Environmental Engineering Reference
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so close to the average that individual variations can be neglected.
Thus we replace the actual MF Hamiltonian of the i th ion with the
configurationally-averaged MF Hamiltonian and, considering a type 1
ion ( c i = 1), obtain
H MF ( i )
H MF ( i )
cf =
(5 . 6 . 3)
( J 1 i 2 J 1 i
H 1 ( J 1 i )
)
·
J
( ij )
{
c
J 1 j
+(1
c ) γ
J 2 j }
.
j
From this equation, and the similar one for c i =0,wemaydetermine
the MF values of the two moments
, and the corresponding
susceptibilities χ 1 ( ω )and χ 2 ( ω ). For a paramagnetic or ferromagnetic
system these quantities are all site-independent, in the present approx-
imation. We note that (5.6.3) is correct in the case of a paramagnet,
as possible environmental variations on the individual ions are already
neglected in the starting Hamiltonian. The next step is the introduction
of a 2
J 1
and
J 2
2 matrix of susceptibility tensors χ rs ( ij, ω ), where the elements
with r = 1 or 2 are defined in terms of c i J 1 i or (1
×
c i ) J 2 i respectively,
and s = 1 or 2 similarly specifies the other component. We may then
write the RPA equation (3.5.7):
χ rs ( ij, ω )= χ r ( i, ω ) δ rs δ ij +
j
( ij ) χ s s ( j j, ω ) ,
γ rs J
(5 . 6 . 4 a )
s
where
χ 1 ( i, ω )= c i χ 1 ( ω )
c i ) χ 2 ( ω ) ,
;
χ 2 ( i, ω )=(1
(5 . 6 . 4 b )
recalling that c i
= c i (= 0 or 1), and defining
J rs ( ij )= γ rs J
( ij ), with
γ 22 = γ 2 .
γ 11 =1
;
γ 12 = γ 21 = γ
;
(5 . 6 . 4 c )
In spite of the great simplification introduced through the random-phase
approximation, the RPA equation for the alloy is still very complicated,
because χ r ( i, ω ) depends on the randomness, and it cannot be solved
without making quite drastic approximations. The simplest result is
obtained by neglecting completely the site-dependence of χ r ( i, ω ), and
consequently replacing c i in (5 . 6 . 4 b ) by its average value c .Thispro-
cedure corresponds to the replacement of each individual angular mo-
mentum J ri
c ) J 2 i , and it is known as the
virtual crystal approximation (VCA). In this approximation, (5.6.4) may
be solved straightforwardly after a Fourier transformation, and defining
the T-matrices according to
by the average c J 1 i +(1
χ rs ( q )= χ r ( ω ) δ rs + χ r ( ω ) T rs ( q ) χ s ( ω ) ,
(5 . 6 . 5 a )
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