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E. Metral et al.
around the driving frequency. If one observes two particles, one with the exciting
frequency ω c and one with a frequency slightly different, at the beginning, they
oscillate “coherently” (same amplitude and same phase). However, after a while
the particle with the exciting frequency, being resonantly driven, continues to
increase in amplitude as time increases, whereas the other particle with a slightly
different frequency realizes that its frequency is not the same as the driving one
and the “beating” phenomenon is observed for this particle. If one considers the
phenomenon for a time t, the number of particles which still oscillate coherently
decreases with time as 1/t, while their amplitude increases as t, the net contribution
being constant with time.
The origins of the frequency spread that leads to Landau damping have not
been taken into account till now. The case where the frequency spread comes from
the longitudinal momentum spread of the beam is straightforward (for a coasting
beam), because the longitudinal momentum is a constant, which just affects the
coefficients in the equations of motion of the transverse oscillations, and hence their
frequencies. It can be dealt with the same method as in the previous sections, i.e. it
is the distribution function which is important. The same result applies also if one
considers a tune spread that is due to a non-linearity (e.g. from octupole lenses) in
the other plane. However, this result is no longer valid if the non-linearity is in the
plane of coherent motion. In this case, the steady-state is more involved because
the coherent motion is then a small addition to the large incoherent amplitudes that
make the frequency spread, and it is inconsistent to assume that it can be treated
as a linear superposition [92]. One needs to consider “second order” non-linear
terms and the final result is that in this case it is not the distribution function which
matters but its derivative. Using the Vlasov formalism, this result is recovered more
straightforwardly.
4.4.1 Transverse
Considering the case of a beam having the same normalized rms beam size σ =
√
ε in both transverse planes, the Landau damping mechanism from octupoles of
coherent instabilities, e.g. in the horizontal plane, is discussed from the following
dispersion relation [17, 93]
1 = −ΔQ
x
coh
+∞
J x =0
dJ x
+∞
J y=0
dJ y
J x
∂f (Jx,Jy)
∂J x
Q c − Q x
J x , J y
− mQ s
,
(4.33)
with
Q x
J x , J y
= Q 0 + a 0 J x + b 0 J y .
(4.34)
E. Metral et al.
around the driving frequency. If one observes two particles, one with the exciting
frequency ω c and one with a frequency slightly different, at the beginning, they
oscillate “coherently” (same amplitude and same phase). However, after a while
the particle with the exciting frequency, being resonantly driven, continues to
increase in amplitude as time increases, whereas the other particle with a slightly
different frequency realizes that its frequency is not the same as the driving one
and the “beating” phenomenon is observed for this particle. If one considers the
phenomenon for a time t, the number of particles which still oscillate coherently
decreases with time as 1/t, while their amplitude increases as t, the net contribution
being constant with time.
The origins of the frequency spread that leads to Landau damping have not
been taken into account till now. The case where the frequency spread comes from
the longitudinal momentum spread of the beam is straightforward (for a coasting
beam), because the longitudinal momentum is a constant, which just affects the
coefficients in the equations of motion of the transverse oscillations, and hence their
frequencies. It can be dealt with the same method as in the previous sections, i.e. it
is the distribution function which is important. The same result applies also if one
considers a tune spread that is due to a non-linearity (e.g. from octupole lenses) in
the other plane. However, this result is no longer valid if the non-linearity is in the
plane of coherent motion. In this case, the steady-state is more involved because
the coherent motion is then a small addition to the large incoherent amplitudes that
make the frequency spread, and it is inconsistent to assume that it can be treated
as a linear superposition [92]. One needs to consider “second order” non-linear
terms and the final result is that in this case it is not the distribution function which
matters but its derivative. Using the Vlasov formalism, this result is recovered more
straightforwardly.
4.4.1 Transverse
Considering the case of a beam having the same normalized rms beam size σ =
√
ε in both transverse planes, the Landau damping mechanism from octupoles of
coherent instabilities, e.g. in the horizontal plane, is discussed from the following
dispersion relation [17, 93]
1 = −ΔQ
x
coh
+∞
J x =0
dJ x
+∞
J y=0
dJ y
J x
∂f (Jx,Jy)
∂J x
Q c − Q x
J x , J y
− mQ s
,
(4.33)
with
Q x
J x , J y
= Q 0 + a 0 J x + b 0 J y .
(4.34)