Approximate periodically focused solutions to the nonlinear Vlasov-Maxwell equations for intense beam propagation through an alternating-gradient field configuration

Abstract

This paper considers an intense non-neutral ion beam propagating in the z direction through a periodic-focusing quadrupole or solenoidal field with transverse focusing force, -[κ_{x}(s)xe[over ^]_{x}+κ_{y}(s)ye[over ^]_{y}], on the beam ions. Here, s=β_{b}ct is the axial coordinate, (γ_{b}-1)m_{b}c^{2} is the directed axial kinetic energy of the beam ions, and the (oscillatory) lattice coefficients satisfy κ_{x}(s+S)=κ_{x}( s) and κ_{y}(s+S)=κ_{y}( s), where S=const is the periodicity length of the focusing field. The theoretical model employs the Vlasov-Maxwell equations to describe the nonlinear evolution of the distribution function f_{b}(x,y,x^{′},y^{′},s) and the (normalized) self-field potential ψ(x,y,s) in the transverse laboratory-frame phase space (x,y,x^{′},y^{′}). Here, H[over ^](x,y,x^{′},y^{′},s)=(1/2)(x^{′2}+y^{′2})+( 1/2)[κ_{x}( s)x^{2}+κ_{y}(s)y^{2}]+ψ(x,y,s) is the (dimensionless) Hamiltonian for particle motion in the applied field plus self-field configurations, where (x,y) and (x^{′},y^{′}) are the transverse displacement and velocity components, respectively, and ψ(x,y,s) is the self-field potential. The Hamiltonian is formally assumed to be of order ε, a small dimensionless parameter proportional to the characteristic strength of the focusing field as measured by the lattice coefficients κ_{x}(s) and κ_{y}(s). Using a third-order Hamiltonian averaging technique developed by P. J. Channell [Phys. Plasmas 6, 982 (1999)], a canonical transformation is employed that utilizes an expanded generating function that transforms away the rapidly oscillating terms. This leads to a Hamiltonian, H(X[over ̃],Y[over ̃],X[over ̃]^{′},Y[over ̃]^{′},s)=(1/2)(X[over ̃]^{′2}+Y[over ̃]^{′2})+(1/ 2)κ_{f}(X[over ̃]^{2}+Y[over ̃]^{2})+ψ(X[over ̃],Y[over ̃],s), correct to order ε^{3} in the “slow” transformed variables (X[over ̃],Y[over ̃],X[over ̃]^{′},Y[over ̃]^{′}). Here, the transverse focusing coefficient in the transformed variables satisfies κ_{f}=const, and the asymptotic expansion procedure is expected to be valid for a sufficiently small phase advance (σ<π/3=60°, say). Properties of axisymmetric beam equilibrium distribution functions, F_{b}^{0}(H^{0}), with ∂/∂s=0=∂/∂Θ, are calculated in the transformed variables, and the results are transformed back to the laboratory frame. Corresponding properties of the periodically focused distribution function f_{b}(x,y,x^{′},y^{′},s) are calculated correct to order ε^{3} in the laboratory frame, including statistical averages such as the mean-square beam dimensions, 〈x^{2}〉(s) and 〈y^{2}〉( s), the unnormalized transverse beam emittances, ε_{x}(s) and ε_{y}(s), the self-field potential, ψ(x,y,s), the number density of beam particles, n_{b}(x,y,s), and the transverse flow velocity, V_{b}(x,y,s). As expected, the beam cross section in the laboratory frame is a pulsating ellipse for the case of a periodic-focusing quadrupole field or a pulsating circular cross section for the case of a periodic-focusing solenoidal field.