Adiabatic thermal equilibrium theory for periodically focused axisymmetric intense beam propagation

Abstract

An adiabatic equilibrium theory is presented for an intense, axisymmetric charged-particle beam propagating through a periodic solenoidal focusing field. The thermal beam distribution function is constructed based on the approximate and exact invariants of motion, i.e., a scaled transverse Hamiltonian and the angular momentum. The approximation of the scaled transverse Hamiltonian as an invariant of motion is validated analytically for highly emittance-dominated beams and highly space-charge-dominated beams, and numerically tested to be valid for cases in between with moderate vacuum phase advances ͑ v Ͻ 90°͒. The beam root-mean-square ͑rms͒ envelope equation is derived, and the self-consistent nonuniform density profile is determined. Other statistical properties such as flow velocity, temperature, total emittance and rms thermal emittance, equation of state, and Debye length are discussed. Numerical examples are presented, illustrating the effects of the beam perveance, emittance, and rotation on the beam envelope and density distribution. Good agreement is found between theory and a recent high-intensity beam experiment performed at the University of Maryland Electron Ring ͓S. Bernal, B. Quinn, M. Reiser, and P. G. O'Shea, Phys. Rev. ST Accel. Beams 5, 064202 ͑2002͔͒.

Dynamics of charged particles in an adiabatic thermal beam equilibrium

“…Recently, adiabatic thermal beam equilibria have been discovered in a periodic solenoidal magnetic focusing field [6][7][8] and an AG quadrupole magnetic focusing field [8,9]. The measured density distribution [5] matches that of the adiabatic thermal beam equilibrium in a spatially varying solenoidal magnetic focusing field [6,8].…”

“…The importance of this result is twofold: First, the elimination of chaotic particle motion provides a further numerical proof that the scaled transverse Hamiltonian defined in Eq. (25) in [6] is a very good approximate constant of motion. This approximate constant of motion and the exact contact of motion of the canonical angular momentum assure that the motion of charged particles is approximately integrable in the fourdimensional phase space of the adiabatic thermal beam equilibrium.…”

Simulation of periodically focused, adiabatic thermal beams

“…However, agreement between the theory and the experiment was found only for a short propagation distance that was less than one focusing period [12,13]. The lack of agreement between the theory and the experiment may be due to the fact that the beam was not created under the conditions required for the thermal beam equilibrium.…”

“…Recently, adiabatic thermal beam equilibrium has been predicated theoretically in a periodic solenoidal magnetic focusing field [11][12][13]. In general, an adiabatic thermal equilibrium corresponds to a spatially-varying equilibrium state in the zero-order approximation of the Boltzmann equation.…”

“…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 [12,13], 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.…”