Single-beam collective effects include incoherent phenomena, concerning the behaviour of a single particle in the electromagnetic field produced by all the others, and coherent interactions of the beam with its surroundings, usually described in terms of coupling impedances. The second group can be further subdivided into single-bunch effects, associated with the broad-band impedance of low-Q structures, and multi-bunch effects, dominated by the narrow-band impedance of high-Q resonators. Landau damping of coherent beam oscillation modes, that takes place providing their tune shifts remain within the incoherent tune spread, can be considered as a bridge between incoherent and coherent phenomena.
The main examples of incoherent effects are synchrotron radiation losses,
direct space charge and Laslett tune shifts due to image currents, as well
as intra-beam scattering (that we shall not discuss here). As we will see,
the magnetic Laslett tune shift at injection in the LHC is of the order of 
: this large tune shift can be compensated by adjusting the tuning
quadrupoles, but sets rather stringent requirements on the equalization of
the bunch populations.
Coherent effects include parasitic losses, associated with the real part of
the longitudinal coupling impedance 
, and complex tune shifts
of the beam oscillation modes. For Gaussian bunches, with r.m.s. bunch
length 
, the power dissipated by the wall
currents in an impedance with a resistive component Re
is
where 
is the number of bunches, 
the bunch
current and 
the angular revolution frequency around the
ring. The quadratic dependence on the bunch current is to be noted.
For a given beam intensity, the coherent oscillation modes are self-consistent solutions of the linearized Vlasov equation. They satisfy an integral equation that can be transformed into an infinite-dimensional eigenvalue problem: the eigenvectors are connected with the power spectrum of the corresponding coherent modes and the eigenvalues are the associated complex tune shifts. Note that the exact eigenvectors, as well as the eigenvalues, depend both on the coupling impedance and on the longitudinal distribution of the bunches. The latter is also a function of the current, owing to the potential well distortion.
In the limit of weak beam intensity, each mode is characterized by a number 
, defining its azimuthal dependence in the longitudinal phase space. For
longitudinal modes (
) the unperturbed tunes are 
, while for transverse head-tail modes (
) the unperturbed tunes are 
, where 
denotes the
synchrotron tune and 
the betatron tune. When observed at a fixed
location around the ring, the signal corresponding to each oscillation mode
consists of discrete spectral lines at frequencies
Here m denotes the coupled-bunch mode number 
.
To specify uniquely a given coherent mode, one should also assign its radial
dependence in the longitudinal phase space. For vanishing beam intensities,
modes with the same azimuthal number n and different radial dependence
have the same tune 
, but this degeneracy is progressively removed
for increasing beam currents. However, if the tune shifts remain much
smaller than 
, different radial modes will couple only when
they belong to the same azimuthal family n: such regime of weak beam
intensity is governed by the so-called Sacherer integral equation. A further
simplification is obtained by neglecting the possible coupling between
radial modes with the same n (e.g., when the problem can be exactly or
approximately diagonalized analytically) and expressing the complex tune
shift of the most prominent radial mode in terms of its effective
impedance [1], measuring the degree to which the impedance
overlaps the mode spectrum. Then the complex tune shift for longitudinal
modes is approximately given by
where 
and 
are the harmonic number and peak
rf-voltage, respectively, R is the average machine radius, 
the full bunch length and the effective longitudinal impedance is
defined by
Here 
denotes the mode power spectrum and, for
Gaussian bunches, it is
Similarly, the complex tune shift for transverse modes is approximately given by
where 
is the average betatron function, 
the beam
energy in volts and the effective transverse impedance is defined by
In the following, we shall not consider the chromatic shift 
associated with the chromaticity 
and the slippage factor 
.
From Eqs. (1) and (2), we see that the imaginary
part of the effective impedance is responsible for (real) coherent tune
shifts and can lead to collective instabilities owing to mode coupling or to
loss of Landau damping, while the real part of the effective impedance is
related to the instability rise time. In the following, we show that the
bunch population in the LHC is limited to about 
protons,
owing to suppression of Landau damping for longitudinal high-order modes at
7 TeV. The transverse resistive wall instability has a rise time longer than
100 revolution periods and can thus be cured by feedback. However the LHC
impedance budget is not yet complete and requires more detailed calculations.
