DYNAMICS AND THERMODYNAMICS OF A ONE DIMENSIONAL NON-LINEAR LATTICE IN THE CONTINUUM LIMIT

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DYNAMICS AND THERMODYNAMICS OF A ONE DIMENSIONAL NON-LINEAR LATTICE IN THE

CONTINUUM LIMIT

S. Behera, A. Khare

To cite this version:

S. Behera, A. Khare. DYNAMICS AND THERMODYNAMICS OF A ONE DIMENSIONAL NON-

LINEAR LATTICE IN THE CONTINUUM LIMIT. Journal de Physique Colloques, 1981, 42 (C6),

pp.C6-314-C6-316. �10.1051/jphyscol:1981691�. �jpa-00221628�

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JOURNAL DE PHYSIQUE

CoZZoque C6, supp Zdment au n ' 1 2, Tome 42, ddcernbre 1981 page C6-314

DYNAMICS AND THERMODYNAMICS OF A ONE DIMENSIONAL NON-LINEAR LATTICE IN THE CONTINUUM LIMIT

S.N. Behera and A.

hare*

I n s t i t u t e of Physics, Bhbaneswar-752007, India

*

Department of Physics, Manchester University, Manchester, U. K.

A b s t r a c t , - The dynamics of a non-linear one dimensional l a t t ce, with t h e o n - s i t e p o t e n t i a l Y(+ = (t1/8) Cosh 4 9

- ⁵

^Cosh^2&

-

( s2/8) i s considered i n t h e continuum l i m i t . Exact c l a s s i c a l k i n k s o l u t i o n s a r e obtained; f o r

3

⁴^2, i n which c a s e t h e p o t e n t i a l has t h e double w e l l £ o m , The f r e e energy of t h e system i s c a l c u l a t e d using t h e ground s t a t e eigen value of t h e S chrodinger l i k e equation f o r t h i s p o t e n t i a l .

1, I n t r o d u c t i o n

.-

I n r e c e n t y e a r s e x a c t c l a s s i c a l s o l j . ~ t i o n s of one dimensional non-linear e q u a t i o n s have found a p p l i c a t i o n s i n v a r i o u s branches of condensed m a t t e r physics1. The non-1 i n e a r problems which a r e of p a r t i c u l a r i n t e r e s t t o l a t t i c e dynamics a r e t h e Toda l a t t i c e ² and t h e l a t t i c e s having o n - s i t e p o t e n t i a l s w i t h more than one degene- r a t e minima, such a s M e ++and

@'

f i e l d t h e o r i e s 3 '

*;

t h e l a t t e r c a s e s being t h e continuum r e p r e s e n t a t i o n s of t h e corresponding l a t t i c e problems. I n a l l t h e s e cases t h e r e e x i s t e x a c t l a r g e amplitude c l a s s i - c a l s o l u t i o n s c a l l e d ' S o l i t o n s ' (kinks) b e s i d e s t h e w e l l known s m a l l amplitude harmonic v i b r a t i o n s (phonons). It has been shown by

3 4

Krumhansl and S c h r i e f f e r f o r t h e c a s e of t h e

(h

f i e l d theory t h a t a t low temperatures both t h e phonons a s w e l l as the k i n k s a r e well

d e f i n e d e x c i t a t i o n s of t h e system and hence c o n t r i b u t e t o t h e f r e e energy. They f u r t h e r i d e n t i f i e d a p a r t of t h e e x a c t ( c a l c u l a t e d with-

in t h e WKB approximation) f r e e energy with t h a t of a n i d e a l gas of t h e k i n k s . S i n c e t h e n t h i s i d e n t i f i c a t i o n has undergone much r e f i n e - ment and rigour5. I n t h e

4 '

f i e l d t h e o r y t h e o n - s i t e p o t e n t i a l b e i n g of a double w e l l n a t u r e s e r v e s as a model f o r second o r d e r phase t r a n - s i t i o n s . I n a n e a r l i e r p u b l i c a t i o n we4 considered t h e +'-field t h e o r y

(where t h e o n - s i t e p o t e n t i a l has t h r e e minima) as a model f o r t h e f i r s t o r d e r phase t r a n s i t i o n . Xn c o n t r a s t t o t h e

($

^4-theory, t h i s h a s t h e advantage t h a t t h e free energy of t h e system can be c a l c u l a t e d e x a c t l y , However, we f a i l e d t o i d e n t i f y t h i s e x a c t r e s u l t w i t h t h a t of t h e i d e a l kink-gas phenomenology.

In t h e p r e s e n t p a p e r we r e p o r t t h e dynamics of y& another one-

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1981691

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dimensional l i n e a r l a t t i c e , which can serve a s a model f o r second o r d e r phase t r a n s i t i o n and f o r which it i s p o s s i b l e t o c a l c u l a t e t h e e x a c t f r e e energy.

2. Dynamics of t h e Mo4el.- I n t h i s model t h e on-site p o t e n t i a l i s of t h e form

VC+) = ^(52/8) C 0 ~ 1 \ + + - - 5 6 ~ h 2 4 - ( 5 ~ / 8 )

~t can be e a s i l y checked t h a t

v(+)

has minima a t

4 9 0

^-fay

'5 ^,2

and

Cisb 24 = 2 / 5

^{$ 0 ~}

5 ^{< 2}

For t h e l a t t e r condftion t h e r e are two degenerate minima. The v a l u e s of t h e p o t e n t i a l a t t h e minima a r e

and

I n t h e continuum l i m i t the equation of motion of t h e corresponding c l a s s i c a l f i e l d theory i s given by

where

and Co being t h e v e l o c i t y and t h e maximum sound v e l o c i t y . It can

I n t h e l i m i t of S

-+ +_

^og

,

eqn. ( 6 ) gives

t a M ~ + ( t e ) = + L ( ~ - ~ ( ~ + T I J ~

⁽⁷^I

which a r e t h e values of

(p

corresponding t o t h e degenerate minim.

Hence, t h e s o l u t i ~ n s given by eqn

.

⁽⁶¹a r e t h e kink s o l u t i o n s having energy

EK = 2

m~ Co ( 8

where

m,= ( 4 ~ 1 ~ ~ ) [$

^{( 1}

49) Besides t h e l a r g e amplitude kink s o l u t i o n s given by eqn.(6) t h e r e

win

a l s o e x i s t small amplitude harmonic s o l u t i o n s (phonons) around t h e p o t e n t i a l minima.

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C6-3 16 JOURNAL DE PHYSIQUE

3. S t a t i s t i c a l Mechanics of t h e Model.- The f r e e energi) of t h e system c a n be from a knowledge of t h e ground s t a t e of +he

Schrodinger l i k e equation

where t h e temperature dependent e f f e c t i v e mass i s given by

, ( cl= ^{k B ~ )}

⁽¹¹⁾

For t h e p o t e n t i a l given by eqn. (1)

,

equation (10) can be f a c t o r i z e d as

t o y i e l d t h e e x a c t ground s t a t e 6 e i g e n v a l u e

and t h e e i q e n f u n c t i o n

Hence t h e f r e e energy p e r unit length of the system becomes

Thus t h e f r e e energy can be e v a l u a t e d e x a c t l y . However a s i n t h e c a s e of t h e

@

6 -problem, f o r t h i s p o t e n t i a l a l s o it is n o t p o s s i b l e t o i d e n t i f y t h e e x a c t f r e e energy with a phonon p a r t and t h a t of an i d e a l kink gas.

References

.-

1. ~ishop,A.R, and Schnider, T., Eds, " S o l i t o n s i n Condensed-Matter Physics" S p r i n g e r o v e r l a g ( B e r l i n 1978).

2, Toda,M. Prog. Theoret Phys. Suppl. 85, 174 (1970).

3. K W a n s 1 , J . A . and S c h r i e f f e r , J.R. Phys. Rev.Bu, 3535 (1975).

4. Behera,S .N. and Khare,A, Pramana

15.

245 (1980).

5. Curie,J-;F. e t . a l . Phys. Rev. B a r 477 (1980).

6. Razavy,M.Am. J . P h y s . 4 8 , 285 (1980).