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E. Metral et al.
Finding the eigenvalues and eigenvectors of a complex matrix by computer can
be difficult in some cases, and a simple approximate formula for the eigenvalues is
useful in practice to have a rough estimate. This is known as Sacherer’s (longitudinal) formula [52]. Sacherer’s formula is also valid for coupled-bunch instability
with M equally-populated equally-spaced bunches, assuming multi-bunch modes
with only one type of internal motion. In the case of gaps between bunch trains, a
time-domain approach is usually better suited.
As the bunch intensity increases, the different longitudinal modes can no longer
be treated separately and the situation is more involved. In the longitudinal plane, the
microwave instability for coasting beams is well understood. It leads to a stability
diagram, which is a graphical representation of the solution of the dispersion
relation (taking into account the momentum spread) depicting curves of constant
growth rates, and especially a threshold contour in the complex plane of the driving
impedance (see Sect. 4.4) [57]. When the real part of the driving impedance is
much greater than the modulus of the imaginary part, a simple approximation,
known as the Keil-Schnell (or circle) stability criterion, may be used to estimate
the threshold curve [58]. For bunched beams, it has been proposed by Boussard to
use the coasting-beam formalism with local values of bunch current and momentum
spread [59]. A first approach to explain this instability, without coasting-beam
approximations, has been suggested by Sacherer through Longitudinal ModeCoupling (LMC) [60]. The equivalence between LMC Instabilities (LMCI) and
microwave instabilities has been pointed out by Sacherer and Laclare [53] in the
case of broad-band driving resonator impedances, when the bunch length is much
greater than the inverse of twice the resonance frequency. Furthermore, due to the
potential well-distortion, a bunch is more stable below transition than above [53,
61, 62]. Typical pictures of LMCI are shown in Fig. 4.6 [63] and a comparison with
macroparticle tracking simulations, which revealed a good agreement, is discussed
in Ref. [64]. Experimentally, the most evident signature of the LMCI is the intensitydependent longitudinal beam emittance blow-up to remain just below the threshold
[65], as revealed also by macroparticle tracking simulations (see Fig. 4.7 [64]).
4.3.2 Transverse
A similar analysis as the one done for the longitudinal plane can be done in the
transverse plane [53, 63]. Following the same procedure, the horizontal coherent
oscillations (over several turns) of a “water-bag” bunch (i.e. with constant longitudinal amplitude density) interacting with a constant inductive impedance are shown
in Fig. 4.8.
The main difference with the longitudinal plane is that there is no effect of the
stationary distribution and the bunch spectrum is now centered at the chromatic
frequency f ξ = Q x0 f 0 ξ /η, where f 0 is the revolution frequency and ξ is the relative
chromaticity. The sign of the chromatic frequency is very important and to avoid
the head–tail instability (of mode 0) it should be slightly positive, meaning that
E. Metral et al.
Finding the eigenvalues and eigenvectors of a complex matrix by computer can
be difficult in some cases, and a simple approximate formula for the eigenvalues is
useful in practice to have a rough estimate. This is known as Sacherer’s (longitudinal) formula [52]. Sacherer’s formula is also valid for coupled-bunch instability
with M equally-populated equally-spaced bunches, assuming multi-bunch modes
with only one type of internal motion. In the case of gaps between bunch trains, a
time-domain approach is usually better suited.
As the bunch intensity increases, the different longitudinal modes can no longer
be treated separately and the situation is more involved. In the longitudinal plane, the
microwave instability for coasting beams is well understood. It leads to a stability
diagram, which is a graphical representation of the solution of the dispersion
relation (taking into account the momentum spread) depicting curves of constant
growth rates, and especially a threshold contour in the complex plane of the driving
impedance (see Sect. 4.4) [57]. When the real part of the driving impedance is
much greater than the modulus of the imaginary part, a simple approximation,
known as the Keil-Schnell (or circle) stability criterion, may be used to estimate
the threshold curve [58]. For bunched beams, it has been proposed by Boussard to
use the coasting-beam formalism with local values of bunch current and momentum
spread [59]. A first approach to explain this instability, without coasting-beam
approximations, has been suggested by Sacherer through Longitudinal ModeCoupling (LMC) [60]. The equivalence between LMC Instabilities (LMCI) and
microwave instabilities has been pointed out by Sacherer and Laclare [53] in the
case of broad-band driving resonator impedances, when the bunch length is much
greater than the inverse of twice the resonance frequency. Furthermore, due to the
potential well-distortion, a bunch is more stable below transition than above [53,
61, 62]. Typical pictures of LMCI are shown in Fig. 4.6 [63] and a comparison with
macroparticle tracking simulations, which revealed a good agreement, is discussed
in Ref. [64]. Experimentally, the most evident signature of the LMCI is the intensitydependent longitudinal beam emittance blow-up to remain just below the threshold
[65], as revealed also by macroparticle tracking simulations (see Fig. 4.7 [64]).
4.3.2 Transverse
A similar analysis as the one done for the longitudinal plane can be done in the
transverse plane [53, 63]. Following the same procedure, the horizontal coherent
oscillations (over several turns) of a “water-bag” bunch (i.e. with constant longitudinal amplitude density) interacting with a constant inductive impedance are shown
in Fig. 4.8.
The main difference with the longitudinal plane is that there is no effect of the
stationary distribution and the bunch spectrum is now centered at the chromatic
frequency f ξ = Q x0 f 0 ξ /η, where f 0 is the revolution frequency and ξ is the relative
chromaticity. The sign of the chromatic frequency is very important and to avoid
the head–tail instability (of mode 0) it should be slightly positive, meaning that