2 Beam Dynamics
35
Fig. 2.14 Coupled betatron oscillations for 1/|Q H − Q v | turns
2.3 Liouville’s Theorem
Now let us return to the ‘mainstream’ of transverse dynamics. Liouville’s theorem is
a conservation law that applies to the area occupied by a number of particles plotted
in phase space.
We should think of a beam of particles as a cloud of points within a closed
contour in a transverse phase space diagram (Fig. 2.16). Liouville’s theorem tells
us that this area within the contour is conserved. The contour is usually, but not
always, an ellipse. In Fig. 2.5 we came across such an elliptical contour—the locus
of a particle’s motion plotted in phase space (x,x ) and we call its area, the emittance.
We could also think of it as a limiting contour enclosing all the particles in the beam
which we would again call the emittance—not of the particle but of the beam as a
particle ensemble.
We express beam emittance in units of π mm·milliradians. According to
Liouville the emittance area will be conserved as the beam passes down a transport
line or circulates in a synchrotron whatever magnetic focusing or bending operation
we do on the beam—provided that only conservative forces are taken into account.