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COMPUTER SIMULATION OF THE STRUCTURE OF bcc/hcp AND bcc/9R MARTENSITE
INTERFACES
G. Barcelo, A. Crocker
To cite this version:
G. Barcelo, A. Crocker. COMPUTER SIMULATION OF THE STRUCTURE OF bcc/hcp AND
bcc/9R MARTENSITE INTERFACES. Journal de Physique Colloques, 1982, 43 (C4), pp.C4-179-
C4-184. �10.1051/jphyscol:1982421�. �jpa-00222135�
JOURNAL DE PHYSIQUE
CoZZoque C4, suppZ6ment au n o 22, Tome 43, d6cembre 1982 page C4-179
COMPUTER SIMULATION OF THE STRUCTURE OF bcc/hcp AND b c c / 9 R MARTENSITE.
INTERFACES
G.N. ~ a r c e l o * and A. G. Crocker
Department o f Physics, U n i v e r s i t y o f Surrey, GuiZdford, Surrey GU2 5 X H , EngZand
(Accepted 9 August 1982)
Abstract.- The s t r u c t u r e s o f two i n t e r f a c e s o f m a r t e n s i t i c transformations i n Cu Zn based a l l o y s have been i n v e s t i g a t e d using computer s i m u l a t i o n techniques.
A new i n t e r a t o m i c p o t e n t i a l has been developed which i s assumed t o represent a l l i n t e r a c t i o n s between atoms i n t h e parent bcc phase and t h e product hcp and 9R phases. Stable r e l a x e d s t r u c t u r e s o f bcc/hcp and bcc/9R i n t e r f a c e s have been found. I n b o t h cases t h e i n t e r f a c e migrates i n t o t h e bcc phase d u r i n g t h e r e l a x a t i o n process. The boundary i n the bcc/hcp computer model i s broad i n v o l v i n g about 6 (110) bcc planes o f atoms, whereas t h a t i n t h e bcc/9R model i s s i m p l e r i n v o l v i n g o n l y 3 planes. This suggests t h a t t h e 9R product s t r u c t u r e m i g h t be p r e f e r r e d i n p r a c t i c e .
I n t r o d u c t i o n . - The growth o f a r n a r t e n s i t i c phase i s c o n t r o l l e d i n p a r t by t h e atomic s t r u c t u r e o f i t s i n t e r f a c e w i t h the p a r e n t m a t e r i a l . U n f o r t u n a t e l y i t i s v e r y d i f f i c u l t t o o b t a i n d i r e c t experimental data on t h e s t r u c t u r e s o f i n t e r f a c e s and t r a d i t i o n a l t h e o r e t i c a l models are based on continuum o r phenomenological approaches which provide o n l y macroscopic i n f o r m a t i o n . However computer s i m u l a t i o n u s i n g i n t e r - atomic p o t e n t i a l s has, i n r e c e n t years, become a w e l l - e s t a b l i s h e d method o f i n v e s t i - g a t i n g t h e s t r u c t u r e s , energies and i n t e r a c t i o n s o f c r y s t a l d e f e c t s ( 1 ) . I n p a r t i - c u l a r i t has been w i d e l y used t o study t h e p r o p e r t i e s o f g r a i n boundaries. The e x p e r t i s e gained i n t h i s work now makes i t f e a s i b l e t o s t a r t t o examine t h e equi- l i b r i u m s t r u c t u r e s o f interphase boundaries i n c l u d i n g m a r t e n s i t e i n t e r f a c e s .
The authors are p a r t i c u l a r l y i n t e r e s t e d i n t h e f a s c i n a t i n g range o f phenomena associated w i t h t h e m a r t e n s i t i c transformations which occur i n Cu Zn and Cu Zn based a l l o y s ( 2 ) . I g n o r i n g t h e e f f e c t s o f ordering, t h e p a r e n t phase i s body centred cubic and the product has a f a u l t e d hexagonal c l o s e packed s t r u c t u r e , known as 9R, i n which t h e s t a c k i n g o f close packed planes f o l l o w s t h e sequence ABCBCACAB. However by applying a p p r o p r i a t e stresses t o t h e product phase t h e f a u l t s may r e a d i l y be removed t o generate a p e r f e c t hcp s t r u c t u r e ( 3 ) . I t has been deduced t h a t t h e 9R and hcp s t r u c t u r e s have s i m i l a r energies. This suggests t h a t t h e i n t e r f a c e might p l a y an important r o l e i n t h e choice o f t h e s t r u c t u r e o f t h e product phase. I n order t o i n v e s t i g a t e t h i s proposal t h e bcc/hcp and bcc/9R transformations a r e being i n v e s t - i g a t e d using t h e computer s i m u l a t i o n method. A new i n t e r a t o m i c p o t e n t i a l has been developed t o perform t h i s work and p r e l i m i n a r y r e s u l t s on t h e s t r u c t u r e s o f t h e two i n t e r f a c e s have been obtained.
The P o t e n t i a l . - An e q u i l i b r i u m e m p i r i c a l two-body, i n t e r a t o m i c p o t e n t i a l $ ( r ) was s p e c i a l l y developed f o r t h i s p r o j e c t and i s shown i n F i g . 1. I t c o n s i s t s of e i g h t piece-wise continuous cubic polynomials, o r s p l i n e s , $ i ( r ) and terminates a t zero slope a t t h i r d nearest neighbour d i s t a n c e o f t h e bcc s t r u c t u r e . I t i s g i v e n by
where
For r
<
r,, $ ( r ) i s represented by t h e Born-Mayer p o t e n t i a l $,(r) = A exp(- B r ) ."on leave from : centrb Atomico Bariloche, comisi6n Nacional de ~ n e r g i a ~ t b m i c a S.C. d e Bariloche-R.N. (8400) Republica ~ r g e n t i n a
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1982421
JOURNAL DE PHYSIQUE
The s p l i n e s are l i n k e d smoothly together, w i t h $ ( r ) and i t s f i r s t and second d e r i v a - t i v e s continuous a t t h e knot p o s i t i o n s r. which are i n d i c a t e d i n F i g . 1. The c o e f f i c i e n t s A.
.
were matched, u s i n g exp$essions given by Johnson ( 4 ) and M i l l e r ( 5 ) t o t h e l a t t i c e l d a r meter a,, t h e e l a s t i c constants c,,, c,
c,, and the vacancy formation energy Eev o f t h e h i g h temperature bcc phase.A:
t h e computer s i m u l a t i o n method r e s u l t s i n c r y s t a l s a t e a u i l i brium a t absolute zero temperature, t h e values o f ao, c,,, c,,, c and E F ~ adopted were chosen f o r a Cu -48%Zn a l l o y f o r which t h e Ms temperature i s8.
I n p a r t i c u l a r a was taken t o be 2.9464 ( 6 ) , c,,
c,,, c were extrapolased from r e s u l t s from a !ange o f a l l o y s and taken t o be 0.j604, 0.6746 and 0.5162 eV A-3 r e s p e c t i v e l y (7-11) and i n a s i m i l a r way EF was deduced t o be 0.58 eV (12-18). The above e l a s t i c constants were r e s o l v e d i n r o e l e c t r o n i c and i o n i c c o n t r i b u t i o n s using a procedure developed by Fuchs (19). The i o n i c components, t o which t h e p o t e n t i a l i s i n f a c t f i t t e d , were found t o be c = 0.4805 eV f i - j and c,, = c,, = 0.4202 eV so t h a t t h e Cauchy pressure i s :At-o. The constants o f t h e Born-Mayer p o t e n t i a l were taken t o be A = 13, and B = 5. The o n l y way i n which t h e p r o p e r t i e s o f t h e product s t r u c t u r e s were i n t r o d u c e d i n t o t h e p o t e n t i a l was by ensuring t h a t the energies per atom were i d e n t i c a l f o r t h e bcc s t r u c t u r e and an hcp s t r u c t u r e o f i d e a l a x i a l r a t i o .I n t h e a p p l i c a t i o n s t o be described i n t h i s paper i t was assumed t h a t t h e above p o t e n t i a l i s v a l i d f o r t h e parent bcc phase and t h e product hcp and 9R phases. I n a d d i t i o n t h e nearest neighbour d i s t a n c e i n t h e c l o s e packed planes of t h e product phases was taken t o be t h e same as i n the p a r e n t s t r u c t u r e . I t i s t o be noted t h a t t h i s s i n g l e p o t e n t i a l was used f o r t h e i n t e r a c t i o n s between a l l atoms i n t h e a l l o y , no d i s t i n c t i o n being made between Cu-Cu, Cu-Zn, Zn-Zn and o t h e r bonds. This meant t h a t o r d e r i n g o f t h e atomic species c o u l d n o t be accounted f o r , so t h a t f o r example t h e product phase had the 9R disordered r a t h e r than t h e 18R ordered s t r u c t u r e . The Model.- Computer s i m u l a t i o n of t h e s t r u c t u r e s o f t h e i n t e r f a c e s were c a r r i e d o u t using a model c r y s t a l c o n t a i n i n g 240 atoms. I n i t i a l l y a model r e p r e ~ e n t i n g t h e parent bcc phase was constructed i n t h e form o f a r e c t a n g u l a r block w i t h (111 ), (1T2) and (110) faces. t h e r e were 3 (111) planes, 24 ( I f ? ) planes and 20 (110) planes which were l a b e l l e d -9 t o 10. P e r i o d i c boundary c o n d i t i o n s were used f o r t h e (111) and
(172) faces b u t t h e (110) faces were f i x e d . The i n t e r f a c e was assumed t o be p a r a l l e l t o (110) and the boundary c o n d i t i o n s i m p l i e d t h a t these planes were e f f e c t i v e l y o f i n f i n i t e e x t e n t . An a p p r o p r i a t e number o f a d d i t i o n a l f i x e d (110) planes were added t o t h e model so t h a t t h e boundary atoms o f t h e computational c e l l i n t e r a c t e d w i t h a f u l l quota o f neighbours.
S i n g l e c r y s t a l s o f t h e product hcp and 9R-structures were generated from t h e bcc model by transforming (110) planes i n t o (1011) planes o f hcp and t h e correspond- i n g ( 1
i
5 ) p l a n e s of 9R (20). T h i s was achieved by d i s p l a c i n g t h e (110) planes p a r a l l e l t o themselves i n t h e [ l i p ] d i r e c t i o n , by a p p l y i n g a uniform s t r a i n i n the[TI01 d i r e c t i o n , and f i n a l l y by s h u f f l i n g atoms i n the [1101 and [IT11 d i r e c t i o n s . However, because a small model w i t h p e r i o d i c boundary c o n d i t i o n s was used i t was necessary t o use i n t e r p l a n a r spacings o f 0.838a f o r hcp and 0.811a f o r 9R, i n order t o match the f o u r c e l l s i n t h e bcc (110) plane w i t h t h r e e hcp and two 9R c e l l s as shown i n F i g . 2. For i d e a l s t r u c t u r e s the spacing o f these planes would be 0.817a.
The models o f the i n t e r f a c e s were generated by transforming (110) planes 1 t o 10 o f t h e p a r e n t bcc model. Because o f t h e way i n which t h e a x i a l r a t i o s o f t h e product phases were selected, t h e two i n t e r f a c e s were coherent.
The models o f t h e t h r e e s i n g l e c r y s t a l s and the two i n t e r f a c e s were r e l a x e d using l a t t i c e handling techniques known as DEVIL o r i q i n a l l y developed a t AERE, Harwell and described p r e v i o u s l y (21). An IBM 360/195 computer was used.
Results.- A l l f i v e models s t u d i e d i n t h i s i n v e s t i g a t i o n , the bcc, hcp and 9R s i n g l e m s and t h e bcc/hcp and bcc/9R i n t e r f a c e s r e l a x e d t o s t a b l e e q u i l i b r i u m
s t r u c t u r e s . The energies o i t h e f i v e r e l a x e d s t r u c t u r e s were t h e same t o w i t h i n 4 per cent, i n d i c a t i n g t h a t , d e s p i t e t h e necessary m o d i f i c a t i o n s t o t h e s t r u c t u r e s o f t h e product phases, the s i n g l e p o t e n t i a l t h a t was used gave a s a t i s f a c t o r y represent- a t i o n o f t h e t h r e e phases near t h e i r t r a n s f o r m a t i o n temperatures. However a t t h i s stage i t i s n o t meaningful t o quote i n t e r f a c i a l energies. This i s p a r t l y because t h e
r I I I I I I
0.05
- -
@(r) 0.00. K4
b2id
-0.05
-
- 2[iol
21(b)
-0.10
-
[iio]1 I I I I
1.0 1 . 2 1 . 4
r/a (c)
F i g . 1
.-
E m p i r i c a l i n t e r a t o m i c F i g . 2.- P r o j e c t e d atomic p o t e n t i a l $ ( r ) i n eV f o r Cu Zn a l l o y s . s t r u c t u r e s o f (a) t h e (110) bcc The l o c a t i o n s o f knots K i between plane, ( b ) t h e (1011) hcp plane, s p l i n e s a r e i n d i c a t e d on t h e r / a a x i s , and ( c ) t h e (175) 9R plane, which a being t h e nearest neighbour distance. a r e used i n t h e computer models.The n o t a t i o n l P , 6H
,
4H3 i s explained i n t h e t e x t .F i g . 4.- P r o j e c t e d atomic arrangements o f planes
-
3 t o+
1 of t h e bcc/9R i n t e r f a c e .C4-182 JOURNAL DE PHYSIQUE
p o t e n t i a l i s non-equilibrium f o r t h e hcp and 9R product s t r u c t u r e s so t h a t volume dependent c o n t r i b u t i o n s t o t h e energy must be considered. Also d u r i n g t h e r e l a x a t i o n pvocess t h e i n t e r f a c e s tended t o m i g r a t e towards t h e bcc phase. This was a very encouraging aspect o f t h e r e s u l t s , r e v e a l i n g t o some e x t e n t t h e g l i s s i l e nature o f the boundaries. However i t meant t h a t t h e p r o p o r t i o n s o f parent and product phase remaining i n t h e model was r a t h e r i l l - d e f i n e d , so t h a t t h e r e l a t i v e c o n t r i b u t i o n s t o t h e t o t a l energy o f t h e two phases and o f t h e i n t e r f a c e between them was u n c e r t a i n .
The r e s u l t s t h a t were obtained on t h e s t r u c t u r e o f t h e i n t e r f a c e s a r e much more r e l i a b l e . They a r e however complex, so t h a t o n l y a s i m p l i f i e d p r e s e n t a t i o n o f t h e wealth o f i n f o r m a t i o n contained i n t h e atomic co-ordinates provided by t h e computer s i m u l a t i o n can be attempted. I n p a r t i c u l a r o n l y t h e l o c a t i o n s o f t h e atoms i n t h e (110) bcc planes and t h e p r o j e c t e d l o c a t i o n s i n t h e corresponding hcp and 9R planes w i l l be shown. I n p r a c t i c e t h e hcp and 9R nlanes a r e o f course f a r from f l a t and i n a d d i t i o n planes a r e t r a n s l a t e d r e l a t i v e t o each o t h e r d u r i n g the r e l a x a t i o n process.
The d e t a i l s o f these displacements w i l l be g i v e n elsewhere.
The s t r u c t u r e o f t h e i n t e r f a c e between the bcc and hcp phases i s i l l u s t r a t e d i n Fig. 3. I n t h i s case d u r i n g t h e r e l a x a t i o n t h e i n t e r f a c e migrated i n t o t h e bcc phase and became d i f f u s e . Thus plane
+
1, t h e f i r s t plane o f t h e hcp phase i n t h e i n i t i a l s t r u c t u r e , remained e s s e n t i a l l y unchanged b u t planes 0 t o-
5 o f t h e bcc phase were d i s t o r t e d appreciably t o take up i n t e r m e d i a t e s t r u c t u r e s . Plane-
6 was c l o s e t o t h e o r i g i n a l bcc s t r u c t u r e . Hence i n F i g . 3 p r o j e c t i o n s o f planes 1 t o-
6 a r e shown.It i s convenient t o describe t h e change o f s t r u c t u r e across t h e i n t e r f a c e i n terms o f t h e pseudo two-dimensional polygonal c e l l s shown i n t h e diagrams. The p e r f e c t hcp s t r u c t u r e o f Fig. 2(b) has s i x hexagonal c e l l s each o f which has t h e area o f two u n i t c e l l s , I t may thus be represented by 6H2. Plane
+
1 has t h e same s t r u c t u r e . Plane 0 has f o u r o f these hexagons and f o u r parallelograms o f u n i t area. I t i s thus represented by 4H2+
4P1. Plane-
1 has t h e same s t r u c t u r e b u t the polygons a r e more d i s t o r t e d . The s t r u c t u r e o f plane-
2 i s more complex being 3H,+
4P1+
lP, where P, i n d i c a t e s a p a r a l l e l o g r a m w i t h t w i c e t h e area o f P,. S i m i l a r l y plane-
3 i sc h a r a c t e r i s e d by lH, + 1H3
+
l P 1+
2P3 and plane-
4 by IH,+
lP,+
lPg. F i n a l l y t h e s t r u c t u r e o f plane-
5 although s t i l l d i s t o r t e d can be represented most s a t i s f a c t o r i l y by a s e t of l i n e s r a t h e r than by polygons. These l i n e s may be considered t o bei n f i n i t e l y l o n g parallelograms so t h a t the s t r u c t u r e i s d e f i n e d by lPm. To be c o n s i s t - e n t w i t h the o t h e r planes however t h e t o t a l number o f u n i t c e l l s described by t h e n o t a t i o n should be 12 so t h a t s t r i c t l y 12n-lP ( n + m) i s a more accurate n o t a t i o n f o r plane
-
5 and indeed t h e bcc s t r u c t u r e o f ~ i ~ ! 2 ( a ) .The s t r u c t u r e of t h e i n t e r f a c e between the bcc and 99 phases i s shown i n F i g . 4.
Again t h e boundary has migrated i n t o t h e bcc phase b u t t h e i n t e r f a c e i s narrower than f o r t h e bcc/hcp case. The s t r u c t u r e s o f planes
+
1 t o-
3 a r e shown i n t h e f i g u r e and using t h e above n o t a t i o n can be represented by 4H3, 4H,+
4P1, 2H2+
2P1+
2P,,2H
+
2P1 t 2P3, lP, r e s p e c t i v e l y . Note t h a t i n t h i s case t h e number o f polygonal c e f l s i s even I n each case r e f l e c t i n g t h e f a c t t h a t two 9R c e l l s are matched t o f o u r bcc c e l l s . A model o f one-half o f t h i s s i z e matching cne 9R c e l l t o two bcc c e l l s c o u l d c l e a r l y have been used r e s u l t i n g i n a t o t a l o f 6 r a t h e r than 12 u n i t c e l l s i n each o f t h e diagrams. However t h e l a r g e r model provides a u s e f u l check on t h e r e l a x - a t i o n procedure and as shown i n F i g . 4, the l e f t and r i g h t - h a n d sides o f t h e model d i d i n f a c t r e l a x i d e n t i c a l l y .I n t h e s i m u l a t i o n s o f both t h e bcc/hcp and bcc/9R i n t e r f a c e s l a r g e displacements o f atoms occurred. These displacements c o u l d have been i n d i c a t e d i n F i g s . 3 and 4 b u t would have complicated t h e diagrams. However t h e i r e x t e n t can be appreciated by r e a l i s i n g t h a t i n t h e unrelaxed models planes n
<
0 had t h e bcc s t r u c t u r e of F i g . 2(a).I n a d d i t i o n t h e r e l a x a t i o n s tended t o d i s p l a c e t h e (110) bcc planes and t h e correspond- i n g hcp and 9R planes p a r a l l e l t o themselves. This r e s u l t e d i n an i n - p l a n e t r a n s l a t - i o n o f t h e i n t e r f a c e , b u t as t h e boundaries are spread over several planes t h e d e t a i l s a r e complex and w i l l n o t be described here.
Conclusions
( 1 ) I t has been demonstrated t h a t t h e DEVIL s u i t e o f computer programs can be used successfully t o i n v e s t i g a t e t h e s t r u c t u r e of complex interphase boundaries.
Fig. 3.- Projected atomic arrangements of planes
-
6 to 1 of the bcc/hcp interface.C4-184 JOURNAL DE PHYSIQUE
(2) A new e m p i r i c a l i n t e r a t o m i c p o t e n t i a l has been developed f o r copper-zinc based a l l o y s which can be used s a t i s f a c t o r i l y t o describe t h e h i g h temperature bcc phase and t h e m a r t e n s i t i c hcp and 9R phases.
(3) By a l l o w i n g s l i g h t d i s t o r t i o n s o f t h e hcp and 9R product s t r u c t u r e s q u i t e small b u t r e a l i s t i c computer models w i t h p e r i o d i c boundaries have been generated of t h e bcc/hcp and bcc/9R i n t e r f a c e s .
( 4 ) The r e l a x e d s t r u c t u r e s o f s i n g l e c r y s t a l s o f bcc, hcp and 9R and o f bcc/hcp and bcc/9R b i c r y s t a l s a r e a l l s t a b l e .
(5) The r e l a x e d energies o f t h e f i v e models were s i m i l a r , i n d i c a t i n g t h a t mutual transformations a r e l i k e l y and t h a t i n t e r f a c i a l energies a r e small.
( 6 ) Both t h e bcc/hcp and bcc/9R i n t e r f a c e s migrated d u r i n g t h e r e l a x a t i o n process i n t o t h e bcc phase, i n d i c a t i n g t h a t t h e boundaries a r e g l i s s i l e .
(7) The i n t e r f a c e i n t h e bcc/hcp model i s broad, i n v o l v i n g about 6 sheets of atoms, whereas t h a t i n t h e bcc/9R model i s s i m p l e r i n v o l v i n g o n l y 3 sheets. This suggests t h a t t h e 9R t r a n s f o r m a t i o n m i g h t be p r e f e r r e d .
( 8 ) The present r e s u l t s are considered t o be encouraging and i t i s proposed t o extend t h e p r o j e c t by u s i n g improved p o t e n t i a l s and l a r g e r models. A f u l l account of t h e work i s being published.
The authors acknowledge h e l p provided by J. I. Akhter and I.Q. Malik, K. M i l l e r and M. Balanzat.
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