Chapter 8
The Periodic Transport
In the case of the periodic transport over long distances, the desire is not
so much to give a special shape to the beam as the beam exits, but, even
much more simply, to just contain the beam. This is of key importance
in all devices in which the beam repeatedly passes through the same (or a
very similar) structure. We may wonder whether this again translates into
the requirement that a certain matrix element vanish, but as we shall see, this
is not quite the case.
Actually it is rather straightforward to formulate a necessary condition on
the linear matrix: it is not allowed to have any eigenvalue of magnitude greater
than unity. If the eigenvalue is real, the argument is simple: if this were the
case, any particle that has its coordinates lined up with the corresponding
real eigenvector will after one period end up on the same line, but all its
coordinates would have increased by a factor equal to the eigenvalue.
If on the other hand the eigenvalue is complex, there is another eigenvalue
that is conjugate and hence has the same magnitude. Similar to the eigenvalues, also the eigenvectors are conjugates of each other. Now simply consider
the sum of the two eigenvectors, which is real; sending this sum through the
matrix multiplies the first eigenvector by the first eigenvalue, and the second
one by the conjugate, resulting in a sum that is again real and increased in
size by the magnitudes of the eigenvalues.
In both cases, coordinates grow exponentially in time, and so eigenvalues that are even only a tiny amount above unity in magnitude are detrimental. Of course the nonlinear effects also influence the motion and break the
purely exponential pattern, but all experience shows that it is not possible
to correct linear instability with nonlinear means; in practice, things usually
work quite to the contrary.
8.1 The Transversal Motion
8.1.1 The Eigenvalues
Because of emittance preservation due to Liouville’s theorem, the fact that
eigenvalues greater than unity are prohibited means that, in fact, all eigen189
DOI:10.1201/b12074-8
The Periodic Transport
In the case of the periodic transport over long distances, the desire is not
so much to give a special shape to the beam as the beam exits, but, even
much more simply, to just contain the beam. This is of key importance
in all devices in which the beam repeatedly passes through the same (or a
very similar) structure. We may wonder whether this again translates into
the requirement that a certain matrix element vanish, but as we shall see, this
is not quite the case.
Actually it is rather straightforward to formulate a necessary condition on
the linear matrix: it is not allowed to have any eigenvalue of magnitude greater
than unity. If the eigenvalue is real, the argument is simple: if this were the
case, any particle that has its coordinates lined up with the corresponding
real eigenvector will after one period end up on the same line, but all its
coordinates would have increased by a factor equal to the eigenvalue.
If on the other hand the eigenvalue is complex, there is another eigenvalue
that is conjugate and hence has the same magnitude. Similar to the eigenvalues, also the eigenvectors are conjugates of each other. Now simply consider
the sum of the two eigenvectors, which is real; sending this sum through the
matrix multiplies the first eigenvector by the first eigenvalue, and the second
one by the conjugate, resulting in a sum that is again real and increased in
size by the magnitudes of the eigenvalues.
In both cases, coordinates grow exponentially in time, and so eigenvalues that are even only a tiny amount above unity in magnitude are detrimental. Of course the nonlinear effects also influence the motion and break the
purely exponential pattern, but all experience shows that it is not possible
to correct linear instability with nonlinear means; in practice, things usually
work quite to the contrary.
8.1 The Transversal Motion
8.1.1 The Eigenvalues
Because of emittance preservation due to Liouville’s theorem, the fact that
eigenvalues greater than unity are prohibited means that, in fact, all eigen189
DOI:10.1201/b12074-8