Single-parameter characterization of the thermal equilibrium density profile for intense non-neutral charged particle beams

The present analysis considers an intense non-neutral ion beam with characteristic axial velocity V_b = β_bc and directed kinetic energy (γ_b - 1)m_bc² propagating in the z direction through an applied focusing field which produces a transverse focusing force F_foc = -γ_bm_bω _βb²(xê_x + yê_y) on a beam ion (smooth focusing approximation). For a thermal equilibrium distribution function F_b(H_⊥⁰) = const × exp(-H_⊥⁰/T_b), it is shown that the normalized radial density profile πr_b²n _b⁰(r)/N_b, plotted as a function of r/r _b, depends on a single dimensionless parameter, δ_b = N_bZ_b²e²/2γ_b ²T_b, which is a measure of the normalized beam intensity. Here, N_b ≡ ∫ dx dy n_b⁰(r) is the number of beam ions per unit axial length and r_b² ≡ N_b^-1 ∫ dx dy r²n_b²(r) is the mean-square beam radius. "Universal" profiles for the beam density are presented for a wide range of system parameters. The present results are readily extended to the case of a cylindrical non-neutral plasma column confined by a uniform axial magnetic field B₀ê_z.