Truncated thermal equilibrium distribution for intense beam propagation

Abstract

An intense charged particle beam with directed kinetic energy (γ_{b}-1)m_{b}c^{2} propagates in the z direction through an applied focusing field with transverse focusing force modeled by F_{foc}=-γ_{b}m_{b}ω_{β⊥}^{2}x_{⊥} in the smooth-focusing approximation. This paper examines properties of the axisymmetric, truncated thermal equilibrium distribution F_{b}(r,p_{⊥})=Aexp⁡(-H_{⊥}/T[over ^]_{⊥b})⊕(H_{⊥}-E_{b}), where A, T[over ^]_{⊥b}, and E_{b} are positive constants, and H_{⊥} is the Hamiltonian for transverse particle motion. The equilibrium profiles for beam number density, n_{b}(r)=∫d^{2}pF_{b}(r,p_{⊥}), and transverse temperature, T_{⊥b}(r)=[n_{b}(r)]^{-1}∫d^{2}p(p_{⊥}^{2}/2γ_{b}m_{b})F_{b}(r,p_{⊥}), are calculated self-consistently including space-charge effects. Several properties of the equilibrium profiles are noteworthy. For example, the beam has a sharp outer edge radius r_{b} with n_{b}(r≥r_{b})=0, where r_{b} depends on the value of E_{b}/T[over ^]_{⊥b}. In addition, unlike the choice of a semi-Gaussian distribution, F_{b}^{SG}=Aexp⁡(-p_{⊥}^{2}/2γ_{b}m_{b}T[over ^]_{⊥b})⊕(r-r_{b}), the truncated thermal equilibrium distribution F_{b}(r,p) depends on (r,p) only through the single-particle constant of the motion H_{⊥} and is therefore a true steady-state solution (∂/∂t=0) of the nonlinear Vlasov-Maxwell equations.