energy loss of very high charge beams in plasma: theory j. rosenzweig, n. barov, m. thompson, r....

Energy Loss of Very High Charge Beams in Plasma:

Theory

J. Rosenzweig, N. Barov, M. Thompson, R. Yoder

UCLA Dept. of Physics and AstronomyICFA Workshop, Sardinia, July, 2002

J. Rosenzweig, ICFA 2002

Background 1: Blowout regime PWFA• Modern plasma wake-field

accelerators (PWFA) run in the “blow-out” regime (1991)

• Very nonlinear plasma electron response; total rarefaction

• Conditions:

• Measure nonlinearity in response: # of beam e-/plasma e- in cubic plasma skin-depth

Plasma wake response (blowout regime PIC sim.)

0

0.02

0.04

0.06

0.08

0.1

2 2.2 2.4 2.6 2.8 3

r (cm)

z (cm)

v

b

€

nb >n0

€

σ r <kp−1

€

σz <kp−1

€

˜ Q ≡Nbkp

3

n0

=4πkpreNb

<<1, linear regime

>1, very nonlinear ⎧ ⎨ ⎩

Fundamental quantity for theory…


What is Q?

• Ratio of beam density to plasma density

• Measure of field amplitude.• General measure of nonlinearity (blowout).• Dimensionless analysis is powerful tool…

€

nb

n0

≈≡Q

kpσr( )2

~


Some notable characteristics of nonlinear plasma response

• Relativistic plasma motion• Electromagnetic wave

trapped in e--rarefied region

• Artifacts in both physics and modeling– Density spike at wave

turning – Noise inside of beam

region – Electric field spike

(sensitive to mesh)

XOOPIC simulations a la D. Bruhwiler


Measuring the field amplitude from the spike not desirable

• The good focal qualities of the blow-out regime fall apart in the spike region (defocusing!)

• Beam loading eliminates the spike– Very little stored energy

in spike region (narrow)

• The spike can be many times the “useful” field amplitude

-2

-1

0

1

2

-6 -3

W

z

W

r

n

b

0 3

.

Figure 3

k

p

ξ

6 9

W

z

(0)/mc ω

p

, 2W

r

( σ

r

)/k

p

σ

r

mc ω

p

,n

b

/n

,b max

Beam loading in blowout PWFA


Recent experiments in nonlinear regime at FNAL and

SLAC

€

˜ Q =1.5−3.5 for E157/E162 and FNAL/UCLA experiments!

Energy spectra from UCLA experiment at FNAL A0 (E890),12 MeV energy loss in 8 cm of 1E14/cc plasma (~ liquid Li!)

Energy gain observed limited by spectrometer


Background 2: Scaling of PWFA with respect to plasma density

• From 1993, Rosenzweig, et al.*, noted in simulations that for blowout regime the maximum field excited in PWFA scaled as coherent Cerenkov radiation

• For finite bunch length, we must choose

• Thus we have scaling with bunch length

• Why is this linear “Cerenkov” scaling ~ valid in blowout?

€

eEz,dec=e2Nbn k( )−1

n k( )dk∫ ⇒=e2Nbkp

2

€

kpσ z ≤2

€

eEz,dec≅4e2Nb

σ z2

*J.B. Rosenzweig, in Proceedings of the 1992 Linear Accelerator Conference, (AECL-10728, Chalk River, 1993).*J.B. Rosenzweig, et al., Nuclear Instruments and Methods A 410 532 (1998).*N. Barov, J.B. Rosenzweig, M.E. Conde, W. Gai, and J.G. Power, Phys. Rev. Special Topics – Accel. Beams 3 011301 (2000).


Recent PWFA scaling exercises

• S. Lee, et al., have performed scaling study in context of the “afterburner” concept

• Scaling with constant bunch charge, changing bunch length, • Linear scaling found? Maybe…

€

kpσ z =const.

Replot

0

0.5

1

1.5

2

2.5

3

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

2D simulation (norm)3D simulation (norm)

σz ( )mm

?

?


Background 3: Fundamental issues

• Energy loss in plasma — extend linear theory? • Single particles

– Jackson/Landau (calculate energy loss from transverse current, radiation… we will mimic)

• High charge, point-like bunches– Chen, et al., original PWFA

• Logarithmic divergence with beam transverse size• Relevance to astrophysics

– Intense pulse behavior– Single particle energy loss


Dimensionless theory of bunched beam energy loss in plasma

• Proceed through H-fields (connection to transverse currents)

• Convective force, continuity

• Normalize time and space to and• Normalize charge and current to plasma density, speed of light• Normalize fields to wave-breaking

• Norm. variables use tilde ~

€

r ∇ ×

r H =

4πc

r J +

1c∂

r E

∂t

€

r ∇ ×

r E =−

1c∂

r H ∂t

€

∂r p

∂t+

r v ⋅

r ∇ ( )

r p =−e

r E +

1c

r v ×

r H

⎡ ⎣ ⎢

⎤ ⎦ ⎥

€

∂n∂t

+r ∇ ⋅ n

r v ( )=0

€

ωp = 4πe2n0 /me

€

kp =ωp /c

€

EWB=mecωp /e


Second order differential eqn.

with useful relations

€

∂˜ E z∂˜ r

=˜ J r

€

∂∂τ

˜ E r − ˜ H φ( )=−˜ J r

Note we are now looking at magnetic field role in driving “electrostatic” wave…

Induction


Linear theory

Delta-function beam in assumed (Green ftn in linear world). Finite radius - disk beam.

Governing equation;

Integrate over , H-field is momentum kick

€

˜ a =kpa

€

∂ 2 ˜ H φ∂˜ r 2

+1˜ r

∂ ˜ H φ∂˜ r

−˜ H φ˜ r 2

− ˜ H φ =˜ Q

π˜ a 2δ(τ)δ(˜ r −˜ a )

€

τ =ωp t−z/c( )

€

∂ 2H∂˜ r 2

+1˜ r ∂H∂˜ r

−H˜ r 2

−H=˜ Q

π˜ a 2δ(˜ r −˜ a )

Nonrelativistic term

€

H= ˜ H φε−

ε+∫ dτ = ˜ E r

ε−

ε+∫ dτ


Linear solutions

Integrate to obtain longitudinal field just behind beam

€

˜ E z(˜ r )τ=ε+

= H(˜ ′ r )d˜ ′ r ∞

˜ r ∫ =

˜ Q π˜ a 2

1−˜ a K1 ˜ a ( )I 0 ˜ r ( )˜ a I1 ˜ a ( )K0 ˜ r ( )

⎧ ⎨ ⎩

Same result as given by Chen, Katsouleas, Ruth, etc…


Limiting energy loss cases

Wide beam

€

˜ E z(˜ r )τ=ε+

≅˜ Q

π˜ a 2

€

˜ a >>1

€

eEz τ=ε+≅

4e2Nb

a2 =4πeΣbso

As expected from simple charge-plane model (1D limit)

Narrow beam

€

˜ a <<1

€

˜ E z(˜ r )τ=ε+

≅˜ Q

π˜ a 21−˜ a K1 ˜ a ( )[ ] ≅

˜ Q 2π

ln2˜ a

⎛ ⎝ ⎜

⎞ ⎠ ⎟ −0.577...

⎡

⎣ ⎢

⎤

⎦ ⎥ and

€

eEz τ=ε+≅2e2kp

2Nb ln1.123kpa

⎛

⎝ ⎜

⎞

⎠ ⎟


Comments on linear case

• Narrow beam is close to Cerenkov scaling, except…

• Logarithmic divergence with transverse beam size– Removal by Debye shielding? No, Debye

shielding applies to maximum distance a charged particle’s field exists in equilibrium, not minimum distance of effect in fast transient.

– Caused by divergence of transverse current induced at small impact parameters. Mitigated by relativistic saturation (transparency)?


Nonlinear case

General response of charged particle to E and H fields for infinitesimal length beam (be very careful!)

€

˜ p r =H same as NR!

€

˜ p z =12

˜ p r2

But…density increase accompanies longitudinal kick (continuity equation)

€

˜ v r =H

1+H2 +14H2

=H

1+12H2

Induced radial velocity

€

˜ n = 1−˜ v z( )−1 =1+1

2H2


Transverse current is identical to NR case!

• Differential equation unchanged• Energy loss for a zero-length beam is the same as NR case• Two effects cancel (Snowmass effect, where first presented)

– Saturation of radial velocity (rel. mass increase)– Enhancement of density (Snowplow effect)

€

˜ J r =˜ n ̃ v r = 1+12H

2( )⋅

H1+1

2H2

=H


Check with deposited energy spread

Field energy density right behind beam (only long. E), and

mechanical energy density (watch for snowplow “group velocity” effect)

€

d ˜ U d˜ z

=2π 1+ ˜ p r2 + ˜ p z

2 −1[ ]˜ n 0

∞

∫ 1−˜ v z( ) ˜ r d˜ r +2π 12

0

∞

∫ ˜ E z2(˜ r )˜ r d˜ r

=π H2

0

∞

∫ ˜ r d˜ r + ˜ E z2( ˜ r )˜ r d˜ r

0

∞

∫⎡

⎣ ⎢ ⎤

⎦ ⎥ .

€

d ˜ U d˜ z

≅˜ Q 2

2π˜ a 2K0

2 ˜ r ( )+K12 ˜ r ( )[ ]

˜ a

∞

∫ ˜ r d˜ r .

=˜ Q 2

2π˜ a 21− ˜ a K1 ˜ a ( )I 0 ˜ a ( )[ ] ≅ 1

2˜ Q ̃ E z(˜ r )

τ=ε+

Checks! (familiar 1/2)


Logarithmic divergence revisited

• View as Coulomb logarithm• Maximum impact parameter is skin-depth• Minimum impact parameter from uncertainty

principle,and maximum p• Energy loss of point-like particle

• Quantum effects at both limits (Compton/plasmon)• Fundamental for cosmic ray propagation…

€

bmax=2/kp

€

ln bmax/bmin( )

€

bmin≅h

mec2γ

€

dUdz

≅q2kp2 ln 0.794 γ

meckph

⎛

⎝ ⎜

⎞

⎠ ⎟ ≅q2kp

2 ln 5 γλp

λc

⎛

⎝ ⎜ ⎞

⎠ ⎟


Comparison with simulation

• Need to extend results to realistic finite length beam

• Self-consistent 2D PIC, narrow beam • Examine average energy loss

– Connect with -beam– Measures efficiency of wave excitation

• Examine peak accelerating field – Connect with previous work (S. Lee, et al.)– Dangerous (spike!)

€

kpσz =1.1

€

kpa=0.2

€

(kpσ z =0.11 for snowplow study)


Snowplow observed in simulation

MAGIC simulation shows clear snowplow.This effect does not occur in linear model.


Linear to extremely nonlinear cases examined, with MAGIC and

OOPICOOPIC surface plot of longitudinal E-field.

Measure average deceleration, peak acceleration…


Simulation results

• Compare with predictions of linear theory

• Fields saturates at high charge – Snow-plow loses to relativistic

“transparency”– Fields only a few times wave-

breaking are possible• Peak does not saturate as fast

– We are misled by the spike • Little field growth for• Implications for experiments…

€

˜ Q >20 0.001

0.01

0.1

1

10

100

0.01 0.1 1 10 102 103

Sim. energy lossSim. peak accelerationLinear energy lossLinear peak acceleration

Q~


High charge experiments

• Experiments transitioning for Q~2 to near 100

• Peak field still near to “scaling” for E164

• E164 -> E164* loses a factor of 3 off linear scaling (this is worse if you look at the useful field, not the spike).

0

0.2

0.4

0.6

0.8

1

1.2

1.4

0.01 0.1 1 10 102

103

Forces normalized to linear prediction

Peak accelerationAverage deceleration

Q~

E162 FNAL/UCLA

E164

E164*

~


Conclusions

• Linear “scaling” of energy loss (and accelerating field) into nonlinear regime explained– Scaling persistence well into nonlinear regime– Use of spike in accel. field not a good measure

• Point-particle limit understood?• Recent experiments already have • Pulse compression critical (shown at FNAL)• Next generation (at SLAC) may have

– Expect fall-off in scaling

€

˜ Q =1.5−3.5

€

˜ Q ~100

energy loss of very high charge beams in plasma: theory j. rosenzweig, n. barov, m. thompson, r....

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