Adiabatic Thermal Equilibrium of an Axisymmetric

Charged-Particle Beam

C. Chen, K. Samokhvalova, and J. Zhou

Plasma Science and Fusion Center

Massachusetts Institute of Technology

Symposium on Recent Advances in Plasma Physics

--- In Celebration of Ronald C. Davidson's 40 Years

of Plasma Physics Research and Graduate Education

June 12, 2007

C. Chen 206/12/07

ce

re

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re

re

Thermal rigid-rotor equilibrium in a uniform magnetic field

r

nb(r)

z

Davidson and Krall, 1971

Trivelpiece, 1976

C. Chen 306/12/07

Applications of high-brightnesscharged-particle beams

– Large Hadron Collider (LHC)

– Spallation Neutron Source (SNS)

– High Energy Density Physics (HEDP)

– International Linear Collider (ILC)

– Photoinjectors

– High Power Microwave Sources

C. Chen 406/12/07

Periodic focusing channels

S

S

S

S

N N

N

N

S/2 S/2

I

I

Periodic Quadrupole Field Periodic Solenoidal Field

C. Chen 506/12/07

Periodically focused beam equilibria

• Kapchinskij-Vladimirskij (K-V) equilibrium in an alternating-gradient (AG) magnetic quadrupole focusing channel

– I.M. Kapchinskij, and V. V. Vladimirskij, Proc. Int. Conf. High Energy Accel. (CERN, Geneva, 1959), p. 274

– Delta function distribution in transverse ‘energy’

• Generalized KV distribution in an axially varying, linear focusing channel

– F.J. Sacherer (Ph.D thesis, UC Berkeley, 1968)

• Rigidly rotating equilibrium in a periodic solenoidal magnetic focusing field

– C. Chen, R. Pakter and R. C. Davidson, Phys. Rev. Lett. 79, 225 (1997)

– KV-like distribution

• Issues – Non-physical– Beam halos– Chaotic-particle motion

Tnb

11

2

2

2

2

b

y

a

xT

ax

by

z

C. Chen 606/12/07

Chaotic phase space in a KV beam

-2.0 -1.0 0.0 1.0 2.0x/a

-2.0

-1.0

0.0

1.0

2.0

(S/

)1/2X

'

(a)

Qian, Davidson and Chen (1994)Pakter, Chen and Davidson (1999)Zhou, Chen, Qian (2003)

C. Chen 706/12/07

Thermal equilibrium in a periodic solenoidal magnetic field

New: Thermal equilibrium in a periodic solenoidal magnetic field

• Experiment: Recent experiment at UMER demonstrated that the beam focused by a solenoid has a bell-shaped profile– S. Bernal, B. Quinn, M. Reiser, and P.G. O’Shea, PRST-AB 5, 064202

(2002).

rnb(r)z

C. Chen 806/12/07

Kinetic and warm-fluid theories

selfzyyxx qcPqAcPqAcPcmH 22242

0 Vbn

pcc

qnmnextself

selfbbb

BVBVVV

- constants

0,

pxp

BvE

xv f

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Constants of motion

AngularMomentum

GeneralizedEnergy

Equation of state (adiabatic)

Transverse velocity

Distribution function Beam density

const1111

xy PyPxP

const21

121

21

21

21

selfyx PPyx

21

22

21 2

ˆ rzwzr

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zrr brzbrms

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,,C

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zrC

zrnBb

self

brms

rmsth

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b 22

2

,

2

,

2

2

,4

24exp,

Kinetic theory (Zhou, et al., 2007)

Warm-Fluid Theory (Samokhvalova, et al., 2007)

C. Chen 906/12/07

Self-Consistent Density Distribution

zTkzrq

zrKr

zrC

zrnBb

self

brms

rmsth

rmsthbrms

b 22

2

,

2

,

2

2

,4

24exp,

perveancefocusing parameterrms beam radius thermal rms emittance

Poisson’s equation

Beam rotation

bself nq 42

Envelope equation

zrr

zzbrms

bbcbb 2

20

21

zr

Vr

zr

Kzrz

dz

zrd

brms

bbbrmsth

brms

brmszbrms

3

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2

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bb

b

mV

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constVm

zrzTk

bb

brmsBrmsth

2

22

,2

Density

C. Chen 1006/12/07

Beam envelope and beam density

800 sS z 104

ˆ,

rmsth

KSK

Szsz 2cos132 0 0b

cold beam

warm beam

2

2

04 qn

Tk

b

BbD

C. Chen 1106/12/07

UMER edge imaging experiment*

• 5 keV electron beam focused by a short solenoid. • Bell-shaped beam density profiles• Not KV-like distributions

*S. Bernal, B. Quinn, M. Reiser, and P.G. O’Shea, PRST-AB, 5, 064202 (2002)

C. Chen 1206/12/07

Density profile comparison for 5 keV, 6.5 mA electron beam

-10 -5 0 5 10x (mm)

0.0

0.5

1.0

1.5

No

rma

lize

d D

en

sity Experiment

Theory

s=6.4 cm

-10 -5 0 5 10x (mm)

0.0

0.5

1.0

1.5

No

rma

lize

d D

en

sity

s=11.2 cm

Experiment

Theory

-10 -5 0 5 10x (mm)

0.0

0.5

1.0

1.5

No

rma

lize

d D

en

sity

s=17.2 cm

Theory

Experiment

Experiment

z=6.4cm z=11.2cm z=17.2cm

C. Chen 1306/12/07

• Established kinetic and warm-fluid equilibrium theories for charged-particle beams in periodic solenoidal focusing channels, extending Davidson’s seminal work on the rigid-rotor thermal equilibrium.

• Adiabatic.

• Applicable for both high and low intensities.

• Found excellent agreement between our theory and the UMER experiment.

• Future plans:

• Study chaotic particle motion and halo formation in thermal-equilibrium beams in periodic solenoidal focusing channels.

• Establish thermal equilibrium theory of charged-particle beams in periodic quadrupole focusing channels.

• Develop bunched beam equilibrium theory in rf accelerators.

• Pursue high-brightness beam applications.

Conclusion