conventional acceleration 87
if we assume that the following condition is satisfied
λ
e =
z
q
with q = 0, 1, 2, ...
(5.16)
2
Therefore, only certain well-defined wavelengths λ are
present in the cavity.
Near the resonant wavelength, the resonant cavity behaves like an oscillator with a high quality factor Q, allowing it
to build up high voltages that can be used for particle acceleration. The cavities are often modeled as electrical oscillators,
with their Q-value determined by losses of equivalent individual coils, capacitors and resistances of the circuit model.
5.3.2 Pill-box cavity
An enclosed section of a waveguide (either rectangular or
cylindrical) forms the simplest RF cavity, called a pill-box cavity.
The conventionally accepted classification of the modes
in pill-box cavities separates the cases of transverse electric or
TE modes (zero electric field along the axis) and transverse
magnetic or TM modes (zero magnetic field along the axis).
As Eq.5.16 suggests, many modes can exist in a cavity, as
defined by the corresponding dimension of the pill-box, and
the integer number of the mode. The corresponding integer
indexes are used to identify a particular mode. In a rectangular pill-box, a mode can be called TE klm or TM klm where
the integer indexes indicate the number of half-wavelength
variations across the corresponding dimension (x, y, z) of the
cavity.
Cylindrical pill-box cavities are very common in accelerators. An example of a cylindrical pill-box cavity with holes
for the beam is shown in Fig.5.22. The classification of the
modes in cylindrical cavities is very similar, TE klm or TM klm ,
but in this case the indexes refer to the polar coordinates (ϕ,
r and z).
Examples of modes in cylindrical pill-box cavities, with
longitudinal electric fields and no variation over ϕ (TM 0lm ),
which are therefore suitable for use as an accelerating cavity,
are shown in Fig.5.23.
5.3.3 Quality factor of a resonator
The quality factor of a resonator — Q — is defined as the ratio
of the energy stored in the cavity to the energy dissipated per
oscillating cycle, divided by 2π
W
Q =
s
W
= ω
s
(5.17)
W d
P d
]
70
]
70
FIGURE 5.23
Examples of pill-box cylindrical cavity modes with electric field lines shown.
K
U
]
FIGURE 5.22
Cylindrical pill-box cavity.
if we assume that the following condition is satisfied
λ
e =
z
q
with q = 0, 1, 2, ...
(5.16)
2
Therefore, only certain well-defined wavelengths λ are
present in the cavity.
Near the resonant wavelength, the resonant cavity behaves like an oscillator with a high quality factor Q, allowing it
to build up high voltages that can be used for particle acceleration. The cavities are often modeled as electrical oscillators,
with their Q-value determined by losses of equivalent individual coils, capacitors and resistances of the circuit model.
5.3.2 Pill-box cavity
An enclosed section of a waveguide (either rectangular or
cylindrical) forms the simplest RF cavity, called a pill-box cavity.
The conventionally accepted classification of the modes
in pill-box cavities separates the cases of transverse electric or
TE modes (zero electric field along the axis) and transverse
magnetic or TM modes (zero magnetic field along the axis).
As Eq.5.16 suggests, many modes can exist in a cavity, as
defined by the corresponding dimension of the pill-box, and
the integer number of the mode. The corresponding integer
indexes are used to identify a particular mode. In a rectangular pill-box, a mode can be called TE klm or TM klm where
the integer indexes indicate the number of half-wavelength
variations across the corresponding dimension (x, y, z) of the
cavity.
Cylindrical pill-box cavities are very common in accelerators. An example of a cylindrical pill-box cavity with holes
for the beam is shown in Fig.5.22. The classification of the
modes in cylindrical cavities is very similar, TE klm or TM klm ,
but in this case the indexes refer to the polar coordinates (ϕ,
r and z).
Examples of modes in cylindrical pill-box cavities, with
longitudinal electric fields and no variation over ϕ (TM 0lm ),
which are therefore suitable for use as an accelerating cavity,
are shown in Fig.5.23.
5.3.3 Quality factor of a resonator
The quality factor of a resonator — Q — is defined as the ratio
of the energy stored in the cavity to the energy dissipated per
oscillating cycle, divided by 2π
W
Q =
s
W
= ω
s
(5.17)
W d
P d
]
70
]
70
FIGURE 5.23
Examples of pill-box cylindrical cavity modes with electric field lines shown.
K
U
]
FIGURE 5.22
Cylindrical pill-box cavity.