Muon Colliders ‘ December 2004 Optimization of adiabatic buncher and phase rotator for Muon Accelerators A.Poklonskiy (SPbSU, MSU), D.Neuffer (FNAL) C.Johnstone (FNAL), M.Berz (MSU), K.Makino (MSU)

Muon Colliders ‘ December 2004 Adiabatic buncher + (  ) Rotator (David Neuffer) Drift (90m) –  decay, beam develops  correlation Buncher (60m) (~333Mhz  200MHz, 0  4.8MV/m) –Forms beam into string of bunches  Rotator (~12m) (~200MHz, 10 MV/m) –Lines bunches into equal energies Cooler (~50m long) (~200 MHz) –Fixed frequency transverse cooling system Replaces Induction Linacs with medium-frequency RF (~200MHz)

Muon Colliders ‘ December 2004 Longitudinal Motion (2D simulations) Drift Buncher (  E) rotator Cooler System would capture both signs (  +,  - )    

Muon Colliders ‘ December 2004 Adiabatic Buncher overview Array of RF cavities Fix RF frequency at the end to 200 Mhz  1.5m, desired central energy and total length of the buncher RF phase is set to be 0 for reference energy through all buncher Find the condition for particles velocities to pass last RF in 0 phase (no energy change) and set frequencies in all RFs in buncher to maintain this condition Example: rf : 0.90  1.5m

Muon Colliders ‘ December 2004 Adiabatic Buncher overview Adiabatically increase RF gradient:

Muon Colliders ‘ December 2004  Rotator overview At end of buncher, change RF to decelerate high-energy bunches, accelerate low energy bunches, i.e. rotation in  phase space With central reference particle at 0 phase, set rf a bit less than bunch spacing (increase RF frequency) Places low/high energy bunches at accelerating/decelerating phases Change frequency along channel to maintain phasing Example: rf :  1.517m;

Muon Colliders ‘ December 2004  Rotator overview At end of buncher, choose: –Second reference particle TN –Vernier offset  Example: –T0 = 125 MeV –Choose N= 10,  =0.1 – T10 starts at MeV Along rotator, keep second ref particle at (N +  ) rf spacing –  10 = 36° at  =0.1 –Bunch centroids change: Use Erf = 10MV/m; L=8.74m –High gradient not needed –Bunches rotate to ~equal energies. Example: rf :  1.517m ;

Muon Colliders ‘ December 2004 Key Parameters Drift –Length L D Buncher –Length L B –RF Gradients E B –Final RF frequency RF (L D, L B, RF : (L D + L B )  (1/  ) = RF ) Phase Rotator –Length L  R –Vernier offset, spacing N  R,  V –RF gradients E  R

Muon Colliders ‘ December 2004 Central Energies Optimization Approach This is how rotator could look like in reality This is “transit time factor” (percent of the acceleration from maximum which particle could gain in changing E field): g – length of the cavity, w – cyclic frequency, v – particle’s velocity For  1. We can use kick approximation for the particle energy gain 2. We “forget” about the influence of cavity phases and gradients on all beam particles dynamics (could be added lately)  Study dynamics of the central particles of the bunches separately: T_final = T(n,…)

Muon Colliders ‘ December 2004 Centroids Kinetic Energies From buncher synchronism condition one could derive following relation for kinetic energies of central particles: Puts limits on n_min and n_max => n_bunches!

Muon Colliders ‘ December 2004 Final Centroids Kinetic Energy –From the rotator concept one could derive amount of energy gained by n-th synchronous particle in each RF (kept const by changing frequency) or, more generally –So for final energy n-th bunch central particle has after the ROTATOR consists of m RFs we have

Muon Colliders ‘ December 2004 Evolution of central energies shape T(n,m,…)

Muon Colliders ‘ December 2004 Energies shape in buncher and amount of kick they get in rotator

Muon Colliders ‘ December 2004 Energy Shape Evolution in Rotator

Muon Colliders ‘ December 2004 Objective Functions –The idea of the whole structure is to reduce overall beam energy spread and to put particles energies around some central energy. So we have general objective function:

Muon Colliders ‘ December 2004 Objective function 1 –First, we can set and get

Muon Colliders ‘ December 2004 Different optimized paremters (n vs T_fin), COSY built-in optimizer

Muon Colliders ‘ December 2004 Different optimized paremters (T_0 vs T_fin), COSY built-in optimizer

Muon Colliders ‘ December 2004 Objective Function 2 –As we can use particle’s energies distribution in a beam n energy particles % …

Muon Colliders ‘ December 2004 Objective Function 2

Muon Colliders ‘ December 2004 Modeled Optimization (OBJ1), whole domain search Fixed params: Desired central kinetic energy (T_c) = T_0 in buncher (T_0) = Drift+Buncher length (L_buncher) = Final frequency (final_freq) = Varied params: 1st lever particle (n1) : ==> nd lever particle (n2) : ==> Vernier parameter (vernier) : ==> RF gradient (V_RF) : ==> Number of RFs in rotator (m) : ==> Objective functions: !! ==> = ==> = ==> = ==> =

Muon Colliders ‘ December 2004 Modeled Optimization (OBJ2), whole domain search Fixed params: Desired central kinetic energy (T_c) = T_0 in buncher (T_0) = Drift+Buncher length (L_buncher) = Final frequency (final_freq) = Varied params: 1st lever particle (n1) : ==> nd lever particle (n2) : ==> Vernier parameter (vernier) : ==> RF gradient (V_RF) : ==> Number of RFs in rotator (m) : ==> Objective functions: ==> = !! ==> = ==> = ==> =

Muon Colliders ‘ December 2004 Other possible objective functions We may try to incorporate information about buckets widths and lengths We may combine this objective function with any of the first two with any weight coefficients

Muon Colliders ‘ December 2004 Other Optimization (OBJ1) (asked by David), whole domain search Fixed params: Desired central kinetic energy (T_c) = T_0 in buncher (T_0) = Drift+Buncher length (L_buncher) = Final frequency (final_freq) = Varied params: 1st lever particle (n1) : ==> nd lever particle (n2) : ==> Vernier parameter (vernier) : ==> RF gradient (V_RF) : ==> Number of RFs in rotator (m) : ==> Objective functions: !! ==> = ==> = ==> = ==> =

Muon Colliders ‘ December 2004 Other Optimization (OBJ2) (asked by David), whole domain search Fixed params: Desired central kinetic energy (T_c) = T_0 in buncher (T_0) = Drift+Buncher length (L_buncher) = Final frequency (final_freq) = Varied params: 1st lever particle (n1) : ==> nd lever particle (n2) : ==> Vernier parameter (vernier) : ==> RF gradient (V_RF) : ==> Number of RFs in rotator (m) : ==> Objective functions: ==> = !! ==> = ==> = ==> =

Muon Colliders ‘ December 2004 Summary Model of central energies shape optimization for buncher and phase rotator is proposed. Ready-to-use program is written, It allows to perform optimization on any set of supported parameters (length of the buncher and rotator, final frequency, central energy, E field gradient, phases). We can search for optimal parameters values in any desired range and check some previously chosen params for optimality. (It could take long… ) Some example results are presented

Muon Colliders ‘ December 2004 To do  Check results in (t,E) space as more important (problem: energy is changing  )  Different RF field waveform?  Check optimized parameters for the whole beam distribution (COSY, ICOOL?) Is it really better?  Switch to 3D-motion simulation and optimization