Hiroshima University, Higashi-Hiroshima
HU/AdSM, Higashi-Hiroshima
LBNL, Berkeley, California
Liouville's theorem implies that the six-dimensional phase-space volume occupied by a charged-particle beam is an approximate invariant unless the beam is subjected to dissipative interactions (such as in cooling). Symplectic conditions, in a Hamiltonian system (once again, no dissipation), put constraints upon emittance transfer between the various degrees of freedom. [1] We can, however, even in non-dissipative Hamiltonian systems arrange for partial emittance transfers. This process results in phase space correlations and change in the emittance projections on to various phase planes; namely, the projected emittances in three degrees of freedom are controllable while the direction and amount of a possible emittance flow are not very flexible because of the symplectic nature of Hamiltonian system. In some applications, it is clearly advantageous to optimize the ratios of projected emittances despite the effect of correlations. Since the three emittances are not always equally important, we may consider reducing the emittance of one direction at the sacrifice of the other emittance(s). As a possible scheme to achieve such emittance control, we study a compact storage ring operating near resonance. The basic features of linear and nonlinear emittance flow are briefly discussed with numerical examples. A general discussion touching on some of these matters has been previously presented. [2]
[1] E. D. Courant, Perspectives in Modern Physics, edit R. E. Marshak (1966).[2] H. Okamoto, K. Kaneta and A. M. Sessler, to be published in J. Phys. Soc. Jpn.
BNL, Upton, Long Island, New York
Hiroshima University, Higashi-Hiroshima
LBNL, Berkeley, California
HU/AdSM, Higashi-Hiroshima
Recent theoretical investigations of beam crystallization mainly use computer modeling based on the method of molecular dynamics (MD) and analytical study based on phonon theory [1]. Topics of investigation include crystal stability in various accelerator lattices under different beam conditions, colliding crystalline beams [2], and crystalline beam formation in shear-free ring lattices with both magnets and electrodes [3]. In this paper, we review the above mentioned theoretical studies and, in particular, discuss the development of the phonon theory in a time-dependent Hamiltonian system representing a storage ring of AG focusing. Analytical study of crystalline beam stability in an AG-focusing ring was previously limited to the smooth approximation. In a typical ring, analytical results obtained under such approximation largely agrees with the results obtained with the molecular dynamics (MD) simulation method. However, as we explore ring lattices appropriate for beam crystallization at high energies (Lorentz factor gamma much higher than the betatron tunes) [2,4], this approximation fails. Here, we present a newly developed formalism to exactly predict the stability of a 1-dimensional crystalline beam in an AG focusing ring lattice.
[1] X.-P. Li, et al, PR ST-AB, 9, 034201 (2006). [2] J. Wei, A. M. Sessler, EPAC, 862 (1998)[3] M. Ikegami, et al, PR ST-AB 7, 120101 (2004).[4] J. Wei, H. Okamoto, et al, EPAC 2006.