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MARKOFFIAN SPATIAL PROCESSES AND
SPATIAL CORRELATIONS BETWEEN LOCAL
STRUCTURES
S. Baer
To cite this version:
MARKOFFIAN SPATIAL PROCESSES AND SPATIAL CORRELATIONS BETWEEN LOCAL
STRUCTURES
S. Baer
Department o f Physicat Chernistry, The Hebreu University, JemsaZem 91904, IsraeZ
Ahstract.Local structures at different points in space can be compared when they are defined in terms of a common set of parameters (local structure parameters) which may Vary randomly from one point in space to another. An example of such parameters is provided by the local quasi-lattice, randomly translated and rotated in space. The spatial correlations between local quasi- lattices are denved from a .sparial Fokker-Planck type theory. From these correlations one can denve the radial distribution as well as the bond orienta- tional correlations in the liquid and non-crystalline solid. The theory is extended to include different types of local symmetnes (quasi-lattices) and the spatial transition probability between them.
L The local structure.
The mode1 of liquid or amorphous solid structure which we apply in the following is based on the assumption that one can associate with any point in the system a certain local structure, Say a geomettic figure, charactenzing a real or an idealized spatial configuration of the molecules in the near neighborhood of this point. We specify the local structure by a certain set of parametes - the local structure parameters, which are considered to be random functions of position in space. A con- crete example of a n assumed local structure is the local crystalline structure characterized by a vir- tual lattice
"'
randomly translated and rotated from point to point in space (see fig. 1). The local structure parameters are, the components of the translation s=
(sI,s2,s3) and rotationalR
=(a$,?)of the local lattice with respect to a reference lattice. II. The structural diffusion equations.
Spatial Fokker-Planck type equations
'
are deduced from the assumption that the parameters s , R obey a certain spatial Markoffian process'
along a straight line between any two points r,,rl. Mak- ing certain simplifications we obtain a diffusion type equation : l awhere P
=
P(s.R;r/s,,R,) is the conditional probability density that the local structure parameters have the values s,R at rl=r, given that they have the values s,,R, at ro=O.D,
and Do are "struc- tural diffusion" coefficients, of translation and rotation respectively, of the virtual lattice.V,
is the gradient in s space and L is the differential operator of rotation (the "angular momentum" operator). The solution of (1) has the form :We see from here that translation and rotation are two independenr random spatial processes. For the translation we have :la
JOURNAL
DE
PHYSIQUEFig. 1. Local lattice structures at two points ro and
rl,
shifted and rotated respectively by so,ao and sl,al with respect to a reference lattice.Fig. 2. Comparison of rTg(r)-l] from computer simulation and experirnental data on Ar, with formula (8) for a hcc lattice. D , = 0.018, r , = 0.83(nn distance a = l ) .
where the sum extends over al1 points of the local reciprocal lattice, with position vectors h,
.
and v is the volume of the physical lattice unit cell. As seen from (3), D , determines the extent of spatial correlations of random translarions of local structures. When D,=0 we have infinitel? c~tended correlations and :For the rotations we have :'
where :
and o is the angle of the combined rotation RR~-'. As seen from (5). Do determines the extent of spatial correlations of the random rotation of local structures. When Do=O we have spatially extended orientational order and :
where the sum extends over al1 local lattice points with position vectors a,. The radial distribution is then given by :
where
The double sum extends over al1 pairs of points inside the unit cell and n, is the number of such points. Results obtained from a refined model 1b.c.6: where D,r is replaced by D,<(r-r,), correcting for an exclusion diameter r,, are plotted in fig. 2. The calculated g ( r ) (solid line) is obtained from an assumed hcc virtual lattice and the parameters D,c,r, are chosen to fit data from computer simu- lation and scattering expenments (crosses) for liquid Ar. 7.
IIIb. Bond orientational correlations.
The local bond density spherical tensor components
*"
are defined by :where the sum extends over ail the bonds of the local lattice. A "bond" is here a vector joining two neighbonng lattice points and a k is the position vector of the mid-point of the bond in the lattice. Clk is the set of polar angles of the direction of the bond. Invariant correlation functions are defined by :
where :
The double sum extends over al1 pairs of bonds inside one unit cell. The sum in (11) depends oniy on the parameter D, and not on the degree of orientational order. When Do=O we obtain, in the r
-
CG1 2
limit,
d l ( - )
= C /v.
Their values have been evaluated for a sample of computer sirnulated super-B
cooled liquid Ar.
.
The obtained even 1 ~ ' , ( m ) i have a maximum at 1=6 and decrease monotoni- cally from 1=6 to [=IO. This is in conformity with calculated ~;(a) values for hep, f cc and hcclattices, but in contrast with the icosahedral local structure whose values show a gap at 1=8. IV. "Structural" Master equations.
JOURNAL
DE
PHYSIQUEwhere A is a "transition probabilities" matrix satisfying :
A further extension of the mode1 to include both random transitions to different lattice types and random translations (and rotations ) can be achieved in an obvious way. Here the conditional proba- bility has the form :
and satisfies the combined Master-diffusion equation :
where again A is a transition probability matrix and
DS
is a structural diffusion matrix. ReferencesS. Baer, (a) Physica 87A , 569 (1977): (b) ibid. 91A , 603 (1978); (c) Chem. Phys. 39 , 159 (1979). S. Franchetti. Nuovo Cim. 55B
.
335 (1968).For other models which can
be
interpreted i n terms of a spatial Markoffian process, see e.g. R. Hosemann, Z. Physik 128.
1. 465 (1 950): S.N. Bagchi, Adv. Phys. 19 , 11 9 (1 970).J. Frenkel, Kinetic Theory of Liqiridr (Dover, New York, 1955) Ch. 3.
The factor 2 j + l is missing from eqn. 20 of ref. la.
N.N. Medvedev and Y u Naberukhin, Phys. Chem. Liquids 6 . 137 (1977); ibid. 8 , 167 (1978). J.L. Yamell, M.J. Katz,
