Parametric Resonance in Linacs

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Parametric Resonance in Linacs
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Parametric resonance, a simple picture
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Parametric resonance condition

• Resonance is when the driving frequency equals
  the natural frequency.We have two possible
  driving frequencies.
• For the sum term this requires w1w0w0. This is
  satisfied when w10,
• which gives the usual condition w0w0 for an
  oscillator driven at natural frequency. Nothing
  new from the sum term.
• For the difference term it requires w1-w0w0, or
  w12w0. Resonance when the modulation frequency
  is twice the natural frequency. This is called a
• parametric resonance.

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What conditions produce a parametric resonance?

• There exist oscillators in which external forces
  produce a time variation in the natural
  frequency. (See Mechanics, 3rd edition, Landau
  and Lifshitz, Pergamon Press, pp. 80-84)
• A simple example is an oscillating pendulum whose
  point of support executes periodic vertical
  motion.
• Parametric resonances are important for
  accelerators.
• An important example for ion linacs is the
  beam-dynamics resonance known as the kl2kT
  resonance. (See R.L.Gluckstern, in Linear
  Accelerators, Lapostolle and Septier, 1970, Wiley
  Sons, pp. 799-801, R.L.Gluckstern, Linear
  Accelerator Conf. 1966, LA Rept. 3609, p.250.)

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Longitudinal equation of motion in a linac
including  longitudinal-transverse coupling
(nonrelativistic)
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Including the transverse coupling term for the
longitudinal equation
The transverse coupling term is sometimes ignored
when writing the longitudinal equation of
motion. This is OK near the beam axis wherethe
Bessel function I0 is near 1, but more generally
I0 depends on y.
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Next write transverse equation of motion in a
linac including transverse-longitudinal
coupling
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Summary of the kl2kT resonance in ion linacs

• Coupling of longitudinal and transverse motion
  takes place through phase dependence of the RF
  defocusing and radial dependence of the
  accelerating gradient.
• Simulations show that this resonance causes
  transverse emittance growth.
• The coupling terms decrease rapidly with
  increasing b so this effect is most important at
  low velocities.
• A dangerous scenario is when kl gt2kT initially
  and if kl falls off with b faster than kT until
  the resonance condition is satisfied, or if the
  resonance condition is not passed through rapidly
  enough.

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A longitudinal beam-dynamics constraint on
accelerating gradient encountered for very low
velocities in a conceptual design of a
high-gradient superconducting linac.

• Another example (unpublished) of a parametric
  resonance.

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Conceptual design to see what problems we would
encounter pushing the superconducting linac to
lower velocities

• Our design attempted to extended the
  superconducting linac to very low velocities all
  the way down to the RFQ.
• 5-cell spoke cavities were used in first three
  sections to increase the real-estate accelerating
  gradient at low beta 7-cell cavities (spoke and
  elliptical) were used for entire remaining linac.
  The 5-spoke and 7-spoke would need RD for
  development.
• We found that using high accelerating gradients
  at low velocities produced a longitudinal
  envelope instability. We had to reduce the
  accelerator gradient.
• Design modifications to be described mitigated
  the effect and allowed higher gradients to be
  used.

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• We found that longitudinal rms beam size is
  resonantly driven (parametric resonance) by the
  focusing cavities, when the longitudinal phase
  advance per focusing period of the mismatch
  oscillations equaled 180.

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Longitudinal 1D envelope equation of motion

• Z is longitudinal beam envelope size
• kl02 is smoothed longitudinal phase advance per
  period
• L is period length of array of RF cavities that
  provide the longitudinal focusing.
• a is amplitude of periodic part of cavity
  focusing
• e is the longitudinal emittance
• Kl is longitudinal space-charge term from the 3D
  
• ellipsoid model. It is proportional to the beam
  current.
• The periodic longitudinal focusing from the
  cavities is treated here in a
• quasi-smooth approximation. This means smooth
  approx with a small periodic term to induce an
  instability from the periodic focusing lattice.

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Equation of motion of a longitudinal mismatch
perturbation

• Write ZZ0 as matched solution and introduce a
  small perturbation z that produces mismatch
  oscillations. Then the longitudinal envelope is
  ZZ0z, and we assume zltltZ0.
• Substituting ZZ0z into envelope equation and
  assuming the matched envelope Z0 is uniform and
  Z00, we obtain an equation of motion for the
  mismatch perturbation z.

Notice the oscillator on the left, and the
driving term on the right.
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Equation for a mismatch parametric resonance

• The equation for the mismatch perturbation z is
  that of a driven
• oscillator .
• The right side of the equation for z is the
  product of two sinusoids, a sum term (2p/Lkmm)
  and a difference term (2p/L-kmm) .
• Parametric resonance corresponds to the
  difference term,
• kmm2p/L-kmm, or kmmLp. The modulating frequency
  with 2p phase
• advance per focusing period comes from the
  periodic focusing array.
• Parametric resonance occurs when the phase
  advance per period of the envelope mismatch
  oscillation kmmL is p, i.e. the period of the
  envelope mismatch oscillation is twice the
  period of the focusing lattice.

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Introduce a parameter h that measures the
importance of longitudinal space charge

• Phase advance of longitudinal oscillations per
  unit lengthwithout space charge, kl0
• Phase advance of longitudinal oscillations per
  unit length with space charge , kl
• Define longitudinal space-charge tune-depression
  ratio,
• h kl/kl0 where 0lthlt1.
• h1 is no space charge, h0 is space charge limit
  (where space charge cancels the external
  focusing) .

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Express the resonance condition in terms of the
tune-depression parameter h
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Avoiding the mismatch resonance

• These two equations give the phase advance per
  focusing period kl0L and klL as a function of the
  tune depression h. See next vugraph.
• Resonance is always avoided if kl0Lltp/2. This is
  controlled by limiting the longitudinal focusing
  strength.
• Realizing that the model is approximate, we
  conservatively adopt this kl0Lltp/2 criterion for
  kl0 to ensure we avoid the instability.

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Safe criterion to avoid longitudinal instability
for all beam currents is kl0Lltp/2. This will
limit the average (real estate) accelerating
gradient ltE0Tgt.

• The longitudinal constraint becomes important
  for highcharge-to-mass ratio, low
  velocities(cubic dependence), high frequencies,
  and long focusing periods (quadratic dependence).
  
• Reducing magnitude of phase f below 30 deg
  doesnt help because phase width of bucket
  shrinks causing beam losses.

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Our beam-dynamics approach for overcoming the
constraint on the accelerating gradient

• Make each cryomodule with identical elements and
  as a short FODO lattice with its characteristic
  period L.
• Allow period L to change from one cryomodule to
  the next. -Do not require that focusing period
  must be large enough to span the large space
  between cryomodules.
• Shorten the focusing period L. -Include only one
  cavity and one solenoid per focusing period.
  -For compactness use solenoids instead of
  quadrupoles for transverse focusing.
• Use cavities and solenoids at both ends of
  cryomodule for matching between cryomodules.
• Gradients are still limited by kl0L ltp/2
  requirement but these measures help a lot.

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Example of two cryomodules Cryomodules are short
FODO lattices with different focusing periods.
Each period consists of one cavity and one
solenoid.
Solenoid
Cavity
L1
Cryomodule 2 (b2), period L2
Cryomodule (b1)period L1
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Beam profiles for 8 superconducting sections from
6.7 to 600 MeV after approximate matching between
cryomodules. Matching not perfect but
satisfactory.
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Summary

• The longitudinal instability limits the
  accelerating gradient especially at
  low-velocities (blt0.2). You may not be able to
  operate at these low velocities with the
  accelerating gradient that the cavities are
  technically capable of.
• However for CW applications at these low
  velocities, superconducting may still be more
  attractive than normal conducting.
• Our longitudinal beam-dynamics design approach
  has been to keep
• kl0Llt p/2 and to minimize the focusing
  period.
• The cryomodules form piecewise constant FODO
  lattices where each period contains one cavity
  and one solenoid.
• For 350-MHz proton linac in b range of 0.2 to 0.5
  (20 to 150 MeV) we could use cavity gradients up
  to about 8 MV/m without longitudinal
  beam-dynamics problems.