Hamiltonian formalism for space charge dominated beams in a uniform focusing channel

Abstract

Halo formation for a test particle in a mismatched KV beam is studied. Parametric resonances of the particle Hamiltonian due to envelope modulation are studied with particular emphasis on period 2 resonance which plays dominant role in Halo formation. It is shown that the onset of global chaos exhibits a sharp transition when the amplitude of modulation is larger than a critical value which is a function of a single parameter, , i.e., the ratio of the space charge perveance to the focusing strength.

Particle-beam approach to collective instabilities—application to space-charge dominated beams

“…which agrees with the result of Riabko [6] that 2J r +|p ϕ | = 1 2 , where J r is the radial action. The equation of motion for the particle radial position inside the beam core is (6.8) where the Floquet phase advance dψ = dθ/R 2 has been used.…”

“…In terms of the normalized and dimensionless envelope radius R, together with its conjugate momentum P , the Hamiltonian for the beam envelope in a uniformly focusing channel can be written as [5,6] H e = 1 4π 1) with the potential 2) where µ/(2π) is the unperturbed particle tune, κ = Nr cl /(µβ 2 γ 3 ) plays the role of the normalized space-charge perveance, N is the number of particles per unit length having the classical radius r cl , and β and γ are the relativistic factors of the beam. The normalized K-V equation then reads…”

“…We choose y as the particle's transverse coordinate with canonical angular momentum p y . Its motion is perturbed by an azimuthally symmetric oscillating beam core of radius R. The particle Hamiltonian is [6] …”

Crystalline ion beams in the RF quadrupole storage ring PALLAS

“…The excitation of oscillatory modes of an ion crystal passing through the periodic bending and focusing sections has to be carefully avoided. Numerical simulations yield that to maintain a beam crystal (and, more fundamental, to prevent envelope instabilities of space charge dominated beams [26,27]), the number of repeating focusing sections P (the periodicity of the ring) has to be larger than 2V2-. Q, the storage ring tune Q being the number of radial oscillations per revolution [28].…”

Halos of intense proton beams

“…The above single particle Hamiltonian was used as a starting point for deriving three-dimensional envelope equations in rf linacs, and this was in turn used to develop a procedure, based on a Hamiltonian formulation of the envelope equations, to find matched beams in rf linacs [21]. The concept of an envelope Hamiltonian has also been used in analytical studies of halo formation [22], [23].…”