Instability Dynamics with Electron Cloud Buildup in Long Bunches

1
Instability Dynamics with Electron Cloud Buildup
in Long Bunches

• Mike Blaskiewicz
• BNL RF and Accelerator Physics

April, 2004
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High Intensity Hadron Machines under construction
ORNL SNS J-PARC 3 GeV J-PARC 50 GeV
Primary concern Instability Instability Pressure rise? Instability Pressure rise?
80 injection 38 extraction injection 196 extraction
5.5e-4 5.9e-3 injection 3.2e-4 extraction 4.1e-4 injection 3.3e-6 extraction
14 19 injection 12 extraction 11 injection 5 extraction
Kinetic energy 1 GeV 375 MeV inj 3 GeV ext 3 GeV inj 50 GeV ext
100 125 65
6.3 4.2 22.2
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Evolution of PSR instability, beam current,
stripline difference signal, electron flux at
wall. From R.J. Macek
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Origin of Electron Cloud

• Primary electrons due to losses
• and gas stripping.
• Loss generated electrons more
• likely to multipactor
• Approximate conservation of
• adiabatic invariant (long bunch)
• For sufficient strike energy
• multiplication occurs.
• Secondary Emission Yield

5
Cylindrically Symmetric EC simulations

• Baseline calculation for PSR, very robust EC

6
Effective wakefields for electron cloud
interaction I

• Round, uniform charge distributions
• Force on protons
• Insert in equations of motion

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CSEC estimates for wakes

• Densities for PSR (left) and the
• wakes they generate (bottom)
• Full model includes the variation
• of electron density within the beam
• Calculations using a constant
• electron density equal to the
• central value are also shown

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Is the model accurate?

• The CSEC code was modified to allow for Nearly
  Cylindrically Symmetric ECs
• The electron density in x,y coordinates was
  expanded as
• This was then used to calculate the potential due
  to the electrons (electrostatic)
• The potential vanishes at the pipe wall
• The wake was started by introducing an offset in
  the proton centroid that
• was small compared to the beam radius
• The electric field due to the electrons was
  averaged over a disc of radius
• centered on the pipe axis

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NCSEC wakefield calculations

• PSR with 8 uC and stainless steel wall.
  Simulations for 9 turns shown

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Smooth transverse distribution with 8 uC

• Electron density
• Same voltage between pipe center and wall as in
  previous case
• Effective frequency is lower than small amplitude
  value.
• 9 turns plotted for electron density (in beam)
  and wakefield

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Burst at tail gives slightly larger values

• For the smooth beam the peak value of the wake is
  larger than the central model
• but net effect is close to central model estimate.

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Uniting wakes and beam dynamics I

• The previous calculations took about 10 minutes
  per turn for a single EC slice.
• There need to be about 10 slices per betatron
  period to obtain leading order corrections for
  detuning with betatron amplitude.
• The EC interaction is intrinsically serial, as
  opposed to parallel.
• The central wake model will be used to model the
  EC effect for the PSR.
• The other coherent force is space charge.
• Simple pair wise interaction is approximated as
• Derivatives are exact. Longitudinal part is
  finely binned and smoothed.
• Algorithm is O(Nmacro) using summation laws for
  sinusiods.

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Uniting wakes and beam dynamics II

• The beam size varies with the beta functions.
• For PSR (and SNS) the spread in the electron
  frequency from this effect yields a
• quality factor Q?2. Arrival time is used as the
  longitudinal coordinate in the bunch.
• Proton beam is modeled in the smooth
  approximation. Azimuth is the evolution
  variable. True lattice should yield very similar
  results.

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Preliminary checks

• Scatter plot of tune along the bunch and
  histogram for a central slice.
• The distribution is quite similar to the exact
  results for a Gaussian
• cross section and the actual Coulomb
  force.
• Cousineau et.al. PRSTAB 074202 show comparable
  (smaller) tune shifts for PSR

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Instability calculations

• PSR just beyond threshold, 270ns bunch length,
  5e5 particles, 0.5ns.
• With linear space charge and the same average
  tune shift there is strong instability
• Stable with half the RF voltage for no space
  charge

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PSR Stability Scaling Law, from R.J. Macek

• Maximum number of stored protons scales linearly
  with RF Voltage.
• Very weak dependence on bunch length (major
  theoretical challenge)
• In earlier work, shortening bunches increased
  maximum stored protons
• Significant conditioning after a month of
  operation

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Modeling the scaling law

• The effective electron density as a function of
  bunch/gap length is crucial.
• How do we dead-reckon this? (Compare pink and
  red!)
• Threshold estimates for future machines require
  caution.

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half the voltage, half the intensity

• Not surprisingly, the variation of electron
  density with intensity matters

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Simulations for SNS with 32 uC (2 MW)

• SNS simulations for single harmonic RF with 2
  nC/m, Q3, no burst.
• Same bunch length, peak current and rms momentum
  spread as design.
• Linear space charge, correct rms emittance.
• Coasting beam estimate is 37 kV.

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Conclusions

• Numerical calculations of the EC wakes show that
  the true value is between the central model and
  the full model.
• A better analytic result would be useful.
• For smooth transverse distributions the electron
  tune spread causes the actual wakefield to decay,
  and the central model is pretty good.
• Threshold estimates for PSR give voltages about
  twice the observed values, this is largely due to
  including nonlinear space charge forces.
• It is important to understand WHY space charge is
  effective in bunched beams.
• This is especially true since simple 2-D
  transverse models show small nonlinear space
  charge effects.
• For SNS the design calls for a uniform density
  beam and linear models are appropriate. The SNS
  still looks like it will be stable, though the
  effect of the debuncher cavity is still included
  in the simulations.

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Electron Cloud Generation

• Primary electrons are generated via losses and
  collisions with gas
• Loss created electrons at the pipe wall produce
  many secondaries
• Model loss rate proportional to instantaneous
  current
• Along with secondary emission there is also the
  possibility of reflection
• Reflection can be elastic or rediffused
• For true secondaries

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Average pressure rise from electron impact

• Desorption yield desorbed molecules
  per incident electron
• Average electron current per square centimeter
  microamps/cm2
• Average pumping speed liters per meter
  per second
• Pipe radius decimeters
• For uniform pumping the pressure is related to
  the pumping speed and
• the outgassing rate via PSqA where qA is in
  Torr-liter/meter/second
• 22.4760 Torr-liter 6.022e23 atoms
• Putting everything together
• For RHIC so
  for P1.e-6 we get

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PHOBOS Pressure rise

• For PHOBOS there is a 12 meter long, beryllium
  pipe with diameter D7.2cm
• Within the pipe the pressure obeys
• 
  Torr-liter/sec/cm
• For a round pipe at room temp
  cm-liter/sec
• 
  M28 for CO, D7.2 for
  PHOBOS
• The source term is the same.
• Assume uniform electron flux (independent of
  longitudinal position, s)
• Then
  the pressure difference between the
• 
  center of the pipe and the ends
• For PHOBOS at high intensity we need small
  desorption (baking/conditioning) or small SEY
  (coating/conditioning) or solenoids. Current
  plan is to install NEG pipes outside the IR. If
  the multipacting depends strongly on s, this will
  help.