Review of Accelerator Physics Concepts -...
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Review of Accelerator Physics Concepts
Jeff Eldred
Beam Loss Machine Protection Course
01/23/2017 USPAS
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Overview
1. Betatron Motion
2. Nonlinearities
3. Off-Momentum Particles
4. RF Capture
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Betatron Motion
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We can solve the linear Hill’s equation:
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Linear Focusing
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Transfer Matrices
The final position and slope is a linear combination of the
initial position and initial slope. We can use matrices:
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Example: FODO Cell
The transfer matrix for a sequence of elements can be
obtained by multiplying the matrices for the components.
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TWISS (Courant-Snyder) Parameters
The transfer matrix for a stable ring can be parametrized:
The transfer matrix for a general transfer matrix:
For example, we can calculate TWISS for a FODO ring:
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TWISS Plots
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TWISS & Betatron Oscillation
The general transfer matrix can be decomposed:
An inverse transformation, a rotation, and transformation.
This allows us to understand what the TWISS parameters
mean for the particle motion:
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Betatron Oscillation
Emittance:
Spot size:
Relativistic Adiabatic
Damping:
Wille
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Betatron Phase-space
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Betatron Motion
Bartolini
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Betatron Phase Advance
Betatron Tune:
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The correlation between X1 and X2, can be used to see the
betatron phase advance between those points.
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Two BPM Measurement
S.Y. Lee
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Nonlinearities
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Betatron Tune Resonance
Perturbation will accumulate if tune is a
fraction corresponding to the symmetry
of the applied fields.
Barletta S.Y. Lee
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Tune Diagrams The tune is carefully
picked to avoid
resonances.
The tune for the beam
occupies a finite space:
- Laslett space-charge
tune spread.
- Amplitude-dependent
tune shift.
- Tune-change during
acceleration.
- Chromaticity
- beam-beam effects
Barletta
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Phase-space Distortions
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Nonlinear Decoherence
Kain
Injection errors,
instabilities, and
sudden lattice changes
may cause a phase-
space mismatch.
Tune spread causes
the beam to fill-out
along the phase-space
contours.
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Off-Momentum Particles
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Dispersion
Dispersion:
Spot Size:
Barletta
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Chromaticity
Chromaticity:Change in tune with momentum:
Barletta
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Sextupoles & Chromaticity Correction
Chromaticity is tune
dependence on
momentum.
Sextupoles provide
tune-shift depending
on position offset.
Dispersion is position
offset dependence on
momentum.
FNAL Rookie Book
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RF Capture
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Energy in one pass through cavity
FNAL Rookie Book
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Phase-Slip Factor η
The arrival time of the particle depends on the momentum:
Higher momentum particles may arrive earlier or later than
lower momentum particles:
We can write the change in phase per unit time using the
phase-slip factor:
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Longitudinal Focusing
Synchrotron Freq.
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Oscillatory & Slipping Motion
The further off momentum, the faster the slipping motion.
Lost particles rapidly decohere from each other.
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• Particles in the bucket can be accelerated by adiabatically
changing the RF frequency.
• Particles outside the bucket are left behind, decelerated
relative to the moving reference frame.
RF Acceleration
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X. Kang
SY. Lee
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Transition Crossing