1 EPIC SIMULATIONS V.S. Morozov, Y.S. Derbenev Thomas Jefferson National Accelerator Facility A....

1

EPIC SIMULATIONS

V.S. Morozov, Y.S. DerbenevThomas Jefferson National Accelerator Facility

A. AfanasevHampton University

R.P. JohnsonMuons, Inc.

Operated by JSA for the U.S. Department of Energy

Muon Accelerator Program - Winter Meeting, March 1, 2011

Muons, Inc.

Outline

• Concept of Parametric-resonance Ionization Cooling (PIC)

• PIC linear optics requirements

• Epicyclic twin-helix channel for PIC– Magnetic optics design– Possible practical implementation

• G4beamline simulations of twin-helix channel– Cooling with wedge absorbers followed by regions of static electric field– Effect on the orbit and its compensation– Timing of rf cavities– Cooling with wedge absorber and rf cavities

• Conclusions and future plans

2Operated by JSA for the U.S. Department of Energy


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• Parametric resonance induced in muon cooling channel

• Muon beam naturally focused with period of free oscillations

• Wedge-shaped absorber plates combined with energy-restoring RF cavities placed at focal points (assuming aberrations corrected)– Ionization cooling maintains constant angular spread– Parametric resonance causes strong beam size reduction– Emittance exchange at wedge absorbers produces longitudinal cooling

• Resulting equilibrium transverse emittances are an order of magnitude smaller than in conventional ionization cooling



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PIC Concept

• Resonant dynamics: angular spread grows while beam size shrinks

• Absorbers keep angular spread finiteAbsorbers

Optics to restore parallel beam envelope

x

x

x

x

xx const

Beam envelope without absorbers

PIC Principle



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• Equilibrium angular spread and beam size at absorber

• Equilibrium emittance

(a factor of improvement)

w

Absorber plates Parametric resonance lenses

/ 8

x x

xx

xx const

22

3 ( 1)

2e

a

mZ

m

1

2 3a aw

3( 1)

4e

n

mZ w

m

3 2 3

acc

abs

w

PIC Schematic



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PIC Channel Optics Requirements



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• Horizontal free oscillations’ period x equal to or low-integer multiple of vertical free oscillations’ period y

• Oscillating dispersion– small at absorbers to minimize energy straggling– non-zero at absorbers for emittance exchange– large between focal points for compensating chromatic and spherical aberrations

• Correlated optics: correlated values of x, y and dispersion period D

x = n y = mD , e.g. x = 2 y = 4D or x = 2 y = 2D

• Fringe-field-free design

• Practical fringe-field-free approach• Periodic solutions of source-free Maxwell equations in vacuum

• Harmonic of order n given by

• Total field

0, 0B B

0

1 1

1 1 010

0

2( , , ) [ ( ) ( )]cos( [ ])

n

n n nn n

n nnkz

BB z I nk I nk n kz

nk

0

1 1

1 1 010

0

2( , , ) [ ( ) ( )]sin( [ ])

n

n n nn n

n nnkz

BB z I nk I nk n kz

nk

0

1 1

010

0

2( , , ) 2 ( )cos( [ ])

n

n n nn nz nn

kz

BB z I nk n kz

nk

n

n

B B

Helical Harmonics



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• Consider two dipole helical harmonics (n = 1) of equal strengths with equal-magnitude and opposite-sign wave numbers k1 = -k2 = 2/

• Field periodic with = 2/k, • Vertical field only in horizontal plane

1 2 1 20 0, ( , ) ( , )ikz ikz

z zb b e b e b b

1 2

1 2

1m0.74 T0

d d

q q

b bb b

Twin Helix



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• Vertical field only in horizontal plane Periodic orbit in horizontal plane• Horizontal and vertical motion uncoupled• Region of stable transverse motion in both planes

Periodic Orbit and Dispersion



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• Superimpose straight quad to redistribute horizontal and vertical focusingD = x = 2y = 4 x = 0.25, y = 0.5• Down side: cannot satisfy correlated optics conditions for both charges

Adjusting Correlated Optics



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• Dispersion: • Chromaticity: • Scaling pattern:

max / 0.098 mx aD p x p / 0.646, 0.798x x yp p

2/ , / / , , , , constd y a x x yB p B x p x D

Dispersion and Chromaticity



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Layer of positive-helicity helical conductors with cos azimuthal current dependence

Layer of negative-helicity helical conductors

Normal quad

Possible Practical Implementation



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Layer of positive-tilted loops with cos z longitudinal current dependence

Layer of negatively-tilted loops

Normal quad

• Adopt existing technology?

Possible Practical Implementation



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G4beamline Simulations



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Going In Coming Out

• No absorber and no RF• 105 100 MeV/c - through 100 periods of “twin helix” with correlated optics• Initially parallel beam uniformly distributed with 10 10 cm square

Dynamical Aperture Test



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• 2 cm thick Be wedge absorber with 0.3 thickness gradient• 1 m helix period, absorbers placed every 2 periods (x = 0.25) at points with

3 cm dispersion for appropriate distribution of cooling decrements• Timing of rf cavities is not straightforward, absorbers are followed by regions of

static electric field adjusted to compensate energy loss of 2 cm Be• Energy recovery regions are short (2 cm) to decouple from transit time effects

and reduce optics perturbation• In practice, as much space as possible should be taken up by absorbers and rf

Absorber / Energy Recovery Model



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• 103 200 MeV/c muons, uniform x = y = 2 mm, x = y = 50 mrad, p/p = 2.7% • Beam started along the “unperturbed” periodic orbit, however, the orbit has changed

due to absorbers / energy recovery, this causes initial mismatch• Stochastic processes are off

Beam Cooling Simulation



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• Periodic momentum changes

Longitudinal Cooling



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• Particle tracked over many periods until cooling makes it converge to new periodic orbit• Particle observed at the same point within 2 period (~2 in front of each absorber)

New Periodic Orbit



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Cooling Process in Phase Space



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Starting point

New periodic orbit

• Conceptually different picture when electric field is adjusted to restore the original momentum corresponding to correlated optics

Near Correlated Optics



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Steady oscillations established

• Particle still cools but the phase space splits into two islands• Stable resonance?

Phase Space View



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• Similar picture at 250 MeV/c• Electric field is tuned to give periodic momentum close to 250 MeV/c

Correlated Optics at 250 MeV/c



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Phase Space View



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• Blue: trajectory with magnetic field only • Red: trajectory with absorbers / energy recovery

Periodic Trajectory in Phase Space



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• Reference particle becomes unstable, rf cavities’ timing set manually, s = 30• The trajectory is not perfect but exhibits the same characteristic behavior

Tracking Single Particle with RF Cavities



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Phase Space View



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• 103 200 MeV/c muons, uniform x = y = 6 mm, x = y = 50 mrad, p/p = 2.7%, t = 0.04 ns

• Stochastic processes are off

Beam Cooling with Absorbers / RF



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Longitudinal Cooling



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• Cooling channel’s correlated optics is well-understood

• The basic model with wedge absorbers and rf cavities is in place

• Cooling simulations initiated

• Next steps– Phase space dynamics with absorbers and rf in the correlated optics case

needs to be understood– Induce parametric resonance probably using lumped quadrupoles– Turn stochastic processes on– Look into aberration compensation, there is a well-understood approach to

correcting at least chromatic aberrations– Compare final emittances in case of conventional ionization cooling and PIC

Conclusions and Future Plans



Muons, Inc.

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