Resonances and Equipartition in High Intensity Beams

1
Resonances and Equipartition in High Intensity
Beams
I. Hofmann, GSI Oak Ridge, May 17, 2001
1. Thermodynamics and Linacs 2. Collective
Resonance Model 3. Application to high-current
linac projects (CERN,
SNS) Acknowledgment J. Qiang, R. Ryne, Los
Alamos, O. Boine-Frankenheim, G. Franchetti,
GSI F. Gerigk, CERN
2
Acceleration leads to anisotropy

• Beams from source plasma are isotropic and
  Maxwell-Boltzmann
• Longitudinally colder by acceleration
• Distribution changes by scraping, capturing in RF
  buckets, change of focusing, mismatch etc.
• Anisotropic beams may result - do they persist as
  such under space charge?

3
Does the system return to equipartition?
Reisers book this equipartitioning process is
particularly strong in high-current linacs -gt
design linac as equipartitioned!
Kishek, OShea, Reiser, PRL 85, 2000 beams
naturally want to equipartition
Why and how?
1) Collisional relaxation
Linacs are too short!
4
Is thermodynamics applicable to beams?
Coulomb range

• Thermodynamic limit for systems with
  long-range forces questionable need
  sizegtgtrange
• for Coulomb interaction only Debye screening can
  lead to a short-range force
• lDebye /a n / n0 ltlt 1 only at space charge
  limit, hence usual bunch does not fall into this
  category
• real one-component plasmas with thermodynamics
  valid in Penning traps (ONeill, Physics Today,
  February1999)

5
Arnold The only computer experiments worth
doing our those that yield a surprise
2) Nonlinear resonances
Fermi-Pasta-Ulam s classical experiment on
MANIAC-I chain of N-1 (periodic) mass points to
study equilibrium statistical mechanics
E .Fermi, J.Pasta, S. Ulam and M. Tsingou,
LASL-report, 1955
recurrence
why recurrence, and no energy in modes 632
? Chirikov resonance overlap needed for chaos
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3) Collective resonance model of anisotropic beams
I.H., IEEE Trans. Nucl. Sci. NS-28, 1981
(borrowed idea from Harris modes in infinite
plasmas) R.A. Jameson confirmed in linac
simulations (gt1981) I.H., Phys. Rev. E 57
(1998), I.H., J. Qiang, R. Ryne, PRL 86
(2001)

• Resonances evolving from collective motion
  space charge non-linearity
• Analytical perturbation model in 2D
  (KV-distribution)
• Predictions confirmed by 2D and 3D simulation
  closed gap to real world
  linacs
• Recently also discussed in context of halo
  formation in rings (2D)

7
What is an anisotropic equilibrium solution?

• Isotropic equilibria can be expressed as function
  of the 2D or 3D Hamiltonian (constant focussing)
  f f0 (H0)
• With anisotropy we use as solution in 2D the
  anisotropic KV-distribution (the only one
  known!)
• f0 d (H0x T H0y - mg nx2a2/2)
• with
• H0x (px2 m2 g2 nx2x2)/(2 mg)
• H0y (py2 m2 g2 ny2y2)/(2 mg)
• which implies uniform density within (x/a)2
  (y/a)2 lt1 and
• the anisotropy parameter T E x/ E y nx2a2/
  ny2b2 exnx / eyny
• The unperturbed equilibrium distribution is
  described by 3 parameters, which can be chosen as
  ex/ ey , nx/ny, ny/ny0 (in 1D only n/n0 in fully
  3D need 5!)

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Vlasov equation to describe evolution of
collisionless beams in 2D
The time-dependent Vlasov equation df/dt 0 can
be linearized and solved with Poissons equation
for the perturbed charge densities (space charge
potentials as polynomials)
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Eigenmodes are multipole oscillations in density

• for l3,4 densities are non-uniform and space
  charge potentials correspond to sextupole or
  octupole magnets
• the theory predicts exponential growth of
  infinitesimal initial perturbations for certain
  parameters

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Dispersion Relation has been solved for l 2, 3,
4
The coherent frequency sw/nx is solved as
function of the 3 parameters any/nx, ha/b,
spwp/nx with wp2q2N/(e0pmg3 ab)
Im sgt0 instability
Example l3, even
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Charts for given ey/ex indicate regions with
instability
ey/ex 2
ny-2nx0
3rd

• large areas free from instabilities, depending
  on tune ratio ny /nx
• for strong tune depression sea of instability
  (nx /n0x lt0.3)

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2ny-2nx0
ny-3nx0
Calculated in I.H., Phys. Rev. E 57
(1998) Further studies since rich in phenomena

4th
13
31
22
EP
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Applied to Linac fully 3D simulation

• using IMPACT code (R. Ryne, J. Qiang, LANL) in
  constant focusing
• (I.H., J. Qiang, R. Ryne, PRL 86 (2001))
• initial rms-matched waterbag distribution
  (realistic)
• 2-10 million simulation particles on 64x64x64
  grid

T4.1 2.8
ez/ex 2
T5.9 1.6
22
21
I.H., J. Qiang, R. Ryne, PRL 86 (2001)
13
Initial KV-beam subject to 3rd order resonant
instability
14
3D unexpected departure from initial transverse
equipartition
x
z
z
y
x,y
15
focusing strength in z and x, y equal -gt sc
linac
ez/ex 2
T2.3 -gt 1.3
22
We recently started (motivated by SNS) a refined
and more systematic scan of parameter space
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Studied 22 in detail using 2D and ny /ny00.8,
ex/ey 2
Montague 1968 split tunes in rings to avoid
2Qx-2Qy0 (IH et al., PRL acc.)
KV- theory
ny /ny00.8
KV
Non-oscillating Re w 0
KV
WB
WB overshoot beyond EP
WB
nx /ny
nx0 /ny01
KV exponential growth takes time if it grows
from noise! helps in linacs!
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Simulation is strongly supporting the thesis
Stopbands of unstable non-oscillating (only!) KV
modes characterize behavior of real beams -gt
basis for our resonance charts
Likely reason oscillating KV-instabilities are
Landau-damped in real beams
ny /ny00.5 shows same profile for relative
emittance change
New charts including rates (0.053 betatron
periods)
ex/ey 2
rates
Non-oscillating
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Recent scans in parameter space on 40 Galaxie
processors at BNLto check if hidden further
resonances (with G. Franchetti)
ex/ey 2
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21
13
ex/ey 2
Clear evidence that no resonances higher than 4th
order! (except for KV-beam, where 5th order 32
and 23 were found )
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Ring space charge issues near 22
resonanceconstant focusing
KV
WB
ex/ey 1
ny /ny00.95

• Discussed in paper by D. Jeon et al.,
• Recently explained by A. Fedotov et al. in terms
  of these KV modes in SNS lattice

halo
Actual exchange cannot be derived from a single
particle approach (ring Montague/1968, linac
Lagniel/1999)
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In 22 resonance different emittances cause
resonant exchange
ex/ey 1.2
Returns to ex/ey 1
WB
Relates to tune splitting needed to enforce or
avoid exchange after injection
KV-theory non-oscillating modes
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Systematic comparison for SNS-parameters
Nominal lo/tr emittance ratio 1.2
EP
EP
EP
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Some questions

• resonances vanish if equipartitioned - near
  equipartition largest resonance-free regions
• effect of periodic focusing (probably small)
• need to check which stop-bands are really
  dangerous over linac length
• linac design
• consider possible upper and lower bounds of
  emittance

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CERN SPL and the Neutrino Factory
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The SPL Design
45 keV 7 MeV
120 MeV 1.08 GeV
2.2 GeV
13m
78m
334m
357m
2 MeV
18MeV
237MeV 389MeV
-
b
b
b
H
RFQ1 chop. RFQ2
RFQ1 chop. RFQ2
RFQ1 chop. RFQ2
DTL
CCDTL
RFQ1 chop. RFQ2
0.52
0.7
0.8
LEP-II
dump

b
b
Source Low Energy section
DTL
Superconducting
1
Superconducting low-
785 m
Stretching and
collimation line
PS / Isolde
Accumulator Ring
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Chart for the SPL (emittance ratio of 2)
growth rate
SPL IIb
case 3
case 2
case 1
emittance ratio2
F. Gerigk/CERN
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3D IMPACT full linac simulation confirms 22
resonance
ez
ez
ex,y
ex,y
ez
ez
ex,y
ex,y
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Mismatch 1.3 rx, 0.7 ry
ez
ez
ex,y
ex,y
ez
ez
ex,y
ex,y
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Tested on SNS (preliminary)
Have raised nominal lo/tr emittance ratio from
1.2 to 2
Found predicted exchange of emittances ratio
returns to 1.4, but 30 increase of both
transverse emittances in DTL further effects (?)
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To explore mismatch we conduct scans on charts
for different emittances first in 2D
ex/ey 1 MM1.3/1.3
22
breathing mode
X-max/X-rms
Need to explore also resonant behavior of
emittance and halo growth due to mismatch
2D-scans -gt 3D -gt Linac code
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Conclusions

• Collective resonance model KV-based charts well
  confirmed by simulation in 2D and 3D
• Only low-order resonances of concern (3rd and
  4th) free areas of non-equipartitioned beam in
  between!
• For sc linacs mainly 22 resonance (nz0 / nx0
  1) of concern suppressed by EP, but also weak
  longitudinal focusing in DTL
• exchange not necessarily reversible!
• need to include mismatch driven emittance growth
• Require realistic modeling of mismatch

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Linac space-charge resonancesmain candidate is
avoided in SNS for nominal emittance ratio 1.2
EP
ez/ex 2
kx/kox
ez
kz/kx
ez/ex 1.2 Nominal SNS
Emittance exchange
ex
IMPACT code (Qiang/Ryne, LANL) in constant
focusing
Analytical Resonance Chart
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Examples by R. Jameson
longitudinal
transverse
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3D stability charts

• Found very similar topology of stable and
  unstable regions
• confirm resonance pattern
• all unstable insea of instability
• checked some vs. semi-Gauss

ez /ex2
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Choice of parameters

• 10 high energy muons/year ? 4 MW beam power on
  target
• 2.2 GeV above pion production threshold, few
  space charge problems in accumulator
• 2.2 ms pulse length ? 660 injection turns for
  accumulator (no instabilities)
• 75 Hz repetition rate (? 13.3ms) ? below
  mechanical resonances of SC LEP cavities
• 5/8 micropulse structure ? lossfree ring injection

? 11mA pulse current, 18 mA bunch current, 16.5
duty cycle
13.3 ms
5.60 cm
? linac macropulse
.80 cm
2.2 ms
22.7 ns
1.30 cm
14.2 ns
.80 cm
linac micropulse ?

5 bunches
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emittance for 1.3 rx,y, 0.7 rz
ez
ez
ex,y
ex,y
ez
ez
ex,y
ex,y
37
tune ratios
full current
full current
zero current
zero current
full current
full current
zero current
zero current
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Discussion
T E z/ E x ez,nnz / ex,nnx 1
equipartition - can this be satisfied
easily? Reisers book nz0 (b0g0) -3/2
nx0 (sx0 / b0l) b0 nz0 / nx0
b0-1/2 g0 -3/2 decreases with
acceleration -gt beam gets transversely hotter
maintaining equipartition requires reducing sx0
with energy as 1/(b0g03 ) 1/2 -gt bore radius
should increase - undesirable -gt nz0 should
increase (frequency doubling)? dont enforce
equipartition!
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focusing in z and x, y equal -gt sc linac
T2.3 -gt 1.3
ez/ex 2
nz/nx