Hamiltonian formalism for solving the Vlasov-Poisson equations and its applications to periodic focusing systems and coherent beam-beam interaction

Tzenov, Stephan I.; Davidson, Ronald C.

doi:10.1103/physrevstab.5.021001

Cited by 16 publications

(21 citation statements)

References 13 publications

(12 reference statements)

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“…Such a distribution, due to its highly inverted population in phase space, of course is of very limited practical interest. While Hamiltonian averaging techniques have been developed [31][32][33][34] that justify the smooth-focusing approximation and thereby permit investigation of a whole class of (approximate) beam equilibria, these averaging techniques typically require sufficiently small vacuum phase advance (s vac , 60 ± , say) and other approximations for their validity. Therefore, whether or not there exist periodically focused non-KV solutions to the Vlasov-Maxwell equations remains a question of continued fundamental importance, which we examine in this paper for an intense sheet beam propagating through a periodic focusing field.…”

Section: Introductionmentioning

confidence: 99%

Kinetic description of intense beam propagation through a periodic focusing field for uniform phase-space density

Davidson

Qin

Tzenov

et al. 2002

Phys. Rev. ST Accel. Beams

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The Vlasov-Maxwell equations are used to investigate the nonlinear evolution of an intense sheet beam with distribution function f b ͑x, x 0 , s͒ propagating through a periodic focusing lattice k x ͑s 1 S͒ k x ͑s͒, where S const is the lattice period. The analysis considers the special class of distribution functions with uniform phase-space density f b ͑x, x 0 , s͒ A const inside of the simply connected boundary curves, x 0 1 ͑x, s͒ and x 0 2 ͑x, s͒, in the two-dimensional phase space ͑x, x 0 ͒. Coupled nonlinear equations are derived describing the self-consistent evolution of the boundary curves, x 2 . The resulting model is shown to be exactly equivalent to a (truncated) warm-fluid description with zero heat flow and triple-adiabatic equation of state with scalar pressure P b ͑x, s͒ const͓n b ͑x, s͔͒ 3 . Such a fluid model is amenable to direct analysis by transforming to Lagrangian variables following the motion of a fluid element. Specific examples of periodically focused beam equilibria are presented, ranging from a finite-emittance beam in which the boundary curves in phase space ͑x, x 0 ͒ correspond to a pulsating parallelogram, to a cold beam in which the number density of beam particles, n b ͑x, s͒, exhibits large-amplitude periodic oscillations. For the case of a sheet beam with uniform phase-space density, the present analysis clearly demonstrates the existence of periodically focused beam equilibria without the undesirable feature of an inverted population in phase space that is characteristic of the Kapchinskij-Vladimirskij beam distribution.

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Section: Introductionmentioning

confidence: 99%

Kinetic description of intense beam propagation through a periodic focusing field for uniform phase-space density

Davidson

Qin

Tzenov

et al. 2002

Phys. Rev. ST Accel. Beams

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“…23,28,29 The averaged Hamiltonian allows one to look for more realistic distribution functions that are close but not exactly at equilibrium. These methods were developed primarily for application to linear focusing systems.…”

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confidence: 99%

Near equilibrium distributions for beams with space charge in linear and nonlinear periodic focusing systems

Sonnad

Cary

2015

Physics of Plasmas

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A procedure to obtain a near equilibrium phase space distribution function has been derived for beams with space charge effects in a generalized periodic focusing transport channel. The method utilizes the Lie transform perturbation theory to canonically transform to slowly oscillating phase space coordinates. The procedure results in transforming the periodic focusing system to a constant focusing one, where equilibrium distributions can be found. Transforming back to the original phase space coordinates yields an equilibrium distribution function corresponding to a constant focusing system along with perturbations resulting from the periodicity in the focusing. Examples used here include linear and nonlinear alternating gradient focusing systems. It is shown that the nonlinear focusing components can be chosen such that the system is close to integrability. The equilibrium distribution functions are numerically calculated, and their properties associated with the corresponding focusing system are discussed. V C 2015 AIP Publishing LLC.

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“…Examples include: detailed analytical and nonlinear perturbative simulation studies of collective processes, including the electron-ion two-stream instability [2][3][4][5][6][7], and the Harrislike temperature-anisotropy instability driven by T ⊥b T b [8][9][10][11]; development of a selfconsistent theoretical model of charge and current neutralization for intense beam propagation through background plasma in the target chamber [12][13][14][15]; development of a robust theoretical model of beam compression dynamics and nonlinear beam dynamics in the final focus system using a warm-fluid description [16]; development of an improved kinetic description of nonlinear beam dynamics using the Vlasov-Maxwell equations [2,[17][18][19][20], including identification of the class of (stable) beam distributions, and the development of…”

Section: Nonlinear Beam Dynamics and Collective Processesmentioning

confidence: 99%

Overview of Theory and Modeling in the Heavy Ion Fusion Virtual National Laboratory

Davidson¹,

Kaganovich²,

Lee³

et al. 2003

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Hamiltonian formalism for solving the Vlasov-Poisson equations and its applications to periodic focusing systems and coherent beam-beam interaction

Cited by 16 publications

References 13 publications

Kinetic description of intense beam propagation through a periodic focusing field for uniform phase-space density

Kinetic description of intense beam propagation through a periodic focusing field for uniform phase-space density

Near equilibrium distributions for beams with space charge in linear and nonlinear periodic focusing systems

Overview of Theory and Modeling in the Heavy Ion Fusion Virtual National Laboratory

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