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

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

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

Momentum Range for the Model The 3 to 10 GeV protons have β  = to The 3 to MeV eˉ have β  = to Range is smaller because the model has fewer cells, causing too large an edge focusing at low energies. Proton ring has 66 cells, with ° 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.

Lattice Cell for Electron Model bd(-) BF(+/-) BD(+) BF(+/-) bd(-) ( cm) ° 7.0° ° 7.0° -7.0° Lengths and angles for the MeV orbit

Parameters for the Electron Model Maximum orbit length of five unit NFFAG cell = m. Minimum orbit length of five unit NFFAG cell = m. Circumference for the 27 lattice cells = to m. bd magnet field range = to gauss. BF magnet field range = to gauss. BD magnet field range = to 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 MeV =

Lattice and Magnet Parameters Kinetic energy (MeV) Full ε n ((π) mm mr) Maximum β v (m) Maximum β h (m) Maximum D h (m) Max. v beam size (mm) Max. h beam size (mm) Max. chamber height (mm) 4.5 Max. orbit separation (mm) 39.5 Magnet v x h gap size (mm) 10.0 x 45.0

Acceleration Frequency at Reference Energies Proton Driver (h =40) Electron Model (h =3) T(GeV) F(MHz) T(MeV) F(MHz) x x 12……… x 12……… x 12……… x 12……… x 12……… x 12……… x 12………. Code for e - model to be modified for more accuracy. (Smaller frequency range for electrons)

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 ° to °. 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.

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 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

Comparison of Model with Proton Driver For the proton driver at a reference energy of 6.5 GeV, for example: α 1 = α 2 = 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.

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.

Space Charge Effects Model needs to simulate the proton driver space charge effects. Assume electrons in a single, MHz beam bunch. Assume Hofmann-Pederson, longitudinal beam distributions. Find bunching factors for  Q= 0.1 at compression & injection. Find bunch area for δ = ± , 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.

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

3 MeV Injection Parameters For N = ,  = & gaussian transverse distributions,  Q = 0.1 at a bunching factor of B f = The bunch phase and time extents needed at MHz are :  φ = ± 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 = , α = For the injection, when h =3, η sc = and  p/p = ± V = volts during injection with h =3.

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 Needed per drift tube are ~ 50 V (inj.) and 164 V ( compr.). 50 Ω load 25 Ω input feeder 50 Ω load eˉeˉ

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 m (too large for the DL hall?) Increased bend angles restrict the range to 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.