1 electron cloud cyclotron resonances for short bunches in a magnetic field * c. m. celata a, miguel...

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Electron Cloud Cyclotron Resonances for Short Bunches in a Magnetic Field *

C. M. Celataa, Miguel A. Furman, J.-L. Vay, and Jennifer W. Yub

Lawrence Berkeley National LaboratoryBerkeley, CA 94720, U.S.A.

•Work supported by Office of Science, U.S. Dept. of Energy contract DE-AC02005CH11231

a Presently also a visitor at California Institute of Technology, Pasadena, CA, USAb Presently an undergraduate student at Cornell University, Ithaca, NY, USA

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We used POSINST to Simulate e- Cloud Buildup in x-y Slices of the ILC e+ Damping Ring Wiggler

Geometry

round vacuum chamber,perfectly conducting

beamz

computational plane

antechamber

x

y

By vs. z

Bwiggler

€

ωc =qB

meγ

B used is spatially uniform-- i.e, ideal dipole

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POSINST uses certain assumptions...

• Beam does not evolve in time (OK for short times, e.g., buildup)

• Beam electric field is transverse only (because beam velocity is c)

• Beam magnetic field neglected (ve small)

• Electrons generated according to phenomenological models

secondaries: Furman-Pivi* model

The force of the electrons on each other as it evolves in time is calculated self-consistently by a Particle-in-Cell algorithm.

* M.A. Furman and M.T.F. Pivi, PRST-AB 5, 124404 (2002).

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Cloud Buildup Calculations were done using ILC Damping Ring Parameters

“Wiggler”:

By 1.6 T; Bx = Bz = 0 (ideal dipole)

Vacuum Chamber:

R = 2.3 cm (vacuum chamber radius)

Antechamber full height = 1 cm (on +x side only)

Beam:

2 x 1010 e+ per bunch

9 GeV

x = 0.112 mm, y = 4.6 m, z = 6 mm

bunch spacing: 6.15 ns

Electron Production:

photon reflectivity = 1

peak SEY @ normal incidence = 1.4 at 195 eV

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For a Given B, the Average Electron Density

Builds Up over Time, then Plateaus

QuickTime™ and aTIFF (LZW) decompressor

are needed to see this picture.

Ave

rage

Ele

ctro

n D

ensi

ty (

/m3 )

Time (sec)

Equilibrium Average Density

B = 1T

500 bunch passages

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The cloud density is higher at certain B values



Note: runs were only done at “+” signs. The +’s were connected by lines for visibility. Thus not all peaks are shown.

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Peaks in density fall where n = ωcyclotron/ωbunch = an integer

Note: runs were only done at “+” signs, so not all peaks are shown. When simulation is done at n=integer there is always a peak if n<103.

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How it Works

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3gyro orbitof e– with x > 0

favoredphase (270°)

v

x

z

1y

Fx is always toward the center

Fy

Fx

vacuum chamber

B

xbeam kick

electron

– before beam kick– after beam kick

vv

B

x

z

y

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Resonance condition

ωcyc/ ωb = n:

n = integer, b=bunch spacing.

Single particle tracking clearly shows the effect.

€

B = n2π me

qτ b

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How the Resonance Increases the Cloud Density

Higher p

1. increases electron energy

2. increases the angle from the normal () when

the electron hits the wall

Both effects increase the secondary electron yield (SEY) of the electron (see next slide).

Note: Because B is spatially uniform, all electrons in the system are in resonance.

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Increased v and decreased cos() are seen in POSINST results

–– n=12–– n=11.5

–– n=12–– n=11.5

is the angle between the electron velocity and the normal to the wall surfact at impact. SEY increases as cos() decreases.

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Electrons are spread through the chamber for the resonance case

n=12 n=137.8

The high-field(no resonance) case shows the characteristic “stripes” pattern seen in many experiments. At resonance the electrons are much more widely distributed in x.

Color contour plots of electron density averaged over entire simulation

vacuum wall

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At resonance both the x and y beam kicks are important to increasing the energy

In what part of the chamber is the beam force most effective?

Assume r >> x, r >> y. Then E 1/r. Contours of constant Ex and Ey are:

So at resonance, more electrons can pick up the energy needed to make secondaries.

x

y

constant Ey

constant Ex

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At High B there is no effect. Why?



Note: runs were only done at “+” signs. The +’s were connected by lines for visibility. Thus not all peaks are shown.

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Why no density peaks at high B?

As B increases,

cyc becomes comparable to time for beam to pass, so

v changes direction during beam kick

Beam kick integrates over cycle no effect

For effective resonance, require:

€

B << 2πmec

qlb

lb = bunch length

We believe this is why this resonance has not been mentioned before-- other work was for cases with longer bunches and higher fields.

This is indeed the range of B where the peaks decrease and disappear for our simulations.

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Results of a Simple Analytical Model are Illuminating

Positron beam line densityQuickTime™ and a

TIFF (Uncompressed) decompressorare needed to see this picture.

QuickTime™ and aTIFF (Uncompressed) decompressor

are needed to see this picture.E-field from positron beam

QuickTime™ and aTIFF (Uncompressed) decompressorare needed to see this picture.= 2D Bassetti-Erskine field


are needed to see this picture.Eqs. of motion for a single electron

(assumed nonrelativistic)

Assumptions:

• Beam kick is always that for electron’s position at t=0 - Appropriate for portion of electrons’ phase space (not near x=0, y-amplitude small)

• Electrons don’t hit wall, i.e.: - Short time

- 0 < Rchamber (so B not very small)

Model:

Gives:

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And solve ...





Take d/dt of #3 above and combine with #1





Solve to first order in :

Solution:

exponential falloff of effect as cyclotron period beam extent in time



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Simple analytic approach - 2

Can then obtain solutions for vx(t), x(t) and z(t). Also y(t), but for this intensity electrons hit the wall in a few bunch passages.

Results:

Equations show that:

• Amplitude and energy grow on resonance (n=integer, because then A(K,n) grows in time)

• For n=integer, electrons soon “forget” their initial conditions & become synchronized with

the beam

• If n ≠ integer, no amplitude growth

• Resonance suppressed by exponential factor when ωt 1

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At low B there are special features

n = (cyclotron frequency)/(bunch frequency)

Peaks have a minimum at or near the resonance, and peak height decreases from n=4 to n=1.

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Explanation of low-B Features

“Double peaks”

POSINST data show:

at n=2 electron lifetime longer than at nearby field with peak density

energy higher, but fewer hit the wall per sec

fewer secondaries formed

This occurs after space charge important. We don’t understand lifetime yet.

Note: Higher energies at resonance not too high to be effective at making

secondaries. Avg SEY is slightly higher for resonance case.

Decreasing peak height from n=4 to n=1

Appears to be due to large gyroradius electrons hitting wall (gyroradius ~ ).

€

1

B

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New 3D Results Show Resonance Effects

WARP-POSINST simulation: ideal dipole, 10 cm longWe see the phase of the cyclotron motion coming into phase with the beam position, and the density at equilibrium agrees with POSINST results.

n=12 after 2 bunch passages

black = primary electronsred = secondaries

At z location of beam, vs come into phase all vz’s close to zero for electrons that have been in the system for a few bunch passages. Distance between min and max spread in vz = distance beam travels while the electrons execute 1/4 cyclotron period = 3.8 cm. Distance matches the simulation.

Also see vx spread maximize where the vz spead minimizes and vice versa, as expected.

beam is blue line

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3D Wiggler Calculations are Essential but Challenging-- We are starting on these now

• ExB drift e– s different z (and B) go in (and out) of resonance.

Resonance may affect more e–s, but each gains less energyWhat is the sign of the effect?

• Use correct 3D field:

Bx and Bz, and variation in By across the chamber.

Can do a lot with POSINST.

• width of the resonances (ILC DR wigglers) 10 G need z resolution ~ 2 m! Grid cells asymmetric (350:1:1), leading to possible error, or could instead make huge runs by resolving x and y to m scale.

• Time step must be ~ 1 x 10-11 s to resolve beam and cyclotron motion.

Need to simulate 3D effects:

But it is hard:

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Experimental Observation

The resonances have been seen by Pivi et. al. in the PEP-II chicane experiments (see EPAC proceedings).

Density peaks have correct spacing, but details aren’t understood.

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The Resonances and CESR-TA

Photon reflectivity, bunch spacing, & bunch length are somewhat different from what I have been investigating for ILC. This will change:

Electron Distribution:

Reflectivity runs for PEP-II chicane showed clear resonances with

reflectivity of 0.1, as did ILC runs for reflectivity of 1.0.

Resonance Spacing:

14 ns bunch spacing gives resonance spacing of 26 G;

4 ns gives 89 G spacing.

Number of Resonances:

Resonances should go to about n=70 for the 14 ns case

n=20 for the 4 ns case.

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Preliminary Results for Ecloud Density in CESR dipole for SEY peak = 1.6

Ave

rage

Den

sity

(/m

3 )x 1011

n=11.5

n=12.0

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Conclusions

• These resonances will occur for accelerators with short bunches in wigglers, low-field dipoles, and in fringe fields of magnets.

• The resonances produce an increase in the electron cloud density that is not huge (factor of 3), but it is periodic with the wiggler periodicity and therefore also with the beam centroid motion. Thus it could possibly cause resonant effects on the beam. But the number of electrons near the beam is less.

• Differences in spatial electron distribution at resonance affect wall heating and electron diagnostic placement.

• 3D calculations will be very important in showing what effect this resonance has on the electron cloud magnitude in the wiggler, and in dipole fringe fields.

We are starting on 3D calculations with the WARP code

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Backup Slides

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Dependence of the secondary emission yield (SEY) on , energy for our parameters

= angle of the electron velocity to the normal of the vacuum wall

1 electron cloud cyclotron resonances for short bunches in a magnetic field * c. m. celata a, miguel...

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