The role of the focusing field profile on the stability of periodically focused particle beams

Abstract

Magnetic compound refractive lens for focusing and polarizing cold neutron beams Rev. Sci. Instrum. 78, 035101 (2007) The role of the focusing field profile on the stability of periodically focused particle beams In this paper, the role of the focusing field profile on the stability of periodically focused particle beams is investigated, paying special attention to the transport within the new regions of stability found recently for vacuum-phase advances well above 90°͓R. Pakter and F. B. Rizzato, Phys. Rev. Lett. 87, 044801 ͑2001͔͒. In particular, a solenoidal focusing field profile that goes from a smooth sinusoidal-like function to a sharp-edged step-funcion as a continuous parameter is varied is considered. It is shown that the new regions are always present, but may be very sensitive to changes in the focusing field profile. Specifically, as the focusing field becomes more localized, the new regions become narrower, occur at higher vacuum phase advances, and present a larger number of nonlinear resonances and chaos in the beam envelope phase space. Although in all the cases analyzed here it was found that there is a relatively thick layer of regular trajectories isolating the matched solution from the chaotic region, self-consistent simulations show that envelope phase-space chaos may affect beam dynamics, leading to some small emittance growth.

Equilibrium and Stability of Off-Axis Periodically Focused Particle Beams

“…Let us call attention to the interesting fact that the centroid motion and the envelope dynamics are uncoupled in this case. In other words, centroid dynamics does not affect the known stability results for the envelope dynamics [4,5,9,11,17] and is not affected by the latter as well. One should keep in mind that for good beam confinement both centroid and envelope have to be stable.…”

“…The beam envelope that determines the outer radius of the equilibrium beam around the centroid is shown to obey the familiar envelope equation [6,8,15,16], being independent of the centroid motion. An example of the Vlasov equilibrium is discussed in detail to show the possibility of finding beam solutions for which the extensively studied envelope equation [4,5,[9][10][11]17] is stable, whereas the centroid motion is unstable, revealing the importance of the centroid motion to overall beam confinement properties.We consider a free, continuous charged-particle beam propagating with average axial velocity b cê z through a periodic solenoidal focusing magnetic field described bywhere r xê x yê y , r x 2 y 2 1=2 is the radial distance from the field symmetry axis, s z b ct is the axial coordinate, B z s S B z s is the magnetic field on the axis, the prime denotes derivative with respect to s, c is the speed of light in vacuo, and S is the periodicity length of the magnetic focusing field. Since we are dealing with solenoidal focusing, it is convenient to work in the Larmor frame of reference [8], which rotates with respect to the laboratory frame with angular velocity L s qB z s=2 b mc, where q, m, and b 1 ÿ 2 b ÿ1=2 are, respectively, the charge, mass, and relativistic factor of the beam particles.…”

“…The beam envelope that determines the outer radius of the equilibrium beam around the centroid is shown to obey the familiar envelope equation [6,8,15,16], being independent of the centroid motion. An example of the Vlasov equilibrium is discussed in detail to show the possibility of finding beam solutions for which the extensively studied envelope equation [4,5,[9][10][11]17] is stable, whereas the centroid motion is unstable, revealing the importance of the centroid motion to overall beam confinement properties.…”

Centroid motion in periodically focused beams

“…We presently extend the technique applied in a previous analysis performed on envelope stability alone, 7 and, as mentioned earlier, draw the stability boundaries ␣ = ± 1 now simultaneously for both envelope and centroid equations in order to map the profusion of stable and unstable regions of the complete envelope-centroid system.…”

“…Pure envelope dynamics has been the subject of recent studies searching for stable operational regions in a parametric plane defined by the focusing field profile and intensity, which are relevant control parameters for this sort of system. 7 Drawing attention to solutions with the same periodicity as the focusing lattice-we call these solutions matched solutions-previous results point to the fact that stable regions exist and are separated from each other by a series of openings where the matched solutions are either unstable or simply do not exist. [8][9][10] Unstable regions of envelope dynamics are immediately discarded from the set of operational regions for beam transport, so we are interested in how the stable envelope regions respond to the centroid dynamics.…”