Halo formation in Linacs

1
Halo formation in Linacs

• A.V. Fedotov and R.L. Gluckstern

ORNL/SNS Workshop June 25-26, 2001
2
Acknowledgments

• This work summarizes studies of beam halo done at
  DSAT group at University of Maryland (1997-99).
• Halo studies were driven by the APT project at
  LANL.
• The studies were done together with
  R. Ryne and S. Kurennoy (LANL).
• Collaborators T. Wangler, J. Barnard, S.
  Lund, J. Qiang, N. Pichoff.

3
Definition of Beam Halo

• ICFA Workshop on Beam Halo (Wisconsin 1999)
• In general, one refers to HALO as long as there
  are tails outside the beam core.
• Halo extent can be of any size still HALO
• Definition of Halo as particles outside several
  sigmas is not good.
• Definition is NOT important
• What really matters is the source of halo
• Parametric Halo (P.H.) parametric resonance
  between individual
  particles collective modes
• P.H. is believed to be an important source of
  halo in Linacs
• P.H. may exist in circular machines. Its
  existence will be strongly machine dependent
  not necessarily the main source of halo

4
Parametric Halo
Uniform density

Q depressed tune m mismatch parameter k -
space-charge perveance a beam radius p
frequency of beam oscillations

• Key mechanism parametric resonance
• The dominant resonance (21) p2Q

5
P.H. Transverse (phase space)
Typical stroboscopic plot in the transverse phase
space for 125 particles with low angular
momenta for a KV beam. (tune depression
0.8) (mismatch parameter 0.7)
6
P.H. 2D 3D

• 2D
• Analytic studies for a KV beam were found to
  give excellent agreement with corresponding
  computer simulations which were then extended to
  other 2D distributions.
• The location of P.H. radius gave some
  confidence that a beam pipe wall could be placed
  far enough from the beam to avoid intercepting
  the halo particles.
• 3D
• Attention then shifted to short 3D beam
  bunches of ellipsoidal shape with c/a
  length/width ratio in the range 2 - 4.
• Both transverse and longitudinal modes were
  capable of generating P.H.

7
P.H. - Longitudinal
The signature of the longitudinal halo was the
same as that of the transverse halo (a peanut
diagram in the phase-space projection). The
transverse and longitudinal mismatch and tune
depression parameter space was extensively
explored with numerical simulations.
8
P.H. - Longitudinal (phase space)
Phase-space diagram of a longitudinal
halo. (c/a3, hx.65, hz.49, mz1.4)
9
Longitudinal halo extent for different
mismatches (tune depression dependence)
10
Longitudinal halo extent for different beam sizes
(tune depression dependence)
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Dependence of halo intensity on mismatch
(c/a3, hx.65, hz.49) m1.1, m1.2, m1.3,
m1.4
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Dependence of halo intensity on tune depression
(c/a3, m1.2) hz0.87, hz0.65, hz0.49,
hz0.32
13
Dependence of halo onset on mismatch
(c/a3, hx.65, hz.49) m1.1, m1.2, m1.3,
m1.4
14
Dependence of halo onset on tune depression
(c/a3, m1.2) a) hx0.79, hz0.65 b) hx0.65,
hz0.49 c) hx0.53,
hz0.39 d) hx0.45, hz0.32
15
Low tune depression region dependence
onmismatch
(c/a3, hx.93, hz.87, t900) m1.2, m1.3,
m1.4, m1.6
16
Time evolution
(c/a3, m1.6, hz.87) t250, t350, t450, t550
17
Halo structure low density cut
a) no cut b) with low density cut allowed to
study onset of various order resonances and halo
structure
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Main features halo extent

• Transverse Longitudinal
• An important quantity is the ratio of the
  halo radius to that of the matched distribution
• Halo extent primarily governed by mismatch
• Small dependence on tune depression does exist
  (different for KV and non-uniform distributions).
• Longitudinal halo
• In addition to features similar to transverse
  halo, there is significant increase of halo
  extent for severe tune depressions

19
Extent of transverse halo
Extent of the transverse halo for the 6D
stationary distribution
20
Extent of longitudinal halo
Extent of the longitudinal halo for the 6D
stationary distribution
21
Extent of longitudinal halo non stationary
distributions
Extent of the longitudinal halo for the 6D
uniform distribution
22
Halo extent

• Halo extent is minimized by matching control
• Transverse halo extent is larger than that of
  longitudinal halo
• For very large mismatch parameter
• Transverse (xmax/xrms) can be as big as 5 8.
• Longitudinal (zmax/zrms) can be as big as 4
  6.
• Non-uniform distributions largest extent
• Gaussian largest extent
• No mismatch threshold on halo formation for
  non-uniform distributions.

23
Halo intensity

• Transverse and Longitudinal
• Strong dependence on mismatch
• Very weak dependence on tune depression (except
  for a very severe tune depression, below 0.5).

24
Halo onset

• Strong dependence on tune depression
• For elongated bunches Longitudinal Halo develops
  earlier than Transverse since longitudinal tune
  is more depressed than transverse.
• Strong dependence on mismatch parameter

25
Coupling effect stationary distribution
Coupling effect for the 6D stationary
distribution with zero transverse mismatch and
50 longitudinal mismatch (c/a2, hz.45, hx.55)
26
Coupling effect uniform distribution
Coupling effect for the 6D uniform distribution
with zero transverse mismatch and 50
longitudinal mismatch (c/a2, hz.45, hx.55)
27
Coupling effect Gaussian distribution
Coupling effect for the 6D Gaussian distribution
with zero transverse mismatch and 50
longitudinal mismatch (c/a2, hz.45, hx.55)
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Coupling halo extent

• Mismatch in one direction leads to both
  longitudinal and transverse halo.
• With relatively small mismatch in all directions
  one can get noticeable halo.
• Coupling effect becomes important for almost
  spherical beam with c/a
• Example Beam (c/a2) is perfectly matched
  transversely but has strong mismatch
  longitudinally - quick transverse halo and
  significant beam loss if beam pipe acceptance is
  not big enough.

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Other issues

• Loss from the RF bucket due to longitudinal halo
  stabilization of P.H. by non-linear RF.
• Loss from RF bucket could not be a problem.
  However, careful longitudinal matching is
  necessary to minimize impact on transverse halo
  when coupling is important.
• Periodic focusing channel apart from the
  instabilities due to the structure driven
  resonances, a close resemblance to the
  continuous focusing channel results.

30
Coulomb scattering halo extent

• Some numerical studies suggested that small angle
  single Coulomb scattering may not be negligible
  in high-intensity linacs
• Because of the importance of this question of
  high-intensity linacs under design, rigorous
  treatment of this effect was required.
• Self consistent 3D distribution was used to
  estimate the extent and rate of halo formation
  analytically.
• N. Pichoff numerical simulations for
  non-stationary distributions.
• Shell thickness/extent around the beam
  (populated due to scattering) depends on tune
  depression and could be significant even for
  matched equipartitioned beams. When the beam is
  nonequipartitioned or the beam with stationary
  distribution is rms mismatched, halo extent can
  be big and comparable to the one of the
  Parametric Halo.

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Coulomb scattering - rate

• Rate of this process was estimates for various
  stationary f(H0-H)n and non-stationary
  (N. Pichoff) distributions.
• Simple expression in terms of beam parameters
  were obtained for various n
• For typical beam parameters (APT) the
  fraction of ions which leaves the beam per unit
  length
• Relatively singular distribution n-1/2 -
  10-11/km
• Distribution with n0
  - 10-14/km
• Distributions with n 0
  - 10-15/km
• 
  Conclusion
  
• Effect of Coulomb collisions on halo
  development in high-current ion
  linear accelerators is not important.

32
Practical Note

• Parametric Halo is a resonant mechanism
• Thus at least few betatron oscillation (depressed
  by space charge) are required. As result, this
  mechanism is more applicable for transport
  systems with sufficient/large number of zero
  current betatron periods. Large - will depend on
  tune depression and mismatch parameter from 100s
  to just a few.
• One has to check how phase advance is changing
  along the system length and whether this change
  is adiabatic enough to support the resonant
  mechanism.

33
Parametric Halo in Linacs Rings

• Difference of halo formation between linacs and
  rings
• Linacs P.H. could be an important source of
  halo
• Rings 1. P.H. is possible should be studied
  for each individual machine
• 2. Machine resonances
• Modest mismatches
• P.H. (Linac) strong tune depressions - very
  quick development
• P.H. (Rings) low tune depression very slow
  development
• In Rings Rate of P.H. development is the most
  important question.

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Summary

• Development of parametric halo in ideal Linacs is
  well understood. Developed models in conjunction
  with multiparticle simulations give confidence in
  predictions of halo formation.
• Predictions applications to real accelerators
  are being tested in LEDA experiment.
• If developed P.H. could be the largest source
  of halo formation in Linacs since it can lead to
  a large emittance increase.
• An important question to understand is formation
  of Parametric Halo in the vicinity of
  space-charge coupling resonances (I. Hofmann et
  al.).