Synchrotron Motion
243
For f = 500 MHz, R c = 0.2295 m. In realistic cavity designs, the actual
shape of the cavity is often more spherical than cylindrical. Yet the overall
dimension is not very far from this crude estimate.
For r R c , which is usually where the beam is, the field can be approximated by the lowest order term of the Taylor expansion, which is
E z = 1 E 0 cos (ωt) , B θ = 1 −
E 0
c
x 01 r
2R c
sin (ωt) .
(10.1)
Note that J 0 (x) = 1+O
x
2
and J 1 (x) = x/2+O
x
3
for x 1. As a result,
the effect of B θ is much weaker than that of E z on the beam. Furthermore,
the focusing effect of the magnetic field is usually negligible compared to main
focusing elements, the quadrupole magnets in the ring.
To illustrate this, let us look at an example. Let us consider again the case
of a 500 MHz cavity. Assuming that E 0 = 20 MV/m, which is not far from
the breakdown limit of copper at this frequency, we obtain the peak gradient
of the magnetic field, which is 0.35 T/m. Normal conducting quadrupoles, on
the other hand, can have field gradient up to 20 T/m. Furthermore, there are
usually tens to hundreds of quadrupoles with lengths between 0.2 and 1 m in
a ring, whereas there are at most a handful of cavities with lengths usually
below 0.5 m (around 0.3 m for a 500 MHz pillbox cavity). As a result, the
integrated gradient of the cavities is on the order of up to perhaps 1/1000
that of the quadrupoles.
Recently, the dipole mode (TM 110 ) has also been used to kick the beam
transversely. The field of TM 110 is
E z = E 0 J 1
x 11 r
R c
cos θ cos (ωt) ,
E r = 0, E θ = 0,
B z = 0,
B r =
E 0
c
R c
x 11 r
J 1
x 11 r
R c
sin θ sin (ωt) ,
B θ =
E 0
c
J 0
x 11 r
R c
−
R c
x 11 r
J 1
x 11 r
R c
cos θ sin (ωt) ,
where the relation J
1 (x) = J 0 (x) − J 1 (x)/x is used to obtain B θ . Note
that x 11 = 3.832. For the same cavity, the frequency of the TM 110 mode
is x 11 /x 01 ≈ 1.6 times that of the TM 010 mode. In order to get a clearer
physical picture of the effect of the field on the beam, let us again perform
Taylor expansion around the origin and keep only the leading term. The field
is
E z = 1 E 0
x 11 r
2R c
cos θ cos (ωt) ,
B r = 1
E 0
2c
sin θ sin (ωt) , B θ = 1
E 0
2c
cos θ sin (ωt) .
243
For f = 500 MHz, R c = 0.2295 m. In realistic cavity designs, the actual
shape of the cavity is often more spherical than cylindrical. Yet the overall
dimension is not very far from this crude estimate.
For r R c , which is usually where the beam is, the field can be approximated by the lowest order term of the Taylor expansion, which is
E z = 1 E 0 cos (ωt) , B θ = 1 −
E 0
c
x 01 r
2R c
sin (ωt) .
(10.1)
Note that J 0 (x) = 1+O
x
2
and J 1 (x) = x/2+O
x
3
for x 1. As a result,
the effect of B θ is much weaker than that of E z on the beam. Furthermore,
the focusing effect of the magnetic field is usually negligible compared to main
focusing elements, the quadrupole magnets in the ring.
To illustrate this, let us look at an example. Let us consider again the case
of a 500 MHz cavity. Assuming that E 0 = 20 MV/m, which is not far from
the breakdown limit of copper at this frequency, we obtain the peak gradient
of the magnetic field, which is 0.35 T/m. Normal conducting quadrupoles, on
the other hand, can have field gradient up to 20 T/m. Furthermore, there are
usually tens to hundreds of quadrupoles with lengths between 0.2 and 1 m in
a ring, whereas there are at most a handful of cavities with lengths usually
below 0.5 m (around 0.3 m for a 500 MHz pillbox cavity). As a result, the
integrated gradient of the cavities is on the order of up to perhaps 1/1000
that of the quadrupoles.
Recently, the dipole mode (TM 110 ) has also been used to kick the beam
transversely. The field of TM 110 is
E z = E 0 J 1
x 11 r
R c
cos θ cos (ωt) ,
E r = 0, E θ = 0,
B z = 0,
B r =
E 0
c
R c
x 11 r
J 1
x 11 r
R c
sin θ sin (ωt) ,
B θ =
E 0
c
J 0
x 11 r
R c
−
R c
x 11 r
J 1
x 11 r
R c
cos θ sin (ωt) ,
where the relation J
1 (x) = J 0 (x) − J 1 (x)/x is used to obtain B θ . Note
that x 11 = 3.832. For the same cavity, the frequency of the TM 110 mode
is x 11 /x 01 ≈ 1.6 times that of the TM 010 mode. In order to get a clearer
physical picture of the effect of the field on the beam, let us again perform
Taylor expansion around the origin and keep only the leading term. The field
is
E z = 1 E 0
x 11 r
2R c
cos θ cos (ωt) ,
B r = 1
E 0
2c
sin θ sin (ωt) , B θ = 1
E 0
2c
cos θ sin (ωt) .