hence the variation of v(k) is given by dv = .pe-GF)

dk e-GF2

The derivative dv/dk is the group velocity, which can be associated with the coherent propagation speeds while the second derivative d2v/dk is characteristic of the dispersion and can be associated with the diffusion coefficients. It can be shown that for the rectilinear case d2v/dk at k = 0 exactly returns the diffusion coefficient derived by Hasselman and Wibberenz (1970). Kota (1999) presents some examples to illustrate the method for various kinds of scattering. It was assumed that the dependence of was as given in Eq. 2.23.6. Clearly, q = 0, a = 0 describes isotropic scattering, a ± 0 implies dominant helicity, whilst q ~ 1 represents hemispherical scattering. Kota (1999) consider both rectilinear and focusing geometries with a constant focusing length L.

2.23.4. Dispersion relations for isotropic pitch-angle scattering

The simplest scattering is isotropic pitch-angle scattering. Then the eigenfunctions at k = 0 are the spherical harmonics, whilst the eigenvalues are Vj = j(j + 1)/2r (j = 0, 1, 2, ...). The variations of vo(k) and v1(k) as function of k are shown in Fig. 2.23.2.

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