6.3 Prolate spheroids; the polarizable quadrupole model

The calculation of the polarizabilities of prolate spheroids is analogous to that of oblate spheroids, given in the previous section. Consider a prolate spheroidal particle, see fig. 6.3, with a dielectric constant ? surrounded by the ambient, with a dielectric constant ? _a. The z-axis is chosen as the axis of revolution, with z = 0 in the center of the spheroid. This is the long axis and has a length 2 a ? ₀. For the prolate spheroid the two foci lie on this axis at a distance a from its center. The two short axes have the lengths 2 a( ? ² ₀ ? 1) ^1/2 (1 ? ? ₀ < ?), in terms of the elongation parameter ? ₀. The limit ? ₀ ? ?, a ? 0, with a ? ₀ = R again corresponds to a sphere with radius R. The limit ? ₀ ? 1 corresponds to a needle with length 2 a. The prolate spheroidal coordinates are defined by [37]

with

Here ? ₁ and ? ₂ are the distances of the point ( x, y, z) to the two foci and ? describes the orientation of the plane through ( x, y, z) and the z-axis with respect to the x ? z plane. The value range is 1