muon colliders ‘2004 14 december 2004 optimization of adiabatic buncher and phase rotator for muon...

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14 December 2004 Muon Colliders ‘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)

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14 December 2004 Muon Colliders ‘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)

14 December 2004 Muon Colliders ‘2004

Adiabatic buncher + () Rotator (David Neuffer)

• Drift (90m) decay, beam develops correlation

• Buncher (60m) (~333Mhz200MHz, 04.8MV/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)

14 December 2004 Muon Colliders ‘2004

Longitudinal Motion (2D simulations)

Drift Buncher

(E) rotator Cooler

System would capture both signs (+, -)

14 December 2004 Muon Colliders ‘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 : 0.901.5m

RF

L

14 December 2004 Muon Colliders ‘2004

Adiabatic Buncher overview

• Adiabatically increase RF gradient:

mMVzL

zz

L

zzC

L

zzBzE

D

DDDRF /8.4)( 2

2

2

2

01.0

150

5.1111

1 Lc

nTn n

cn

,111

1

)( LLRF

cnnT

vvLttt RF

RFc

c

11 n,

14 December 2004 Muon Colliders ‘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 : 1.4851.517m;

14 December 2004 Muon Colliders ‘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 77.28 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.

RRFR zEeTzT )sin()0()( 101010 Example: rf : 1.4851.517m;

14 December 2004 Muon Colliders ‘2004

Key Parameters

• Drift

– Length LD

• Buncher

– Length LB

– RF Gradients EB

– Final RF frequency RF (LD, LB, RF: (LD + LB) (1/) = RF)

• Phase Rotator

– Length LR

– Vernier offset, spacing NR, V

– RF gradients ER

14 December 2004 Muon Colliders ‘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,…)

2

2sin

...d

v

dv

FTT

97.0...86.0...333...200,1000...50 FTTMhzMeVT

14 December 2004 Muon Colliders ‘2004

Centroids Kinetic Energies

• From buncher synchronism condition one could derive following relation for kinetic energies of central particles:

nTn n

cn

,111

1

111

1)1,,(

20

cc

c

cc

n

WnT

Puts limits on n_min and n_max => n_bunches!

14 December 2004 Muon Colliders ‘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

12

121 2sin),,,,(

nn

nnEnnEnT RFRF

)()(,...),( nTmnTmnT

)2sin(),,,(N

nENEnT RFRF

14 December 2004 Muon Colliders ‘2004

Evolution of central energies shape T(n,m,…)

14 December 2004 Muon Colliders ‘2004

Energies shape in buncher and amount of kick they get in rotator

14 December 2004 Muon Colliders ‘2004

Energy Shape Evolution in Rotator

14 December 2004 Muon Colliders ‘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:

2)),(( cn TmnTcI

14 December 2004 Muon Colliders ‘2004

Objective function 1

– First, we can set and getnCn ,1

21 ),...),(( cTmnTI

14 December 2004 Muon Colliders ‘2004

Different optimized paremters (n vs T_fin), COSY built-in optimizer

14 December 2004 Muon Colliders ‘2004

Different optimized paremters (T_0 vs T_fin), COSY built-in optimizer

14 December 2004 Muon Colliders ‘2004

Objective Function 2

– As we can use particle’s energies distribution in a beam

22 )),(( cn TmnTcI

nc

n energy particles %---------------------------------------------- -12 963.96 1023 17.050000 -11 510.85 692 11.533333 -10 374.64 537 8.950000 -9 302.98 412 6.866667 …

14 December 2004 Muon Colliders ‘2004

Objective Function 2

22 )),(( cn TmnTcI

14 December 2004 Muon Colliders ‘2004

Modeled Optimization (OBJ1), whole domain search Fixed params:

Desired central kinetic energy (T_c) = 125.0000000000000 T_0 in buncher (T_0) = 125.0000000000000 Drift+Buncher length (L_buncher) = 90.00000000000000 Final frequency (final_freq) = 200000000.0000000Varied params: 1st lever particle (n1) : -4.000000000000000 ==>

1.000000000000000 2nd lever particle (n2) : -3.000000000000000 ==>

27.00000000000000 Vernier parameter (vernier) : 0.1000000000000000 ==>

0.3000000000000000 RF gradient (V_RF) : 1.000000000000000 ==>

10.00000000000000 Number of RFs in rotator (m) : 1.000000000000000 ==>

10.00000000000000Objective functions:!! 188624.7593430086 ==> 7434.672341694457 =

181190.0870013142 21918840.88332907 ==> 1643615.384258914 =

20275225.49907015 215.4599659295154 ==> 238.7961711271633 = -

23.33620519764796 31921138.21770231 ==> 15260635.43728683 =

16660502.78041548

14 December 2004 Muon Colliders ‘2004

Modeled Optimization (OBJ2), whole domain search Fixed params:

Desired central kinetic energy (T_c) = 125.0000000000000 T_0 in buncher (T_0) = 125.0000000000000 Drift+Buncher length (L_buncher) = 90.00000000000000 Final frequency (final_freq) = 200000000.0000000Varied params: 1st lever particle (n1) : -7.000000000000000 ==>

1.000000000000000 2nd lever particle (n2) : -6.000000000000000 ==>

14.00000000000000 Vernier parameter (vernier) : 0.1000000000000000 ==>

0.4000000000000000 RF gradient (V_RF) : 1.000000000000000 ==>

10.00000000000000 Number of RFs in rotator (m) : 1.000000000000000 ==>

10.00000000000000Objective functions: 552558.8685890788 ==> 331194.3462835634 = -

221364.5223055154!! 1120051685.347023 ==> 740898336.4248258 = -

379153348.9221967 790.5115736637570 ==> 714.6396495910632 = -

75.87192407269379 1614049125.098601 ==> 1105871829.498042 = -

508177295.6005592

14 December 2004 Muon Colliders ‘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

14 December 2004 Muon Colliders ‘2004

Other Optimization (OBJ1) (asked by David), whole domain searchFixed params:

Desired central kinetic energy (T_c) = 125.0000000000000 T_0 in buncher (T_0) = 125.0000000000000 Drift+Buncher length (L_buncher) = 75.00000000000000 Final frequency (final_freq) = 100000000.0000000Varied params: 1st lever particle (n1) : -3.000000000000000 ==>

1.000000000000000 2nd lever particle (n2) : -2.000000000000000 ==>

4.000000000000000 Vernier parameter (vernier) : 0.1000000000000000 ==>

0.2000000000000000 RF gradient (V_RF) : 1.000000000000000 ==>

10.00000000000000 Number of RFs in rotator (m) : 1.000000000000000 ==>

10.00000000000000Objective functions:!! 920802.1870498012 ==> 716144.2078318644 = -

204657.9792179369 2906605812.448390 ==> 2289340374.907225 = -

617265437.5411658 -1154.766872358291 ==> -1012.150761595708 =

142.6161107625823 2905272325.918894 ==> 2288315925.743026 = -

616956400.1758685

14 December 2004 Muon Colliders ‘2004

Other Optimization (OBJ2) (asked by David), whole domain searchFixed params:

Desired central kinetic energy (T_c) = 125.0000000000000 T_0 in buncher (T_0) = 125.0000000000000 Drift+Buncher length (L_buncher) = 75.00000000000000 Final frequency (final_freq) = 100000000.0000000Varied params: 1st lever particle (n1) : -3.000000000000000 ==>

0.000000000000000 2nd lever particle (n2) : -2.000000000000000 ==>

5.000000000000000 Vernier parameter (vernier) : 0.1000000000000000 ==>

0.4000000000000000 RF gradient (V_RF) : 1.000000000000000 ==>

10.00000000000000 Number of RFs in rotator (m) : 1.000000000000000 ==>

10.00000000000000Objective functions: 920802.1870498012 ==> 716325.0523539253 = -

204477.1346958759!! 2906605812.448390 ==> 2288276641.662843 = -

618329170.7855473 -1154.766872358291 ==> -1015.263140233114 =

139.5037321251762 2905272325.918894 ==> 2287245882.418927 = -

618026443.4999671

14 December 2004 Muon Colliders ‘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

14 December 2004 Muon Colliders ‘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

2

1

s

s v

dst