CENTRAL RESEARCH INSTITUTE FOR PHYSICSBUDAPEST

г и Т и т 1 э г

K F K I - 1 9 8 1 - 4 3

T. SIKLÓS

T H E T H E R M O D Y N A M I C G R E E N ' S F U N C T I O N M E T H O D IN T H E S E L F - C O N S I S T E N T P H O N O N T H E O R Y

O F A N H A R M O N I C C R Y S T A L S

1H u n g arian ‘Academy o f ‘ Sciences

CENTRAL RESEARCH

INSTITUTE FOR PHYSICS

BUDAPEST

KFKI-1981-4 3

THE THERMODYNAMIC GREEN'S FUNCTION METHOD IN THE SELF-CONSISTENT PHONON THEORY OF ANHARMONIC CRYSTALS

T. Siklós

Central Research Institute for Physics H-1525 Budapest 114, P.O.B. 49, Hungary

To be published in

Acta Universität is Lodr.ienqis

HU ISSN 0368 5330 lUBN 903 371 825 2

A self-consistent phonon theory of lattice dynamics based on the thermodynamic double-time Green's function method is reviewed. The theory is applied for the investigation of the simplest model, the anharmonic linear chain with nearest neighbour central force interaction.

АННОТАЦИЯ

Теория самосогласованного фононного поля сформулирована на основе мето­

да двухвременных термодинамических функций Грина. Теория применена для рас­

смотрения простейшей модели - ангармонической линейной цепочки с централь­

ным парным взаимодействием ближайших соседей.

KI VONAT

A self-consistent fonon-tér elméletet a kétidős termodinamikai Green- függvények módszerének a felhasználásával foglaljuk össze. Az elméletet ez­

után a legegyszerűbb model, az anharmonikus lineáris lánc tárgyalására al­

kalmazzuk, a közvetlenül szomszédos atomok párkölcsönhatását feltételezve.

1. INTRODUCTION

The theory of lattice dynamics founded on the classical works by Debye, Bo r n and Kármán in its mo s t simple approxi m a t i o n

/in the harmonic approximation/ is c o n sidered to be a well established theory capable to describe m a n y of the physical

properties of the crystals in terms of independent normal m o d e s - -phonons. For more precise d e s cription the a n harmonicity of

lattice vibrations or the interaction b e t w e e n the phonons s hould be taken into acc o u n t and usually the o r d i n a r y pertur b a t i o n

theory, cons i d e r i n g the cubic and qua r t i c interaction is q u i t e appropriate for this purpose /see e.g. [ 1 J, [2], [3], [4]/.

The investigations of the past y e ars showed, however, that this approach c annot be a pplied in c e r t a i n cases: near the phase transition points, e.g. m e l t i n g point; for the q u a n t u m c r y s t a l s with large zero-point energy; for the light impurities w i t h small binding e n e r g y etc., when the a n h a rmonic effects are not small /see, e.g. [5]/.

Thus some m o d i f i c a t i o n of the well e s t a b l i s h e d Born-Kármán theory of lattice dynamics is needed in c o n s i d e r i n g the h i g h l y anharmonic crystals: the q u a n t u m crystals and the crystals at high temperatures, T > (0,3-0,5) Tm , where T^ is the m e l t i n g

temperature. In order to o b t a i n a theory w h i c h is convenient for these h i g h l y anharmonic crys t a l s it is n e c e s s a r y to take into acc o u n t all orders of the anharmonic i n t eractions applying a s e l f - consistent method. This ver y natural idea of introducing the s elf-consistent c o l l e c t i v e m o des was a lready p r o p o s e d by Bor n [6].

In r e c e n t years Born's idea was r e d i s c o v e r e d and the self- -cons i s t e n t phonon t h e o r y (SCPT) of anharmonic crys t a l s was e l a b orated s i m u l t a n e o u s l y and in d e p e n d e n t l y by several authors by a var i e t y of techniques. In one of these a p p r oaches the SCPT is based on the variational principle, the most ele g a n t t r eat­

me n t of w h i c h is given by W e r t h a m e r [7]. A selective resummation of d i a g r a m m a t i c p e r t u r b a t i o n t h e o r y was used in an o t h e r group of papers, the most d e t a i l e d d e s c r i p t i o n of this appr o a c h being p r e s e n t e d in Choquard's book [8]. The SCPT based on the t h e r m o ­ dynamic doub l e - t i m e Green's function m e t h o d was proposed

i n d e p e n d e n t l y in [9] , [10]. It was shown in [7], [11] that all of these three variants of the SCPT are equivalent.

Today the SCPT is c o n s i d e r e d to be a well estab l i s h e d

t heory and it has been app l i e d for the i n v e s t i g a t i o n of d y n a m i ­ cal, t h e r m o d y n a m i c a l and elastic properties of v arious crystals

/see e.g. [5 ] / .

3

In the present p a p e r the SCPT is f o r mulated in general br i e f l y using the t h ermodynamic doub l e - t i m e Green's function method (12]. In the next Section the H a m i l t o n i a n and the

e q u i l i b r i u m conditions for an anharmonic c rystal are discussed and the method of G reen's functions is introduced. In Section 3 the SCPT is formulated in a rather simple but general way on the basis of the i r r educible Green's function. In Section 4 the SCPT is applied for the investigation of the properties of an anharmonic linear c h ain in the first o r d e r of SCPT. Some c o n clusions are p r e s e n t e d in the last Section.

2. DESCRIPTION OF A N H A R M O N I C CRYSTALS

2.1 The Hamiltonian

Let us consider a crystal in the a d i a b a t i c approxi m a t i o n [1] when it can be d e s c r i b e d by the Hamiltonian:

i2

H - E Ж 7 + U < V

1 1

/2.1/

w i t h the local potential energy U(R^) d e p e n d i n g onl y on the coordinates R ^ = R ^ = R a (^) of the at o m of type к = 1 ,2,...,г in the unit cell i; a=(x,y,z); p^=p^=-itiv” is the m o m e n t u m operator and M.=M is the mass of the к-th type of atom. For the anharmonic crystal the e q u i l i b r i u m p o s i t i o n s of atoms

x^=<R^>=x (k ) are temperature dependent and should be obtained

fro m the e q u i l i b r i u m conditions. Let us apply an e x t e r n a l static f i eld w i t h forces F ± a cting on atoms at R ±

H, - - E F R. - - E F “ r\ /2.2/

1 i 1 1 sa 8 8

F r o m the e q u a t i o n of m o t i o n for the m o m e n t u m o p e r a t o r in the H e i s e n b e r g r e p r e s e n t a t i o n

p ^ t ) “ e i>3Ct р ±е 1L - H + /2.3/

one gets the e q u i l i b r i u m c o n d i t i o n s in the form

a s < P i ( t ) » - < 6 i . i P i ] > - щ /2.4/

F r o m t h e r m o d y n a m i c a l c o n s i d e r a t i o n s and Eq. /2.4/ follows an e q u a t i o n for the stress t e n s o r

°CXÍ3 “ V I X 8 F S - I £ x “ < V» u < * t ) >

or for the pres s u r e

P - - * E a

3 aa

a - 1 7 E o(»i) >.

as

/2.5/

/2.6/

w h ere V is the volume of the c rystal of N unit cells. The s t a t i s t i c a l average in Eqs. / 2 . 4 /-/2.6/ is taken o v e r the c a n o n i c a l ensembles

< A > = Sp{e"ß^A}/Sp{e'ß^}> 8 - A ; • /2.7/

The lattice parameters can be o b t a i n e d also d i r e c t l y from the p a r t i t i o n function:

R > = i

R i > ß 9 F. £n Sp<e“ **}. ^/2.8/

Now introducing the d y n a m i c a l d i splacements of the atoms ví^=R^-x^==ua (^) , the H a m i l t o n i a n /2.1/ can be w r i t t e n as

H = l i 2 M .

l

U (x. )

o' i'

l

n=l _1_

nl 1. 1. ,n u

r ■ u ,

n /2.9/

w h ere the coefficients of the T a y l o r e x p a n s i o n

a. a

= 7. . . . V U (x.) = ф / 0 n

.n 1 n О ' 3/ A. к.... x , к

i 1 n n

/2.9а/

are symmetric functions of the index (l...n) and satisfy several conditions w h i c h follow f r o m the invariance of the lattice under translations and rotations [l].

2.2 The Green's functions

Various d y n a mical and t h e r m o d y n a m i c a l p r o p erties of the enh a r m o n i c crystal can be d i s c u s s e d in terms of the Green's func­

tion /GF/. F o l l owing [ 9 ] , [10], [l2] let us c o n s i d e r the t h e r m o ­ dyn a m i c GF [1 3] :

с ±1' (t“t /)—<<u± (t), u ± '(t')»-j §£ e“±U)(t“t ')<<í i / V >>(1)

— 00

/2.10/

in usual notat i o n s [l4] . The s p e c t r a l r e p r e s e n t a t i o n for it h a s the fo r m

x

2TT ^ (ев“ ' - 1 ) а . Л » ' ) ,

w-ш J

/2.11/

w h e r e the F o u r i e r t r a n s f o r m J , .(w) for the c o r r e l a t i o n

*3 function

< u i (t) Uj> “ dto io)t _ / \ e J i;j(w)

2 ТГ ^/2.12/

is real and has the p r o p e r t i e s

J ± j(w) = = e *"ßU)ji j (“ w ) =

“ (е0а,-1) 1 [-2ImGi;. (w+ie)] , /2.13/

(e-*-0+ )

since the d i s p l a c e m e n t o p e r ators u^ are hermitian. The GF /2.11/ obeys the sum rules [l4] :

codw[- -■ Im G i;j(w+ie)]» 6 £ ,

• CO 00

w3da)[- i im G ij (w+ie)] - V ^ U ( R ^ >.

— 00 1 3

/2.14/

/2.15/

In the d i s c ussion of the a n h a r m o n i c p r o p e r t i e s of the lattice an ( п / п О “p o i n t GF of the type < < A n (t)j A n / ( t f)>>, where

7

A n (t) - { ^ ( t ) . . .un (t) - < u 1 ...un >} and

A n '(fc') = í u ^ C t ' ) . . . ^ (ti) - .un,>) w i l l appear for

w h i c h the r e p r e sentations sim i l a r to E q s . /2.11/ - /2.13/ hold.

In the t r a n s l a t i o n a l l y i n v a riant lattice the GF /2.10/

d epends only on the dif f e r e n c e of the coor d i n a t e s and the F o u r i e r trans f o r m a t i o n for it can be w r i t t e n in the form:

0 ) ^“

I

$jj'

*a / v

e<5j K> , , ( < ■ ) -i 3

/2.16/

Here for each w a v e v ector q = {q^„..,qN } the set of p o l a r i z a t i o n vec t o r s e+.(<); j = {l...3r} sat i s f y i n g the o r t h o n o r m a l i t y and

43

clo s u r e conditions:

I S S . (k)

L q3

к , a

I

3

/2.17/

are introduced.

The p h y s i c a l m e a n i n g of the retar d e d GF /2.16/ follows from linear response t h e o r y [.14]: the e n e r g y of p h o n o n - l i k e e x c i t a t i o n s at given (5,j) m e a s u r e d by i n e l a s t i c n e u t r o n sca t ­ tering, are d e f i n e d by the i m a g inary pa r t of the GF:

e “ - Im Gj_j,(q,w+ie) . /2.18/

The posi t i o n and the w i d t h of the m a x i m u m of /2.18/ give the e n e r g y and the inverse life-time of the e x c i t a t i o n s respectjvel-

ly. T h e long-wave length (q-*-o) limit of the s t a t i c (w=o) solf- - e n e r g y of the GF defines the i s o thermal e l a s t i c constants [15] .

T h e r e f o r e the d y n a m i c a l p r o p e r t i e s of the lattice are w e l l d e f i n e d by the GF /2.16/ and a d i r e c t com p a r i s o n b e ­ t w e e n theory and e x p e r i m e n t is possible.

2.3 The free e nergy a n d the internal energy of the anharmonic crystal

T o discuss the t h e r m o d y n a m i c a l p r o p e r t i e s of the an- h a r m o n i c c r y s t a l its free energy s h o u l d be calculated. The most e l e g a n t w a y for doing this is to integrate the GF over the

f o r m a l coup l i n g cons t a n t X [l6]. F o r the a n h a r m o n i c lattice Hamiltonian, Eq. /2.9/ X can be i n t r o d u c e d in the form:

T h e n for the free e n e r g y

F(X) - - I *nSp{e"0H(Xh - - i AnZ(X)

one obtains the e q u a t i o n

W *

ifa**

эх

3HL (X)

Э X X*

/2^.21^/

T o express F in terms of GF, Eq. /2.10/, c o n s i d e r an equation of m o t i o n for the GF w i t h the H a m i l t o n i a n /2.19/

9

I - »Jjl íjj, - « ±1. +

j

СО

+ 1 • П"2

Хп

(n-l)í

I

2

.

= <$.^/ + I (ш) •

j

/2.2 2/

A f t e r i n t e g r a t i n g the i m a g inary p a r t of it using Eqs. /2.12/

and /2.13/ one gets the right h a n d side of /2.21/ a n d the free e n e r g y in the form

F - F (X=o)

ЭН.

dX ‘ г г * \ "

, T n ( Í (“ Л г ^ К - 2 1 “ G i j ( “ + i e > >

Л oo •»

К & J 7 C Í * ( X *>

о C i j

/2.23/

where in the last line the i n t e g r a t i o n over the co m p l e x v a r i ­ able z is p e r f o r m e d along the contour C of two strai g h t lines: (-oo+ic)-*■ (°°+ie) and (°°-ie)-*-(-°°-ie) . D e f o r m i n g the contour C to circle the i m a g i n a r y axes of z one obtains, by c o u n t i n g the residues from the poles of (e®z-l) 1 at

zn = ( 2 n i n / 3 ), the same result as in [17] b a s e d on the imaginary time GF [lfi].

The i n t e r n a l e n e r g y of the a n h a r m o n i c crystal

apart fro m b e i n g c a l c u l a t e d t h e r m o d y n a m i c a l l y from the free energy /2.23/ c a n be o b t a i n e d in a m o r e d irect w a y by w r i t ­ ing it in the f o r m

E = < H > * = < T > + < U (x\+u± ) > , /2.24/

w here the a v e r a g e kinetic e n e r g y is e a s i l y expressed in terms of GF w i t h the h e l p of Eqs. /2.12/, /2.13/ ass

< T >

w dto cot h

№ у

2 L i

l

i

M.

2M,

►2

[-ilm G (u+ie)] .

The a verage p o t e n t i a l e n e r g y can be w r i t t e n in the form of a c u m u l a n t expansion:

< U ( x . + u . ) > *» <exp{ Уи . V , } > и (x.) = i r ' r 6 i i o x

” e x p { I Ы Í <V - V c V - - V u o < * i ) '

/2.20/

n=2 1.. . n

where the c u m u l a n t s < u . . . . u >c can a l s o be d efined from the

l n

GF as it w i l l be shown in the next Section. Then from a given a p p r o x i m a t i o n for the s e l f - e n e r g y П ^ ( ш ) and GF in

/2.23/, /2.25/ and /2.26/ one o btains a cor r e s p o n d i n g a p ­ p r o x i m a t e v a l u e for the t h e r m o d y n a m i c a l functions.

11

T h e r e f o r e the d y n a m i c a l as wel l as the t h e r m o d y n a m ­ ical p r o p e r t i e s of anh a r m o n i c c r y s t a l s can b e i n v e s t i g a t e d by m e a n s of G F . This p r o c e d u r e g r e a t l y s i m p l i f i e s the c a l ­ c u l a t i o n and a l l o w s one to p e r f o r m th e m in a u n i f i e d s e l f ­ - c o n s i s t e n t manner.

3. SELF-CONSISTENT PHONON THEORY

3.1 I r r e d u c i b l e GF

C o n s i d e r an e q u a t i o n of m o t i o n for the GF /2.10/

by d i f f e r e n t i a t i n g it t w ice with r e s p e c t to the time t and p e r f o r m i n g the n e c e s s a r y co m m u t a t i o n s . T h e n for the F o u r i e r t r a n s f o r m of the GF one gets

M .ш

i G ijLM

i. + У —■ ^У

^., < < u , ,

ii L nl u ll. . .n 1 n=l 1. . ,n

-*

. u

n

I v

w h ere the s y m m e t r y p r o p e r t i e s of /2.9а/ w e r e taken i n t o a c ­ count. There is a large class of n - p o i n t /mult i - p h o n o n / GF in /3.1/ that d e s c r i b e s an u n c o r r e l a t e d p r o p a g a t i o n of pho n o n s in an a v e r a g e d p h o n o n field. This c l a s s should be summed up not o n l y for the s i m p l i f i c a t i o n of further c a l c u l a ­ tions b u t als o for p h y s i c a l reasons: in h i g h l y a n h a r m o n i c crys t a l s atoms m o v e not in a s t a t i c field b u t rather in the d y n a m i c p o t e n t i a l of their v i b r a t i n g n e i g h b o u r s and this r e n o r m a l i z a t i o n should be taken i n t o acc o u n t from the b e g i n ­ ning. T h e r e f o r e we i n t r oduce the i r r e d u c i b l e /or cumulant/

GF £13] that h a v e no d i s c o n n e c t e d p a r t s of ave r a g e f i eld r e ­ n o r m a l i z a t i o n

• • iU U. , >>

1 n 1 i

irr . .-*■ + 1^ . .

“ <<u, . . .u u, , » -

1 n 1

n-1 - I c

m=>l

/3.2/

n ‘m < a ,...u > « 2 , ...u |2.,>>lrr

n m+1 n 1 m 1 i

He r e the s y m m e t r y w i t h r e s p e c t to p e r m u t a t i o n s of the c o m ­ m u t i n g o p e r a t o r s u -^...и^ has b e e n taken i n t o a c c o u n t and the c o r r e s p o n d i n g c o e f f i c i e n t m = C ™ = nl/ml(n-m)l w a s

introduced. T h e GF /3.2/ can n o t be r e d u c e d to lower order ones by the u s u a l d e c o u p l i n g p r o c e d u r e [14] and so it d e ­ scribes the c o r r e l a t i o n s b e t w e e n n - p a r t i c l e v i b rations. The d e f i n i t i o n /3.2/ can b e r e w r i t t e n in the f o r m of the d e c o m ­

p o s i t i o n for the n - p o i n t GF as t h e sum of the i r r e d u c i b l e ones:

<< u ... u ,u .,>>

n' 1

n

=

I

L n m + 1 n 1

m = l

. u

m ' > >rrr /3.2а/

By u s i n g a s p e c t r a l r e p r e s e n t a t i o n of the type g i v e n in Eqs. /2.12/, /2.13/ o n e gets f r o m /3.2а/ the d e c o m p o s i t i o n for the n - p o i n t c o r r e l a t i o n f u n c t i o n s in terms of the ir­

r e d u c i b l e /or c u m u l a n t / ones:

n-1

<U1* * - V " E Cn-1 . <Um+2*,,Un> (<ul’ *-um+l>C)

m=»l /3.3/

In o b t a i n i n g Eq. /3.3/ some o b v i o u s c h a n g e s of i n d i c e s have b e e n p e r f o r m e d in Eq. /3.2а/.

13

S u b s t i t u t i n g n o w /3.2а/ into the e q u a t i o n /3.1/

and p e r f o r m i n g the s u m m a t i o n at first ove r n for each m - -particle i r r e d u c i b l e GF and then over a l l m, one obtains:

|(M i“ 2 { ij ‘ *ij> +

* Í J r I n « 5 . . . . S J S . , » i r r ,

^ ~ n * T

vs 1 n 1

/3.4/

w h e r e the r e n o r m a l i z e d i n t e r a c t i o n has the form:

'S«

Ф-1« • • П .n, l'. . .n' < U 1'** *u rf> e < v i* • ,vn u ^x i + u i ^ :

“ V i-* * Vn exp{ ) n'=

The c u m u l a n t e x p a n s i o n in the las t line /as w e l l as in /2.26//

follows from the e q u a t i o n [l9] :

1 г С 1 /-у \

—r*r L ^ Uy. ■ . u § * V у . . . V J U (x , ) •

n i < 1 n I n o x

JL • • • XI

ЭЛ

°° , n V i_.

L nl n = o

l ^Ф

n 1. ,n

/3.6/

that can be e a s i l y s olved in the form /2.26/ by i n t r o d u c i n g the e x p a n s i o n /3.3/ and p e r f o r m i n g the s u m m a t i o n over n and m in the same m a n n e r as in /3.4/.

To o b t a i n the e q u a t i o n of m o t i o n for the n -point irre d u c i b l e GF in Eq. /3.4/ let us d i f f e r e n t i a t e the operator u^Ct') w i t h r e s p e c t to t'. The n t a king int o a c c o u n t the i d e n t i t y

< [ u 1 ...un# 1 р ±/]>1ГГ = 0 (n > 2)

that f o l l o w s f r o m the d e f i n i t i o n /3.2/, and i n t r o d u c i n g the same d e c o m p o s i t i o n as in Eq. / 3 . 2 a / for the o p e r a t o r s

{vi ,(t' )... it № ) ) on the r i g h t hand side of the (n,n' )-point GF, o n e gets

2 - irr

1 ^M j/ ш < < u i* • .цп I u j» > > (t)

j ’

00 •

" ^ n'T ^ *i'l' .. .n'

n ' -2 Г...П'

/3.7/

w h e r e

is the (n,n') - p o i n t GF i r r e d u c i b l e b o t h in the n - p o i n t and n' -point parts of it.

N o w it is c o n v e n i e n t to d e f i n e the z e r o order GF by the e q u a t i o n

I(Miw2 6ij - Ф± .) (to) “ 0±1" /3.8/

j

w h i c h d e s c r i b e s the u n d a m p e d v i b r a t i o n s /or o n e - p h o n o n p r o ­ p a g a t i o n / in an a v e r a g e p h o n o n field. Then s o l v i n g the matrix e q u a t i o n s /3.4/ and /3.7/ w i t h the h e l p of Eq. /3.8/ one

gets for the GF t

G i i'(«) “ + I P j j ' M /3.9/

jj'

w h e r e t h e s c a t t e r i n g m a t r i x is d e f i n e d by

P j j ' (w)“ £ Í I F T ^ * j l . . . n G í - . n , l ' . . . n ' (w)*j' l'...n', n , n ' - 2 l...n

- 15 -

1'... n' /3.10/

T h e n the o n e - p h o n o n GF can be w r i t t e n in the for m of the D y s o n e q u a t i o n

Jii'

(ш)

- *ii' * n ii‘ <“ »

-1 /3.11/

w h e r e the s e l f - e n e r g y operator TI^»(u))ie g i v e n by the proper p a r t (p) of the s c a t t e r i n g m a t r i x /3.10/: П ' (w) ' (“>) • A c c o r d i n g to Eqs. /3.9/ and /3.11/ П ^ ' s a t i s f i e s the e q u a ­ tion:

p ii»(w) - + I n± j (w) G®j#(u») /3.12 /

j j'

Hence, the s e l f - e n e r g y operator h a s the same form as Eq. /3.10/ w here the ( n , n * ) - p o i n t GF is r e p l a c e d b y its p r oper par t K. ./ » ( « ) - G ^ i r r 'p ^1, , ( w ) . к(ш) ас- c o r d i n g to Eq. /3.12/ can not be cut i n t o two p i e c e s by c u t ­ ting only one G°-line; G ° is d e f i n e d b y Eq. /3.8/.

The n - p o i n t i r r e d u c i b l e GF in /3.7/ can a l s o be w r i t t e n in terms of K, . p t (w) if one us e s e q u a tions

/3.8/ - /3.12/:

< < u , . . .u I u . •> > irr в 1 n 1 i ш

& 1 Vl'...»' e ü ! . a . . ,3.

j n<=2 1'. ..n'

13/

I V j ' W I

У n'»2 1'. . . n'

Ф-дгр f (w)

3 1 . ..n l...n, 1 ...n 1

n' J

T h e n f r o m the s p e c t r a l r e p r e s e n t a t i o n , E q s . /2.11/ - /2.13/

for the c u m u l a n t p a r t of the c o r r e l a t i o n f u n c t i o n s one gets:

■ V

00

f dm 1 J e ^ - J

_—*1™ u2 - • -un 1 (D+ieJ ^/3.14/

00 oo

21m

d t I ^{Т И К} 1

П — 3 • П

< u 1 (t)u1<> < u 2 (t) . . .un (t) l u ^ .n'

urí irr,p

w h e r e a t w o - t i m e p r o p e r i r r e d u c i b l e c o r r e l a t i o n function, c o r r e s p o n d i n g to K. „ ,,, , (t), has been introduced.

• <П f <L • • • П

T h u s the o n e - p h o n o n GF Eq. /3.11/ as w e l l as the c u m u l a n t s /3.14/ in the r e n o r m a l i z e d i n t e r a c t i o n /3.5/ are w r i t t e n in terms of the ( n , n ' ) - p o i n t GF K, ,, _,(w) “

i.» «П / X • * •П

= < < u-^. . • u n I u^, . . . un , > > шГ Г ' T h e e q u a t i o n s o b t a i n e d are e x a c t b u t u n c l o s e d a n d t h e r e f o r e some a p p r o x i m a t i o n s to the K, ., ,(w) s hould be c o n s i d e r e d in order t o obtain

l...n, i ...n

a s e l f - c o n s i s t e n t s y s t e m of equations.

3.2 F i r s t order or r e n o r m a l i z e d h a r m o n i c a p p r o x i m a t i o n In the first order o f the SCP T /SCI/ o n l y the r e ­ n o r m a l i z a t i o n of p h o n o n s in t h e s e l f - c o n s i s t e n t f i e l d is taken i n t o account. Thus the S C I is o b t a i n e d by n e g l e c t i n g all the t e rms w h i c h c o n t r i b u t e to the d a m p i n g /or c o r r e l a ­ tions/ of phonons. In that c a s e the s e l f - e n e r g y opera t o r П (to) in the GF /3.11/ and the c u m u l a n t s /3.14/ for n > 3

ii'

17

should b e put e q u a l to zero. T h e r e f o r e the SCI GF is equal to the zero order one /3.8/ w i t h the r e n o r m a l i z e d pseudo- h a r m o n i o f o r c e - c o n s t a n t m a t r i x

•ij5“ 7 i7 :j exP<5 I <“ l“2>7l7 2 )Uo (*l) 5 V l7 j U l(xi>

1» 2

/3.15/

The s y s t e m of e q u a t i o n s ge t s c l o s e d by the e q u a t i o n for the p a i r - c o r r e l a t i o n func t i o n in /3.15/. F r o m the spec t r a l r e p ­ r e s e n t a t i o n Eqs. /2.12/, /2.13/ and /2.16/, one gets

<u“ u?,>

1 . (<) e® . (к' ) -I N L

43i / * . M K'

2ы

3w . c o t h —

“ i q ( x s - x s ,)

в 9 /3.16/

43

w h e r e the frequencies and the p o l a r i z a t i o n vectors

^ q j ( K ) are defined by the e q u a t i o n

u>2 . e a .(<)- 1 . (<' )— ---- 43 q ] v ‘ 43 4 //m m 1

3s' / M < М к'

6 S /3.17/

The f r e e energy /2.23/ in S C I is o b t a i n e d by u s i n g the m e a n - f i eld s e l f - e n e r g y operator / 2 . 2 2 / г

П $ * > я

- 3

1,2

<UlU2>7l72 )Uo (íti) /3.10/

So one finds

F, - F +

1. О

á l т п , Ф ( х ) < 5. 3 , >

X L

± з ч / 1 з о

ij

- F 0 +

5 ^ )

I

/3.19/

The trial f o r o e - c o n s t a n t m a t r i x in /2.19/ has n o t been

s p e c i f i e d yet. F r o m the s e l f - c o n s i s t e n c y c o n d i t i o n Ф°. should be p u t equal to Ф ^ ' in /3.15/ since the latter one d e f i n e s the s p e c t r u m of e x c i t a t i o n s in SCI a c c o r d i n g to /3.17/. Then for the fre e e nergy o n e gets

Зит*. . ßk’cn

F 1 “ U l(x <) + £ E Än(2 sinh — — ) “ 4 l cot h ~ ~2— ' /3.19а/

q j 2 43

w h i a h c o i n c i d e s w i t h t h a t o b t a i n e d from the v a r i a t i o n a l a p p r o a c h

iD

F o r the i n t e r n a l e n e r g y /2.24/ one easily o b t a i n s fr o m Eqs. /2.25/

and /2.26/ In the SCI

ßüT- .

E 1 “ 4 ^ wqj COth 2^* + ^ x i ^ ‘ /3.20/

q j

T h e r e f o r e in the SCI a p p r o x i m a t i o n one t reats the vib r a t i o n s of an a n h a r m o n i c crystal as a s y s t e m of n o n i n t e r a c t i n g pseudoharmoni.c p h o n o n s w i t h the á-fu n c t i o n type b e h a v i o u r for the p honon s p e c t r u m

/2.18/. This a p p r o x i m a t i o n s i m p l i f i e s the c a l c u l ations; but d u e

19

to this a p p r o x i m a t i o n all the odd terms in the a n h a r m o n i c i n t e r ­ a c t i o n /2.9a/ a r e missing. O n e s h o u l d n o t hop e to o btain a

q u a n t i t a t i v e d e s c r i p t i o n in SCI: even in the limit of w e a k an- n a r m o n i c i t y the results d o not c o i n c i d e w i t h those of the o r d i ­ nary p e r t u r b a t i o n theory /see, e.g.[20]/.

3.3 Second o rder of SCPT

S ince the s e l f - e n e r g y oper a t o r П ^ , (w) in the GF /3.11/

on a ccount of /3.10/ is p r o p o r t i o n a l to the s econd order of the renor malized a n h a r m o n i c inte r a c t i o n /3.5/ one s h ould c a l c u l a t e the

(n,n')-point G F in the l o west order. This is d o n e by t a k i n g a c ­ c o u n t only the u n c o r r e l a t e d p r o p a g a t i o n of n=n' "dressed" ph o n o n s and results in the f o l l owing a p p r o x i m a t i o n for the ( n ,n')-point two-time c o r r e l a t i o n f u n c t i o n (n > 2):

un( t ) | u r . u ,

n n 1 6

nn'

lL -> ->

n < u i (t)u.,> . i=l

/3.21/

N o w e m p l o y i n g the spectral r e p r e s e n t a t i o n for the ( n , n * ) - p o i n t GP one o btains for the s e l f - e n e r g y o p e r a t o r in the second order of the SCPT / S C 2 /

d “ ' (ee“ '-l) Ш-Ш'

dt -iw't x 2 TT e *

/3.22/

w h e r e V j я э /эх^ and V_., a respectively.

iu ( x i+ u i ) > < V i, u ( x i, + u ±,)>,

S /Эх., a r e a c t i n g on u ( x ) and u(x,,

J I X

)

F o r the c u m u l a n t s /3.14/ one gets in the same SC2 a p p r o x i m a ­ tion /3.21/:

. -> c ~

< u , . . . u > ~

1 n

* 2 Im

n -у -У -У -y

d t П {£ < u ± (t ) u i, > V ± , } < u ( x ±, + u i<) > • i “ l if

I 3.23/

The e q u a t i o n for the p a i r c o r r e l a t i o n f u n c t i o n

< u ± (t)u;,> -

' e 8“-l oo

iut - - Im{o)2M i6i;, ~ 4^ , - n^j\uH-ie)} /3.24/

closes the s y s t e m of s e l f - c o n s i s t e n t e q u a t i o n s /3.5/, /3.22/-

■/3.24/. In the p r e s e n t form the SC2 s y s t e m of e q u a t i o n s can ho app l i e d for a n h a r m o n i c c r y s t a l s w i t h s t r o n g r e p u l s i v e .inter­

ac t i o n s s i nce only fully r e n o r m a l i z e d v e r t i c e s Ф /3.5/

X • • • n

a ppear in the e q u a t i o n s /3.22/, /3.23/, as in the H o r n e r theory [21], But ju s t d u e to the full r e n p r m a l i z a t i o n of the vert.ices the s y s t e m of e q u a t i o n s is r a ther u n t ractable. To s o lve it one should either i n t r o d u c e a trial s h o r t - r a n g e c o r r e l a t i o n f u n c ­

tion in Eq. /3.5/:

<U(xi+ui )>=exp{ I ~r l ‘'un>Cvi*

n-3 X • « • n

•vn }íii (xi h g s r (xi )ü1(x.),

/3.25/

or, e m p l o y i n g some cut - o f f p r o c e d u r e for the strong r e p u l s i v e p a r t of the i n t e r a c t i o n e x p a n d /3.25/ in p o w e r s of cumulants:

21

< u ( x ± +

* { 1 + I in* ^ < u 1 ...un>-C V 1 ...Vn + ... }U1 (xi ) n=3 1 . . .n

- + Д и 2 (х1 ) + ...

/3.26/

The r e n o r m a l i z e d p o t e n t i a l e n e r g y in the p s e u d o h a r m o n i c a p ­ p r o x i m a t i o n

U 1 (xi ) = exp I < u ± u* V ±V.j) и о (х± ) /3.26а/

is cal c u l a t e d by i n t e g r a t i o n w i t h a G a u s s i a n function.

We shall n o t dis c u s s the p r o b l e m of hard core i n t e r ­ a ction her e since an e l e g a n t p r e s e n t a t i o n of it is given by

Horner [21] and c o n s i d e r only the c u m u l a n t e x p a n s i o n in Eq. /3.26/

IV

up to s econd order in A^ 2 (x^). In thi s a p p r o x i m a t i o n the r e ­ no r m a l i z e d v e r t i c e s /3.15/*

(2

1... n v r - - V V * i > + a ü2 (*í)> /3.27/

can be c a l culated on ac c o u n t of Eqs. /3.23/, /3.26/ by i t e r a ­ tion. In the c l a s s i c a l limit of high t e mperatures, ®шт а х < < 1' the i n t e g r a t i o n over time in c u m u l a n t s /3.23/ using the s p e c t r a l r e p r e s e n t a t i o n can e a s i l y be do n e w i t h the r e s u l t

and

<V*1‘ * *^n>C ^ ""е1П £1, <uiui' >Vi' ) < и (^± + u ± )>,

o° _

л и ^ х ^ - е I

(I < V V > V i ' ) u 1 ( x i ).

n - 3 ii'

/3.28/

/3.29/

The t h e r m o d y n a m i c a l p r o p e r t i e s in the S C 2 a p p r o x i m a t i o n can be o b t a i n e d from the i n t e r n a l e n e r g y /2.24/, w h e r e the a v e r ­ age k i n e t i c energy /2.25/ is c a l c u l a t e d by u s ing GF /3.11/

w i t h the s e l f - e n e r g y o p e r a t o r /3.22/ and the a v e r a g e p o t e n ­ tial e n e r g y /2.26/ is g i ven b y /3.26/. T o c a l c u l a t e the second order c o r r e c t i o n to the free e n e r g y in /2.23/ one should in­

t roduce first the i r r e d u c i b l e GF in /2.22/ as d o n e in the S e c t i o n 3.1. Then by i n t e g r a t i o n over f r e q u e n c i e s using Eq.

/3.14/ one gets:

A F. f u 2 I < 2 , 2 > v j v j i ö ^ A . í p + ij

" An

+ z T ^ D T l < u i - “ V xv r - - 7 n u i<x 'x i ) }

n — 3 1.. . n л

w h e r e the first term is d u e to the second order c o r r e c t i o n to the r e n o r m a l i z e d p s e u d o h a r m o n i c m a t r i x фУ.'(Л) as in /3.27/, The А - d e p e n d e n t function < u ^ . . . u n >^, A U ^ (A ,x.j) a n d U^(A,x.) are g i v e n by /3.23/, /3.26/, and /3.26а/ r e s p e c t i v e l y with every power of u^ m u l t i p l i e d b y A. After the i n t e g r a t i o n over A w i t h <UjU.>, <u.(t)u.> b e i n g i n d e p e n d e n t of A, o n e gets

j D

oo

d t I ~ ( I < u i ( t ) u i ,>ViV. ,)n U 1 (x.) U 1 (xJ_) =

AF Im

n=3 i i ‘

/3.30/

2 ^ nl ^ < U l * “ *un >C V l"*V n ^ l ^ X i^ “ 2A^ 2 ^ X i^ * l...n

In the c a l c u l a t i o n of the f i r s t two terms F +A^f .. in the free

о 1

e n e r g y e x p a n s i o n one s hould e m p l o y /3.19/ wit h the SCI f r e q u e n ­ cies b eing r e p l a c e d by the SC2 freq u e n c i e s defined as the m a x i m a of the i m a g i n a r y p a r t of the GF in the SC2 a p p r o x i ­ m a t i o n a c c o r d i n g to /2.18/.

23

The p r o p o s e d cumulant e x p a n s i o n s for the average p o t e n ­ tial energy /3.26/ and the free e n e r g y /3.30/ with the GF

/3.11/ and e q u a t i o n s /3.22/ - /3.24/ give the same r e s u l t s as other m e t h o d s based on d i a g r a m m a t i c .techniques or o n the variational a p p r o a c h proposed by W e r t h a m e r [7] /see e.g. [5]/.

4. THE A N H A R M O N I C L I N E A R CHAIN

In the pre s e n t Section a s i m p l e model of the crystals, the a n h a rmonic linear chain w i t h nearest n e i g h b o u r i n t e r a c t i o n will be i n v e s t i g a t e d briefly in the first order of t h e SCPT

[22], [23]. In this case we can o b t a i n a simple e x p l i c i t solution w h i c h helps to clarify some a spects of the SCPT.

4.1 The s e l f - c o n s i s t e n t system o f e q u a tions

Let us c o n s i d e r an a n h a rmonic linear chain of l e n g t h L w h i c h consists of N+l identical a t oms w i t h mass M. T a k i n g into a ccount only nearest n e i g hbour interaction, the H a m i l t o n i a n in the a d i a b a t i c a p proximation [1] reads:

V t =

N H + H

1 =

I

+2

Í n

N

i l

<p(R -

_{V i }}

w h e r e p^ and R^ are the m o m e n t u m and p o s i t i o n o p e r a t o r s for the n - t h atom. The interaction p o t e n t i a l b e t w e e n the n e i g h b o u r i n g a t oms is d e n o t e d by <p( R - R . ) . In the case of a o n e - d i m e n s i o n a l

-1 n n-1

c h a i n the e f f e c t of the external forces can be d e s c r i b e d by

the external te n s i o n P w h i c h acts o n the ends of the chain:

H1 ' P <RN - Ro )= P ” , (Rn - Rn-l> • I * - 2 ' n=l

It is c o n v e n i e n t to intro d u c e the e q u i l i b r i u m separation l bet w e e n the n e i g h b o u r i n g atoms a n d the rela t i v e d i s placement o p e r a t o r s by t h e following d e f inition:

R -n R . = <R -R , > + u

n-1 n n-1 n - u

n-1 a + u -

n u

n-1 /4.3/

w h e r e the s t a t i s t i c a l a verage < ... > is c a l c u l a t e d for the e q u i l i b r i u m s t ate of the system d e s c r i b e d by the H a m i l t o n i a n

/4.1/:

< ... > = Sp { e ^ 9... } /Sp{ e (e = kT) . /4.4/

The e q u i l i b r i u m s e p a ration l in the o n e - d i m e n s i o n a l case can be o b t a i n e d from the equation

P

tp(R -R .

n n-i <p'U) ^{> /4.5/}

w h i c h shows t h a t the ave r a g e force a c ting on an a r b i t r a r y atom in the e q u i l i b r i u m p o s i t i o n is e q u a l to zero.

It is c o n v e n i e n t to introduce e x p l i c i t e l y the d i s placement o p e r a t o r s in the H a m i l t o n i a n /4.1/ b y the F o u r i e r transformation:

<p(R) = I <p(q) eiqR ; q

0(q) = l

T J J / _

/ d R <p(R) e iq .

—L/2

/4.6/

In this r e p r e s e n t a t i o n the H a m i l t o n i a n of the linear c h a i n

25

/4.1/ takes the form:

9 t = I n

I I Ф(Ч) ei<3*

n q

iq(V V i )

+ H. /4.7/

For the c a l c u l a t i o n of the c o r r e l a t i o n function of nearest ne i g h b o u r s and the frequency of the lattice v i b r a t i o n we apply the m e t h o d of t h e r m o d y n a m i c d o u b l e - t i m e GF [13] . We use the following o n e - p h o n o n G F :

G

nn

/4.8/

in usual notations [14].

To o b t a i n the e q u a t i o n of m o t i o n for the GF /4.8/ we d i f f e r e n ­ tiate it twice w i t h respect to time t a n d employ the equa t i o n of m o t i o n for the He i s e n b e r g o p e r a t o r s un (t) and pn (t) •

In this manner we get:

2 H2

M i

2-=-

) =

)

+

,,2 nn' nn'

I

q

iq « { e

iq(W i >

^ n+1 n

u . (t 1 ) » . n'

/4.9/

j*

The m u l t i p h o n o n GF on the r.h.s. of Eq. /4.9/ describes I an uncorrelated p r o p a g a t i o n of phonons in an aver a g e d p honon

field. We use the first o r d e r or r e n o r m a l i z e d h a r m o n i c a p p r o x i ­ m a t i o n of the SCPT here, in w h i c h the p r o c esses c o n n e c t e d with

the damping of p h o n o n s are not considered, but the renormali-

zation of the energy of phonons in the p h o n o n s e l f - consistent field is taken into account. In this app r o x i m a t i o n the m u l t i ­ phonon GF can be written in the form:

iq(u-u ,) » ,

« e ; u , » = У — r « { iq(u -u n)}S ; u , »

n' L. s! u n n-1 n'

s=l

00

~ У -~r < { i q (u -u . ) }S ^ > s « iq (u -u . ) ; u , » =

L , s! 1 n n-1 ^ n n-1 n'

s=l

= < e

iq(u -u .)

^ n n-1

> i q « (u -vi ,) ; u , »

n n-1 n

/4.10/

For the calc u l a t i o n of the c o r r e l a t i o n function on the

r.h.s. of Eq. /4.10/ we use the same approximation. We introduce the following function:

Xq(u -u . )

F (\) = < e П > ; F(o) = 1 . /4.11/

D i f f e r e n t i a t i n g it on X and u sing the similar a p p r o x i m a t i o n s as in Eq. /4.10/ we get:

3F(\)

- Г Г = < ’ ' V V i 1 e

Xq(u -u . )

^ n n-17

> =

= < q(u -u 1 ) У

^ n n-1

s=l

(q\ ) b , 4 s „ (u -u .) >

s! n n-1

~ X q 2< ( % ‘V ] . ) 2 > *

The i n t egration of this e q u a t i o n over X from X = o u s :

H l ' V V i ’ „ - ^ 2<(W i )2> - ^ 2ц2

< e > = e = e ,

/4.12/

to \=i gives

/4.13/

27

where we take into account that the corre l a t i o n function of n earest neighbour atoms does not d epend on n:

ü 7 - < (v r % > 2 > - •= < W i > 2 >

N o w we introduce the F o u r i e r t r a n s f o r m for the GF /4.8/:

oo G , (t-t') = / do

nn Vri '

-io(t-t')

2n G

Ли)

nn' /4.15/

and take into a c c o u n t that it depends onl y on the dif f e r e n c e of lattice sites (n-n')s

G ,(o) = rjjr

У

nn' MN f

к

ik£(n-n')

/4.16/

Then Eq. /4.9/ t a kes the form:

о G. (о ) = 1 + к

2 2

+ 2^ I <p(q)

/4.17/

where Eqs. /4.10/, /4.13/ have been used. The s o l u t i o n of Eq.

/4.17/ reads

Gr (o )

2 2 о

-о.

к

/4.18/

as in the harmonic a p p r o x i m a t i o n e x cept for the r e n o r m a l i z a t i o n of the force constant:

2 4f(0Д ) . 2 ki. f 0 Д 2 _ 2 2

к M 2 f ok ok /4.19/

where oQk is the harmonic frequency of vibration and f stands

for the harmonic force constant. The r e n o r m a l i z e d force constant f(0,i) accor d i n g to Eq. (13) can be w r i t t e n as:

2 2

f(6,£) = J I <p(q) eiq£(iq)2 e U = ^ф" (£) , /4.20/

q

w here we introduced the self-consistent potential 2~2

< <p(Rn ~ R n _ 1 ) > ~ (p (Ä ) = I (p(q) eiq£ e 24 U = q

- I ± ( Ф f Ф (2в)<») ■ /4.21/

s=o

2 2 In o b t a i n i n g Eq. /4.21/ we d e c o m p o s e d the function exp(-^ q u ) into the series of u ~ T and i n t egrated it o v e r q. The s e l f - c o n s i s ­ tent potential can al s o be w r i t t e n in the form

oo 2 a .■■1 .—

ф(£) = j dx e X 2 (p(JH-x/u ) , /4.22/

where x = R / / u 2 . It is easy to see that in Eq. /4.22/ the r e n o r ­ m a l i z a t i o n of the p o t e ntial due to the vi b r a t i o n s of the atoms is taken into a c c o u n t by a v e r a g i n g it o v e r the small region

/~j~ 2

R ~ / u « £ with the Gaussian function e x p ( - x /2) w h i c h descr i b e s the effect of the p h o n o n self-consistent field. O w i n g to this

function onl y the shape of the potential o(R) at the b o ttom of the potential well is of importance.

The c o r r e l a t i o n function of nearest n e i g h b o u r s in Eqs. /4.21/, /4.22/ can be o b t a i n e d from the spectral t h e o r e m [13] [14] for the G F :

< V V - é / -ÖTtrr < - 2 Im Gnn'(brt-15) I • - oo e -1

/4.23/

29

Fro m Eq. /4.18/ we get:

, ,2 4

<

(V un-l>

У

2“k

G1k 0001 20 G)ok 1 Ч 2

dcp sin ф coth a sin cf

nf a > 2 г

о

where in the s e c o n d line of Eq. /4.24/ we have replaced the sum over к by-an integral over cp = ki,/2 . The m a x i m u m v a lue of the vibrational f r e q u e n c y of the c h ain in the harmonic approxi m a t i o n is denoted by о =(4f/M) and т = ö/o T stands for the reduced

OL» Ob

temperature. In the high t e m p e r a t u r e (t » 1) and low t e m p e r a ­ ture (1 « 1 ) limit the integral in Eq. /4.24/ b e c o m e s

— a2 u2 = п T { 1 + (— ) } + 0 (t 3) (t » 1 ) , /4.25/

CO T T

oL

— 2

a u2 = 1 + 4 — (— )2 + 0( ) ( i « 1) . /4.26/

"oL J a

In a d d i t i o n to the tempe r a t u r e i the p r o p e r t i e s of the linear chain are de t e r m i n e d also b y the length of the chain L = N£ or b y the external tension P . According to Eqs. /4.5/, /4.21/ these pa r a m e t e r s satisfy the following equation:

P = - i <cp'(R-R . ) > = - i ip'U) . /4.27/

2 n n-1 2

The internal e n e r g y is given in o u r ap p r o x i m a t i o n by the equation

*

l E i =

“ Ж í T 0001 <"к,20) + I * <1)

Í { <pU ) + f(0,A) u2 } . /4.28/

Then for the free energy of the a n h a r m o n i c linear c h ain according to Eq. /3.19/ in our a p p r o x i m a t i o n one gets:

^ F1 = Í H { 2 Sinh (“k /29) } + j { <^pU ) - f(0#^ä) ^u2 } . /4.29/

к

In this w a y we have a closed s y s t e m of s e l f - c o n s i s t e n t

equations /4.19/, /4.20/, /4.21/ or /4.22/, /4.24/, /4.27/, /4.28/, /4.29/ which d e t e r m i n e the dynamical, t h e r m o dynamical a n d elastic properties of the a n h armonic linear c h a i n in the r e n o r m a l i z e d harmonic approximation. This s e l f - c o n s i s t e n t system of e q u a tions is determined by the s e l f - consistent p o t e ntial /4.21/ o r /4.22/, w h i c h can be o b t a i n e d if the form of the interaction p o t e n t i a l

in the Hami l t o n i a n /4.1/ is known.

4.2 The s e l f - c o n s i s t e n t system of e q u a t i o n for the M o r s e p o t e n ­ tial

Let us take the M o rse potential as a m odel p o t e n t i a l -a(R-r ) 2

<p (R ) = D { [ e - 1 ] - 1 } , /4.30/

where rQ is the a verage distance b e t w e e n the n e i g h b o u r i n g atoms in the harmonic approximation: <p'(ro )=o and D is the d e p t h of

31

the potential: <p(rQ ) = - D . The force c o n s t a n t in the harmonic a pproximation is g i v e n by f = 1/2 <pM (r ) = D a .2

■%

Applying the e x p a n s i o n of Eq. /4.21/ or taking the integral of Eq. /4.22/ we get the following e x p r e s s i o n for the self-

-consistent potential:

-2ar x 2y -ar x y/2

ф(^х) = D { e ° e - 2 e ° e } , /4.31/

w h e r e у = a2 u2 = (a rQ )2 (u2/r2 ) and x = (i,/rQ ) - 1 • In Fig. 1.

F i g . 1. T h e s e l f - a o n s i s t e n t p o t e n t i a l q>(x)/D

the self-consistent p o t e n t i a l ip(x)/D is p r e s e n t e d for some values of у . Since у d e p e n d s on the temperature and, as we will see, in the qu a n t u m limit on the e nergy of ze r o - p o i n t vibrations,

the larger values of у mean h igher temperature or larger energies of the zero-point vibrations. The diagrams of the s elf-consistent p o t e ntial in Fig. 1 show the thermal expansion of the crystals and the decrease of the b i n d i n g e nergy of lattice atoms when the t e m perature or the e n e r g y of the zero-point v i b r ations

increases. So for larger у v a l u e s the s e l f - c o n s i s t e n t potential suffic i e n t l y d iffers from the i n t eraction poten t i a l that means that the b e h a v i o u r of the c r y s t a l s d i f f e r s u f f i c i e n t l y from that cal c u l a t e d in the harm o n i c approximation.

Eq. /4.5/ for the M o r s e p o t e n t i a l reads:

P* = 4 {

-2ar x

e о e

2y -ar x y/2

о 1 ,

e e } f /4.32/

w h ere we i n t r oduced the r e d u c e d tension P = (4/arQ ) P(rQ /D) . We note, that the introduced reduced tension di f f e r s from that used in works [22], [23]. The r e n o r m a l i z a t i o n factor of the frequency, a c c o r d i n g to Eqs. /4,19/, /4.20/, for the Morse p o t e n t i a l takes the form:

a2(y) = | { P * + e y [ l + / l + P * e Y ] } .

T aking into account /4.31/ we can rewrite Eqs.

/4.25/, /4.26/ as equations for y:

tt/2

\ ct(y) у = / d (p sin <p coth ( a— ) , о

a2 (у) У = T* { 1 + i ^ )2 } T » 1 ,

2 о

\a(y) у = 1 + IL- ( i )2

/4.33/

/4.24/,

/4.34/

/4.35/

T « 1 /4.36/

33

where X = (п D /со т ) is the d i m e n s i o n l e s s coupling cons t a n t and

OJLi

T * = 0/D = тп/\ is the r e d u c e d temperature.

The expressions for the e q u i l i b r i u m sep a r a t i o n of n e i g h ­ bou r i n g atoms £ , the i n t e r n a l energy /4.28/ and the free energy

/4.29/ in the c a s e of M o r s e potential can be w r i t t e n as follows

/4.37/

/ 4 .38/

/4.39/

w h ere

a

к

The system o f self-c o n s i s t e n t e q u a t i o n s /4.32/, /4.33/, /4.34/ determines the p r o p e r t i e s of the an h a r m o n i c linear chain, when Л,т and P ■ft or £ are given, see Eqs. / 4 . 3 7 /-/4.39/.

Let us n o w consider the equation /4.34/ in the high t e m p e ­ rature limit, Eq. /4.35/. T a k i n g into account o n l y the first term in the r.h.s. of Eq. /4.35/ and using /4.33/ the s e l f - consistent equa t i o n can be w r i t t e n as follows

F(y) = 1 -- { P* + e ^ [ 1 + / l + P*e-^ ] } = о . /4.40/

2T*

The d e p endence of the solution o f Eq. /4.40/ on the reduced

‘X- -X*

temperature T and r e d u c e d tension P is g i v e n in Fig. 2.

F i g . 2. T h e r e a l s o l u t i o n o f t h e s e l f - c o n s i s t e n t

e q u a t i o n f o r т » 1

■X* *

At sufficiently low temperature: T <Tc = 0 , 5 7 8 , and pressure:

P < Pc = 0,222 there are several real s o l u t i o n s for y(T), the therm o d y n a m i c a l l y stable ones are shown b y the full lines. The lowest line, у^(Т) ^ 1 c o r responds to a c r y s t a l l i n e state wit h small vibrational motions. In the limit o f small a n h a r m o n i c i t y /or lower temperature/, this solution g i v e s the h a r m o n i c c o r r e ­ lation function. But as T* -> T*(P) /denoted by the full dots in Fig. 2./ y^(T) b e c o m e s unstable: Ts (P) is the i n s tability temperature. As c a n be seen from Fig. 2. and from the PV d i a g r a m

corresponding to Eq. /4.32/ or /4.37/ (tor ar =6), shown in Fig.3.

p*

F i g . 3. T h e P V - d i a g r a m o f t h e ( i n h a r m o n i c l i n e a r c h a i n

f o r г » 1

at T* = T*(P) there is a first order p h ase t r a nsition that drives the chain from the state w i t h у = y-^(T) to some o t h e r state with у = y2(T) , shown by the u p p e r full line in Fig. 2. The latter is also a "crystalline" state /due to the r e s t r i c t i o n s imposed by the theory/ but the v i b r a t i o n s of the atoms are rather large, У 2(Т)>1 and are d e f i n e d by the external p r e s s u r e : i n the limit P+ о the solution y2(T) ■*■<*>. At s u fficiently high pressure P >Pf;

*

or cor r e s p o n d i n g hi g h temperature, T >Tc this type of vibra- -H-

tional i n s tability disappears. The two solutions at ?c coincide and there is onl y one stable solution for the c o r r e l a t i o n func­

tion y(T) for P*>P* . P h y s i c a l l y it m e a n s that the external

forces become more e f f i c i e n t than the interatomic one s and the former determine the lattice dynamics: th e y are s t rong-enough to localize the atomic vibrations.

At low t e m p e r a t u r e s ( t « 1) the s o l u t i o n of the self- -con s i s t e n t equation /4.36/ for the c o r r e l a t i o n function y(\,T) leads to the same res u l t s [24] as in the hig h temperature limit.

The on l y difference is that at low t e m p e r a t u r e s the amplitude of atomic v i b rations is given m o s t l y by the zero-point energy, propor t i o n a l to К . As a result, a h i g h l y a n h a rmonic chain wi t h small coup l i n g constant \<\c =l,207 can be unstable even at T=OK, the critical value of the tension in this case is Pc =0,037.

In Fig.4. idle insta b i l i t y temperature tgis p r e s ented as a function of

F i g . 4. T h e d e p e n d e n c e o f t h e i n s t a b i l i t y t e m p e r a t u r e T s o n t h e r e d u c e d t e n s i o n P * f o r s o m e v a l u e s o f t h e d i m e n s i o n l e s s c o u p l i n g c o n s t a n t \ i n t h e l o w

t e m p e r a t u r e l i m i t т « 1

!

Í