Accelerator Physics Topic VII Coupled Bunch Effects

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Accelerator PhysicsTopic VIICoupled Bunch
Effects

• Joseph Bisognano
• Engineering Physics
• Synchrotron Radiation Center
• University of Wisconsin-Madison

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Coupled Bunch Instabilities

• We have discussed instabilities internal to a
  single bunch of charged particles
• Typically in a storage ring or linear accelerator
  there are trains (finite or cw) of bunches
  separated by nanoseconds to maybe milliseconds
• Say we have a resonant structure at 300 MHz, with
  an angular frequency of 2?(300) ?2 GHz
• If it has a Q of 20,000 (typical of Cu), its
  fields ring for 20,000/2 GHz10 microsecond if
  the Q were 2 109 more typical of superconducting
  RF, the ringing would last a full second
• So a sequence of bunches can talk to each other
  through resonant structures
• Whereas low Q impedances have a large bandwidth
  and can see the peak current, these high Q
  structures have a narrow bandwith and only see
  the average current.
• In other words, broadband impedances generate
  peak current limitations in accelerators,
  narrowband impedances generate average current
  limitations

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Bunch Spectrum
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Robinson Instability

Following A. Hoffman, CERN77-13
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Robinson/cont.
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Robinson/cont.
R
R-
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Robinson/cont.
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Robinson/cont.
Damping or antidamping
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Robinson Conclusions
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Robinson Stability Condition
Above transition
Below transition
-

-

?r
?0
?r
?0
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Coupled Bunch Instabilities
phase definition change

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Coupled Bunch/cont.
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General Phase Relationship
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Normal Modes N4
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Spectrum/cont.
4 4 3 1 2 2 1 3 4 4
3 1 2 2 1 3 4 4
-4 -3 -2 -1 0
1 2 3 4
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Growth Rates
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Fixes
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Mode Coupling
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Mode Coupling at SRC
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Transverse Phenomena
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Transverse Coupling
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Deflecting Modes
Particle on axis doesnt see Ez , doesnt deposit
energy Particle off axis can excite mode through
Ez But deflection is constant through derivative
of Ez
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Resonant Wakefield
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Beam Breakup in Linear Accelerators

• In a linac there the higher order cavity modes
  produce the same basic resonant self-interaction,
  both longitudinal and transverse
• For relativistic linacs, the longitudinal motion
  is more frozen than in a storage ring, which
  has bending. So transverse effects are often the
  limiting factor in linacs
• For transverse effects, the primary difference in
  the dynamics is number of times the same bunch
  sees a given cavity HOMs
• Straight linac once, amplification
• Recirculated linac several times, instability
  with finite threshold
• Storage ring infinite times, zero threshold
  unless some form of damping present
• In linacs, these effects are call Beam Breakup

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Regenerative Beam Breakup

• Basic mechanism a train of bunches excites a
  transverse deflecting mode of a single cavity
• Feedback loop
• Say, HOM has small excitation
• Even a bunch perfectly aligned on axis will
  receive a transverse kick
• If energy is low and structure long, a
  significant deflection will occur while the bunch
  is in the cavity
• The offset bunch is now in a region of
  longitudinal electric field and can deposit
  energy into mode
• Go to next bunch
• We have a feedback loop that can go unstable
  unless the cavity losses (more with lower Q)
  exceed the gain of the loop
• An honest instability

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Regenerative Beam Breakup
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Threshold Condition
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Cumulative BBU Amplification
1 2 3
4 5
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Cumulative BBU/cont.

• Cavity 1 Bunch will coherently excite cavity,
  later bunches will receive transverse kick
• Cavity 2 Bunch will enter cavity 2 with an extra
  offset cavity 2 experiences an enhanced
  excitation
• Cavity N DITTO
• Overall, initial offset causes growing excitation
  of subsequent cavities which can increase offset
  downstream Amplification
• Since there is no closure of loop, there is no
  instability as such

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Cumulative Beam Breakup

• Typically bunching frequency and transverse HOM
  frequency are not harmonically related
• So, there can be a large transient, but the
  equilibrium excitation can be rather small. For
  a pulsed linac, however, the transient can cause
  beam loss, limiting currents to 100 mA
• For CW operation with equally spaced bunches, the
  excitation settles down to a DC value that can be
  steered away

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Multipass Beam Breakup

• A new feature of SRF linacs is the possibility
  of recirculation, and even energy recovery
• SRC allows CW operation and the beam can pass
  through the linac several times
• The cumulative beam breakup amplifier now has
  its feedback loop closed and at high enough gain
  there can be instability
• Limited the first generation of SRF linaces to 10
  microamps average currents when HOM Qs were in
  the 10,000,000 range
• In some ways its a combination of cumulative and
  regenerative BBU

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Multipass BBU Mechanism

• Displaced bunch excites a HOM
• Following bunches deflected
• Recirculation optics transforms kick into a
  displacement
• Displaced bunch further excites HOM in same
  cavity
• Again threshold occurs when excitation rate
  exceeds damping rate

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Beam Breakup Mechanism
Initial noise excitation of cavity mode kicks
particle bunch
beam on pass n
cavity
On subsequent pass, bunch enters off axis and
coherently excites cavity mode
Beam on pass n1
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CEBAF
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Jlab FEL
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Multipass BBU Theory
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Multipass BBU Theory/cont.
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Multipass BBU Theory/cont.
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Multipass BBU Theory/cont.
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Multipass BBU Theory/cont.
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Multipass BBU Theory/cont.
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Multipass BBU Theory/cont.
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Multipass BBU Theory/cont.
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Multipass BBU Theory/cont.
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Simulation transient and steady state below
threshold (cumulative-like)
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Simulation instability
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Longitudinal Multipass BBU Theory
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Longitudinal Multipass BBU Theory
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Longitudinal Multipass BBU Theory
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Longitudinal Multipass BBU Theory
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Longitudinal Multipass BBU Theory
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Longitudinal Multipass BBU Theory
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Longitudinal Multipass BBU Theory
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Longitudinal Multipass BBU Theory
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Longitudinal Multipass BBU Theory
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Longitudinal Multipass BBU Theory
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Longitudinal Multipass BBU Theory
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Longitudinal Multipass BBU Theory
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Longitudinal Multipass BBU Theory
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Longitudinal Multipass BBU Theory
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General Scaling with Q

• For a single cavity, the threshold scales like
  1/Q
• For several cavities at the same resonance
  frequency, the threshold scales like 1/Q times
  weighted sum over the transport optics
• But HOMs have a distribution in frequency from
  construction errors which
• Decreases the peak value of the weighted sum
• Changes the 1/Q dependence to something more
  typically like 1/?Q until frequency spread is so
  large that the cavity modes dont overlap
  significantly

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Typical Frequency Distribution Scaling
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Typical Q Scaling
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Energy Recovery Linacs
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Current Limits

• Typically, a storage ring light source will store
  a few hundred milliamps
• Progress in electron source development make CW
  guns at the 100 milliamp level reasonable to talk
  about with emittance performance comparable and
  even better than storage rings
• Progress in HOM damping has made current limits
  at the 100 milliamp level obtainable
• So, an energy recovery linac should be able to
  produce storage ring levels of current with
  better emittance

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Is It Worth Recirculating an ERL DriverMore
Than Once
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Energy Recovery Linac (ERL)
superconducting
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Issues

• Good
• Save money SRF, RF, cryo
• Beams of different energy right there
• Bad
• Costs of more magnets
• Beam breakup
• Coherent synchrotron radiation (CSR)
• emittance growth
• energy spread growth
• CSR instability
• Space charge
• Weaker focusing head tail could be worse
• RF constraints off phase choices
• Layout of compressors
• Just different
• Site constraints shorter and wider
• Multipass harder, but are limits limiting and are
  the savings significant

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One up/one down Current Limits with Arc Optics
Variation
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Two up/two down Current Limits with Arc Optics
Variation
300
mA
200
100
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Homework Problems Topic VII

• Topic VII-1 Consider a storage ring with 4
  equally spaced bunches.
• a) Derive an eigenvalue problem for longitudinal
  coupled bunch motion for 4 point-like bunches
  undergoing small oscillations. Assume the
  instability is excited by a single higher order
  mode at some frequency. (Proceed as follows
  calculate the voltage produced by an individually
  oscillating bunch. Second, calculate the
  perturbatin induced by this voltage on any of
  the four bunches.)
• b) Show that the eigenmodes reduce to those
  discussed in lecture
• c)Discuss what happens if one of the bunches is
  missing i.e., there is a gap.
• Topic VII-2 Consider a charge Q passing rQ off
  axis through a cavity and a charge e passing time
  ? behind
• For a transverse deflecting mode, write an
  expression for the energy loss of the charge Q
  and the transverse deflection of charge e if it
  were on axis, and also if it were at rQ
• In physical terms, why would isochronous
  transport in a two-total-pass recirculated linac
  prevent longitudinal beam breakup? What would be
  the analogous transport constraint to prevent
  transverse beam breakup? What are the
  limitations of such solutions?