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...

Results are presented for electron beam transport experiments in a 1-m-long straight section consisting of a solenoid and five short printed-circuit quadrupoles. A linear computer code for rms envelope matching, SPOT, is used for channel design, while final simulations with more realistic elements are obtained with a 2 1 2 D version of WARP, a particle-in-cell code. Reasonable agreement is found between calculations and the effective beam envelope obtained from pictures of the beam on a movable phosphor screen. The results validate, within experimental errors, the use of short magnetic quadrupoles for the transport of space-charge dominated beams. The straight section constitutes the prototype matching section for an electron recirculator to be built at the University of Maryland. [S1098-4402(98)

%e present in this Letter a novel experimental method that demonstrates the time reversability of the 1D cold-fluid equations for a space-charge dominated beam. A bunch of charged particles with an initially rectangular line-charge and velocity profile experiences edge erosion during transport through a focusing system due to the nonlinear, longitudinal space-charge forces at the edges. By imposing on the eroded beam pulse a linear velocity gradient at a proper location one can restore the rectangular line-charge profile downstream.This technique could be used to offset edge erosion and assure a rectangular pulse shape for efficient acceleration and uniform power flow in the interaction region for applications of intense, high-power beams.

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