1. Electron Model for a 3-10 GeV, NFFAG Proton Driver G H Rees, RAL

2. Proton Driver: Linac, Booster & NFFAG Layout 3 GeV RCS booster 66 cells 200 MeV Hˉ linac 10 GeV NFFAG Hˉ collimators H°, Hˉ

3. Aims of the Electron Model The aims of the model are to check the following: The effects of the non-linear magnetic fields Constancy of tune at 25 reference energies The halo growth during 50 Hz acceleration The range of momentum for the acceleration Adiabatic bunch compression to < 1 ns rms Longitudinal space charge limited operation The effect of  Q = 0.1 at the injection energy The effect of  Q = 0.1 for compressed bunch

4. Momentum Range for the Model The 3 to 10 GeV protons have β  = 4.197 to 11.658. The 3 to 5.44632 MeV eˉ have β  = 6.871 to 11.658. Range is smaller because the model has fewer cells, causing too large an edge focusing at low energies. Proton ring has 66 cells, with 5.4545° bend per cell. Electron ring has 27 cells,with 13.33° bend per cell. An eˉ ring with more cells leads to very low B fields. The β  range is chosen for the compression studies.

5. Lattice Cell for Electron Model bd(-) BF(+/-) BD(+) BF(+/-) bd(-) 17.5 5.5 5.0 5.5 5.0 11.0 ( cm) 5.0 5.5 5.0 5.5 17.5 -7.0° 7.0° 13.333° 7.0° -7.0° Lengths and angles for the 5.446 MeV orbit

6. Parameters for the Electron Model Maximum orbit length of five unit NFFAG cell = 0.88000 m. Minimum orbit length of five unit NFFAG cell = 0.87385 m. Circumference for the 27 lattice cells = 23.594 to 23.76 m. bd magnet field range = - 259.71 to - 439.78 gauss. BF magnet field range = - 90.44 to + 439.78 gauss. BD magnet field range = + 580.63 to + 418.84 gauss. Horizontal and vertical cell betatron tunes = 4/13, 3/13. Horizontal and vertical ring betatron tunes = 8 4/13, 6 3/13. Value of gamma transition at 5.446 MeV = 15.3635

7. Lattice and Magnet Parameters Kinetic energy (MeV) 3.0000 5.4465 Full ε n ((π) mm mr) 10.0000 10.0000 Maximum β v (m) 0.7120 1.5127 Maximum β h (m) 0.8190 0.6057 Maximum D h (m) 0.0790 0.0626 Max. v beam size (mm) 2.0468 2.2824 Max. h beam size (mm) 2.5112 2.4458 Max. chamber height (mm) 4.5 Max. orbit separation (mm) 39.5 Magnet v x h gap size (mm) 10.0 x 45.0

8. Acceleration Frequency at Reference Energies Proton Driver (h =40) Electron Model (h =3) T(GeV) F(MHz) T(MeV) F(MHz) 10.0 14.90741 5.446 3 x 12.571026 8.8 14.89598 5.1 3 x 12………. 8.0 14.88475 4.8 3 x 12………. 7.2 14.86920 4.5 3 x 12………. 6.2 14.84049 4.2 3 x 12………. 5.0 14.78232 3.9 3 x 12………. 3.6 14.64220 3.3 3 x 12………. 3.0 14.52924 3.0 3 x 12………. Code for e - model to be modified for more accuracy. (Smaller frequency range for electrons)

9. Electron Model Lattice Studies Fix path length at top energy, but change the bend angles: bd: - 7° to - 8°, BF: 7° to 7.5°, BD 13.333° to 14.333°. Reduce number of reference energies (δ) from 25 to 17. There are two complete designs for the model; final choice is related to bunch compression.

10. Non-linear Cell, Longitudinal Effects For p = p o (1 + δ), L = L o (1 + δ (α o + α 1 δ + α 2 δ 2 + …..) Non-linear lattice program gives L(+δ), L(-δ) and α o (=  t -2 ) Hence it is possible to solve for α 1 and α 2. F o is the frequency of one of the 25 (p o, C o ) reference orbits.  F is the change in frequency between the orbits. C = ring circumference = C o (1 + δ (α o + α 1 δ + α 2 δ 2 +,..) h = 3, F = h β(δ) c / C(δ),  =  F/F o = (  o -2 - α o ) δ + (  o -2 (α o - 1 - β o 2 /2) - α 1 ) δ 2 - (α 2 +  o -2 (α o (1 + β o 2 /2) + α 1 + 5β o 2 /2 - β o 4 /2 +  o -4 )) δ 3

11. Comparison of Model with Proton Driver For the proton driver at a reference energy of 6.5 GeV, for example: α 1 = - 27.94037 α 2 = - 681.8648. The values of α 1 and α 2 at the compression energy need to be compared for the proton driver and the two electron model designs, before selecting the final electron model.

12. Betatron Tunes for the Model Reference energies are at 2 to 3% momentum intervals. Tunes at ref. energies are 4/13, 3/13, as in proton driver. Ring has 27 cells, 26 of which have Q h = 8 and Q v = 6. At a reference energy, betatron tunes vary with amplitude. Orbits near reference energies don’t have zero chromaticity. The tunes vary with energy, but return to reference values.

13. Space Charge Effects Model needs to simulate the proton driver space charge effects. Assume 5 10 9 electrons in a single, 37.713 MHz beam bunch. Assume Hofmann-Pederson, longitudinal beam distributions. Find bunching factors for  Q= 0.1 at compression & injection. Find bunch area for δ = ± 8 10 -3, T < 1 ns rms, at compression. Find the rf voltage needed to achieve the bunch compression. Find rf voltage needed at injection for  Q = 0.1 and η sc < 0.4.

14. Bunch Compression Parameters For N = 5 10 9,  = 11.658 & gaussian transverse distributions,  Q = 0.1 at a bunching factor of B f = 0.0322. The bunch phase and time extents needed at 37.713 MHz are:  φ = ± 0.45 and  T = ± 1.899 ns (~ 0.76 ns rms). Assume  p/p = ± 8 10 -3 as for compressed proton driver bunch. Bunch area A (  E,  T ) = 0.282233 10 -3 eV sec. If Z wall cancels η sc, then V = hηπ (Ach / (Rα) ) 2 / (128  E o ). V = 112.5 volts for compression with h= 3.

15. 3 MeV Injection Parameters For N = 5 10 9,  = 6.675 & gaussian transverse distributions,  Q = 0.1 at a bunching factor of B f = 0.0989. The bunch phase and time extents needed at 37.713 MHz are :  φ = ± 1.159 and  T = ± 4.89 ns (~ 1.96 ns rms). η sc not cancelled; V(1 - η sc ) = hηπ (Ach / (Rα) ) 2 / (128  E o ). Vη sc = Neh 2 g / (2ε o RF  2 ), F = 0.9564, α = 0.2455. For the injection, when h =3, η sc = 0.1117 and  p/p = ± 5.5 10 -3. V = 33.98 volts during injection with h =3.

16. 37.713 MHz RF System The use of three, broad band mini-drift tubes is proposed. These are located in cells 1, 4 and 7 (120° in phase apart). Each drift tube is 0.29 m long in a 0.35 m straight section. The phase extent is ± 6.59° at the rf harmonic of three. The voltage reduction factor due to the short  φ is 0.2281 Needed per drift tube are ~ 50 V (inj.) and 164 V ( compr.). 50 Ω load 25 Ω input feeder 50 Ω load eˉeˉ

17. Summary An eˉ model for the NFFAG proton driver is now proposed. It has 27 cells in comparison with the 66 of the proton driver. The ring circumference is 23.76 m (too large for the DL hall?) Increased bend angles restrict the range to 3 - 5.4463 MeV. The cell tunes at each reference energy are as for the driver, but there are more edge effects in the electron models. Design aims for the eˉ model are met, but development of a 3D space charge code is needed for tracking of acceleration.