Single bunch stability to monopole excitation

Abstract

We study single bunch stability with respect to monopole longitudinal oscillations in electron storage rings. Our analysis is different from the standard approach based on the linearized Vlasov equation. Rather, we reduce the full nonlinear Fokker-Planck equation to a Schrödinger-like equation which is subsequently analyzed by perturbation theory. We show that the Haissinski solution [Nuovo Cimento Soc. Ital. Fis. 18B, 72 (1973)] may become unstable with respect to monopole oscillations and derive a stability criterion in terms of the ring impedance. We then discuss this criterion and apply it to a broadband resonator impedance model. [S1098-4402(99)

Strongly Nonlinear Stochastic Processes in Physics and the Life Sciences

“…The reason for this is that the dynamics of individual particles is affected by the wake-field which can be calculated from an appropriate expectation value of the bunch particle probability density ( , ). As a result, in various studies it has been proposed that ( , ) satisfies a strongly nonlinear Fokker-Planck equation of the form (29) (see, e.g., [89][90][91][92][93][94][95][96][97][98]). A useful model, in particular for the purpose of theory development, is the Haissinski model [95,96,[99][100][101][102][103][104] for which analytical solutions of the reduced density ( 1 ) can be derived.…”

From nanoparticle to nanocable: Impact of size and geometrical constraints on the optical modes of Si/SiO2 core/shell nanostructures

“…3͑c͒, the m = 0 mode actually corresponds to a uniform charge distribution across the cylindrical surface and the longitudinal charge oscillations in such a configuration has a monopole character. 24,27 For nanocables, the IP excitation is contributed by a summation of all the different modes. Nevertheless, when the diameter of the Si core is reduced, the number of m Ͼ 0 modes will be greatly reduced, and the m = 0 mode becomes dominant in the very thin wires.…”

Single bunch monopole instability

“…Since the off-diagonal terms of n;k are small and the others quickly converge to zero, a good approximation for the roots can be found by truncating the matrix M. If we truncate it to the lowest nontrivial rank 2 then zero determinant occurs for = , 1 …”

“…The approach is summarized below and the details can be found in [1]. We assume small deviation from the Haissinski solution.…”

“…s0 t; (1) where z is the position of a particle with respect to the bunch centroid (z 0 in the head of a bunch), is the relative energy spread, ! s0 is the synchrotron frequency, and the subscript "0" refers to zero-current quantities.…”