136
E. Metral et al.
The interplay between Landau octupoles and beam–beam long-range interactions
can be either beneficial or detrimental depending on the sign of the Landau
octupoles current (see Sect. 4.6) [108] and this effect has to be careful taken into
account in the CERN LHC to be able to push its performance.
Another destabilising effect currently under investigation at the CERN LHC (and
which could explain some long latencies observed in the past, of the order of few
minutes or even tens of minutes) is the effect of noise, whose detrimental effect was
predicted in 2012 [109] and confirmed experimentally in 2018 [110].
Some work is being done to try and use the nonlinear optics as a path to high
intensity, providing “infinite (transverse) Landau damping” [111], or electron lenses
[112] or Radio Frequency Quadrupoles (or similarly second order chromaticity)
[113–115]. The latter two methods are believed to be more efficient than Landau
octupoles at high energy due to the adiabatic damping and the associated significant
reduction of the transverse beam sizes.
4.4.2 Longitudinal
When the bunch is very small inside the RF bucket, the motion of the particles
is linear and all the particles have the same (unperturbed, maximum) synchrotron
frequency ω s0 . By increasing the bunch length the incoherent synchrotron frequency
spread S is increased (the maximum synchrotron frequency spread is obtained when
the bunch length is equal to the RF bucket length as in this case the synchrotron
frequency of the particles with the largest amplitude is equal to 0: the synchrotron
frequency spread S is equal to ω s0 in this case). In the presence of an impedance,
the coherent synchrotron frequency of the dipole mode ω c11 , which is equal to
the low-intensity synchrotron frequency ω s0 without synchrotron frequency spread
(due to the compensation between the incoherent and coherent tune shifts), moves
closer and closer to the incoherent band (stable region). The two possible cases are
represented in Fig. 4.20 (using the rigid-bunch approximation), which is similar to
what was obtained by Besnier (who considered a parabolic distribution function,
which introduces some pathologies in the stability diagram due to its sharp edge)
([116], and references therein): the case of a capacitive impedance below transition
or inductive impedance above transition corresponds to U > 0 (the coherent
synchrotron frequency shift of the dipole mode has been written Δω c11 = U − jV)
and the incoherent synchrotron frequency shift (due to the potential-well distortion)
is Δω s
i < 0 (and thus ω s < ω s0 ), and the case of a capacitive impedance above
transition or inductive impedance below transition corresponds to U < 0 and Δω s
i > 0
(and thus ω s > ω s0 ). Motions ∝e jωt are considered, which means that the beam
is unstable when V > 0 (V is called the instability growth rate). The usual case
where the resistive part of the impedance is small compared to the imaginary part
is assumed, i.e. V << |U|. Beam stability is obtained when ω c11 enters into the
incoherent band. In both cases, the stability limit is reached for k = 4, i.e. S = 4|U|,
which is Sacherer’s stability criterion for the dipole mode.
E. Metral et al.
The interplay between Landau octupoles and beam–beam long-range interactions
can be either beneficial or detrimental depending on the sign of the Landau
octupoles current (see Sect. 4.6) [108] and this effect has to be careful taken into
account in the CERN LHC to be able to push its performance.
Another destabilising effect currently under investigation at the CERN LHC (and
which could explain some long latencies observed in the past, of the order of few
minutes or even tens of minutes) is the effect of noise, whose detrimental effect was
predicted in 2012 [109] and confirmed experimentally in 2018 [110].
Some work is being done to try and use the nonlinear optics as a path to high
intensity, providing “infinite (transverse) Landau damping” [111], or electron lenses
[112] or Radio Frequency Quadrupoles (or similarly second order chromaticity)
[113–115]. The latter two methods are believed to be more efficient than Landau
octupoles at high energy due to the adiabatic damping and the associated significant
reduction of the transverse beam sizes.
4.4.2 Longitudinal
When the bunch is very small inside the RF bucket, the motion of the particles
is linear and all the particles have the same (unperturbed, maximum) synchrotron
frequency ω s0 . By increasing the bunch length the incoherent synchrotron frequency
spread S is increased (the maximum synchrotron frequency spread is obtained when
the bunch length is equal to the RF bucket length as in this case the synchrotron
frequency of the particles with the largest amplitude is equal to 0: the synchrotron
frequency spread S is equal to ω s0 in this case). In the presence of an impedance,
the coherent synchrotron frequency of the dipole mode ω c11 , which is equal to
the low-intensity synchrotron frequency ω s0 without synchrotron frequency spread
(due to the compensation between the incoherent and coherent tune shifts), moves
closer and closer to the incoherent band (stable region). The two possible cases are
represented in Fig. 4.20 (using the rigid-bunch approximation), which is similar to
what was obtained by Besnier (who considered a parabolic distribution function,
which introduces some pathologies in the stability diagram due to its sharp edge)
([116], and references therein): the case of a capacitive impedance below transition
or inductive impedance above transition corresponds to U > 0 (the coherent
synchrotron frequency shift of the dipole mode has been written Δω c11 = U − jV)
and the incoherent synchrotron frequency shift (due to the potential-well distortion)
is Δω s
i < 0 (and thus ω s < ω s0 ), and the case of a capacitive impedance above
transition or inductive impedance below transition corresponds to U < 0 and Δω s
i > 0
(and thus ω s > ω s0 ). Motions ∝e jωt are considered, which means that the beam
is unstable when V > 0 (V is called the instability growth rate). The usual case
where the resistive part of the impedance is small compared to the imaginary part
is assumed, i.e. V << |U|. Beam stability is obtained when ω c11 enters into the
incoherent band. In both cases, the stability limit is reached for k = 4, i.e. S = 4|U|,
which is Sacherer’s stability criterion for the dipole mode.