Thermal equilibrium of bunched charged particle beams

Abstract

The Maxwell–Boltzmann distribution of a bunched charged particle beam is the state toward which every other distribution will relax. For beams with lifetimes much shorter than the time required for relaxation to equilibrium, it is the distribution at injection that minimizes the emittance growth due to relaxation toward equilibrium. Three-dimensional thermal distributions are found numerically for the case of linear external focusing forces acting on an axially symmetric bunched beam in a conducting pipe. Equations are derived for current loss into the conducting channel due to particle thermal motion in the equilibrium distribution. Relations between parameters such as perveance, emittance, space-charge tune depression, bunch radius, and bunch length are given over a wide range of conditions from short to long bunches and from space-charge dominated to emittance-dominated beams. Comparison is made with previous results of radial density profiles for unbunched beams and line-charge density profiles for bunched beams [Phys. Rev. Lett. 71, 2911 (1993)].

Implications of the electrostatic approximation in the beam frame on the nonlinear Vlasov–Maxwell equations for intense beam propagation

“…At the high beam currents and charge densities of practical interest, of particular importance are the effects of the intense self fields produced by the beam space charge and current on determining the detailed equilibrium, stability and transport properties, and the nonlinear dynamics of the system. Through analytical studies based on the nonlinear Vlasov-Maxwell equations for the distribution function f b (x, p, t) and the self-generated electric and fields E s (x, t) and B s (x, t), 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 [10][11][12][13][14][15][16][17][18][19][20][21][22][23][24][25][26][27][28]. In almost all applications of the Vlasov-Maxwell equations to intense beam propagation, the analysis is carried out in the laboratory frame, and various simplifying approximations are made, ranging from the electrostatic-magnetostatic approximation [29], to the Darwin-model approximation [30][31][32][33][34][35] which neglects fast transverse electromagnetic perturbations.…”

Single-parameter characterization of the thermal equilibrium density profile for intense non-neutral charged particle beams

“…Extensive numerical investigations of Eqs. (6) and (7) have been presented in the literature [1,2,[11][12][13][14][15][16][17] which characterize the equilibrium solutions typically in terms of two parameters corresponding to the temperature T b and the on-axis number densityn b , or scaled versions thereof. The purpose of the present work is to show that appropriately normalized radial profiles for n 0 b ͑r͒ and f 0 ͑r͒ can be characterized in terms of a single dimensionless parameter d b defined by…”

“…In addition, for present purposes, the particle motion in the beam frame is assumed to be nonrelativistic, and we consider the class of intense non-neutral beam equilibrium solutions ͑≠͞≠t 0͒ to the nonlinear Vlasov-Maxwell equations which are axisymmetric ͑≠͞≠u 0͒ about the beam axis and are continuous in the axial direction with ≠͞≠z 0. Denoting the four-dimensional transverse phase space by ͑x, y, x 0 , y 0 ͒, where x 0 dx͞ds and y 0 dy͞ds are (dimensionless) transverse velocities, and s b b ct is the normalized time variable, it is readily shown that distribution functions f 0 b ͑x, y, x 0 , y 0 ͒ of the general form [1,[11][12][13] …”

“…For a thermal equilibrium distribution function A detailed understanding of the influence of spacecharge effects on the equilibrium and stability properties of intense charged particle beams is increasingly important for applications of high-intensity accelerators and transport systems to basic scientific research, heavy ion fusion, spallation neutron sources, waste transmutation, and tritium production [1][2][3][4][5][6]. In the beam frame, such intense non-neutral beams [1-13] share many properties in common with laboratory-confined non-neutral plasmas [1,[14][15][16][17][18][19], including thermal equilibrium properties, with density profile shape that exhibits a sensitive nonlinear dependence on space-charge intensity [1,2,[11][12][13][14][15][16][17][18][19]. For the case of a thermal equilibrium distribution function , and an inference (through a "best-fit" determination of d b ) of the temperature T b .…”

“…In the beam frame, such intense non-neutral beams [1][2][3][4][5][6][7][8][9][10][11][12][13] share many properties in common with laboratory-confined non-neutral plasmas [1,[14][15][16][17][18][19], including thermal equilibrium properties, with density profile shape that exhibits a sensitive nonlinear dependence on space-charge intensity [1,2,[11][12][13][14][15][16][17][18][19]. For the case of a thermal equilibrium distribution function F b ͑H 0 Ќ ͒, standard analyses [1,2,[11][12][13] of the nonlinear Vlasov-Maxwell equation for an intense cylindrical beam typically characterize the equilibrium density profile n 0 b ͑r͒ R dx 0 dy 0 F b ͑H 0 Ќ ͒ in terms of two parameters corresponding to the (transverse) temperature T b and onaxis densityn b , or scaled versions thereof. Here, r is the radial distance from the beam axis.…”