Determination of the geometry factor for longitudinal perturbations in space-charge dominated beams

Abstract

We present the results of an experimental investigation of the parametric dependence of the geometry factor g associated with line-charge perturbations in space-charge dominated beams. The experiment consists of a novel method of launching localized space-charge waves and measuring simultaneously the wave velocity and the radius a of an electron beam propagating through a periodic focusing channel with pipe radius b. We find that the g factor obeys the relation g 2ln(b/a). This result is supported by theoretical analysis, and is also in agreement with previous theoretical work. The experimental technique can be used for any type of beam, whether space charge dominates over emittance or not.with r being the radial position within the beam. This relation implies that the g factor, as well as the field F" is a maximum with a=1 on the axis, and reduces parabolically to a minimum with a=O on the beam edge. Averaging the field over the beam cross section yields a=0.S. Hence, there is the question as to which value of a should be used. Indeed, Neil and Sessler raised this question from the very beginning: "This involves some average of E, over the beam cross section but. . . the precise average required is not clear. Because F, , varies slowly across the beam, we will. . . employ E, (r =0), although E, (r=a) is probably more accurate. " In the literature following this early work most authors (e.g. , The geometry factor g is an important parameter in longitudinal beam dynamics, which is discussed in most accelerator and beam physics books [1][2][3][4][5], as well as in other literature on plasma physics and fusion energy research, microwave theory and devices [6-8], etc. It relates the longitudinal electric field associated with a perturbation in a beam with the line-charge density variation. Under the long-wavelength limit this relationship can be expressed in the formwhere A~(z, t) is the perturbed line-charge density, eo is the permittivity of free space, and y is the Lorentz factor. For a round, unbunched beam of radius a in a pipe of radius b the g factor can be represented by the general, long-wavelength formula g =21n(b/a)+ a, where a is a constant for which difTerent values (1, 0.5, and 0) can be found in the literature. Neil and Sessler, in their original work [9], treated longitudinal instabilities of beams in particle accelerators. They used a uniformbeam model with constant radius a, and derived the relation [3,4]) use the value u=i though the average value of a =O.S would be more appropriate. The constant-radius assumption applies to emittance-dominated beams, as in circular accelerators. However, such beams have a Gaussian profile, and it is not clear to what extent the results of the uniform-beam model are valid. For spacecharge dominated beams, the uniformity of the particle density is a good approximation, but the radius does not remain constant so that Eq. (3) is not applicable. These questions concerning the theoretical models have motivated our investigation. So far, to the best of our knowledge, the...

Experimental Observations of Soliton Wave Trains in Electron Beams

“…Measurements with small initial perturbations confirm theoretical predictions and earlier experimental results [18], observing a slow and fast wave in the beam frame with identical sound speeds:…”

“…3, q and m are the charge and mass of beam particles, g is the geometry factor [18], λ 0 is the unperturbed beam line charge density. When the perturbation amplitude is large (say > 20% of beam current), the linear approximation no longer stands, and the particles on the crest travel faster than the ones on the trough, so that the wave will eventually steepen.…”

Longitudinal density modulation and energy conversion in intense beams

“…2). Such perturbations have been investigated for use in beam diagnostics [22,23]. The behavior of these waves also depends on the strength of the current, density, and velocity perturbations and on the shape of the initial disturbance.…”

“…For example, space-charge waves are widely thought to have potential as a beam diagnostic to measure quantities that appear in the sound speed [22,23,39]. The presence of two counter-propagating waves of similar shape will lead to interference effects [8], which may enhance the precision of such techniques.…”

Analytical Solutions for the Nonlinear Longitudinal Drift Compression (Expansion) of Intense Charged Particle Beams

“…In the present analysis, we employ a one-dimensional warm-fluid model [86][87][88]91] to describe the longitudinal nonlinear beam dynamics with average electric field given by the g-factor model with e b E z = −e 2 b g∂λ/∂x [79][80][81][82][83][84][85]. For example, for a space-charge-dominated beam with flat-top density profile in the transverse plane, g 2 ln(r w /r b ) [85].…”