## 1. Introduction

[2] Quasilinear diffusion theory has been widely used in modeling radiation belt dynamics [e.g., *Lyons and Thorne*, 1973; *Kennel and Petschek*, 1966; *Horne et al.*, 2005], which is a crucial part of space weather specification and prediction. The quasilinear theory describes interactions between charged particles and small amplitude broadband waves in terms of a diffusion equation and diffusion coefficients. The general form of the quasilinear diffusion coefficients, written in velocity space (*v*_{⊥}, *v*_{∥}), with a uniform background magnetic field and a distribution of plasma waves with arbitrary wave normal angles was described by *Kennel and Engelmann* [1966] for nonrelativistic particles and by *Lerche* [1968] for relativistic particles. Here *v*_{⊥}(*v*_{∥}) is the velocity component perpendicular (parallel) to the background magnetic field. *Lyons* [1974] expressed the diffusion coefficients of *Kennel and Engelmann* [1966] in terms of pitch angle (*α*) and energy (*E*), which was widely adopted subsequently by the radiation belt community. Several numerical codes [*Glauert and Horne*, 2005; *Albert*, 2005; *Shprits and Ni*, 2009; *Xiao et al.*, 2009, 2010] have recently been developed, following *Lyons* [1974], to calculate the quasilinear diffusion coefficients and model radiation belt dynamics. We will use *Kennel and Engelmann* [1966] to represent quasilinear diffusion theory discussed above, even though it is only strictly applicable for non-relativistic electrons.

[3] The quasilinear diffusion coefficients for parallel propagating waves have been reduced to a set of closed analytical forms for computational convenience [*Summers*, 2005]. However, as later noted by *Albert* [2007], the starting point of this analysis, namely equation (1) of *Summers* [2005] has an extra factor of 2 compared with that of *Kennel and Engelmann* [1966], and correspondingly the diffusion coefficients of *Summers* [2005] are two times larger than those of *Kennel and Engelmann* [1966], *Lyons* [1974], *Glauert and Horne* [2005] and *Albert* [2005]. This difference in the value of the diffusion coefficients has serious implications because the expressions from both *Summers* [2005] and *Kennel and Engelmann* [1966] are widely used, even though the “factor-of-two” discrepancy has not been resolved.

[4] *Liu et al.* [2010a, 2010b] used a test particle simulation to calculate diffusion coefficients of interactions between relativistic electrons and electromagnetic ion cyclotron (EMIC) waves generated from a hybrid simulation. Good agreement has been found between the test particle results and the quasilinear theory of *Summers* [2005] (but see a discussion in Section 3). As an independent and complementary approach, we use a whistler wave field created from a summation of plane waves in a test particle simulation both to verify the validity of the quasilinear theory and to investigate the factor of 2 difference between *Summers* [2005] and *Kennel and Engelmann* [1966]. In the course of the analysis, we also attempt to clarify several basic concepts that can potentially cause confusion in the calculation of the diffusion coefficients.