The particle transport equation which describes radial diffusion combined with quasi-linear diffusion by waves is analytically intractable and is usually dealt with by fixing a priori either the radial transport term or the wave diffusion term. For the case of electromagnetic cyclotron waves we simplify by taking moments of this equation, thereby generating three equations for the total number density ⟨N⟩ of the energetic particles, their total energy ⟨NE⟩, and their total perpendicular energy ⟨NE⊥⟩. The quasi-linear conservation equations for ⟨NE⟩ and ⟨NE⊥⟩ allow us to estimate the total wave intensity in terms of the particle moments and their radial derivatives and thus to express wave diffusion entirely in terms of these moments. The resulting moment transport equations for the particles describe relaxation of ⟨NE⟩ toward a critical value ⟨NE⟩c (or equivalently a critical value &bgr;c of the plasma &bgr;); when ⟨NE⟩ is less than ⟨NE⟩c, there can be no finite amplitude waves. From ⟨NE⟩c we can estimate a critical flux value which is never greater than the stably trapped flux limit (STFL) of Kennel and Petschek. We show that the STFL controls the anisotropy of the energetic plasma, while ⟨NE⟩c controls the loss rate. As one expects, the effective loss rate is greater for ⟨NE⟩ than for ⟨N⟩, since particles may lose energy before they are precipitated.