The design of the LHC beam screen results from an optimization of its
geometrical, mechanical, thermal and electromagnetic properties [18]
. In particular, the thickness of the inner copper layer is constrained to
small values to minimise eddy current forces in case of a magnet quench and
to large values to reduce the low-frequency resistive wall impedance,
responsible for transverse coupled-bunch instabilities. An original proposal
to have only four copper strips, instead of a uniform copper coating, would
have solved the problem of quench forces; however the ohmic losses due to
image currents induced by the beam in the uncoated high-resistivity regions
(about 50% of the screen surface) would have been unacceptable [11].
Indeed, the resistive losses in stainless steel are about 
times larger than in copper at
cryogenic temperatures (assuming 
m) and, over most of the bunch spectrum, image currents
can be computed just by solving the electrostatic two-dimensional problem
with boundary conditions independent of the wall resistivity.
In case of uniform copper coating, the resistive wall losses for a square
liner of side a are the same as the losses in a circular liner of radius 
. A numerical solution of the electrostatic problem shows that these
losses are also the same for the LHC square liner with rounded corners,
having radius of curvature equal to 
. Therefore, using the formula for
a circular liner of radius 
mm, the power loss is given by
where we have assumed a screen temperature of 
K and a nominal
bunch population 
particles,
corresponding to a total beam current of 536 mA. This result does not
change appreciably if the anomalous skin-effect is taken into
account [11].
The numerical solution of the electrostatic problem also shows that the
ratio between the image current density induced at the centre of the rounded
corners and that for a circular inscribed liner is almost exactly 
.
Therefore a high-resistivity region of small azimuthal extent 
(e.g., a weld with resistivity 
), located at one
of the rounded corners increases the ohmic losses by
For two welds at the top and bottom of the LHC screen, each having a width 
mm, the corresponding resistive wall losses are increased by
14%.
To obtain an upper bound for the power loss through the pumping slots, we
consider circular holes of diameter equal to the slot width 
mm, covering the same fractional surface 
% of a circular
screen with radius 
mm. Therefore, we neglect the geometric
reduction factor of 
for the induced image current at the slot
position and the further reduction of power loss through slots compared to
that through circular holes, confirmed by recent measurements by F. Caspers.
However, even for an infinitely thick wall, 
waves with cut-off
frequency 
can propagate through a slot of length 
. The
issue of how efficiently these waves can be excited by the beam and the
consequent tolerances on the slot alignment with respect to the beam axis
remain to be clarified. The power loss through circular holes of radius 
, covering a fractional surface f in a circular screen of radius
b, thickness t, outer resistivity 
surrounded by an outer
circular pipe of radius 
and resistivity 
, is
given by [19]
where the function 
is associated with attenuation from
the `inside' to the `outside' of the hole through the circular wave guide of
radius 
and length t equal to the hole depth:
Assuming the same resistivity 
m for the outer surface of the beam screen and of
the cold bore, for a screen thickness t=1 mm and a cold bore radius 
mm, the power loss through circular holes is 
kW.
To help reducing the fraction of this power dissipated at the cold bore, it
is foreseen to arrange microwave absorbers attached to the outside surface
of the LHC screen (not touching the 
K surface of the cold
bore). Since the attenuation length of the coaxial region between beam
screen and cold bore would than become shorter, these microwave absorbers
can also significantly reduce the coherent build-up of the TEM waves
travelling in synchronism with the beam and thus the corresponding power
loss through the pumping slots.
Table 10: Summary of parasitic losses for LHC at 7 TeV.
The parasitic losses for LHC at top energy are summarized in Tab. 10.
We have included an upper bound for the power loss due to coherent
synchrotron radiation (possibly largely overestimated) and for the leakage
of electromagnetic energy through a gap 
mm between the
sliding contacts of the bellows [7]. The latter is certainly a
rather pessimistic assumption, in view of the new bellows design including
spring fingers at the entrance of the gap. The parasitic losses in the
shielded bellows have been estimated using the broad-band resonator
parameters of Tab. 2. However, as in the case of the
monitor tanks, the broad-band resonators are not meant to model the real
part of the longitudinal impedance; the corresponding losses represent
therefore only an upper bound.