168
E. Metral et al.
Self-consistent multi-particle simulations form a complementary study, but
require large computing resources. Furthermore, the fields produced by the beams
must be computed in a self-consistent form before every collision [186].
The 0-mode is found at the unperturbed tune as expected. The π-mode is shifted
by 1.2–1.3ξ . The precise value depends on the ratio of the horizontal and vertical
beam sizes [183]. We have seen before the incoherent tune spread (footprint) the
individual particles occupy and we know that it spans the interval [0.0,1.0]ξ , starting
at the 0-mode.
Here one can make an important observation: under the strong–strong condition
the π-mode is a discrete mode outside the incoherent spectrum [184, 185]. This
has dramatic consequences for the stability of the beams. A coherent mode that is
outside an incoherent frequency spectrum cannot be stabilized by Landau damping.
Under these conditions the coherent beam–beam effect could drive the dipole
oscillation to large amplitudes and may result in the loss of the beam. Observations
of the coherent beam–beam effects have been made at PETRA [181]. Beam–beam
modes have been observed with high intensity coasting beams in the ISR [187], and
recently in a bunched hadron collider at RHIC [188].
Coherent beam–beam modes can be driven by head-on collisions with a small
offset or by long-range interactions. In the first case and for small oscillations,
the problem can be linearized and the theoretical treatment is simplified. The
forces from long-range interactions are very nonlinear but the numerical evaluation
is feasible. Since the coherent shift must have the opposite sign for long-range
interactions, the situation is very different. In particular the π-mode from long-range
interactions alone would appear on the opposite side of the 0-mode in the frequency
spectrum [185, 186]. Both, the incoherent and the coherent spectra include both
types of interactions.
4.6.10 Compensation of Beam–Beam Effects
For the case the beam–beam effects limit the performance of a collider, several
schemes have been proposed to compensate all or part of the detrimental effects.
The basic principle is to design correction devices which act as non-linear “lenses”
to counteract the distortions from the non-linear beam–beam “lens”. For both headon and long-range effects schemes have been proposed
• Head on effects:
– Electron lenses
– Linear lens to shift tunes
– Non-linear lens to decrease tune spread
• Long-range effects:
– At large distance: beam–beam force changes like 1/r
– Same force as a wire!
E. Metral et al.
Self-consistent multi-particle simulations form a complementary study, but
require large computing resources. Furthermore, the fields produced by the beams
must be computed in a self-consistent form before every collision [186].
The 0-mode is found at the unperturbed tune as expected. The π-mode is shifted
by 1.2–1.3ξ . The precise value depends on the ratio of the horizontal and vertical
beam sizes [183]. We have seen before the incoherent tune spread (footprint) the
individual particles occupy and we know that it spans the interval [0.0,1.0]ξ , starting
at the 0-mode.
Here one can make an important observation: under the strong–strong condition
the π-mode is a discrete mode outside the incoherent spectrum [184, 185]. This
has dramatic consequences for the stability of the beams. A coherent mode that is
outside an incoherent frequency spectrum cannot be stabilized by Landau damping.
Under these conditions the coherent beam–beam effect could drive the dipole
oscillation to large amplitudes and may result in the loss of the beam. Observations
of the coherent beam–beam effects have been made at PETRA [181]. Beam–beam
modes have been observed with high intensity coasting beams in the ISR [187], and
recently in a bunched hadron collider at RHIC [188].
Coherent beam–beam modes can be driven by head-on collisions with a small
offset or by long-range interactions. In the first case and for small oscillations,
the problem can be linearized and the theoretical treatment is simplified. The
forces from long-range interactions are very nonlinear but the numerical evaluation
is feasible. Since the coherent shift must have the opposite sign for long-range
interactions, the situation is very different. In particular the π-mode from long-range
interactions alone would appear on the opposite side of the 0-mode in the frequency
spectrum [185, 186]. Both, the incoherent and the coherent spectra include both
types of interactions.
4.6.10 Compensation of Beam–Beam Effects
For the case the beam–beam effects limit the performance of a collider, several
schemes have been proposed to compensate all or part of the detrimental effects.
The basic principle is to design correction devices which act as non-linear “lenses”
to counteract the distortions from the non-linear beam–beam “lens”. For both headon and long-range effects schemes have been proposed
• Head on effects:
– Electron lenses
– Linear lens to shift tunes
– Non-linear lens to decrease tune spread
• Long-range effects:
– At large distance: beam–beam force changes like 1/r
– Same force as a wire!