emittance oscillations in long-pulse induction linacs bruce carlsten los alamos national laboratory...

Emittance Oscillations in Long-Pulse Induction Linacs

Bruce Carlsten

Los Alamos National Laboratory

May 25, 2011

U N C L A S S I F I E DOperated by Los Alamos National Security, LLC for NNSA

Outline

Comments about DARHT

Long-term emittance decrease in long-pulse induction linacs

Emittance oscillations and their thermalization

Nonlinear focusing forces

Discussion


Long-Pulse Induction Linacs

LANL and LBNL (with LLNL) collaborated on DARHT-2 accelerator.

DARHT-1 was thoroughly designed and tested at ITS (only experimental demonstration of the centrifugal space-charge force)

DARHT-2 was on a faster schedule THOR test facility with DARHT-2 cell


DARHTLong design effort on LCLS - short turn on

Short design effort on DARHT-2 – long turn on

There was a lot of new engineering for DARHT-2, but also unpredicted beam physics


Simulations

LBNL – AMBER (Fawley and Vay)MRC – SPROP (Hughes)LANL – SLICE (Carlsten)

All codes used equivalent physics – Gauss law for radial electric field, Ampere’s law for diamagnetic field, some other features

Typically saw remarkable emittance decrease of about a factor of 10 (1000 mircons down to 150 microns) – very unique and interesting beam physics

2222 rrrrnormalized

90% emittances, use radial to avoid effect of solenoids

3.5-MeV, 4-kA diode


DARHT Enabled Study of Exciting Beam Physics

Centrifugal Space-Charge Force Measurement

CSCF is extra force term:

CSCF cancels potential depression:

Ion-hose suppression in induction cells

Halo interception in beam cleanup zone (BCUZ) (Vlasov push)

Rdr

deFr

221

dr

d

m

e

mRe

m

vBe

r

vr

dt

d

2

20

2


Beam Physics We Want To Understand Today

1. Why does the emittance oscillate?

2. Why is there a gradual emittance decrease?

3. What causes these final stable oscillations?


1. Why Does the Emittance Oscillate?

Initial nonlinearities come from:Anode holeMagnetic field nonlinearitiesDensity non-uniformity

Spherical aberration from anode hole acerbated by beam

Without beam With beam

Minor hollowing of beam density

Nonlinearities in initial phase space, minor wavebreaking

These two set the initial conditions on beam oscillations


1. Why Does the Emittance Oscillate?Beam wants to oscillate about a stationary state

Stationary state in space-charge dominated regime has uniform density – a non-uniform beam (zero emittance) will oscillate between a hollow and a peaked distribution. Below is phase space and density at first emittance maximum:


1. Why Does the Emittance Oscillate?The excess free energy (above the minimum needed for the stationary state) is left over to make the emittance. We can calculate the excess free energy by calculating the beam energy and subtracting the beam energy of the stationary state, using Reiser’s prescription for nonlinear free energy:

2,

,, Esc

extBscEscexttotalW

WWWWW

drrVrW ext

a

ext )(0

drrEW r

a

Esc2

0

0, 2

2

2

1rm

N

WU total

excess

This approach provides a surprising accurate estimate of the maximum emittance in the plots (within 5% when include correlations)


1. Why Does the Emittance Oscillate?These oscillations are same mechanism as emittance compensation, but with radial variations instead of axial variations:

“Thin lens” compensation


1. Why Does the Emittance Oscillate?Emittance compensation in a drift show wavebreaking also:

Focusing lens at 28 cm

These particles dominate the final emittance

Hanerfeld, Herrmannsfeldt, and Miller, 1989 PAC


1. Why Does the Emittance Oscillate?Emittance compensation in a uniform focusing field:(Exact same equations for the radially nonuniform case)

0

sKK

K

Kseq

ˆ

eq

02

22

eq

eqeqKKK

02 K

Transverse eqn of motion for slice edge

Equilibrium particle radius for a slice

Do a perturbation expansion

Slice expansion is oscillatory with a frequency that only depends on the external focusing

3300

2/12/2 A

plasma I

IKz

33

2ˆA

sI

IK Depends on axial

position of slice

K – external focusing, doesn’t depend on axial position of slice


1. Why Does the Emittance Oscillate?S&R used a Cauchy transform to extend this to an accelerating beam:

where:

Worth pointing out this solution is substantially different. There are oscillating terms, but also now a constant acceleration term in the slice divergence. S&R addresses this with an “invariant envelope” trajectory.

C&P identified that rf focusing at the cathode provides control of and , which help line up the phase space ellipses.

ySe

dy

d 2

2

2

ye

2

202

sin

/8/ EcBz

2

33

2)(ˆ

2

)()(

s

A

KI

IS

)lnsin(

4

1/

2

)lncos(

4

1/

02

3

02

S

S


1. Why Does the Emittance Oscillate?If we keep the next order, we can predict when the phase space oscillations get out of phase:

02

22

eq

eqeqKKK

z cos10

2/122/1 2

eq

rmsK

Expand to second order

rmseq J

Jr

J

Jrmseq

rms 120

22/1 1

2

12)(

J

JKr rms

30

1

12

J

Jz

z

rmsplasma

damping

Particle oscillations out of phase after about 30 periods

Final equipartitioning at 70% of maximum emittance

J is current density up to r


1. Why Does the Emittance Oscillate?Boersch effect is very slow (thermalization due to Coulomb scattering), emittance growth of ~ 100 microns over 1 km at 5 MeV, 4 kA:

log24

2223

Aun

I

Iecaz

Wavebreaking can also lead to thermalization (Reiser talks about wavebreaking in ¼ betatron period). Not true for low-emittance electron beams, wavebreaking occurs when a beam is focused if the charge density drops to < ½ of the average charge density within that point (Oscar Anderson). We can equivalently equate an emittance that is needed for particles to overcome the potential barrier of the beam and wavebreak:

/)/(2 Aedgebw IIr

For a 4-kA, 4-MeV beam, this is about 10000 microns. For a 1-GeV, 100-mA, 1-mm radius H beam, this is about 0.1 micron.


2. Why is There a Gradual Emittance Decrease?The initial phase space curvature is not consistent with the initial beam density profile (doesn’t become flat). We can fix that with nonlinear forces in intense beams:

r

mvBevBvBveeE

dt

rdm extzdiar

2

)()(

rrzmc

eEr

mc

eEr

dt

drr

dt

d zr 2

22

Radial eqn of motion

rEvc

erEv

c

e

r

mvBBev

eEmvr zaradiaext

ra

2

222

2

2

32 1

For a uniform density beam, radial variation in diamagnetic field cancels (to first order) change in potential depression

This is the dominant nonlinear term, tailor beam focusing to work out phase space nonlinear “kink”

We are defining divergences relative to the axial velocity on axis (r=0)


2. Why is There a Gradual Emittance Decrease?So we adjust the magnetic field profile to use the nonlinear focusing to straighten out the phase space as the beam is accelerated to 16 MeV


3. What Causes the Final Stable Emittance Oscillations?There is a beam halo from periodic wavebreaking (from every radial bounce in). Halo particles are emittance dominated with an oscillation wavenumber half the space-charge oscillation of the core:

32.78 m53.7 micron

33.40 m85.1 micron

34.03 m102.8 micron

34.67 m98.54 micron

35.30 m53.98 micron

Minimum emittance occurs when the two distributions are lined up (2:1 resonance).

There is a minor trade between the focusing and minimizing the beam halo.


3. What Causes the Final Stable Emittance Oscillations?In principle, we can design a magnetic field match that leads to a nice radial profile spiraling into the axis (not oscillating) and get rid of the emittance oscillations. Let’s assume the beam is emittance dominated:

0)(

rzKr

r f

0zu 0)(

ruKu

rr f

ujeu

rr

2/10

22

4

1

uK f

20rnorm

ujrr

2

1

Change of variables:

Nice beam radial function (one of many):

Leads to reasonable magnetic field profile:


3. What Causes the Final Stable Emittance Oscillations?Using the matched solution, we can squeeze the beam to get the necessary conditions into the field profile (angle in phase space), entire beam is emittance dominated:

Final emittance drops to about 40 microns (10 microns rms!)

Comparison: thermal emittance is 50 microns from a 0.1 eV temperature, 8-inch cathode


Final Comments

• Non-thermalized beam preserves structure remarkably long

• Provides significant capability to tailor phase space using variety of nonlinear effects

• Insufficient diagnostics to measure these emittance

• Emittance dominated by downstream transport and bunch slicing

emittance oscillations in long-pulse induction linacs bruce carlsten los alamos national laboratory...

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