EFFECT OF THE HEAT BATH ON ACTIVATED RATE PROCESSES IN SOLIDS

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EFFECT OF THE HEAT BATH ON ACTIVATED

RATE PROCESSES IN SOLIDS

R. Bauer

To cite this version:

EFFECT OF THE HEAT BATH

ON

ACTIVATED RATE PROCESSES IN SOLIDS

R .

BAUER

Max-Planck-Institut fur Metallforschung, Institut fur Physik,

Heisenbergstr. 1, 0-7000 Stuttgart 80

und

Institut fur Theoretische und Angewandte Physik der

Universitat Stuttgart, Pfaffenwaldring 57

0-7000 Stuttgart 80, F.R.G.

A b s t r a c t

-

The e f f e c t o f heat-bath c o u p l i n g on t h e c o n d i t i o n s o f access t o

sadale-point c o n f i g u r a t i o n s o f vacancy jumps i n s o l i d - s t a t e d i f f u s i o n i s i n -

v e s t i g a t e d by analysing, i n t h e framework o f quantum s t a t i s t i c s , t h e i n t e r p l a y

between i ) t h e r a t e o f phonon-phonon c o l l i s i o n s caused by anharmonicity, r e -

presenting t h e b a t h r a t e , i i ) t h e resonant mode o f t h e d e f e c t and i t s l i f e -

time determined by t h e a d d i t i o n a l anharmonicity caused by t h e defect, and

i i i ) t h e r a t e o f b a r r i e r r e l a x a t i o n o f t h e mu1 t i v a r i a t e deformable b a r r i e r .

A t low temperature where damping i s small, t h e r e s u l t i n g jump r a t e c l o s e l y

approaches the r a t e l a w o f Vineyard (1957) c h a r a c t e r i z e d by t h e f u l l y r e -

l

axed saddle p o i n t .

With i n c r e a s i n g temperature, damping o f t h e resonant mode increases,

a l l o w i n g b u t f o r a p a r t i a l r e l a x a t i o n ; jumps occur a t reduced r a t e over non-

e q u i l i b r i u m c o n f i g u r a t i o n s o f t h e saddle p o i n t w i t h increased m i g r a t i o n en-

t h a l p y and, caused by l a r g e s p a t i a l g r a d i e n t s o f t h e energy, w i t h enlarged

values o f m i g r a t i o n entropy and a c t i v a t i o n v01 ume. The t y p i c a l non-Arrhenius

form of t h e r a t e r e s u l t i n g from temperature-dependent damping a l l o w s t o i n -

t e r p r e t s e l f - d i f f u s i o n i n many metals by jumps o f monovacancies only.

Anharmonic damping o f t h e resonant mode i n v o l v e s e s s e n t i a l l y

l

ow-fre-

quency phonons n o t contained i n t h e spectrum o f c r y s t a l l i t e s w i t h up t o

1000 atoms. Computer modellinq o f .jump r a t e s thus underestimates t h e r e l e -

v a n t damping and y i e l d s alwayi the-low-temperature approximation, t h a t i s

t h e Vineyard r a t e .

I

-

INTRODUCTION

The temperature dependence of experimental d a t a of t h e t r a c e r s e l f - d i f f u s i o n c o e f f i - c i e n t of m e t a l s u s u a l l y e x h i b i t s curved Arrhenius p l o t s l g DT v s 1/T, / l / ; t h e c u r v a t u r e i s s t r o n g f o r bcc m e t a l s l i k e Nb, Ta, MO, W and Na while it i s small b u t e x p e r i m e n t a l l y w e l l e s t a b l i s h e d f o r t h e f c c m e t a l s Cu, Ag, Au, N i , P t , /l-3/. This c u r v a t u r e may be caused e i t h e r by t h e c o n t r i b u t i o n s :D of a t l e a s t two m i g r a t i n g t y p e s of i n t r i n s i c atomic d e f e c t s i w i t h s u i t a b l y d i f f e r i n g a c t i v a t i o n p a r a m e t e r s , o r by c u r v a t u r e of D: = f l a* c l

r l

of one d e f e c t o n l y , due most l i k e l y t o i t s jump r a t e

.

,

'

l

The jump r a t e of atomic d e f e c t s has s o f a r been e s t i m a t e d by t r a n s i t i o n s t a t e t h e o r y (TST) /4,5,6/ and i t s e x t e n s i o n s /7,8/ based on t h e assumption of t h e e q u i l i b r i u m d i s t r i b u t i o n of s t a t e s and implying t h e r e l a x e d Vineyard s a d d l e p o i n t SPVin. The a c t i v a t i o n e n t h a l p y of t h i s s a d d l e p o i n t depends b u t l i t t l e on tempera- t u r e /g/; hence TST s u p p o r t s two-defect models of s e l f d i f f u s i o n /10,11,12/.

Any r a t e t h e o r y which i s more complete than TST h a s t o e v a l u a t e t h e d e v i a t i o n s from t h e e q u i l i b r i u m d i s t r i b u t i o n . T h i s a f f o r d s c o n s i d e r a t i o n of t h e coupling t o t h e h e a t b a t h , which on t h e one hand e s t a b l i s h e s a temperature B-I = kgT and thdrmal f l u c t u a t i o n s p r o v i d i n g f o r hopping of atomic d e f e c t s , and on t h e o t h e r hand imposes damping o r d i s s i p a t i o n on t h e dynamical motions. These two e f f e c t s of a h e a t b a t h a r e i n s e p a r a b l y i n t e r c o n n e c t e d l i k e e i t h e r s i d e of one c o i n , a s it

i s

expressed i n

C10-28

JOURNAL

DE

PHYSIQUE

the fluctuation-dissipation theorem, cf. /13/. TST neglects dissipation and establi-

shes thereby an upper bound for any rate. Besides the early work of Kramers /14/ on

this topic, several hundred papers have appeared in the past eight years, cf. /IS/

for a review, dealing often with reactions in molecules immersed in a solvent. The

present note discusses the elements to treat the jump rate of atomic defects in crys-

tals for the example of the monovacancy in metals.

I1

-

THE ONE-DIMENSIONAL PROBLEM

One-dimensional

rigid

barrier

We consider a particle with mass M in a

potential U

(X)

,

Fig.

1,

describing a meta-

stable well at

X =

0 and a barrier with

wb=

(-

$

&

U

1

I2

height

ub

at xb. The curvatures in the

bottom and at the barrier are measured

by wo and wb

> 0,

respectively. Coupling

to a heat bath provides the inverse tem-

perature

B

and imposes a friction

<(X)

or

3

% M '

a damping rate y(x)

=

<(x)/M. TST yields

Fig. 1. The metastable well with

the rate of escape

basic parameters ub, wO,

wb.

we use w

=

2n v throughout.

either

wb nor y are contained in this rate law. Kramers

/14/ found that in the limit of low friction (I) the thermal activation in the well

is the rate-limiting process and the rate is proportional to

y o

s

y(0): with the

transmission coefficient r

E

r/TTST, he obtained

b

rI

=

y0Bu /v0 for yo

*

0.

(2)

For intermediate and high friction (II), activation of the particle to the barrier

is so fast that esca e over the barrier is the rate-determining process. There, damp-

ing at the barrier yg

s

y(xb) is essential in reducing the equilibrium rate TTsT.

With the normalized barrier damping a

E

yb/2wb the result is

b

yielding r l ~

*

1/2a

=

wb/yb for a

> 1

and rII

2

0.8 for a

=

1/6. w~ and wb are the

friction-induced and the bare transmission frequencies of the barrier, respectively.

This expression, formerly often judged as an approximate interpolation formula, is

now well established: it appears as the classical limit of the quantum-statistical

rate for linear coupling to the bath and for Ohmic or viscous damping /16,17/, and

it is obtained for non-Markovian dynamics /18,13/ where a memory-renormalized

2

re-

places a.

The complete solution combining the regimes I and I1 for arbitrary but uniform

damping y

=

const(x) is worked out in

/13/,

approximate solutions are given in

/19,20/. In regime I,

ub

in the exponent is replaced by Ul(y) with U1

*

ub

for y

*

0

and U1

*

0

if rI approaches unity /13/. This decrease of the activation energy in

the transition region is a new result not obtained in the Kramers limits.

I11

-

THE MULTI-DIMENSIONAL CASE

The rate constituting the diffusion coefficient of a monovacancy is the jump rate of

one nearest neighbour of the vacancy over the barrier formed mainly by one gate of

four gate atoms in the £cc lattice and by two consecutive gates of three atoms each

in the bcc lattice. Various saddle-point configurations SPc are distinguished by c;

the equilibrium saddle point is SPVin. The infinitesimal potential at extreme points

is described by mass weighted normal co-ordinates

qA,

q and normal mode frequencies

1.L

wX,

W

where A

and

p

count modes in the ground state at the bottom of the well and

p o t e n t i a l l e a d i n g t o t h e damping parameters y X , yU /21/

and a corresponding e x p r e s s i o n f o r y The V p ) . and V;:). depend on t h e n-th d e r i - v a t i v e of t h e p o t e n t i a l about t h e wet; bottom and t h e b a r r i e r t o p , r e s p e c t i v e l y . For t h e s t a b l e modes t h e c o n t r i b u t i o n s t o t h e y a r e i n t e r p r e t e d a s n-phonon c o l l i s i o n s which f o r T > TE (= E i n s t e i n t e m p e r a t u r e ) a r e

-

T ( n - 2 ) . The y ~ a r e t h e phonon widths o r t h e i n v e r s e of t h e l i f e t i m e s T A . The e n t i r e t y of modes

X

o r r e p r e s e n t t h e h e a t b a t h coupled by t h e a n h a r m o n i c i t i e s of t h e p o t e n t i a l t o t h e modes involved immediate- l y i n t h e dynamics of t h e jump process. I n t h e p e r f e c t l a t t i c e , t h e o r y /22/ and ex- periment /23/ i n d i c a t e yX $ 0.5

wA

a t T = Tm and yX

-

0.05

w~

f o r T + 0 . The vacancy i n t r o d u c e s s t r o n g a d d i t i o n a l a n h a r m o n i c i t i e s , e s p e c i a l l y a t t h e b a r r i e r , where on symmetry grounds t h e even c o e f f i c i e n t s a r e e s s e n t i a l f o r t h e t r a n s m i s s i o n damping: yg z B , T ~

+

D,T'. The normalized b a r r i e r damping a c = yg/2wE depends t h u s on t h e c o n f i g u r a t i o n c and on T.

The TST r e s u l t f o r t h e vacancy h a s been given by Vineyard /5/

N N

rTST = VVin exp(-BUVin),

v

_Vin = a v / w v _{A=l X ~ = 2 p '}

Again n e i t h e r

w$

nor any damping parameter do appear i n t h i s e x p r e s s i o n .

The regime I of low damping y ~ <<

w~

where v i b r a t i o n a l energy t r a n s f e r i n t h e w e l l

i s

r a t e l i m i t i n g i s p o s s i b l y r e a l i z e d f o r T + 0. Assuming t h a t Nc modes

X

have t o be e x c i t e d t o reach SPc, t h e reduced r a t e i n Nc dimensions i s /24,25/

going t o t h e Kramers l i m i t f o r Nc = 1 . Since @Uc >> 1 a t low T , t h e r a t e i s dramati- c a l l y i n c r e a s e d with r e s p e c t t o t h e one-dimensional c a s e and t h e turnover between I and I1 /25/ should h a r d l y be observable i n s o l i d - s t a t e d i f f u s i o n ; t h e TST r a t e i s expected t o h o l d a t low T u n t i l a, s t a r t s t o i n c r e a s e .

Regime I1 where b a r r i e r damping reduces s u c c e s s f u l b a r r i e r c r o s s i n g s i s governed by a,. Expanding ( 3 ) we o b t a i n

which f o r f i x e d c i s an e x p o n e n t i a l whose p r e - f a c t o r d e c r e a s e s a s T-* o r T-h with i n c r e a s i n g temperature. The magnitude of t h e p r e - f a c t o r depends on t h e r a t i o of t h e harmonic t o t h e anharmonic p a r t s of t h e b a r r i e r and v a r i e s with c . Consider t h e bcc l a t t i c e w i t h i t s double-gate s t r u c t u r e of t h e b a r r i e r of t h e vacancy jump. I n t h e r e l a x e d s t a t e c = Vin it t e n d s t o be c l o s e t o a r e c t a n g u l a r b a r r i e r w i t h very small wb and l a r g e yb. T h i s case e x h i b i t s a s t r o n g r e d u c t i o n of rl= by damping due t o nu- merous r e c r o s s i n g s of t h e d i f f u s i n g p a r t i c l e b e f o r e it moves away from t h e b a r r i e r . For t h e unrelaxed o r p a r t i a l l y r e l a x e d b a r r i e r with Uc > UVin W:

i s

i n c r e a s e d and

C10-30

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DE

PHYSIQUE

t e d w i t h a n h a r m o n i c i t y and c u r v a t u r e o f t h e s a d d l e s u r f a c e .

ACKNOWLEDGEMEWS. The a u t h o r would l i k e t o t h a n k H. S c h u l t z f o r c o n t i n u o u s c r i t i c a l d i s c u s s i o n d u r i n g o n e y e a r . He a p p r e c i a t e s comments by D . L a z a r u s , C.P. F l y n n , G. DeLorenzi, A.V. G r a n a t o ; R. Benedek, J . N . Mundy, N.L. P e t e r s o n and R.W. S i e g e l ; H. G r a b e r t and U . Weiss. The work i s s u p p o r t e d by t h e B u n d e s m i n i s t e r f u r Forschung und Technologie.

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