Emittance growth from merging arrays of round beamlets

Abstract

The cost of an induction linac for heavy ion fusion (HIF) may be reduced if the number of channels in the main accelerator is reduced. There have been proposals to do this by merging beamlets (perhaps in groups of four) after a suitable degree of preacceleration. This process, which results in r.m.s. emittance growth, occurs in two stages. The first stage occurs instantly when the beamlets, no longer separated by electrodes, enter the merging region. At this point one has a collection of beamlets whose centers are displaced from the central axis by distances δ xi , δ yi . The mean square emittance, now calculated for the whole collection, is ϵ xi 2 = ϵ x1 2 + 〈 δ x 2〉 V x 2 and similarly for the y direction. Here ϵ x1 is the emittance of one undisplaced beamlet, V x is the r.m.s. thermal velocity and ϵ xi is the initial emittance of the composite beam. This first stage of emittance growth is easy to calculate and has obvious scaling properties. The second stage of merged beam emittance growth mostly occurs in about one-quarter of a plasma period, although the full development may take much longer. In this stage, space charge forces cause transverse accelerations. The maximum increase in mean square emittance is proportional to the excess electrostatic energy (free energy) in the array when the merging begins. It tends to dominate the first stage for strongly depressed initial tunes. The relatively complex calculations involving the second stage are the main concern of this paper. In some designs it may be desirable to reduce the emittance growth below that produced by a basic 2 × 2 array. For this a general understanding of the free energy is helpful. Therefore we investigate three factors affecting the normalized free energy U n of an array of charged interacting beamlets: (1) the number of beamlets N in the array; (2) the ratio η of beamlet diameter to beamlet center spacing; (3) the shape of the array. For circular arrays we obtain an analytic expression for U n n as a function of N and η. If η is held constant, it shows that U n ∼ N −1 in the large- N limit, i.e. U n would become arbitrarily small in this idealized case. We show that this is not true for square or rectangular arrays, which have larger free energy with a lower limit determined by the non-circular format. Free energy in square arrays can be reduced by omitting corner beamlets; in the case of a 5 x 5 array the reduction factor is as large as 3.3.