310
J. Seeman et al.
where v ph is the phase velocity of the wave excited at operating frequency and
v p corresponds to the velocity of the charged particle. Ultra-relativistic case with
v p = c, the speed of light, is considered below. In this case, the beam line with a
slope c (strait line in Fig. 7.6b as ω = v p k z ) must intersect the dispersion curve
at the operating point: (f 0 ; ϕ 0 ) on the Brillouin diagram in order to satisfy the
synchronism condition (7.12). The slope of the dispersion curve provides another
important parameter of the wave propagating in the structure, the so called group
velocity:
v g =
∂ω
∂k z
,
(7.13)
which can also be expressed using the cell stored energy U and power flow through
the cell iris aperture P z :
v g =
P z d
U
.
(7.14)
The stronger the coupling between the cells the higher the group velocity and the
faster energy propagates along the structure. Combining Eqs. (7.9, 7.10 and 7.14)
yields expression for the power flow along the travelling-wave structure which is
needed to maintain an accelerating gradient E acc = V acc /d:
P z = v g
E 2
acc
ωR /Q
,
(7.15)
where R
= R/d is the shunt impedance per meter length. For a given working
point (f 0 ; ϕ 0 ), the group velocity, the Q-factor and the R-upon-Q fully describe the
accelerating properties of the cell. In so-called constant impedance, the geometry
of all cells is identical and the three above parameters are identical in all cells.
In practice, the so-called constant gradient structures are used. In these structures,
the geometry of the cells is tapered in order to maintain E acc (z) ≈ const. This
is achieved by reducing the group velocity along the structure to compensate the
reduction in power flow along the structure which is caused by two terms: ohmic
losses according to Eq. (7.15) and power gained by the beam of a current I = qf b ,
where q is the bunch charge and f b is the bunch repetition frequency. In this case, the
energy conservation law yields an equation for the power flow along the structure:
dP z
dz
= −
P z ω
v g Q
− E acc I.
(7.16)
J. Seeman et al.
where v ph is the phase velocity of the wave excited at operating frequency and
v p corresponds to the velocity of the charged particle. Ultra-relativistic case with
v p = c, the speed of light, is considered below. In this case, the beam line with a
slope c (strait line in Fig. 7.6b as ω = v p k z ) must intersect the dispersion curve
at the operating point: (f 0 ; ϕ 0 ) on the Brillouin diagram in order to satisfy the
synchronism condition (7.12). The slope of the dispersion curve provides another
important parameter of the wave propagating in the structure, the so called group
velocity:
v g =
∂ω
∂k z
,
(7.13)
which can also be expressed using the cell stored energy U and power flow through
the cell iris aperture P z :
v g =
P z d
U
.
(7.14)
The stronger the coupling between the cells the higher the group velocity and the
faster energy propagates along the structure. Combining Eqs. (7.9, 7.10 and 7.14)
yields expression for the power flow along the travelling-wave structure which is
needed to maintain an accelerating gradient E acc = V acc /d:
P z = v g
E 2
acc
ωR /Q
,
(7.15)
where R
= R/d is the shunt impedance per meter length. For a given working
point (f 0 ; ϕ 0 ), the group velocity, the Q-factor and the R-upon-Q fully describe the
accelerating properties of the cell. In so-called constant impedance, the geometry
of all cells is identical and the three above parameters are identical in all cells.
In practice, the so-called constant gradient structures are used. In these structures,
the geometry of the cells is tapered in order to maintain E acc (z) ≈ const. This
is achieved by reducing the group velocity along the structure to compensate the
reduction in power flow along the structure which is caused by two terms: ohmic
losses according to Eq. (7.15) and power gained by the beam of a current I = qf b ,
where q is the bunch charge and f b is the bunch repetition frequency. In this case, the
energy conservation law yields an equation for the power flow along the structure:
dP z
dz
= −
P z ω
v g Q
− E acc I.
(7.16)