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INTENSITY LIMITATIONS (Space Charge and Impedance)
M. Zobov
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Is it important?
J.Bosser et al., NIM 441 (2000) 1-8
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SELF FIELDS AND WAKE FIELDS
In a real accelerator, there is another important
source of e.m. fields to be considered, the beam
itself, which circulating inside the pipe,
produces additional e.m. fields called
"self-fields
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These fields depend on the current and on the
charges velocity. They are responsible of many
phenomena of beam dynamics

• energy loss
• shift of the synchronous phase and frequency
  (tune)
• shift of the betatron frequencies (tunes)
• energy spread and emittance degradation
• instabilities.

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• What do we mean with space charge?
• It is net effect of the Coulomb interactions in
  a multi-particle system
• Space Charge Regime gt dominated by the self
  field produced by the particle distribution,
  which varies appreciably only over large
  distances compared to the average separation of
  the particles gt Collective Effects

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Example 1. Relativistic Continuous Uniform
Cylindrical
Gausss law
Linear with r
Amperes law
L. Palumbo, JUAS
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Lorentz Force

• has only radial component
• is a linear function of the transverse coordinate

The attractive magnetic force, which becomes
significant at high velocities, tends to
compensate for the repulsive electric force.
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Transverse Incoherent Effects
We take the linear term of the transverse force
in the betatron equation
The betatron shift is negative since the space
charge forces are defocusing on both planes.
Notice that the tune shift is in general function
of z, therefore there is a tune spread inside
the beam.
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Incoherent Tune Shift
The tune shift for unbunched beams in a perfectly
conducting vacuum chamber of half-height h,
between perfect magnetic pole pieces at a
distance g from the axis
For bunched beams we have to take into account a
bunching factor B defined as the ratio of average
to peak current
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Estimates for EDM Machine
P 0.7 GeV/c g 1.0674
P 1.5 GeV/c g 1.2804
! To be compared with
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1.
2. Optics functions change by changing the tune.
This leads to the size changes, i.e. collective
effects
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Wake Potentials
there can be two effects on the test charge
1) a longitudinal force which changes its
energy, 2) a transverse force which deflects
its trajectory.
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Impedance of a Step (S. Kheifets and S. Heifets)
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Limits on Impedances
The wake fields can act as a positive feedback
leading to instabilities. Nonlinearities damp
them (Landau damping)
Longitudinal
Transverse
Similar to DAFNE ?
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REFERENCES

• L. Palumbo, Space Charge Effects and
  Instabilities, JUAS, 2003.
• M. Zobov and A. Gallo, Instabilities,
  http//www.lnf.infn.it/acceleratori/dafne/seminary
  /dafne_zobov.pdf
• L. Palumbo, V. Vaccaro and M. Zobov, Wake Fields
  and Impedance, CAS CERN 95-06, 1995