Betatron Stability in Field-Modulated Compaction-Managed Mirror-Bend Achromats

JLAB-TN-03-039 20 October 2003 Betatron Stability in a Field-Modulated Compaction-Managed Mirror-Bend Achromat

D. Douglas

Abstract We give an example of a field-modulated compaction-managed mirror-bend achromat with reasonable betatron properties.

Introduction The homogenous-field compaction-managed mirror-bend achromat has been described in some detail elsewhere [1]. In the example provided during that discussion, all dipoles were assumed to be excited to the same field, only linear compaction terms were treated, and betatron stability was conjectural. Further investigation has led to the recognition – for certain parameter sets – that it is desirable to operate the various dipoles of the achromat at different excitations. This is particularly true when very compact layouts (with less than 3 m gross diameter) are desired. Using this additional free parameter, an example of a “field modulated” compaction-managed mirror-bend achromat has been generated to investigate the betatron stability of such systems.

In this example, an achromat with a gross diameter of 2.5 m was constrained to provide first- and second-order compactions appropriate to energy recovery with energy compression during deceleration at 20o before trough from ~100 MeV to ~7 MeV using a 750 MHz RF system.

Design Process and Results The compactions required for acceleration or deceleration with RF waveform curvature are readily established using simple arguments [2]. Given these, the bend angles and drift lengths defining the magnet pole-face contours of a mirror bend can readily be evaluated at numerous momenta for any desired selection of nominal dipole radii (fields) using straightforward spreadsheet based optimization tools [3]. For simplicity, the example given here was assumed to be mirror symmetric, though this is not required.

With the pole face geometry in hand, linear optics can be evaluated and beam envelopes propagated at any desired set of momentum offset from the reference orbit geometry/momentum. This can similarly be done using spreadsheet based models [4], though in the following example bend radii and angles and drift lengths for each of several momenta were input by hand into a standard optics code (DIMAD) as a cross check of the optimization computation. Good agreement was obtained, with spreadsheet- generated pole face rotation angles yielding only mm/mrad-level remnant dispersions and DIMAD optimization of pole-face rotations leading to only fractions of degree changes in the initial values required for numerically exact dispersion suppression.

Once numerically exact pole face rotation angles were established, the off-momentum reference orbit was evaluated (Figure 1) and beam envelopes propagated to give the

1 of 4 JLAB-TN-03-039 20 October 2003 image of a fixed object phase space (Figure 2). Orbit length was checked against the nominal required variation as well (Figure 3). The orbit behavior, by construction, was as expected; of greater interest is the fact that the betatron behavior is quite good over a reasonably large (10%) momentum interval. The conjectural betatron stability of Reference [1] is thus supported in this example.

Figure 1: Field-modulated compaction-managed mirror-bend achromat layout.

0.5 0.45 0.4 0.35 0.3 0.25 0.2 0.15 0.1 0.05 0 -3 -2.5 -2 -1.5 -1 -0.5 0

Figure 2: Lattice functions through mirror bend at various momentum offsets.

Figure 2a: Horizontal envelopes Figure 2b: Vertical envelopes

horizontal envelope vertical envelope

3 10 5% 9 5% 2.5 4% 8 4% 3% 7 3% ) ) 2 m m 6 ( ( 2% 2%

x x 1.5 a 5 a 1% 1% t t e e 4 b b 0% 0% 1 -1% 3 -1% 0.5 -2% 2 -2% -3% 1 -3% 0 -4% 0 -4% 0 1 2 3 4 -5% 0 1 2 3 4 -5% s (m) s (m) Figure 2c: Horizontal dispersion Figure 2d: Phase advance vs. momentum

horizontal dispersion phase advance

1 1.2 0.9 5% 1 0.8 4% e

0.7 3% c

) 0.8 n a

m 0.6 v ( 2%

d phase_x x a 0.6

a 0.5 1% t e phase_y e 0.4 s b

h 0.4 0.3 -1% p 0.2 -2% 0.2 0.1 -3% 0 -4% 0 0 1 2 3 4 -5% -0.06 -0.04 -0.02 0 0.02 0.04 0.06 s (m) dp/p

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Table 1 summarizes some of the parameters characterizing this achromat. Figure 4 provides the required bend and entry/exit angles vs. orbit radius (directly proportional to momentum) for each dipole.

Table 1: Design parameters

Full (FEL Exhaust) Energy 99 MeV Extraction Energy 6 MeV Linac operating phase 160o (20o before trough) Linac on-crest energy gain (99-6)/cos 20o MeV = 98.96 MeV

M56 required for energy compression 0.198006812 m

T566 required for curvature correction 0.8466128 m Gross diameter 2.5 m Nominal bend radius exterior dipoles 0.4 m interior dipole 0.3 m Bend angles Figure 4 Pole face rotation angles Figure 4

Figure 4: Bend angles vs. bend radius for mirror bend dipoles

140 120 100 bend angle, outboard dipoles ) 80 g e d (

60 inboard dipole bend angle e l g 40 n a inboard entry/exit angle,

d 20 outboard dipoles n e b 0 entry/exit angles, inboard dipole -200.25 0.3 0.35 0.4 0.45 -40 -60 bend radius (m)

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References [1] D. Douglas, “Compaction Management for Mirror Bend Achromats”, JLAB-TN- 03-032, 24 September 2003. [2] D. Douglas, A. Hutton, Ch. Leeman, and D. Neuffer, “Analytic Modeling and Lattice Scaling Relations for FEL Driver Accelerators”, CEBAF-TN-95-015, 21 March 1995. [3] Simply proceed as in reference [1], but allow the radii to differ from dipole to dipole. [4] D. Douglas, “An Alternate Paradigm for High-Level Application Development”, JLAB-TN-97-047, 12 December 1997.

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