A NEW LOCAL ORDER PARAMETER FOR AMORPHOUS METALLIC STRUCTURES : DEFINITION AND POSSIBLE MEASUREMENT

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A NEW LOCAL ORDER PARAMETER FOR

AMORPHOUS METALLIC STRUCTURES :

DEFINITION AND POSSIBLE MEASUREMENT

P. Allia, F. Vinai

To cite this version:

A NEW L O C A L ORDER PARAMETER FOR AMORPHOUS METALLIC STRUCTURES

:

DEFINITION A N D P O S S I B L E MEASUREMENT

P . Allia and F. Vinai

I s t i t u t o BZettrotecnico NazionaZe "G. Ferrarisf', Torino, G q p o NazionaZe S t r u t t u r a deZZa Materia d e Z CNR, U.R. 24, 1-10125 Torino, I t a l y

R6sum6

-

Un nouveau paramstre d'ordre local, mi, e s t introduit pour d6crire l a distortion des cellules unitaires d'une structure mstallique amorphe. Les principales caract6ristiques de ce paramstre sont decrites. Une relation e s t & t a b l i e entre <m> e t l a distortion quadratique moyenne de l a structure ; les corr6lations avec l e trainage nap6tique sont discut6es.

Abstract

-

A new local order parameter, m., i s introduced t o describe the de- gree of angular distortion of the unit c e l i s i n metallic glass structures. The properties of m. are reviewed. A simple relation between < m > and the r . m . s . distortion of tAe amorphous structure is given. The possible connections be- tween<m >and a measurable property (the aftereffect of the i n i t i a l magnetic permeability) are discussed.

I - INTRODUCTION

Quite recently, considerable attention has been devoted t o the study of site-depen- dent, single-atom order parameters, describing the geometry of the configxations of nearest neighbor atoms around any atomic s i t e . Most of these parameters were spe- c i f i c a l l y proposed t o study various computer simulations of single-size sphere pack- ings, whose analysis may be helpfhl t o get information on the atomic structures of amorphous metallic alloys. Examples of such quantities are provided by the bond- -0rientational order parameters defined by Steinhardt e t al 111 and by the distortion coefficient of Delaunay tetrahedra introduced by Kimura and Yonezawa 121. Other para- meters of the same type, although originally derived i n a s i m i l a r way, have l a t e r been linked t o macroscopic, measurable parameters of glassy metals. This is the case of the second moments of the distribution of atomic-level stresses, widely used be Egami and coworkers 131, which have been shown t o be related t o quantities playing a central role i n the theory of the glassy s t a t e , such a s the glass transition tempera-

ture. In any case, a l l these parameters contain i n f o m t i o n on the angular part of the correlation function between p a i r s of nearest neighbows. This property allows one t o investigate the types of short-range symnetries existing i n amorphous metals, by examinating the geometry of polyhedra defined a t each atomic s i t e , a s f o r instance the convex polyhedra whose vertexes are coincident with the z . nearest neighbors of a given atom i , o r , equivalently, the corresponding (dual) ~ o r o h o i polyhedra 141. Both types of structures can be considered a$ non-traslationally invariant "building blocksll of amorphous metallic structures. The relevance of approaches involving any of the forementioned order parameters l i e s i n the f a c t that the angular dependence of

C8-318 JOURNAL DE PHYSIQUE

z . = 11, 12, 14, 15 t o a regular Frank-Kasper. polyhedron 17 1 , i .e. t o a particularly s&tric arrangement of the neighboring atoms on a sphere centered a t i .

Specifically, these polyhedra can be viewed as composed of slightly distorted tetrahedra having a c o m n vertex a t i . In a l l these cases, m. is very close t o zero (when zi = 12, the Frank-Kasper polyhedron is the icosahedron: and m. = 0 ) . This f a c t indicates that the most significant variations of m. are linked toldistortions from high-symnetry configurations, rather than t o changeslof z . . This i s true also f o r the

1

average value m

,

which i s related t o the r . m . s . distortion of the structure. Such a property has been verified so f a r only by means of numerical calculations of the effect of a random distortion of bond directions from the configuration corresponding t o the Frank-Kasper polyhedron which minimizes m. 181. Here, an approximate analytic relation between m. and the distortion is given.& us start from Eq. ( l c ) , and make use of the relatio;

Now, l e t

5

. ( o ) ,

. . .

,

represent the bond directions corresponding t o the undistorted ~ r a n k - ~ a s ~ e $ polyhedron i n a properly chosen reference system. By developing

Z ( Q . . ) i n a Taylor s e r i e s i n the distortions A n . . (

A

0 . .,A+ . . )

,

<q21p> can be a t t e A J 1~ IJ 1~

(0) d L

+Zjk

zlP(nij

)(-+lo

] - A ~

+

..-

( 3 )

2 2 2

a+

i k

where

A

=

<

Aai .

>

= <A$.

>.

Notice t h a t the average values of both

A

e

. .

and

A(ij

are

d m .

Eq. ( 3 ) cdnbe written a s 1 J

where6 is the Kronecker's d e l t a function, and

P O Finally 2 TI ( 0 ) 2 < m > ( i ) =

7

-

2; 2

xl[

'l lo ( i )

+

F~ (i)

-

A

I

( 6 ) whereF = S + T + q

1 1 1. lo(0)*

ul,

S1 = 1; S

,

T 1 =

I,,

T 1, Eq. ( 6 ) describes the

variation of<m>wlth lncreaslng distortion %r Frank-Kasper polyhedra having z . ver- texes. Let us i l l u s t r a t e t h i s behavior i n two relevant cases. When z . = 12,

1

of the system.

I1 - A NEW ANGULAR ORDER PARAMETER

Different types of high-symmetry, close-packed atomic configurations, different from the ones usually present in ordered structures, can be singled out in the Voronoi po- lyhedron analysis of typical computer simulations. Some of these are a comnon feature of all considered models, not depending on the specific choice of the potential used in the simulation. In particular, it appears that fivefold symmetry occurs with high frequency in the analysis of relaxed models of single-size sphere packings 12,41

,

in- dicating that the icosahedral configuration and its modifications play a not negligi- ble role among the various short-range structures of amorphous metallic systems. It has been stressed that regular, undistorted Voronoi polyhedra corresponding to spe- cific point group syrnnetries are scarcely found in close packed structures 141. A significant amount of distortion of these polyhedra is indeed intrinsically present. A good characterization of these distortions is then needed in any complete descrip- tion of amorphous metallic structures. To this purpose, we have defined a purely an- gular order parameter, m. which is a direct measure of the degree of distortion of a

1'

given environment from a short-range order configuration of high symmetry. This pa- rameter may be defined as

where z . is the local number of nearest neigbors, 0 is the angle between the j-th and thelk-th bond direction between the atom i a n t its neighbors j

,

k

,

the P (cos 0. ) functions are Legendre polynominals. Finally, 9 = Z . Z (Q. .)

,

where

1\1 J

tRe Zy ?&ctions are 5(9) tesseral harmonics (real hctions) wiih =

1

(4) 151

,

and the s u m t i o n 1. is performed over the n.n. bond directions Q . . in an arbitrary frame. Notice that; whi e the value of q clearly depends on th?e choice of this reference system,

₁

_rl

h

1P

1u (and therefore m. A ) is a rotational invariant. Eq. (lc)

guarantees that

rn.>d!

Now, m. attains its minlmum value, rn. = 0, only when z . = 12, the twelve bond directions

n:.

corresponding to the axes &f fivefold symmeth of a regular icosahedron. As a conkdquence, m. may be considered as an absolute measure of the deviation of a given envimment fro; the icosahedral s-try. In addition, for any other particular type of short-range order configuration which may occur in metallic glasses (e.g., f.c.c. or h.c.p. 13-atom clusters), m. always increases, with increasing distortion, from the value corresponding to thelregular arrangement of atoms. As a matter of fact, m. is not only sensitive to an arbitrary distortion of a starting, high symmetry con&guration, but also a) to geometrical transformations from a particular type of short-range syrranetry to another one, without necessarily involving changes of the local topology (as for instance in the case of a transformation between different 13-atom clusters: icosahedron - f.c.c. - h.c.p.

161); and b) to transformations involving a variation of 2 . . However, for any set of

z. bond directions (with z. varying between 11 and 15), it is possible to show the

1.

C8-320 JOURNAL DE PHYSIQUE

The behavior of<m>with

A

( i n units TI )

,

predicted by Eq. ( 6 ) , is shown i n Fig. 1

( f u l l l i n e s ) . The dots represent the exact values as obtained from numerical calculations. For both values of z . the agreement i s very good up t o

A

--

0.1

n

.

The arrow, representing the estimatedldegree of r . m . s . distortion of a particular model structure 131, is well within the region described by the present approximation.

Figure 1 - Behavior of m with r . m . s . distortion

A

f o r two s e t s of polyhedra with

z . = 1 2 and z . = 14 vertexes ( a and b, respectively). The undistorted

( A

=lo) configurktions are the corresponding Frank-Kasper polyhedra.

Lines: approximate formula (Eq. ( 6 ) ) . Dots: exact values, from numerical calculations.

I11

- RELATION WITH

A MEASURABLE QUANTI'IY

A recent theory (Allia and Vinai, unpublished), based on the consideration of the effect of the short-range syrranetries on the local magneto-elastic energy of ferromag- netic metallic glasses, relates<m>to the aftereffect of the magnetic permeability observed i n these systems. This e f f e c t , consisting i n a reversible decay of the ini-

tial magnetic permeability p. a f t e r any rearrangement of the domain structure of un- saturated samples of s o f t &etic materials, can be easily observed i n all Fe

-

and

still rather controversial (9,101, the experimental results indicate that at least a part of the effect arises from the presence of a magnetostrictive coupling between clusters of atoms (stress defects, according to the picture by Egami and coworkers

/

31 ) , and spontaneous meetization 191. The aftereffect, or at least a part of it, can in fact be shown to be expressed as

where ), is the isotropic magnetostriction constant of the material, wfiile<h >is S

another constant, related to A and to an average value of the dipolar term

09

the local magnetic anisotmpy, f (a?, by

r - T 2

where<z>is the average number of nearest neighbors, a the average n.n. distance, I the saturation magnetization, C y a n average shear elastic constant. Equation (7) sugz gests that the observed dependence of Ap/p on the quenching rate 111) and its irre- versible reduction with the& treatments 1121 may be interpreted also by taking in- to account the variations of<m)l i.e. of the average distortion of the amorphous stmcture, according to the derived relation (Eq.(6)). Preliminary data concerning the behavior with annealing of the aftereffect in Co-based amorphous alloys give in- dications (Allia and Vinai, unpublished) that irreversible changes of<m>play indeed some role in determining the room temperature intensity of the aftereffect in samples submitted to thermal treatments. If the relevance of the structural term<m>will be

confirmed by future measurements, the aftereffect of the magnetic permeability of amorphous fermmagnets would become a physical property of striking interest in the study of the structure of these glasses, owing to its intrinsic relation with the disorder of the atomic bond directions in non-crystalline metallic systems.

REFERENCES

111 Steinhardt, P.J., Nelson, D.R., Ronchetti, M., Phys. Rev.

B28

(1983) 784.

) 2

1

Kimura, M.

,

and Yonezawa, F., J. Non-Cryst

.

Solids

61-62

(1984) 535. (31 Egami, T., and Srolovitz, D., J. Phys.

F12

(1982) 2141.

141 Srolovitz, D., Maeda, K., Takeuchi, S., Egami, T., Vitek, V., J. Phys. (1981) 2209.

15) Hutchings, M.T., in "Solid State Physics", (Ehrenreich, H., Seitz, F., llumbull, D., editors), (1964) 227.

(61 Corb, B.W., OIHandley, R.C., Megusar, J., Grant, N.J., Phys. Rev. Lett.

51

(1983) 1386.

(71 Frank, F.C., and Kasper, J.S., Acta Cryst.

11

(1958) 184.

18

1

Allia, P.

,

and Vinai

,

F.

,

in "Magnetic Excitations and Fluctuations1', (bvesey, S.W., Balucani, U., Borsa, F., Tognetti, V., editors) Springer, Berlin 1984 p. 110.

191 Allia, P., and Vinai, F., Phys. Rev.

B26

(1982) 6141. [I01 K m ~ l l e r , H., Phil. Mag.

B48

(1983) 127.

(111 Allia, P., Luborsky, F.E., Sato Turtelli, R., Soardo, G.P., Vinai, F., IEEE

Trans. Magnetics MAG17 (1981) 2615.