Truncated thermal equilibrium distribution for intense beam propagation

An intense charged particle beam with directed kinetic energy (γ_b - 1)m_bc² propagates in the z direction through an applied focusing field with transverse focusing force modeled by F_foc = - γ_bm_bω _β⊥²_x in the smooth-focusing approximation. This paper examines properties of the axisymmetric, truncated thermal equilibrium distribution F_b(r, p_⊥) = A exp(-H_⊥/T̂_⊥b,) ⊕ (H_⊥- E_b), where A, T̂_⊥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) = ∫ d²pF_b(r, p_⊥), and transverse temperature, T_⊥b(r) = [n_b(r)]^-1 ∫ d² p(p_⊥²/2γ_bm _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/T̂_⊥b. In addition, unlike the choice of a semi-Gaussian distribution, F _b^SG = A exp(-p_⊥²L/ 2γ_bm_bT̂_⊥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.