RMS envelope matching of electron beams from “zero” current to extreme space charge in a fixed lattice of short magnets

We present detailed calculations of RMS-envelope matching over a broad range of beam intensities for the University of Maryland Electron Ring (UMER). Containment of beams from zero current to extreme space charge, all without changing the strength of external focusing in the periodic lattice, is possible thanks to the high density of quadrupoles in UMER. In turn, the small-aspect ratio of the UMER magnets results in gradient or field profiles that are “all edges,” thus requiring special treatment when constructing accurate hard-edge models. Further, the results of matching calculations, for both symmetric and asymmetric FODO (alternating gradient) schemes, are compared with calculations from simple general expressions valid in the uniform-focusing approximation of the periodic lattice. Finally, some aspects of the source-to-FODO matching calculation/optimization problem are discussed, together with sensitivity studies of the matching solutions under realistic conditions. The examples from the UMER project, which include experimental results, emphasize the practical aspects of beam envelope matching.

Figure 1

Matching/injection section and first ring diagnostics chambers (RC) in UMER. All focusing/bending elements, including the injection dipole (D0) and quadrupoles, were DC operated at this stage of UMER.

Figure 2

On-axis gradient (per Amp) profile of UMER air-core, printed-circuit quadrupole (2nd generation): (a) calculated, analytical, and hard-edge (HE) profiles, and (b) calculated, PMQ-equivalent gradient, and hard-edge (HE) profiles.

Figure 3

Characterization of effective hard-edge model of (2nd generation) UMER magnetic quadrupole: (a) effective lengths in the focusing and defocusing planes and average values as a function of quadrupole current, and (b) effective average strength as a function of quadrupole current [see Eq. (12)].

Figure 4

Characterization of short solenoid in UMER matching section: (a) measured and analytical fit function [Eq. (13)] of on-axis $B_z$ profile, and (b) normalized solenoid strength and hard-edge models.

Figure 5

Beam envelopes in a unit FODO cell in UMER for ${\sigma }_0=72.92°$. Periodic-matched solutions from the envelope code SPOT are shown for extreme space-charge dominated (top) and emittance-dominated (bottom) electron beams (see Table ). The focusing function is proportional to the gradient $g(s)$ of Eq. (3).

Figure 6

Asymmetric FODO matching: beam envelopes in UMER for ${\sigma }_{0x}=57.2°$, ${\sigma }_{0y}=73.4°$. Periodic-matched solutions from the envelope code SPOT are shown for a extreme space-charge dominated beam (10 keV, 100 mA, $60\mu \mathrm{m}$-edge emittance).

Figure 7

Envelope-matching calculations for a 100 mA, 10 keV, $ϵ=60\mu \mathrm{m}$ electron beam in UMER. The matching section consists of one solenoid and seven PC quadrupoles (see Fig. 1). Three solutions for ${\sigma }_0=72.92°$ are shown (SPOT code with smooth-gradient quadrupoles). The maximum value on the vertical axis corresponds to the vacuum pipe radius. The numbers in parentheses indicate the peak strengths (in ${\mathrm{m}}^{-2}$) of the solenoid and first quadrupole (see Table  for complete parameters).

Figure 8

Envelope-matching calculations for ${\sigma }_0=72.92°$ (from SPOT) vs experimental beam dimensions for a 100 mA, 10 keV, $ϵ=60\mu \mathrm{m}$ electron beam in UMER. The matching lattice consists of a short solenoid and six PC quadrupoles (see also last two columns in Table ). The focusing function (one polarity shown) appears at the bottom, including the equivalent hard-edge representation. The inset illustrates the envelope calculated over 1 m along the periodic lattice, which starts at Q7.

Figure 9

UMER RMS-envelope matching with a short solenoid and seven PC quadrupoles (the periodic lattice starts at Q7). Electron beam parameters are: 7.2 mA, 10 keV, and $ϵ=16\mu \mathrm{m}$. Envelope calculations for ${\sigma }_0=72.92°$ (from SPOT). The use of the “reference trajectory” for matching is explained in the Appendix. One transverse component of the focusing function is shown at the bottom, including the equivalent hard-edge function (see also third column in Table ).

Figure 10

Two solutions for UMER RMS-envelope matching with a short solenoid and a number of PC quadrupoles (see also Table ). Electron beam parameters are: 0.55 mA, 10 keV, and $ϵ=6.0\mu \mathrm{m}$. One transverse component of the focusing function is shown at the bottom in each case, including the equivalent hard-edge function. (a) Solution employing all six PC quadrupoles, and (b) solution with four PC quadrupoles (Q1 and Q2 are off). In (a), the envelopes that result from a simple hard-edge model (not the one at the bottom) are also shown.

Figure 11

UMER RMS-envelope matching with a short solenoid and seven PC quadrupoles. Electron beam parameters are: 100 mA, 10 keV, and $ϵ=60\mu \mathrm{m}$. Calculated beam envelope for ${\sigma }_{0x}=57.2°$, ${\sigma }_{0y}=73.4°$, highly asymmetric (from SPOT). One transverse component of the focusing function is shown at the bottom. The average beam radii indicated by the horizontal broken lines are $a_x=14.1\mathrm{mm}$, and $a_y=9.0\mathrm{mm}$ (see also Fig. 6).

Figure 12

Measured beam transverse dimensions (X: horizontal, Y: vertical) over 24 FODO periods in UMER for a 7.2 mA, 10 keV, electron beam. The horizontal dashed lines represent calculated beam dimensions for ${\sigma }_0=72.92°$ at the fluorescent screen diagnostics (FSD). FSD are available at every other FODO cell (ring chambers labeled “RC” in Fig. 1) at the plane indicated by the vertical dotted line in the inset. More details of similar experiments can be found in 46, 47.