As shown in Fig. 1 of Ref. [11], the LHC beam screen has a square
cross section with rounded corners; the radius of the inscribed circle is 
mm and the radius of curvature of the rounded corners is almost
exactly equal to 
. The stainless steel screen has a thickness of 1 mm
and its inside is coated with a copper layer of thickness t=50 
m and
residual resistance ratio 
. Since the screen is at a
temperature 
K, the corresponding copper resistivity is 
m at injection and 
m at top energy, respectively,
taking due account of the magnetoresistance effect in a magnetic field 
Tesla at injection and 
Tesla at 7 TeV. The stainless steel
resistivity is 
m.
As sketched in Fig. 1 of Ref. [12], in the beam screen there are 500
rectangular slots per meter. Each slot has a length 
mm and a width
mm, with rounded edges; the corresponding fractional surface of beam
screen covered by slots is 4.3%, which is sufficient for vacuum pumping
considerations. The slots are staggered along two rows located at a distance
of 
from the middle of each screen side, i.e., just where the rounded
corners touch the sides of the square, and their arrangement is such that
each cross section of the screen contains four slots. The induced image
currents at the slot positions have been numerically computed to be smaller
by a factor 
, compared to the case of a round screen with radius b.
This gives a geometric reduction factor 
for
the longitudinal slot impedance and we assume the same geometric factor also
for the transverse impedance, although the analytic result for a perfectly
square screen cross section is about 15% lower. A further reduction factor 
comes from the finite thickness of the beam screen.
Finally, the aspect ratio 
of the LHC slots
corresponds [13] to a relative polarizability 
compared to the case of round holes with radius 
. In
conclusion, the total coupling impedances of the pumping slots can be
obtained by computing the impedances of 
millions of round holes, in a thin round screen inscribed in
the square LHC liner, and then multiplying the result by a global reduction
factor F
This yields
The latter corresponds to a weighted transverse impedance 
. The corresponding tune shifts at
injection, for a nominal bunch population 
,
are shown in Tab. 6.
Table 6: Coherent tune shifts at injection due to pumping slots.
In addition to these inductive impedances, the pumping slots may give rise
to high-Q trapped modes with frequencies slightly below the screen cut-off
frequency and associated with narrow-band impedances having high peak values
of 
[13]: according to
preliminary estimates [14], the maximum acceptable value of 
for beam stability is around 15 
Therefore, the
present base-line design of the LHC beam screen includes a 10%
randomization of the slot lengths, leading to a reduction of their effective
quality factors and of the corresponding narrow-band impedance by a factor
20. The slot spacing should also be randomized, to damp possible
higher-frequency travelling modes associated with the periodic slot pattern.