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Hence the mass operator under QCA-CP is no.f dr/J j A system of nonlinear equations were developed for the case of dielectric spheres under QCACP [Tsang and Kong, 1980]. 3.6 Low-Frequency Solutions In this section the effective propagation constants under EFA, QCA, EFA-CP and QCA-CP will be calculated for spherical particles in the lowfrequency limit. Both the real part of the effective propagation constant J(T and the imaginary part of the effective propagation constant Ie will be studied. We consider cases where J(i J(,. with J( = J(T + iJ(i. In solving the integral equations of the various approximations, the leading term of the real part and the leading term of the imaginary part will be kept while the other terms can be safely discarded. The introduction of the coherent potential into the effective field approximation and the quasi-crystalline approximation results in a nonlinear equation for the mean Green's function. This is analogous to the relation between nonlinear approximation and the bilocal approximation of random medium, where it was found that the difference between the results of the two approximations is insignificant if the real part of the effective propagation constant J(,. is approximately equal to k m . For the case of random discrete scatterers, the permittivity of the particles can be appreciably different from that of the background medium. When the particles are densely distributed, the real part of the effective wavenumber J(" is appreciably different from k. Because of the difference between J(T and k, the introduction of any nonlinear approximation can significantly change the effective propagation constant. Thus the results of EFA-CP differ significantly from EFA, and the results of QCA--CP differ significantly from QCA.

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Effective Field Approximation (EFA)

The Lippmann-Schwinger equation for the transition operator is (5.3.97) In space representation, (rlU Ir') = U (r) 7 (rlr') where

The Scale tab of the Transform pane, shown in Figure 4.34, allows you to set the ScaleX and ScaleY properties of the ScaleTransform setting for the control. These settings control the horizontal and vertical sizing, on a one-based scale, of the control during animation. The result of setting these options is the following XAML statement inside the <TransformGroup> block of the control:

(5.3.98)

and a is the radius of the particle of permittivity Es . To solve (5.3.97), a mixed representation (rITIP) == T m (r, 15) can be used. Applying to (5.3.97)

gives (5.3.99) Since the particles are spherical, the singularity of the dyadic Green's function will be separated out in the following manner (5.3.100)

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FIGURE 4.34 Setting the Scale Transform settings of a control in the Transform pane of the Properties tab in Expression Blend for the resulting effect

33 101 100 64 110 73 139 135 337 306 258 118 156 258

r = - r'.

where P S stands for principal value with a spherical exclusion volume at At low frequencies, p can be set equal to zero in (5.3.99). We further let T m(r)

= Tm(r,p = 0).

The Skew tab of the Transform pane, shown in Figure 4.35, allows you to set the AngleX and AngleY properties of the SkewTransform setting for the control. These settings control the horizontal and vertical skew, in pixels, of the control during animation. The result of setting these options is the following XAML statement inside the <TransformGroup> block of the control:

Tm(r) = U(r) 1-

~~) Tm(r) + U(r)

dr' PS Go(r, r') Tm(r')

FIGURE 4.35 Setting the Skew Transform settings of a control in the Transform pane of the Properties tab in Expression Blend for the resulting effect

(5.3.101)

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Tm(r)={Tm r'~a (5.3.102) o r' > a Substitute (5.3.102) into (5.3.101) and set r = O. The integral in (5.3.101) is of the same form as that (4.3.320,) and can be evaluated readily.

The integral equation in (5.3.101) then reduces to the form a (1.'2 - k 2) . 21 T, = (k 2 - k 2) '8 T m + T m (k 82 - k 2)_ 0 drre th m 8 3k2 3

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