where p and j = v///v are particle momentum and pitch-angle cosine, and

Dup = (1/2)(AMp)/At and Dun = (1/2)(a^)2/At are components of the momentum diffusion tensor in the CR kinetic equation. In general, the diffusion tensor components Dnju(ju,p) and Dnp(u,p) are obtained by taking ensemble averages of the first-order corrections owed to wave fields to the helical particle orbit. For parallel cold-plasma waves, they have been calculated by Steinacker and Miller (1992). Vainio et al. (2003a), however, show that the bulk speed V (p) in Eq. 2.20.8 may be estimated without knowing the detailed form of these coefficients. They study the interaction of the CR particle with a single resonant wave mode with phase speed w. The interaction between the particle and the wave component can be viewed in the wave frame, where the wave's magnetic field is static making the scattering elastic. Thus we may write the equation Ap' = 0 for the wave-frame momentum, p= p

which leads to

if terms of the order (w/c) are neglected. Thus in this case we may write for the ratio of the momentum-diffusion-tensor components

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