The single-beam collective effects expected in LHC were stated
in [1] and reviewed in [7] with an updated
knowledge of the machine impedance. They encompass the single-bunch
coherent motion, discussed later, and the coupled-bunch oscillations.
A detailed analysis of the latter [8, 9], driven by
the narrow-band impedance due to resistive wall and HOM's in the
super-conducting and feedback cavities, shows that an active feedback system with
a half bandwidth of 1.5 MHz and a gain corresponding to a damping time of 10 ms
is sufficient at injection to damp all dipole modes, provided the Q values
of the cavity HOM's are in the order of 
.
At collision energy, a further increase in
bandwidth or gain is necessary to correct these multi-bunch modes.
The evaluation of the coherent tune shift for each transverse
oscillation mode relies on the impedance budget of the machine and on
the broad-band resonator models assumed. The results corresponding to
the impedance discussed in [7], with the ultimate
bunch population of 
protons and
an average betatron function of 89.1 m,
are summarized in Table 1
together with new calculations for the collision energy.
| Mode | 450 GeV | 7000 GeV |
 |  |  |
 |  |  |
 |  |  |
 |  |  |
 |  |  |
Table 1: Coherent tune shifts of the transverse head-tail modes
for the ultimate bunch population of protons
and a Gaussian longitudinal distribution
with an r.m.s. bunch length 
cm at injection
and 
cm at top energy.
Note that the effective broad-band impedance
times the average betatron function is 
M
at injection, when the capacitive space charge impedance dominates,
while at top energy the effective impedance is inductive
M and the coherent tune shifts are negative.
The negative tune shift of the rigid dipole mode
at injection is caused by the inductive low-frequency impedance associated
with strip-line monitors and abort kickers.
The direct space charge provides a spread of the incoherent tunes in
the order of 
at injection energy and is therefore
sufficient for Landau damping of the higher-order head tail modes
(other then the rigid dipole mode).
With a 
dependence, this spread decreases rapidly to reach less than
at collision energy. Given the safety factor required, it
can be verified from Table 1 that
this `natural' tune spread becomes insufficient for Landau damping
of the higher-order head tail modes at top energy,
before the beams are put in collision
(with a resulting beam-beam tune spread of about 
).
It has been foreseen to damp the rigid dipole mode by an active
feedback. The minimum octupole scheme shall provide a betatron
frequency spread to damp the higher-order modes which would be
unstable otherwise.
Whatever the quality of the electronics, however,
active damping is likely to reduce the reliability and complicate the
measurement of the betatron tunes.
As a safety measure, we propose that the full octupole scheme shall be
able to damp the rigid dipole mode at collision
energy, corresponding to a real coherent tune shift of .
The imaginary coherent tune shifts for multi-bunch dipole modes
are an order of magnitude smaller [9]
while, as shown in Fig. 2, the stability limit for imaginary
tune shifts is at worst 4 times lower than for real tune shifts.
The full octupole scheme therefore ensures Landau damping
also against multi-bunch instabilities at collision energy.
At injection, assuming damping of the dipole mode by the feedback
system, the direct space charge tune spread is sufficient to damp the
higher-order coherent modes.