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HIGH TEMPERATURE DEFORMATION OF MgO.1.8Al2O3 SPINEL SINGLE CRYSTALS OF
AXIAL ORIENTATION
R. Duclos
To cite this version:
R. Duclos. HIGH TEMPERATURE DEFORMATION OF MgO.1.8Al2O3 SPINEL SINGLE CRYS- TALS OF AXIAL ORIENTATION. Journal de Physique Colloques, 1981, 42 (C3), pp.C3-49-C3-58.
�10.1051/jphyscol:1981305�. �jpa-00220696�
JOURNAL DE PHYSIQUE
Colloque C3, supplément au n°6, Tome 42, juin 1981 page C3-49
HIGH TEMPERATURE DEFORMATION OF MgO • 1 . 8 A 1203 SPINEL SINGLE CRYSTALS OF
< 1 1 1 > AXIAL ORIENTATION
R. Duclos
Laboratoire de Structure et Propriétés de l'Etat Solide* Université des Sciences et Techniques de Lille, 59655 Villeneuve d'Ascq Cedex, France
Résumé.- On a déformé en compression à vitesse constante des échantillons de spinellëMgO-l,8Al„0, orientés selon un axe <111>. Pour cet axe de com- pression, le facteur de Schmid des systèmes de glissement {110} <110> est nul; on espérait ainsi favoriser une déformation de nos échantillons par glissement de dislocations dans des plans {lll}. En fait les résultats ob- tenus sont en bon accord avec un mécanisme de déformation où la majeure par- tie de celle-ci proviendrait non pas du glissement des 'dislocations mais de leur montée. On les compare alors avec ceux obtenus dans le cas d'échantillons pratiquement équimolaires pour lesquels la déformation par montée a déjà été établie.
A b s t r a c t . - <\I1> specimens of MgO-1.8Al„0- spinel have been deformed a t con- s t a n t s t r a i n r a t e i n the temperature range 1420-1630°C. For a <111> compression axis the Schmid factor of {110} <110> s l i p systems i s zero and specimen defor- mation was expected to occur by d i s l o c a t i o n glide in { i l l } p l a n e s . In fact our r e s u l t s are consistent with a deformation process due in main p a r t r a t h e r to d i s l o c a t i o n climb than to d i s l o c a t i o n g l i d e ; they are then compared to those obtained in the case of nearly equimolar s p i n e l for which deformation r e s u l t s from d i s l o c a t i o n climb.
1 . I n t r o d u c t i o n . - D i s l o c a t i o n g l i d e i n s p i n e l s M g 0 . n A l20 . , ( w i t h n^2) c a n o c c u r on {110} a s w e l l a s on {111} p l a n e s / 1 , 2 / ; n e v e r t h e l e s s t h e two s l i p s y s t e m s ( {110} <110> a n d { i l l } <110> ) h a v e n o t t h e same c r i - t i c a l r e s o l v e d s h e a r s t r e s s ; t h u s t h e { i l l } <110> o n e i s o n l y o b s e r v e d when t h e Schmid f a c t o r r a t i o S {111} / S {110} i s h i g h e n o u g h s o t h a t i t f a v o u r s s t r o n g l y t h e {111} p l a n e s .
T h i s c o n d i t i o n i s of c o u r s e s a t i s f i e d f o r a <111> o r i e n t a t i o n o f t h e c o m p r e s s i o n a x i s f o r w h i c h
S {111} / S {110} = O . 2 7 / 0
a n d o n e s h o u l d e x p e c t i n t h i s c a s e t h a t d e f o r m a t i o n r e s u l t s from d i s - l o c a t i o n g l i d e i n { i l l } p l a n e s .
I n o r d e r t o s t u d y { i l l } g l i d e , we h a v e d e f o r m e d <111> s a m p l e s o f c o m p o s i t i o n r a t i o n = 1 . 8 . I n f a c t e x c e p t a t t h e o u t s e t o f d e f o r m a t i o n , o u r r e s u l t s a g r e e w i t h a d e f o r m a t i o n p r o c e s s d u e i n m a i n p a r t r a t h e r t o d i s l o c a t i o n c l i m b t h a n t o d i s l o c a t i o n g l i d e ; t h i s p a p e r d e a l s w i t h t h e p o s s i b l e e x p l a n a t i o n s o f t h i s b e h a v i o u r .
Associe au C.N.R.S.
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1981305
C3-50 JOURNAL DE PHYSIQUE
I t p r e s e n t s f i r s t l y r e s u l t s o f c o n s t a n t s t r a i n r a t e t e s t s (a-&
c u r v e s , d i s l o c a t i o n s u b s t r u c t u r e ) f o r a < I l l > C.A. which a r e t h e n ana- l y s e d and d i s c u s s e d by t a k i n g i n t o a c c o u n t o b s e r v a t i o n s made f o r o t h e r C.A. o r i e n t a t i a n s ( n = 1 . 8 ) o r f o r t h e c o m p o s i t i o n r a t i o n = 1.1.
2.- E x p e r i m e n t a l procedure.- Samples o f < I l l > o r i e n t a t i o n o f approxima- t i v e s i z e 2 x 2 x 7.5 mm 3 have been compressed i n a i r on an I n s t r o n t e s t i n g machine i n 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 range: 1420
-
1630°C, 4.6-
2 3 x 10-'s-l. A more complete d e s c r i p t i o n o f t h e compression de- v i c e u s e d h e r e i s r e l a t e d i n /3/.A f t e r d e f o r m a t i o n t h e specimen f a c e s ( {110) and {112) ) a r e ob- s e r v e d by o p t i c a l microscopy t o r e v e a l t h e p r e s e n c e of p o s s i b l e s l i p l i n e s u s i n g t h e Nomarski c o n t r a s t t e c h n i q u e . The d e f o r m a t i o n s u b s t r u c - t u r e i s s t u d i e d a t a macroscopic s c a l e by X-ray t o p o g r a p h s (Berg B a r r e t t t e c h n i q u e / 4 / )
.
3. E x p e r i m e n t a l s r e s u l t s . - 3 . 1 . M e c h a n & c a i - ~ r o p e r t i e s . - F i g u r e 1 i l l u - s t r a t e s ( G - E ) c u r v e s o b t a i n e d f o r t h e l o w e s t s t r a i n r a t e
( t
= 4.6 x 10-6 -1s ) . A f t e r i n g e n e r a l a y i e l d p o i n t , t h e a m p l i t u d e o f which i n - c r e a s e s w i t h d e c r e a s i n g t e m p e r a t u r e s , a p l a t e a u i n flow s t r e s s i s ob- s e r v e d ; t h e same s h a p e i s e q u a l l y found f o r t h e h i g h e r s t r a i n r a t e s .
-6 -1 F i g . 1.- < I l l > C.A.: a-E curves for & = 4.6 x 10 s
The s t r a i n r a t e and t e m p e r a t u r e dependence of t h e flow s t r e s s h a s been s t u d i e d i n t h e p l a t e a u by assuming a r e l a t i o n between o , and T a s
E = A o e x p n
-
Q / ~ T (1)c u r r e n t l y o b s e r v e d d u r i n g h i g h t e m p e r a t u r e d e f o r m a t i o n and e q u i v a l e n t t o :
o = (;/iX)lIn e x p Q/nkT ( 2 )
where Q i s an a p p a r e n t a c t i v a t i o n e n e r g y .
The s t r a i n r a t e s e n s i t i v i t y o f p l a t e a u x h a s been d e t e r m i n e d e i t h e r by s t r a i n r a t e jumps o r by p l o t t i n g t h e l o g a r i t h m o f t h e p l a t e a u s t r e s s v e r s u s l o g a r i t h m o f s t r a i n r a t e ; from 1420 t o 1630°C i t h a s been found t o b e r e l a t i v e l y i n s e n s i t i v e t o t e m p e r a t u r e and e q u a l t o :
The a p p a r e n t a c t i v a t i o n e n e r g y Q h a s been d e t e r m i n e d by p l o t t i n g Log ~ / o v e r s u s r e c i p r o c a l t e m p e r a t u r e a s i n f i g u r e 2. One o b t a i n s ~ . ~ f o r ' Q t h e v a l u e
Q = 5.8
+
0.5eV.Fig. 2.- Determination o f activation energy : E / O ~ ' ' ( i ~ arbitrary units) versus 1 L T ( K ) .
c3-52 JOURNAZ, DE PHYSIQUE
I t must be n o t e d t h a t t h e s e v a l u e s o f n and Q a r e n o t v e r y d i f f e - r e n t from t h o s e o b t a i n e d i n t h e c a s e o f < l o o > c r e p t samples o f compo- s i t i o n n = 1 . 8 /5/ o r n = 1.1 / 6 / .
3 . 2 D i s l o c a t i o n s u b s t r u c t u r e s . - 3 . 2 . 1 S l i p l i n e s . - The o b s e r v a t i o n a f t e r d e f o r m a t i o n o f specimen f a c e s h a s n o t a l l o w e d u s t o r e v e a l s l i p l i n e s e x c e p t n e a r t h e sample ends: t h e y c o r r e s p o n d t o { 1 1 0 1 p l a n e s up t o 1 5 7 0 ° C and t o (1111 p l a n e s a t 1 6 3 0 ° C .
Moreover t h e g e n e r a l s h a p e o f s t r a i n e d samples remains q u i t e s i m i l a r t o t h e u n s t r a i n e d o n e s and d o e s n o t b r i n g any i n f o r m a t i o n on t h e d e f o r m a t i o n mode.
3 . 2 . 2 Berg B a r r e t t topographs.- I n this p a r a g r a p h w e s t u d y on o n e hand t h e s t r a i n i n f l u e n c e on t h e d e f o r m a t i o n s u b s t r u c t u r e o f a specimen deformed a t 1 4 2 0 ° C and on t h e o t h e r hand t h e t e m p e r a t u r e e f f e c t a t a b o u t 5% s t r a i n .
( i ) F i g u r e 3a p r e s e n t s a t y p i c a l view o f s u b s t r u c t u r e o b t a i n e d j u s t b e f o r e t h e upper y i e l d p o i n t ( a r r o w A i n f i g u r e 1 ) . I t i s c l e a r l y s e e n t h a t d e f o r m a t i o n s t a r t s from a s grown s u b g r a i n b o u n d a r i e s ; however t h e r e a l s o t h e d e f o r m a t i o n mode i s d i f f i c u l t t o s p e c i f y .
-6 -1.
F i g , 3 . - S t r a i n e f f e c t og d i s d o c a t $ n s u b s t r u c t u r e a t T = 1420°c, E = 4.6 x 10 s
,
[I 1 1 1 C.A. v e r t i c a l , (1 12) f a c e s , g = [404];bars = 1 m. a , b and c r e f e r t o s t r a i n s i n d i c a t e d b y a r r o w s A,B and C i n f i g u r e 1 .
A t t h e upper y i e l d p o i n t ( a r r o w B i n f i g u r e 1) t h e d e f o r m a t i o n and c o n s e q u e n t l y t h e s u b s t r u c t u r e i s n o t v e r y homogeneous t h r o u g h t h e whole sample; it c o n s i s t s o f g l i d e bands and/or p o l y g o n i z a t i o n bounda- ries which do n o t l i e i n s i m p l e c r y s t a l l o g r a p h i c p l a n e s ( F i g . 3 b ) .
A s t h e s t r a i n p r o c e e d s (arrow C i n f i g u r e 1)
,
t h i s d i r e c t i o n a l s u b s t r u c t u r e d i s a p p e a r s and i s r e p l a c e d by a n o t h e r o n e made w i t h d i s - l o c a t i o n s h e e t s c r u d e l y g a t h e r e d i n t o randomly o r i e n t e d s u b g r a i n boun- d a r i e s ( F i g . 3 c ) ; i n some p a r t s o f t h e specimen t h e l a c k o f t h e s e boun- d a r i e s c a n b e n o t e d .(ii) I n 5% s t r a i n e d samples t h e t e m p e r a t u r e o n l y i n d u c e s a s l o w e v o l u t i o n o f t h e s u b s t r u c t u r e and d o e s n o t produce any s h a r p t r a n s i t i o n o f t h e t y p e o b s e r v e d f o r
< l o o >
specimens /5,7/.A t 1460°C ( F i g . 4 a ) b o u n d a r i e s a r e o n l y v i s i b l e i n some p l a c e s , t h e c e l l u l a r s u b s t r u c t u r e h a s a l r e a d y p r a c t i c a l l y v a n i s h e d .
Above t h i s t e m p e r a t u r e t h e o v e r a l l p i c t u r e remains t h e same, d e s - o r i e n t a t i o n s i n t h e kpecimen simply f a d e o u t a s t e m p e r a t u r e i n c r e a s e s
( ~ i g . 4b and c ) .
I n summary Berg B a r r e t t t o p o g r a p h s show t h e f a s t d i s a p p e a r a n c e o f a d i r e c t i o n a l s u b s t r u c t u r e a t r e l a t i v e l y l o w t e m p e r a t u r e and micro- s t r u c t u r e o b s e r v a t i o n s f a i l t o produce c l e a r e v i d e n c e s o f g l i d e e v e n t s .
F i g . 4.- Temperature e f f e c t on d i s l o c a t i o n substruc_ture a t 5% s t r a i n i = 4.6 x 10-6s-1;
[ill]
C . A . v e r t i c a l , (1 12) f a c e s ,3
= 14041; b a r s = 1 mm, a : T = 1460°c, b: T = 1500°c, c: T = 1630°c.c3-54 JOURNAL DE PHYSIQUE
4 . D i s c u s s i o n . - One o f t h e major p o i n t s t o u n d e r s t a n d c o n c e r n s t h e de- f o r m a t i o n mode o f specimens. With t h i s aim i n mind we compare t h e above e x p e r i m e n t a l r e s u l t s t o t h o s e o b t a i n e d f o r t h e same c o m p o s i t i o n r a t i o on
< l o o >
and <110> specimens where d e f o r m a t i o n r e s u l t s unambi- g u o u s l y from d i s l o c a t i o n g l i d e . Numerous d i f f e r e n c e s a p p e a r :( i ) i n t h e p r e s e n t c a s e , no workhardening i s shown beyond y i e l - d i n g w h a t e v e r be t h e t e m p e r a t u r e an? s t r a i n r a t e , i n s t r o n g c o n t r a s t t o t h e two o t h e r o r i e n t a t i o n s
( < l o o >
C.A. /3/; <110> C.A. f i g u r e 5 ) i n which a t low t e m p e r a t u r e (T % 1420°C) t h e workhardening, f o r analogous s t r a i n r a t e s , c a n r e a c h v a l u e s a s h i g h a s p/200;( i i ) s l i p t r a c e s c a n b e o b s e r v e d f o r
< l o o >
and < l l O > o r i e n t a t i o n s i n t h e whole r a n g e of t e s t e d t e m p e r a t u r e s (1350-1630°C) / 2 / ;-6 - 1 F i g . 5.- <110> C . A . : 0-& curves f o r = 6 x 10 s
.
( i i i ) d i s l o c a t i o n s u b s t r u c t u r e , p a r t i c u l a r l y a t low t e m p e r a t u r e ( T Q, 1420°C) i s v e r y d i r e c t i o n a l i n t h e c a s e o f <110> and
< l o o >
s p e c i - mens and composed o f s u b g r a i n b o u n d a r i e s l e f t on t h e l o c a t i o n o f g l i d e bands ( F i g . 6 ) .These d i f f e r e n c e s s u g g e s t u s t h a t d e f o r m a t i o n s h o u l d r e s u l t h e r e from d i s l o c a t i o n climb. T h i s h y p o t h e s i s i s n o t u n r e a l i s t i c and c a n b e s u p p o r t e d by f u r t h e r arguments :
( i ) we can c a l c u l a t e , from o u r e x p e r i m e n t a l c o n d i t i o n s (T,o)
,
t h es t e a d y s t a t e c r e e p r a t e r e s u l t i n g from p u r e c l i m b a c c o r d i n g t o t h e Na- barro-Weertman model /8/ and assuming t h a t d i s l o c a t i o n c l i m b i s con- t r o l l e d by oxygen d i f f u s i o n .
Fig. 6.- Typical dislocation substructure of <110>
specimen deformed a t T; 1 4 1 0 ° ~ , ' & = 0.051 [ l l o ] C . A . vertical, (001) face, g = [044]; bar = 1 m.
With t h e f o l l o w i n g p a r a m e t e r s :
T = 1420°C; o = 220 MPa i . e . ac = 145 MPa ( t h e climb Schmid f a c t o r on {110} p l a n e s i s Sc = 0 . 6 7 ) ; Dox (1420°C) 2 4 x 1 0 - ~ ~ m ~ . s - l /9/; i2 = 2 x 10-29m3 ( t h e r e l e v a n t oxygen i o n i c volume) and b = 2.9 x 10-1°m
( i t i s assumed a s i n
/ l o /
t h a t p a r t i a l s c l i m b i n d e p e n d e n t l y ) t h e c r e e p r a t e i s e s t i m a t e d from /8/ a ti . e . t h e same o r d e r o f magnitude t h a n t h e s t r a i n r a t e u s e d i n t h i s t e s t
( E
= 4.6 x 1 0 ~ ~ s - l ).
( i i ) Values o f s t r e s s - s t r a i n r a t e s e n s i t i v i t y , n , and a c t i v a t i o n e n e r g y , Q , a l t h o u g h t h e y a r e s l i g h t l y h i g h e r t h a n t h e e x p e c t e d v a l u e s , a r e n o t i n d i s c r e p a n c y w i t h t h i s model.
( i i i ) i f i t i s c l a i m e d t h a t t h e d e f o r m a t i o n of o u r < I l l > samples f o r n = 1 . 8 , r e s u l t s from d i s l o c a t i o n c l i m b , a s i t h a s been shown t o b e t h e c a s e f o r specimens o f c o m p o s i t i o n n = 1.1 / 6 / , t h e n t h e s e two t y p e s of samples o u g h t t o show s i m i l a r m e c h a n i c a l b e h a v i o u r s .
S i n c e samples o f c 0 m p o s i t i o n l . l we have deformed /11/ had n o t a
< I l l > o r i e n t a t i o n , comparison h a s t o c o n s i d e r t h e r e s o l v e d s t r e s s e s a.Sc, a t t h e same r e s o l v e d s t r a i n rate
&/sc,
on t h e assumed t110) c l i m b p l a n e s (Sc b e i n g t h e c l i m b Schmid f a c t o r ) v e r s u s s t r a i n . T h i s i s doneC3-56
i n f i g u r e 7.
JOURNAL DE PHYSIQUE
Fig.7.- Climb s t r e s s 0, = U.S, on the {110} planes versus s t r a i n f o r <100>.and
! I l l > samples (n = 1.8) and <loo> samples (n = 1 . I ) ; resolved s t r a i n r a t e E9 =
€ I S , = 2.8 x 10-~s-!
I t i s s e e n t h a t < 1 1 1 > specimens of composition n = 1 . 8 and
< l o o >
specimens of composition n = 1.1 do compare very w e l l . On t h i s f i g u r e we have a l s o drawn f o r comparison ( U c - ~ ) c u r v e s f o r
< l o o >
samples o f composition n = 1 . 8 where deformation i s produced by d i s l o c a t i o n g l i d e ; t h e s e curves s e p a r a t e c l e a r l y from t h e t w ~ o t h e r s .o ow ever
we do n o t c l a i m t h a t s l i p does n o t occur a t a l l d u r i n g deformation. I n p a r t i c u l a r t h e d i s l o c a t i o n s u b s t r u c t u r e a t t h e upper y i e l d p o i n t ( F i g . 3 b ) i s c o n s i s t e n t w i t h g l i d e . We have checked t h a t a p p l i e d s t r e s s e s a t t h i s p o i n t a r e adequate f o r a c t i v a t i n g ( 1 1 1 ) < 1 1 0 >s l i p systems by comparing ( F i g . 8) t h e r e s o l v e d s h e a r s t r e s s e s o b t a i n e d f o r t h e s e g l i d e systems t o t h e i r c . r . s . s . deduced from < 1 1 0 > (a-E) curves measured i n s i m i l a r c o n d i t i o n s (Fig. 5 ) . The two curves can be superposed showing t h a t e f f e c t i v e l y g l i d e can occur i n t h i s p a r t of de- formation curves. This a l s o shows t h a t , e s p e c i a l l y a t low temperature, t h e s t r e s s i n t h e p l a t e a u does n o t reach t h e c . r . s . s . of l 1 1 1 ) < 1 1 0 >
s l i p systems.
The problem now i s l e f t t o understand why g l i d e does n o t c o n t i n u e when deformation proceeds. I n f a c t , w h i l e f o r t h i s o r i e n t a t i o n of com- p r e s s i o n a x i s , t h e Schmid f a c t o r of { 1 1 0 3 < 1 1 0 > s l i p systems i s z e r o , t h e climb Schmid f a c t o r o f t h o s e { 1 1 0 ) p l a n e s i n which d i s l o c a t i o n can climb and which a r e n o t i n zone w i t h t h e C.A. i s 0 . 6 7 . So due t o t h e
h i g h s t r e s s e s needed f o r a c t i v a t i n g g l i d e i n (111) p l a n e s i t t u r n s o u t t h a t d i s l o c a t i o n climb becomes e a s i e r and more c o m p e t i t i v e t h a n d i s l o - c a t i o n g l i d e .
F i g . 8 . - Comparison of t h e r e s o l v e d s h e a r s t r e s s on ( 1 1 1 ) < ] l o > s l i p systems a t t h e upper y i e l d p o i n t of 0-& c u r v e s f o r < I l l > specimens w i t h t h e c . r . s . s . of t h e s e s l i p systems deduced from G-E c u r v e s f o r <110> samples, f o r e q u i v a l e n t s h e a r s t r a i n r a t e s ? .
One c a n sum up t h e d e f o r m a t i o n of < 1 1 1 > samples a s below; a s e d g e d i s l o c a t i o n s ( t h e l e s s mobile d i s l o c a t i o n s ) a r e produced t h e s t r a i n r a t e r e s u l t i n g from t h e i r climb become h i g h enough t o accomodate t h e imposed s t r a i n r a t e ; t h i s c a u s e s t h e s t r e s s t o d r o p making t h u s f u r t h e r g l i d e more d i f f i c u l t .
O b s e r v a t i o n o f
I1111
g l i d e needs t h e r e f o r e a Schmid f a c t o r (S.F) r a t i o , ( 1 1 1 1 g l i d e S.F. / { 1 1 0 } c l i m b S.F., i n f a v o u r o f g l i d e ; t h i s i s t h e c a s e f o r t h e < 1 1 0 > C.A. w i t h :5 . Conclusion.- T h i s s t u d y h a s shown two major p o i n t s :
( i ) when d i s l o c a t i o n g l i d e i s h i n d e r e d , non e q u i m o l a r s p i n e l can e x h i b i t a mechanical b e h a v i o u r a s h a r d a s t h e one o f e q u i m o l a r o r n e a r - l y e q u i m o l a r s p i n e l ;
( i i ) c o n s e q u e n t l y t h e h i g h e r p l a s t i c i t y o b s e r v e d w i t h t h e i n c r e a - s i n g o f c o m p o s i t i o n r a t i o r e s u l t s from f a c t o r s f a v o u r i n g a h i g h e r g l i d e m o b i l i t y .
JOURNAL DE PHYSIQUE
R e f e r e n c e s
/1/ L e w i s , M.H., P h i l o s . M a g .
17
( 1 9 6 8 ) 4 8 1/2/ D u c l o s , R . , D o u k h a n , N . , E s c a i g , B . , accepted f o r p u b l i c a t i o n by P h i l o s . M a g .
/3/ D u c l o s , R . , C r a m p o n , J . , Mem. S c i . R e v . M e t L X X V I I ( 1 9 8 0 ) 1 5 1
/ 4 / D u c l o s , X . , blem. S c i . R e v . M e t LXXV ( 1 9 7 8 ) 1 9 9 /5/ D o u k h a n , N . , D u c l o s , R., E s c a i g , B . , J . P h y s i q u e
C o l l o q .
34
( 1 9 7 3 ) C9-379/ 6 / D u c l o s , R . , D o u k h a n , N . , E s c a i g , B . , J . M a t e r . S c i . 1 3 ( 1 9 7 8 ) 1 7 4 0
/ 7 / D o u k h a n , N . , D u c l o s , R . , E s c a i g , B . , J . P h y s i q u e C o l l o q . 3 7 ( 1 9 7 6 ) C7-566
-
/a/
W e e r t m a n , J . , T r a n s . A.S.M.61
( 1 9 6 8 ) 6 8 1/ 9 / A n d o , K . , O i s h i , Y . , 5. C h e m . P h y s .
61
( 1 9 7 4 ) 6 2 5/ l o /
E s c a i g , B . , R e v u e P h y s . A p p l .14
( 1 9 7 9 ) 4 6 9/11/ D o u k h a n , N . , D u c l o s , R . , E s c a i g , B., J . P h y s i q u e C o l l o q . 4 1 ( 1 9 8 0 ) C6-139
