
of second order; no first order terms are allowed by the
selection rules on the n's. For this diagram, Equation 35
becomes
F h0 1t p gnd e kdI
From the computer program and the diagram,
V V &Kcl(1) =VK(-K 30)
28
D/ () -I
From the computer program, the only contributing choices
of frequency sets and their vertex contributions and
factorial multipliers are shown in Table I.
TABLE I
2ND ORDER CONTRIBUTING FREQUENCIES,CONTRIBUTIONS, AND FACTORIAL
MULTIPLIERS
InteractionCoefficients Factorial
Multipliers(2) (1) (E)
no(2)n(l)(f s) no()[ (fs) ]
no (2) n (l)(-f.,s ) n0 (0) [04; (f ,s) r ]-1
Substituting into Equation 42 from Equation 43 and from
Table I gives
29
(m) NE f k V (eL- ) I ))
V h~~t'~dfh' S)4W(4 5) . i)(4 ' 4(.f IS#)](4)
As it stands, this equation represents several terms in
the susceptibility. Each of the n's can take on the values
+1, and f can take on any allowed value except 0. The
mode labels s and s' can each take on 1 or 2. The equation
may therefore be summed over the n's and s's and L's. The
sums over n0 (2) and no (0) give similar results to that
of Equation 19, while the sum over n(1)(f,s) and n(G)(-f,s')
is
I ( )+ I(fs)4fs) + (-f') 4 ff s')
+ ~cy ~+ ^)1
Making these substitutions into Equation 44 and summing
over f, s, s' gives
30
AY eo 2 (f~s 7f s61%.
[0
The third diagram involves two pairs of anharmonic
interactions whose associated lattice frequencies are
assumed to be distinct. From the computer program, the
contributing frequency sets and their vertex contributions
and factorial multipliers are listed in Table II. The
corresponding anharmonic coefficients and denominators
can be read from the diagrams and all quantities sub-
stituted into the equation represented by the diagram. The
result is
(3) +l)(3) 4)
Nam nn t
(4/7)
where k, -k have been used for (f,s), (-fs'), re-
spectively.
H
CY9
4 0
H HU-
Zo
C)
F-Z01
d 0r-4,.4
0 PO4J orHU 4JCr-
4-
e-4
oi
C)
14
04*-4
*1
zt-4
0
C
-
o-f
00
C
C
L
Ic
o
-Ift
0
'4
0
d
31
4
0
;,CSC C C
/"(Nj
- -
32
Just as in the case of LxA each bracketed term may
be summed over all f's, all s's, and all n's to include
all diagrams of this type. The result is
3< 14i ) V L I4~~ -'
(3( )
If it is first recognized that the summation indices are
arbitrary, then it is evident that Equation 48 contains
Equation 20 multiplied by an identical factor of second
order in the anharmonic coefficient
Evidently, succeeding diagrams of higher order will result
in the formation of a geometric series with the leading
term
x the.V (atioV
and ithes frto cgie httesumto nie r
aritaythniti eidn tatEuainz8 otan
Eqa i_____uli___e_____________al (actorof secon
ordr n heanarmni ceficen
33
This series sums to the value
pwhere rF2 is the bracketed [ ] term in Equation 49.
expression for F' 2 has poles at z = 4_ (W + 4
These can be taken account of by writing
+
giving
r14 (.where
A 4t,(e U): +Z V IeIL
x f El' [((4 "L.V 1L__ _ __ _
__ _______I)< 0 f f( .
(so)
The
(si )
(62.)
and
it353)
s~)
NY ( 4 %:-+, T
I
Nam
y I (ZI-)
34
Putting Equation 52 into Equation 50 and rationalizing the
denominator gives
CHAPTER BIBLIOGRAPHY
1. Balescu, R., Statistical Mechanics of Charged Particles,Vol. IV of Monographs in Statistic'l Physics, 4 vols.,edited by I igogine~TInterscience, New York, 1963).
2. Henin, F., I. Prigogine, C. C. L. George and F. Mayne,"Kinetic Equations of Quasiparticle Descriptions,"Physica, 32, 1828 (1966).
3. Prigogine, I., Non-Equilibrium Statistical Mechanics,Vol. I of Monographs- in~TaTistica1-Physics, 4 vols.,edited by 1. Prigogine(Interscience, New York, 1962).
35
CHAPTER III
NUMERICAL EXAMPLE
For nearest-neighbor interactions (see Appendix C)
V (; )O)S t.s I
elm; p (0-k1/o3
) 55
Thus
V t);c)O
/o3
(.I ) S*sI
S =
which is independent of f and s.
Putting Equation 58 into Equations 53 and 54 of
Chapter II gives
~ N3 +
4-~~# Pr(k;A
36
and
(.5-1)
)s s
S
(57)
(s)
4J t)( ; (-1 s):009 n'(rKIN,)
37
and
The sums over K in Equation 59 and Equation 60 can be
replaced by integrals over A2 _x]2:
E/FV 4 f al-a ~ (+-)F( l), (/)
where 6 N is the product of 3 (dimensions), 2 (ions per
unit cells), and the total number of unit cells N, and the
frequency spectrum Gf(+-)(i2) is the fraction of squared
frequencies in the interval (f12. jA 2 + dq 2 ), and F is
any function of the .4)'s and K 's. The function G+-) 2
is related to the frequency spectrum g(#)(.j), the fraction
of frequencies in the interval (.fi, ..l.+ dn.), by
2G (A2)= g(+-)(.) Using Equation 61 in Equation 59
gives for 44j2, the real part ofF'2
3h2exp res n fr ,
The expression for J2, the imaginary part of [2, is
3 3 6i I') ea dd%&f)(z )l2L (6) 3 )
38
which is simply
.
N 6-,;.0()
Montroll (2, 3) has developed techniques for deter-
mining the frequency spectra of simple cubic lattices,
and presents the results in several graphs. These spectra
are for the frequency set 4)x or 4/, , though, and the
frequency spectra for the sum and difference 4 4J., are
needed. The following paragraphs derive a general relation
to accomplish the change of variables.
Formally,
A change of variables allows one of the integrations to
be performed. For g(4-'4), take
Then
Since
(a)Lb) K
39
Integrating over a gives
~( t ~ ) L db 4ff0-( 4 b /____
A second change of variable y 4 )k+ t.K+ b puts
2
Equation 70 into the more convenient form
Similar changes of variables for g(A/I(-tv-s) gives the
result
Closed form solutions for g and g(- are not avail-
able. Approximate forms obtained by numerical methods
have been determined by Mazur (1) and reported by Montroll
(3) for a similar crystal model to that used herein.
Fig. 2 shows the spectra G (.,W 2) and G (42) .The
spectra have been normalized by multiplication by the
largest frequency W.2 and are plotted against the
normalized squared frequency (&/ V )2.
40
104
4'
1~I
0.1
Fig. 2--Plot of squared frequency spectrum 0. 0 )Vs.
squared frequency ratio ( e/.)) for a diatomic simple
cubic lattice.
Fig. 3 shows the frequency spectra g (+(4j) and
g() (4)z) obtained through the relation g(4A) = 2W G(A)2).
These spectra have been used with Equations 72 and 73
to determine an approximate form for the sum and dif-
ference frequency spectra using the computer program
0
I
OR
2 40
41
Iw"16
0 I
0.1 0.5
III'ItIIIIItIti tI I IIi *Iv 'I II II II I
L I
I 0
010
Fig. 3--Plot of frequency spectra g ~)(.. ) and
g C+) (&)----- vs. frequency ratio 4/4.. for a diatomicsimple cubic lattice.
described in Appendix J. The resulting frequency spectra
for g( 4)1c+ ty_*) are plotted in Fig. 4.
0
0
w
I.1
4/A)
42
Z
+1 I
o.(.o. 9\-oI -' I -
Fig. 4--Plot of frequency spectra f 4. -- and{I, 4. 1 ------ vs . fr equency r at io 4a//..
According to Equations 63 and 64, the frequency spectrum
needed is
(7I
43
This spectrum is readily obtained from Fig. 4 and is
plotted in Fig. 5. Multiplying the spectrum of Fig. 5
by the constant
S32T r
inN*-elgives .
Similarly, multiplying the spectrum of Fig. 5 by
TT
2 2and integrating with respect to .f.2 gives 4\-.)2
Another segment of the computer program of Appendix J
was used to perform this integration. Fig. 6 shows
the result of the integration, plotted as a function
of ( . )2q
Returning now to Equation 55, the full expression
for the susceptibility, and putting into a more con-
venient form for plotting:
In this form, /400 2 is the curve plotted in Fig. 5,
multiplied by 4)V/4 , and A 74,' is the curve of
Fig. 6, multiplied by In ),. The real part of the
susceptibility given by Equation 76 is plotted in Fig. 7
44
.O
o.01.0 '.5-
Fig. 5--Plot of squared frequency spectrum a )vs.squared frequency ratio ..f?'.
and the imaginary part in Fig. 8, aside from the constant
factor NE.I/Myw, For purposes of illustration,
Ah /k), was taken to be 0.7. Shown for reference in
Figs. 7 and 8 is the result for A)Q = 0, corresponding
to a purely harmonic crystal.
-Imommom
wlo
rg
ft
17
4'
4'
I~
i0-
V
I / Iaow~ 3
Fig. 6--Plot of the normalized squared frequency
displacement Tri4*/4J* vs. the squared frequency ratio
45
1.1
-JL----
lift
a
No
\)ko
46
10
oI
0
oI
S!
(/4.
Fig. 7--Plot of the normalized real part of the
susceptibility(,,,(X vs. the squared frequency ratio (/,
47
30
00ni s i
anharmonic susceptibility
harmonic susceptibility
Fig. 8--Plot of the normalized imaginary part of thesusceptibilityQ(Ivs. the squared frequency ratio (z/z1)) .
CHAPTER BIBLIOGRAPHY
1. Mazur, P., unpublished thesis, University of Maryland,1957.
2. MontrolJ, E. W., "Theory of the Vibrations of SimpleCubic Lattices with Nearest Neighbor Interactions,"Proceedings of the Third Berkeley Symposium on Mathe-matical StatistiZcs~ aindProbability, Vol. iiT~Berkeley,Calitornia, Univers ot Calitornia Press, 1956).
3. , "Theory of Lattice Dynamics in theHarmonic Approximation," Supplement 3 to Solid StatePhysics (Academic Press, New York, 1963).
48
CHAPTER IV
DISCUSSION OF RESULTS
The principal results of this paper are expressed
by Equation 55 of Chapter II and by the numerical results
of Chapter III. The results obtained are similar to
those obtained by other methods and compare favorably
with measurements of absorption and dielectric properties
on real crystals. In addition, the variation of these
properties with temperature has been shown to be linear,
also in agreement with real crystals.
The theoretical results obtained in this paper are
quite similar to those obtained by Maradudin and Wallis
(3, 4) and Cowley (1) using other mathematical approaches.
Compare, for example, Equation 50 of this dissertation to
Equation 5.3 of reference (4) and Figs. 7 and 8 to Fig. 9,
which is a reproduction from reference (1, p. 193).
Similarly, the numerical results of Chapter III may be
compared to the experimental results of Genzel, Happ, and
Weber (2, p. 327), reproduced in Fig. 10.
This paper has treated the dielectric properties of
ionic crystals using an extremely simple lattice model,
yet has obtained results in substantial agreement with the
results of other approaches using more complex models. The
49
21
-2
REAL
.OoF
1./
II I I I
0 2 4 6 8 0I I2 4 6 8
12FREQUENCY (10 cps)
Fig. 9--The contributions to the real and imaginaryparts of the dielectric susceptibility of KBr at 300* K.
I IWT WL
NoCIIR
R
-T
I I I I I
10
0'
~eI'U
10p
-210
-310,
C1
C
0
0 100 200 300 400 500S(cm')
Fig. 10--Thie infrared E,(w) spectrum (IR) and theRaman spectrum (R) of NaCl.
50
IMAGINARY
A---- B
4
3
2LitfL-
--IA.'
I I I - -t I
IV
51
good agreement with the properties of real crystals suggests
that the method allows the most significant terms in a
perturbation expansion to be picked out in a more straight-
forward way than with other methods. It might therefore
be expected to be useful for the treatment of other crystal
transport properties. Extension to more complex lattices
and ionic forces would allow an even closer fit of theory
to experiment, although at a substantial increase in
analytical complexity.
CHAPTER BIBLIOGRAPHY
1. Cowley, R., "The Lattice Dynamics of an AnharmonicCrystal," Advances in Physics 12, 421 (1963).
2. Genzel, L., H. Happ, and R. Weber, Zeitschrift furPhysik 154, 13 (1959).
3. Wallis, R. F. and A. A. Maradudin, "Lattice Anharmonicityand Optical Absorption in Polar Crystals I. TheLinear Chain," Physical Review 120, 442 (1960).
4. "Lattice Anharmonicityand-Uptical Absorption in Polar Crystals II. ClassicalTreatment in the Linear Approximation," PhysicalReview 123, 777 (1961).
52
APPENDIX A
PROOF OF THE GENERALIZED GREEN'S FORMULA FOR
THE LIOUVILLE OPERATOR
The formula to be proved is
where and are functions of time and have the
properties
T) + re -~ )0) j , | A, -
(T--0 e) o(A2)
and the operator is
where
and Hk isth3;HiotxniatnT.
and H is the Hamiltonian.
53
54
Making the substitution of A3 and Al gives the relation to
be proved:
fI'1fdafdJf O(jO II?
l4fdf Vi a (A#
The left-hand side of A4 may be rewritten in the form
Defining and writing out in AS gives
0 w.4a. 9
(A)Integrating by parts on &, in the first term of the
expression under summation and on in the second term
gives
where the primes on the integrals indicate the omission of
9,orT .
55
Since the function f obeys the same boundary con-
ditions as (/, and4X separately, the first term under
the summation must vanish. The second and fourth terms
cancel, leaving
Now
4X T-T"eTPIUU 0P f 1C++e HAIO
so that vanishes for Tc = 0.
Since vanishes for 1-+0o , only the first term in A8
is non-zero, and the left-hand side of A4 has been reduced
to the right-hand side. This proves the generalized Green's
formula.
APPENDIX B
GREEN'S FUNCTION SERIES DEVELOPMENT
The Green's function of the Liouville operator
is defined by that solution to the equation
for which the causality condition
C ( ;01 )=0 ) L(i 4)
is satisfied. This section will develop a perturbation
series solution of Bl for the Liouville operator com-
posed of a larger part, , and an incremental part,
For 0 =0, the solution to BI is .Writing
out Bi explicitly for this operator gives
and the X operators depend on the variables (a).
56
57
To proceed to a solution, multiply B4 on the left by
and integrate over and t. The result
is
Making use of the fact that Zis a differential operator,
the first term on the left hand side of BS can be inte-
grated by parts:
The integrated portion vanishes because of the periodic
boundary conditions on ., and the remaining portion can
make use of the adjoint Green's function
-( to obtain
By its definition as the anti-causal solution,
use of the adjoint Green's function reduces B7 to
58
-t ")- f dq6S0 Cd
(O 5)
Substituting B9 and B6 into B5 and solving for q (pi 04:
(3,o)An infinite series expansion for can be obtained by
repeated substitution of the left hand side into the inte-
gral on the right hand side. The result is
For effects linear in X , all terms past the second are
dropped, leaving
where the arguments of the integrand are given in B10.
Now the Green's functions are all functions of the
time differences of their arguments. Defining the new time
variables
'S z
59
and substituting into B12 gives
~~~i( (40 r)(O) dLi0~ (:c~~pThis equation is recognized as a convolution integral in
time. Taking the Fourier transform of both sides of the
equation, and introducing the resolvent R(i)as the
transform of the Green's function gives
R(0 ')4 m ) M #0 Z).Y R(O ' )
This equation can now be solved in iterative form by
repeated substitution of the left hand side into the last
term of the right hand side. The result is the series
2f o f..R7.A (RI)
This can be written in a generalized symbolic form as an
operator
rz
60
01
Specific representations of R()are then determined by
matrix elements
APPENDIX C
NORMAL COORDINATES AND ACTION-ANGLE VARIABLES
In the following, a crystal of finite size, but con-
taining a large number of ions will be assumed. The
position of the unit cell in which the ion is located
will be labelled by the triple integer set
which counts the number of cells between the corner of the
crystal and the cell in question. Corresponding to the
set P are the primitive translation vectors 4., , 43
of the lattice, so that the position of a given unit
cell can be specified by the vector ' ,4 4 P,.4
For definitiveness, the crystal will be assumed to con-
sist of N1 , N2 , N3 unit cells in the 4,)144 directions,
respectively, a total of N = N1*N 2 .N3 unit cells. The
resulting rhombohedral parallelepiped will have edges
measuring
L,= N , oI 1 , ,- 0.%,L, , s .
Each unit cell will be assumed to contain just two ions
whose equilibrium positions with respect to the corner of
the unit cell are given by the vectors -.:/ .
61
62
Thus the position of a given ion is the vector sum
Finally, the instantaneous displacement of the o(ft ion in
the pth unit cell will be labelled by
O(I( P)
and the mass of the ion by A O(= /1z
The collective motion of all ions in the crystal can be
expressed in terms of normal coordinates through the
trans formation
where ( fjs) is the time dependent amplitude and
the eigenvector of the normal mode associated with
type ions, labeled by the wave vector and the mode S.
tn. is the mass of the a( type ion.
Cyclic boundary conditions will be assumed to avoid
problems arising at the boundaries of the crystal. This
requires that
where N1 , N2 , and N3 are unit cells counted along the
Bravais lattice directions, and n = (N1 ,N2 ,N3 ). Substi-
tuting normal coordinate expansion gives
63
Since the fas) are abritrary, each term in the series
on the left must be equal to its corresponding term on
the other side of the equation. Thus
> r - )(g
Now a reciprocal lattice for the crystal can be defined by
the reciprocal lattice vectors
4LL
Since 9,-bjf= Sj / , and in view of C2, evidently
any vector of the form
f: .2tkb, f______b%. atrK b 3
A(, Af 3"
where the K1 are integers will satisfy the relation C8.
However, the range of f must be restricted to give the
correct number of degrees of freedom for the crystal. In
restricted form, then,
64
o - r bP3. k , A < /
/VV'3~N< All.
- Nx< k,< /
Using C2 and C10, the expression can also be
written
S-r = n / .(eli)
Hamiltonian
The complete Hamiltonian of the crystal is just the
sum of the kinetic energy and the potential energy.
Assuming that the potential energy can be expanded in a
Taylor series in the displacements of the ions from their
equilibrium positions,
- 0 ++114 Z4ofPP.
Pr,?
where the summations extend over all repeated symbols.
The terms of third order and higher will be assumed to be
small compared to those of lower order, and will be
ignored for the moment. The term of first order in the
displacements must be zero if the crystal is to be
65
stationary, and the constant term may be set to zero.
remaining term is the "Harmonic potential,"
At-x4
The
(c' 13)
For the "Harmonic crystal," then, the Hamiltonian is
)f I (ci/'f)p p'
The equation of motion for one of the ions is
Ja il.lo 1
A$
r Ak
Oval
i ar]
(rc/,)elf.
Substituting the expression on the right hand side of C15
into C14 gives
R Im ( ILr(p (e 14)
Putting in normal coordinate expansions for the dis-
placements:
57
04 si:
(c 17')
ISI)
0%0
)< 11Lt
('fs 7:c'')Z f c
I a &ftME-no-W ( , 51)
66
From C9, the sum over p is
(C 3&)
But each of these series is a geometric series which sums
toKA.
(cl 9)0
The numerator is zero for any value of k, making the sum
zero unless the denominator vanishes, also. Since k=0 is
the only value for which this is true, the original ex-
pression is easily summed:
Substituting into C17 and summing over gives
To further reduce this expression, return to equation C15
and put in normal coordinate expansions:
67
Pr,Combining terms:
But since {s) is a normal mode,
where s) is the frequency of oscillation of the normal
mode. Substituting C24 into C23 gives
~ (f 35f-sjz f ( s Q)+3Z ~v'f
X A( * M 0.
Since f~fs) is arbitrary, the expression in curly
brackets must be zero for each f and s, and thereby becomes
the set of equations determining the normal mode fre-
quencies 6U( . Next multiply the expression in curly
brackets by L'vv*(f and sum over o and j:
v X~~ Fv (,5)*V/( f,) eo
Pr' (c(f)
68
Next replace s by s' in C25 and multiply the resulting
bracketed expression by F ^t(.s1* and sum over o( and
Subtracting C27 from the complex conjugate of C26 gives
AP(f'
Pr '
=0.(cA0)
Since summation indices are arbitrary, p may be inter-
changed with p , o with c4 and j with j' in the last
term. When this is done., the result is
F (f ist) [ ~ ~'~fs]~ir~c~
al Y vv
AAO'A>0.PfI(c,;1)
69
The expression in brackets is zero since A and
e. %, A. 9 are functions only of the difference
P--' .Thus whether p or PIis summed over is immaterial,
and the two terms cancel. Equation C29 therefore reduces
to
102'P s)] V, +Ip)=&. (C30)
If s=s', the equation is obviously satisfied. However,
if s/s', then the bracketed expression cannot be zero,
because 4<ffs'):4Yfs) is contrary to the assumption that
s and s' are different modes. Therefore, it must be true
that
E)s (e3/)
When s=s', the choice of normalization of the eigenvector
is arbitrary. The usual choice that is made is
Equation C25 gives this defining relation for the fre-
quencies:
f=/ (c ) A )
70
If the frequency is changed from f to -f, the expression
becomes
(cu)where Aj(-(4s)= v(-fs) has been assumed.
If the complex conjugate of C33 is taken, the result is
(C 35)
If / ,-
the two expressions will be identical.
Equation CS defines the ion displacements in terms
of the normal coordinates. If the complex conjugate of
CS is taken, the result is
(d 37)
But since must be real, this requires , ,,or
(c3 ')Changing the summation index on the right from f to -f gives
71
Using C36 reduces this to
Ignoring the trivial case -(f-sjO then gives
Substituting C24, C32, and C41 into C21 gives
For some purposes, a new set of variables is con-
venient. These are "action-angle" variables, and are re-
lated to the normal coordinates through the transformation
equations
c~fs:(~s/~YWfs) C )/L 0A4f4 fT-I
(C 3)"J" is the action variable and " " is the angle variable,
and are the generalized momentum and position, respectively.
Both are real variables.
72
Substituting C43 into C42 and multiplying out and
combining like terms gives
Changing the summation index in the second term:
Since the entire range of f is summed over, the order of
summation is immaterial and the two terms of equation C44
are identical. The harmonic Hamiltonian in action-angle
coordinates therefore reduces to
The complete Hamiltonian for the crystal consists of
the sum of the Harmonic Hamiltonian just determined plus
the higher terms in the series expansion of the crystal
potential. The next term in this series is the cubic
potential term, which from C12 is
C Z ~ ~ cB7
or, substituting normal coordinates,
)4
73
t t I. Ip plapl
Yz~Qo +f'fr) (cot
+- + - a reciprocal lattice
0 otherwise.
Defining new cubic potential coefficients for the normal
coordinates
if ) lw IL 90r)(.I
y( ( )ot4
XA~f fhF-f4 f )I (J f
V(ccs)
Substituting action-angle variables for the normal
coordinates gives eight terms which may be written
with l 4
vector
v(f
gives
6/6c I
= 8 yxzof
(-f ISO)0-40 ) I
74
U% 7 +e' 0 Te, ) T~ (eTOs)
X Ce.
However, it will be more convenient to deal only with
positive "f" values for the J's. If each term has the sign
of f changed whenever e is negative, the result of the
summation will be the same since all values are still
summed. This change, though, has the effect of transferring
the change of sign to the V's, so that
CC
where(4 E
Nearest neighbor cubic (central)
The cubic potential term for a general lattice is
1 (c53)P? fviI A J
75
In an alternating, simple cubic lattice with central forces,
all cross-terms vanish (1) so that
OLp pp/pV( IXts'
1-lift
ocr
(cs/)
The cubic part of the force on a particle o(:) pl:o in the
j direction is thus
0f'f"
(e s5)
But for nearest neighbor interactions, this force can also
be written
(pt. * )13Lj'~0X4-
Expanding CSS:
a
lo* dT .+0
A,0 2,0(c 60-6)
Im Iow 4z
Alla f
f 10
6
OLA
4- '7C,
OeX0X003~+ 'lX
0 0-/
+U jg
I 0
+ 1IL
Al ,,
Poo
0 0 04 i
Comparing term-by-term with C56 shows that
(3"
0I
13
00 200.
4) 0 -I
Bill
I x 0000
0'-
(00
0 I
/ .a p' I
76
0- 0
IL 2.
8oB
OX -
(C57)
58)
+r A*
a
"l ,f
77
In a similar way, the force on the particle o , 0
is
j 3 ' (c17)
or
FjC, ( . * i)K ( a4 0
so that
1$
000U a tio 0)0
000
oil
0a
001
01 0
-mm6NW A
(C&I)
((4o)
I-was
zzI ao
# 0
-i- I lp
-*C')
- ti
78
Thus
where ( 0 or + 1, depending on the combi-
nation of the 4 parameters on which it depends. Simi-
larly,
with E given by previous expressions.
Substituting normal coordinates gives
K0 of C
P f f 10%0
79
Def ining
V (I'Y ~~prPO( 1siio4)Iv ) ( n) ( P, qit )
(s)v )-0jC ~
allows A to be written
The anharmonic coefficients required in Chapter III
are v>S ) ?
V( )V(*4Q) ,VG ( 9 fv). c
For the Montroll model (2) of a simple cubic lattice in
which motions in the three coordinate directions are
independent, and for which only nearest neighbor forces
are considered, the number of terms in C65 contributing
to each of the V's is reduced considerably.
The eight non-zero terms (excluding common factors)
are (dropping the superfluous coordinate superscript and
omitting common factors):
80
- i, (0),1) ) vi (sAl) , (- ,')e
v; (0,1) v'.(f,s)Vf, (f), g) e~AreN
x~o,1 Yi f(r~s~s(-Cr)Ie " (
where use has been made of CS and the fact that the
ions in a simple cubic lattice are equally spaced in any
mutually perpendicular lattice directions. Also, the
superfluous coordinate superscript has been dropped.
Making the substitutions into C67 gives
>, .N0) s Y/,
or
81
V) -
S sa
From Appendix D, the normaliza,1( (cf.9)
tion of the V5 gives
27(c 70)
But for the independent coordinate motion of the present
model,
Therefore
V2 ;(0) =)and
( ' 4&
Maradudin and Wallis (3) show that the eigenvectors of the
linear chain can be written
V, (K~,a) 1r OfK 1/ z' C05
(71)
(c1~
where
C'A I\ 3)n (7T N/,> f
Y OV (01 m, 7mom ,)
IS)
(Vi~o ,)f= e/,
Ij Lvo,>J = M/Im. .
2[r/4.(0,)] =[f2/;(O, Ol 20= n /mg .
V, (0, 1) = (M/M ,) A
-- n/r')--. ( 14 /A , '1/4:10,10(1)
071
V, ( K 1 1) = --- 45 ol C I(
( X( h~n '1 s,-1 c 05 (Tr KIN, 273)
82
Since the equations of motion for the linear chain are
identical to those of one of the independent motions of
the present model, their results may be used directly.
There are four combinations of the mode labels Ss
for the Vs in C49, namely 1, 1; 2, 2; 1, 2; 2, 1.
Evaluating
for each of these, using C 73,gives
I: Cos o0h (- ino -(..$in 4 4)Cos(I 0
1 ,* sin o( Cosa 1 o. 9 ~n
eos4 ~s~ (..~I1o()$~l K
in 0 o 4s O I~C
-3 f-I (C 74)using C u-.anL C7 if Ci 4 gives
0 f . 70)
31V(?04 - 7 l,
The linear chain solution also yields values for
the normal frequencies CQ y c
namely
( 'C 7)
T T r. .10 ., 1.11 r *7 -) . .
(wJ fz : /2.(, ) -.
where 4j: M
ei- /r7
Multiplying C74 and C?? gives
4a'C 1 3.%
some C'-'b ITrn K/11
'9f Pn rr K/A/m, m
and
83
% 2 ' "f/ ncrY
4 1 4C77
and
in /
( c 79)
4w4l. 4)61L.= A)o 4t = c2 a-lo /M
CHAPTER BIBLIOGRAPHY
1. Leibfried, G. and W. Ludwig, "Theory of AnharmonicEffects in Crystals," Solid State Physics 12, 275(1961).
2. Montroll, E. W., "Theory of the Vibrations of SimpleCubic Lattices with Nearest Neighbor Interactions,"Proceedings of the Third Berkeley Symposium on Mathe-matical Statistics andTProbability,Tvol. III~(niver-sity oF Callifornia Press, Berkeley, California, 1956).
3. Wallis, R. F. and A. A. Maradudin, "Lattice Anharmonicityand Optical Absorption in Polar Crystals I. The LinearChain," Physical Review 120, 442 (1960).
84
APPENDIX D
SOLUTION OF THE EQUATIONS OF MOTION
The potential energy of a crystal having only harmonic
forces can be written
Pr'
401(9 / (01)
where repeated indices indicate summation.
The equation of motion for the jth component of the
o(p particle is
s2Zpp
.4,
(D-Z2)
P 1where
phas been used.
Substituting normal coordinates
into Eq. D2 gives
(P3)
85
oy
.00
86
=(' .()p4Z( 0 f (t f )4 ) ,(04)
Since is a normal mode,
t(ts)r - (es) $Y~fWs)P (Os)
Making this substitution in equation Eq. D4 and rearranging
terms gives
Since 5) in general, the expression in brackets
must vanish:
plo,
X'i'r' -=.
(D7)For the special case f = 0, the exponential term becomes
unity:
87
, O s 'Pf,
PA.
Pp'
J)= - -- I.
4,tp 'C$el?-
)
.1 .
A C &
For a cubic crystal, symmetry demands that
41 = I OAC A7 .PPI
Furthermore, for a diatomic crystal, the translational
invariance of the potential demands that
cc A2.L A~oIo(, P)=a ' 10
A]
Combining these results gives the relations
2 PA owAPp'
=m-,0 A I =21
Now
7/'f ( (oa)={08
so A d
(D:)
0. (oIZ)
(013)
88
Substituting into Eq. D8 gives
6&(s) , 'y+,A t) /a,,(=os)+A (h J.
N/O
This equation is in reality six equations, one for each
combination of the three coordinates j and the two particle
types ok . All six can be satisfied simultaneously only
if the determinant of coefficients of the -10o) vanishes:
4.)jI 0 0 0
o0 0 rn0
0= 0 0
(i.. 0 0 W - 0 0
O n~' M -?ir
0 C)0 0
4% er C. =A /M, ,M,. rs
The lattice frequency notation (f) = (0) has been dropped
for convenience. The solution is triply degenerate:
89
4 - 4 . ~
w= (0,z) 0
4)12 / (0, /)=
The eigenvectors follow by substituting
For
A (i He):A(D4)
into Eq. D14
4, (0 )'00/r.y (o 2
Forw o, ) A/m,
S/. vio,) + - (i,/0) v;(o, i) o--vy
(0'?)
The triple degeneracy of the solution is a consequence
of the cubic-symmetry, and implies that the direction of
90
one of the eigenvectors for each mode may be chosen arbi-
trarily.
The normalization of the eigenvectors is
-~ (~~u):
For Fo, f1f'o,)3 '
so
Thus
I
-a__
Similarly,
21 Ut(nJ
and
o2 (rx~o)
(045
2 -i;4)7 J ( ) '+ 1.(o,) *)'
ii rm o1] =
211Ii l,{p0
0 4c
(1.3
(oii)
APPENDIX E
ELECTRIC DIPOLE MOMENT AND ELECTRIC FIELD
PERTURBATION
Electric Dipole Moment
The jth component of the linear electric dipole
moment arising from the displacement of a given ion from
its position of equilibrium is
11 I -: E, p
where is the charge carried by the ion.
In terms of normal coordinates
if(0 m1
Summing over all unit cells and bases gives the jth
component of the instantaneous total dipole moment of the
crystal:
Now
I, e'z=
XJ. (E 3)
(Eq)
91
(,E I)
92
Making this substitution and summing over f gives
M 2 6. N/rn.)*i S)1/.;{Oj S). ES
The solution of the harmonic equations of motion in
appendix D gave the following relations between the v's:
v'(0, ;L) f -( , 0j;Lo ) (4
When these relations are substituted into the corresponding
terms in E5, the result is
t1=~ (N) (0I)LE,(Y)v /o, e) - E1(h) v'$(o,1)/]
Now if = - -( , then the entire second term
vanishes and the first becomes
If the vector of unit length 7,' =(M/M) Vi/ )is defined,
then
or(Ei1)
Electric Field Perturbation
The interaction energy of the crystal with a uniform
electric field is
it')alp
or
Thus the electric field perturbation operator is
r4E)= *
, (o,1)(
The transform of ) is
where the notation ( o )') = F . .- /,I) has been
introduced.
93
(Eo)
0,C,
cc (Allm) (0,
A-loom 0 E (
cz (Nlpn 6: 60,(E I ;L)
E
APPENDIX F
NOTES ON THE TRANSFORMATION TO THE BASIS
(e~~~~~ tn -(n~ i ' f,S ) e (f ,s).
General
For simplicity, treat first in one dimension:
<19 =(Arr (F/)
The double fourier series expansion of a periodic
function of two angle variables G and e can be written
Fe .h'e'
Multiplying by
o gives
ff F(l&)
and integrating over & and
d) w(e'
.. ( n")e''-"
,
94
(F3)
th&G -
go" (;L,7)Imm z ., m t ff We
= 21r < -n > s<..owl3
95
so that
.i (n - '')
(F' )(F5)Defining
shows that
F(e q' 7T) Z F t'|i
If P( ')is integrated over
am.. (F
, the result is
/' F /,mf1
i ( e'''x eL fd9 e
3rr< o Fl>op (F 7
For an operator which depends on only one set of angle
variables, the matrix element is defined by
fd9e (Fg)
a..I=m ( ; rr'f(
a0, I =
F(e, e')ded=()
= 2rz i'/F/hIcfpikf
e~e')dede'=
<&/F|0h> (.ff'
96
For the 2N dimensional problem being worked, the
transformation formuli are
(
F =fd=(de .'
h'l~n =(.Y)~ fdoe
Flo.eF 160, a
if do de' F(oo') (?,r)A<oI F />
for the basis vectors
<& :(-dl
r 0 has been used as a shorthand notation for
and the set symbolism Ifni
I)>0has been simplified to
Transformation of q(0,1)
From Appendix C, the normal coordinate
terms of action-angle variables is
(f s) in
i09(.f S)
'..o4- (F 12) ( S
in e
i V r)
(Flo)
(F/I)
')e
i n (f ) S (f~s) I*
(F12)
VI.
97
Thus
<o/p(oil)/ F>
xfdfee
The integration over all @(f,5) except G(O,/) gives
7fr' 'f 5)J , where the prime denotes the omission
of r) (o j) . The remaining integral over e/o, 1) is
(F/ 5)
Defining
.fSm w 0 35= I
0 {o ,s1)
cr, T(o.)J)
W (0, i
allows a more compact expression for the matrix element to
be written:
(o )
and
C4)Y4, 4 I)
6F3
(F13)
y(1:19(abr) .2A/ [T o 1/2Qo o1)
i G (0.4 0 + -C ( 1
- i IO~) ih(o'Ofofcl 0(0 ,1) re1.0(11
arr S~~,)1 o o, >]
( F/ 6)
98
ect(F
Transformation of q(0,1)
From Appendix E, the operator q(0,1) is a shorthand
notation for the reduced poisson bracket
too,,(0,) j''(0 -)-
eo =q 0,I)..
The transformation of this operation is therefore
< 01, >T) ~J ________
Performing the G differentiation in the first term and
integrating the second term by parts gives
X n o - o)1;0 )(
Using F13 to convert q(0,1) to action-angle notation and
grouping terms gives
99
%.I-'<h Too ai3T..
xd F0i 6 &-
(F~
where the prime again indicates the omission of e,The 8 integrations give
a Al3h{s h--)4}+(
AJ.f 1 (01)
t (o ):1
)
(F-;)
f 1 , s-*
Kr) 'I~(o~iIn>: i~./~'I ~A Ye (j, 41*
Transformation of
The anharmonic operator is
Thus
(Tr)
100
c ate(>j) j&G;4,) L ;j
(F. i)
where cL is the perturbation on the potential energy
associated with the cubic anharmonic interaction. If the
potential energy is written as a Taylor series expansion
4 1
Pt'pP
and the expression transformed to normal coordinates, the
cubic term UC becomes
where f { <4 ii )(N 1 if - = 0 or a
reciprocal lattice vector
= 0 otherwise.
101
Substituting action-angle variables as in F 12 and defining
VI( ~)$ C
(z(t *I' )
allows the perturbation energy to be written
_______ l1
64:1cif
RU
do
)
A straightforward rearrangement of summations gives the
equivalent form
(,~C~jj7A(l
e 110( 1 '
(Fr)or, in shorthand notation,
(f7)
(F)
er T (f7'4
m
ow I"OR-.- (v on% 7 t
Al'r'f"
s
)
i c
V ef fJT2I
The formal expression for the transformation of
is
xi I4 TC
From F29 and F30
41Ve K' [T Jh1 T)~
X e Col,
([&9 (Ki J: S(v k1) +J(kr) 4 (v, K')
Integrating by parts on G gives for the second term in
F31
doeIc
Jdo
102
(F30)
(F31)
(Fsz)
(F33)
n Al > ) -aA/ f i fbi 09
'IX iki =(;Lff
izr)&
Ole.
de
JTx
L 'Ka,
103
Putting F32 and F33 into F31 and summing over k gives
h 1- ) i/gh (
XhIe n j fdo e e el
where I7
Finally, integrating over all & gives
(135)where C C unless k is one of the three k's on which
depends.
Transformation
Since neither HHnorthey may be removed from
leaving
of
z
the
F At)
are functions of Gmatrix element Kh IF'Io)
Z eI e~ /O>
to be evaluated. The exponential in the matrix element
may be expanded in a series
- W
( F3 )
h -4)3q)
104
Only the cubic anharmonic potential is to be retained,
giving for the matrix element
KI F^/o> rZ <'< lo >.(F3S0)
The first matrix element in brackets is
< h 0 -:VhI
(F: 3)
The second is
\ h1 0>(a7 T )Q. 'e 27VfcklJL /7.C
(F4o)
Only the exponential is G dependent, so that the integral
gives
< \e tl 0> 2 Vre 4 jIfl-11 T/2-74 pi e(F 1)
with eK = 0 unless (eK
is one of the set e KI on
which V depends.
The complete matrix element of the distribution is thus
X T-S Ico .) . ( F42.)
APPENDIX G
DERIVATION OF THE EXPLICIT FORM OF R
The harmonic Green's function is the causal solution
x C quo ;4'' =( + q (0t; 4,)), H (C-7
of
) 0 FO, 3 I
Since the Harmonic Hamiltonian is e aT,
the Poisson bracket reduces to
I eofg P F OL .1.e 1gTiI. 1. 7
ic.
(q4,-)
Using this in Gl gives
c + ) k O ei (c3)Taking the Fourier transform w.r.t. (t-t') gives
f(d -t')
v2,. 6 7i ? !1-v
105
f d(i.Y)(4'4)
= S(M' )5(H') ,1 <4
aslop
106
Integrating the first term on the left hand side by parts:
f tt---) =H (i t-t -'1)
The second term on the left hand side can be treated
by reversing the order of integration and differentiation:
f Oe
Evaluating the right hand side of G4 and substituting G5
and G6 gives
- ~ f- 4 ic 2 6 M 7T St-, )t&.. ' ( ,
The transformation to the new basis will now
be made. The second term can be written out explicitly
and integrated by parts:
h.>'7T
7r) R"(j ,e 8
107
a I ( LAf I R i(9 -4h'1)
(,-I)
The right hand side of G7 is also easily evaluated:
(j.//-/J..j)Jd'
-_ 7P' (j---Y.#) S (A,-4. . (',o)
With these two simplifications, G7 in the new
representation becomes
Solving for
( A'IP)~(&~
Using this result, evidently
71' (T~.Y-k. -) f(i noten)4
gives
(c,1i)lc~'r (*Tit--x.) (t--) k.f
<w/, k
-i <hh/R /h>j4ic4)S <h'/R1
fn>
h h )
0
<h' \ &0Ih>
where 0 is another operator.
rI /4>"<n-/ 13>
T S (n' h , 1h
IL
WX, me ?)
108
APPENDIX H
REPRESENTATIVE OPERATORS FOR <hI IX A4/ r>
The particular representation of the cubic anharmonic
operator h> will be applied to a general term
of the form 71',, fte.4'^To'and conclusions drawn as to
the effect of applying a series of such operators in
sequence.
The explicit expression was derived in Appendix F
and is
The primary interest here lies in the operations involving
the J's. The operator is therefore redefined to break out
that part not involving J's directly:
< hx> <: ,C 3 >,f( k 1)
Where
ILK(I3 x 4)
109
and
The reduced operator (hIX In> will now be
applied to the term of the form
TT=TTwhere k stands for both the lattice frequency f and the
mode label s.
Each of the differential operations gives
TTJ 7Pic "TW
The other terms are obvious and combine with the differ-
ential operations to give
IL
(HB)
110
(13)
1too
111
Each application of the operator <h0J fh> will be
referred to as an "interaction," with each interaction
involving three J's.
The nine terms resulting from an interaction are
grouped in three sets, with the three terms in each set
corresponding to the operations /[in which just
one of the three J's is involved, then the qr terms re-
sulting from r successive interactions can be grouped into
3r terms. Each term is made up of a product of factors,
one from each interaction. Each factor is of the form
) V. ( 0 r) e (,)4
TAeJf-S) (He6)
where , , and TC are the three J's involved in the
interaction, and the superscript labels the par-
ticular value of the quantities at each interaction.
Equation 33 of the main section shows that the
sequence of anharmonic interactions is followed by
integrating over all the J's.
It is postulated that the result of integrating one
of the terms made up as described at H6 over a particular
J is to change the factors involving that Ja into a
numerical coefficient which depends on the sequence
112
of interactions that follow it. Furthermore, the entire
term is multiplied by a factorial which also depends on
the sequence in a similar way.
The postulated result to be proved is
(H?)
X [other coefficients, and factors not involving
Ja], with ,>O, rr, 0 , P:O1, 2*-- ,where ra is an interaction in which Ja was the
variable chosen to be operated on by the differential
operation of Eq. H6, and r-a is an interaction in which
Ja is one of three J's involved, but was not the J chosen
for the differential operation. Note that interactions in
which Ja is in no way involved need not be included.
Also, the coefficients associated with intermediate inter-
actions can be determined independently, if the postulate
holds.
113
The postulate will be proved by induction, with the
proof in three steps:
Step 1. Show that the postulate of H7 holds for
Step 2. Show that it holds for 2.
Step 3. Show that if the postulate holds for /,then it holds for / + I.
Proof of step I.
The postulate for / I is
C4 L.hA
X [other factors not involving Ja],
with PO+-(lO) -
Each of the interactions r-a contains Ja (see Eq. H2).
The result of applying this interaction n times is to
multiply the integrand by Jam/ 2 x factors not involving
Ja. The integral over Ja therefore becomes
where Eq. H5 has been used.
114
Now
J rc-(3w4II e g~f = '
Thus if + o w..
r, then the integration gives
I ( i#%po4
(uhIn )
which reduces upon factoring and combining terms to give
which is the desired result. Step I is therefore com-
plete.
Proof of step 2_.
The postulate for q = 2 is
Ed J, '"A, e (Im)C'O,
XE[ ce+ (h 13)
IP s O e, l . ..
(Hio)
I. (rn + 4)oot e4qlm [ PA,*
J6
M
PA.
ill)
115
(coefficients and factors not involving Ja),
with PJ-4-L(i-.):: OI;),=..
Putting in the expression for Ap% (omitting terms and
factors not involving Ja) in the integrand gives
Performing the operations in the right-most parenthesis
using HS gives
(X.. 4 PC,1 ~1 I~
Rearranging terms puts Eq. H15 into the form
In this form, the application of H5 is easily done, giving
xA- f>j jP L
(I) ~ P6,
Putting in Jam/ 2 for A ? and accumulating terms of
equal power in Ja gives
Integrating over Ja gives
IP J
4, -0 6 i
j
116
AV (II)3. 1I
4W04
d4p YAP
117
Accumulating terms shows that all terms involving pa vanish,except in the factorial, leaving
The expression is the same as was postulated. Step 2
proof is therefore complete.
Step 3.
This step must prove that if
fdTA2zA3-LfAe ~ T&4(
X [other factors not involving
Ja],
then
dT4 AOL A,~
P 44
X (other factors not involving
Ja)
Performing the first ra operation in Eq. H22 gives
fdJA rA :Tf, A-*'A e.P+
118
or
Both integrals in Eq. H23 are of the form specified in
Eq. H21. Using Eq. H21, then, gives
Combining the two terms gives
This expression agrees with Eq. H22, and therefore the
proof of step 3 and of the postulate of Eq. 117 is com-
plete.
The postulate having been proved, it must now be
used to define representative operators for the complete
119
expression of Eq. Hi, including successive applications
of the complete operatorAh and the final
integration over all J. What is required then, is a
simplified expression for
fdT <h'I cotI, > JWJT e~From H2,
According to the postulate of H7, when each J is integrated
over, a factorial #14 rM-JP-.
is generated for each of the 3r terms resulting from r
successive applications of< r . Also, each
operation produces an interaction coefficient
multiplying the term. The complete integral may there-
fore be written
120
f dJ TI>T \
XI(r nr i r:
~()~ s j (9jK .'K
where
and
where k(j) is the particular k chosen at the jth inter-
action ,jIC6is the set of three k's entering the jth
interaction K' is the number of interactions re-
maining at the jth in which Jk takes part, but is not
chosen, is the number of interactions (including
the jth) remaining in which Jk is the chosen mode.
Evidently the st is the same as the subscript
set t ( e f e -r e4oSimilarly, the e(.% included in C are the same as one
of the 4 in I') and also of
The summation resulting from the integral is therefore
superfluous. The final result is therefore
121
fdfF4I/ >1TJ'e/
TT2
cfC6')* f I
P1 "iJ) (j)eK (4 11i)
APPENDIX I
COMPUTER PROGRAM TO SORT OUT N-SEQUENCES
AND EVALUATE INTERACTION COEFFICIENTS
A FORTRAN computer program has been written to examine
all possible sequences of n's of a unique character up to
a certain order and to determine the interaction coefficients
of all allowed sequences. The computer program output
has a diagrammatic appearance and is referred to as a
"diagram" in what follows. The following selection rules
are built into the program. These selection rules result
from the character of the anharmonic potential in the
coordinates used. Selection rules:
1. three and only three n's may change in an
anharmonic interaction.
2. The sum of the k's of the three nk's changing
in an interaction must be zero ("Umklapp" processes are
excluded).
3. Each complex diagram begins on the right with
the diagram fragment corresponding to one of the distri-
bution function series expansion, followed immediately
by the first part of the electrical field interaction
vertex. This vertex changes n0 by +1, leaving all other
123
n's unchanged, and can be treated as a special form of
the anharmonic interaction.
4. Each complex diagram ends on the left with a null
diagram, preceded by the second half of the electrical
field interaction vector, which again brings about a
change of +1 in n0, leaving other n's unaffected. This
interaction may also be treated as a special case of
the anharmonic interaction.
Operation of the program follows the flow chart
shown in Fig. 11. Program operation begins by reading in
the desired number of vertices to be examined and the
maximum number of a particular lattice frequency which
can enter an interaction. After initialization, the
first interaction is considered, working from left to
right. Because of rule 4, the electrical field inter-
action is trivial, so operation proceeds with the first
anharmonic interaction. One of the allowable set of
changes is chosen and each change paired with a lattice
frequency to construct a change vector. Three and only
three n's must change. After the change vector is
selected, it is applied to the present set of n's (the
"occupation number vector") to form a new trial occu-
pation number vector.
The new trial occupation vector (ONV) is then examined
for compliance with the selection rules on allowable
S tg r t
ReadJN
ReadRN
Initializearrays
Initis1izecounters
47
FIN =cIN+ 1
yes IIN>)JN? IN -
no
yesb IN =0? Stop
no
-ons t-uc'tChoose a Q. ne wset of di& gram ofe's lower order
All e's ysu se d?
no
d
Fig. 11--Flow chart of diagrain evaluation program
124
125
d e
Construct no CalculateC 1 IN =J N? IC's
ces
All ,'s Yes ONV= yes All yesUs dDFV? IC's = 0?
o ( b E noo
Con. ,0ct d Print CV,ONV, IC'SIL
Nj > RN ed
Choose f soef Ye s the tTf= 0 New f yes
L, -ld ? old f?
yesif 0? Too man esd
NV 09
C) 0
FiIg. 11 (cont.)
126
frequency sets entering an interaction (rule 2), and for
the possibility of ultimately matching a distribution
function diagram fragment (rule 3). Only if the trial
ONV meets these rules does further evaluation of it take
place. If it fails to meet either, program flow returns to
construct a new change vector (CV). The new CV is formed
either with a new pairing of the same set of changes
with different frequencies or, if the frequency set is
exhausted, a new set of changes is selected. Note: to
reduce redundancy, only positive ONV components are con-
sidered. Each diagram has a "mirror image" which is
also allowed.
Once a new ONV is found which meets the selection
rules, the vertex coefficient (VCo) for each of the 3n
vertex combinations is calculated. If all the VCots of
a given new ONV are zero, the trial ONV is dropped and
program control transfers to construct a new CV and ONV.
If a chain of VCo's is non-zero, the ONV is retained,
the CV, ONV, and VCo's are printed out, and program flow
transfers to increment the vertex counter prior to seeking
an acceptable ONV of higher order.
When acceptable ONV's with non-zero VCo's of the
maximum order to be considered have been found, a simu-
lation of the anharmonic coefficients, the denominator
accompanying it, and the factorial multipliers for each
127
of the k's participating in the diagram are printed
out.
If at any point, no acceptable ONV of a given order
can be found, the vertex counter is decreased and new
CV's constructed at that level to attempt to build up a
new diagram of lower order. When all possible combinations
of CV's have been tried, the program stops.
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APPENDIX J
PROGRAM TO CALCULATE FREQUENCY SPECTRA AND
SUSCEPTIBILITY COMPONENTS
A FORTRAN computer program has been written to do the
transformation from squared frequency spectra G(4 /1) to
ordinary frequency spectra g(W ) and vice versa. It also
does the integration necessary to calculate the real part
of the anharmonic contribution to the denominator of the
susceptibility (Equation 55 of Chapter II).
Program flow is quite straightforward, following
the flow chart in Fig. 12. The program begins by reading
in the number of samples to be input for the squared
frequency spectrum, and the number of samples corre-
sponding to the frequency o . It then reads the given
number of cards containing the values of G( 4/4' )
and begins calculation.
The first calculation converts G[( x// ) 3 to
g ( 9/&, ) by the simple relation
Symmetry of G[( 4/w, ) ] about ( A/i/S ) 1/2 is
assumed so that G[l - ( /w/y, )x ] = G[( 4 / , ) 'I
150
Start
ReadN, NMAX
Read
Calculateg (4 /'o)
Calcula teg (%V.-Ad...) ,9
g (, 1, 4 .k; )
CalculateS G (.W) ,
Calculateaj
Read4, 4
Calculate
x )V
Calculate
CStop
Fig. 12--Flow Chart of Program to Calculate FrequencySpectra and Susceptibility
151
- m - - muktMAW "
152
and therefore
I (41)/'t.) = ( k/2) U/-5)/4 v.)'
The ranges of definition of g ( t)//, ) and g"(Ai%/4. )
do not overlap.
Next, the numerical integrations are performed to
determine
if4 C1j ) aa }
These are
('~~)44c) 4'.,4/
andffj4 4 1 44'es
No, 410
The ranges of the summations extend over all n's for which
neither frequency spectrum is zero.
Next the frequency spectrum fLC (a)is calculated:
where =4, 4) .kand ~ ( 4)#)
4'.
IO
153
This spectrum, aside from a constant factor, is justY ,
the imaginary part of the anharmonic contribution to the
denominator of the susceptibility.
Next, another numerical integration is performed to
obtain the real part of the anharmonic contribution to
susceptibility denominator
The last calculations to be made are those of the real
and imaginary parts of the rationalized susceptibility.
The ratio Y is read in and the normalized sus-
ceptibility calculated:
24.I
hor [j ....(/4)14A)4/)+ /*
The harmonic susceptibility is also calculated by
evaluating norm for 4y =0.
154
... PROGRAM TO DO PFCTRAL DFNSTTY FUNCTION CALCULA-
C TTONJS FOP FnTSSFRTATION. PROGPAYMMFD HY
c HOJMAP KENNFDY, JANUARY 10, 1974...
DIMENSION ?(40) *GH(21) *GL(150) *G(GP(.171) .GGN(171) .* GG(30 0 ).,;?-(41), PS (41)
PFAL NUMD AT A G?/40*0 .0/i GH/?1*0 . 0/, GL,/150*0 .0/ , GGP/l71*0 . 0/9,
* GGtNA/1 71*0.0/,GGS/300*0.0/.GS2/41*0.0/ ,PS/41*0.0/
CC ...PEAD SOUARED DISTPIUTION FUNCTION..C
0 RFAD(c.1.FND=600) JGJMAXI FORMAT(?T)
p F A n( C,.?;,) ( (-,(j) 1J= I * J )
? FOPMAT((10.4)WPI TF (6,15)
15 FORMAT (1I TNPIJT SQUARFD DF//)'RITE( 6.17) (J4G2 ( J) , J:-,IJG)
17 FOPM AT(15.F10.4)FJG=JCG- IRJMA X=JMAX-IYL=SOPT (FJG/FJMAX)XH=ROPT(l.-XL**?)DX=.05*(Q.-XH)
FNMAX =./DxFNM?=F NM A A**?
SIN=0.J GM = F J 6nO 80 ~<=2, JGM
90 SIN=ST N+G? (K)STN=';tIN+0 .* *( G? (I +G? (JG3))
S IN=0 .Y-FJMAX/SIND)O A8 K=1.,J-,
P1 G?(K)=G?(K)*STN
C ... CAL C1LATc' (F (-K)r
4TTF 1 3)3 FoRMAT( 1 F FOP W(-K) //
* tNGL X GL IL )
TL=lNGL=1X=0.0GL (1) =0.0
100 WQT T 0-.) NG)LoX 9GL (NGL).I L
SFOWRAA T (T*.F9.3,FR.4,15X= X+ r)X
W=FJMAX* (X**?)TF(W.GrT.FJG) Go TO ?00NGL=N GLL+ I
155
TL=TFTX(W) +1FR=W-.FLOAT( IL) +1.GL (NGL ) =2.*X* (FR*G?( IL+1) + (1.-FP) *G2 ( IL) )GO TO 100
C ... CALCULATE fF(+K) ...C
200 x=XH-fxNGH=OWRITE (6*6)
6 FOPMAT(*1OF FOR W(+K)#//* NGH X GH TL*)
P 10 X=X+DQTF(X.GT.1.) GO TO 300W=FJMAX*(1.-X**2)NGH=NGH+l
TL = T F I X (W) +FR=W-FLOAT (IL) +1.GH (NGH) =2.*X* (FR*G2 ( IL+1) + (1.-FR) *2 ( IL )WRITE(6*5) NGH.XGH(NGH),ILGO TO 210
CC ... CALCULATF DF( W(+K)+W(-K)C
300 WRITE(6,7)7 FOPMAT(01DF FOP W(+K)+W(-K) //
* K Y GGP )Y=XH-xKGP=0
310 KGP=KGP+l
G(;P(KGP)=0 . 0Y=Y+nxKGL=KGP+ 1TF(KGL.GF.NGL+NGH+I) GO TO 33000 315 KGH=1, NGHKGL=KGL-1TF(KGL.GT.N L)o GTO 315IF(KGL.LF.0) GO TO 316OG=GL(KGL)*GH(KGH)GGP(KGP)=GP(KCGP) +oG
315 CONTINUE316 CONTINUE
GGP (KOGP) =nX*GGP (KGP)WPTTF(U.F) KGPYGGP(KGP)
A FOPMAT(I5.F9.3, F10.3)GO TO 310
CC ... CALCULATE DF ( WC+K)-W(-K))...
310 WPTTE(69 )Q FOPMAT(t1OF FOR W (+K)-W(-K) //
* IKGN Y GGN)
Y=XH-XL-DXK G N= ()
340 KGN=KCN+ 1GGN (KGN) =0.0Y=Y+r)XK GL =N GL-KGNTF(K(I+NIGH.LF.0) GO TO 36000 350 KOH=IqN(HKG7L=KGL+l
TF(KGL.LE.0) GO TO 3c0F(K GL .GT.NGL) GO TO 351
DG=GL (KGL) *GH (KGH)GGN(KGN) =G(N (KGN) +DG
350 CON T INUE351 CONTThNIE
GGN (KCN) =0X*GGN (KGN)WRITE (6.*) KGNYqGGN(KGN)GO TO 340
*...CALCIJLATF SOUARF ODF VERSUS W FOP SUM ANDDTFFFRENCF DF t S...
360 WRITF(6*10)
10 FORMAT (-1SOUAPFD F VS W, FOP (W(+K)+W(-K)) AND01( (+K) -W (-K)) /
* fKGS Y G(GS KGS KGH*)
Y=XH-XL-XKGH=1-NGLKGS=0
365 (KGS=KGS+lKGH=KGH+1IF(VGH.rT.KcP) GO TO 370Y=Y+DX
GGS(KGS)=0.0TF(KGS.LT.KGN) GGS(KrGS)=GGS(KGS) +G;CN(KGS)
TF(KGH.GT.0) G(S(KrS)=GGS(KGS)+GGP(KGH)GGS0(K0GS)=0.5*G(S(KGS) /YWRITF(6*11) KGSYGGS(KGS)*KGSKGH .
11 FOPMAT(15,Fq.3, E11.392I5)G0 TO 365
... CALCULATF SOtAPFD DF VERSUS W**? FOP SUM ANDDIFFFPFNCE OF S...
370 CONTINUEWRITF (69,16)
16 FORMAT(1W**2 TIMES LAST DF*AGAIN;T Wu**//* KGS 7 lGSILAt)
07= ( Y **2- (N'-XL ) **?) *0 .O? 27= ( XH-XL ) **?-r)7
156
CC
C
C
C
157
KGS?=03Aa Z=Z+.n7
Y=(SOPT(7)-(XH-XL))/DX
IL=TFIX(Y)+1IF(TLI-1.GT.KGS) GO TO 390KGS2=<GS?+ 1FR=Y-FLOAT(IL)+1.GS2 (KGS?) = (FR*GG ( IL+1) +(1.-FR)-*GGS (IL) ) *7WRTTF(",11) KGS?.7,GS2(KGS2),TLGO TO 380
390 CONT INUEKGS?=KGS2-1
CC ...CALCULATF INTEGRAL OF 1/?7-A)*GS?...C
WRITE(6.14)14 FORMAT(*ITNTFGRATION OF P(1/(Z-A)) AND GS2*//
* NP A PS f)
NP=ODA=4.*DZA=-r)ADO 401 NP=1,40
A=A+f)A
7=(XH-XL)**?-D7PS (Ne) =0.000 400 K=l.KGS?7=7+D7DEN=7-AIF(7-A.FO.0) OFN=1.F-ADG=GS? (K) /0ENos (NP) =PS (NP) +0G
400 CONTINUEPS (NP) =D7*PS (NP)WRITE(6,13) NPA ,PS(NP)
13 FORMAT(TSFIO.2, F14.3)401 CONTINUE
C ...CALCULATE PEAL AND IMAnINARY PARTS OF NORMALIZEDC SIJSCFPTIRILITY PLUS HARMONIC SUSCFPTIBTLITY...C
READ(5,18) WOWG18 FORMAT(PG10.4)
WRITF(6.?l) WO.WG?l FOMA'T(1Xo.?G12.4)
WPITE(6.19)1P FORMAT (*1 Lf.7X ,M4,1I1X, CHTdA8 , CHIHAPi
* .AXtGAMMAA/)CG= (W('/WQ) **2CR=CG/3.14159A=-r)A
158
LG=-14O 01 L=1*40
A=A+AL =L G+f4PFNH=1.-ATF(DFNH.EQ.O.) DFNH=1.F-ANUM=B.-A+CP*PS(L)DEN=NUM**?GAM=0.0IF(LG.LF.41.AN.LG.GE.1) GAM=CG*GS2(LG)f) EN O=DEN +GAM**2CH I =NUMA/rE\CHIHAP=1./PFNHGAM= GA M4 /rFNWPTTF (6.20) L*AqCHICHIHAR,GAM
501 CONTINUE20 FO9MAT(1XT5,4r1?.4)
WR1TF (6.??)2 FOPMAT(*1 )
GO To 40400 STOP
EN P
BIBLIOGRAPHY
Books
Balescu, R., Statistical Mechanics of Charged Particles,Vol. IV ot Monographs in StatiSTical PTysics, 4 vols.,edited by I. Prigogine~TInterscience, New York, 1963).
Born, M. and K. Huang, Dynamical Theory of Crystal Lattices(Oxford University Press, London, 17Y4).
Burnstein, E., "The Intrinsic Infrared and Raman LatticeVibration Spectra of Cubic Diatomic Crystals," LatticeDynamics, Supplement 1 to Journal of Physics andChemistry of Solids (Permagon Pres7S New York~~T955).
Lorentz, H. A., The Theory of Electrons (Reprinted by DoverPublications,New York,1952).
Montroll, E. W., "Theory of the Vibrations of Simple CubicLattices with Nearest Neighbor Interactions,"Proceedings of the Third Berkeley Symposium onMathematical~KtaT-Fstics and Probability, Vol~~III(University ot Calitornia~Fress, Berkeley, California,1956).
, "Theory of Lattice Dynamics in the HarmonicApproximation," Supplement 3 to Solid State Physics(Academic Press, New York, 1963).
Prigogine, I., Non-Equilibrium Statistical Mechanics, Vol. Iof Monograpjis in Statistical Physics, 4 vols.,editedby 1. Prigogine~(Interscience, New York, 1962).
Articles
Adler, S. L., "Quantum Theory of the Dielectric Constantin Real Solids," Physical Review 126, 413 (1962).
Barnes, B., Zeitschrift fur Physik 75, 723 (1932).
Barnes, R. B., R. R. Brattain and F. Seitz, "On theStructure and Interpretation of the Infrared AbsorptionSpectra of Crystals," Physical Review 48, 582 (1935).
159
160
Blackman, M., "Die Feinstruktur der Reststrahlen," Zeitschriftfur Physik 86, 421 (1933).
Born, M. and M. Blackman, "Uber die Feinstruktur derReststrahlen," Zeitschrift fur Physik 82, 551 (1933).
Cowley, R. A., "The Lattice Dynamics of an AnharmonicCrystal," Advances in Physics 12, 421 (1963).
Dick, B. G., Jr. and A. W. Overhauser, "Theory of theDielectric Constants of Alkali Halide Crystals,"Physical Review 112, 90 (1958).
Ehrenreich, H. and M. H. Cohen, "Self-Consistent FieldApproach to the Many-Electron Problem," PhysicalReview 115,, 786 (1959).
Genzel, L., H1. Happ and R. Weber, Zeitschrift fur Physik154, 13 (1959).
Henin, F., I. Prigogine, C. C. L. George and F. Mayne,"Kinetic Equations of Quasiparticle Descriptions,"Physica 32, 1828 (1966).
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Unpublished Materials
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