SpaceCharge Instabilities in Linacs

1
Space-Charge Instabilities in Linacs
2
What limits the beam current in ion linacs?

• This is a very important topics for anyone
  designing linacs. For example, it is important
  for beam dynamics design of RFQs.
• In general the beam current is limited by the
  focusing provided to confine the beam to within
  the aperture, balancing the defocusing effects of
  space charge and emittance. There are formulas
  that have been derived for this.
• In addition, there are some instabilities that
  must be avoided. That is what we will discuss.

3
What instabilities are important without space
charge?

• The equation of transverse motion for periodic
  focusing arrays with linear quadrupoles or
  solenoids is known as Hills equation.
  xK(s)x0, where K(s)K(sL) is periodic with
  period L.
• Hills equation gives stable motion when phase
  advance per focusing period is s0lt180 deg. This
  is an instability that affects strongly-focused
  beams periodic focusing arrays.
• The other main instability without space charge
  is the parametric resonance when kl2kT. This
  affects nonrelativistic beams.
• Usually these two instabilities can easily be
  avoided. But avoiding instability for beams with
  space charge imposes additional restrictions.

4
The most important instability with space charge
is the transverse- envelope instability

• This beam instability is a parametric resonance
  driven by the periodic-focusing lattice and
  mismatch oscillations of the beam.
• Free energy is available from the beam mismatch.
• See Martin Reiser, Theory and Design of Charged
  Particle Beams, Wiley VCH

5
Transverse collective modes in RF linacs

• Collective modes are charge-density oscillations
  in the beam. They are excited by beam mismatches
  and by the quadrupole or solenoid
  periodic-focusing lens array.
• Pure transverse modes are described by their
  azimuthal symmetry, such as monopole (1st order)
  or breathing mode, quadrupole (2nd order),
  sextupole (3rd order), octupole(4th order),
• These modes can be either stable or unstable, and
  if unstable can generate significant emittance
  growth or beam losses.

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Characterizing beams with space charge in linear
periodic focusing arrays

• The importance of space charge in the beam is
  measured by the phase advances per focusing
  period s and s0. These parameters are the same
  for every particle in the beam s the
  transverse phase advance per focusing period in
  degrees of the equivalent uniform beam including
  space charge. s depends on the net effect of both
  focusing and space charge.s0 phase advance
  per focusing period of the beam without space
  charge. s0 depends on only the focusing.The
  equivalent uniform beam is a uniform density beam
  with the same current and same rms properties as
  the real beam (which generally is not uniform).

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Another useful quantity is the tune-depression
ratio s/s0
Range of tune-depression ratio is 1 s/ s0 gt0
s/ s0 1 corresponds to no space charge, only
focusing affects the beam. s/ s0 0 corresponds
to extreme space-charge limit where space charge
is large enough to exactly cancel the focusing
force.
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Most important instability for beams with space
charge is the envelope instability of the
quadrupole mode.

• The stability criterion for the envelope
  instability depends on both s0, and s.
• Instability occurs when s0 gt90 deg and s lt90 deg.
  This is more restrictive than the Hills equation
  instability.
• Simulations show that the envelope instability
  generates rapid and significant emittance growth
  and beam loss and must be avoided.

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Must avoid s0gt90 deg for high current beam design

• When s0 gt90 deg and slt90 deg, the envelope
  instability gives an unstable beam. It blows up
  in just a couple transverse oscillation periods.
• The envelope instability has been confirmed
  experimentally at LBNL and Maryland.
• The general practice for conservative design is
  to limit the external focusing strength so that
  s0 lt90 deg.

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Sextupole instability

• The sextupole instability can be excited when s0
  gt60 deg and s lt60 deg.
• This is a weak instability. Usually no observed
  emittance growth in simulations.
• Generally there is no design requirement from the
  sextupole instability to limit s0.
• However, at PAC09 the SNS people reported large
  emittance growth from the sextupole instability,
  attributed to their use of quadrupole lenses with
  dipole corrector magnets that gave a very large
  dodecapole component (factor of 3 higher than
  expected).
• This gave halo and beam loss in SNS a factor of
  20 higher than expected. Can be corrected by
  reducing s0 from 60 deg to 50 deg.
• (See Y. Zhang, C.K.Allen, J.D.Galambos, J.Holmes,
  J.G.Wang, Beam Transverse Issues at the SNS
  Linac, PAC2009, Vancouver.)

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Conclusions about transverse envelope
instabilities

• The quadrupole or envelope instability can be
  avoided by keeping s0lt90 deg.
• The sextupole instability can be avoided by using
  high quality quadrupoles especially with small
  dodecapole component. Sextupole is also avoided
  by keeping s0lt60 deg, but that restricts the
  focusing strength.
• All other transverse modes are too weak to be of
  concern.

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Another type of collective instabilityAnisotropy
instability leads to emittance transfer between
transverse and longitudinal degrees of freedom
Caused by space charge, and transverse-longitudina
l coupling. Free energy is available when there
are different temperatures for different degrees
of freedom.
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Space-charge coupling Instabilities for
anisotropic beams

• Anisotropic beams are important for linacs
  because the transverse and longitudinal beam
  parameters are usually different. Thus, beam
  bunches are anisotropic.
• Even without space charge recall that the
  transverse and longitudinal motions are coupled
  through nonlinear effects., i.e. kl2kT.a)
  dependence of transverse RF defocusing on the
  longitudinal phase in the transverse
  equation.b) dependence of transit-time factor
  on radial displacement of beam in longitudinal
  equation (through the I0 Bessel function).

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Gluckstern (1966) showed that there is a
parametric instability even with no space charge
when kl2kt (or sl2st)

• The wave number k means phase advance per unit
  length, and skL where L is the period is the
  phase advance per focusing period.
• This coupling resonance is important when there
  is strong longitudinal focusing. You have to
  avoid this parametric resonance.

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Physics with space charge

• Note that binary collisions of the particles play
  no significant role in the physics. The
  space-charge force, a smoothed or average effect
  over all the particles in the bunch, is what
  matters.
• When space charge is important, the physics is
  controlled by collective anisotropy resonances,
  which causes emittance transfer between planes.
• I. Hofmann et al., Space charge resonances in
  two and three dimensional anisotropic beams,
  Phys. Rev. ST-AB, 6, 024202 (2003)

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Stability Plots of I. Hofmann, et al.

• The anisotropy resonancies are observed in a plot
  of the space-charge tune-depression ratio nx/n0x
  (ordinate), a measure of the importance of space
  charge, versus the longitudinal to transverse
  tune ratio (abcissa) nz/nx.
• The tune ratio nz/nx allows identification of
  the anisotropy resonances. Resonances can occur
  when tune ratio is a ratio of integers.

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Stability plots

• In Hofmanns publications the symbol n is
  sometimes replaced by the symbol k. Both
  represent the wave number or phase advance per
  unit length.
• The stability plots are shown for constant values
  of the emittance ratio ez/ex, which is
  interpreted as the emittance ratio of the initial
  beam if the emittances change.

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Stability plots (continued)

• The most prominent stop bands are located near
  tune ratios kz/kxnz/nx1/3, 1/2, 1, and 2, not
  all of which are always present.
• Equations of motion

Coupled equations of motion for x
(transverse) and z (longitudinal) showing kx and
kz phase advance per unit length in x and z. The
functions f and g include space charge.
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Stability plots (cont.)

• The stability plots show contours of constant
  calculated growth rates.
• The contours identify calculated stop bands
  (regions of exponential growth) that lie near
  tune ratios with integer tunes.

20

• This shows stability plot for collective
  anisotropy resonances with analytically-calculate
  d stop bands in plot of transverse tune
  depression versus tune ratio for ez/ex2.
• Dashed line corresponds to equipartitioning
  where the ½ resonance is suppressed and no
  emittance transfer occurs.
• Resonances in this case are near 1/3, 1 and 2.

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Energy equipartitioning and the energy anisotropy
parameter

• For emittance transfer between x and z requires
  unequal temperatures in
• x and z. Energy equipartitioning means equal
  temperatures in all three
• degrees of freedom, and therefore no emittance
  transfer is possible

• Define energy-anisotropy parameter Ta as ratio
  of average kinetic energies
• in any two degrees of freedom x and z. Ta1 gives
  equipartitioning.

• Equipartitioning is introduced, not because
  beams necessarily evolve to
• equipartitioned states, which they do not, but
  because equipartitioned beams (Ta 1) have no
  free energy for emittance transfer.

22
Stability plot for ez/ex2 showing CERN SPL
design trajectory points. Significant emittance
transfer in simulation when trajectory overlaps
the kz/kx1 stop band.
SPL short Trajectory stays out of kz/kx1 stop
band. The simulation shows no emittance
transfer Case 2 Trajectory overlaps with kz/kx1
stop band. Simulation shows significant
emittance transfer. (Longitudinal decreases from
0.75 to 0.56 mm-mrad.Transverse emittances
increase from 0.4 to 0.5 mm-mrad.) SPL full Very
little overlap and very little emittance transfer
as expected.
23
Stability plot for ez/ex1.4 showing the SNS
linac trajectory (above) and for ez/ex1.3 for
the proposed ESS linac (below).
SNS trajectory (above) passes through stopbands
near 1/3, 1/2, and 1. Simulation shows emittance
growth only for kz/kx1. Transverse emittance
growth is 27. Longitudinal growth is 3.
Other sources of nonlinearity affect
longitudinal. ESS (below) has no emittance change
in simulation..
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How to minimize emittance transfer

• Simulations show that RMS emittance transfer is
  insignificant in nonequipartioned beams if the
  kz/kx1 stop band is avoided. The other stop
  bands are too weak to matter.
• However, if the kz/kx1 stopband cannot be
  avoided, an emittance ratio near unity
  (approximate equipartitioning ) would limit free
  energy for emittance transfer.
• In other words, significant emittance transfer
  requires overlap with kz/kx1 stopband and an
  emittance ratio not near unity.

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As space charge becomes stronger

• As space-charge tune depression becomes stronger
  (smaller tune-depression ratios and stronger
  space charge) the stop-band widths increase and
  overlap.
• The thermodynamic picture that anisotropic beams
  approach energy equipartitioning applies only
  close to the space-charge limit where stop bands
  completely overlap.
• Then emittance transfer occurs at all tune
  ratios.

26
A equipartitioning argument for many years has
been resolved.

• Some argued that a nonequipartitioned beam would
  always equipartition at the expense of unwanted
  emittance transfer.
• They argued that the beam had to be
  equipartitioned to prevent emittance transfer.
• But, Hofmann et al. studies showed that
  equipartitioning is not a necessary condition to
  prevent emittance transfer.
• You can prevent emittance transfer if you can
  simply avoid the kz/kx1 resonance, or minimize
  the time the beam is on that resonance.

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Operating near the space-charge limit without
emittance growth is very difficult

• Near the space-charge limit, other emittance
  growth mechanisms with free energy from beam
  mismatch or nonlinear field energy will dominate,
  and equipartitioning will not help.
• Thus, maintaining a bright beam at tune
  depression near the space charge limit remains a
  significant challenge.

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Summary of space-charge instabilities in linacs

• To avoid the envelope instability require s0lt90
  deg.
• To avoid the sextupole instability, minimize the
  dodecapole component of the quadrupole magnets,
  or require s0lt60 deg.
• To prevent emittance transfer avoid kz/kx1
  stopband or that is not possible, limit free
  energy for emittance transfer by approximate
  equipartitioning.
• The simple thermodynamic picture that anisotropic
  beams always approach energy equipartitioning
  applies only close to the space-charge limit
  where stop bands completely overlap. No one
  should operate there anyway!