Bunched-Beam Phase Rotation-Variation and 0ptimization

David Neuffer,A. Poklonskiy

Fermilab

2

Talk Intro

“Proprietary” documents:http://www.cap.bnl.gov/mumu/study2a/

3

0utline Introduction

Study 2 Study 2A

“High-frequency” Buncher and Rotation Concept Matched cooling channel

Study 2A scenario Match to Palmer cooling section Obtains up to ~0.2 /p

Variations Be absorber (or H2, or …) Shorter rotator (52m 26m), fewer rf frequencies Short bunch train (< ~20m) Optimization ….

4

R&D goal: “affordable” e, -Factory

Improve from baseline: Collection

– Induction Linac “high-frequency” buncher

Cooling– Linear Cooling Ring

Coolers(?) Acceleration

– RLA “non-scaling FFAG”

+ e+ + + e

– e– + e + and/or

5

Study 2 system Drift to develop Energy- phase

correlation

Accelerate tail; decelerate head of beam (280m induction linacs (!))

Bunch at 200 MHz

Inject into 200 MHz cooling system

E

c

6

Adiabatic buncher + Vernier Rotation

Drift (90m) decay;

beam develops correlation

Buncher (60m) (~333200MHz) Forms beam into string of bunches

Rotation(10m) (~200MHz) Lines bunches into equal energies

Cooler(~100m long) (~200 MHz) fixed frequency transverse cooling system

beam Drift Buncher

Rotator

Cooler

Overview of transport

Replaces Induction Linacs with medium-frequency rf (~200MHz) !

7

Adiabatic Buncher overview

Want rf phase to be zero for reference energies as beam travels down buncher

Spacing must be N rf

rf increases (rf frequency decreases)

Match to rf= ~1.5m at end of Rotator

Gradually increase rf gradient (linear or quadratic ramp):

Example: rf : 0.90~1.4m m5.11010L

11L rf

1tot

010tot

Captures both (+, -)

8

Rotation At end of buncher, change rf to

decelerate high-energy bunches, accelerate low energy bunches

Reference bunch at zero phase, set rf less than bunch spacing

(increase rf frequency)

Places low/high energy bunches at accelerating/decelerating phases

Can use fixed frequency or

Change frequency along channel to maintain phasing

“Vernier” rotation –A. Van Ginneken

rf : ~1.41.5m in rotation;

Nonlinearities cancel:T(1/) ; Sin()

Z(N) – Z(0) = (N + δ) λrf(s)

9

Key Parameters General:

Muon capture momentum (200MeV/c?) 280MeV/c? Baseline rf frequency (200MHz)

Drift Length LD

Buncher – Length (LB)

Gradient VB (linear OK)

Rf frequency: (LD + LB(z)) (1/) = RF

Phase Rotator-Length (LR)

Vernier, offset : NR, V

Rf gradient VR

COOLing Channel-Length (LC)

Lattice, materials, VC, etc.

Matching …

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Study 2a Cooling Channel Need initial cooling channel

(Cool T from 0.02m to 0.01m)

Longitudinal cooling ?

Examples Solenoidal cooler (Palmer) “Quad-channel” cooler 3-D cooler

Match into cooler Transverse match

– B=Const. B sinusoidal– Gallardo, Fernow & Palmer

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Palmer Dec. 2003 scenario Drift –110.7m Bunch -51m

V’ = 3(z/LB) + 3 (z/LB)2 MV/m (× 2/3) (85MV total)

(1/) =0.0079 -E Rotate – 52m – (416MV total)

12 MV/m (× 2/3) P1=280 , P2=154 V = 18.032

Match and cool (100m) V’ = 15 MV/m (× 2/3) P0 =214 MeV/c 0.75 m cells, 0.02m LiH

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Study2AP June 2004 scenario Drift –110.7m Bunch -51m

V(1/) =0.0079 12 rf freq., 110MV 330 MHz 230MHz

-E Rotate – 54m – (416MV total) 15 rf freq. 230 202 MHz P1=280 , P2=154 V = 18.032

Match and cool (80m) 0.75 m cells, 0.02m LiH

“Realistic” fields, components Fields from coils Be windows included

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Simulation of Study 2Ap

Drift(110m)

Bunch(162m)

E rotate(216m)

Cool(295m)

System would capture both signs (+, -) !!

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Variation –Be absorbers Replace LiH absorbers with Be

absorbers suggested by M. Zisman (0.02m LiH 0.0124m Be)

Performance somewhat worse Cooling less(tr ~0.0093; LiH has 0.0073)

Best is ~0.21/p within cutsafter 80m cooling (where LiH has ~0.25 at 100m)

Be absorbers could be rf windows

H2 gas could also be used Gas-filled cavities (?)

Mu Capture

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0 40 80 120 160 200 240 280 320

All mu's

e_t < 0.15

e_t<0.3

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Cool (to 100m)Rotate(26m)Bunch

(51m)Drift (123.7m)

0.00E+00

1.00E-01

2.00E-01

3.00E-01

4.00E-01

5.00E-01

6.00E-01

7.00E-01

0.00E+00 4.00E+01 8.00E+01 1.20E+02 1.60E+02 2.00E+02 2.40E+02 2.80E+02 3.20E+02

e_t < 0.30

e_t< 0.15

All mu's

Shorter bunch Rotator Drift –123.7m (a bit longer) Bunch -51m

V’ = 3(z/LB) + 3 (z/LB)2 MV/m

(1/) =0.0079

-E Rotate – 26m – 12 MV/m (× 2/3) P1=280 , P2=154 V = 18.1 (Also P1=219 , P2=154, V = 13.06)

Match and cool (100m) V’ = 15 MV/m (× 2/3) P0 =214 MeV/c 0.75 m cells, 0.02m LiH

Obtain ~0.22 /p

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Try with reduced number of frequencies Change frequency every

6 cells (4.5m) Buncher (11 freqs.):

294.85, 283.12, 273.78, 265.04, 256.04, 249.13, 241.87, 235.02, 228.56, 222.43, 216.63 MHz

Rotator (6 freqs) 212.28, 208.28, 205.45, 203.52,

202.34, 201.76 Cooler (200.76 MHz)

Obtains ~0.2 /p (~0.22 /p for similar continuous

case - 105 frequencies) Not reoptimized ….

Phasing within blocks could be improved, match into cooling…

Cool (to 100m)Rotate(26m)Bunch

(51m)Drift (123.7m)

Mu Capture

0

0.1

0.2

0.3

0.4

0.5

0.6

0.00E+00 5.00E+01 1.00E+02 1.50E+02 2.00E+02 2.50E+02 3.00E+02

m

All mu's

e_t < 0.15

e_t<0.3

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Short bunch train option Drift (20m), Bunch–20m (100 MV)

Vrf = 0 to 15 MV/m ( 2/3) P1 at 205.037, P2=130.94 N = 5.0

Rotate – 20m (200MV) N = 5.05 Vrf = 15 MV/m ( 2/3)

Palmer Cooler up to 100m Match into ring cooler

ICOOL results 0.12 /p within 0.3 cm

Could match into ring cooler (C~40m) (~20m train)

Cool (to 100m)

Rotate(20m)Bunch

(20m)

Drift (20m)

60m

40m

95m

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FFAG-influenced variation – 100MHz 100 MHz example

90m drift; 60m buncher, 40m rf rotation

Capture centered at 250 MeV

Higher energy capture means shorter bunch train

Beam at 250MeV ± 200MeV accepted into 100 MHz buncher

Bunch widths < ±100 MeV

Uses ~ 400MV of rf

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Lattice Variations (50Mhz example)

Example I (250 MeV) Uses ~90m drift + 100m

10050 MHz rf (<4MV/m) ~300MV total

Captures 250200 MeV ’s into 250 MeV bunches with ±80 MeV widths

Example II (125 MeV) Uses ~60m drift + 90m

10050 MHz rf (<3MV/m) ~180MV total

Captures 125100 MeV ’s into 125 MeV bunches with ±40 MeV widths

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Variations, Optimizations

Shorter bunch trains ? (for ring cooler, +-- collider: more ’s lost?)

Longer bunch trains (more ’s, smaller E)

Remove/reduce distortion ? Different final frequencies

(200,88,44Mhz?) Number of different RF frequencies

and gradients (6010 ok?) Different central momenta

(200MeV/c, 300MeV/c …?) Match into cooling channel,

accelerator Transverse focusing (solenoidal

field?) Mixed buncher-rotator? Cost/perfomance optimum

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Control Theory Approach

),())()(,()( 100 kkk

ku uxxsZsssZ - impulse effect model

rn RUuRxTtuxtfdt

xd ,],,0[),,,(

u – control function (incorporate lattice parameters)

- continuous model

T

o M

TT

M

tt

uTut

dxxgdtdxtuxtuI,,

)())(,,()( - quality functional

Seek for u such that it will minimize (maximize) some functional of this type describing some properties of the beam we want to maintain during its propagating through the lattice (1st part) and at the end (2nd part). is the set of coordinates of the beam particles at the time t under control functions u

utM ,

or

0M

0x

utM ,

),,( 0 uxtx

A. Poklonskiy - MSU

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Control Theory Approach

E central

T

t

ifM

If we define penalty functions on a rectangle as

],[],[ 2121 bbaa

112

111

212

211

211

1

,)(

,)(

],[,0

)(1

1

axxak

axaxk

aax

xq

q

122

212

222

222

212

2

,)(

,)(

],[,0

)(2

2

bxxbk

bxbxk

bbx

xq

q

and the quality functional

i M

fiff

iff

if

xdMxxCMxxCI )),((),(( 022011

we can perform optimization using control theory methods (gradient?)

A. Poklonskiy - MSU

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A. Poklonskiy - Summary

Model of buncher and phase rotator was written in COSY Infinity

Simulations of particle dynamics in lattice with different orders and different initial distributions were performed

Comparisons with previous simulations (David Neuffer’s code, ICOOL, others) shows good agreement

Several variations of lattice parameters were studied

Model of lattice optimization using control theory is proposed

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Summary High-frequency Buncher and E Rotator

simpler and cheaper (?) than induction linac system

Performance better (?) than study 2,And

System will capture both signs (+, -) !(Twice as good ?) Good R&D model for +-- Collider.

Method could be baseline capture and phase-energy rotation for any neutrino factory …

To do: Optimizations, Best Scenario, cost/performance

…