Thermal distribution of relativistic particle beams with space charge

Abstract

The Maxwell-Boltzmann ("thermal") distribution constitutes the natural thermodynamic equilibrium state for a charged particle beam, and knowledge of its properties is therefore of fundamental importance. The Boltzmann relation for the particle density has a nonanalytic form when the space-charge force is included. We use numerical integration to determine the transverse and longitudinal density profiles for a relativistic beam in a linear focusing system at different temperatures T± and 7V The calculated profiles are related to space-charge tune depression, rms width, perveance, and emittance of the beam.PACS numbers: 41.85.Ew, 52.25.Wz Many advanced charged particle beam experiments and applications, such as high-power microwave sources, free electron lasers, linear accelerators for heavy-ion inertial fusion, spallation neutron sources, radioactive waste transmutation, high-energy colliders, and other uses, require very high beam intensity so that the beam dynamics depend strongly on the particle density profile. It is therefore of fundamental interest to know the equilibrium state of the charged particle beam for a given situation. Thermodynamically, this equilibrium state is best described by a Maxwell-Boltzmann ("thermal") distribution with different transverse and longitudinal temperatures (T ± and T\\) since in practice many beams are not equipartitioned. Many effects lead to coupling between T± and T\\ [1]; we deal here with cases where the coupling is small. When space-charge forces are significant the equilibrium density profiles have a nonanalytic form and must be found numerically, which explains why the thermal distribution has received less attention in the literature on beam theory than it deserves. Lawson [2] has published numerical results for the radial density profiles of a continuous nonrelativistic thermal beam in a linear focusing channel. In our work reported here we extend Lawson's results by including the relativistic factor y 2 , correcting an error, and correlating the density profiles with space-charge tune depression, perveance, and emittance of the rms equivalent uniform (K-V) beam. In addition, we determine the line-charge density profiles for a bunched beam with linear longitudinal focusing forces for different longitudinal temperatures and relate the results to the tune depression and other parameters of the rms equivalent parabolic bunch.The equilibrium distribution /of a group of charged particles in a focusing channel can be found from the Vlasov equation. We assume that the potential can be written as the sum of a transverse potential </>±(r) and a longitudinal potential </>\\(z) in cylindrical coordinates, and that the beam and focusing system are uniform, or "smooth." Each of these potentials is the sum of a selfcomponent [0j_ 5 (r) and </>\\ s (z)] and an external focusing component [(/>± e (r) and (j>\\ e (z)].The Maxwell-Boltzmann distribution has the form f^foQxpi -H/kBT), where H is the single-particle Hamiltonian, kg is Boltzmann's constant, and T is the temperature. I...

Generation of initial kinetic distributions for simulation of long-pulse charged particle beams with high space-charge intensity

“…The thermal equilibrium distribution has been studied extensively in nonneutral plasma physics [30,110] and in accelerator physics by Reiser [31,111] and others [32,67,68,112,113]. Here we review previous results in a format that allows easy contrast to other continuousfocusing distributions (see Appendices D and E) while presenting extensions needed for practical implementation of Vlasov simulation loads using standard inputs for accelerator physics.…”

Adiabatic thermal equilibrium theory for periodically focused axisymmetric intense beam propagation

“…9,10 In the present work, the Hamiltonian for singleparticle motion is analyzed to find the approximate and exact invariants of motion, i.e., a scaled transverse Hamiltonian ͑nonlinear space charge included͒ and the angular momentum, from which the beam equilibrium distribution is constructed. For continuous beams with long pulses, the longitudinal energy spread is small such that the longitudinal motion can be treated as "cold" and decoupled from the transverse motion, which is kept nonrelativistic.…”

Emittance Measurement of Axisymmetric High-Current Electron Beam Confined by a Longitudinal Magnetic Field

“…The numbers from 1 to 8 are the same as those of Lawson and Reiser's curves. [11][12][13] Curves 4a, 4b, 4c, 5a and 5b have been added, because they are close to the measured result that will be shown later. Table I …”