The equilibrium concentration of vacancies in ordered solid solutions

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BRIEF COMMUNICATIONS AND LETTERS TO THE EDITOR
THE EQUILIBRIUM CONCENTRATION OF VACANCIES
IN ORDERED SOLID SOLUTIONS
L. E. Popov, M. D. Starostenkov,
and N. S. Golosov UDC 669.018
In [1-3] it was shown that atomic long-range order should have a substantial effect on the equilibrium
concentration of vacancies in a solid solution. Here it was suggested that the presence of a vacancy does
not affect the equilibrium state of long-range order in its vicinity. However, since the atoms which are
located at sites adjacent to a vacant site do not have Z but Z - 1 neighbors (where Z is the coordination
number), the appearance of an irregular position near the vacaney is associated with a lower increase in
the eonfigurational energy of the erystal. The probability that a site adjacent to a vacant site (such sites
will henceforth be designated as a' and 8' sites as distinct from the ~ and /? sites in the remainder of the
solid solution) will be substituted by an irregular atom should therefore be expected to be greater than it
would be in the absenee of the vacancy, i.e., there will be some relaxation of the long-range order near the
vaeaney. The present report gives an assessment of the change in the energy of vacancy formation due to
such relaxation of the long-range order in a B2 superlattiee.
In the calculation it is assumed that the vacancies are sufficiently remote from each other that they
do not have common neighbors. There are no double vacancies. The eonfigurational energy of the crystal
containing the vacancies can then be represented in the form E = E 0 + Ev, where E 0 is the energy of the
homogeneous part of the crystal and E v is the total configurational energy of the parts of the erystal which
are disturbed by the presence of the vacancies. If it is assumed that the relaxation of the long-range order
only takes place in the first coordination sphere near the vacancy,
e~ = z (z-- 1) {n~ ~) (p~ va~ p(~ ~ cx~ ,~ ,~ ~,~, ,~
+ BBVBB+ P]BVAB) + n~ '--AA" AA + --BB ~ BB + P~BVAB)], (1)
where Z is the coordination number" P(~/) is the probability of finding pairs of atoms x and y eontatned in
'
xy
the first coordination sphere of a vacancy loeated at a ,/ site; and 4 c~) and n(P) are the number of vacant
c~ and /3 sites, v
For simplicity we will restrict ourselves to the case of an alloy withthe stoichtometric composition
AB. In addition, we will assume that VAA = VBB. If these conditions are satisfied the concentrations of
the vacancies at the sites of the two sublattices n(v c~) and ~fi) will be identical [3]. Owing to the completely
symmetrical nature of the solid solution with respect to the two components the relaxation of the long-range
order near the vacancies at the a and fl sites will also be identical. From this it follows that, although each
vacancy is surrounded by only ~ sites or only fi sites, the state of the long-range order in the vicinity of
all the vacancies can in this case be described by a single long-range order parameter, e.g. :
Pl c~') is the probability of the substitution of the ~ ' site by an A atom.
where
By expressing p(c~) p(a) etc., in terms of the long-range order parameter ~' for the vacancies in
AA' BB'
the first coordination spheres and the corresponding parameter ~7 for the homogeneous solid solution by
means of relations in the form P(~) = P(Afi')P(c~) (which are only valid in the approximation which does not
take account of coorelation) [2] we obtain
V. D. Kuznetsov Siberian Physicoteehnical Institute. S. M. Kirov Tomsk Potyteehnieal Institute.
Translated from Izvestiya Vysshikh Uchebnykh Zavedenit Fizika, Vol. 12, No. 1, pp. 123-125, January, 1969.
Original article submitted May 6, 1968.
9 1972 Consultants Bureau, a division of Plenum Publishing Corporation, 227 g/est 17th Street, New York
N. Y. 10011. All rights reserved. This article cannot be reproduced for any purpose whatsoever without
permission of the publisher. A copy of this article is available from the publisher for $15.00.
92

~f_E
a7
~t ~2 ~$ ~4 tZ5 tM ~t2 05
09 I~
Fig. 1. The temperature de-
pendence of the change in
the formation energy of a
vacancy arising from relaxa-
tion of the long-range order
in its vicinity.
and
Ev=--f6nv[W(l + ~) -[- VBI~] ,
(3)
where
= 2VAB-- VAA -- Ve~, ~d ,,~ = ,~ + n~)
The number of different permutations for the atoms between the ~'
and fi' sites wilI be
where, for example, N(. c~') is the number of A atoms at c~' sties By ex-
pressing N(ff ), NB~ ), etc., in terms of the long-range order parameter by
means of equations of the following form
N~A~') --_ Z-~ p~Ar Z-~ (I +.~ ")
r/7)
N~')~Zn---vPf~')--Z
(I
,
2 B -- ~ --~') etc.
and combining Eqs. (3) and (4), we obtain an expression for the free energy of the disturbed part of the solid
solution F v = E v - kT In W v. The equilibrium value V can be obtained from the condition for minimum free
energy 3Fv/0~ 7 = 0, which after the necessary rearrangements takes the following form
W
--3,5- i
l -- ( _ e ~r (5)
1 + -c~'
Hence
--3,5 W ~t
~,
= 1 -- e ~r (6)
_3,5 W___. ~
1 +e ~r
The ~ values can be obtained from the equation 0F/0~? = 0 [2], which is the condition for minimum free en-
ergy in the homogeneous solid solution9
The change in the formation energy of vacancies arising from relaxation of the long-range order in
their vicinity is equal to
0
E~,
--
Ev
(7)
AEy
=
rt v
Ev(0)is the binding energy of the atoms located in the first coordination spheres of the vacancies with
where
their nearest neighbors in the absence of relaxation. From Eqs. (7) and (3) it is easily seen that
~Ey=- 14,q~ W. (8)
From Fig. 1 it is seen that the AEf value calculated from Eq. (8) passes through a maximum at T ~ 0.7
T C. The maximum value is AEf ~ 09 W. Thus, the assessment of the effect of long-range order on the en-
ergy of vacancy formation in the B2 superlattice made in [1-3] is much too high, at least in the range of
temperatures between 0.4 T C and 0.95 T C.
This calculation is also valid for the case of bcc antiferromagnetics in which the "right" and "left"
spins are ordered in accordance with a superlattice of the B2 type.
1.
LITERATURE CITED
E. T. Nesterenko and A. A. Smirnov, Voprosy Fizika Metallov i Metallovedeniya, AN UkrSSR, No. 3,
152 (1952).
93

2o
3.
M.A.Krivoglaz and A. A. Smirnov, The Theory of Ordered Alloys [in Russian], Fizmatgiz, Moscow
(1958).
A. Girifalco, J. Phys. and Chem. of Solids, 25, 323 (1964).
94