Synchrotron Motion
245
It is clear that the transit time factor is the ratio of the energy gain of a RF
cavity to that of a DC (direct current) gap of the same field. The relation
between the transit time factor and the length of the cavity is shown in Fig.
10.2. It is obvious that T → 1 as l → 0 which means that, for constant voltage
between the gap, the shorter the gap, the closer the energy gain to that of
the DC gap. Yet the electric field breakdown limit of the material determines
the maximum field that can be achieved, thus the energy gain is proportional
to lT, which is in turn proportional to sin(πl/β 0 λ). Therefore, the maximum
energy gain for the case of constant field is obtained when l = β 0 λ/2. For
electron storage rings such as those of the synchrotron light sources, β 0 ≈ 1.
So we have l = λ/2, which corresponds to the fact that the time an electron
takes to pass through the cavity equals half of the period of the oscillation.
The transit time factor is T = 2/π = 0.637. For a 500 MHz cavity, we have
l = 0.3 m.
In addition, other issues such as RF power efficiency also have to be taken
into account. The most used parameter measuring the efficiency is called the
shunt impedance, which is defined as
R s =
(ΔV )
2
P d
,
where
ΔV = E 0 lT cos (φ 0 ) ,
which is the voltage across the accelerating gap and P d is the power dissipated
in the wall. As a result, realistic normal conducting cavities are more or less
spherical in shape, minimizing the total surface area, with nose cones around
the beam axis to reduce the length (acceleration gap) of the cavity, maximizing
the voltage across the gap. For superconducting cavities, the dissipated power
is much smaller and hence the main goal of design optimization shifts to
minimizing the peak field on the surface for a given on-axis field to reduce the
risk of costly quench. The resulting shape is basically the bell shaped cavity,
which is preferable for other practical reasons as well.
10.2 The Phase Slip Factor
Toward the end of the previous section, we studied the energy gain per pass
of one particle. In this section, we will study the energy gain of a particle
over many passes in a circular accelerator. Since the cavity is always designed
for a given accelerator, there is at least one particle (the reference particle)
that comes back to the cavity at the same phase (synchronous phase φ s )
every turn. For an arbitrary particle, it may not come back to the cavity
at the same phase since it may have different energy. Although an arbitrary
245
It is clear that the transit time factor is the ratio of the energy gain of a RF
cavity to that of a DC (direct current) gap of the same field. The relation
between the transit time factor and the length of the cavity is shown in Fig.
10.2. It is obvious that T → 1 as l → 0 which means that, for constant voltage
between the gap, the shorter the gap, the closer the energy gain to that of
the DC gap. Yet the electric field breakdown limit of the material determines
the maximum field that can be achieved, thus the energy gain is proportional
to lT, which is in turn proportional to sin(πl/β 0 λ). Therefore, the maximum
energy gain for the case of constant field is obtained when l = β 0 λ/2. For
electron storage rings such as those of the synchrotron light sources, β 0 ≈ 1.
So we have l = λ/2, which corresponds to the fact that the time an electron
takes to pass through the cavity equals half of the period of the oscillation.
The transit time factor is T = 2/π = 0.637. For a 500 MHz cavity, we have
l = 0.3 m.
In addition, other issues such as RF power efficiency also have to be taken
into account. The most used parameter measuring the efficiency is called the
shunt impedance, which is defined as
R s =
(ΔV )
2
P d
,
where
ΔV = E 0 lT cos (φ 0 ) ,
which is the voltage across the accelerating gap and P d is the power dissipated
in the wall. As a result, realistic normal conducting cavities are more or less
spherical in shape, minimizing the total surface area, with nose cones around
the beam axis to reduce the length (acceleration gap) of the cavity, maximizing
the voltage across the gap. For superconducting cavities, the dissipated power
is much smaller and hence the main goal of design optimization shifts to
minimizing the peak field on the surface for a given on-axis field to reduce the
risk of costly quench. The resulting shape is basically the bell shaped cavity,
which is preferable for other practical reasons as well.
10.2 The Phase Slip Factor
Toward the end of the previous section, we studied the energy gain per pass
of one particle. In this section, we will study the energy gain of a particle
over many passes in a circular accelerator. Since the cavity is always designed
for a given accelerator, there is at least one particle (the reference particle)
that comes back to the cavity at the same phase (synchronous phase φ s )
every turn. For an arbitrary particle, it may not come back to the cavity
at the same phase since it may have different energy. Although an arbitrary