BOND-ANGLE DISTRIBUTION FUNCTIONS IN
METALLIC GLASSES
J. Hafner
To cite this version:
J. Hafner. BOND-ANGLE DISTRIBUTION FUNCTIONS IN METALLIC GLASSES. Journal
de Physique Colloques, 1985, 46 (C9), pp.C9-69-C9-78. <10.1051/jphyscol:1985908>. <jpa00225269>
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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 icosahedral 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 characterisation (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 threedimensional 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 - interstice and centre - interstice correlation functions /3/; and (c) bond-angle correlation functions /4-6/. The first two methods offer essentially topological information - 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-orientational 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 fourdimensional 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/.
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1985908
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 pseudopotent 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
der 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
alloys
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 ~-?-gyd 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 /2U 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.
T h i s encouraged us t o p e r f o r m a m o l e c u l a r dynamics s i m u l a t i o n o f amorphous Mg4pZn30.
Our approach f o l l o w s q u i t e c l o s e l y t h e s t r a t e g y proposed by Weber and S t i l l i n r
/27,28/.
The p r e s e n t m o l e c u l a r dynamics s t u d y has been performed f o r a system of
800 p a r t i c l e s (5b0 Mg and 240 Zn atoms) c o n f i n e d w i t h i n a rhombic dodecahedron a t
f i x e d d e n s i t y . P e r i o d i c boundary c o n d i t i o n s were a p p l i e d . The atoms were i n i t i a l l y
p l a c e d a t random i n t h i s ce1 1, t h e Newtonian e q u a t j p 9 s o f m o t i o n were s o l v e d u s i n g
t h e V e r l e t a i g o r i t h m w i t h a t i m e increment o f 3x10
S. C l a s s i c a l dynamical t r a j e c t o r i e s o f 10 t i m e s t e p s were generated t o d e t e r m i n e t h e p r o p e r t i e s o f t h e a l l o y a t
each temperature ( a f t e r r u n s s u f f i c i e n t l y l o n g t o e q u i l i b r a t e t h e system). A l t h o u g h
each m o l e c u l a r dynamics r u n conserves energy, t h e k i n e t i c energy and t h u s t h e temp e r a t u r e can be v a r i e d between t h e r u n s by s c a l i n g t h e momenta o f a l 1 t h e a p r t i c l e s .
The p r o p e r t i e s of t h e a l l o y were c a l c u l a t e d f r o m i n s t a n t a n e o u s c o n f i g u r a t i o n s samp l e d t y p i c a l l y every 200 t i m e steps. To r e a c h e q u i l i b r i u m w i t h r e s p e c t t o t h e d i s t r i -
-
Fig.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 1i n 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 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/.
2''
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.
An amorphous s t r u c t u r e may be produced e i t h e r by a s t e p w i s e m o l e c u l a r dynamics
c o o l i n g / 2 9 / o r by t h e "mapping" procedure proposed by Weber and S t i l l i n g e r /28/:
an i n s t a n t a n e o u s c o n f i g u r a t i o n o f t h e l i q u i d ( t h e k i n e t i c energy i s t o t a l l y removed
f r o m t h e system) i s s u b j e c t e d t o a steepest-descent "quench" on t h e energy hypersurf a c e . T h i s procedure a l l o w s t o l o c a t e t h e s t r u c t u r a l l y r e l e v a n t minimum c l o s e s t t o
t h e high-temperature c o n f i g u r a t i o n . Weber and S t i l l i n g e r have shown t h a t t h i s
I1mapping" on t h e p o t e n t i a l energy minima produces a s i g n i f i c a n t enhancement o f t h e
s h o r t - r a n g e o r d e r p r e s e n t i n t h e l i q u i d system. The "quenchedl' c o r r e l a t i o n f u n c t i o n s
a r e q u i t e independent f r o m t h e temperature b e f o r e t h e quench, as l o n g as i t i s above
JOURNAL DE PHYSIQUE
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
: histogram - c a l c u l a t e d using t h e molecular dynamics "quench"
Mg Zn
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 i q 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 favourab 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 essent 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 rc 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 nearestneighbour 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/.
MO?!!
inaormation may be obtained by l o o k i n g a t t h e p a r t i a l bond-angle d i s t r i b u t i o n
i . e . t h e p r o b a b i l i t y o f measuring a g i v e n bond-angle e a t o n l y one
f u n c t i o n s fi(0),
Amorphous
I
Amorphous
'C
n
1
-
Fig.3
lntermetallic
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
the
Indeed a l 1
t h e i d e a l icosahedral bond angles o f 0=63.5 , @=116.5 , and o= 180
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
t h e maximum o f t h e
partial pair correlation function 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 nnumbers
~
a r e 3.00 t o 3.35
( 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
(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
i n s t e a d o f s t a t i c r e l a x a t i o n ( t h e r e b y i n t r o d u c i n g CSRO) ? Fig.5 shows t h a t the
t o t a l and t h e 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 o f t h e g l a s s a r e l e f t v i r t u a l l y unchanyed, those o f t h e 1 i q u i d a l l o y a t T=800 K a r e o n l y s l i g h t l y broadened
(however, we have t o remember t h a t t h e MD-quench was performed a t a f i x e d d e n s i t y
- t h a t o f t h e g l a s s - so t h a t t h e l i q u i d i s e f f e c t i v e l y under compression, a r e l a x a t i o n o f t h e volume would probably y i e l d a stronger r e d u c t i o n o f t h e b o n d - o r i e n t a t i o na1 o r d e r ) . T h i s suggests t h a t the t o p o l o g i c a l s h o r t range order (TSRO) and t h e CSRO
a r e o n l y v e r y weakly coupled f o r t h i s a l l o y . A v e r y weak c o u p l i n g between TSRO and
28
.
i,
$
k
h
JOURNAL DE PHYSIQUE
Fig.4
-
P a r t i a l bond-angle d i s t r i b u t i o n functions f . ( o ) in ( a ) glassy Mg
(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
t h e i n t e r m e t a l l i c compound Mg51Zn20.
Fig.5
-
Total and p a r t i a l bond-angle d i s t r i b u t i o n s in l i q u i d (continuous l i n e ) and
amorphous (histogram) model s t r u c t u r e s of Mg70Zn30 produced by molecular
dynamics simulation.
Zn
(63 do
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 trh 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
similar.
IV
-
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 exp 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 icosaheu r a l bond-orientational order i n m e t a l l i c glasses ?
The i d e a l c o o r d i n a t i o n p a t t e r n i s t h e icosahedron, whose b a s i c f e a t u r e i s t h e p a c k i n g
o f f i v e s l i g h t l y d i s t o r t e d icosahedra i n t h e form o f a f i v e - f o l d bi-pyramid (Fig.6).
R e p l a c i n g t h e f i v e - f o l d AB-axis by a s i x - o r a f o u r - f o l d one i n t r o d u c e s n e g a t i v e ,
r e s p e c t i v e l y p o s i t i v e 72' d i s c l i n a t i o n l i n e s a l o n g AB /15,17/. The c a n o n i c a l Frank-
6-FOLD
BI PYRAMID
4 - FOLD
BlPYRAMlD
t--\''\/
\
z 12
l COSA HEDRON
hedra. A f t e r Nelson / I V .
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C9-76
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,
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
and f o u r
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 canon 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 sa 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 bondangles 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 .
s t r u c t u r e i n terms of
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
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
-
.
Fiy.7
-
-
Coordination shels o f t h e canonïcal Frank-Kasper polyhedra ( l e f t s i d e )
and o f t h e canonacal aernal holes ( r i g h t ) and t h e i r r e p r e s e n t a t i o n as
l i n k s o f -(+) 72 d i s c l i n a t i o n l i n e s . The open c i r c l e s represent t h e
p a r t i c l e s i n t h e center o f t h e c o o r d i n a t i o n s h e l l s . A f t e r Nelson /15/.
z=10
I
I
i
l
I
I
I
I
I
II
I
1
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, vert i c a l bars and s o l i d 1 i n e ) , compared 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
180
120
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 o 100
Thus i t seems t h a t a c e r t a i n number of
t i o n i n t h e range between 0,130
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.
.
We have analyzed the bond-angle d i s t r i b u t i o n f u n c t i o n s i n r e a l i s t i c computer-generaand Ca Mg
and i n t h e i n t e r m e t a l t e d models o f t h e simple-metal glasses Mg Zn
l i c compound iilg Zn
We f i n d t h a t t h e b&?h $Be tota178nd3?he p a r t i a l bond-angle
d i s t r i b u t i o n s 8#--t6?'glasses and o f t h e c r y s t a l a r e a s t o n i s h i n g l y s i m i l a r T h i s
sugyests s i m i l a r i t i e s i n t h e t o p o l o g i c a l short-range order i n t h e amorphous and i n
t h e c r y s t a l l i n e s t r u c t u r e s . We have analyzed t h e computer-generated bond-angle d i s t r i b u t i o n s i n terms of l o c a l icosahedral b o n d - o r i e n t a t i o n a l order. The r e s u l t s i n d i c a t e t h e presence of a l a r g e number of near-icosahedral and defected icosahedral
.
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C9-78
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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