0 1 2 3 4 K$_{z}$ 1 2 3 4 K$_{x}$ 0.1 0.2 0.3 δ 0 1 2 3 4 K$_{z}$ 1 2 3 4 K$_{x}$ 0.1 0.2 0.3 δ FIG. 1: Growth rate δ in terms of K for v 0 /c = 0 . 1 and H = 0 . 6. Top: Exact calculation. Bottom: Longitudinal approximation. very simple longitudinal approximation which, in the classical case, has been found worthy over the full wave-vectors spectrum except ne ar the perpendicular direction [16, 17]. The dispersion equation for longitudinal waves read s [18], where f ± is the distribution function of each b eam. With f ± ( v ) = δ ( v$_{x}$ ) δ ( v$_{y}$ ) δ ( v$_{z}$ ∓ v 0 ), the result reads ϵ (Ω) = 1 − F (Ω) = 0, where in terms of the non-dimensional parameters, where ω$_{p}$ = 4 πn 0 q $^{2}$/m is the electron plasm a frequency (note that the plasma frequency is here defined for a density n 0 different fro m the beam electronic densities n 0 / 2). The 0 = 1 − ∑ ± 4 πq $^{2}$n 0 / 2 m ∫ f ± ( v ) d $^{3}$v ( ω − k · v ) 2 − ( ℏ k 2 / 2 m ) $_{2}$, (7) F (Ω) = 1 2 [ 1 $^{(Ω}$− K$_{z}$ ) 2 − 1 $_{4}$H 2 ( K 2 x + K 2 $_{z}$) 2 + 1 (Ω + K$_{z}$ ) 2 − 1 $_{4}$H 2 ( K 2 x + K 2 $_{z}$) 2 ] , (8) K = k v 0 ω$_{p}$ , Ω = ω ω$_{p}$ , H = ℏ ω$_{p}$ mv 2 0 , (9) 4