REFERENT / • r

IC/68/72

INTERNATIONAL ATOMIC ENERGY AGENCY

INTERNATIONAL CENTRE FOR THEORETICAL

PHYSICS

DISPERSION THEORYFOR AN ELECTRON - PHONON SYSTEM

IN A MAGNETIC FIELD

A. MADUEMEZIA

1968MIRAMARE - TRIESTE

IC/68/72

IKTSBHATIOITAL ATOMIC EBEBGT AGEKCY

ESTERITATIONAL CEBTHE FOR THEORETICAL PHTSICS

LISPEESIO1T THEORY FOE AIT ELBCTHOff-IPHOlfOS STSTBM IB" A

MAGUETIC FE3IJ) *

Awele Kad-uemezia

SH JUKI ABB - TRIESTE

August 1968

* To be submitted to "The Physical Review".

** On leave from Department of Physios , Univers i ty

of Ghana, Legon, Ghana*

ABSTRACT

A (non-relativistio) local field theory is constructed for

a system of phonons interacting with electrons in a degenerate

electron gas situated in a high magnetio field. This construction

involves a definite prescription for the renormalization of the

eleotron-phonon vertex so as to remove a singularity which it

possesses at zero phonon momenta. Within the framework of this

theory, the forward dispersion relation for high frequency phonons

interacting with electrons in a normal metal is derived. The lower

limit for the dispersion integral explains the (experimentally

observed) phenomenon of "giant quantum oscillations" in ultrasonic

attenuation in metals. It indicates that there is a non-zero

frequency threshold for the onset of this phenomenon. The cut on

the phonon energy real axis is consistent with the physically reason-

able condition that for positive energy phonons, phonon absorption

ooours only when the eleotron and phonon "oollide head—ontf.

-1-

DISPERSION THEORY FOR AN ELECTRON-PHONON SI STEM Iff A

MAGNETIC FIELD

1. INTRODUCTION

A considerable amount of work has been done on the application

of quantum field theoretic methods to statistical phyaios. The

results obtained and the general techniques employed have been

summarized in a number of exoellent monographs. Among these, one

one should mention the books of Abrikosov, Gorkov and Dzyaloshinski \

2) 3)of Pines J and of Kadanoff and Baym, to name only a few. One could

certainly say that, in contrast to the situation in high-energy physios,

quantum field theory generally does give good results in statistical

physics.

There is one aspect of the theory of quantized fields that

does not seem to have been exploited to the full in statistical physios.

This is the area known as dispersion theory, or the derivation of

Kramera-Kronig-type relations, that is, Cauchy integral rolations.

This paper is ooncerned principally with the derivation of

such relations for an electron-phonon system which has the added

complication of being situated in a magnetic field.

The physical motivation for such a study is the following:

It has been observed that when certain metallic crystals are placed

in a magnetic field and an ultrasonic wave is propagated through

them, a number of distinct attenuation processes take place. Those

are usually classified as geometric resonances, cyclotron resonances,

De Haas-van Alphen oscillations, open-orbit resonances, and giant

quantum osoillations. The first four of these are classical or semi-

classical effects and have been satisfactorily explained as such.

The last effect, as the name implies, appears to be a specifically

quantum effeot. If the temperature is low enough, then as the

magnetio field is varied, the amplitude of the transmitted wave

exhibits spike-like osoillations. An explanation has been offered

by Gurevioh, Skobov and Firsov in terms of a certain seleotion

rule on eleotron wave vectors. The dispersion theoretic approach

-2-

has the advantage thatf although i t ia quite complicated, i t is fully

quantum meohaaioal and explores both the process of phonon absorption

and that of phonon emission. It indicates that there is always a net

absorption of phonons. I t also throws a lot of light on the physical

mechanism of phonon absorption.

This ultrasonic absorption effect is a low temperature effeot -

typically, T » 2-4°K. This enables us to use a T - 0 approximation*

This effectively transfonus the many-body problem to a two-body problem

and we do not have to take Giaba (or ensemble) averages in computing

"vacuum" expectation values. We propose to incorporate the effects of

finite temperature in a eequel to this work.

Keeping in mind our motivation for this work, we shall assume

that the interacting electron-phonon system of interest is created

when a beam of phonons, specified by a current vector £ (^"O enters

a solid, eventually leaving i t at a modified intensity. The external

source of phonons i s , typically, an ultrasonio wave generated by a

transducer in contact with the crystal. This source then establishes

a dominant single-frequency phonon speotrum in the solid, enabling us

to disregard the thermal phonons completely.

One expects multiphonon processes to be important in this

situation* Suoh processes have been studied by Engelsberg and

Sohrieffer , by way of Dyson's equations and the electron Green's

functions•

In the problem studied here, the most important effects are

due to the grouping of electron energies into so-called Landau levels

in a magnetic field.

We have found i t necessary to construct an effectively local

interaction for the eleotron-phonon system. This then ensures a re-

normalizable theory. This programme was effected - in spite of the non-

local nature of the electron-ion interaction, assumedCoulombian - by

a suitable definition of the phonon field and of the electron-phonon

coupling funotion. The lat ter now incorporates the effeots of normal

and Omklapp processes and, in a renormalized form, also includes the

shielding effeots produced on the interaction potential by the dressing

of electrons by phonons.

-3-

The general f ield theoretic methods we use are described in1 the took of Bogolubov

refer as S3 in a l l that follows.

detail in the took of Bogolubov and Shirkov ' t o whioh we shall

2 . THE ELECTRON FIELD

We assume a simple Bravais l a t t ioe and use the quasi-freeelectron model for a metal, with an isotropio energy surfaoe E(k) ink-spaoe. The eleotron effective mass i s then given hy m * ^•^lid^./d^z)

where we have assumed that the magnetic f ie ld B, i s in the z-direotion.

Ke assume that B - JjjJ i s large (or a l ternat ively that thetemperature i s suff iciently low), so that

j U b/'mc ^ > ^T"j (2.1)

k - Boltemann's oonetant.

However, B must not be large enough to produce interband transitions.Therefore we have a single band theory. Using natural units fc- c •• 1,we set eB/mo - eB/m • « „ say.

The wave equation for the eleotrons is

where A is the vector potential,

'1 0\> (2.3)

o-iyg is the eleotron g-faotor, and the spin function "X is given by X - (Q[

for "spin-up" and 7C- (,) for "spin-down".

If one takes into aocount open orbit effeots, g may be as

large as 200. w

Eg.. (2.2) has a solution

-4-

(2.4)

where

1 \V* A TV y

The wave function V^(T) i s degenerate with reBpeot to the parameter

since the energy levelB are given by

with oj equal to + 1 for spin—up and —1 for spin-down •

In a system in whioh the momentum vector p i s a good quantummimber, a momentum state I b) transforms under translations T(a)according to

In the system under study, p is in general conserved only up to the

z-oomponent of a reoiprooal lattice vector K . One oould then say

that k is (modulo K ) a good quantum number* The other componenta

of k are not good quantum numbers even in this sense.

A state of given k therefore may be presumed to transformSG

under T(a) aooording to

\K>(2.6)

where A

Bay, and the reciprocal lattice vector K. depends parametrically on

kg. This is consistent with the description of scattering processes

involving TTmklapp processes. Under circumstances in which these U-

prooesses are rare, the phase faotor 7j> becomes unity and ve have

(2.7)

The eleotron field is quantized as follows; We define a "vacuum"

(or ground state) as a single Slater determinant built up of states

(2.4) with n, k£ and a^ taking such values that

E (k ) < -/^(T) ** /+

where/i(T) is the ohemioal potential at the (small) temperature Tt

and jm- JU{0),

The vaouum so defined ia full of electrons obeying the Pauli

prinoiple. I t oontains neither holes nor phonons.

For the non-interaoting many-electron system we construct

conjugate fields y(i} f fix) as follows:

' (2-9)

where e denotes (here and always) the set {m,o^ ;ky^ } E. Cl«z)and \f ( i ) are the wave funotions of (2.4)»

For E(kz)>/4, ^ (^e) oreates one of the sets of eleotrons of

z-momentum k?, spin direction o^, and situated in the n-th Landau level.

1s(k£) annihilates such an eleotron.

For E(kE.)<Cyu t these two operations apply to holes instead

of electrons. The anticommutation rules are

(2.10)

either jE(p2)>/< (e lect rons)

wr E(p s )< /< (ho l e s ) .

(All other antiootnmutatora vanish) .

3 . THE ELECTROH-PHOHOH IHTERACTIO i TEE PHOTON FIELD

The eleotron-phonon interact ion i s obtained as the f i r s t

order tern in the expansion of the eleotron-ion (Couloinbio) i n t e r -

action with respeot to changes in ion co-ordinates . Thus i f H i s

the interaction potential}

where r, is the co-ordinate of the i- th electron,

R , is the general co-ordinate of the oL-th ion

and R is the equilibrium oo-ordinate of the a-th ion.

The eleotron-phonon interaction is then

For simplioity, we are assuming that we have a monatomio i-valent ion,

i . e . , an ion of oharge + Ze

(3.3)

where SI is the volume of the crystal.

Let

- 7 -

(3.4)

where £** is a unit polarization vector in the direction /A {/Am 1,2,3).

We irish to convert H^^ into an integral of the form

where (r) is a suitably defined phonon field, and g(r,) is a coupling

function.

the representation of the y's given "by (2,9), we have

(3.6)

The foilowing relation is true:

where N is the number of ions in the crystal and K is, as usual, a

reoiprooal lattioe vector.

Hence,

//

6

-8-

2. >* I

(3.8)

The ^ Integral is zero or small unless

or (3.9)

- A ' -Sinoe K takes positive and negative values alike, the two oonditione

are the same. Therefore we have

W . - S

— ^ A

K -

(3.10)

vhere

Ve may now define a free (vector) phonon field

(p(t)u

in vhioh U>(Q) is the phonon frequency*

(f(r) is quantised in the usual way by introducing oreation

and annihilation operators 1)1, (cp and b^,(q), respectively.

on vhioh the commutation relations

" - (3.13)

(all others zero) are imposed. Then

c= Z (fit? + f*(Z)

We shall need to use the amplitudes w , to , as such, later in the text.

-10-

We also define a coupling function

(3.17)This coupling function is local in configuration space, "but involves

the phonon momentum cj, an axial (i.e.^along the iB-axis) eleotron

current veotor

i'A ~ fi 4 0.18)

and a prescription for describing normal and {/-processes - the former

corresponding to K - 0 and the latter to E / 0

Using (3.11), (3.17) and (3.18), one may now write the inter-

action (3.10) as

The coupling function g( ,) is singular for processes in which p' •= p ,

i.e., at zero phonon momentum. We shall remove this singularity by

a renormalization of the electron and phonon fields and of the electron-

phonon vertex. Physically, such a renormalization corresponds to taking

into aocount the polarization of the electron field on the one hand

and the overall damping of the external phonon current on the other,

induced by ionic notion.

We assume, as is usual in field theory, that the renormalizedi t T-l'

fields If/ j Cp and vertex t are simply related to the corresponding

unrenormalieed quantities aooording to w

fix) —* j

Hz)(3.20)

-11-

Since there is no derivative coupling in our theory, we may

define an interaction Lagrangian '

L (*> (3.21)

where the interaction Hamiltonian H. Ax) now includes terms due toint *«*/

external electromagnetic and phonon fields, and a term due to phonon-

phonon interaction. He shall, for simplicity, ignore the effects of

the external electromagnetic field, since these effects are expected

to be inconsequential. It is also clear that phon cm-phonon inter-

actions are not of interest in this work* Therefore we shall work

with the Lagrangian density.

(3.22)

where £u> is "the external phonon current density, i . e . , the average

flux density of acoustic energy

where $> i s the density of the orys ta l ,

u0 i s the amplitude of sound wave o sc i l l a t i ons ,

•$ i s the group veloci ty of sound,

and Q i s 2TT times the sound frequency;

If along with the renormalization of the fields and the

vertex, the coupling function is also renormalized

a ^ ? ' (3.24)and the external phonon ourrent J« is transformed according to

> T ' 7 'A

then the coupling is unohanged and the Lagrangian is unchanged but

takes the form

-12-

) = J

The prime inZ indicates the ren ormal i zat ion of the vertex. Fhysioally,the J« transformation corresponds to a renormalization of the soundspeed s.

We can work out a rather explioit form for Z. as follows:

In the absence of U-processes (i.e. ;with .K • 0) the Lagrangian(3.26) may he written

X.) -= J fi> T - ' '

1 -^ ' r * (3.27)

where

is a form factor normalized so that

(3.29)

where K is an arbitrary function of some momentum parameters, withthe property that i t s value is 1 at p1 • f> _ and g is the eleotron-phonon coupling oonstant, to be determined experimentally by way ofthe dispersion relation whioh we derive in Sec* 7.

I t follows that

co SL

i.e.,

^ 9 J Z/i. t v) *A t'H+*')• * '* > i- 5 "i

S (*"-•£'

Ke must therefore have

vhere A is a constant momentum veotor.

Eenoe

and

vhere ve hare written out explicitly the dependenoe of H on the

constant momentum veotor A . A convenient choice for "St which

satisfies the condition N - 1 at p'-p , is

where n i s an in teger .

The combination g(x) E(A) i s now free from any s ingular i t ies

at p1 - p , and we would say that the vertex i s now renonaalized.

4 . THE S-KATRIX

Let HQ "be the unperturbed Hamiltonian of the eleotron-phononsystem and l e t H - HQ + H. . b e the to ta l Hamiltonian. The chemicalpotential /J Is zero for phonons.

Let N he the number of par t ic les in the system, and l e t

- <' V ( IJ •> ^o - ~ > ( " (4.1)

Then for any operator <9, t ranslat ions in time are generated

according to the formula '

Q-(t) - e> # e (4#2)

Thus

The S-matrix is then defined as

(4.4)

-15-

where z » (z»t) and T is the usual time ordering operator. The ex-

ponential function is a shorthand for the power series in ,' L(t)dt ,

The S-aatrix is functionally differentiate with respect to

electron and phonon fields. The functional differentiation with respect

to a quantized field is to "be understood in the limiting sense speoified

in BS • First we note the relationship

IL - i J - ^ 1 &>JL*'"* (4.5)

where ^(x) is any one of the fields. It is important to note that

differentiations of the Lagrangian function are performed with respect

to the funotionsl// (x),^/ (x) and ? (x) which already contain summations

over spin and polarization. "~

Using the series expansion of 5 in terms of jL(z)dz. and the

anticommutation relations for the 's» we arrive after some straight-

forward algebra at the following relations: (x - (*»*) )<-0

*« (4.7)

and

2

For the phonon field, we have the corresponding relations,

- 1 6 -

5 . THE SCATTERING ^TPLITDDE ABD RELATED PONCTIOITS

We are considering the e l a s t i c process shown in F ig . 1, in

which the wavy l ine denotes a phonon and the s t ra igh t l ine an electron,

1 • E las t io sca t t e r ing of phonons from elect rons

There are a number of assumptions one usual ly considers reasonable in

a f ie ld theory such as t h i s . Ve s t a te these "below in the form of

eiz pos tu la tee , P . , . . . , ? , .

P. There ex i s t s a vacuum 1 0 > defined as in Sec. 2 .

P . There ex i s t s t a t e s I n ) of n non-interact ing pa r t i c l e s (e lectrons

and phonons) generated from the vacuum by successive operation

with the appropriate creation operators for e lec t rons , holes

and phonons* These s t a t e s are vectors of a cer ta in l i nea r space.

- 1 7 -

P.. The states 1 n> transform according to a unitary representation

2) of a group M_ of translations. I!_ is the direct product

of two groups, KG.»» M-, t whose elements act on the phonon

states and the electron states, respectively. We shall assume

that KGf is the group of arbitrary translations in configuration

space and in time, whereas TAn is the group of translations

along the e-axis and in t ine. Thus the momentum operators

k,p z generate translations a according to

* i i \ / (5.1)

where Ik pj> is a state of definite phonon momentum k and

definite electron momentum y^ .

P. The completeness relation

holds for the combined electron-phonon system, (m denotes

al l discrete numbers, and m - p - k^» 0 corresponds to the

| 0)) . In (5.2) i t is assumed that the energies satisfy

E (P,)>/< for electrons, E (\>,)<.MtoT holea, &>(k)>0for

phononB and E«(p ) • ju by definition.

P, The stability condition

(5.3)

holds for the vacuum state and for states containing one

real particle or one stable combination. This condition

would have to be modified if partioles were allowed to have

large but finite lifetimes, i .e . , if quasi-particleB are

used instead of particles.

We also have orthonormality,

(5.4)

-18-

where OL}8 are discrete and p}c continuous.

The causality condition holds in the form

L L J1L n t

I (5.5)

for t x < t y .

or a 2 ( t x - t y ) 2 - (

vhere a is the speed of sound.«

This ia stronger than the usual space-like condition and

defines what we shall later call the forward sound cone*

It follows of course that (5.5) holds for o2(t - t )2 - (x-y)2<0i y ~ -~

where o is the speed of light.

How the scattering amplitude f for the process of Pig. 1 is

given "by

(5.6)

where we have used (and shall always use) the following ab'breviationst

K \ •

(5.7)

-19-

On the condition that the letter p always refers to eleotron energy-momenta and cj_ to phonon momenta, the following abbreviations willalso *be understood.

f\ r'J ; (5.8)

1

x c £7^ , t ~

and

7 v

•* 0 (^(p)-f- <^>(cl) - fcff>'} to(<Lft - * ~ ' (5.11)

It is perhaps in order to remark that the use of the phonon a-vector

as a parameter characterizing a phonon state depends on our tacit

assumption that the positions of ions are not appreciably affected by

the magnetic field. The ordinary translation operator, with a domain

restricted to the space spanned by the phonon states, commutes with

the Hamiltonian. The cj-representation space of the phonons remains

isotropio. This is the same as saying that only electrons interact

directly with the magnetic field, while phonons do not. Energy is

conserved for phonons and electrons. Only overall z~momentum is

conserved, and this possibly only up to an arbitrary reciprocal lattice

vector. There is a net transfer, between the magnetic field and the

eleotron-phonon system, of the component of momentum transverse to the

magnetio field.

. 20 -

We shall now use a contraction scheme involving the formulas(4.6) - (4.13), and postulate P2, to reduce the scattering amplitudeffa to a matrix element of the functional derivative

between electron s t a t e s . We have

(5.12)

where we have used (4 .13) .

Using Pc and (3.13), we have

(5.13)

We now oomnrute b«, (q ) with S using (4.12) and obtain

* +

- 2 1 -

where the vector t** is "dotted" to the -t— nearest to i t .dp

Comparing (5.I4) with (5«6), and using (5.1l)» we have

' (Q X-Q. W . > ° M , , /•

(5.15)

It follows that in deriving the dispersion relation, it would be

sufficient to study the function

Other functions that will he useful are

/ 6

-22-

•=.!.. ,» - . . . , . * .

and

where

^

»

is a current.

It follows from these definitions that

(5.22)

6 . TRANSITION TO MOMBNTDM SPACE

In order t o move in to momentum space, we have to know the

rules for space-time translations. Suppose f(y) denotes a trans-

lation "by (jf,^o ) aa^- K « H -/iN is the (reduoed) Hamiltonian. Then

for an energy eigenstate | A > ,

2y / 9 (6a)Also

Y (6.2)

Sinilaxly

- 2 3 -

Using (5»l)t (5»5) and (6.2), WB see that for any of the F's,

(6.4)

where a « i(x+y) and

and p'jjrj are energy momenta as in (5*9).

If ire define an operator PQ- «+ that interchanges o and o1

10) ~ i-and also -,+ at the same time, so that '

. -f • '

then

(6.7)

and

^ ) •.' (6.8)

When ne go to momentum spacet

-y J -y (6.9)

-fe.Jt - X t

ire obtain

() f T

-24-

and

^f.» \ —= / . ( 6 . 1 2 )

More ore r

I * c ^ J

(6.13)

where we have written k/ \ for k in the exponential in (6.13) in

order to indicate that k i s an electron energy-momentum parameter'K i s a phonon parameter. Carrying out the (x - y) integration,

we hare

(6.14)

-25-

Thus T+ vanishes unless

(6.15)

Prom momentum conservation, we have

Henoe (6 .15) "beoomes

(6.16)

In the same way, T+ vanishes unless

E(k2) + W(A) - iMf'z ) + E(pz) - u)(H,)-u,(q/)) , (6.17)

and energy conservat ion B( b1 ) + w( Q , ' ) " E (P X ^ + **(%) o o n v e r t 8 " t n i s "t

) - W( «j» ) . (6.18)

How because E(k ) is an intermediate state energy, we have for given

n and &„ , the inequality

It follows that

E n a (kB) - E ^ (p ) - - (*KX) + w(^) ) > 0 (6.20)

which is impossible if w(q)>Ot as i t is for real phonons.

We have thus proved that T+- 0 if the matrix element is takenbetween real partiole states of energies satisfying the inequalities

Prom (5*22) and (6.9) we then have

T<y<" ~ T - > -

for W(k) > 0 •

Similarly

_26-

A .«!,»»*•

for w(k) < 0 .

7 . FORWARD DISPERSION RELATIONj CATJCHT RELATIONS

I t i s convenient to work in the 'b r iok-wal l ' frame of

reference • ^

p a + P ; - . 0 (7.1)

2 i 2Henoe p z - p^

From energy conservation.

Henoe

for no spin-flip • (7.4)

ri + I-) for spin-flip . (7.5)

From momentum conservation^

for forward scattering,

- 27 -

Under this oondition,therefore, the briok-wall frame i s , as far as the

ss-axis is concerned, the rest frame of the electron.

Let us consider the "spin-flip" and the "no spin-flip" cases

separately. For "no spin-flip",

4>( ' ) - w(cp \ 0 according as ^ \r\ . (7.7)

In order to resolve the ambiguity in this relationship, we shall carry

out some numerical estimates*

A typical experimental value of o)(a) ( i .e . , the frequency of

the ultrasonio wave) is 2 x 10 o/s (See Ref. 14). A typical value ^

of B is 104 Gauss or more. Hence «H Z 2 x lCr1o/e,i,eM

or

It follows, a fortiori, that

5 < &>n . ( 7 . 9 )

Hence in (7.4)t VQ muot liave Sn - 0 in practice. Tt also foil own

from (7.9) that the np\n-flip interaction is most improbable. \U'.

shall therefore ignore i t henceforth. Thus

' (7.10)

Let us oonsider the limits of the phonon wave length and frequency.

It is obvious from (7.8) that we must place an upper hound on W if

this condition, and also the condition (7.9), are to be satisfied,

the former at least in the weaker form U £ WTT. In deriving dis-

persion relations, we would at some stage need to go to the limit

tj_>oo# prom the praotical point of view, i t is not conceivable

that we would generate ultrasound of infinite frequency, since sound

waves are matter waves, with a frequenoy limitation set by the in-

ertia and elastic constants of the vibrating Bystem. It is reasonable

to suppose,therefore,that by "intending to infinity", we mean "<u

being sufficiently large"t in the mathematical sense of the word.

-28-

To "be definite, we shall arbtrarily set this upper bound to be

U) ~ UL. Thus for B « 10^ Gauss or over, W - IT i 10 tfc/s or"n max.over. Microwave sound of frequencies CJ » 1CP Kc/s has been pro-

duced experimentally , so that the upper end of our frequency

spectrum can be reached in practice. The artificial condition we

have imposed on U) ensures that the set of relations (7.8) - (7.10),

which are rather essential for the subsequent analysis, are pre-

served for all frequencies of interest. Furthermore, we may now

estimate the bounds on phonon wave vectors, so as to justify our

neglect of (/-processes.

The velocity s of sound in a metal iB of the order of 4 x lCr

cm/sec. Hence with i*i ~ ir x 10 c/s and (J « s I a I , we havemax ' x

x 1.° 6 m ~ (7.11)

The lattice constant for a typioal metal (e.g., sodium) is—8

» 10 om. Hence wave vectors in the first Brillouin zone are such

that

1 - ' (7.12)

Thus our phonon wave vectors are well within the first Brillouin zone,

provided one does not attempt to apply this theory to semi-metals and

the like. The External) phonon spectrum ia given by U> - B\<^ \ so we

shall write

% " + & !t* f where (e I » 1"~ S

Thus

f (A *T \ ,<'*' J Z1 (7.13)

We remove the ambiguity in sign in (7.13) by considering the sym-metrized and antisymmetrized funotions

-29-

1 . - «> ;

£0

to e • *t

(7.14)

In order to derive the dispersion relations, we have to effect

an analytic continuation into the complex D plane. When we do that,

we may have difficulties with the exponent in (7.14). For

(7.15)

Thus, with either of the two signs in (7.15)» there are, for a given

value of t, values of x in 0^. | x |<°° for which t ± (e-x)/s<0 and

the integral blows up. The same argument applies to T^L, ( [n ^o')) .

This is where we have to introduce the concept of a forward

sound cone . He anticipated this situation in the formulation of the

(strong) causality condition Pg.

Thus F T(x,t) vanishes outside the forward sound oone shown

in Pig. 2.

Forward sound cone

Forward light cone

Fig. 2. Forward sound cone. This oone lies within the

conventional forward light oone.

- 3 0 -

:$

Inside the forward sound pone, t - A'£ > 0 and P T / 0,a

I t fOIIOWB that STr((d,e) is an analytic function of di in

the region Imw>0.

Similarly, 3T4((J,e) is analytio in the region In>w<0.

We now oonstruot

f for lmu)>o(7.16)

STa(w,e) for Imio<O

and

S i V e ) .

^MFf) (7.17)

(.7.18)

(7.19)

Using (6.14) and the momentum condition (6.15)» "«re have, after

carrying out the x,t,k^ and w1 integrations,

- 3 1 -

(7.20)

where 6) • «(-^ a) - ini t ial phonon frequency,

Ix to tlie fir8t term»in the second term

s X

and 6J(A) •» a • 0 for single electron states.

The intermediate state I K, 2 ^ m> consist f irst ly of single-

electron stateB, with energies B such that E >/"1(» i(lifi/2)wH), then

of _ single-electron, single-phonon states, and then of a continuumphonons

beginning with a single electron and two/. However, since

for a phonon, we have E(K, ) + N6)(A) t, for any number of phonons N.

Therefore the branch point ia at the threshold for single-electron

intermediate states with spin down.

The matrix elements <Ty3'liq(0) I 0/ between one-electron Btates

and the vacuum are required to be zero by electron charge conservation.

One can further show that all matrix elements <mk | o (O)jnp ^

vanish identically if ^mk | , ^ p (are real-electron states, but one

of them is not a one-eleotron state. Thus the intermediate state

contains only one-real-eleotron states.

Since there are no eleotron-phonon bound states at any

energy, we may assume that the cut in the complex energy plane is

due to the phonon spectrum. If multi-electron states existed, their

thresholds would successively be embedded in this cut.

-32-

or of eleotronsFor normal metals, i t ie known that bound states of eleotrone/

17)and boles do not exist. As is well known, ' bound states of the

latter type, excitone, are important only in semi-conductors and

insulators, while bound electron states, e,g.,"Cooper pairs" for

two eleotrons, are important in superconductors.

The single electron terms in (7.20) differ from zero only

when

E (-'I&L) + «-E(p ) - 0 (7.21)B Z !S

and/or

E (-S«L) - W-E(pJ - ° (7-22)i.e.,

2 s / a 7 n (7.23)

for forward scattering, where

With the upper aign in (7-23),

6> - a. - 0 or - ^J- cos

With the lower sign in (7.23),

or (7.26)

With the aid of the formulas

1(7.27)

e (xn) - o , g»(xn) / o ,we rewrite the single partiole delta funotion in (7.20) in the form

-33-

Let

(7.28)

Henoe

- 3 o r l ;

2The single-«leotroEi terms in (7.20) may now be written

%

(7.29)

(7.30)

•+

(7.31)

The first term in (7»3l) corresponds to a process in which an electron

emits a phonon, Fig. 3.

t— —• e

» Emission of one phonon hy electron,

-34-

The second term in (7.31) corresponds to a process in vhioh anelectron absorbs a phonon, Fig. 4.

E(p ) + w - E ( £ e | )

Fig. 4. Absorption of one phonon by electron.

It i s seen from (7.3l) that the two processes shown in Figs. 3 and 4

have opposite phases. Furthermore the amplitude for the process of

Fig. 4 is three times the amplitude for the prooess of Fig. 3» if

we assume that the magnitude of \ <i | j \y | is the same for both.

Therefore we have the result that therei-_isi an overall attenuation of

the ultrasonic wave.

The two roots displayed in (7.26) (or in (7.25)) give rise to

two distinct dispersion relations for forward scattering.

The Koot 6) - 0 gives a dispersion relation with a cut starting

from the origin. This would be the ordinary dispersion relation for

phonons behaving like photons . The specific effects of the mag-

netio field are given by the second root

1 ->n S

r PL

It is clear that 3T and ST represent the function ST.,

whioh is analytic in the region Imti) / 0 with cuts along the real

axis

-35-

We now formulate Cauchy's integral theorem for ST.(<j) af ter

we have determined i t s rate of increase at I c o i ^ 0 0 . This we now

proceed to do.

We have to investigate the ^independence of J«, (f>(x), g(x), S

and L ( i ) . Prom (3.17)» we see that g(x) ~l/v. Also

e )£

~ (constant + J ) S+ , (7.32)

since . Jy - % p ^ f e ^ - M . constant (3.23 )

(by oonstruotion, in spite of i ts apparent U) -dependence1.) .

V - 0 — | < /V°W > ' (7.33!V VTo find the o-dependence oftofx), and hence of the S-matrix S, we

'vuse the fact that the value of a is set by the incident "beam, whilethe restoring force is set by the constants of the solid. The re-

storing force is therefore constant. The ^-dependence of the lattice-

site displacement Qu(<p follows from the relation,? ~ 1

Force ~ to x d isplacement , i .e . , Q^(^) *" ^ 2 .

Henoef (x) » uK C j,) - « .

Therefore the Lagrangian density goes as

and

S - T e L^ goes as (constant + — + — 2 H

higher terms j.

Hence from (7.32) and (7.33) we have ST.(^) ~ (constant +

A (1 + — + higher terms)) , due to the assumed stability of single

particle states by which

Hence ST. (to) goes as (constant +— + higher orders). I t follows that

one subtraction is necessary in writing the dispersion relations.

We have to write a dispersion relation for

-36-

where ImflJ0 • 0 and -

We use the contour indicated in Pig. 5<

Pig. 5 , Contour of integration in phonon energy plane,

Wo then have for ax real

ST.

t (u) -

Hence

(7.34)

(7.35)

and

CO - COO

Im a) / 0 , Imii1 -

ai)

' / >•

I(7.36)

-37 -

8. FORtfAKD DISFSRSIOT RSLATIOKj FHTSICAL AKPLITUBSS

Fi r s t , l e t UB look at the absorptive part of the scat ter ing

amplitude,, We reca l l that

If we regard T i t as a matrix whose rows are labelled by (£'a}

and whose columns are labelled by ((x',o)» then we may define amatrix, Hermitian adjoint to T » , (k) , by

Then (8,1) becomes

Going over to the variables CJ, je, we have

T /Hence

and

are Hermitian matrices.

By adding (8.5) and (8.6), we have

Hence (7.36) becomes

-38-

" l S " Z r ~\Z" • ' " ; " " '• • ' " ; " "" • ' • " • •• ••" »

....... ' 41t

¥e now have to eliminate the region of negative energies.

Prom (8.1) and (8 .5 ) , (8 .6) ire have

and

A, /"-») - ~ fr V A...^/") (6.10)

Thus l e t t i n g a ) -+-a* in (8 .8) , we have

v V""1"'(8.11)

Adding together (8,8) and P times (8.11), we have

Henoe

/

The scat ter ing amplitude Tdui(d)) dependB on the spins tx,a*,

on momenta Pa*»p of the electron, and on the polarizations &<, u •

and momenta o, Qf of the phonon. I t i s invariant under rotat ions

round the e^axis and to inversions along the z-axis . Therefore,

we have to find the Btruoture of ^^ai03) such that

-39-

«/. & ^ -iL 9/ ^) e

9= e T

(8.13)

where T is a space conversion operator operating along the

s-axis, and L is the total angular momentum operator for the z—

direction* One might say that with a spin value equal to •§ for the

electrons and a "spin" value 1 for the phonons (/i - 1, 0, - l ) , the

eigenvalues of L are 3/2, l/2, - l /2, -3/2.

In calculating cross-sections, one is usually not interested

in the polarization states of the phonons. Therefore we have to

sum over polarizations of final phonons and average over polarizations

of the incident phonons. Measurements are, however, often made with

incident phonons of a definite polarization, for example with longi-

tudinal sound waves*

Thus we have the amplitude

; M< _ _ ^ A

(u>) ~=- a 2- fc ' (O.14J

in which the sum over^tf1 and the average over yU have been carried

out. f will now depend only on t> ' ,&, , oi , oL1, and — e . In particular

i t will have two components f,, fp corresponding to spins •£, - ^ ,

respectively. I t will also have a spin-independent term. We may

therefore write

T(8.15)

where the f irst term is spin-independent and corresponds to scatter-

ing with no spin-flip, and the seoond term is spin dependent. It

corresponds to scattering with spin-flip, whioh we have shown is

energetically improbable. Hence

-40-

We now use the optioal theorem in the form

LO

where tf(cu) is the total cross-section and s is the speed of sound.

Using f for D in (8.12), we have

N t T L W / - "7 • i .«.*r ! fj)' — CO (8 l8)

irhere

and

- JC^o) - a IT'S I « o 1 1 - • * * > .. 1 . v •*•

o(8.19)

• constant.

In laboratory co-ordinates, 0 is the angle "between the ma^ietio

field and the wave vector of the incident phonons.

We identify g with the electron-phonon coupling constant

introduced in Eq. (3.29).

Eq.. (8.18) is the required dispersion relation for forward

scattering of phonons.

9. CONCLUSION

One important deduction one makes from the d i spers ion r e l a t i o n

(8.18) is that there is a non-zero frequency threshold for the on-

set of the phenomenon of giant quantum oscillations in ultrasonic

absorption. One could make a rough estimate of this threshold for

zinc, one of the metals for which these osoillatione have been

observed

-41-

s -

e -Then f., «

frigidity modulus

* density

1.1 x lO~2m(

0

2ms2

- (from

2.25 x

Itef. 20

6-fold axis)

(i.e.,

126 Hc/s

para l le l

10 cm/sec

for B normal

propagation)

to crystal

*

27rft

This is lower than the frequency of 220 Ko/s at which thia effect

has "been observed. To tils extent therefore, one oould say that

this theory is not incorsistent with experiment. Experiments are

usually performed with 6.1 fixed and B varying, and not vice versa.

It would be interesting to devise an experiment in which B is fixed

and ot is varied over certain values. Admittedly, thia would be a

muoh more difficult arrangement to set up. The frequency threshold

is a minimum for parallel propagation and recedes to infinity as

we approach propagation transverse to the magnetic field. Finally,

let us relate the implications of this threshold with the theory

of Gurevich et al ^K

These authors use the energy conservation law (for n - n1 )

(9.1)' / • •

where | k

so thatk z - (>)/a oos e • (9.2)

If we expand (9.1) fully, and not just to first order in kz,

we have

i

Thus at tu » ttfn, P™ • 0 and one oould say that the quantum osoil lat ion21)takes plaoe as the minimum of each Landau level traverses the

Fermi level in the general d r i f t of the levels under a smoothly

varying magnetic f i e l d . Furthermore, forof>6J]L, P-^ 0 * which we

-42-

interpret as meaning that the phonon and eleotron collide "head-on"for the absorption process. ThUB starting from the situation

i? and energy conservation, f f > O ; /

we get, for 6*> UL

This ia a reasonable physical situation.

The problems that remain to be solved include corrections

for the possible non-isotropy of the energy surface E(k), the in-

corporation of finite temperature effects and the study of processes

involving1 a ohange in the Landau level parameter n.

ACKNOWLEDGMENTS

The author wishes to thank Professors Abdus Salam and

P. Budini for hospitality at the International Centre for

Theoretical Physics, Trieste and the IAEA for an Assooiateship

that made his stay at the Centre possible.

- 4 3 -

REFEEEKCSS ANI> FOOTNOTES

1) A.A. Abrikosov, L .P . Gorkov and I . E . Dzyaloshinski ,

"Methods of quantum f i e l d theory in s t a t i s t i c a l phys ics" ,

(P ren t i ce -Ha l l , 1TJ (1963) ) .

2) D, Fines, "Elementary excitations in solids", (Benjamin,

SI (1963)).

3) L.P. Kadanoff and 0, Baym, "Quantum stat is t ical mechanics",

(Benjamin, NY, (196?)).

4) V.L. Gurevich, V.G. Skobov and Yu. A. Firsov, Soviet Phys .-

JETP j3, 552 (1961).

5) S. Engel3berg and H.R. Schrieffer, Phys. Hev. 131, 993 (1963),

6) N.N. Bogolubov and D. Shirkov, "Introduction to the theory

of quantized fields", (interecience, NT (1959) eh. 9).

7) M.H. Cohen and E.I. Blount, Phil. Mag. $, 115 (i960).

8) N.N. Bogolubov and D. Shirkov, Op. cit. p.227.

9) A.A. Abrikoeov, L.P. Gorkov and I .E . Dzyaloshinski,

Op. cit. p.IO4.10) We shall always maintain the pairings (Q', + ) and (a , - ) , i.e.,

always associate + with <£' and - with q , ~

11) In order to obtain (6.12) we need to use the relations

l (q x - o -x, ^ -+ t f hf h - b) "*-

-44-

12) The intermediate state must have at least one electron.This follows from charge conservation.

13) J . Higelvoord, "Dispersion relations end causal description",

(North-Holland Publishing Co., Amsterdam (1962)).

14) T. Shapira and B. Lax, Phys. Letters 12,166 (1964)}

Phys. Rev. 138, A1191 (1965).

15) See,e.g.,lT. Tepley and H.W.P. Strandberg, Bull. Am. Phys.

Soc. j?, 15 (1964).

16) tfe must note that the summation index m includes the three

discrete quantum numbers of the system n,cr. , k and the

number of particles H in each state . The number of states

g(n) in a given Landau level n with momentum in the interval

d k of k , say, is easily calculated thus,

for k ^ k ^k° + dk ., we have in the presence of the

magnetic field

How dN/dk - volume/(lirf ~ V/(2,TT)3,

Hence

ffJ

where k • -r oos9 , k

L - [ t t H (n - J ) ]* 5 X

Hence g(n) -

Hence the sura X. i s the same as

-45-

17) JJ I . Ziman, "Principles of solid state theory",

Cambridge University Press (1964) p. 162

18) Cf. the wall-knowi Kramers-Kronig relation

P f oo'1 <r-r^')~ e J- J i .

We

19) A.P. Korolyuk and T,A. Prusohak, Soviet Phys*-JETP,, 1201 (1962).

20) B. Lax, Rev. Mod. Phys. ^£» 122 (1958).

21) This, one must note, invalidates the "seleotion rule" theory

of Gurevioh et a l , (Ref. 4)»

A SET.1P68 . ,- 4 6 -

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