• E. Startsev, R.C. Davidson, H. Qin
  PPPL, Princeton, New Jersey

In electrically neutral plasmas with strongly anisotropic distribution functions, free energy is available to drive different collective instabilities such as the electrostatic Harris instability and the transverse electromagnetic Weibel instability. Such anisotropies develop naturally in particle accelerators and may lead to a detoriation of beam quality. We have generalized the analysis of the classical Harris and Weibel instabilities to the case of a one-component intense charged particle beam with anisotropic temperature including the important effects of finite transverse geometry and beam space-charge. For a long costing beam, the delta-f particle-in-cell code BEST and the eighenmode code bEASt have been used to determine detailed 3D stability properties over a wide range of temperature anisotropy and beam intensity. A theoretical model is developed which describes the essential features of the linear stage of these instabilities. Both, the simulations and analytical theory, clearly show that moderately intense beams are linearly unstable to short-wavelength perturbations, provided the ratio of the longitudinal temperature to the transverse temperature is smaller than some threshold value.

• H. Qin, R.C. Davidson
  PPPL, Princeton, New Jersey

Single-particle dynamics in a time-dependent focusing field is examined. The existence of the Courant-Snyder invariant* is fundamentally the result of the corresponding symmetry admitted by the oscillator equation with time-dependent frequency.** A careful analysis of the admitted symmetries reveals a deeper connection between the nonlinear envelope equation and the oscillator equation. A general theorem regarding the symmetries and invariants of the envelope equation, which includes the existence of the Courant-Snyder invariant as a special case, is demonstrated. The symmetries of the envelope equation enable a fast algorithm for finding matched solutions without using the conventional iterative shooting method.

*E.D. Courant and H.S. Snyder, Ann. Phys. 3, 1 (1958). **R.C. Davidson and H. Qin, Physics of Intense Charged Particle Beams in High Energy Accelerators (World Scientific, 2001).

• R.C. Davidson, S.R. Hudson, I. Kaganovich, H. Qin, E. Startsev
  PPPL, Princeton, New Jersey

In application of heavy ion beams to high energy density physics and fusion, background plasma is utilized to neutralize the beam space charge during drift compression and/or final focus of the ion beam. It is important to minimize the deleterious effects of collective instabilities on beam quality associated with beam-plasma interactions. Plasma electrons tend to neutralize both the space charge and current of the beam ions. It is shown that the presence of the return current greatly modifies the electromagnetic Weibel instability (also called the filamentation instability), i.e., the growth rate of the filamentation instability greatly increases if the background ions are much lighter than the beam ions and the plasma density is comparable to the ion beam density. This may preclude using underdense plasma of light gases in heavy ion beam applications. It is also shown that the return current may be subject to the fast electrostatic two-stream instability.

• H. Qin, R.C. Davidson, S.R. Hudson, E. Startsev
  PPPL, Princeton, New Jersey

The collective effects in high-intensity 3D bunched beams are described self-consistently by the nonlinear Vlasov-Maxwell equations.* The nonlinear delta-f method,** a particle simulation method for solving the nonlinear Vlasov-Maxwell equations, is being used to study the collective effects in high-intensity 3D bunched beams. The delta-f method, as a nonlinear perturbative scheme, splits the distribution function into equilibrium and perturbed parts. The perturbed distribution function is represented as a weighted summation over discrete particles, where the particle orbits are advanced by equations of motion in the focusing field and self-consistent fields, and the particle weights are advanced by the coupling between the perturbed fields and the zero-order distribution function. The nonlinear delta-f method exhibits minimal noise and accuracy problems in comparison with standard particle-in-cell simulations. A self-consistent 3D kinetic equilibrium is first established for high intensity bunched beams. Then, the collective excitations of the equilibrium are systematically investigated using the nonlinear delta-f method implemented in the Beam Equilibrium Stability and Transport (BEST) code.

*R.C. Davidson and H. Qin, Physics of Intense Charged Particle Beams in High Energy Accelerators (World Scientific, 2001). **H. Qin, Physics of Plasmas 10, 2078 (2003).