BOND-ANGLE DISTRIBUTION FUNCTIONS IN METALLIC GLASSES

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BOND-ANGLE DISTRIBUTION FUNCTIONS IN

METALLIC GLASSES

J. Hafner

To cite this version:

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JOURNAL

DE

PHYSIQUE

Colloque

C9,

supplément au n012,

T o m e 46, décembre 1985

page C9-69

BOND-ANGLE DISTRIBUTION FUNCTIONS IN METALLIC GLASSES

J. Hafner

I n s t i t u t fUr Theoretiscke Physik, l'eckniscke Universitçlt Wien, Karlsplatz 13, A 1040 Wien, Austria

Abstract -

Bond-angle distri bution functions have been calculated for

realistic models of metallic glasses. They suggest a defected icosa-

hedral short-range bond-orientational order and a close analogy of the

short-range topological order in the amorphous and in the crystalline

states.

1 -

INTROùUCTION

One of the basic problems in the study of non-crystalline structures is their charac-

terisation (both by way of theory and experiment) beyond the statistical information

on interparticle distances and coordination numbers contained in a set of partial

pair correlation functions.Although this information is sufficient to express the

equation of state and even to describe chemical short range order, it is clear that

the pair correlation function - which is a projection ont0 one dimension of three-

dimensional information -

is insufficiently sensitive to topological short-range

order. Several geometrical characterisations of the non-crystalline structures have

been explored: (a) The Voronoi polyhedron analysis /1/ and its general isation for

multi-component systems, the radical plane method /2/; (b) the interstice

-

inter-

stice and centre

-

interstice correlation functions /3/; and (c) bond-angle corre-

lation functions /4-6/. The first two methods offer essentially topological informa-

tion -

face and edge statistics of the polyhedra in the former case - which is often

diff icul

t to interpret (small deviations from the ideal polyhedron shape frequently

lead to a complete change of the face and edge statistics) and require extensive

computations.

Interest in bond-orientational order has been revived recently by the remarkable

observation of the existence of perfect icosahedral near-neighbour bond-orientational

order in Al-Mn "quasi-crystals" (quasi-crystals have quasi-periodic (incommensurate)

rather than periodic translational order) /7,8/.

Short-range icosahedral order is

believed by many authors to be a characteristic feature of metallic glasses /9-11/

and supercool ed

1 iquids /12,13/. Ne1 son and CO-workers /10,14,15/ have developed

a Landau description of short-range icosahedral order in supercooled liquids and

metallic glasses. It is argued that perfect icosahedral order is related to an ideal

icosahedral crystal (the polytope {3,3,5}

)

which cjonsists of the closest packing

of 600 tetrahedra (120 particles) on the surface S of a four-dimensional hypersphere.

Supercooled liquids and metallic glasses are viewed as defected states of bond-orien-

tational order: regions of short-range icosahedral I3,3,51 order are broken by an

array of disclination lines which are forced in during the mapping of the ideal four-

dimensional structure on the flat three-dimensional real space as a consequence of

the impossibil ity of fil1 ing a three-dimensional space with an icosahedral network /16/.

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C9-70 JOURNAL D E PHYSIQUE

The c r y s t a l l i n e Frank-Kasper phases /17/ a r e an ordered a r r a y o f t h e s e d i s c l i n a t i o n 1 ines, t h e metal 1 i c g l a s s e s a r e considered as a d i s o r d e r e d , entangl ed network o f these l i n e s ( t h e entanglement o f t h e d i s c l i n a t i o n l i n e s p l a y s an i m p o r t a n t r o l e i n e x p l a i n i n g t h e m e t a s t a b i l i t y o f g l a s s e s ) . T h i s s u b s t a n t i a t e s an e a r l i e r s u g g e s t i o n o f t h e p r e s e n t a u t h o r t h a t t h e f o r m a t i o n o f m e t a l l i c g l a s s e s and t h e f o r m a t i o n of t o p o l o g i c a l l y close-packed i n t e r m e t a l l i c compounds such as t h e Frank-Kasper phases (and hence t h e i r s t r u c t u r e s ! ) a r e c l o s e l y connected /Il/.

The a i m o f t h e p r e s e n t paper i s t o a n a l y z e t h e b o n d - o r i e n t a t i o n a l o r d e r i n r e a l i s t i c computer-generated moddls o f m e t a l l i c glasses i n terms o f t h e i r bond-angle d i s t r i - b u t i o n f u n c t i o n s . The c o n s t r u c t i o n o f t h e s t r u c t u r a l models i s based on pseudopoten- t i a l d e r i v e d i n t e r a t o m i c f o r c e s and s t a t i c e n e r g y - m i n i m i z a t i o n and m o l e c u l a r dynamics techniques

-

t h i s i s b r i e f l y reviewed i n Sec.11. Bond-angle d i s t r i b u t i o n f u n c t i o n s f o r amorphous and l i q u i d Ca-Mg and Mg-Zn a l l o y s d e r i v e d f r o m b o t h t h e s t a t i c and t h e dynamic s i m u l a t i o n s a r e presented i n Sec.111. The r e s u l t s i n d i c a t e a l a r g e number o f n e a r - p e r f e c t and o f d e f e c t e d i c o s a h e d r a l near-neighbour c o o r d i n a t i o n s , b u t a l s o a s u b s t a n t i a l number of caped pentagonal p r i s m a t i c c o n f i g u r a t i o n s ( t h e y may be viewed as twinned c o n f i g u r a t i o n s o f t h e icosahedron). Thus t h e b o n d - o r i e n t a t i o n a l o r d e r t u r n s o u t t o be s u r p r i s i n g l y s i m i l a r t o t h a t i n t h e c r y s t a l l i n e i n t e r m e t a l l i c compound Mg Zn /18/, which i s produced f r o m t h e amorphous phase by a polymorphous c r y s t a i 1 9 a t S 8 n process /19/.

II

-

MODELLING THE STRUCTURE OF METALLIC GLASSES

I n general t h e c o n s t r u c t i o n o f any s t r u c t u r a l model f o r an amorphous s o l i d w i l l p r o - ceed by t h e f o l l o w i n g steps: ( a ) S e l e c t i o n o f a s e t o f i n t e r a t o m i c p o t e n t i a l s

-

de- r i v e d from t h e e l e c t r o n i c s t r u c t u r e o f t h e m a t e r i a l under c o n s i d e r a t i o n i f p o s s i b l e , e m p i r i c a l o t h e r w i s e ; ( b ) c o n s t r u c t i o n o f an i n i t i a l s t a r t i n g s t r u c t u r e , and ( c ) r e - finement o f t h i s i n i t i a l model by s t a t i c energy m i n i m i z a t i o n /20/ o r a dynamic r e l a - x a t i o n u s i n g m o l e c u l a r dynamics o r Monte Car10 t e c h n i q u e s .

For Our simple-metal a l l o y s r e l i a b l e i n t e r a t o m i c p o t e n t i a l s may be d e r i v e d f r o m p s e u d o p o t e n t i a l t h e o r y /11,21/. As t h e s t a b l e i n t e r m e t a l l i c compounds i n t h e s e a l l o y systems (Ca-Mg, Mg-Zn) a r e o f a t o p o l o g i c a l l y close-packed t y p e , a dense random packing o f hard spheres (DRPHS) i s considered as an adequate s t a r t i n g s t r u c t u r e . The s t a t i c energy m i n i m i z a t i o n y i e l d s e x c e l l e n t r e s u l t s f o r amorphous Ca Mg a l l o y s as can be seen f r o m t h e comparison o f t h e t h e o r e t i c a l curves w i t h t h e ~-?-gy d Q f f r a c - t i o n experiments o f N a s s i f e t a l /22/, see F i g . 1 . Note t h a t t h i s i s p r o b a b l y t h e o n l y t h e o r e t i c a l p r e d i c t i o n o f an amorphous s t r u c t u r e which i n v o l v e s no a d j u s t a b l e p a r a - meters - n o t even t h e d e n s i t y has been t a k e n f r o m experiment. I n f a c t t h e p a r t i a l s t a t i c s t r u c t u r e f a c t o r s had a l r e a d y been p u b l i s h e d

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when t h e d i f f r a c t i o n e x p e r i - ment was performed.

The s i t u a t i o n i s more d i f f i c u l t f o r Mg Zn

.

Again t h e r e i s good agreement between t h e r e s u l t o f a s t a t i c c l u s t e r - r e l a x a t j 8 n ? % l c u l a t i o n / 2 i / and t h e n e u t r o n d i f f r a c - t i o n experiments /23/, b u t t h e X-ray d a t a /24,25/ show a weak, though d i s t i n c t p r e - peak i n d i c a t i n g a c e r t a i n degree o f chemical s h o r t - r a n g e o r d e r (CSRO). T h i s i s a f e a t u r e t h a t a s t a t i c r e l a x a t i o n s t a r t i n g f r o m a c h e m i c a l l y random c o n f i g u r a t i o n w i l l never be a b l e t o reproduce. However, we have r e c e n t l y presented a thermodynamic v a r i a t i o n a l t e c h n i q u e f o r c a l c u l a t i n g CSRO i n l i q u i d and supercooled l i q u i d a l l o y s /26/

,

w i t h good r e s u l t s f o r Mg70Zn30 and even more s t r o n g l y o r d e r i n g a l l o y s such as Li-Mg and Li-Pb.

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F i g . 1

-

S t a t i c s t r u c t u r e f a c t o r S(q) and reduced p a i r c o r r e l a t i o n f u n c t i o n G ( R ) (weighted f o r X-ray d i f f r a c t i o n ) o f amorphous Ca Mg F u l l 1 in e

-

s t a t i c r e l a x a t i o n c a l c u l a t i o n ( i n c l u d i n g temperature e f f ? c t

2''

v i a t h e c a l c u l a t e d p a r t i a l Debye-Waller f a c t o r s /21/), crosses experiment /22/.

b u t i o n o f t h e two chemical species, t h e d i f f u s i o n r a t e must be s u f f i c i e n t l y h i g h . I n Our case t h e system was e q u i l i b r a t e d a t temperatures r a n g i n g between 2000 K and do0 K, i . e . w e l l above t h e l i q u i d u s temperature.

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Fig.2

-

Composite s t a t i c s t r u c t u r e f a c t o r S(q) (X-ray weighted) f o r amorphous Mg Zn : histogram

-

c a l c u l a t e d using t h e molecular dynamics "quench" pr6!?ed8e described i n t h e t e x t ( i n c l u d i n g temperature broadening), t h i n l i n e - thermodynamic v a r i a t i o n a l c a l c u l a t i o n (see Ref. /26/ f o r d e t a i l s ) , crosses and c i r c l e s

-

experiment /24,25/.

t h e 1iqu.idus temperature. E f f e c t s of the c o o l i n g h i s t o r y a r e expected t o show up i f undercooled 1 iq u i d s a r e quenched.

Fig.2 demonstrates t h a t a weak prepeak shows up both i n t h e molecular dynamics quench and i n t h e thermodynamic v a r i a t i o n a l c a l c u l a t i o n and t h a t i t compares q u i t e favoura- b l y w i t h experiment. I n t e g r a t i n g over t h e f i r s t peaks o f t h e p a r t i a l p a i r c o r r e l a - t i o n functions y i e l d s a Cargill-Spaepen short-range order parameter o f t = 0 . 1 3 5 . The CSRO found i n t h i s system can be t r a c e d back t o an i n t e r a c t i o n between unlike-neighbour p a i r s which i s s l i g h t l y stronger than t h e average i n t e r a c t i o n between l i k e - neighbour p a i r s /26/. I n Ca-Mg on t h e other hand the p a i r p o t e n t i a l s a r e e x a c t l y a d d i t i v e . However, s u b s t a n t i a l CSRO i s expected i n h e t e r o v a l e n t Ca- and Mg-based glasses.

I n t h e f o l l o w i n g we study the bond-angle d i s t r i b u t i o n s i n these model s t r u c t u r e s . III

-

BOND-ANGLE DISTRIBUTIONS

The bond-angle d i s t r i b u t i o n f u n c t i o n f ( 0 ) measures t h e p r o b a b i l i t y t h a t t h e d i r e c - t i o n s from a c e n t r a l atom t o two o f i t s neighbours form an angle a. f ( o ) i s essen- t i a l l a r a d i a l average over t h e t r i p l e t c o r r e l a t i o n f u n c t i o n g3(R1,R2,e) over t h e nearest n e i ~ h b o u r c o o r d i n a t i o n s h e l l :

where D i s t h e maximum d i s t a n c e between two nearest neighbour atoms ( t a k e n t o be t h e d i s t a n c e where t h e f i r s t minimum i n t h e p a i r c o r r e l a t i o n f u n c t i o n occurs). I n p r a c t i c e f ( o ) i s v e r y e a s i l y c a l c u l a t e d from t h e known coordinates and t h e nearest- neighbour t a b l e o f t h e model c l u s t e r .

We begin by c o n s i d e r i n g the bond-angle d i s t r i b u t i o n s o f t h e s t a t i c a l l y r e l a x e d models (see Fig.3J. For both Ca1fiMg30 t h e c a l c u l a t e d d i s t r i b u t i o n shows a prominent peak near e s5d

,

a broad max um near e d 1 0

-

115' which has a s l i g h t shoulder a t t h e low-angle side, and a r a t h e r f l a t maximum around es150°. T h i s bond angle d i s t r i b u - t i o n i s v e r y s i m i l a r t o t h a t c a l c u l a t e d f o r t h e c r y s t a l l i n e i n t e r m e t a l l i c compound Mg Zn2 /1&.46/.

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I Amorphous

A m o r p h o u s

'C

1 n

l n t e r m e t a l l i c

Fig.3

-

T o t a l bond-angle d i s t r i b u t i o n f u n c t i o n s f ( B ) f o r s t a t i c a l l y r e l a x e d models o f amorphous Ca and Mg70Zn30 and f o r t h e i n t e r m e t a l l i c compound Mg51Zn20 ( a f t e r 7 ' 2 ? ? / 6 / ) .

atomic species of t h e component atoms (Fig.4J. f ( o ) i n c r y s t a l l i n e Mg Zn has t h r e e r a t h e r sharp peaks a t a 6 0

.

o % l l 4 - l l a anan ot$30°, i . e . ver8 c l a i e

28

t h e t h e i d e a l icosahedral bond angles o f 0=63.5

,

@=116.5

,

and o= 180

.

Indeed a l 1 Zn atoms have àcosahgdral c o o r d i n a t i o n polyhedra i n Mg Zn /18/. f ( 8 ) has t h r e e maxima a t m 5 7

,

110

,

and 145O, t h a t near 110' being ?Ath@ assymmk!&ric. I n terms o f t h e p a r t i a l bond-angle d i s t r i b u t i o n s , t h e s i m i l a r i t y between t h e c r y s t a l l i n e and t h e amorphous s t r u c t u r e s i s even more s t r i k i n g . The f . ( o ) ' s o f t h e glasses a r e j u s t broadened v e r s i o n s o f t h e corresponding d i s t r i b u t i o n t u n c t i o n s i n t h e c r y s t a l . Not unexpectedly, t h e broadening (and hence t h e d i s t o r t i o n o f t h e bond-angles) i s more important around t h e m i n o r i t y atoms (Zn i n t h i s case).

The s t a t e d s i m i l a r i t y between t h e short-range order i n t h e c r y s t a l l i n e and i n t h e g l a s s y s t r u c t u r s - i s a l s o c o n s i s t e n t w i t h t h e a n a l y s i s o f t h i n t e r a t o m i c distances: t h e Zn-Zn distances i n t h e c r y s t a l range from 2.71 t o 3.07

i ,

t h e maximum o f t h e p a r t i a l p a i r c o r r e l a t i o n f u n c t i o n g (R) o f t h e m e t a l l i c g l a s s oc u r s a t R=2.88

8 ,

f o r t e Mg-Mg distances t h e c o r r e s p 6 ~ 6 f n ~ numbers a r e 3.00 t o 3.35

k

( c r y s a l ) and 3.04

$

( q l a r s ) ; f o r t h e M -Zn distances 2.60 t o 3.20

8

( c r y s t a l ) and 2.92

h

(glass).. Uoes t h e l o c a i t o p o l o g i c a l order change i f t h e mode1 s t r u c t u r e i s produced by dynamic

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Fig.4

-

P a r t i a l bond-angle d i s t r i b u t i o n f u n c t i o n s f . ( o ) in ( a ) g l a s s y Mg Zn (model s t r u c t u r e produced by s t a t i c r e l a x a t j o n of a DRPHS) and

(63

do

t h e i n t e r m e t a l l i c compound Mg51Zn20.

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CSRO i s t o be expected, as t h e p a i r p o t e n t i a l O (R) ( t h e average p a i r p o t e n t i a l O,, (R) c o u p l e s t o t h e l o c a l f l u c t u a t i o n s i n t h e Mgan number d e n s i t y , t h e o r d e r i n g p&!ential O (R) c o u p l e s t o t h e l o c a l c o n c e n t r a t i o n f l u c t u a t i o n s , @ (R) p r o v i d e s t h e c r o s s - c o 6 b l i n g ) i s v e r y weak i n Mg-Zn (see /26/ f o r d e t a i l s ) .

~ f b r

t h e dynamic r e l a x a t i o n , t h e p a r t i a l c o o r d i n a t i o n numbers i n t h e g l a s s and i n t h e c r y s t a l a v e r y s i m i l a r .

I V - BOl'iD-ANGLE DISTRiBUTIONS AND ICOSAHEDRAL OROER

Sachdev and Nelson /IO/ have shown t h a t t h e Landau d e s c r i p t i o n o f m e t a l l i c g l a s s e s p r e d i c t s peaks i n t h e s t a t i c s t r u c t u r e f a c t o r o f t h e g l a s s a t p o s i t i o n s determined by t h e symmetries o f t h e i d e a l , curved-space i c o s a h e d r a l c r y s t a l . Peak p o s i t i o n s a r e ex- p e c t e d t o be r e l a t e d t h r o u g h Q

/ Q

- 1.7, Q / Q =2.0

-

i n good, b u t n o t e x c e l l e n t agreement w i t h t h e p o s i t i o n s d e t e r m f n e i - f r o m e i s i i e h experiment o r computer m o d e l l i n g . Can a c o n s i d e r a t i o n o f t h e bond-angle d i s t r i b u t i o n s b r i n g f u r t h e r e v i d e n c e f o r icosahe- u r a l b o n d - o r i e n t a t i o n a l o r d e r i n m e t a l l i c g l a s s e s ?

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C9-76 JOURNAL

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Kasper polyhedra w i t h c o o r d i n a t i o n numbers Z=14,15,and 16 a r e l i n k s o f two, three, and f o u r

-

72 d i s c l i n a t i o n l i g e s , t h e canonical Bernal holes w i t h c o o r d i n a t i o n numbers 2=10,3, and 8 l i n k +72 d i s c l i n a t i o n l i n e s (Fig.7, Frank and Kasper g i v e a p r o o f t h a t no t r i a n g u l a t e d c o o r d i n a t i o n polyhedra w i t h Z=13 o r Z = l l a r e p o s s i b l e ) . I n t h e f o l l o w i n g we s h a l l sttempt t o analyze t h e bond-angle d i s t r i b u t i o n s c a l c u l a t e d from t h e computer-generated models i n terms o f t h e bond-angles i n these c o o r d i n a t i o n polyhedra. Fig.d shows t h e bond-angle d i s t r i b u t i o n s o f t h e icosahedron, o f t h e cano- n i c a l Frank-Kasper polyhedra and o f t h e canonical Bernal holes. They a r e weighted according t o t h e occurence o f t h e c o r r e s p o n d i n j c o o r d i n a t i o n numbers i n t h e mode1 s t r u c t u r e s (Z=13 ~ o o r d i n a t i o n s a r e d i s t r i b u t e d e q u a l l y between t h e Z=12 and Z=14 polyhedra, i n tne same manner we proceed f o r t h e Z = l l c o o r d i n a t i o n s ) and .:olded w i t h a i a u s s i a n d i s t r i b u t d o n those w i d t h a t haIf-maximu8 increasea w i t h t h e bond angle

( i t i s taken t o be 6 at"0=60° and increases t o 18 a t 0=180

-

t h i s should account f o r t h e f a c t t h a t t h e bond-angles between atoms which a r e n o t nearest neighbours on t h e surface o f t h e c o o r d i n a t i o n polyhedron a r e more e a s i l y d i s t o r t e d ) . The r e s u l t s agrees q u i t e w e l l w i t h t h e c a l g u l a t e d bond-angle d i s t r i b u t i o n , except f o r a s l i g h t s h i f h of t h e maximum near 0%60 t o lower angles and a too pronounced minimum near 01.90

.

The former probably means t h a t Our simple assignment o f t h e c o o r d i n a t i o n numbers underestimates t h e number o f Z=14 and Z=15 polyhedra

-

t h i s should be v e r i - f i e d by a Voronoi polyhedron a n a l y s i s . The second p o i n t appears t o be more serious. As none of t h e canonical c o o r d i n a t i o n polyhedra has a s u b s t a n t i a l number o f bond- angles around 90

,

i t seems t h a t Our a n a l y s i s i n terms o f t h e canconïcal c o o r d i n a t i o n polyhedra alone misses an e s s e n t i a l p o i n t .

h i g a s h i e t a l / I d / have a l s o analyzed the c r y s t a l l i n e Mg Zn s t r u c t u r e i n terms of i n t e r p e n e t r a t i n g c o o r d i n a t i o n polyhedra. O f t h e atomic s?kes28ccupied by t h e Zn-atoms a l 1 b u t one possess a s l i g h t l y d i s t o r t e d icosahedral c o o r d i n a t i o n polyhedron ( t h i s shows up very d i s t i n c t l y i n t h e p a r t i a l bond a n g l e - d i s t r i b u t i o n , c f . F i g . 4 ) . The

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z = 1 0

I I i l I I I I I II 1 I

Fig.3

Bond-angle d i s t r i b u t i o n s i n t h e canonical Frank-Kasper polyhedra, i n t h e icosahedron, and i n t h e canonical Berna1 holes ( t o p ) , weighted according t o t h e i r occurence i n t h e model s t r u c t u r e and Gaussîan-broadened (bottom, ver- t i c a l bars and s o l i d 1 in e )

,

com- pared w i t h t h e a c t u a l bond-angle d i s t r i b u t i o h f u n c t i o n o f t h e model s t r u c t u r e (See t e x t ) .

O 60 120 180

0 (deg

s i t e s occupied by t h e Mg atoms fa11 i n t o two classes: Mg2, Mg4, Mg6, and Mg9 ( i n t h e n o t a t i o n o f Hiyashi e t a l ) occupy t h e centers o f pentagonal prisms w i t h p o l a r atoms ( n o t e t h a t t h i s c o o r d i n a t i o n polyhedron may be viewed as a twinned c o n f i g u r a t i o n o f t h e icosahedron, the t w i n n i n g i n t r o d u c e s a l a r g e number o f square faces). The r e - maining Mg atoms a r e surrounded by r a t h e r i r r e g u l a r polyhedra, most o f these s i t e s have a c o o r d i n a t i o n number Z=14, t y p i c a l l y they possess a l a r g e number o f t r i a n g u l a r and a small number (one t o t h r e e ) o f square faces. The a n a l y s i s o f t h e i r bond angle d i s t r i b u t i o n s shows t h a t both hypes o f golyhedra make a r e l a t i v e l y l a r g e c o n t r i b u - t i o n i n t h e range between 0,130 t o 100

.

Thus i t seems t h a t a c e r t a i n number of these non-icosahedral c o o r d i n a t i o n p a t t e r n s s u r v i v e s i n t h e glass.

(11)

C9-78

JOURNAL

DE

PHYSIQUE

coordinations,but some non-icosahedral c o o r d i n a t i o n p a t t e r n s characterist'ic of t h e compound seem t o s u r v i v i v e .

ACKI~OWLEDGEMENTS

I t i s a pleasure t o thank D.Nelson f o r s t i m u l a t i n g conversations, as w e l l as f o r sending p r e p r i n t s o f unpublished work

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