Guiding-center Vlasov-Maxwell description of intense beam propagation through a periodic focusing field

Abstract

This paper provides a systematic derivation of a guiding-center kinetic model that describes intense beam propagation through a periodic focusing lattice with axial periodicity length S, valid for sufficiently small phase advance (say, s , 60 ± ). The analysis assumes a thin ͑a, b ø S͒ axially continuous beam, or very long charge bunch, propagating in the z direction through a periodic focusing lattice with transverse focusing coefficients k x ͑s 1 S͒ k x ͑s͒ and k y ͑s 1 S͒ k y ͑s͒, where S const is the lattice period. By averaging over the (fast) oscillations occurring on the length scale of a lattice period S, the analysis leads to smooth-focusing Vlasov-Maxwell equations that describe the slow evolution of the guidingcenter distribution functionf b ͑x,ȳ,x 0 ,ȳ 0 , s͒ and (normalized) self-field potentialc͑x,ȳ, s͒ in the fourdimensional transverse phase space ͑x,ȳ,x 0 ,ȳ 0 ͒. In the resulting kinetic equation forf b ͑x,ȳ,x 0 ,ȳ 0 , s͒, the average effects of the applied focusing field are incorporated in constant focusing coefficients k x sf . 0 and k y sf . 0, and the model is readily accessible to direct analytical investigation. Similar smoothfocusing Vlasov-Maxwell descriptions are widely used in the accelerator physics literature, often without a systematic justification, and the present analysis is intended to place these models on a rigorous, yet physically intuitive, foundation.

Near equilibrium distributions for beams with space charge in linear and nonlinear periodic focusing systems

“…These units are identical to those used in Ref. 28 where is shown that for thermal equilibrium distributions, a beam under constant focusing is specified by a single dimensionless parameter. In the present system of units, this dimensionless parameter is given by…”

“…23,28,29 The averaged Hamiltonian allows one to look for more realistic distribution functions that are close but not exactly at equilibrium. These methods were developed primarily for application to linear focusing systems.…”

Kinetic description of intense beam propagation through a periodic focusing field for uniform phase-space density

“…Such a distribution, due to its highly inverted population in phase space, of course is of very limited practical interest. While Hamiltonian averaging techniques have been developed [31][32][33][34] that justify the smooth-focusing approximation and thereby permit investigation of a whole class of (approximate) beam equilibria, these averaging techniques typically require sufficiently small vacuum phase advance (s vac , 60 ± , say) and other approximations for their validity. Therefore, whether or not there exist periodically focused non-KV solutions to the Vlasov-Maxwell equations remains a question of continued fundamental importance, which we examine in this paper for an intense sheet beam propagating through a periodic focusing field.…”

Weibel and Two-Stream Instabilities for Intense Charged Particle Beam Propagation through Neutralizing Background Plasma

“…(37) provides a convenient unit in which to measure the growth rate of the Weibel instability in the subsequent numerical analysis of the general dispersion relation (26). In the subsequent analysis of the dispersion relations (26) and (29), it is useful to define …”