Ae/

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

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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).

Korff, S. A. and G. Breit, "Optical Dispersion," Review ofModern Physics 4, 471 (1932).

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Unpublished Materials

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