Confinement and Equilibrium of Bunched Pencil and Annular ... · Confinement and Equilibrium of...

APS DPP 2001 Intense Beam Theoretical Research Group

Confinement and Equilibrium of Bunched Pencil and Annular Beams *

M. Hess and C. Chen MIT - Plasma Science and Fusion Center

Presented atAPS DPP 2001

Long Beach, CAOct. 29- Nov. 2, 2001

*Research supported by DOE.


Outline of the Poster

•Experimental Evidence of Beam Loss

•Bunched Pencil Beam Model and Confinement Criterion (PPM

Averaging Theory)

•Comparison of Averaging Theory to Full Beam Dynamics

•Comparison of Criterion to SLAC PPM Klystrons

•Bunched Annular Beam Model and Confinement Criterion

•Comparison of Criterion to Phillips Lab. RKO and UNM BWO

•Conclusion


l Pencil Beam Microwave Sources

l SLAC Periodic Permanent Magnet (PPM) Klystrons:

l 50 MW-XL-PPM: Measured 0.8% beam power loss(D.Sprehn, et al, Proc. LINAC98, p. 689,1998)

l 75 MW-XP: Measured greater beam power loss than 50 MW (D.Sprehn, et al, SPIE Proc., p. 132, 2000)

l Klystrino: Still under development (G. Scheitrum, 2000)

Experimental Evidence


l Annular Beam Microwave Sources

l Phillips Laboratory, Relativistic Klystron Oscillator (RKO)

l Measured rf pulse shortening (< 100 ns pulse width) (K. Hendricks, Phys. Rev. Lett., 76,pp. 154-157)

l Univ. of New Mexico, Backward Wave Oscillator (BWO)

l Measured rf pulse shortening (~ 50 ns pulse width) and substantial beam loss (F. Hegeler, et al., IEEE Trans. Plasma Sci. 26, 275, 1998.)

Experimental Evidence


l Green’s functions can provide a powerful technique for the modeling of high-power microwave sources and particle accelerators.l Both electrostatic and electromagnetic interactions with

realistic boundary conditions can be modeledl Better resolution than PIC codesl Allows for analytical modeling of applications (e.g.

klystrons and BWOs)

l Using a Green’s function, we self-consistently model pencil thin bunched beams and annular bunched beams. l Find conditions for beam confinementl Find a mechanism for beam loss

Advantages of Green’s Functions


2-D Configuration

3-D Configuration

L

raa r

x

y

z

rB

Diagram of Pencil Beam Model

Vz Vz


,Lz

z,L

zz,

Lr

r,L

rr,

La ′

=′=′

=′==πππππ

α22222

( ) ( ) ( )

( )

( ) ( ) ( ) ( ) ( )[ ] ( )( ) ( ) ( ) ( ) ( )[ ] ( )

′>−′

′<′−′×

=′ ∑ ∑∞

−∞=

∞

−∞=

′−′−

rr,nKrnIrnKnIrnI

rr,nKrnIrnKnIrnI

nIee

Lx;xG

lllll

lllll

n l l

ilzzinD

αα

αα

αθθ 12

3Solution

Definition

( ) ( ) ( )'''43

2 zzrrr

G LD −−−−=∇ δθθδδπ

( ) ( )∑∞

−∞=−−=−

nL nLzzzz '' δδ

03 ==arDG

Green’s Function


Beam-Wall Interaction

l By Lorentz transforming to the beam rest frame, the problem becomes completely electrostatic

l Using the Green’s function to compute the induced surface charge on the wall, we calculate the electric field acting on each bunch due to the beam-wall interaction :

( ) ( ) ( ) ( )( )

( ) ( ) ( )( ) r

n n l l

lll

b

bself enI

rnIrnInKnI

rnIrnInKr

rL

eNrE

′−

′−

−−= ∑ ∑∑

∞

=

∞

=

∞

=1 1 10

0002222 42

4α

αα

ααγ

πr

2-D 3-D (Effect of Bunching)

21

122

bb

bb,

Lr

r,La

βγ

γπ

γπ

α−

===


Beam-Wall Interaction (cont.)

l Transformation back to the laboratory frame yields:

selfb

selflab EE

rrγ= self

bbselflab EB

rrr×−= βγ

l These fields take into account the electromagnetic wakefields observed in the lab frame due to longitudinal motion of all the charges

l The fields do not include effects from transverse motion


Hamiltonian Analysis

l The self-fields may be expressed in terms of potentials and incorporated into a single particle relativistic Hamiltonian:

( ) ( )k

tkzcosQEQcMAQAQPcH rfself

labextself

lab

ωφ

−−++−−= 422rrr

selflab

selflabE φ∇−=

rrselflab

selflab AB

rrr×∇=

l The rf traveling wave term is included to model fields in an rf accelerator.

l For bunched beams in a klystron drift tube section, we may assume the rf term is due to space charge waves


Hamiltonian Analysis (cont.)

Generating Function: ( ) ( ) zphz Ptvzt,P,zF ′′ −= kvph ω=

Canonical Transformations: zph PFtvzz ′∂∂=−=′ zFPP zz ∂∂== ′

tFHH ∂∂+=′

⊥′+′=′ HHH ||Decompose Energy: ⊥′>>′ HH||

( )zksink

QEPvMcH zphb|| ′−−=′ ′

2γ

2

22

21

b

selfext

rb

QA

cQ

rP

PM

Hγφ

γ θθ +

−+=′⊥


Averaged Equations of Motion: Periodic B

l We assume that the transverse time scale is much slower than the longitudinal time scale through the PPM field

beb

r

NmP

dtdr

γ=

∂

∂−−=

reN

cmBNe

rNmP

dtdP self

b

b

e

rmsb

beb

r φγγ

θ2

22

3

2

41

Radial Motion Equation:

Radial Momentum Equation:

Using: [ ]zkcosrB

Aext0

0

2=θ

,Hdzk

Hk

∫ ⊥⊥ ′=′02

0

0

2

π

π

22

222

2

22

421

b

selfrms

rb

Qc

rBQrP

PM

Hγφ

γθ +

++=′⊥


Confinement Criterion I (Pθ ≠ 0)

cmeB

,La

Nm

e

e

rmsc

b

b

ep =

= ω

πγπ

ω 2

22 4Define:

0.0 0.5 1.02Pθ/meωca2

0.0

0.5

1.0

2ωp2 /ω

c2

a/γbL = infinitya/γbL = 0.3a/γbL = 0.15a/γbL = 0.08


l The upper bound on the self field parameter when confined orbits exist occurs at Pθ = 0. Analysis yields:

Criterion II (Self Field Parameter Limit)

(Brillouin Density Limit)l The 2-D case:

l The 3-D case:

12

2

2

≤c

p

ω

ω

( ) ( )∑∞

=

+≤

1 10

2

2

1

12

n

c

p

nInIn

αααω

ω


0.0 0.2 0.4 0.6 0.8 1.0 1.2r/a

-0.2

-0.1

0.0

0.1

0.2

2πP

r/mγ b

0Lω

L

0.0 0.2 0.4 0.6 0.8 1.0 1.2r/a

-0.2

-0.1

0.0

0.1

0.2

2πP

r/mγ b

0Lω

L

0.0 0.2 0.4 0.6 0.8 1.0 1.2r/a

-0.2

-0.1

0.0

0.1

0.2

2πP

r/mγ b

0Lω

L

0.0 0.2 0.4 0.6 0.8 1.0 1.2r/a

-0.2

-0.1

0.0

0.1

0.2

2πP

r/mγ b

0Lω

L

Comparison of Full and Averaged Dynamics (SLAC 50 MW PPM Klystron)

No Rf Field

Rf Field


Periodic Permanent Magnet (PPM) Focusing Klystron Amplifier*

*from SLAC Web Site.


Table of Parameters for SLAC PPM Klystrons

PARAMETER 50 MW-XL 75 MW-XP KLYSTRINO

f (GHz) 11.4 11.4 95

bI (A) 190 257 2.4

bγ

Brms (T)

1.83

0.20

1.96

0.16

1.22

0.29

a (cm) 0.48 0.54 0.04

α 0.75 0.77 1.15

expAc

b

IaIc22

28ω 0.19 0.28 0.35

crAc

b

IaIc22

28ω 0.238 0.244 0.366

efNI bb =

1

222 −

==bb c

afLa

γ

πγ

πα

fc

L bβ=

Ac

b

c

p

IaIc22

2

2

2 82

ωω

ω=

bb

ebbA

kA

ecm

I

βγ

βγ

17

3

≈

=


0.0 1.0 2.0 3.0 4.0 5.0 6.0α=(2πa /γbL)

0.0

0.5

1.0

1.5

2ωp2 /ω

c2

b

ac

a. 50 MW-XL-PPM

b. 75 MW-XP

c. Klystrino

Comparison of Theory to Experiment


Diagram of Periodic Annular Beam Model

lPeriodically spaced flat bunches have arbitrary density profiles

lWe assume azimuthal symmetry

lWe solve for a criterion for fluid equilibrium (steady state)

a

L

x

y

z

rB

Vz


Annular Beam Equilibrium

( ) ( ) ( )∑∞=

−∞=−−=

k

kz kLtVzrrn δσDensity Equation:

0=rV

and using Continuity and Force Balance Equations yields:

( ) ( )

+±=

rmreE

recb

self

b

cb 2

411

2 ωγγω

ω( ) ( )rrrV bωθ = where:

( )e

rmrE ecbself

4

2ωγ−≥ everywhere inside the beam

(1)

(3)

(2)


Bessel Expansion and Computing Criterion

( ) ( )∑∞

==

100

mmm arjJr σσ

( ) ( ) ( )∑∞

=−=

1001 22

mmbmmbb

self aLjcotharjJeNrE γσγπ

( ) erNaLEr bself 22−=Γ

( ) ( ) ( )∑∞

==Γ

1001

322

mmmm jcotharjJ

ra

r απσαπ

La bγπα 2=

Density Expansion:

Electric Field:

Define Γ:

Confinement implies:maxc

p

Γ≤

122

2

ω

ω


Numerical Example

0.0 0.2 0.4 0.6 0.8 1.0r/a

-1.0

0.0

1.0

2.0

3.0

4.0

σ

0.0 0.2 0.4 0.6 0.8 1.0r/a

-60.0

-30.0

0.0

30.0

60.0

Γ

l σ is a quadratic density distribution centered at r/a=0.8

l expansion uses 200 modes

020402049 22 .,. cpmax ≤≈Γ ωω


Functional Invariance of Space Charge Limit

0.00 0.05 0.10 0.15 0.20 0.25δ/a=(ro-r i)/a

0.00

0.01

0.02

0.03

0.04

0.05

2ωp2 /ω

c2

QUADRATICTENT

l density functions centered at r/a=0.8

l quadratic and tent function limits differ by <1%


Observation of Electron Density Profile in BWO*

*F. Hegeler, et al., IEEE Trans. Plasma Sci. 26, 275 (1998).

Electron Beam

Laser Beam

Slow-Wave Structure

Experimental Setup

MeasuredElectronDensity


Table of Parameters for BWO and RKO

PARAMETER RKO BWO

f (GHz) 1.3 9.4

bI (kA) 10.0 3.0

bγ

B0 (T)

2.0

0.8

1.7

2.0

ir (cm) 6.60 0.90

or (cm) 7.10 1.15

a (cm) 7.65 1.28α

1.20 1.83

expAc

b

IaIc22

28ω 0.0021 0.0045

crAc

b

IaIc22

28ω

0.016 0.059


Conclusions

l Modeled bunched pencil beams and bunched annular beams

with a Green’s function technique

l Established confinement criterion on the space-charge limit for

bunched pencil beams in a PPM field by using an averaging

technique. Verified the validity of the bunched beam

confinement criterion by comparing averaged and full

dynamics

l Space-charge limit for pencil beams agrees well with SLAC

PPM klystron parameters. 75 MW-XP may observe greater

beam loss because it is operating above limit.


Conclusions (cont.)

l Established a bunched annular beam space-charge limit

l Phillips Lab. RKO and Univ. of New Mexico BWO experiments

are both operating below the annular beam space-charge limit.

This limit may not be the reason for the observations of beam

loss and rf pulse shortening.

l Further study of the annular beam model is needed. A stability

analysis of the equilibrium may enforce a more stringent

confinement criterion. Multi-particle simulations are needed to

supplement the fluid theory.


a

L

x

y

z

rB

Vz

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