Intense nonneutral beam propagation through a periodic focusing quadrupole field II--Hamiltonian averaging techniques in the smooth-focusing approximation
- Plasma Physics Laboratory, Princeton University, Princeton, New Jersey 08543 (United States)
This paper considers a compact Paul trap configuration to model the transverse nonlinear dynamics of an intense charged particle beam propagating through a periodic focusing quadrupole magnetic field in the collisionless regime. A long non-neutral plasma column (L>>r{sub p}) is confined axially by applied dc voltages V=const. on end cylinders at z={+-}L, and transverse confinement of the particles in the x-y plane is provided by segmented cylindrical electrodes (at radius r{sub w}) with applied oscillatory voltages {+-}V{sub 0}(t) over 90 deg. segments. Here, V{sub 0}(t+T)=V{sub 0}(t), where T=const. is the oscillation period. Neglecting axial variations ({partial_derivative}/{partial_derivative}z=0), the Hamiltonian describing the transverse motion (assumed nonrelativistic) of a particle with charge q and mass m near the cylinder axis (r{sub p}<<r{sub w}) is given by H{sub perpendicular}(x,y,x,y,t)=(m/2)(x{sup 2}+y{sup 2})+(m/2){kappa}{sub q}(t)(x{sup 2}-y{sup 2})+q{phi}{sub s}(x,y,t), where {phi}{sub s}(x,y,t) is the self-field electrostatic potential, and {kappa}{sub q}(t){identical_to}8qV{sub 0}(t)/{pi}mr{sub w}{sup 2} is the (oscillatory) quadrupole focusing coefficient due to the applied field. Using a third-order Hamiltonian averaging technique [R. C. Davidson, H. Qin, and P. J. Channell, Physical Review Special Topics on Accelerators and Beams 2, 074401 (1999)], a canonical transformation is employed that utilizes an expanded generating function that transforms away the rapidly oscillating terms in the Hamiltonian H{sub perpendicular}(x,y,x,y,t). Formally, {epsilon}=|{kappa}{sub q}|T{sup 2}/(2{pi}){sup 2}<1 is treated as a small dimensionless parameter, where {kappa}{sub q} is the characteristic (maximum) amplitude of the applied quadrupole field, and the canonical transformation is carried out correct to order {epsilon}{sup 3}. This leads to a Hamiltonian, correct to order {epsilon}{sup 3} in the 'slow' transformed variables Here, the transverse focusing coefficient in the transformed variables satisfies {omega}{sub q}{sup 2}=const., leading to enormous simplification in the analysis of the nonlinear Vlasov-Poisson equations for F(X-tilde,Y-tilde,XI1;,Y-dot-tilde,t) and {phi}{sub s}(X-tilde,Y-tilde0009,.
- OSTI ID:
- 21210368
- Journal Information:
- AIP Conference Proceedings, Vol. 498, Issue 1; Conference: Workshop on non-neutral plasma physics III, Princeton, NJ (United States), 2-5 Aug 1999; Other Information: DOI: 10.1063/1.1303712; (c) 1999 American Institute of Physics; Country of input: International Atomic Energy Agency (IAEA); ISSN 0094-243X
- Country of Publication:
- United States
- Language:
- English
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Related Subjects
APPROXIMATIONS
BEAM-PLASMA SYSTEMS
BOLTZMANN-VLASOV EQUATION
CANONICAL TRANSFORMATIONS
CHARGED PARTICLES
ELECTRIC POTENTIAL
FOCUSING
HAMILTONIANS
MAGNETIC FIELDS
NONLINEAR PROBLEMS
OSCILLATIONS
PERIODICITY
PLASMA
PLASMA CONFINEMENT
POISSON EQUATION
QUADRUPOLES
TRAPS