Warm-fluid stability properties of intense non-neutral charged particle beams with pressure anisotropy

Abstract

The macroscopic warm-fluid model developed by Lund and Davidson [Phys. Plasmas 5, 3028 (1998)] is used in the smooth-focusing approximation to investigate detailed electrostatic stability properties of an intense charged particle beam with pressure anisotropy. The macroscopic fluid-Maxwell equations are linearized for small-amplitude perturbations, and an eigenvalue equation is derived for the perturbed electrostatic potential δφ(x,t), allowing for arbitrary anisotropy in the perpendicular and parallel pressures, P⊥0(r) and P‖0(r). Detailed stability properties are calculated numerically for the case of extreme anisotropy with P‖0(r)=0 and P⊥0(r)≠0, assuming axisymmetric wave perturbations (∂/∂θ=0) of the form δφ(x,t)=δφ̂(r)exp(ikzz−iωt), where kz is the axial wavenumber, and Imω>0 corresponds to instability (temporal growth). For kz=0, the analysis of the eigenvalue equation leads to a discrete spectrum {ωn} of stable oscillations with Imωn=0, where n is the radial mode number. On the other hand, for sufficiently large values of kzrb, where rb is the beam radius, the analysis leads to an anisotropy-driven instability (Imω>0) provided the normalized Debye length (ΓD=λD⊥/rb) is sufficiently large and the normalized beam intensity (sb=ω̂pb2/2γb2ωβ⊥2) is sufficiently below the space-charge limit. Depending on system parameters, the growth rate can be a substantial fraction of the focusing frequency ωβ⊥ of the applied field.

Three-dimensional stability theory of a continuous beam with axisymmetric Kapchinskij-Vladimirskij distribution

“…The instability was then linked to the one investigated by Harris [17] for plasmas with anisotropy in velocity distribution. These findings and many related fine papers published afterward [18][19][20][21][22][23][24][25][26][27][28][29] mark success in exploring the intense beam stability. However, to date, the rigorous three-dimensional stability analysis based on the kinetic theory is still left in an incomplete stage and further progress remains to be pursued.…”

“…Applying Eqs. (21), (25), (61), and (C18), one can also show that the cold-beam limit for the electric potential inside the beam is given by r / I m kr-another familiar result.…”

“…(21) into the right-hand side (RHS) of Eq. (15) and carrying out the lengthy calculation outlined in Appendix A yield…”

Analytical Solutions for the Nonlinear Longitudinal Drift Compression (Expansion) of Intense Charged Particle Beams

“…In the present analysis, we employ a one-dimensional warm-fluid model [86][87][88]91] to describe the longitudinal nonlinear beam dynamics with average electric field given by the g-factor model with e b E z = −e 2 b g∂λ/∂x [79][80][81][82][83][84][85]. For example, for a space-charge-dominated beam with flat-top density profile in the transverse plane, g 2 ln(r w /r b ) [85].…”

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

“…Through analytical studies based on the nonlinear Vlasov-Maxwell equations for the distribution function f b ͑x, p, t͒ and the selfgenerated electric and fields E s ͑x, t͒ and B s ͑x, t͒, and numerical simulations using particle-in-cell models and nonlinear perturbative simulation techniques, considerable progress has been made in developing an improved understanding of the collective processes and nonlinear beam dynamics characteristic of high-intensity beam propagation in periodic focusing and uniform focusing transport systems [1,. Theoretical progress has also been made in the development and application of macroscopic fluid models for the description of intense beam equilibrium and stability properties [39][40][41][42]. Nonetheless, given the complexity of a detailed description of intense beam propagation based on the nonlinear Vlasov-Maxwell equations, it remains important to develop simplified kinetic models of beam propagation through periodic focusing systems, particularly models which are analytically tractable and robust in describing beam propagation over large distances.…”