LOCAL ATOMIC STRUCTURE OF AMORPHOUS METALS

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LOCAL ATOMIC STRUCTURE OF AMORPHOUS

METALS

T. Egami, K. Maed, D. Srolovitz, V. Vitek

To cite this version:

T. Egami,

K. Maed,

D. Srolovitz,

V. Vitek.

LOCAL ATOMIC STRUCTURE OF

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JOURNAL DE PHYSIQUE CoZZoque C8, suppzdrnent au n 0 8 , Tome 41, a o c t 1980, page

C8-272

L O C A L ATOMIC STRUCTURE O F AMORPHOUS METALS

T. ~~ami', K. ~aeda', D. Srolovitz and V. Vitek

Depa2'trnent o f Materials Science and Engineering, U n i v e r s i t y o f PennsyZvania, P h i l a d e l p h i a , Pa. 19104, U.S.A.

* a ~ s o Max-PZanck I n s t i t u t fiir Metallforsclnazg, I n s t i t u t J'iir Physik, 7000 S t u t t g a r t 80, F.R.G.

Abstract.- The local parameters are introduced to describe the local atomic structure of amorphous metals. They define the structural defects which facilitate the explanation of various properties, including the volume change by annealing.

1. Introduction

The atomic structure of amorphous metals has been discussed traditionally in terms of the radial distribution function (RDF) and the polyhedron ana- lysis (1-3). In particular the RDF is most frequent- ly used to describe the structure, since it is the only structural quantity which can be directly deter -mined by experiment.However, the RDF fails to describe the variation in the local structure, since it is a quantity averaged over the entire volume of the sample. For instance, as shown in Fig. I, the same total RDF first peak can result either when the probability is identical for each atom, and the local variation represents purely the statistical

the average of the locally varying quantities. In spite of the importance of the local structural variations in understanding the properties of amorphous solids, we have not until recently possessed effective means to describe such a local structure. However, several local parameters have been introduced in ref.(4,5) and in this paper we discuss their physical meaning, describe the structural defects defined by these parameters, and consider t le role of defects in elucidating various physical properties, focusing in particular on the interpre-

tation of the macroscopic Free volume in terms of these defects.

fluctuation (case (a), uncorrelated structure), or when the peak is narrower and the peak position is different for each atom, so that the local environment has a collective fluctuation (case (b), correlated structure). It is often uncritically assumed

that the case (a) is valid for amorphous structure, site i but actually the case (b) is more realistic.

The variation in the local structure, the fact

that the atomic environment of each atom is differ- _Site_j ent, is in a sense the most fundamental structural

characteristic of amorphous solids, since most of

the physical properties are determined to a large ( a )

(b)

excent by the short range local structure, and to a The first peak of RDF* for (a) uncorrelated structure and (b) correlated structure.

much less extent by the long range order. The macro- I

scopic physical properties we observe are most often

I

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2 . D e f i n i t i o n o f t h e l o c a l parameters

-.

The f i r s t l o c a l parameter i n t r o d u c e d by Egami, Maeda and V i t e k ( 4 ) i s t h e atomic l e v e l s t r e s s t e n -

s o r ( 6 ) which can be e v a l u a t e d b y c o n s i d e r i n g t h e change o f t h e e n e r g y o f t h e s y s t e m due t o a p p l i c a t i o n o f a u n i f o r m s t r a i n upon t h i s s y s t e m and t h u s does n o t r e q u i r e any i d e a l r e f e r e n c e s t r u c t u r e . I f t h e

i n t e r a c t i o n between t h e atoms i s d e s c r i b e d b y a cen- t r a l f o r c e p a i r p o t e n t i a l @ , t h e

CYB

component o f t h e s t r e s s t e n s o r a t t h e p o s i t i o n o f atom i i s

-+

where r i s t h e v e c t o r between i - t h and j - t h atom;

i j

r?' are t h e CY and I3 components o f t h i s v e c t o r and

1 1 r . . i t s magnitude. $2. i s t h e l o c a l a t o m i c volume 1 3 which may b e d e f i n e d a s 4n

a

3 hii =

-

3 i ( 2 ) where

a.

i s t h e e f fe c t i v e atomic r a d i u s r i j w ( r i j )

-

a = J ( j : a l l i n t e r a c t i n g n e i g h b o r s ) ( 3 ) i _{2: w ( r .}_.) j 11 w ( r i j ) = r i j -2

-

2

The w e i g h t i n g f a c t o r r _{i j} was i n t r o d u c e d because i t i s p r o p o r t i o n a l t o t h e s o l i d angle t o v i e w an atom a t t h e s e p a r a t i o n r i j ' Another s e t o f t h e l o c a l parameters a r e t h e s i t e symmetry c o e f f i c i e n t s E : ' ~ ( i ) a t t h e p o s i t i o n o f an atom i d e f i n e d by e x p a n d i n g t h e t o t a l e n e r g y E w i t h r e s p e c t t o t h e d i s p l a c e m e n t Ar. o f t h e atom i i n t o t h e f o l l o w i n g s e r i e s :

where 8 and

@;

a r e t h e azimuthal a n g l e s d e t e r m i n i n g

-+ _m

t h e d i r e c t i o n o f Ari and Y R i s t h e s p h e r i c a l har- monics.

These parameters cannot b e , a t p r e s e n t , d e t e r - mined d i r e c t l y by an e x p e r i m e n t , and can o n l y be c a l c u l a t e d f o r model s y s t e m s . We c o u l d , o f c o u r s e ,

c o n s i d e r o t h e r l o c a l parameters. For example, l o c a l a t o m i c volume ( 3 ) i s one p o s s i b i l i t y . Furthermore, f o r each p h y s i c a l p r o p e r t y i t i s p o s s i b l e t o d e f i n e a l o c a l parameter, such as t h e l o c a l Debye tempera- t u r e f o r l a t t i c e v i b r a t i o n s ( 7 1 , t h e l o c a l e f f e c t i v e mass f o r e l e c t r o n i c s t r u c t u r e , o r t h e l o c a l s p i n - wave s t i f f n e s s f o r a magnetic s y s t e m . A l l t h e s e

l o c a l parameters a r e , however, l i k e l y t o be s t r o n g - l y c o r r e l a t e d t o each o t h e r . I t i s t h e n n a t u r a l t o choose one o f t h e l o c a l parameters a s t h e p r i n c i p a l parameter, and d i s c u s s a l l o t h e r parameters i n r e - l a t i o n t o i t . We suggest t h a t such a s u i t a b l e para- meter i s t h e s t r e s s t e n s o r ( e q . l ) , and p a r t i c u l a r l y t h e l o c a l h y d r o s t a t i c s t r e s s .

An example o f t h e s p a t i a l d i s t r i b u t i o n o f t h i s pa- rameter i s shown i n F i g . 2 ( 4 ) f o r a c r o s s s e c t i o n o f a computer model o f an amorphous metal c o n s t r u c t e d by Maeda and Takeuchi ( 8 ) . A s t r o n g c o r r e l a t i o n be-

tween p. and t h e atomic volume hi. e x i s t s , a s seen i n F i g . ) which shows t h e p l o t o f p v s d f o r t h e above model f o r each atom.

3 . S t r u c t u r a l d e f e c t s d e f i n e d by t h e l o c a l parameters

The v a l u e s o f pi c a l c u l a t e d f o r t h e model amor- 2 1 / 2 phous s t r u c t u r e ( 5 ) a r e r a t h e r l a r g e , w i t h <p.>

3

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JOURNAL DE PHYSIQUE

Fig.2 Atomic level pressure of amorphous iron. Arrows toward the right, tension; the left, com- pression. The length of the arrow corresponds to the magnitude.

8 9 10 11

n(A3)

Fig.3 Microscopic pressure-volume relationship. The solid line is the least square fit by the quadratic equation shown above.

n-type (tensile stress) local density fluctuations

(LDF)(5,9) However, as pi always shows a continu- ous distribution there is no unique way of identifi- cation of the extent of these regions (a statistical definition is given in ref. 5). In this sense, these regions are not structural defects which can be un- ambiguously defined geometrically, but have to be re- garded as elementary units of the structural irregu- larity. On the other hand, if they are treated as structural defkcts, their behaviour strongly paral- lels the behaviour of the crystalline defects. In particular these two basic defects are somewhat ana- loguous to the interstitial (p-type) and the vacancy

(n-type), but they are more diffuse. The atomic transport can then be understood by their diffusion (p-type and n-type defects contribute in an opposite way to the transport, similarly as electrons and holes in semiconductors). Plastic deformation is most likely to be localized in the shear zone between the pairs of p and n-type defects. Their local motion may provide the physical basis for the two- level tunnelling state (10) and may contribute to the anelasticity, anomalous specific heat, and magnetic after-effect (11). The recombination of the n-p pairs explains the structural relaxation, and indeed the calculated RDF from the relaxed regions minus RDF from the defect regions strongly resembles the change in the RDF observed during the structural relaxation

( 5 ) . Furthermore, these defects provide important

insight into the structural stability and the compo- sitional dependence of the structure (9). Because of the limitation in space, we discuss here only their role in determining the macroscopic free volume.

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LDF), but also in the compressed high density regions (p-type LDF) which could be viewed as the "anti-free volume". The n-type and the p-type LDF contribute to the total volume in an opposite way and thus although the creation or annihilation of an n-p pair leads to a volume change owing to the anharmonicity of the potential, this volume change is always very small. The macroscopic free volume can be defined as

A . Q ~ =

a

-

~ ( p )

(6)

where

d

is the average atomic volume and

J(;)

is the atomic volume for the atoms with the average

-

hydrostatic stress p

.

These quantities can be evaluated using the data of Fig.3 and they are sumari- zed in Table I. The total microscopic free volume can be defined as

2 1/2

AD^

=

<(n

-

si)

> (7) and when evaluated using data of Fig.3, a m / Q =

0.067. Thus the macroscopic free volume is much smaller than the total microscopic free volume and these two quantities are not even linearly related.

The experimentally determined volume change during the structural relaxation is about 0.4% (14)

while the observed change in the RDF due to annealing is as much as 10% for the third to fifth peaks

( 15 ) . These two observations cannot be reconciled

in the framework of the free volume theory. They are, however, readily explained in terms of the p-type and n-type defects, since the macroscopic volume change corresponds to A

nM

if a s"bstantia1 portion of the defects are annealed out, while the change in

m

the RDF is more closely related to a1 which is much larger than A

fl

.

It is, therefore, very much mis- leading to try to estimate the defect density from the macroscopic volume change.

This research was supported by the National Science Foundation, MRL Program under Grant NO. DMR76-80994. One of the authors (Egami) would like to thank the Max-Planck-Institut for the hospitality during his sabbatical leave.

Table I

References

+ Present address: The Institute for Solid State Physics, University of Tokyo, Tokyo 106, Japan.

J.D. Bernal, Proc. Roy. Soc.

E ,

299 (1964). G.S. Cargill,III, Solid St. Phys.,z,227 (1975). P.H. Gaskell, J.Non-Cryst. Solids,z,207 (1979). T. Egami, K.Maeda and V.Vitek, Phil.Mag.,in

press. D. Srolovitz, K. Maeda, V.Vitek and T. Egami, unpublished.

M. Born and K. Huang, "Dynamical Theory of Crystal Lattices" (Oxford, 1954).

T. Egami, unpublished.

K. Maeda and S. Takeuchi, J. Phys. F,?, L283 (1979).

9 . T.Egami,in "Metallic Glasses" ed. H.-J. Giinthe-

rodt and H. Beck (Springer,Berlin, 1980)in press 10. P.W. Anderson, B.I. Halperin and C.M. Varma,

Phil. Mag.,

5,

1 (1972).

1 1 . N.Moser and H.Kronmiiller, J. Magn. Magnet. Mater. in press.

12. D. Turnbull and M.H. Cohen, J.Chem.Phys.

34,

120 (1961).