44
E. Wilson and B. J. Holzer
Fig. 2.22 The limiting trajectory for a particle in a ‘moving’ or accelerating bucket when the
stable phase is not zero
of this longitudinal movement in phase space is closed and over many turns the
average deviation from the synchronous energy is zero. This phase stability depends
upon the fact that δE is positive when φ − φ s is small and positive [13, 14].
When a particle reaches the non-linear part of the r.f. wave and over the top of
the wave, it will still be restored and oscillate about the stable phase provided it
does not reach and pass the point where it receives less incremental energy than the
synchronous particle. On this non-linear part of the curve the motion is no longer
an ellipse but is distorted into a fish-shape but its trajectory is still closed and stable.
However, if a particle, C, oscillates with such large amplitude that it falls below
the synchronous voltage, an increase in φ will cause a negative which in turn
causes φ to move further away from the synchrotron particle. This particle is clearly
unstable and will be continuously decelerated. There is a particle which, starting at
φ = π − φ s , would trace out a limiting fish-shaped trajectory which is the boundary
or separatrix between stable and unstable motion. The region within this separatrix
is called the r.f. bucket and is shown in the lower half of Fig. 2.22. Formulae for the
calculation of the parameters of moving buckets are to be found in [15].
Let us look more carefully at the argument that a particle, arriving late because
of its lower energy, would see a higher RF voltage from the rising waveform and,
accelerated to a higher velocity, would catch up with the synchronous particle.
Dispersion may make the situation more complicated. Giving the errant particle
more energy will speed it up but may also send it on an orbit of larger radius.
The path length that the particle, B, must travel around the machine, or more
correctly, the change in path length with momentum, must depend upon the
dispersion function. The closed orbit will have a mean radius:
R = R 0 + D
Δp
p
.
(2.62)
Close to the velocity of light where acceleration can increase momentum but not
velocity, the longer path length will more than cancel the small effect of velocity
and the particle, instead of catching up with its synchronous partner, will arrive
E. Wilson and B. J. Holzer
Fig. 2.22 The limiting trajectory for a particle in a ‘moving’ or accelerating bucket when the
stable phase is not zero
of this longitudinal movement in phase space is closed and over many turns the
average deviation from the synchronous energy is zero. This phase stability depends
upon the fact that δE is positive when φ − φ s is small and positive [13, 14].
When a particle reaches the non-linear part of the r.f. wave and over the top of
the wave, it will still be restored and oscillate about the stable phase provided it
does not reach and pass the point where it receives less incremental energy than the
synchronous particle. On this non-linear part of the curve the motion is no longer
an ellipse but is distorted into a fish-shape but its trajectory is still closed and stable.
However, if a particle, C, oscillates with such large amplitude that it falls below
the synchronous voltage, an increase in φ will cause a negative which in turn
causes φ to move further away from the synchrotron particle. This particle is clearly
unstable and will be continuously decelerated. There is a particle which, starting at
φ = π − φ s , would trace out a limiting fish-shaped trajectory which is the boundary
or separatrix between stable and unstable motion. The region within this separatrix
is called the r.f. bucket and is shown in the lower half of Fig. 2.22. Formulae for the
calculation of the parameters of moving buckets are to be found in [15].
Let us look more carefully at the argument that a particle, arriving late because
of its lower energy, would see a higher RF voltage from the rising waveform and,
accelerated to a higher velocity, would catch up with the synchronous particle.
Dispersion may make the situation more complicated. Giving the errant particle
more energy will speed it up but may also send it on an orbit of larger radius.
The path length that the particle, B, must travel around the machine, or more
correctly, the change in path length with momentum, must depend upon the
dispersion function. The closed orbit will have a mean radius:
R = R 0 + D
Δp
p
.
(2.62)
Close to the velocity of light where acceleration can increase momentum but not
velocity, the longer path length will more than cancel the small effect of velocity
and the particle, instead of catching up with its synchronous partner, will arrive