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

NASA Astrophysics Data System (ADS)

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, -[?x\\(s\\)xe^x+?y\\(s\\)ye^y], on the beam ions. Here, s=?bct is the axial coordinate, \\(?b-1\\)mbc2 is the directed axial kinetic energy of the beam ions, and the (oscillatory) lattice coefficients satisfy ?x\\(s+S\\)=?x\\( s\\) and ?y\\(s+S\\)=?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 fb\\(x,y,x',y',s\\) and the (normalized) self-field potential ?\\(x,y,s\\) in the transverse laboratory-frame phase space \\(x,y,x',y'\\). Here, H^\\(x,y,x', y',s\\)=\\(1/2\\) \\(x'2+y'2\\)+\\( 1/2\\)[?x\\( s\\)x2+?y\\(s\\)y2]+?\\(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 ?\\(x,y,s\\) is the self-field potential. The Hamiltonian is formally assumed to be of order ?, a small dimensionless parameter proportional to the characteristic strength of the focusing field as measured by the lattice coefficients ?x\\(s\\) and ?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~,Y~,X~',Y~',s\\)=\\(1/2\\) \\(X~'2+Y~'2\\)+\\(1/ 2\\)?f\\(X~2+Y~2\\)+? \\(X~,Y~,s\\), correct to order ?3 in the ``slow'' transformed variables \\(X~,Y~,X~',Y~'\\). Here, the transverse focusing coefficient in the transformed variables satisfies ?f=const, and the asymptotic expansion procedure is expected to be valid for a sufficiently small phase advance (?beam equilibrium distribution functions, F0b\\(H0\\), with ?/?s=0=?/??, are calculated in the transformed variables, and the results are transformed back to the laboratory frame. Corresponding properties of the periodically focused distribution function fb\\(x,y,x',y',s\\) are calculated correct to order ?3 in the laboratory frame, including statistical averages such as the mean-square beam dimensions, \\(s\\) and \\( s\\), the unnormalized transverse beam emittances, ?x\\(s\\) and ?y\\(s\\), the self-field potential, ?\\(x,y,s\\), the number density of beam particles, nb\\(x,y,s\\), and the transverse flow velocity, Vb\\(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.

Davidson, Ronald C.; Qin, Hong; Channell, Paul J.

1999-07-01