the kink picture of dislocation mobility and dislocation … › 5f46 › 66475185f3... · in terms...
TRANSCRIPT
THE KINK PICTURE OF DISLOCATION MOBILITY
AND DISLOCATION-POINT-DEFECT
INTERACTIONS
A. Seeger
To cite this version:
A. Seeger. THE KINK PICTURE OF DISLOCATION MOBILITY AND DISLOCATION-POINT-DEFECT INTERACTIONS. Journal de Physique Colloques, 1981, 42 (C5), pp.C5-201-C5-228. <10.1051/jphyscol:1981531>. <jpa-00221073>
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Submitted on 1 Jan 1981
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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, Ins t i tu t fur Physik, m d Universittit Stuttgart, Ins t i tu t fur theoretische und angewandte Physik, Postfach 800665, 0-7000 Stuttgart 80, Germany
Abstract.- The paper gives a coherent account of the theory of d i s loca t ion mobil i ty i n terms of kink-pair formation and kink migration. Three leve l s of descr ip t ion a r i s e i n a n a t u r a l way, namely those of the motion of s t r a i g h t d i s loca t ions , of kink-pair formation on d i s loca t ions , and of kink migration. The i n t e r r e l a t i o n s h i p s between these leve l s ("hierarchies") a r e discussed. The i n t e r a c t i o n with phonons and foreign atoms ("impurities") i s t rea ted on the t h i r d l e v e l , t h a t of the kink mobility.
Experimental information on the proper t i es of kinks may be obtained from measurements of d i s loca t ion 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 theory 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 loca t ion ve loc i ty i n Ge) and from body-centred cubic t r a n s i t i o n metals (y re laxa t ion = kink-pair for- mation on screw d is loca t ions ; dislocation-enhanced Snoek e f f e c t ; Snoek- K6ster re laxa t ion i n Nb and Ta).
1 . Introduct ion and General Background.- The "discovery" of d i s loca t ions during the
period 1929-34 provided the explanat ion f o r the observation t h a t i n many c r y s t a l l i n e
mate r ia l s the c r i t i c a l shear 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 several
orders of magnitude lower than the t h e o r e t i c a l shear s t reng th oth as calculated from
the model of per fec 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 the resis tance-to-gl ide due t o the d i s c r e t e nature of c r y s t a l s was not complete-
l y removed by the movement of s t r a i g h t d i s loca t ions running p a r a l l e l t o one of the
major c rys ta l lographic d i rec t ions ( t h i s was the model used throughout i n the e a r l y
days of d i s loca t ion theory) . Based on what i s now ca l led the PRANDTL-DEHLINGER-
FRENKEL-KONTOROVA model [1,6,7] an attempt t o est imate the s t r e s s required to move a
s t r a i g h t "dis locat ion" i n a c r y s t a l i s already 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 loca t ion models and t h e i r
re la t ionsh ips t o the 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.)
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1981531
C5-202 JOURNAL DE PHYSIQUE
A t the suggestion of E.OROWAN, i n 1939 R.PEIERLS [g] performed the f i r s t de-
t a i l e d ca lcu la t ion of the resolved shear s t r e s s required t o move a s t r a i g h t d i s -
loca t ion through a c r y s t a l i n the absence of thermal f luc tua t ions . This s t r e s s is
now ca l led the 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 loca t ion on a
given g l i d e plane i s a minimum i n the so-called "Peier ls va l leys" and a maximum on
the "Peier ls h i l l s " (comp. Fig.1). The o r i g i n a l est imate of P e i e r l s ( f o r a h i s t o r i -
c a l descr ip t ion see [10]) was corrected and improved by var ious authors; f o r the
methods used and the r e s u l t s obtained the reader i s re fe r red t o the review l i t e r a t u r e
[]l-141. The bes t ca lcu la t ions ava i lab le ind ica te t h a t i n many instances the ob-
served c r i t i c a l shear s t r e s s e s , a t l e a s t a t not too low temperatures, a r e sub-
s t a n t i a l l y lower than the calculated P e i e r l s s t r e s s e s .
I n the post-war development of d i s loca t ion theory i t became soon c l e a r t h a t
d i s loca t ion l i n e s must be considered f l e x i b l e . W.SHOCKLEY [l51 pointed out t h a t to-
gether with the exis tence of the P e i e r l s p o t e n t i a l t h i s may lead t o the formation
of *, i . e . , shor t segments of d i s loca t ion l i n e s i n which these cross over from
one P e i e r l s va l ley t o a neighbouring one (comp. Fig. l ) , and estimated t h a t kinks
may form i n thermal equilibrium. A.SEEGER [12,16] recognized t h a t the thermally
ac t iva ted formation of p a i r s of kinks of opposite s ign ( o r i g i n a l name "double kinks",
i n the following b r i e f l y ca l led kink p a i r s ) under the ac t ion of an o s c i l l a t i n g
applied s t r e s s might lead t o a re laxa t ion 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 the re laxa t ion process observed by P.G.BORDON1 [17,18] i n various deformed
f c c metals below room temperature might be due t o t h i s process. This suggestion was
developed f u r t h e r i n numerous theore t ica l and experimental papers (see, e .g . , [19,20])
and appears now to be general ly accepted as the cor rec t explanat ion of the Bordoni
re laxa t ion . From a h i s t o r i c a l point of view i t i s i n t e r e s t i n g t o note t h a t q u i t e
a l a rge number of a l t e r n a t i v e explanations involving d i s loca t ions have been proposed
during the l a s t twenty-five years (see, e . g . , [20,211) but t h a t the f i r s t explanation
i n terms of disLocation theory proved t o be c o r r e c t (BORDONI himself [l81 suggested
an explanat ion not involving d i s loca t ions) .
According t o the kink-pair formation model the Bordoni re laxa t ion i s an i n t r i n s i c
d i s loca t ion property d i r e c t l y r e l a t e d to the exis tence of the 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 provides the bes t information f o r est imating the numerical
value of the P e i e r l s s t r e s s 0 or the height of the P e i e r l s b a r r i e r s of d i s loca t ions P lying p a r a l l e l t o c rys ta l lographic d i r e c t i o n s . Recently, the modification of the
thermally ac t iva ted kink-pair formation process by atomic defects ( i n p a r t i c u l a r
by foreign i n t e r s t i t i a l atoms) has become the subject of much i n t e r e s t , a s i s w i t -
nessed by t h e present conference.
In add i t ion t o kinks t h a t a r e formed by thermal a c t i v a t i o n we have to consider
geometrical kinks 1221. These a re kinks t h a t a r e present even a t the lowest tempera-
tu res i n d i s loca t ion l i n e s t h a t a r e approximately p a r a l l e l t o low-index c rys ta l lo -
graphic d i rec t ions and f ixed by anchoring points not lying i n the same P e i e r l s
Peierls hill valley hill
anchor point g 1
geometri- cal kink
I anchoring
point kink double kink pair
kink pair under applied stress G
Fig . ] : Top: The la t t i ce -per iod ic 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 loca t ion as a funct ion of i t s displacement U. The mean value of U(u) i s the d i s - loca t ion l i n e en- ergy Ed. The d i s - placement uo of a s t r a i g h t dis- loca t ion under an appl ied s t r e s s U < up i s indicated (b = dis loca t ion s t reng th = modu- l u s of Burgers vec tor ) . Bottom: Various d i s loca t ion con- f igura t ions on an g l ide plane with P e i e r l s va l leys ( 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 th ree h ie ra rch ies of models and energy b a r r i e r s
l
hierarchy (model)
I
I1
111
defec t o r process considered
motion of s t r a i g h t
d i s loca t ions
kink-pair formation
kink migration
overcoming
t h e o r e t i c a l
shear s t reng th
a t h
P e i e r l s p o t e n t i a l
(of 1st kind)
kink p o t e n t i a l
( P e i e r l s p o t e n t i a l
of 2nd kind)
remaining la t t i ce -per iod ic energy b a r r i e r
P e i e r l s p o t e n t i a l
(of 1st kind)
kink p o t e n t i a l ( P e i e r l s
p o t e n t i a l of 2nd kind)
-
C5-204 JOURNAL DE PHYSIQUE
va l ley (comp.Fig.1). They account f o r the f a c t t h a t d i s loca t ions approximately
p a r a l l e l t o one of the major c rys ta l lographic d i rec t ions may contr ibute t o the modu-
l u s defec t even a t temperatures well below t h a t of t h e i r Bordoni re laxa t ion . Theoret ical reasoninganalogous t o t h a t leading t o the exis tence of the 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 loca t ions demonstrates t h a t the p o t e n t i a l energy of a kink
i n a d i s loca t ion must be a per iod ic funct ion of the kink pos i t ion along the dis-
loca t ion l i n e with a period equal to t h a t of the 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 per iod ic p o t e n t i a l i s ca l led "kink po ten t ia l " o r , i n order to emphasize the
close analogy t o the 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 loca t ion l i n e s , "Peier ls po-
t e n t i a l of t h e second kind".
We see 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 ie ra rch ies of angels of
Dionysius the Areopagite we have a d iv i s ion of our models i n t o three orders , 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 loca t ion l i n e s , and t h a t of E. To
these th ree orders of models correspond three orders of b a r r i e r s to be overcome (by
the motion of the defec t s of the next lower o rder ) , namely the t h e o r e t i c a l shear
s t reng th of a p e r f e c t c r y s t a l %, the P e i e r l s p o t e n t i a l (of the f i r s t kind) , and the
kink p o t e n t i a l . These re la t ionsh ips a r e summarized i n Table I.
The subdivis ion i n t o the models and the la t t i ce -per iod ic b a r r i e r s l i s t e d i n
Table I has considerable advantages f o r the discussion of physical phenomena
associated with d i s loca t ions , s ince t h e i r bas ic f e a t u r e s may of ten be understood by
r e f e r r i n g to one of t h e three hierarchies of Table I only. E.g., many bas ic fea tures
of the deformation of c r y s t a l s by g l i d e may be explained i n terms of d i s loca t ions
without paying a t t e n t i o n to the exis tence of the P e i e r l s b a r r i e r s . Most observations
on the Bordoni re laxa t ion 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
kink-pair formation. I f the kink p o t e n t i a l is la rge enough, geometrical kinks may
give r i s e t o a d i s t i n c t re laxa t ion phenomenon associated with kink migration
[19,23-251.
Notwithstanding the impressive successes of the s i m p l i s t i c approach j u s t out-
l ined , a deeper understanding of the observed phenomenon can o f t e n only be achieved
i f i t i s reca l led t h a t each of the t h r e e h i e r a r c h i e s o f Table I can a t bes t give a
p a r t i a l represen ta t ion of a complex physical s i t u a t i o n .
The present paper aims a t developing, i n a coherent fashion, the theory of
kink-pair formation and kink migration, and attempts to apply it to the in te rpre -
t a t i o n of measurement of d i s loca t ion 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 ca re fu l ly d i s t inguish between phenomena t h a t may be understood be reference
t o only one of the models of Table 1,with the aspects represented by the o ther
models coming i n t o play only through phenomenological parameters, and those re-
qu i r ing a more general (" inter-hierarchic") approach.
2. The In te rac t ion between Kinks.- We consider here kinks on the same dis loca t ion
l i n e . The s implest physical p ic tu re we may use i s the descr ip t ion of the d i s loca t ion
l i n e a s a s t r i n g , character ized by the d i s loca t ion l i n e tension Sd and the d i s loca t ion
mass per u n i t length, yd. For a d d i t i o n a l s impl ic i ty we consider only de-
v ia t ions from s t r a i g h t l i n e s , measured by the displacement U of the d i s loca t ion
from a P e i e r l s va l ley . The motion of a d i s loca t ion l i n e i s then governed by the
p a r t i a l d i f f e r e n t i a l equation
In ( 1 ) z denotes the s p a t i a l coordinate i n the d i r e c t i o n of the P e i e r l s va l ley , t
the time, b the d i s loca t ion s t reng th , 0 the appl ied resolved shear 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 funct ion of i t s displacement.
For the purpose of ca lcu la t ing the l i n e tension Sd i t s u f f i c e s t o replace U(u) by
i t s average value Ed, the l i n e energy of the d i s loca t ions (Fig. 1). We then have L261
where 8 i s the angle between the Burgers vector of the d i s loca t ion l i n e and the
d i r e c t i o n of the P e i e r l s va l leys .
For s t a t i c so lu t ions u(z) Eq . ( l ) s impl i f i es t o
Eq.(3) possesses the s o l u t i o n uo = constant corresponding t o a r i g i d displacement
of a s t r a i g h t d i s loca t ion l i n e , where
The maximum value of a a t which ( 4 ) s t i l l has a r e a l so lu t ion defines the P e i e r l s
s t r e s s Op (comp. Fig. l ) .
Among the so lu t ions of (3) f o r a = 0 i s t h a t f o r a s i n g l e kink centred a t
z = z I n i m p l i c i t form i t i s given by 0'
U
where the s igns correspond t o p o s i t i v e o r negat ive kinks, respec t ive ly . For "smooth"
p o t e n t i a l s f o r which the maximum value Umax coincides with U(a/2), where a i s the
d i s tance between neighbouring P e i e r l s va l leys and the height of a s i n g l e kink, we
may define the kink width a s
Within t h e presen 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 the "line-tension
model") the energy of a s ing le kink is given by u=a
C5-206 JOURNAL DE PHYSIQUE
Under an app l i ed shear s t r e s s U(O<]Ul<a ) Eq.(3) possesses a s o l u t i o n which P
corresponds t o a p a i r of kinks i n uns tab le equilibrium.The f o r c e s exer ted on the
kinks by the app l i ed s t r e s s , f abo, tend t o move the kinks a p a r t . A conf igura t ion
of unstable equilibrium e x i s t s i n which these f o r c e s j u s t balance t h e a t t r a c t i o n bet-
ween the kinks of opposi te s igns . The enthalpy d i f f e r e n c e between a kink p a i r i n
t h i s conf igura t ion 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 the
saddle-point enthalpy f o r the formation of a kink p a i r , HkD. Within the p resen t
framework we have u=umax
H = 2(2sd)112 1 kp
[u(u) - U(uo) - (U - U ~ ) ~ C J ] " ~ du ,
where the upper l i m i t of the i n t e g r a t i o n fol lows 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 experiments 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 , (10)
where c l and c2 a r e numerical f a c t o r s t h a t depend on the choice of the p o t e n t i a l
U(u) . For t h e so-cal led Eshelby p o t e n t i a l l ) [27]
these f a c t o r s a r e given by L281
C , = 2c2 = 3-'12/2 , the enthalpy of format ion of a s i n g l e kink by
-1 14 Hk = 3 a ( a b 0 ~ ~ ~ / 2 ) " ~ ,
and t h e kink width by
wk = 2a2 Sd/ 3Hk
c o n t r a s t t o t h e s inuso ida l p o t e n t i a l
U(u) = U(o) + abop [ l - cos(2'rru/a)l /2'rr (11)
the Eshelby p o t e n t i a l (12) does n o t desc r ibe i n an a n a l y t i c a l way t h e ~ e r i o d i c i t y of
U(u) bu t i s wel l s u i t e d f o r t r e a t i n g the e f f e c t of an app l i ed shear s t r e s s on the
i n t e r a c t i o n of a air of c l o s e kinks. For l a t e r use (Sect. 3) we record f o r the
s inuso ida l p o t e n t i a l ( 1 1)
Eq. ( l ) may be used t o determine the v i b r a t i o n a l spectrum associated with a
s ing le kink o r with the saddle-point configurat ion of a kink p a i r and t o ca lcu la te
the entropy of kink formation. For d e t a i l s and r e s u l t s the reader i s re fe r red to 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 kinks 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 the "veloci ty of sound" of small perturbations propagating
along a d i s loca t ion l i n e is
Since (1) is Lorentz-invariant, we may define the 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 preceding t h e o r e t i c a l descr ip t ion corresponds t o l i n e I1 of Table I. We
have disregarded the a tomis t ic s t r u c t u r e of the c r y s t a l i n the d i r e c t i o n of the
P e i e r l s val leys(1ine 111 of Table I ) . This has t h e consequence t h a t the energy of a
kink i s independent of i t s pos i t ion z along the d i s loca t ion . The f a c t t h a t kinks
a r e shor t segments of an otherwise s t r a i g h t d i s loca t ion embedded i n a three-dimen-
s iona 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 the
d i s loca t ion l i n e tension Sd and t h e e f f e c t i v e mass ydper u n i t d i s loca t ion length.
For some purposes, i n p a r t i c u l a r f o r the ca lcu la t ion of the kink--kink i n t e r a c t i o n
a t large separa t ions , t h i s i s not s u f f i c i e n t , s ince i t does not adequately describe
the e l a s t i c i n t e r a c t i o n between kinks.
It i s easy to show t h a t f o r l a rge separat ions the e l a s t i c i n t e r a c t i o n energy
must vary a s the inverse dis tance between kinks on the same d i s loca t ion [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 shear modulus G and Poisson 's
r a t i o v the force between kinks of height a a t separat ions X >> a reads [13,22,27,
29,301
=& 2 -[(l+v)cos2 a 2 0 + (l-2v)sin2 01 (18) 4~ 1-v 2x2
where 8 denotes the angle between the d i s loca t ion l i n e and the C signs r e f e r t o kinks
of equal o r opposite s igns , respec t ive ly . The genera l iza t ion of (12) t o an i so t rop ic
e l a s t i c i t y i s [31, 321 a2 F = sel ( 18a)
O 2 2 Here SE1 is t h e prelogarithmic f a c t o r of the e l a s t i c p a r t of the d i s l o c a t i o n l ine-
tension. I t may be w r i t t e n i n the form
where f i s independent of the d i s loca t ion s t reng th b. For a given g l ide plane
f=f(Cijkl,@) depends only on the 2nd-order e l a s t i c constants Cijkl of the c r y s t a l
and on the characterof the d i s loca t ion l i n e .
C5-208 JOURNAL DE PHYSIQUE
abcx abcx
Fig.2: The t o t a l enthalpy of two kinks of opposi te s i g n ( f u l l l i n e ) and the i n t e r a c t i o n enthalpy H i n t (X) a s a func t ion of the kink separa t ion X. The en- thalpy of kink-pair fo r - mation Hkp is ind ica ted ; the kink separa t ion X I corresponds t o unstable equi l ibr ium.
0 l l XI kink separatio" X
i s given by -abos. The superpos i t ion of these two p o t e n t i a l ene rg ies ( f u l l l i n e )
leads t o a maximum a t X l
x1 = (asE1/2b upr2 9 (20)
corresponding t o a kink-pair format ion enthalpy
Eq.(21) corresponds t o t h e f i r s t term of a "multipole expansion". At smal l
separa t ions and l a r g e o i t ceases t o be v a l i d . Then the e l a s t i c i n t e r a c t i o n between
the kinks depends n o t on ly on a and b but i n a d d i t i o n on the shape of t h e kinks
( c f . [221). I n th$ p resen t context , however, i t is no 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 kink separa t ions the e l a s t i c i n t e r a c t i o n i s small compared
with t h a t given by Eq.(8) of the l ine - t ens ion model.
The stress 3 which separa tes t h e range of v a l i d i t y of (8) and (21) may be
found by equa t ing (10) and (21). The Eshelby p o t e n t i a l (12) gives us [28]
I f we choose, e .g . , sd/SZ1 = 4.1, Eq. (22) has t h e s o l u t i o n ;/c$ = 0.14, ln(3*4-+/;)
= 5. Fig. 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 case . kp
A d i f f e r e n t choice of sd/SZ1 o r U(u) does n o t change t h e func t iona l form of t h i s re-
l a t i o n s h i p appreciably. For o + u p Eq.8 e x h i b i t s a "quasi-universal" behaviour i n the
Fig.3: The re la t ionsh ip between the resolved shear s t r e s s a ( i n u n i t s of the P e i e r l s s t r e s s op) and the en- thalpy of formation of a kink p a i r H ( i n u n i t s of the formation enthalpy kp of a s ing le kink, Hk) according t o Eq.(lO) ( l ine- tension 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 the predicted re la t ionsh ip ( f u l l l i n e ) changes over
0 1 from Eq.(lO) t o Eq.(21) i s indicated. '-',p"%
where the choice of U(u) a f f e c t s only the numerical value of c3 ( f o r (12): 2) c3 = (6 /5) (2/3) 5j ' = 0.73).
For small kink separat ions i t i s not possible t o define the kink separat ion i n
a unique manner. In Fig.2 t h i s i s indicated by the dashed curve. Fortunately, the
physical conclusions t o be drawn from the H (0) re la t ionsh ip a r e unaffected by kp
t h i s uncertainty.
2. The Kink-Pair Formation Rate.- The ca lcu la t ion of the r a t e of formation of kink
p a i r s on an otherwise s t r a i g h t d i s loca t ion l i n e of u n i t length, F, i n the following
b r i e f l y ca l led the kink-pair formation r a t e , i s beset by the problems of r a t e theory
i n general . A theore t ica l framework capable of giving p r a c t i c a l r e s u l t s and being
appl icable under a l l physical condit ions does not e x i s t . The two most successful
approaches a r e ( i ) an adaption of the t r a n s i t i o n s t a t e theory, o r i g i n a l l y developed
f o r the ca lcu la t ion of chemical reac t ion r a t e s and going back t o the work of
H.PELZER and E.WIGNER [34], and ( i i ) the d i f fus ion theory based on the work of
H.A.KRAMERS [ 3 5 ] . For a general discussion of the theory of kink-pair formation
r a t e s t h e reader i s re fe r red to A.SEEGER and P.SCHILLER [13 ] .
The common idea behind both approaches i s t h a t the degrees of freedom of the
e n t i r e c r y s t a l a r e t rea ted by the usual 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 describing tke change i n the kink-kink separat ion X
and the mode associated with the t r a n s l a t i o n of a kink p a i r of fixed separa t ion X
along the d i s loca t ion l i n e . By well-known arguments the maximum i n the energy-
dis tance diagram of the f i r s t of these modes ( F i g . 2 ) corresponds t o the saddle-point
i n the many-dimensional configurat ional space separat ing a kink-free d i s loca t ion
from the region containing a p a i r of kinks of opposite s ign. The two theore t ica l
approaches mentioned above d i f f e r i n the assumptions made concerning the way i n
which the system crosses the above-mentioned saddle-point.
2 ) ~ e take the opportunity t o c o r r e c t an e r r o r i n r e f . [28]: As pointed out by E.MANN,
i n Eq.(ba) t h e term -3fi2 must be replaced by -3fi4. This e r r o r has no o ther conse-
quence f o r t h e main p a r t of [28] but a f f e c t s the Appendix of [28], i n p a r t i c u l a r
the exponent i n t h e re la t ionsh ip (23) of the present paper. The exponent 5/4 given
above i s i n agreement with t h a t recen t ly obtained by MORI and KATO i331.
C5-210 JOURNAL DE PHYSIQUE
(i) The t r a n s i t i o n s t a t e theory makes the assumption t h a t once the system has
reached the saddle po in t with a v e l o c i t y po in t ing from the "no-kink" region t o the
"kink-pair" region, kink-pair formation takes p lace . The p r o b a b i l i t y t h a t the system
"back-crosses" t h e saddle po in t i s neglected. The kink-pair formation r a t e i s then
given
exp [-H (O)/kT] h2 kP
denote the energy eigen-values 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 frequency mm
assoc ia ted with an unkinked d i s l o c a t i o n ,
those assoc ia ted with d i s l o c a t i o n s con ta in ing a kink p a i r i n t h e saddle-point con-
f i g u r a t i o n (3'1 = Planck ' s constant h divided by 2n). As a compensation f o r t h e two
non-vibrat ional modes mentioned above, the sum ( o r product) over m con ta ins two more
terms than t h a t over m".
As mentioned i n Sect .2 , wi th in the framework of t h e l ine - t ens ion model the
f requencies w and w;h,,, which depend on U, may be ca lcu la ted from (1 ) . For the in- m t e r n a l - f r i c t i o n phenomena t o be discussed i n t h e p resen t paper i t i s an admissable
s i m p l i f i c a t i o n t o d i s regard the stress-dependence of w and wuand hence t h a t of the
kink-pair formation entropy. The m'' sums and products may then be r e l a t e d t o the
equ i l ib r ium ( l i n e a r ) dens i ty
( ~ n r n ~ k ~ ) ~ " IISinh(.hwm/2kT) = exp(-Hk/kT) 2 "
h m II: Sinh(*w;. /2kT)
of non-interact ing kinks of a given s i g n , s i n c e the products i n (24) a r e the
squares of those i n (27) . Here U;, r e f e r s t o an i s o l a t e d kink, the product over m'
containing one term l e s s than t h a t over m. For high temperatures (charac te r i zed by
5wm/kT << 1 , hw:/k~ << 1 f o r those w and W" t h a t d i f f e r appreciably from each o t h e r )
t h e e v a l u a t i o n of t h e products l eads t o
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 the Eshelby p o t e n t i a l (12) . I n (28b) ad denotes the in te ra tomic d i s t a n c e a long
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 ives us f o r t h e kink-pair format ion r a t e i n the
t r a n s i t i o n s t a t e theory
Here Hint r e f e r s t o the saddle-point s e p a r a t i o n o f the kinks , (29a) t o t h e s inus-
o i d a l p o t e n t i a l , (29b) t o the Eshelby p o t e n t i a l .
( i i ) The s t a r t i n g equa t ion of t h e d i f f u s i o n theory i s a p a r t i a l d i f f e r e n t i a l
equat ion f o r the p a r t i c l e d e n s i t y p (px ,x , t ) i n t h e phase space of 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 kink-kink s e p a r a t i o n and the momentum px a s s o c i a t e d wi th
i t [35]. The theory is based on t h e phys ica l pictume t h a t , i n a d d i t i o n t o the f o r c e s
g iv ing r i s e t o Hint(x)(Fig.2), t he 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 wi th 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 the Brownian f o r c e s may be des-
c r ibed i n terms of a k ink mobi l i ty pk o r a kink v i s c o s i t y ]/vk. By the ~ i n s t e i r r
Nernst r e l a t i o n s h i p the kink mobi l i ty may be expressed i n terms of t h e k ink 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 cases i n which t h e number of independent 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 equa t ion may be reduced by one.
( a ) I f t he kink mobi l i ty i s so high t h a t t h e o s c i l l a t i o n s of t h e kinks 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 the i n t e r a c t i o n of k inks of oppos i t e s i g n a r e only
l i t t l e inf luenced by the Brownian f o r c e s , o r px and X may be r ep laced by t h e a c t i o n
v a r i a b l e I (E) = pXdx . (31)
where t h e i n t e g r a l has t o be taken a long t h e c losed curve i n phase space correspon-
ding t o a f i x e d energy E . As independent v a r i a b l e one may choose energy and time,
and t h e phys ica l problem may be looked upon a s "d i f fus ion and d r i f t of energy". This
i s the view-point adopted by H.DONTH [36] i n h i s theory o f k ink-pair format ion, a l -
though a s h i s s t a r t i n g p o i n t he d id not use KRAMERS' paper [35] b u t e a r l i e r work by
KOLMOGOROFF [37] and GEBELEIN [38] .
CS-2 12 JOURNAL DE PHYSIQUE
E.MANN [391 has r e c e n t l y shown t h a t under the c o n d i t i o n H >> kT t h e kink- kp
p a i r format ion r a t e f o r h igh k ink m o b i l i t i e s is given by
8Hkwk r =- ( S ) ' " (p:q)? exp ( l l i n t / k ~ ) ,
Dk"k
Here and l a t e r i n (34) the same assumptions f o r t h e c a l c u l a t i o n of the ent ropy terms
have been made a s i n the t r ea tmen t ( i ) .
For temperature-independent k ink d i f f u s i v i t i e s Dk t h e temperature dependence
of (29) and (32) a r e the same. The r e s u l t s of t h e t r a n s i t i o n s t a t e theo ry and t h e
high-mobi l i ty d i f f u s i o n theory ag ree wi th each 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 (33)
( b ) I f t h e e f f e c t of t h e Brownian f o r c e s on t h e k ink v e l o c i t y exceeds those of
t h e "ex te rna l " f o r c e s ( i . e . , i f t h e k ink m o b i l i t y i s s m a l l ) , Kramers' equa t ion
s i m p l i f i e s t o a so-calledSmolukowskiequation ( d i f f u s i o n equa t ion wi th d r i f t term)
wi th the s p a t i a l coord ina te X a s an independent v a r i a b l e . For 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 paper , t h i s is the most important case . I t has been d i scussed i n de-
t a i l by SEEGER and SCHILLER [13], STENZEL [40], ENGELKE [441, and MANN [391. I f t h e
assumption H >> kT i s made t h e kink-pai r format ion r a t e may be expressed i n terms kp
of quadra tu res . The f u r t h e r assumption t h a t t he sadd le p o i n t occur s i n t h e regime of
v a l i d i t y of (21) l e a d s t o the closed-form express ion 1391
where
and Kn(y) denotes t h e MacDonald (modified Hankel) f u n c t i o n of o r d e r n. Important
s p e c i a l ca ses a r e y >> 1, KI (y) " ( ~ r / 2 ~ ) ~ ' ~ e x ~ ( - ~ ) , l ead ing t o [13,401
and y << 1, K1(y) y-l, l ead ing t o [42, 431
-'3/ 2 We no te t h a t i n (36) t h e pre-exponent ia l f a c t o r depends on temperature a s (kT) , whereas i n (37) i t depends on T a s (kT)-'.
3.) The Dis loca t ion Veloci ty . - The v e l o c i t y vd o f a d i s l o c a t i o n proceeding perpen-
d i c u l a r t o the d i r e c t i o n o f i t s P e i e r l s v a l l e y s by success ive format ion o f k ink
p a i r s depends on t h e h e i g h t a of t h e k inks , t h e k ink-pai r format ion r a t e r , and the
distance by which the two members of a kink p a i r separate before they cease t o move.
This may e i t h e r be the case when they meet obs tac les along the d i s loca t ion l i n e s t h a t
a r e impenetrable t o kinks, o r when they annih i la te with o ther kinks on the same d i s -
locat ion l i n e . The d i s loca t ion ve loc i ty w i l l be determined by the shor te r of the two
d i s tances , the separa t ion L of the obstacles o r the separat ion $ of the two kinks
i n a p a i r before t h e i r ann ih i la t ion with o ther kinks, so t h a t we expect X L
The separa t ion % equals the d i s tance t h a t two kinks of opposite s ign d r i f t towards
each o ther during the mean l i fe t ime tk of the kinks. I t i s thus 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 the mean l i f e t i m e tk of the kinks must be equal t o the
average time f o r the nucleat ion of a f resh kink p a i r on a d i s loca t ion segment of
Length %, i . e . 4)
Hence the f i n a l r e s u l t f o r xk reads
I n s e r t i o n of (41) i n t o (38) gives us
with the l imi t ing cases (xk << L)
vd = a(2pkab0r)
and (% >> L)
v d = a L T
In mate r ia l s on which measurements of d i s loca t ion v e l o c i t i e s under the a c t i o n
of a known resolved applied shear s t r e s s U a re ava i lab le (e.g., Ge and S i , see
Sect.5) Eq.(42) may be compared d i r e c t l y with experiment.
3 ) ~ o r s impl ic i ty we disregard the f a c t t h a t d i s loca t ions anchored i n d i f f e r e n t
P e i e r l s va l leys contain a l s o geometrical kinks. Geometrical kinks w i l l be t rea ted
separa te ly a t the end of Sect. 4.
4 ) ~ s was pointed out t o the author by E.MANN, the corresponding expression i n
r e f . [42] i s i n e r r o r by a f a c t o r of two.
C5-2 14 JOURNAL DE PHYSIQUE
I n i n t e r n a l f r i c t i o n experiments the 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 he
equ i l ib r ium d e n s i t y of t he k inks i s mainta ined. We then have
~ ~ ~ ~ ~ t i ~ ~ o f (43) i n t o (41) ~ i v e s us (37) , a s we expect from the assumptionsmade.
4 . ) Re laxa t ion S t r eng th and Relaxat ion Time.- The c a l c u l a t i o n of the i n t e r n a l f r i c -
t i o n and the modulus d e f e c t on the b a s i s of t he model developed s o f a r i s a formi-
dable t a s k . Under c e r t a i n s impl i fy ing assumption a s o l u t i o n was given by H.ENGELKE
[41,44,45] and reviewed elsewhere P92. For t h e p r e s e n t purposes i t s u f f i c e s t o de-
r i v e some s imple e s t i m a t e s capable of b r ing ing 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 ime on temperature T and on the 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 the a r e a A swept ou t by t h e d i s -
l o c a t i o n segments. Following A.SEEGER, H.DONTH and F.PFAFF [46] the 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 wi th
l i n e t ens ion Sd may sweep o u t under t h e app l i ed s h e a r s t r e s s )
may be ob ta ined . I f A denotes the t o t a l l eng th p e r u n i t volume of those 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 rocess and i f ?I denotes the e l a s t i c modulus
a p p r o p r i a t e f o r the mode of deformat ion 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 ~ (45)
We e s t i m a t e the 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 equ i red by a d i s l o c a t i o n
moving wi th the v e l o c i t y v ( L ' ) and sweeping o u t t h e a r e a A i n the manner i n d i c a t e d d i n F ig . 4 . This g ives us w i t h t h e h e l p of (38 )
----- 7------ anchortng anchor~ng
polnt point
Fig.4: A s imple model f o r e s t i m a t i n g t h e r e l a x a t i o n time. F u l l l i n e : D i s loca t ion f u l l y bowed o u t under a p p l i e d s t r e s s U. Dashed l i n e s : In t e rmed ia t e c o n f i g u r a t i o n s corresponding t o d isplacements U of the c e n t r e of t he d i s l o c a t i o n segment. The d i s - l o c a t i o n v e l o c i t y i s assumed t o be given by vd(LT) , wi th L ' (u) i n d i c a t e d i n the f i g u r e .
Among the i n t e r e s t i n g fea tures of (45) is t h a t the re laxa t ion s t reng th i s pre-
d ic ted t o be independent of temperature and ( f o r a given t o t a l Length of d i s loca t ions)
proport ional t o L2. Eq. (46) exh ib i t s a t r a n s i t i o n from a l i n e a r dependence of the
re laxa t ion time f o r small L t o a quadrat ic 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 the temperature dependence of -c.
I n a c r y s t a l there w i l l be i n general a f a i r l y wide d i s t r i b u t i o n of L values.
Following G.SCHOECK 1471 we may est imate the i n t e r n a l f r i c t i o n Q-' and the modulus
defect by making the assumption (which i s not q u i t e c o r r e c t ) t h a t f o r a f ixed length
L we have a Debye re laxa t ion process and i n t e g r a t i n g over the L-dis tr ibut ion. We de-
noteby p ( L ) ~ L the number of d i s loca t ion segments p e r u n i t volume with obs tac le L
separat ion between L and L + dL cont r ibu t ing t o the re laxa t ion process , so t h a t the
d i s loca t ion length per 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 is then given by
and the 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 the 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 evaluated n u m r i c a l l y [47]. - 2
I t gives a maximum value 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 tua-
t ions where the s impl i f i ca t ions (37) and (43) a r e appl icable . Then (46) takes the
form
Eq.(50) shows t h a t a s l imi t ing cases the re laxa t ion time T may e x h i b i t two d i f f e r e n t
dependences on temperature and obs tac le dis tance, namely i n the case p E q ~ << 1
( i . e . , low temperatures)
't .c L ( k ~ ) ' exp(2Hk/kT)/% ,
i n the case ~ T L >> 1 ( i . e . , high temperatures)
C5-2 16 JOURNAL DE PHYSIQUE
The apparent a c t i v a t i o n en tha lp ies
a r e given by
d ln T Heff "
d( l/kT)
i n the case of p E q ~ c < 1 , and by
dlnDk Heff = Hk - - - -
d(l/kT) 2 kT
i n the case p E q ~ > > 1 .
I n those cases i n which Hk i s not small compared with the o ther terms, (51a)
and (52a) show the remarkable fea ture t h a t the a c t i v a t i o n enthalpy is higher a t lower
temperatures. The reason f o r t h i s i s t h a t a t low temperature the d i s tance a newly gene-
ra ted kink has t o t r a v e l u n t i l i t i s stopped a t an obstacle i s independent of tempera-
t u r e , whereas a t high temperatures the kink may recombine with kinks of opposite s ign,
the dens i ty of which increases with increasing temperature.
Another important spec ia l case is t h a t i n which kink-pair formation may be d i s -
regarded a l toge ther (e.g., because of low temperatures) and we have t o consider only
a temperature-independent dens i ty pk of geometrical kinks. This case has been t rea ted
by A.D.BRAILSFORD [23] , C.kTiiTHRICH [25], and A.SEEGER and C.WUTHRICH [ 4 8 ] . I f these
r e s u l t s a r e expressed i n terms of the kink d i f f u s i v i t y , an approximate expression f o r
the re laxa t ion time is
where L denotes again the d i s tance between the obs tac les . The re laxa t ion s t reng th i s
given by
8 ~~b~ A =--F - a 2 ~ 2 p k ~ c o s 2 @ . (55) kT
In (55) $ denotes the angle between the d i r e c t i o n of the P e i e r l s va l ley and the aver-
age d i r e c t i o n of the d i s loca t ion l i n e , which i s determined by the ~ 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 leads t o the i n t e g r a l ( 4 9 ) . In
add i t ion one has t o average over the angle @, taking i n t o account t h a t the densi ty of
geometrical kinks i s given by
pk = s i n $/a (56)
5.)The Kink D i f f u s i v i t y Dk.- The most important parameter i n the general descr ip t ion
of Sect.4 of re laxa t ion processes involving kink-pair formation i s the kink d i f f u s i v i t y
Dk. I t contains the "physics" of the re laxa t ion process. Its ca lcu la t ion requ i res us
t o leave leve l I1 of the h ie ra rch ies of Table I. E.g., Dk may be control led by the
kink p o t e n t i a l c h a r a c t e r i s t i c of l eve l 111, o r i t may be determined by the i n t e r a c t i o n
of k inks wi th 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 can on ly be un-
ders tood q u a n t i t a t i v e l y i f we t ake i n t o account t h a t a k ink i s a p e r t u r b a t i o n on an
otherwise 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 wi th a long-range s t r a i n f i e l d
dec reas ing w i t h the 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 kink. This means
t h a t we have t o go back t o l e v e l I of Table I .
An exhaus t ive t h e o r e t i c a l t r ea tmen t of D would be beyond the scope of the pre- k
s e n t paper . We s h a l l cover only some examples of ( i ) pure c r y s t a l s and ( i i ) i n t e r -
a c t i o n s wi th f o r e i g n i n t e r s t i t i a l atoms.
( i ) Pure Crys t a l s . - I n pure 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 mob i l i t y uk = Dk/kT) of
k inks on d i s l o c a t i o n l i n e s wi th a l a r g e kink p o t e n t i a l is determined by t h e r a t e by
which k inks w i l l overcome t h e kink p o t e n t i a l . A complete t h e o r e t i c a l t r ea tmen t of
t h i s r a t e has n o t y e t been given, b u t we have good reasons t o b e l i e v e t h a t t he ana-
logy w i t h the jumps of atomic 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 the s p i r i t of
t h e t r a n s i t i o n s t a t e theo ry we may in t roduce t h e f r e e en tha lpy of k ink mig ra t ion
a s the d i f f e r e n c e between the f r e e e n t h a l p i e s when the k inks i s i n t h e saddle-point
c o n f i g u r a t i o n and when i t i s i n the p o s i t i o n of s t a b l e e q u i l i b r i u m (excluding the de-
g ree of freedoms l ead ing ove r the sadd le -po in t s i n the second c a s e ) . I f V: denotes
t h e a t t empt f requency wi th which t h e k ink 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
saddle-point and ad t h e pe r iod of t h e k ink 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 theory
g ives us under the assumption >> kT k
A s i m i l a r expres s ion wi th the same temperature dependence may be ob ta ined from the
d i f f u s i o n theory under t h e assumption t h a t t he mob i l i t y of a k ink i n i t s p o t e n t i a l
w e l l between two neighbour ing saddle-points i s h igh .
Simple p h y s i c a l arguments i n d i c a t e t h a t t he h e i g h t of the k ink p o t e n t i a l , t he M k ink mig ra t ion en tha lpy Hk, should depend ve ry s t r o n g l y on t h e n a t u r e of t h e chemical
bond. H; i s s t r o n g l y r e l a t e d t o the width of the k inks . I f t h e k ink width wk i s M much l a r g e r than t h e p e r i o d i c i t y of t he k ink p o t e n t i a l , Hk w i l l be sma l l , s i n c e t h e
c o n t r i b u t i o n s of t h e v a r i o u s atoms n e a r t h e k ink c e n t r e average o u t t o a l a r g e e x t e n t .
The k ink p o t e n t i a l can only be l a r g e i f the k ink width i f comparable w i t h o r sma l l e r
than t h e pe r iod of the k ink p o t e n t i a l ("abrupt k inks" [23,30]). The kink width , i n
t u r n , i s r e l a t e d t o the p e r i o d i c p a r t of t h e P e i e r l s p o t e n t i a l (comp. Eq . (6 ) ) . This
means t h a t we expec t a l a r g e kink p o t e n t i a l only i n d i s l o c a t i o n s and c r y s t a l s t r u c -
Cures wi th h igh P e i e r l s b a r r i e r . For t h i s r eason the a t t empt [23] t o a t t r i b u t e t h e
BORDONI r e l a x a t i o n i n f c c me ta l s t o the motion of ab rup t geomet r i ca l k inks had t o be
r e f u t e d 1131. Es t ima tes by G.SCHOTTKY [49] indicated t h a t i n f c c me ta l s and i n f a c t i n
most d i s l o c a t i o n s i n meta ls the k ink p o t e n t i a l i s s o smal l t h a t an obse rvab le r e l axa -
t i oh due t o the motion of geomet r i ca l k inks cannot be expected a t temperatures above 1K
C5-2 18 JOURNAL DE PHYSIQUE
As p o i n t e d o u t by A.SEEGER and B.~ESTAK [24] t h e r e i s one excep t iona l ca se
where one might e x p e c t , f o r t h e o r e t i c a l r ea sons , a f a i r l y h igh k ink mig ra t ion en tha lpy M .
Hk 1n me ta l s . Th i s excep t ion concerns k i n k s i n screw d i s l o c a t i o n s w i t h Burgers v e c t o r s
b = a <l 11 >/2 i n bcc me ta l s . P .B.HIRSCH [ 5 0 1 recognized t h e p o s s i b i l i t y t h a t because
o f t h e t h r e e f o l d symmetry o f < l 11> axes i n c u b i c c r y s t a l s t hese d i s l o ~ a t i o ~ might
n o t have a wel l -def ined g l i d e p l ane and t h a t t hey may hence be s e s s i l e . For t h e pur-
pose o f t he p r e s e n t d i scuqs ion t h i s i s e q u i v a l e n t t o a h igh P e i e r l s s t r e s s ap. With
c e r t a i n mod i f i ca t ions ( s e e , e . g . , [481) t he view-point of HIRSCH has proved c o r r e c t .
Computer s imu la t ion s t u d i e s by CH.WuTHRICH l511 on an a t o m i s t i c model f o r &Fe suppor t
t h e s u g g e s t i o n of SEEGER and SESTAK 1241 on t h e k ink mig ra t ion en tha lpy i n ao<111>/2
screw d i s l o c a t i o n s i n g e n e r a l terms wi thou t be ing a b l e t o make q u a n t i t a t i v e p red ic -
t i o n s .
I n t he ca se of screw d i s l o c a t i o n s i n bcc me ta l s t h e h igh 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 rcumstances r e l a t e d t o symmetry and n o t t o chemical bonding. (Non-screw
d i s l o c a t i o n s i n bcc me ta l s o r screw d i s l o c a t i o n s w i th o t h e r Burgers v e c t o r s posses 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 th d i r e c t i o n a l
bonds, i . e . , va l ence c r y s t a l s such a s germanium and s i l i c o n .
There i s cons ide rab le expe r imen ta l evidence , s t a r t i n g wi th t h e work of DASH[52]
on d i s l o c a t i o n l i n e s i n S i deco ra t ed wi th Cu and l a t e r s u b s t a n t i a t e d by t r ansmis s ion
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 long <110;. d i r e c t i o n ~ p o s s e s s
h igh P e i e r l s b a r r i e r s [531. The o r i g i n of t h i s i s thought t o l i e i n t h e cova len t na-
t u r e o f bonding i n Ge and S i [54,55]. It is t h e r e f o r e r ea sonab le t o assume t h a t Ge
and S i p o s s e s s n o t on ly h igh k ink format ion e n e r g i e s bu 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 energy b a r r i e r f o r t h e mig ra t ion o f k inks a long d i s l o c a t i o n l i n e s . M
I f t he k ink m i g r a t i o n en tha lpy Hk i s l a r g e enough f o r t he k ink mig ra t ion b a r r i e r
t o show up expe r imen ta l ly , t h e k ink d i f f u s i v i t y (58) w i l l i n gene ra l be sma l l enough
f o r t h e d i f f u s i o n theo ry of k ink -pa i r format ion t o be a p p l i c a b l e . I n S i and Ge the
k ink format ion energy is s o h igh t h a t a t t h e tempera tures a t which d i s l o c a t i o n velo-
c i t i e s a r e measured the c o n d i t i o n
is u s u a l l y f ~ l f i l l e d . ~ ) Th 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)
5 ) ~ a k i n g t h e work of H.SCHAUMBURG [561 on 60' d i s l o c a t i o n s i n Ge a s an example, t h e
h i g h e s t tempera ture used i s about 800 K. With Hk 1 .3 e V (28a ) g i v e s a n e q u i l i b r i u m
d i s t a n c e between k inks of the o r d e r o f magnitude of 1 cm, i . e . , of t h e specimen di -
mensions.
hold, leading t o
rd = 2 Dk a2 b L (pzqf a/kT (60)
The experimental r e s u l t s reported on Ge and S i approach indeed a l i n e a r dependence
of the d i s loca t ion v e l o c i t i e s on s t r e s s a t low s t r e s s e s and high temperatures. From
(60) follows the e f f e c t i v e a c t i v a t i o n enthalpy of d i s loca t ion motion
A t s u f f i c i e n t l y high s t r e s s e s t h e r a t e of kink-pair formation may become so l a r g e t h a t
the kink l i fe t ime i s l imited by the ann ih i la t ion of kinks of opposite s ign. We then
have t o use (42a). Together with (34) and (35) t h i s gives us
- 11 2
vd = 2 s b a u k P? [Y K ~ ( Y ) I (62)
I f y > 0 the s t r e s s dependence predicted by (62) i s s tronger than l i n e a r . The effec-
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 gives us
For a given type of d i s loca t ion se' may be calculated from the e l a s t i c constants , so (1) t h a t from measurements of Heff an: H::; we may determine both the formation enthalpy
Hk and the migration enthalpy H: of kinks. For 60°-dislocations on <I l I > gl ide planes
i n germanium we have sel = 6 ' 10-l0 N (A.KORNER, H.O.K.KIRCHNER, Univers i t l t Wien,
personal communication). From SCHAUMBURG1s L561 observations on such d i s loca t ions
[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 neg l ig ib ly small the kink d i f f u s i v i t y must be e s t i -
mated by considering the i n t e r a c t i o n between kinks and phonons e x p l i c i t l y [13,57,58].
A t low temperatures one expects pk t o decrease with increasing temperature roughly - 1
a s Eph , where E denotes the phonon energy densi ty. ph
6 ) ~ x t r a p o l a t e d t o o = 0 with the help of Eq. ( 2 1).
C5-220 JOURNAL DE PHYSIQUE
A t s u f f i c i e n t l y high temperatures the theory pred ic 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 follows from
(58) f o r kT >> H: . Calculat ions of the absolute magnitude of vk a r e q u i t e d i f f i c u l t . The est imates
of SEEGER and ENGELKE [ 5 7 ] ind ica te t h a t the condit ions f o r the a p p l i c a b i l i t y of the
d i f fus ion theory of kink-pair formation (see Sect . 2) a r e s a t i s f i e d below about one
f i f t h t o one tenth of the Debye temperature.
( i i ) So l id Solutions.- Foreign atoms i n s o l i d so lu t ion may i n t e r a c t with kinks i n
severa 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 kinks and immobile foreign
atoms, and the l a t t e r may a c t as pinning po in t s for the kinks (see, e.g. ( 5 9 ) ) . Here
we consider the case t h a t on t h e time s c a l e of the i n t e r n a l f r i c t i o n o r re laxa t ion
experiments the fo re ign 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 andlor r o t a t i o n a l jump during 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 so lu t ions of fo re ign atoms occupying s i t e s with lower symmetry
than the host l a t t i c e , so t h a t the kink-foreign-atom i n t e r a c t i o n may lead t o a re-
d i s t r i b u t i o n of the symrcetry axes of the foreign atoms. I n o rder t o f i x the ideas,
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 the "heavy" i n t e r s t i t i a l s oxygen, ni t rogen, and carbon
i n bcc t r a n s i t i o n metals . I n severa l t r a n s i t i o n metals with 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 metals V , Nb, and Ta, these foreign atoms have been
demonstrated t o occupy i n t e r s t i c e s of te t ragonal symmetry. The r e d i s t r i b u t i o n between
in te r ' s t i ces with d i f f e r e n t i y o r ien ted te tragonal axes lead t o the well known
re laxa t ion e f f e c t [60]. I n t h i s p a r t i c u l a r example the elementary atomic jumps involved
i n the Snoek e f f e c t and i n the long-range d i f fus ion a r e the same; hence the a c t i v a t i o n
enthalpy of the Snoek e f f e c t , HS, equals t h a t of long-range migration, 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 the s t r a i n - f i e l d surrounding a kink on a d i s loca t ion with
the mobile i n t e r s t i t i a l atoms i n i t s neighbourhood may give an important contr ibut ion
t o the kink v i s c o s i t y Since the 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 kinks on
a <111>/2 screw d is loca t ions i n bcc metals we develop the following 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 case. A q u a n t i t a t i v e , more general treatment w i l l be
given i n a s e r i e s of papers by T.o.OGURTAN1, Z.Q.SUN, and the present w r i t e r .
A t a given d i s tance from a s t r a i g h t screw d is loca t ion along < i l l > the three
possible <loo> axes of the i n t e r s t i c e s with t e t ragona l symmetry a re energe t ica l ly de-
generate. The presence of a kink on the d i s loca t ion destroys t h i s degeneracy and Leads
t o the development of a "Snoek atmosphere1' [62] , i . e . , the three possible <loo> axes
of the i n t e r s t i c e s a r e no longer equal ly populated. When a kink moves along the d i s -
loca t ion l i n e t h i s Snoek cloud exer t s on the kink a res to r ing force which may be re-
presented i n terms of a complex compliance [63]. For a slowly moving kink t h i s leads
t o a dragging force which i s proport ional to the kink ve loc i ty , t o the inverse concen- - 1
t r a t i o n of the i n t e r s t i t i a l s near the d i s loca t ion , Cd , t o DIkT, where D i s the
d i f f u s i v i t y of the i n t e r s t i t i a l atoms, and t o (XI - 12f, where X I and l2 are the
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 ) of t he i n t e r s t i t i a l s .
I n terms of the k ink d i f f u s i v i t y Dk t h i s means t h a t
S ince the 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 ep resen ted by
S D = Do exp(- H /kT) (68)
wi th temperature-independent H' and Do, the temperature dependence of k ink d i f f u s i v i t y
i s desc r ibed by an Arrhenius law with apparent a c t i v a t i o n en tha lpy H M k + H . S
- 1 The appearance of the f a c t o r (Cd) r e q u i r e s some comments. Eq.(67) c l e a r l y be-
comes i n v a l i d when Cd t ends towards zero . Then o t h e r mechanisms l i m i t D k , e . g . those
d i scussed f o r pure c r y s t a l s . As a f i r s t approximation we may superimpose t h e k ink
v i scos 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 the r eg ion where one mechanism
r e p l a c e s ano the r r e c i p r o c a l d i f f u s i v i t i e s should be added.
We t r e a t t h e temperature dependence o f Cd by means of a s imple model used i n a
s i m i l a r con tex 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 the fo re ign .a toms i n t o bulk 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
en tha lpy of b inding Gd. We denote the number of a v a i l a b l e o r occupied bulk s i t e by
nb o r Nb, and the number of a v a i l a b l e o r occupied d i s l o c a t i o n s i t e s by nd o r Nd
( say, p e r u n i t volume). S t r a i g h t forward s t a t i s t i c a l thermodynamics g i v e s us 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 denotes t h e bulk concen t ra t ion , Cd = nd/Nd t h e c o n c e n t r a t i o n of d i s lo -
c a t i o n s . Eq.(70) 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 foreign-atom c o n c e n t r a t i o n i s s o h igh B
t h a t Cb may be considered a s c o n s t a n t . Important l i m i t i n g c a s e s a r e Cbexp(G /kT)>, l , B
corresponding t o a h igh binding en tha lpy and h igh concen t ra t ions , and Cbexp(G /kT)
<< 1 , corresponding t o smal l b ind ing e n t h a l p i e s and smal l concen t ra t ions . I n t h e f i r s t
case Cd reaches i t s s a t u r a t i o n va lue and i s independent of temperature , i n the second
case Cd shows a temperature dependence according t o
There may be s i t u a t i o n s (h igh d i s l o c a t i o n d e n s i t y , h igh b ind ing en tha lpy , low
concen t ra t ion of f o r e i g n atoms) where it i s n o t allowed t o t r e a t Cb a s a cons tan t . We
then have t o so lve
JOURNAL DE PHYSIQUE
under the a u x i l i a r y c o n d i t i o n
wi th the s o l u t i o n
6. App l i ca t ion t o Relaxat ion P rocesses and Comparison wi th Experiments.
( i ) The Bordoni Relaxat ion. - 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 the theory of kink-
p a i r format ion i s t he Bordoni r e l a x a t i o n i n f c c me ta l s (comp. S e c t . l ) . Here H: i s
n e g l i g i b l y sma l l . According t o the d i s c u s s i o n of S e c t . 5 ( i ) the k ink d i f f u s i v i t y i s
e i t h e r temperature independent o r dec reases wi th i n c r e a s i n g temperature . This means
t h a t , depending on t h e l eng th 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
the e f f e c t i v e a c t i v a t i o n energy de r ived from Arrhenius p l o t s of t he r e l a x a t i o n time
a r e given by e i t h e r (51a) o r (52a)7) w i t h a n e g l i g i b l e o r smal l p o s i t i v e va lue of
d I n Dk/d( l /kT) . A d e t a i l e d comparison wi th exper iments i s beyond the scope of t h i s
paper . 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 - re l axa t ion i n p u r e bcc m e t a l s , - Following R.G.CHAMBERS [65] t h e y-relaxa-
t i o n i s de f ined a s be ing r e l a t e d t o the 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
deformat ion. According t o SEEGER and BESTAK [ 2 4 ] the y - re l axa t ion i n bcc me ta l s i s
due t o the format ion of k ink p a i r s on a <111>/2 screw d i s l o c a t i o n s . This i n t e r p r e -
t a t i o n has r e c e n t l y been confirmed i n cons ide rab le d e t a i l .
There i s a l a r g e body of evidence t h a t a t low and in t e rmed ia t e temperatures the
f low-s t r e s s of bcc me ta l s i s c o n t r o l l e d by the motion of the above-mentioned screw
d i s l o c a t i o n s [66]. Recent measurements by ACKERmNN L671 on the temperature and
s t r a i n - r a t e dependence of t h e flow s t r e s s of h igh-pur i ty niobium have confirmed the
kink-pai r format ion theory of the flow s t r e s s L681 i n g r e a t d e t a i l . E.g. , the tempera-
t u r e and s t r a i n - r a t e dependence fo l lowing from t h e expres s ions de r ived of Sec t .2 in -
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 = : could 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 could be der ived from the
d a t a .
The y - re l axa t ion i n h igh-pur i ty Nb was s t u d i e d by deLIMA and BENOIT [69] by
means of i n t e r n a l f r i c t i o n exper iments . A f t e r p l a s t i c deformation they found a maxi-
mum a t about 250 K wi th an e f f e c t i v e a c t i v a t i o n en tha lpy of H e f f = (0.61 + 0.02)eV,
which they i d e n t i f i e d a s a d i s l o c a t i o n r e l a x a t i o n . App l i ca t ion of (51a) g ives us
2Hk + H: = (0.65 t 0.02)eV, i n e x c e l l e n t agreement wi th the value obta ined from
"Since t h e Bordoni r e l a x a t i o n i s r a t h e r s t a b l e a g a i n s t annea l ing i n t e r n a l f r i c t i o n
measurements may be c a r r i e d o u t a t h igh f r equenc ie s ( i n t h e MH range) and h igh
temperatures . I n gene ra l such measurement w i l l p e r t a i n t o t h e regime p E q ~ > > 1 .
ACKERMANN's f low s t r e s s measurements;h7] and thus p rov id ing s t r o n g exper imenta l ev i -
dence t h a t t h e two exper iments p e r t a i n indeed t o the same b a s i c p rocess .
Since t h e q u a n t i t y S:' may be c a l c u l a t e d from the e l a s t i c c o n s t a n t s , the
assignment of t he y - re l axa t ion t o k ink p a i r s on screw d i s l o c a t i o n s may be t e s t e d
f u r t h e r by comparing exper imenta l and t h e o r e t i c a l va lues of a3 b S E ~ . I t t u r n s ou t
t h a t t h e exper imenta l va lue is e i g h t t imes t h a t c a l c u l a t e d f o r k inks of h e i g h t
a = (2/3)'ILao. This i n t u r n i s surmised to be a consequence of the p a r t i c u l a r a to -
m i s t i c s t r u c t u r e of screw d i s l o c a t i o n s i n bcc me ta l s , which may e x h i b i t a symmetry-
breaking e f f e c t and acqu i re a new p rope r ty named "po la r i ty" [ 481 . The occurrence of
"double kinks", i . e . k inks of lowest energy wi th twice the h e i g h t expected from t h e
p e r i o d i c i t y of t he p e r f e c t c r y s t a l s t r u c t u r e , i s thought t o be d i r e c t l y r e l a t e d t o
the "polar i ty1 ' of ao<l11>/2 screw d i s l o c a t i o n s i n bcc me ta l s and t o be wi thout ana- 6 ) logue i n o t h e r d i s l o c a t i o n s .
( i i i ) The Snoek-48ster r e l axa t ion . - Following A.S.NOWICK and B.S .BERRY [ 7 11 we denote
by "Snoek-XOster r e l a x a t i o n " a r e l a x a t i o n p rocess f i r s t d iscovered by SYOEK [60] on
cold-worked i r o n con ta in ing n i t r o g e n a t a temperature above t h a t of t h e Snoek e f f e c t .
I t was s t u d i e d i n d e t a i l by W.KoSTER, L-BANGERT, and R.HAHN [72], who showed t h a t
t he p rocess invo lves an i n t e r a c t i o n between d i s l o c a t i o n s and f o r e i g n i n t e r s t i t i a l
atoms ( n i t r o g e n and carbon i n t h e case of cl-Fe). More s p e c i f i c a l l y , according t o
SCHOECK [47] the r e l a x a t i o n e f f e c t i s due t o the motion of d i s l o c a t i o n s t h a t a r e
dragging the i n t e r s t i t i a l atoms. However, t he a c t i v a t i o n en tha lpy of the SnoekbK8ster
r e l a x a t i o n , H ' ~ , exceeds s u b s t a n t i a l l y t h e mig ra t ion en tha lpy of t h e f o r e i g n in-
t e r s t i t i a l atoms involved. One of the key q u e s t i o n s i n t h e i n t e r p r e t a t i o n of t h e M
Snoek-KGster r e l a x a t i o n i s the o r i g i n of t he d i f f e r e n c e between and H . A.SEEGER [73] has r e c e n t l y proposed t h a t t h e b a s i c p rocess of t h e Snoek-X6ster
r e l a x a t i o n is the format ion of k ink p a i r s i n the presence of mobile i n t e r s t i t i a l
atoms. This theory provides us wi th a c l a s s i f i c a t i o n o f the Snoek-KSster r e l a x a t i o n
according t o the d i s l o c a t i o n s on which the kink p a i r s a r e formed and according t o the
f o r e i g n i n t e r s t i t i a l atoms involved. The " c l a s s i c a l " Snoek-XOster r e l a x a t i o n i s a t t r i -
buted by SEEGER 1731 t o a <111>/2 screw d i s l o c a t i o n s i n t e r a c t i n g wi th t h e "heavy"
i n t e r s t i t i a l s (C,N,O) b u t t h e r e may be analogous r e l a x a t i o n p rocesses involving non-
screw d i s l o c a t i o n s (wi th a lower k ink-pai r format ion en tha lpy) and hydrogen atoms
(wi th a lower mig ra t ion en tha lpy) .
The main d i f f e r e n c e between t h e t h e o r i e s of SCHOECK [47] and SEEGER [73], which
give e s s e n t i a l l y t h e same expres s ions f o r t h e r e l a x a t i o n s t r e n g t h [ comp. (45) l , i s t h a t
i n the l a t t e r theory the expres s ion f o r t he r e l a x a t i o n time con ta ins the k ink
he use of the expres s ion "double kink" f o r t h e k ink c o n f i g u r a t i o n desc r ibed above
r e q u i r e s t o drop "double-kink generation': o r i g i n a l l y coined by t h e p r e s e n t w r i t e r
[ 161, a s a synonym f o r "kink-pair formation".
C5-224 JOURNAL DE PHYSIQUE
format ionenthalpyon s p e c i f i c d i s l o c a t i o n s . The kink-foreign-atom i n t e r a c t i o n i s usual-
l y s t r o n g enough f o r the cond i t ions f o r t he a p p l i c a t i o n o f the d i f f u s i o n theory of
k ink-pai r format ion t o be f u l f i l l e d , s o t h a t the temperature dependence of the Snoek-
K6ster r e l a x a t i o n may be ob ta ined by i n s e r t i n g (34) and (67) i n t o (36). Since the
gene ra l expres s ion f o r ?(T) is r a t h e r complicated we t r e a t a number of s p e c i a l ca ses .
( a ) I f t h e kink-formation en tha lpy i s ve ry smal l compared t o t h e mig ra t ion
en tha lpy of t h e i n t e r s t i t i a l s involved, the c o n d i t i o n p E q ~ >> l w i l l be f u l f i l l e d .
The temperature dependence of the r e l a x a t i o n time i s then given by (52) . Since f o r
Hk z kT the f a c t o r ( k ~ ) ~ ' ~ exp(Hk/kT) i s only weakly temperature dependent, t he tempe-
r a t u r e dependence of t h e r e l a x a t i o n time of the Snoek-Kijster e f f e c t becomes v i r t u a l -
l y the same a s t h a t of t h e d is locat ion-enhanced Snoek e f f e c t [62,74]. This i s reason-
a b l e , s i n c e the lower t h e k ink format ion en tha lpy and the h ighe r t h e temperature , t he
l e s s these two mechanisms a r e d i s t i n g u i s h a b l e . A p o s s i b l e example f o r t h i s s i t u a t i o n
may be a r e l a x a t i o n e f f e c t r e c e n t l y s t u d i e d by U.RODRIAN and H.SCHULTZ [75] on
p l a s t i c a l l y deformed h igh-pur i ty Ta con ta in ing f a i r l y low concen t ra t ions of oxygen.
When the r e l a x a t i o n t imes a s s o c i a t e d wi th t h i s p rocess were r ep resen ted i n an
Arrhenius p l o t T = L exp(Heff/kT) (73)
H e f f = ( l . 12 2 0.04)eV and a pre-exponent ia l f a c t o r TW= 8 ' 1 0 - ' ~ s were ob ta ined . Com-
pa r i son wi th the corresponding q u a n t i t i e s of t he Snoek e f f e c t of oxygen i n Ta,
H' = l . 105 ".01 eV, T = 8 . 6 . 1 0 - ' ~ s L761 shows t h a t du r ing t h e r e l a x a t i o n p rocess
the 0 atoms perform on ly abou t one jump. Thus Cd may indeed be considered a s tempera-
ture-independent and t h e above d e s c r i p t i o n appears a p p r o p r i a t e .
(b) Another s imple case i s t h a t of d i s l o c a t i o n s s a t u r a t e d wi th f o r e i g n atoms,
r e a l i z e d a t h igh i n t e r s t i t i a l c o n c e n t r a t i o n s and h igh f r e e e n t h a l p i e s o f b ind ing t o
the d i s l o c a t i o n s , G:. Then C i s temperature independent even under e q u i l i b r i u m con- d d i t i o n s . The temperature dependence of the r e l a x a t i o n time of t h e Snoek-XEster e f f e c t
i s g iven by e i t h e r
T a T~ exp [ (2Hk + H ~ ) / ~ T I , (74)
( i f p E q ~ << 1 ) o r
T a T~~~ exp [ ( Hk + H ~ ) / ~ T I
( i f pZqL >> I ) . A noteworthy f e a t u r e of (74) and (75) i s t h a t t h e expres s ion (75)
v a l i d a t h ighe r temperatures shows t h e lower a c t i v a t i o n enthalpy.
For a g iven kink-formation en tha lpy and a g iven l eng th d i s t r i b u t i o n of t h e d i s -
l o c a t i o n segments the temperature of which the r e l a x a t i o n p rocess i s observed w i l l be
the lower the sma l l e r t he mig ra t ion en tha lpy fIM of t h e i n t e r s t i t i a l s . I n the bcc
t r a n s i t i o n me ta l s the mig ra t ion en tha lpy of hydrogen i s very much sma l l e r t han t h a t
of t he "heavyv i n t e r s t i t i a l atoms. This means t h a t i n torsion-pendulum o r a f t e r -
e f f e c t measurements the Snoek-Kijster r e l a x a t i o n a s s o c i a t e d wi th hydrogen i s l i k e l y
t o occur i n t h e regime of v a l i d i t y of (74) ( s e e , e .g . , [ 771 ) .
Eq.(74) r e p r e s e n t s the case t r e a t e d by SEEGER C731 and found t o be i n reason-
a b l e agreement wi th t h e exper iments on t h e Snoek-Kiister r e l a x a t i o n of N and C i n a-
Fe. Another example which might be covered by (74) i s t h a t of 0 i n Ta. Here RODRIAN
and SCHULTZ 1751, working a t f r equenc ie s of about 1 Hz, observed a Snoek-Xdster re-
l a x a t i o n wi th an e f f e c t i v e a c t i v a t i o n en tha lpy H e f f = (2.25 + 0.16)eV. The y-re laxa-
t ion,which they f i n d i n pure Ta a t about 390 K, posses ses an e f f e c t i v e a c t i v a t i o n
en tha lpy H e f f = 1.24 rt 0.05 eV. I f we a l low f o r a k ink-migrat ion en tha lpy of
H: = (0.08 P 0.04)eV, we f i n d from t h i s 2Hk = (1 .23 ? 0.09)eV. Together wi th t h e
above-mentioned value f o r t h e mig ra t ion en tha lpy of oxygen i n Ta t h i s g ives us
2H + IIM = ( 2 . 3 3 2 0 . IO)eV, i n good agreement wi th the va lue of (2.35 c O.16)eV k fo l lowing from the H of RODRJAN and SCHULTZ [75]. A c o r o l l a r y of t h i s i n t e r p r e - e f f t a t i o n is t h a t t he f r e e en tha lpy of b ind ing oxygen t o d i s l o c a t i o n s i n Ta i s a t l e a s t
0.6 eV, i n agreement wi th the conc lus ions of RODRIAN and SCHULTZ [75]. A t t h e tempera-
t u r e of the Snoek-4i5ster r e l a x a t i o n the preceding a n a l y s i s g i v e s exp(Hk/kT)=l.5 '10 5
e l so t h a t i t i s indeed p l a u s i b l e t h a t t h e exper iments p e r t a i n t o the regime pk L>>I
a s assumed. B
(C) F i n a l l y we t r e a t t he case Cd.exp(GdlkT) c< 1 and assume, fo l lowing deBATIST
[64], t h a t du r ing t h e de te rmina t ion of the e f f e c t i v e a c t i v a t i o n en tha lpy thermal
e q u i l i b r i u m between d i s l o c a t i o n s and bulk i s mainta ined. We then have
T OC T~ exp [ (2Hk + + H ; ) / ~ T ] (76)
i n the case of p E q ~ << 1 or
B i f p z q ~ >> 1 . (Hd denotes the foreign-atom-dislocation b ind ing en tha lpy . )
Let us compare (76) and (77) wi th the Snoek-Kiister r e l a x a t i o n of oxygen i n
niobium. Here we have accura t e exper imenta l va lues 2Hk + H: = (0.66 f 0.02)eV [comp.
subsec t ion ( i i ) ] and HM = ( 1 . 154 + 0.009)eV 1751. J.DIEHL, T.S.KE, Z.L.PAN, M.WELLER,
J.X.ZHANG, and t h e p r e s e n t w r i t e r have r e c e n t l y s tud ied i n cons ide rab le d e t a i l t h e
Snoek-Kiister r e l a x a t i o n i n t h i s system and compared i t wi th the r e s u l t s r e p o r t e d i n
the l i t e r a t u r e ( f o r t hese s e e 1731).
Two d i s t i n c t r e l a x a t i o n p rocesses wi th the a t t r i b u t e s of the Snoek-Kdster re-
l a x a t i o n a r e observed, v i z . a r a t h e r s t a b l e p rocess a t about 725 K (measuring frequen-
cy f = 1 Hz) wi th a n e f f e c t i v e a c t i v a t i o n en tha lpy Hef f = (1.69 P 0.04)eV and one a t
about 545 K with H e f f = (1.99 + 0.2)eV, which, however, i s l e s s s t a b l e a g a i n s t an-
nea l ing . The sma l l e r of t hese two 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 i e s would be i n ex-
c e l l e n t agreement wi th (74 ) , bu t n o t the l a r g e r one. An i n t e r p r e t a t i o n which e x p l a i n s
q u a n t i t a t i v e l y the occurrence of two d i s t i n c t "Snoek-Kiister r e l a x a t i o n s " n o t only i n
t h i s work bu t a l s o i n the l i t e r a t u r e can be given by assuming t h a t t h e s t a b l e (and
more p r e c i s e l y s t u d i e d ) Snoek-Kiister r e l a x a t i o n i s desc r ibed by (77). From t h i s B
fo l lows a b inding en tha lpy Hd 0.35 eV. This b inding en tha lpy accounts then f o r t he
h ighe r a c t i v a t i o n en tha lpy i n terms of (76) . The i n t e r p r e t a t i o n impl i e s t h a t a f t e r
C5-226 JOURNAL DE PHYSIQUE
cold-work there are dislocations anchored so that p E q ~ << l holds, and that sub-
sequent annealing changes the anchoring points in such a way that at the temperature
of the stable Snoek-Kiister peak the condition p E q ~ >> 1 is fulfilled. This hypothesis
is clearly an interesting subject for further investigation.
(iv) Relaxation by migration of geometrical kinks.- The relaxation time is given by
(54), the relaxation strengthby (55). If (58) is obeyed one expects the frequency v:
to be of the order of magnitude of the Debye frequency.This means that if there is
an observable relaxation strength (i.e., the L values are sufficiently large) the
pre-exponential factor will be several powers of ten larger than the reciprocal
Debye frequency. This feature allows one to distinguish the kink-migration relaxation
fairly easily from the other relaxation processes discussed in this paper, since
these give comparable with or smaller than the reciprocal Debye frequency.
Acknowledgement
The author is indebted to a large number of collaborators for discussions and help.
He should like in particular to acknowledge the support by Dr. E. Mann and Dr. M.
Weller.
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