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⊥²x_⊥ in the smooth-focusing approximation. This paper examines properties of the axisymmetric, truncated thermal equilibrium distribution F_b(r,p_⊥)=Aexp⁡(-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∫d²p(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=Aexp⁡(-p_⊥²/2γ_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.