2 Beam Dynamics
41
the emittance boundary used to be conventionally chosen to be that of a proton
with amplitude 2σ. This would include about 90% (strictly 87%) of a Gaussian
beam where σ is the standard distribution. In an electron machine a 2σ boundary
would be too close to the beam and an aperture stop placed at this distance would
rather rapidly absorb most of the beam as particles redistribute themselves, moving
temporarily into the tails due to quantum emission and damping. The safe physical
boundary for electrons depends on the lifetime required but is in the region of 6σ
to 10σ. The emittance which is normally quoted for an electron beam corresponds
to an electron with the amplitude of σ in the Gaussian projection. We are then free
to choose how many σ’s we must allow. There is consequently a factor 4 between
emittance defined by electron and proton experts.
2.4 Momentum Dependent Transverse Motion
In the previous chapters, we have studied the motion of a particle as it swings from
side to side about the ideal orbit around the synchrotron: the transverse motion.
Nothing is perfect, however, and so we cannot assume that each and every proton
in a large ensemble of up to 10 11 particles will have exactly the ideal momentum.
Instead we expect a certain momentum spread in the beam and therefore we have
to study how transverse motion depends on small departures, p/p 0 , from the
synchronous momentum p 0 .
2.4.1 Dispersion
The central closed orbit of a synchrotron is matched to an ideal (synchronous)
momentum p 0 . A particle of this momentum and of zero betatron amplitude will
pass down the centre of each quadrupole, be bent by exactly 2π by the bending
magnets in one turn of the ring and remain synchronous with the r.f. frequency. Its
path is called the central (or synchronous) momentum closed orbit. In Fig. 2.8 this
ideal orbit is the horizontal axis and we see particles executing betatron oscillations
about it but these oscillations do not replicate every turn. The synchronous orbit,
however, closes on itself so that x and x remain zero.
We now consider a closed orbit which is distorted in the horizontal plane by
non-ideal bends in the dipole. Figure 2.20 shows a particle with a lower momentum
p/p < 0 and which is bent horizontally more in each dipole of a FODO lattice. We
could argue that the total deflection, being more than 2π would cause it to spiral
inwards and hit the vacuum chamber wall. On the other hand there is a closed orbit
for this lower momentum in which the extra bending forces are compensated by
extra focusing forces as the orbit is displaced inwards in the F quadrupoles and
less so in the defocusing in the D’s (Fig. 2.20). We may describe the shape of
this new closed orbit for a particle of unit p/p by a dispersion function D(s). The
41
the emittance boundary used to be conventionally chosen to be that of a proton
with amplitude 2σ. This would include about 90% (strictly 87%) of a Gaussian
beam where σ is the standard distribution. In an electron machine a 2σ boundary
would be too close to the beam and an aperture stop placed at this distance would
rather rapidly absorb most of the beam as particles redistribute themselves, moving
temporarily into the tails due to quantum emission and damping. The safe physical
boundary for electrons depends on the lifetime required but is in the region of 6σ
to 10σ. The emittance which is normally quoted for an electron beam corresponds
to an electron with the amplitude of σ in the Gaussian projection. We are then free
to choose how many σ’s we must allow. There is consequently a factor 4 between
emittance defined by electron and proton experts.
2.4 Momentum Dependent Transverse Motion
In the previous chapters, we have studied the motion of a particle as it swings from
side to side about the ideal orbit around the synchrotron: the transverse motion.
Nothing is perfect, however, and so we cannot assume that each and every proton
in a large ensemble of up to 10 11 particles will have exactly the ideal momentum.
Instead we expect a certain momentum spread in the beam and therefore we have
to study how transverse motion depends on small departures, p/p 0 , from the
synchronous momentum p 0 .
2.4.1 Dispersion
The central closed orbit of a synchrotron is matched to an ideal (synchronous)
momentum p 0 . A particle of this momentum and of zero betatron amplitude will
pass down the centre of each quadrupole, be bent by exactly 2π by the bending
magnets in one turn of the ring and remain synchronous with the r.f. frequency. Its
path is called the central (or synchronous) momentum closed orbit. In Fig. 2.8 this
ideal orbit is the horizontal axis and we see particles executing betatron oscillations
about it but these oscillations do not replicate every turn. The synchronous orbit,
however, closes on itself so that x and x remain zero.
We now consider a closed orbit which is distorted in the horizontal plane by
non-ideal bends in the dipole. Figure 2.20 shows a particle with a lower momentum
p/p < 0 and which is bent horizontally more in each dipole of a FODO lattice. We
could argue that the total deflection, being more than 2π would cause it to spiral
inwards and hit the vacuum chamber wall. On the other hand there is a closed orbit
for this lower momentum in which the extra bending forces are compensated by
extra focusing forces as the orbit is displaced inwards in the F quadrupoles and
less so in the defocusing in the D’s (Fig. 2.20). We may describe the shape of
this new closed orbit for a particle of unit p/p by a dispersion function D(s). The