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 Vb=βbc and directed kinetic energy (γb-1)mbc² propagating in the z direction through an applied focusing field which produces a transverse focusing force ${F}foc={-}{{γ}}b{m}b{{0ex}{0ex}}{{ω}}_{{β}b}²(x{{^}{e}}ₓ+y{{^}{e}}y)$ on a beam ion (smooth focusing approximation). For a thermal equilibrium distribution function ${F}b({H}_{{⊥}}⁰){{0ex}{0ex}}=const×{}exp({-}{H}_{{⊥}}⁰/{T}b)$, it is shown that the normalized radial density profile ${π}{r}b²{n}b⁰(r)/{N}b$, plotted as a function of $r/{r}b$, depends on a single dimensionless parameter, ${{δ}}b={N}b{Z}b²{e}²/2{{γ}}b²{T}b$, which is a measure of the normalized beam intensity. Here, ${N}b{≡}{∫}dxdy{n}b⁰(r)$ is the number of beam ions per unit axial length and ${r}b²{≡}{N}b^{{-}1}{∫}dxdy{r}²{n}b⁰(r)$ 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 B₀^ez.