SIMULATION STUDIES OF ADIABATIC THERMAL BEAMS IN A PERIODIC SOLENOIDAL FOCUSING FIELD

Two-dimensional (2D) particle-in-cell (PIC) simulations are performed to verify earlier theoretical predictions of adiabatic thermal beams in a periodic solenoidal magnetic focusing field. In particular, results are obtained for adiabatic thermal beams that do not rotate in the Larmor frame. For such beams, the theoretical predictions of the rms beam envelope, the conservations of rms thermal emittances, the adiabatic equation of state, and the Debye length are verified in the PIC simulations. INTRODUCTION Adiabatic thermal beam equilibrium was discovered recently in a periodic solenoidal magnetic focusing field [1-3]. In particular, the existence of the adiabatic thermal beam equilibrium was shown in the frameworks of kinetic theory and equivalent warm-fluid theory. In the warm-fluid theory of the adiabatic thermal beam equilibrium [1,2], warm-fluid equations were solved in the paraxial approximation. The equation of state was adiabatic. The rms beam envelope, the density and flow velocity profiles, and the self-consistent Poisson equation were derived. In the kinetic theory of the adiabatic thermal beam equilibrium [3], the thermal beam distribution function was constructed using the approximate and exact invariants of motion, i.e., a scaled transverse Hamiltonian and the angular momentum. By taking statistical averages, all of the equations in the warm-fluid theory were recovered, including the adiabatic equation of state, the rms beam envelope, the density and flow velocity profiles, and the self-consistent Poisson equation. Effects of the beam perveance, emittance and rotation on the beam envelope and density distribution were examined. Good agreement was found [3] between theory and a recent high-intensity beam experiment performed at the University of Maryland Electron Ring (UMER) [4]. The phase space for charged-particle motion in the adiabatic thermal beam equilibrium was analyzed [5] and compared with that of the KV-type beam equilibrium [68]. It was found that the widths of nonlinear resonances in the adiabatic thermal beam equilibrium are narrower than those in the KV-type beam equilibrium. Numerical evidence is presented, indicating the almost complete absence of chaotic particle motion in the adiabatic thermal beam equilibrium. The discovery of the adiabatic thermal beam equilibrium was an important advance in beam physics, overcoming the shortcoming of the KapchinskijVladmirskij (KV) type equilibrium in a periodic solenoidal magnetic focusing field [6-8]. The KV type equilibrium has a singular ( function) distribution in the four-dimensional phase space. Such a function distribution gives a uniform density profile across the beam in the transverse directions, and a transverse temperature profile which peaks on axis and decreases quadratically to zero on the edge of the beam. Because of the singularity in the distribution functions, these beam equilibria are not likely to occur in real physical systems and cannot provide realistic models for theoretical and experimental studies and simulations except in the zerotemperature limit. For example, the KV equilibrium model cannot be used to explain the beam tails in the radial distributions observed in the recent high-intensity beam experiment [4]. In contrast, the measured density distribution matches that of the adiabatic thermal beam equilibrium [3,4]. In this paper, results of two-dimensional (2D) particlein-cell (PIC) simulations are presented, which further validate the theoretical predictions of the adiabatic thermal beam equilibrium. In particular, results are obtained for adiabatic thermal beams that do not rotate in the Larmor frame. For such beams, the theoretical predictions of the rms beam envelope, the conservations of rms thermal emittances, the adiabatic equation of state, and the Debye length are verified in the simulations. PARTICLE-IN-CELL MODEL We study charged-particle dynamics in the adiabatic thermal equilibrium of an intense charged-particle beam propagating with constant axial velocity z bce in the periodic solenoidal magnetic focusing field y x z z z ext y x ds s dB s B e e e B 2 1 , (1) where z s is the axial coordinate, s B S s B z z is the axial magnetic field, S is the fundamental periodicity length of the focusing field, and c is the speed of light in vacuum. The paraxial approximation is made under the following assumptions : 1) S rbrms , where brms r is the rms beam radius, and 2) 1 / 2 3 b b , where 2 2 / mc N q b is the Budker parameter of the beam, q and m are the particle charge and rest mass, respectively, rdr s r n N b b 2 , 0 = const is the number of charged particles per unit axial length, and 2 / 1 2 ) 1 ( b b is the relativistic mass factor. ___________________________________________ *Work supported by DOE Grant No. DE-FG02-95ER 40919, Grant No. DE-FG02-05ER54836, and MIT Undergraduate Research Opportunity Program (UROP). chenc@psfc.mit.edu Proceedings of IPAC2012, New Orleans, Louisiana, USA WEPPR032 05 Beam Dynamics and Electromagnetic Fields D04 High Intensity in Linear Accelerators ISBN 978-3-95450-115-1 3003 C op yr ig ht c ○ 20 12 by IE E E – cc C re at iv e C om m on sA tt ri bu tio n 3. 0 (C C B Y 3. 0) — cc C re at iv e C om m on sA tt ri bu tio n 3. 0 (C C B Y 3. 0) The basic equations in the 2D PIC model are expressed in cgs units as [3]