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3.6 Low-Frequency Solutiolls
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Using the results of (5.3.102) and (5.3.105), the momentum representation of T can be calculated from the mixed representation. In the low frequency limit we have
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T p(15l,152)
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= (15IITI152) = jdre-iPl.T(rITI152) =
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(5.3.107a) = 41Ta 3 /3 is the volume of the particle. The dispersion relation is
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given by det det
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[G~\15) -
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noTp(15,15)] = o. That is
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[pp( _k 2 + fT,n) + (p2 - k 2 - fTm)(iJpiJp + ~p~p)]
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with f = novo as the volume fraction occupied by the particles. The solution for the effective propagation ]{ is
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](2 =
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so that
Quasi-crystalline Approximation (QCA)
We need to solve for C p(15l,152) governed by (5.3.54). Let T p(15l,152) = T,nvol, we have
T,nvaI + fT,n
dp3 Go(153)H(153 - 152)Cp(153,152) (5.3.109)
In the low frequency limit, 152 can be set to zero in (5.3.109). Furthermore, let
Cp(15l,152) = cl
Substituting (5.3.110) into (5.3.109), the integral in (5.3.109) can be put in the space domain. We then have c = Tmv o + fT,; c + cfTm 3k
dr PSGo(r) [g(r) - 1J
Keeping the leading term of the real and imaginary parts of (5.3.111), we have (5.3.112)
The integral in (5.3.112) is of the order O(a:3). Solving for c in (5.3.112), using the expression for Tm as given in (5.3.105), and noting that Re(Tm ) Im(Tm ) and Re(c) Im(c), gives
2 3v k y { 2 (ka)3 y [ c = 1 ~ fy 1 + i 31 _ fy 1 + 47Tn o io
The dispersion relation is, under QCA, det so that
roo drr 2 [g(r) - 1] ]}
[G~l (p) -
nocp(p,P)] = 0
det [pp(-k 2 + noc) + (p2 - k 2 - noc) (Op J p + p p)] = 0 The effective propagation constant K is such that
(5.3.113c) (5.3.113d)
K 2 = k 2 + noc
Effective Field Approximation with Coherent Potential (EFA-CP)
Under EFA-CP, the equation for the modified transition operator
t is
t = u + U Get
To solve (5.3.115). we assume that in the low-frequency limit, G e assumes the same expression as Go with k replaced by K. Hence,
Ge(r,r') = PS Gc(r, r') - 3K2 o(r - r')
The solution of the integral equation in (5.3.115) can be obtained in a manner analogous to that of (5.3.98) through (5.3.104). Thus,
tp(Pl,P2) = tmvol
satisfies the following equation analogous to (5.3.104)
2 _ (k 2 _ k2) _ (k:~ - k )t m '8 3K2
+ tm
3 '(k,2 _ k2)i2Ka
The (k~ - k 2 ) factor in (5.3.118) arises from the potential U, which, as indicated in (5.3.115), is unchanged. Other k's in (5.3.104) are replaced by
3.6 Low-Frequency Solutions
K to give (5.3.118). Solving (5.3.118) gives
z [ 2 = 1 + z/(3K2) 1 + i 91
Ka z ] + z/(3K2)
where z = k; - k 2 . The dispersion relation is, under EFA-CP,
K 2 = k2 + f
tm = k 2 +
'" [ 1 + 3K2
fz _
Ka z ] z 9 1 + 3K2
which is a nonlinear equation for K 2 .
Quasi-crystalline Approximation with Coherent Potential (QCA-CP)
The integral equation is
Cp(lh, P2) = lp(Pl' P2) + no
dfi3 l p(Pl' P3)Gc(P3)H(P3 - P2)Cp(P:3, P2)
The procedure of solution is similar to the case of QCA. Hence, in the lowfrequency limit we have
Cp(Pl,P2) =
and Cobeys the following equation that is analogous to (5.3.112)
Vo tm
+ ftmc
L:{2 + ~iK 1 drr
[g(r) - 1]]
Use the expression of tm as given by (5.3.119), solve c from (5.3.122), and retain only the leading term of the real part and the leading term of the imaginary part of c to get
= 1+
VoZ { .2 .~ Z z(l _ j)/(3f{2) 1 + 1. 9 1\ (L' 1 + z(l _ j)/(3f{2)
.[I + 4,m 1 drr' [g(r) - II] }
The dispersion relation under QCA-CP is
f{2 = k 2 + nJ~ which is a nonlinear equation for f{2.
In the very low frequency limit, the scattering attenuation term that is dependent on particle si~e in (5.3.123) can be neglected, and the mixture formula for Eelf = f{2/ w11o is, on using the first term in (5.3.123), 3f(E s - E)Eelt' Eelt' = f + (5.3.125) 3Eeft' + (E s - 1")(1 - 1)
With simple algebra, it can be shown that (5.3.125) can be expressed in the form
Es)E s - E Es ---------=--'---"s 1 E - E E(1 IE s (E -
f) +
3 Eeff
3Ee ff
+ Es
3.7 QCA-CP for Multiple Species of Particles
In this section, the QCA--CP result is extended to the case where there are multiple species of particles. The different species refer to the fact that particles can be of different shapes, sizes, and permittivities. The different species are denoted by Sj = 1,2, ... , L. Each species can have a distinct size, shape, and permittivity Esj ' The multiple scattering equations become G = Go + Go where for r outside particle j for r inside particle j (5.3.128)
L U;i G
with k Sj = WJllE Sj being the wavenumber in the sj-type scatterers. The QCA-CP equation can be developed in a similar manner. The pair distribution functions for multiple species of particles are described in 8 of Volume II. The average Green's operator must be equal to the coherent potential Green's operator.