The LHC abort kicker magnets are located at a very high-beta region, with 
m, and extend over a total length 
m. They are equipped with a ceramic vacuum chamber of thickness 
mm, width w=40 mm and height h=28 mm. The inside of the chamber has a
titanium coating of thickness d=1 
m to reduce the high-frequency
impedance seen by the beam and to conduct away the static charge [9]. Moreover, two copper plates between the H-shaped ferrites on the
outside of the ceramic chamber reduce the low-frequency impedance, by
carrying most of the beam image currents. Assuming an electric permeability 
for the ceramic and considering the pessimistic case of a
circular pipe with radius 
mm, the coupling impedances at
frequencies well below 
GHz
can be written [10]
where 
is the ratio between the geometric mean of the
pipe radius and the coating thickness, on one side, and the skin-depth 
in the titanium, having resistivity 
m,
The corresponding low-frequency values of 
and 
are
Figure 2: Average beta times
the real part of the transverse impedance 
in M
(a)
and imaginary part of the longitudinal impedance 
in m
(b)
versus the frequency 
in MHz, for the LHC abort kickers.
As shown in Fig. 2, the real part of 
has a peak around
30 MHz
(not far from the bunch frequency
),
corresponding to
Re
M
.
The tune shifts of the transverse coherent modes have
therefore an imaginary part depending on the coupled-bunch mode number m;
its maximum value at injection is 
, corresponding to a very week
coupled-bunch instability of the transverse dipole mode with a rise time of
about 7 sec, easily cured by feed-back. The real parts of the longitudinal
and transverse tune shifts depend very little on m. They are reported in
Tab. 5 for the lowest coherent modes, assuming
:
the only significant contribution is that to the transverse
dipole tune shift 
. The effective
impedances, defined again as the overlap integrals with the bunch power
spectrum 
, are
Table 5: Real parts of the coherent tune shifts at injection due to
abort kickers.
It is worth mentioning that, owing to the shielding by eddy currents, the
maximum surface resistance 
compatible with the
required kicker rise time 
s would be
where we have assumed a safety factor 
between the rise time due to
shielding and the kicker rise time 
, corresponding to more
than 99% of the maximum field after a time equal to 
for
a 
-like variation of the external magnetic field. Therefore, the
associated maximum thickness of the titanium layer could be
At the moment, however, there is no stringent argument in favour of a thicker titanium coating of the ceramic chamber, whose technical realization would also be rather difficult.