THE KINK PICTURE OF DISLOCATION MOBILITY AND DISLOCATION-POINT-DEFECT INTERACTIONS

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AND DISLOCATION-POINT-DEFECT

INTERACTIONS

A. Seeger

To cite this version:

JOURNAL

DE

PHYSIQUE

CoZZoque

CS,

suppZ6ment au nolO, Tome

42,

octobre 1981

page

C5-201

THE KINK PICTURE OF DISLOCATION MOBILITY

AND

DISLOCATION-POINT-DEFECT INTERACTIONS

A. Seeger

Mm-PZanck-Institut fur MetaZlforschung, I n s t i t u t fur Physik,

m d Universittit

Stuttgart, I n s t i t u t fur theoretische und angewandte Physik, Postfach

800665,

0-7000

S t u t t g a r t 80, Germany

A b s t r a c t . - The paper g i v e s a coherent account of t h e t h e o r y of d i s l o c a t i o n m o b i l i t y i n terms of kink-pair formation and kink m i g r a t i o n . Three l e v e l s of d e s c r i p t i o n a r i s e i n a n a t u r a l way, namely those of t h e motion of s t r a i g h t d i s l o c a t i o n s , of k i n k - p a i r formation on d i s l o c a t i o n s , and of k i n k m i g r a t i o n . The i n t e r r e l a t i o n s h i p s between t h e s e l e v e l s ( " h i e r a r c h i e s " ) a r e d i s c u s s e d . The i n t e r a c t i o n w i t h phonons and f o r e i g n atoms ( " i m p u r i t i e s " ) i s t r e a t e d on t h e t h i r d l e v e l , t h a t of t h e kink m o b i l i t y .

Experimental i n f o r m a t i o n on t h e p r o p e r t i e s of kinks may be o b t a i n e d from measurements of d i s l o c a t i o n v e l o c i t i e s , of flow s t r e s s , and of i n t e r n a l f r i c t i o n and modulus e f f e c t . The comparison between t h e o r y and experiment i s i l l u s t r a t e d by examples from valence c r y s t a l s ( d i s l o c a t i o n v e l o c i t y i n Ge) and from body-centred c u b i c t r a n s i t i o n m e t a l s (y r e l a x a t i o n = k i n k - p a i r f o r - mation on screw d i s l o c a t i o n s ; dislocation-enhanced Snoek e f f e c t ; Snoek- K 6 s t e r r e l a x a t i o n i n Nb and Ta).

1 . I n t r o d u c t i o n and General Background.- The "discovery" of d i s l o c a t i o n s d u r i n g t h e p e r i o d 1929-34 provided t h e e x p l a n a t i o n f o r t h e o b s e r v a t i o n t h a t i n many c r y s t a l l i n e m a t e r i a l s t h e c r i t i c a l s h e a r s t r e s s

a.

f o r p l a s t i c deformation by g l i d e i s s e v e r a l o r d e r s of magnitude lower than t h e t h e o r e t i c a l s h e a r s t r e n g t h

oth

a s c a l c u l a t e d from t h e model of p e r f e c t c r y s t a l s . Already i n some of t h e e a r l y work i t was recognized t h a t t h e r e s i s t a n c e - t o - g l i d e due t o t h e d i s c r e t e n a t u r e of c r y s t a l s was n o t complete- l y removed by t h e movement of s t r a i g h t d i s l o c a t i o n s running p a r a l l e l t o one of t h e major c r y s t a l l o g r a p h i c d i r e c t i o n s ( t h i s was t h e model used throughout i n t h e e a r l y days of d i s l o c a t i o n t h e o r y ) . Based on what i s now c a l l e d t h e PRANDTL-DEHLINGER- FRENKEL-KONTOROVA model [1,6,7] an a t t e m p t t o e s t i m a t e t h e s t r e s s r e q u i r e d t o move a s t r a i g h t " d i s l o c a t i o n " i n a c r y s t a l i s a l r e a d y contained i n u.DEHLINGER'~ "Verhakungen" paper of 1929 [ l ] . (A b r i e f h i s t o r i c a l account of e a r l y d i s l o c a t i o n models and t h e i r r e l a t i o n s h i p s t o t h e problem of d i s s i p a t i o n of mechanical energy ( = i n t e r n a l f r i c t i o n ) i n c r y s t a l s has appeared elsewhere

[81.)

A t t h e s u g g e s t i o n of E.OROWAN, i n 1939 R.PEIERLS [g] performed t h e f i r s t de- t a i l e d c a l c u l a t i o n of t h e r e s o l v e d s h e a r s t r e s s r e q u i r e d t o move a s t r a i g h t d i s - l o c a t i o n through a c r y s t a l i n t h e absence of thermal f l u c t u a t i o n s . This s t r e s s i s now c a l l e d t h e P e i e r l s s t r e s s op. The l i n e energy Ed of a s t r a i g h t d i s l o c a t i o n on a given g l i d e p l a n e i s a minimum i n t h e s o - c a l l e d " P e i e r l s v a l l e y s " and a maximum on t h e " P e i e r l s h i l l s " (comp. F i g . 1 ) . The o r i g i n a l e s t i m a t e of P e i e r l s ( f o r a h i s t o r i - c a l d e s c r i p t i o n s e e [10]) was c o r r e c t e d and improved by v a r i o u s a u t h o r s ; f o r t h e methods used and t h e r e s u l t s o b t a i n e d t h e r e a d e r i s r e f e r r e d t o t h e review l i t e r a t u r e

[ ] l - 1 4 1 . The b e s t c a l c u l a t i o n s a v a i l a b l e i n d i c a t e t h a t i n many i n s t a n c e s t h e ob- served c r i t i c a l s h e a r s t r e s s e s , a t l e a s t a t n o t too low temperatures, a r e sub- s t a n t i a l l y lower than t h e c a l c u l a t e d P e i e r l s s t r e s s e s .

I n t h e post-war development of d i s l o c a t i o n t h e o r y i t became soon c l e a r t h a t d i s l o c a t i o n l i n e s must be considered f l e x i b l e . W.SHOCKLEY [ l 5 1 p o i n t e d o u t t h a t to- g e t h e r w i t h t h e e x i s t e n c e of t h e P e i e r l s p o t e n t i a l t h i s may l e a d t o t h e formation of

*,

i . e . , s h o r t segments of d i s l o c a t i o n l i n e s i n which t h e s e c r o s s over from one P e i e r l s v a l l e y t o a neighbouring one (comp. F i g . l ) , and e s t i m a t e d t h a t kinks may form i n thermal e q u i l i b r i u m . A.SEEGER [12,16] recognized t h a t the thermally a c t i v a t e d formation of p a i r s of k i n k s of o p p o s i t e s i g n ( o r i g i n a l name "double kinks", i n t h e f o l l o w i n g b r i e f l y c a l l e d kink p a i r s ) under t h e a c t i o n of an o s c i l l a t i n g a p p l i e d s t r e s s might l e a d t o a r e l a x a t i o n e f f e c t i n i n t e r n a l f r i c t i o n . He proposed t h a t t h e r e l a x a t i o n p r o c e s s observed by P.G.BORDON1 [17,18] i n v a r i o u s deformed f c c metals below room temperature might be due t o t h i s p r o c e s s . This s u g g e s t i o n was developed f u r t h e r i n numerous t h e o r e t i c a l and experimental papers ( s e e , e . g . , [19,20]) and appears now t o be g e n e r a l l y accepted a s t h e c o r r e c t e x p l a n a t i o n of t h e Bordoni r e l a x a t i o n . From a h i s t o r i c a l p o i n t of view i t i s i n t e r e s t i n g t o n o t e t h a t q u i t e a l a r g e number of a l t e r n a t i v e e x p l a n a t i o n s i n v o l v i n g d i s l o c a t i o n s have been proposed d u r i n g t h e l a s t twenty-five y e a r s ( s e e , e . g . , [20,211) but t h a t the f i r s t e x p l a n a t i o n i n terms of d i s L o c a t i o n t h e o r y proved t o be c o r r e c t (BORDONI himself [l81 suggested an e x p l a n a t i o n not i n v o l v i n g d i s l o c a t i o n s ) .

According t o the kink-pair formation model t h e Bordoni r e l a x a t i o n i s an i n t r i n s i c d i s l o c a t i o n p r o p e r t y d i r e c t l y r e l a t e d t o t h e e x i s t e n c e of t h e P e i e r l s p o t e n t i a l . I n most c r y s t a l s i t s t i l l p r o v i d e s t h e b e s t i n f o r m a t i o n f o r e s t i m a t i n g t h e numerical v a l u e of t h e P e i e r l s s t r e s s 0 o r t h e h e i g h t of t h e P e i e r l s b a r r i e r s of d i s l o c a t i o n s

P

l y i n g p a r a l l e l t o c r y s t a l l o g r a p h i c d i r e c t i o n s . Recently, t h e m o d i f i c a t i o n of t h e thermally a c t i v a t e d k i n k - p a i r formation p r o c e s s by atomic d e f e c t s ( i n p a r t i c u l a r by f o r e i g n i n t e r s t i t i a l atoms) has become the s u b j e c t of much i n t e r e s t , a s i s w i t - nessed by t h e p r e s e n t conference.

Peierls hill valley hill anchor point g 1 geometri- cal kink

I

anchoring point

kink double kink

pair

kink pair under applied stress G F i g . ] :

Top:

The l a t t i c e - p e r i o d i c P e i e r l s p o t e n t i a l U(u) of a s t r a i g h t d i s l o c a t i o n a s a f u n c t i o n of i t s displacement U. The mean value of U ( u ) i s t h e d i s - l o c a t i o n l i n e en- e r g y Ed. The d i s - placement uo of a s t r a i g h t d i s - l o c a t i o n under an a p p l i e d s t r e s s U < up i s i n d i c a t e d ( b = d i s l o c a t i o n s t r e n g t h = modu- l u s of Burgers v e c t o r ) . Bottom: Various d i s l o c a t i o n con- f i g u r a t i o n s on an g l i d e p l a n e w i t h P e i e r l s v a l l e y s ( f u l l l i n e s ) and P e i e r l s h i l l s (dashed l i n e s ) .

Table I. The t h r e e h i e r a r c h i e s of models and energy b a r r i e r s

v a l l e y (comp.Fig.1). They account f o r t h e f a c t t h a t d i s l o c a t i o n s approximately p a r a l l e l t o one of t h e major c r y s t a l l o g r a p h i c d i r e c t i o n s may c o n t r i b u t e t o t h e modu- l u s d e f e c t even a t temperatures w e l l below t h a t of t h e i r Bordoni r e l a x a t i o n .

T h e o r e t i c a l reasoninganalogous t o t h a t l e a d i n g t o t h e e x i s t e n c e of t h e P e i e r l s p o t e n t i a l of s t r a i g h t d i s l o c a t i o n s demonstrates t h a t t h e p o t e n t i a l energy of a kink i n a d i s l o c a t i o n must be a p e r i o d i c f u n c t i o n of t h e kink p o s i t i o n a l o n g t h e d i s - l o c a t i o n l i n e w i t h a p e r i o d e q u a l t o t h a t of t h e c r y s t a l l a t t i c e i n t h a t d i r e c t i o n . This p e r i o d i c p o t e n t i a l i s c a l l e d "kink p o t e n t i a l " o r , i n o r d e r t o emphasize t h e c l o s e analogy t o t h e P e i e r l s p o t e n t i a l of s t r a i g h t d i s l o c a t i o n l i n e s , " P e i e r l s po- t e n t i a l of t h e second kind".

We s e e t h a t a s i n t h e c l a s s i c a l system of t h e h i e r a r c h i e s of a n g e l s of Dionysius t h e Areopagite we have a d i v i s i o n of our models i n t o

three

o r d e r s , namely t h a t of p e r f e c t c r y s t a l s , t h a t of s t r a i g h t d i s l o c a t i o n l i n e s , and t h a t of

E.

To t h e s e t h r e e o r d e r s of models correspond t h r e e o r d e r s o f b a r r i e r s t o be overcome (by t h e motion of t h e d e f e c t s of t h e n e x t lower o r d e r ) , namely t h e t h e o r e t i c a l s h e a r s t r e n g t h of a p e r f e c t c r y s t a l

%,

t h e P e i e r l s p o t e n t i a l (of t h e f i r s t k i n d ) , and t h e kink p o t e n t i a l . These r e l a t i o n s h i p s a r e summarized i n Table I .

The s u b d i v i s i o n i n t o t h e models and t h e l a t t i c e - p e r i o d i c b a r r i e r s l i s t e d i n Table I has c o n s i d e r a b l e advantages f o r t h e d i s c u s s i o n of p h y s i c a l phenomena a s s o c i a t e d w i t h d i s l o c a t i o n s , s i n c e t h e i r b a s i c f e a t u r e s may o f t e n be understood by r e f e r r i n g t o one of t h e t h r e e h i e r a r c h i e s of Table I only. E.g., many b a s i c f e a t u r e s of t h e deformation of c r y s t a l s by g l i d e may be e x p l a i n e d i n terms of d i s l o c a t i o n s w i t h o u t paying a t t e n t i o n t o t h e e x i s t e n c e of t h e P e i e r l s b a r r i e r s . Most o b s e r v a t i o n s on t h e Bordoni r e l a x a t i o n a r e accounted f o r , a t l e a s t q u a l i t a t i v e l y , i n terms of k i n k - p a i r formation. I f t h e kink p o t e n t i a l i s l a r g e enough, g e o m e t r i c a l k i n k s may give r i s e t o a d i s t i n c t r e l a x a t i o n phenomenon a s s o c i a t e d with k i n k m i g r a t i o n [19,23-251.

Notwithstanding t h e impressive s u c c e s s e s of t h e s i m p l i s t i c approach j u s t out- l i n e d , a deeper understanding of t h e observed phenomenon can o f t e n only be achieved i f i t i s r e c a l l e d t h a t each of t h e t h r e e h i e r a r c h i e s o f Table I can a t b e s t give a p a r t i a l r e p r e s e n t a t i o n of a complex p h y s i c a l s i t u a t i o n .

The p r e s e n t paper aims a t developing, i n a coherent f a s h i o n , t h e theory of k i n k - p a i r formation and kink m i g r a t i o n , and a t t e m p t s t o apply i t to t h e i n t e r p r e - t a t i o n of measurement of d i s l o c a t i o n v e l o c i t i e s and of i n t e r n a l - f r i c t i o n phenomena. We s h a l l c a r e f u l l y d i s t i n g u i s h between phenomena t h a t may be understood be r e f e r e n c e t o only one of t h e models of Table 1 , w i t h t h e a s p e c t s r e p r e s e n t e d by t h e o t h e r models coming i n t o p l a y o n l y through phenomenological parameters, and those r e - q u i r i n g a more g e n e r a l ( " i n t e r - h i e r a r c h i c " ) approach.

mass p e r u n i t l e n g t h , yd. For a d d i t i o n a l s i m p l i c i t y we c o n s i d e r only de- v i a t i o n s from s t r a i g h t l i n e s , measured by t h e displacement U of t h e d i s l o c a t i o n

from a P e i e r l s v a l l e y . The motion of a d i s l o c a t i o n l i n e i s t h e n governed by t h e p a r t i a l d i f f e r e n t i a l e q u a t i o n

I n ( 1 ) z d e n o t e s t h e s p a t i a l c o o r d i n a t e i n t h e d i r e c t i o n of t h e P e i e r l s v a l l e y , t t h e time, b t h e d i s l o c a t i o n s t r e n g t h , 0 t h e a p p l i e d r e s o l v e d s h e a r s t r e s s , and U(u) the l i n e energy of a s t r a i g h t d i s l o c a t i o n l i n e a s a f u n c t i o n of i t s displacement. For t h e purpose of c a l c u l a t i n g t h e l i n e t e n s i o n Sd i t s u f f i c e s t o r e p l a c e U(u) by i t s average v a l u e Ed, t h e l i n e energy of t h e d i s l o c a t i o n s ( F i g . 1). We t h e n have L261

where 8 i s t h e a n g l e between t h e Burgers v e c t o r of t h e d i s l o c a t i o n l i n e and t h e d i r e c t i o n of t h e P e i e r l s v a l l e y s .

For s t a t i c s o l u t i o n s u ( z ) E q . ( l ) s i m p l i f i e s t o

Eq.(3) p o s s e s s e s t h e s o l u t i o n uo = c o n s t a n t corresponding t o a r i g i d displacement of a s t r a i g h t d i s l o c a t i o n l i n e , where

The maximum v a l u e of

a

a t which

( 4 )

s t i l l has a r e a l s o l u t i o n d e f i n e s t h e P e i e r l s s t r e s s Op (comp. F i g . l ) .

Among t h e s o l u t i o n s of (3) f o r

a

= 0 i s t h a t f o r a s i n g l e kink c e n t r e d a t

z = z I n i m p l i c i t form i t i s g i v e n by

0 '

U

where t h e s i g n s correspond t o p o s i t i v e o r n e g a t i v e k i n k s , r e s p e c t i v e l y . For "smooth" p o t e n t i a l s f o r which t h e maximum v a l u e Umax c o i n c i d e s w i t h U(a/2), where a i s t h e d i s t a n c e between neighbouring P e i e r l s v a l l e y s and t h e h e i g h t of a s i n g l e k i n k , we may d e f i n e t h e kink width a s

Within t h e p r e s e n t t h e o r e t i c a l framework ( b r i e f l y c a l l e d t h e " l i n e - t e n s i o n model") t h e energy of a s i n g l e k i n k i s given by

Under a n a p p l i e d s h e a r s t r e s s U(O<]Ul<a ) Eq.(3) p o s s e s s e s a s o l u t i o n which P

c o r r e s p o n d s t o a p a i r of k i n k s i n u n s t a b l e e q u i l i b r i u m . T h e f o r c e s e x e r t e d on t h e k i n k s by t h e a p p l i e d s t r e s s , f abo, tend t o move t h e k i n k s a p a r t . A c o n f i g u r a t i o n of u n s t a b l e equilibrium e x i s t s i n which t h e s e f o r c e s j u s t b a l a n c e t h e a t t r a c t i o n b e t - ween t h e k i n k s of o p p o s i t e s i g n s . The e n t h a l p y d i f f e r e n c e between a k i n k p a i r i n t h i s c o n f i g u r a t i o n and t h a t of a s t r a i g h t d i s l o c a t i o n l i n e a t u=uo c o n s t i t u t e s t h e s a d d l e - p o i n t e n t h a l p y f o r t h e f o r m a t i o n of a k i n k p a i r , HkD. Within t h e p r e s e n t framework w e have u=umax H = 2(2sd)112

1

kp [ u ( u ) - U(uo)

-

( U

-

U ~ ) ~ C J ] " ~ du

,

where t h e upper l i m i t of t h e i n t e g r a t i o n f o l l o w s from

U(umax)

-

U(uo)

-

(umax

-

uo)bb = 0

.

( 9 )

I n i n t e r n a l f r i c t i o n e x p e r i m e n t s we a r e p a r t i c u l a r l y i n t e r e s t e d i n low s t r e s s e s . For

5/op

<< l Eq. ( 7 ) may be w r i t t e n a s

c l U

H

= 2Hk { l

- -

kp [ l

-

~ n ( c ~ o / ~ ~ ) l +

.

.

.

.

. l

,

( 1 0 ) where c l and c 2 a r e n u m e r i c a l f a c t o r s t h a t depend on t h e c h o i c e of t h e p o t e n t i a l U(u)

.

For t h e s o - c a l l e d Eshelby p o t e n t i a l l ) [27]

t h e s e f a c t o r s a r e g i v e n by L281 C , = 2c2 = 3-'12/2

_,

t h e e n t h a l p y of f o r m a t i o n of a s i n g l e k i n k by -1 14 Hk = 3 a ( a b 0 ~ ~ ~ / 2 ) " ~

,

and t h e k i n k width by wk = 2a2 Sd/ 3Hk c o n t r a s t t o t h e s i n u s o i d a l p o t e n t i a l

E q . ( l ) may be used t o determine t h e v i b r a t i o n a l spectrum a s s o c i a t e d with a s i n g l e k i n k o r w i t h t h e saddle-point c o n f i g u r a t i o n of a kink p a i r and t o c a l c u l a t e the e n t r o p y of kink f o r m a t i o n . For d e t a i l s and r e s u l t s the r e a d e r i s r e f e r r e d t o the l i t e r a t u r e [13,221. The r e s u l t s f o r i s o l a t e d k i n k s w i l l be used l a t e r . From ( 1 ) we may f u r t h e r deduce t h a t t h e " v e l o c i t y of sound" of small perturbations p r o p a g a t i n g a l o n g a d i s l o c a t i o n l i n e i s

Since (1) i s L o r e n t z - i n v a r i a n t , we may d e f i n e t h e r e s t mass

%

of an i s o l a t e d kink according t o

%

= H ~ I C ; =

Y ~ H ~ / s ~

(17) The p r e c e d i n g t h e o r e t i c a l d e s c r i p t i o n corresponds t o l i n e I1 of Table I . We have d i s r e g a r d e d t h e a t o m i s t i c s t r u c t u r e of t h e c r y s t a l i n t h e d i r e c t i o n of t h e P e i e r l s v a l l e y s ( 1 i n e 111 of Table I ) . This has t h e consequence t h a t t h e energy of a k i n k i s independent of i t s p o s i t i o n z a l o n g t h e d i s l o c a t i o n . The f a c t t h a t k i n k s a r e s h o r t segments of an o t h e r w i s e s t r a i g h t d i s l o c a t i o n embedded i n a three-dimen- s i o n a l e l a s t i c continuum ( l i n e I of Table I ) i s taken i n t o account only through t h e d i s l o c a t i o n l i n e t e n s i o n Sd and t h e e f f e c t i v e mass ydper u n i t d i s l o c a t i o n l e n g t h . For some purposes, i n p a r t i c u l a r f o r t h e c a l c u l a t i o n of t h e kink--kink i n t e r a c t i o n a t l a r g e s e p a r a t i o n s , t h i s i s n o t s u f f i c i e n t , s i n c e i t does n o t adequately d e s c r i b e t h e e l a s t i c i n t e r a c t i o n between k i n k s .

I t i s e a s y t o show t h a t f o r l a r g e s e p a r a t i o n s t h e e l a s t i c i n t e r a c t i o n energy must vary a s t h e i n v e r s e d i s t a n c e between kinks on t h e same d i s l o c a t i o n [13,22,27, 29,301. For e l a s t i c a l l y i s o t r o p i c media with s h e a r modulus G and P o i s s o n ' s

r a t i o v t h e f o r c e between k i n k s of h e i g h t a a t s e p a r a t i o n s X >> a r e a d s [13,22,27, 29,301

=&

2

-[(l+v)cos2 a 2 0 + (l-2v)sin2 01 (18)

4~

1-v

2x2

where 8 denotes t h e a n g l e between t h e d i s l o c a t i o n l i n e and t h e C s i g n s r e f e r t o k i n k s of e q u a l o r o p p o s i t e s i g n s , r e s p e c t i v e l y . The g e n e r a l i z a t i o n of (12) t o a n i s o t r o p i c e l a s t i c i t y i s [31, 321 a2 F =

sel

( 18a) O 2 2 Here SE1 i s t h e p r e l o g a r i t h m i c f a c t o r of t h e e l a s t i c p a r t of t h e d i s l o c a t i o n l i n e - t e n s i o n . I t may be w r i t t e n i n t h e form

abcx

abcx

F i g . 2 : The t o t a l e n t h a l p y o f two k i n k s of o p p o s i t e s i g n ( f u l l l i n e ) and t h e i n t e r a c t i o n e n t h a l p y H i n t (X) a s a f u n c t i o n of t h e k i n k s e p a r a t i o n X. The en- t h a l p y of k i n k - p a i r f o r - mation Hkp i s i n d i c a t e d ; t h e k i n k s e p a r a t i o n X I c o r r e s p o n d s t o u n s t a b l e e q u i l i b r i u m .

0

l

l

XI kink

separatio"

X

i s g i v e n by -abos. The s u p e r p o s i t i o n of t h e s e two p o t e n t i a l e n e r g i e s ( f u l l l i n e ) l e a d s t o a maximum a t X

l

x 1 = (asE1/2b upr2 9 (20)

c o r r e s p o n d i n g t o a k i n k - p a i r f o r m a t i o n e n t h a l p y

Eq.(21) c o r r e s p o n d s t o t h e f i r s t term of a " m u l t i p o l e expansion". At s m a l l s e p a r a t i o n s and l a r g e

o

i t c e a s e s t o be v a l i d . Then t h e e l a s t i c i n t e r a c t i o n between t h e k i n k s depends n o t o n l y on a and b b u t i n a d d i t i o n on t h e shape o f t h e k i n k s ( c f . [221). I n th$ p r e s e n t c o n t e x t , however, i t i s n o t worthwhile t o work t h i s o u t i n d e t a i l s i n c e f o r sllall k i n k s e p a r a t i o n s t h e e l a s t i c i n t e r a c t i o n i s s m a l l compared w i t h t h a t g i v e n by Eq.(8) of t h e l i n e - t e n s i o n model.

The stress

3

which s e p a r a t e s t h e r a n g e of v a l i d i t y o f (8) and (21) may be found by e q u a t i n g (10) and ( 2 1 ) . The Eshelby p o t e n t i a l (12) g i v e s u s [28]

I f we choose, e . g . , sd/SZ1 = 4.1, Eq. ( 2 2 ) h a s t h e s o l u t i o n

;/c$

= 0.14, ln(3*4-+/;)

= 5. F i g . 3 shows t h e r e l a t i o n s h i p between

a/%

and H /2Hk f o r t h i s p a r t i c u l a r c a s e . kp

A d i f f e r e n t c h o i c e of sd/SZ1 o r U(u) d o e s n o t change t h e f u n c t i o n a l form of t h i s r e -

Fig.3: The r e l a t i o n s h i p between t h e r e s o l v e d s h e a r s t r e s s a ( i n u n i t s of t h e P e i e r l s s t r e s s op) and t h e en- t h a l p y of formation of a k i n k p a i r H ( i n u n i t s of t h e formation e n t h a l p y kp of a s i n g l e k i n k , Hk) a c c o r d i n g t o Eq.(lO) ( l i n e - t e n s i o n model) and Eq.

(21) (long-range e l a s t i c i n t e r a c t i o n ) . The s t r e s s

3

a t which t h e p r e d i c t e d r e l a t i o n s h i p ( f u l l l i n e ) changes over 0 1 from Eq.(lO) t o Eq.(21) i s i n d i c a t e d .

'-',p"%

where t h e c h o i c e of U(u) a f f e c t s only t h e numerical v a l u e of c 3 ( f o r (12): 2)

c 3 = ( 6 / 5 ) (2/3) 5j

'

= 0.73).

For small kink s e p a r a t i o n s i t i s n o t p o s s i b l e t o d e f i n e t h e kink s e p a r a t i o n i n a unique manner. I n F i g . 2 t h i s i s i n d i c a t e d by t h e dashed curve. F o r t u n a t e l y , t h e p h y s i c a l conclusions t o be drawn from t h e H (0) r e l a t i o n s h i p a r e u n a f f e c t e d by

kp t h i s u n c e r t a i n t y .

2. The Kink-Pair Formation Rate.- The c a l c u l a t i o n of t h e r a t e of formation of kink p a i r s on a n o t h e r w i s e s t r a i g h t d i s l o c a t i o n l i n e of u n i t l e n g t h ,

F, i n t h e f o l l o w i n g

b r i e f l y c a l l e d the k i n k - p a i r formation r a t e , i s b e s e t by t h e problems of r a t e theory i n g e n e r a l . A t h e o r e t i c a l framework capable of g i v i n g p r a c t i c a l r e s u l t s and being a p p l i c a b l e under a l l p h y s i c a l c o n d i t i o n s does n o t e x i s t . The two most s u c c e s s f u l approaches a r e ( i ) a n a d a p t i o n of t h e t r a n s i t i o n s t a t e t h e o r y , o r i g i n a l l y developed f o r t h e c a l c u l a t i o n of chemical r e a c t i o n r a t e s and going back t o t h e work of H.PELZER and E.WIGNER [34], and ( i i ) t h e d i f f u s i o n theory based on t h e work of H.A.KRAMERS [ 3 5 ] . For a g e n e r a l d i s c u s s i o n of t h e theory of k i n k - p a i r formation r a t e s t h e r e a d e r i s r e f e r r e d t o A.SEEGER and P.SCHILLER [ 1 3 ] .

The common i d e a behind both approaches i s t h a t t h e degrees of freedom of t h e e n t i r e c r y s t a l a r e t r e a t e d by t h e u s u a l s t a t i s t i c a l mechanics of harmonic o s c i l l a t o r s with two exceptions: The mode d e s c r i b i n g t k e change i n t h e kink-kink s e p a r a t i o n X

and t h e mode a s s o c i a t e d w i t h the t r a n s l a t i o n of a kink p a i r of f i x e d s e p a r a t i o n X a l o n g t h e d i s l o c a t i o n l i n e . By well-known arguments t h e maximum i n t h e energy- d i s t a n c e diagram of t h e f i r s t of t h e s e modes ( F i g . 2 ) corresponds t o t h e saddle-point i n t h e many-dimensional c o n f i g u r a t i o n a l space s e p a r a t i n g a k i n k - f r e e d i s l o c a t i o n from t h e r e g i o n c o n t a i n i n g a p a i r of k i n k s of o p p o s i t e s i g n . The two t h e o r e t i c a l approaches mentioned above d i f f e r i n t h e assumptions made concerning t h e way i n which t h e system c r o s s e s t h e above-mentioned s a d d l e - p o i n t .

( i )

The t r a n s i t i o n s t a t e t h e o r y makes t h e assumption t h a t once t h e system h a s reached t h e s a d d l e p o i n t w i t h a v e l o c i t y p o i n t i n g from t h e "no-kink" r e g i o n t o t h e "kink-pair" r e g i o n , k i n k - p a i r f o r m a t i o n t a k e s p l a c e . The p r o b a b i l i t y t h a t t h e system "back-crosses" t h e s a d d l e p o i n t i s n e g l e c t e d . The k i n k - p a i r f o r m a t i o n r a t e i s t h e n g i v e n exp [-H (O)/kT] h2 kP d e n o t e t h e energy e i g e n - v a l u e s of t h e harmonic o s c i l l a t o r s of c i r c u l a r f r e q u e n c y

mm

a s s o c i a t e d w i t h a n unkinked d i s l o c a t i o n , those a s s o c i a t e d w i t h d i s l o c a t i o n s c o n t a i n i n g a k i n k p a i r i n t h e s a d d l e - p o i n t con- f i g u r a t i o n (3'1 = P l a n c k ' s c o n s t a n t h d i v i d e d by 2 n ) . As a compensation f o r t h e two n o n - v i b r a t i o n a l modes mentioned above, the sum ( o r p r o d u c t ) o v e r m c o n t a i n s two more terms t h a n t h a t o v e r m".

f o r t h e s i n u s o i d a l p o t e n t i a l ( I I ) , and t o

f o r t h e E s h e l b y p o t e n t i a l ( 1 2 ) . I n (28b) ad d e n o t e s t h e i n t e r a t o m i c d i s t a n c e a l o n g t h e d i s l o c a t i o n l i n e .

I n s e r t i o n of (27,28) i n t o (24) g i v e s us f o r t h e k i n k - p a i r f o r m a t i o n r a t e i n t h e t r a n s i t i o n s t a t e t h e o r y

Here Hint r e f e r s t o t h e s a d d l e - p o i n t s e p a r a t i o n o f t h e k i n k s , (29a) t o t h e s i n u s - o i d a l p o t e n t i a l , (29b) t o t h e E s h e l b y p o t e n t i a l .

( i i ) The s t a r t i n g e q u a t i o n of t h e d i f f u s i o n t h e o r y i s a p a r t i a l d i f f e r e n t i a l e q u a t i o n f o r t h e p a r t i c l e d e n s i t y p ( p x , x , t ) i n t h e phase s p a c e o f t h e s p a t i a l co- o r d i n a t e x d e s c r i b i n g t h e k i n k - k i n k s e p a r a t i o n and t h e momentum px a s s o c i a t e d w i t h i t [35]. The t h e o r y i s based o n t h e p h y s i c a l pictume t h a t , i n a d d i t i o n t o t h e f o r c e s g i v i n g r i s e t o H i n t ( x ) ( F i g . 2 ) , t h e k i n k a r e s u b j e c t t o Brownian f o r c e s due t o t h e i r i n t e r a c t i o n w i t h l a t t i c e v i b r a t i o n s . The e f f e c t s o f t h e Brownian f o r c e s may be des- c r i b e d i n terms of a k i n k m o b i l i t y pk o r a k i n k v i s c o s i t y

]/vk.

By t h e ~ i n s t e i r r N e r n s t r e l a t i o n s h i p t h e k i n k m o b i l i t y may be e x p r e s s e d i n t e r m s of t h e k i n k d i f f u - s i v i t y

D k = k T v k

.

(30)

There a r e two l i ~ i t i n g c a s e s i n which t h e number of i n d e p e n d e n t v a r i a b l e s i n KRAML?RS7 b a s i c p a r t i a l d i f f e r e n t i a l e q u a t i o n may be reduced by one.

( a ) I f t h e k i n k m o b i l i t y i s s o h i g h t h a t t h e o s c i l l a t i o n s of t h e k i n k s i n t h e a t t r a c t i v e p o t e n t i a l formed by t h e i n t e r a c t i o n of k i n k s of o p p o s i t e s i g n a r e o n l y l i t t l e i n f l u e n c e d by t h e Brownian f o r c e s , o r px and X may be r e p l a c e d by t h e a c t i o n v a r i a b l e

I ( E ) = pXdx

.

(31)

E.MANN [391 h a s r e c e n t l y shown t h a t under t h e c o n d i t i o n H >> kT t h e kink- kp p a i r f o r m a t i o n r a t e f o r h i g h k i n k m o b i l i t i e s i s g i v e n by 8Hkwk

r

=- ( S ) ' " (p:q)? exp ( l l i n t / k ~ ) , Dk"k

Here and l a t e r i n ( 3 4 ) t h e same a s s u m p t i o n s f o r t h e c a l c u l a t i o n o f t h e e n t r o p y terms have been made a s i n t h e t r e a t m e n t ( i ) .

F o r t e m p e r a t u r e - i n d e p e n d e n t k i n k d i f f u s i v i t i e s Dk t h e t e m p e r a t u r e dependence o f (29) and ( 3 2 ) a r e t h e same. The r e s u l t s o f t h e t r a n s i t i o n s t a t e t h e o r y and t h e h i g h - m o b i l i t y d i f f u s i o n t h e o r y a g r e e w i t h e a c h o t h e r i f we make t h e i d e n t i f i c a t i o n Dk = ~ w ~ ( B ~ / % ) ' : ' = 8wkco ( 3 3 ) ( b ) I f t h e e f f e c t o f t h e Brownian f o r c e s on t h e k i n k v e l o c i t y e x c e e d s t h o s e of t h e " e x t e r n a l " f o r c e s ( i . e . , i f t h e k i n k m o b i l i t y i s s m a l l ) ,

Kramers'

e q u a t i o n s i m p l i f i e s t o a so-calledSmolukowskiequation ( d i f f u s i o n e q u a t i o n w i t h d r i f t t e r m ) w i t h t h e s p a t i a l c o o r d i n a t e X a s a n i n d e p e n d e n t v a r i a b l e . F o r t h e p h y s i c a l s i t u a t i o n s t r e a t e d i n t h i s p a p e r , t h i s i s t h e most i m p o r t a n t c a s e . I t h a s been d i s c u s s e d i n de- t a i l by SEEGER and SCHILLER [13], STENZEL [40], ENGELKE [441, and MANN [391. I f t h e a s s u m p t i o n H >> kT i s made t h e k i n k - p a i r f o r m a t i o n r a t e may be e x p r e s s e d i n terms

kp

o f q u a d r a t u r e s . The f u r t h e r a s s u m p t i o n t h a t t h e s a d d l e p o i n t o c c u r s i n t h e regime o f v a l i d i t y o f ( 2 1 ) l e a d s t o t h e c l o s e d - f o r m e x p r e s s i o n 1391

where

and Kn(y) d e n o t e s t h e MacDonald ( m o d i f i e d Hankel) f u n c t i o n o f o r d e r n . I m p o r t a n t s p e c i a l c a s e s a r e y >> 1, KI ( y ) " ( ~ r / 2 ~ ) ~ ' ~ e x ~ ( - ~ ) , l e a d i n g t o [13,401

and y << 1, K 1 ( y ) y-l, l e a d i n g t o [42, 431

-'3/ 2

We n o t e t h a t i n ( 3 6 ) t h e p r e - e x p o n e n t i a l f a c t o r depends on t e m p e r a t u r e a s (kT)

,

whereas i n ( 3 7 ) i t d e p e n d s on T a s (kT)-'.

d i s t a n c e by which the two members of a kink p a i r s e p a r a t e b e f o r e they c e a s e t o move. This may e i t h e r be t h e case when they meet o b s t a c l e s a l o n g t h e d i s l o c a t i o n l i n e s t h a t a r e impenetrable t o k i n k s , o r when t h e y a n n i h i l a t e with o t h e r kinks on t h e same d i s - l o c a t i o n l i n e . The d i s l o c a t i o n v e l o c i t y w i l l be determined by t h e s h o r t e r of t h e two d i s t a n c e s , t h e s e p a r a t i o n L of t h e o b s t a c l e s o r t h e s e p a r a t i o n

$

of t h e two k i n k s i n

a

p a i r b e f o r e t h e i r a n n i h i l a t i o n w i t h o t h e r k i n k s , s o t h a t we e x p e c t

X L

The s e p a r a t i o n

%

e q u a l s t h e d i s t a n c e t h a t two kinks of o p p o s i t e s i g n d r i f t towards each o t h e r d u r i n g t h e mean l i f e t i m e tk of t h e k i n k s . I t i s t h u s given by 3)

xk = 2uk a b U tk (39)

I n a s t a t i o n a r y s i t u a t i o n t h e mean l i f e t i m e tk of t h e kinks must be e q u a l t o t h e average time f o r t h e n u c l e a t i o n of a f r e s h kink p a i r on a d i s l o c a t i o n segment of Length

%,

i . e . 4) Hence t h e f i n a l r e s u l t f o r xk r e a d s I n s e r t i o n of (41) i n t o ( 3 8 ) g i v e s us w i t h t h e l i m i t i n g c a s e s (xk << L) vd = a(2pkab0r) and

(%

>> L ) v d = a L T

I n m a t e r i a l s on which measurements of d i s l o c a t i o n v e l o c i t i e s under t h e a c t i o n of a known r e s o l v e d a p p l i e d s h e a r s t r e s s U a r e a v a i l a b l e ( e . g . , Ge and S i , s e e Sect.5) Eq.(42) may be compared d i r e c t l y w i t h experiment.

3 ) ~ o r s i m p l i c i t y we d i s r e g a r d the f a c t t h a t d i s l o c a t i o n s anchored i n d i f f e r e n t P e i e r l s v a l l e y s c o n t a i n a l s o g e o m e t r i c a l k i n k s . Geometrical kinks w i l l be t r e a t e d s e p a r a t e l y a t t h e end of S e c t . 4.

I n i n t e r n a l f r i c t i o n e x p e r i m e n t s t h e a c t i n g s t r e s s i s o f t e n s o low t h a t t h e e q u i l i b r i u m d e n s i t y of t h e k i n k s i s m a i n t a i n e d . We t h e n have

~ ~ ~ ~( 4 3 ) i n t o ( 4 1 ) ~ i v e s ~ t i ~ ~ us ( 3 7 ) , a s we e x p e c t from t h e assumptionsmade. o f 4 . ) R e l a x a t i o n S t r e n g t h and R e l a x a t i o n Time.- The c a l c u l a t i o n o f t h e i n t e r n a l f r i c - t i o n and t h e modulus d e f e c t on t h e b a s i s o f t h e model d e v e l o p e d s o f a r i s a formi- d a b l e t a s k . Under c e r t a i n s i m p l i f y i n g a s s u m p t i o n a s o l u t i o n was g i v e n by H.ENGELKE

[41,44,45] and reviewed e l s e w h e r e P92. F o r t h e p r e s e n t p u r p o s e s i t s u f f i c e s t o de- r i v e some s i m p l e e s t i m a t e s c a p a b l e of b r i n g i n g o u t t h e dependence of t h e r e l a x a t i o n s t r e n g t h and r e l a x a t i o n t i m e on t e m p e r a t u r e T and on t h e o b s t a c l e s e p a r a t i o n L.

The r e l a x a t i o n s t r e n g t h i s p r o p o r t i o n a l t o t h e a r e a A swept o u t by t h e d i s - l o c a t i o n segments. F o l l o w i n g A.SEEGER, H.DONTH and F.PFAFF [46] t h e e s t i m a t e (which i n f a c t c o n s t i t u t e s a n upper l i m i t f o r t h e a r e a t h a t a d i s l o c a t i o n segment w i t h l i n e t e n s i o n Sd may sweep o u t under t h e a p p l i e d s h e a r s t r e s s )

may be o b t a i n e d . I f

A

d e n o t e s t h e t o t a l l e n g t h p e r u n i t volume of t h o s e d i s l o c a t i o n s t h a t p a r t i c i p a t e i n t h e r e l a x a t i o n p r o c e s s and i f ?I d e n o t e s t h e e l a s t i c modulus a p p r o p r i a t e f o r t h e mode of d e f o r m a t i o n employed, we have f o r t h e r e l a x a t i o n s t r e n g t h

A =

A2

b2 ~ 1 1 2 5 ~ ( 4 5 ) We e s t i m a t e t h e r e l a x a t i o n time T by c a l c u l a t i n g t h e time r e q u i r e d by a d i s l o c a t i o n moving w i t h t h e v e l o c i t y v _d ( L ' ) and sweeping o u t t h e a r e a A i n t h e manner i n d i c a t e d i n F i g . 4 . T h i s g i v e s u s w i t h t h e h e l p of ( 3 8 )

---

_7---

anchortng anchor~ng

polnt point

Among t h e i n t e r e s t i n g f e a t u r e s of (45) i s t h a t t h e r e l a x a t i o n s t r e n g t h i s pre- d i c t e d t o be independent of temperature and ( f o r a g i v e n t o t a l Length of d i s l o c a t i o n s ) p r o p o r t i o n a l t o L2. Eq. (46) e x h i b i t s a t r a n s i t i o n from a l i n e a r dependence of t h e r e l a x a t i o n time f o r small L t o a q u a d r a t i c one f o r l a r g e L . This t r a n s i t i o n i s accom- panied by a change i n t h e temperature dependence of -c.

I n a c r y s t a l t h e r e w i l l be i n g e n e r a l a f a i r l y wide d i s t r i b u t i o n of L v a l u e s . Following G.SCHOECK 1471 we may e s t i m a t e t h e i n t e r n a l f r i c t i o n Q-' and t h e modulus d e f e c t by making t h e assumption (which i s n o t q u i t e c o r r e c t ) t h a t f o r a f i x e d l e n g t h L we have a Debye r e l a x a t i o n p r o c e s s and i n t e g r a t i n g over t h e L - d i s t r i b u t i o n . We de- noteby p ( L ) ~ L t h e number of d i s l o c a t i o n segments p e r u n i t volume w i t h o b s t a c l e

L

s e p a r a t i o n between L and L

+

dL c o n t r i b u t i n g t o t h e r e l a x a t i o n p r o c e s s , so t h a t t h e d i s l o c a t i o n l e n g t h p e r u n i t volume i s

n

=

j

pL(L)LdL

.

0

The i n t e r n a l f r i c t i o n measured a t a c i r c u l a r frequency W i s then given by

and t h e corresponding e l a s t i c modulus by

where T(L) i s t o be taken from (46) and where

MU

denotes t h e unrelaxed e l a s t i c modu- l u s . The i n t e g r a l _m

which a r i s e s i f pL(L) CC e x p ( - ~ / i ) and T a L2, has been e v a l u a t e d n u m r i c a l l y [47].

-

2

I t g i v e s a maximum v a l u e I = 2.2 a t (W?) = 7

.

10

,

where

5

=

;

(L).

max

A s remarked above, i n i n t e r n a l f r i c t i o n experiments we o f t e n encounter s i t u a - t i o n s where t h e s i m p l i f i c a t i o n s (37) and (43) a r e a p p l i c a b l e . Then (46) t a k e s t h e form

Eq.(50) shows t h a t a s l i m i t i n g c a s e s t h e r e l a x a t i o n time T may e x h i b i t two d i f f e r e n t dependences on temperature and o b s t a c l e d i s t a n c e , namely i n t h e c a s e

p E q ~

<<

1

( i . e . , low t e m p e r a t u r e s )

The a p p a r e n t a c t i v a t i o n e n t h a l p i e s

a r e g i v e n by

d l n T

H e f f

"

d ( l/kT)

i n the case of p E q ~ c < 1 , and by

dlnDk

Heff = Hk

-

-

- -

d ( l / k T ) 2 kT i n the case

p E q ~ > >

1 .

I n t h o s e c a s e s i n which Hk i s n o t small compared w i t h t h e o t h e r terms, (51a) and (52a) show t h e remarkable f e a t u r e t h a t t h e a c t i v a t i o n e n t h a l p y i s h i g h e r a t lower t e m p e r a t u r e s . The r e a s o n f o r t h i s i s t h a t a t low temperature t h e d i s t a n c e a newly gene- r a t e d kink has t o t r a v e l u n t i l i t i s stopped a t an o b s t a c l e i s independent of tempera- t u r e , whereas a t high temperatures t h e kink may recombine w i t h kinks of o p p o s i t e s i g n , t h e d e n s i t y of which i n c r e a s e s w i t h i n c r e a s i n g temperature.

Another important s p e c i a l case i s t h a t i n which k i n k - p a i r formation may be d i s - regarded a l t o g e t h e r ( e . g . , because of low t e m p e r a t u r e s ) and we have t o c o n s i d e r only a temperature-independent d e n s i t y pk of geometrical k i n k s . This case h a s been t r e a t e d by A.D.BRAILSFORD [ 2 3 ] , C.kTiiTHRICH [25], and A.SEEGER and C.WUTHRICH [ 4 8 ] . I f t h e s e r e s u l t s a r e expressed i n terms of t h e kink d i f f u s i v i t y , an approximate e x p r e s s i o n f o r t h e r e l a x a t i o n time i s

where L denotes a g a i n t h e d i s t a n c e between t h e o b s t a c l e s . The r e l a x a t i o n s t r e n g t h i s given by

8 ~ ~ b ~

A =--F

-

a 2 ~ 2 p k ~ c o s 2 @

.

(55) kT

I n (55) $ denotes t h e a n g l e between t h e d i r e c t i o n of the P e i e r l s v a l l e y and t h e aver- age d i r e c t i o n of t h e d i s l o c a t i o n l i n e , which i s determined by t h e ~ o s i t i o n s of the ob- s t a c l e s .

An L d i s t r i b u t i o n may be handled a s above and l e a d s t o t h e i n t e g r a l ( 4 9 ) . I n a d d i t i o n one h a s t o average o v e r t h e a n g l e @, t a k i n g i n t o account t h a t t h e d e n s i t y of g e o m e t r i c a l k i n k s i s g i v e n by

pk = s i n $ / a ( 5 6 )

of k i n k s w i t h f o r e i g n atoms o r l a t t i c e v i b r a t i o n s . Such i n t e r a c t i o n s c a n o n l y be un- d e r s t o o d q u a n t i t a t i v e l y i f we t a k e i n t o a c c o u n t t h a t a k i n k i s a p e r t u r b a t i o n on a n o t h e r w i s e s t r a i g h t d i s l o c a t i o n l i n e t h a t i s endowed w i t h a long-range s t r a i n f i e l d d e c r e a s i n g w i t h t h e i n v e r s e second power of t h e d i s t a n c e from t h e k i n k . T h i s means t h a t we have t o go back t o l e v e l I of Table I .

An e x h a u s t i v e t h e o r e t i c a l t r e a t m e n t of D would be beyond t h e s c o p e of t h e p r e - k

s e n t p a p e r . We s h a l l c o v e r o n l y some examples o f ( i ) p u r e c r y s t a l s and ( i i ) i n t e r - a c t i o n s w i t h f o r e i g n i n t e r s t i t i a l atoms.

( i ) P u r e C r y s t a l s . - I n p u r e c r y s t a l s t h e d i f f u s i v i t y Dk ( o r m o b i l i t y

uk

= Dk/kT) of k i n k s on d i s l o c a t i o n l i n e s w i t h a l a r g e k i n k p o t e n t i a l i s d e t e r m i n e d by t h e r a t e by which k i n k s w i l l overcome t h e k i n k p o t e n t i a l . A complete t h e o r e t i c a l t r e a t m e n t of t h i s r a t e h a s n o t y e t been g i v e n , b u t we have good r e a s o n s t o b e l i e v e t h a t t h e ana- l o g y w i t h t h e jumps o f a t o m i c d e f e c t s i n c r y s t a l s i s q u i t e c l o s e . I n t h e s p i r i t o f t h e t r a n s i t i o n s t a t e t h e o r y we may i n t r o d u c e t h e f r e e e n t h a l p y of k i n k m i g r a t i o n

a s t h e d i f f e r e n c e between t h e f r e e e n t h a l p i e s when t h e k i n k s i s i n t h e s a d d l e - p o i n t c o n f i g u r a t i o n and when i t i s i n t h e p o s i t i o n o f s t a b l e e q u i l i b r i u m ( e x c l u d i n g t h e de- g r e e o f freedoms l e a d i n g o v e r t h e s a d d l e - p o i n t s i n t h e s e c o n d c a s e ) . I f V: d e n o t e s t h e a t t e m p t f r e q u e n c y w i t h which t h e k i n k v i b r a t e s i n i t s s t a b l e p o s i t i o n towards t h e s a d d l e - p o i n t and ad t h e p e r i o d o f t h e k i n k p o t e n t i a l , t h e t r a n s i t i o n s t a t e t h e o r y g i v e s u s u n d e r t h e a s s u m p t i o n >> kT

k

A s i m i l a r e x p r e s s i o n w i t h t h e same t e m p e r a t u r e dependence may b e o b t a i n e d from t h e

As p o i n t e d o u t by A.SEEGER and B . ~ E S T A K [24] t h e r e i s one e x c e p t i o n a l c a s e where one m i g h t e x p e c t , f o r t h e o r e t i c a l r e a s o n s , a f a i r l y h i g h k i n k m i g r a t i o n e n t h a l p y M . Hk 1 n m e t a l s . T h i s e x c e p t i o n c o n c e r n s k i n k s i n s c r e w d i s l o c a t i o n s w i t h B u r g e r s v e c t o r s b = a <l 11 >/2 i n b c c m e t a l s . P .B.HIRSCH [ 5 0 1 r e c o g n i z e d t h e p o s s i b i l i t y t h a t b e c a u s e o f t h e t h r e e f o l d symmetry o f < l 11> a x e s i n c u b i c c r y s t a l s t h e s e d i s l o ~ a t i o ~ might n o t have a w e l l - d e f i n e d g l i d e p l a n e a n d t h a t t h e y may hence b e s e s s i l e . F o r t h e pur- p o s e o f t h e p r e s e n t d i s c u q s i o n t h i s i s e q u i v a l e n t t o a h i g h P e i e r l s s t r e s s

ap.

With c e r t a i n m o d i f i c a t i o n s ( s e e , e . g . , [481) t h e v i e w - p o i n t o f HIRSCH h a s p r o v e d c o r r e c t . Computer s i m u l a t i o n s t u d i e s by CH.WuTHRICH l511 on a n a t o m i s t i c model f o r &Fe s u p p o r t t h e s u g g e s t i o n o f SEEGER and SESTAK 1241 on t h e k i n k m i g r a t i o n e n t h a l p y i n a o < 1 1 1 > / 2 s c r e w d i s l o c a t i o n s i n g e n e r a l t e r m s w i t h o u t b e i n g a b l e t o make q u a n t i t a t i v e p r e d i c - t i o n s .

I n t h e c a s e o f s c r e w d i s l o c a t i o n s i n b c c m e t a l s t h e h i g h P e i e r l s b a r r i e r i s due t o s p e c i a l c i r c u m s t a n c e s r e l a t e d t o symmetry and n o t t o c h e m i c a l bonding. (Non-screw d i s l o c a t i o n s i n bcc m e t a l s o r s c r e w d i s l o c a t i o n s w i t h o t h e r B u r g e r s v e c t o r s p o s s e s s "normal" P e i e r l s s t r e s s e s . ) The s i t u a t i o n i s d i f f e r e n t i n c r y s t a l s w i t h d i r e c t i o n a l bonds, i . e . , v a l e n c e c r y s t a l s s u c h a s germanium and s i l i c o n .

T h e r e i s c o n s i d e r a b l e e x p e r i m e n t a l e v i d e n c e , s t a r t i n g w i t h t h e work o f DASH[52] o n d i s l o c a t i o n l i n e s i n S i d e c o r a t e d w i t h Cu and l a t e r s u b s t a n t i a t e d by t r a n s m i s s i o n e l e c t r o n microscopy t h a t d i s l o c a t i o n s i n Ge and S i l y i n g a l o n g <110;. d i r e c t i o n ~ p o s s e s s h i g h P e i e r l s b a r r i e r s [531. The o r i g i n o f t h i s i s t h o u g h t t o l i e i n t h e c o v a l e n t na- t u r e o f b o n d i n g i n Ge a n d S i [54,55]. It is t h e r e f o r e r e a s o n a b l e t o assume t h a t Ge and S i p o s s e s s n o t o n l y h i g h k i n k f o r m a t i o n e n e r g i e s b u t t h a t t h e r e may a l s o be a sub- s t a n t i a l e n e r g y b a r r i e r f o r t h e m i g r a t i o n o f k i n k s a l o n g d i s l o c a t i o n l i n e s . M I f t h e k i n k m i g r a t i o n e n t h a l p y Hk i s l a r g e enough f o r t h e k i n k m i g r a t i o n b a r r i e r t o show up e x p e r i m e n t a l l y , t h e k i n k d i f f u s i v i t y (58) w i l l i n g e n e r a l be s m a l l enough f o r t h e d i f f u s i o n t h e o r y o f k i n k - p a i r f o r m a t i o n t o be a p p l i c a b l e . I n S i and Ge t h e k i n k f o r m a t i o n e n e r g y i s s o h i g h t h a t a t t h e t e m p e r a t u r e s a t which d i s l o c a t i o n v e l o - c i t i e s a r e measured t h e c o n d i t i o n

i s u s u a l l y f ~ l f i l l e d . ~ ) T h i s means t h a t a t s u f f i c i e n t l y low s t r e s s e s ( 3 7 ) and (42b)

hold, l e a d i n g t o

rd

= 2 Dk a2 b L

(pzqf

a / k T (60) The experimental r e s u l t s r e p o r t e d on Ge and S i approach indeed a l i n e a r dependence of t h e d i s l o c a t i o n v e l o c i t i e s on s t r e s s a t l o w s t r e s s e s and high temperatures. From

(60) follows t h e e f f e c t i v e a c t i v a t i o n e n t h a l p y of d i s l o c a t i o n motion

A t s u f f i c i e n t l y h i g h s t r e s s e s t h e r a t e of kink-pair formation may become s o l a r g e t h a t t h e k i n k l i f e t i m e i s l i m i t e d by t h e a n n i h i l a t i o n of k i n k s of o p p o s i t e s i g n . We t h e n have t o use (42a). Together w i t h (34) and (35) t h i s g i v e s us

-

11 2

vd = 2 s b a u k P? [Y K ~ ( Y ) I (62) I f y > 0 t h e s t r e s s dependence p r e d i c t e d by (62) i s s t r o n g e r than l i n e a r . The e f f e c - t i v e a c t i v a t i o n energy i s given by

For y >> 1 Eq.(63) becomes

M

Solving (61) and (64) f o r

%

and Hk g i v e s us

For a given type of d i s l o c a t i o n

se'

may be c a l c u l a t e d from t h e e l a s t i c c o n s t a n t s , s o ( 1 )

t h a t from measurements of Heff an: ;::H we may determine both t h e formation e n t h a l p y Hk and t h e m i g r a t i o n e n t h a l p y H: of k i n k s . For 60°-dislocations on < I l I > g l i d e p l a n e s i n germanium we have

sel

= 6 ' 10-l0 N (A.KORNER, H.O.K.KIRCHNER, U n i v e r s i t l t Wien, p e r s o n a l communication). From SCHAUMBURG1s L561 o b s e r v a t i o n s on such d i s l o c a t i o n s

[H!

:: = (3.00 k 0 . 2 0 ) e ~ ~ ) , ::H: = (1.55 i O.OS)eY] we f i n d f o r Ce Hk=(1.39 i O.25)eY and

Hf:

= (0.33 i 0.3)eV.

M

I n c r y s t a l s i n which Hk i s n e g l i g i b l y small t h e kink d i f f u s i v i t y must be e s t i - mated by c o n s i d e r i n g t h e i n t e r a c t i o n between k i n k s and phonons e x p l i c i t l y [13,57,58].

A t low temperatures one e x p e c t s pk t o decrease w i t h i n c r e a s i n g temperature roughly

-

1

a s Eph

,

where E denotes t h e phonon energy d e n s i t y . ph

A t s u f f i c i e n t l y high temperatures t h e t h e o r y p r e d i c t s T - l , i . e . , a tempera- ture-independent kink d i f f u s i v i t y Dk. The same temperature dependence f o l l o w s from (58) f o r kT >> H:

.

C a l c u l a t i o n s of t h e a b s o l u t e magnitude of

vk

a r e q u i t e d i f f i c u l t . The e s t i m a t e s of SEEGER and ENGELKE [ 5 7 ] i n d i c a t e t h a t the c o n d i t i o n s f o r t h e a p p l i c a b i l i t y of the d i f f u s i o n t h e o r y of k i n k - p a i r formation ( s e e S e c t . 2) a r e s a t i s f i e d below about one f i f t h t o one t e n t h of the Debye temperature.

( i i ) S o l i d S o l u t i o n s . - F o r e i g n atoms i n s o l i d s o l u t i o n may i n t e r a c t w i t h k i n k s i n s e v e r a l ways.There may be an a t t r a c t i v e i n t e r a c t i o n between k i n k s and immobile f o r e i g n atoms, and t h e l a t t e r may a c t a s p i n n i n g p o i n t s f o r t h e k i n k s ( s e e , e . g . ( 5 9 ) ) . Here we c o n s i d e r t h e c a s e t h a t on t h e time s c a l e of t h e i n t e r n a l f r i c t i o n o r r e l a x a t i o n experiments t h e f o r e i g n atoms a r e mobile, i . e . t h a t they can perform a t l e a s t one d i f f u s i o n a l a n d l o r r o t a t i o n a l jump d u r i n g a time comparablewith T. We a r e particularly i n t e r e s t e d i n s o l i d s o l u t i o n s of f o r e i g n atoms occupying s i t e s with lower symmetry t h a n t h e h o s t l a t t i c e , s o t h a t t h e kink-foreign-atom i n t e r a c t i o n may l e a d t o a r e - d i s t r i b u t i o n of t h e symrcetry a x e s of t h e f o r e i g n atoms. I n o r d e r t o f i x t h e i d e a s , we s h a l l r e f e r s p e c i f i c a l l y t o t h e "heavy" i n t e r s t i t i a l s oxygen, n i t r o g e n , and carbon

i n bcc t r a n s i t i o n m e t a l s .

I n

s e v e r a l t r a n s i t i o n m e t a l s w i t h bcc s t r u c t u r e , i n p a r t i - c u l a r i n a-Fe and t h e Group-V m e t a l s V , Nb, and Ta, t h e s e f o r e i g n atoms have been demonstrated t o occupy i n t e r s t i c e s of t e t r a g o n a l symmetry. The r e d i s t r i b u t i o n between i n t e r ' s t i c e s w i t h d i f f e r e n t i y o r i e n t e d t e t r a g o n a l axes l e a d t o t h e w e l l known

r e l a x a t i o n e f f e c t [ 6 0 ] . I n t h i s p a r t i c u l a r example t h e elementary atomic jumps involved i n t h e Snoek e f f e c t and i n t h e long-range d i f f u s i o n a r e t h e same; hence t h e a c t i v a t i o n e n t h a l p y of t h e Snoek e f f e c t , HS, e q u a l s t h a t of long-range m i g r a t i o n , HM, of the i n t e r s t i t i a l atoms.

The i n t e r a c t i o n of t h e s t r a i n - f i e l d surrounding a kink on a d i s l o c a t i o n with t h e mobile i n t e r s t i t i a l atoms i n i t s neighbourhood may g i v e an important c o n t r i b u t i o n t o t h e k i n k v i s c o s i t y S i n c e t h e s i t u a t i o n i s p a r t i c u l a r l y simple f o r k i n k s on a <111>/2 screw d i s l o c a t i o n s i n bcc m e t a l s we develop the f o l l o w i n g q u a l i t a t i v e argu- ments [61] f o r t h i s p a r t i c u l a r c a s e . A q u a n t i t a t i v e , more g e n e r a l treatment w i l l be g i v e n i n a s e r i e s of p a p e r s by T.o.OGURTAN1, Z.Q.SUN, and the p r e s e n t w r i t e r .

A t a g i v e n d i s t a n c e from a s t r a i g h t screw d i s l o c a t i o n along < i l l > t h e t h r e e p o s s i b l e < l o o > a x e s of t h e i n t e r s t i c e s w i t h t e t r a g o n a l symmetry a r e e n e r g e t i c a l l y de- g e n e r a t e . The presence of a kink on t h e d i s l o c a t i o n d e s t r o y s t h i s degeneracy and Leads t o t h e development of a "Snoek atmosphere1' [62]

,

i . e . , t h e t h r e e p o s s i b l e < l o o > axes of t h e i n t e r s t i c e s a r e no l o n g e r e q u a l l y populated. When a kink moves a l o n g t h e d i s - l o c a t i o n l i n e t h i s Snoek cloud e x e r t s on the kink a r e s t o r i n g f o r c e which may be r e - p r e s e n t e d i n terms of a complex compliance [63]. For a slowly moving k i n k t h i s l e a d s t o a dragging f o r c e which i s p r o p o r t i o n a l t o t h e kink v e l o c i t y , t o the i n v e r s e concen-

-

1

p r i n c i p a l components of t h e s t r a i n t e n s o r ( e l a s t i c d i p o l e t e n s o r ) o f t h e i n t e r s t i t i a l s . I n t e r m s of t h e k i n k d i f f u s i v i t y Dk t h i s means t h a t

S i n c e t h e d i f f u s i v i t y of t h e i n t e r s t i t i a l atoms i s w e l l r e p r e s e n t e d by S

D = Do e x p ( - H /kT) (68) w i t h t e m p e r a t u r e - i n d e p e n d e n t H' and Do, t h e t e m p e r a t u r e dependence of k i n k d i f f u s i v i t y i s d e s c r i b e d by a n A r r h e n i u s law w i t h a p p a r e n t a c t i v a t i o n e n t h a l p y H

M

k + H . S

-

1

The a p p e a r a n c e o f t h e f a c t o r (Cd) r e q u i r e s some comments. E q . ( 6 7 ) c l e a r l y be- comes i n v a l i d when Cd t e n d s towards z e r o . Then o t h e r mechanisms l i m i t D k , e . g . t h o s e d i s c u s s e d f o r p u r e c r y s t a l s . As a f i r s t a p p r o x i m a t i o n we may superimpose t h e k i n k v i s c o s i t i - e s due t o d i f f e r e n t mechanisms, s o t h a t i n t h e r e g i o n where one mechanism r e p l a c e s a n o t h e r r e c i p r o c a l d i f f u s i v i t i e s s h o u l d be added.

We t r e a t t h e t e m p e r a t u r e dependence o f Cd by means of a s i m p l e model u s e d i n a s i m i l a r c o n t e x t by R.deBAT1ST [64]. The model assumes t h a t we may d i v i d e t h e s i t e s a v a i l a b l e t o t h e f o r e i g n . a t o m s i n t o b u l k s i t e s and d i s l o c a t i o n s i t e s w i t h a f r e e

B

e n t h a l p y o f b i n d i n g Gd. We d e n o t e t h e number o f a v a i l a b l e o r o c c u p i e d b u l k s i t e by nb o r Nb, and t h e number of a v a i l a b l e o r o c c u p i e d d i s l o c a t i o n s i t e s by nd o r Nd ( s a y , p e r u n i t volume). S t r a i g h t f o r w a r d s t a t i s t i c a l thermodynamics g i v e s u s 1641 o r , s i n c e f o r d i l u t e s o l u t i o n s n b << Nb, Here Cb 5 nb/Nb d e n o t e s t h e b u l k c o n c e n t r a t i o n , Cd = nd/Nd t h e c o n c e n t r a t i o n of d i s l o - c a t i o n s . E q . ( 7 0 ) i s p a r t i c u l a r l y u s e f u l i f t h e f o r e i g n - a t o m c o n c e n t r a t i o n i s s o h i g h B t h a t Cb may be c o n s i d e r e d a s c o n s t a n t . I m p o r t a n t l i m i t i n g c a s e s a r e Cbexp(G /kT)>, l ,

B

c o r r e s p o n d i n g t o a h i g h b i n d i n g e n t h a l p y and h i g h c o n c e n t r a t i o n s , and Cbexp(G /kT)

<< 1 , c o r r e s p o n d i n g t o s m a l l b i n d i n g e n t h a l p i e s and s m a l l c o n c e n t r a t i o n s . I n t h e f i r s t c a s e Cd r e a c h e s i t s s a t u r a t i o n v a l u e and i s i n d e p e n d e n t of t e m p e r a t u r e , i n t h e second c a s e Cd shows a t e m p e r a t u r e dependence a c c o r d i n g t o

u n d e r t h e a u x i l i a r y c o n d i t i o n

w i t h t h e s o l u t i o n

6. A p p l i c a t i o n t o R e l a x a t i o n P r o c e s s e s a n d Comparison w i t h E x p e r i m e n t s .

( i ) The Bordoni R e l a x a t i o n . - The c l a s s i c a l f i e l d of a p p l i c a t i o n of t h e t h e o r y of k i n k - p a i r f o r m a t i o n i s t h e Bordoni r e l a x a t i o n i n f c c m e t a l s (comp. S e c t . l ) . Here H: i s n e g l i g i b l y s m a l l . According t o t h e d i s c u s s i o n of S e c t . 5 ( i ) t h e k i n k d i f f u s i v i t y i s e i t h e r t e m p e r a t u r e i n d e p e n d e n t o r d e c r e a s e s w i t h i n c r e a s i n g t e m p e r a t u r e . T h i s means

t h a t , depending o n t h e l e n g t h of t h e d i s l o c a t i o n segments, t h e r e l a t i o n s h i p between t h e e f f e c t i v e a c t i v a t i o n e n e r g y d e r i v e d from A r r h e n i u s p l o t s of t h e r e l a x a t i o n time a r e g i v e n by e i t h e r ( 5 1 a ) o r ( 5 2 a ) 7 ) w i t h a n e g l i g i b l e o r s m a l l p o s i t i v e v a l u e of d I n D k / d ( l / k T ) . A d e t a i l e d comparison w i t h e x p e r i m e n t s i s beyond t h e s c o p e o f t h i s p a p e r . The r e a d e r i s r e f e r r e d t o t h e work of ENGELKE [44,45] and of FANTOZZI e t a l .

[ 201

.

( i t ) The y - r e l a x a t i o n i n p u r e bcc m e t a l s , - F o l l o w i n g R.G.CHAMBERS [65] t h e y - r e l a x a - t i o n i s d e f i n e d a s b e i n g r e l a t e d t o t h e long-range motion of d i s l o c a t i o n s i n p l a s t i c d e f o r m a t i o n . According t o SEEGER and

BESTAK

[ 2 4 ] t h e y - r e l a x a t i o n i n bcc m e t a l s i s due t o t h e f o r m a t i o n of k i n k p a i r s on a <111>/2 s c r e w d i s l o c a t i o n s . T h i s i n t e r p r e - t a t i o n h a s r e c e n t l y b e e n c o n f i r m e d i n c o n s i d e r a b l e d e t a i l .

There i s a l a r g e body of e v i d e n c e t h a t a t low and i n t e r m e d i a t e t e m p e r a t u r e s t h e f l o w - s t r e s s of b c c m e t a l s i s c o n t r o l l e d by t h e motion of t h e above-mentioned s c r e w d i s l o c a t i o n s [66]. Recent measurements by ACKERmNN L671 on t h e t e m p e r a t u r e and s t r a i n - r a t e dependence of t h e f l o w s t r e s s of h i g h - p u r i t y niobium have c o n f i r m e d t h e k i n k - p a i r f o r m a t i o n t h e o r y of t h e f l o w s t r e s s L681 i n g r e a t d e t a i l . E . g . , t h e tempera- t u r e and s t r a i n - r a t e dependence f o l l o w i n g from t h e e x p r e s s i o n s d e r i v e d o f S e c t . 2 i n - c l u d i n g t h e t r a n s i t i o n between t h e d i f f e r e n t H ( U ) laws a t

a

=

:

c o u l d be v e r i f i e d

kp

and t h e q u a n t i t i e s 2 Hk + H: = ( 0 . 6 7

+

0.02)eV and a3 bsE1 c o u l d be d e r i v e d from t h e d a t a .