4 Impedance and Collective Effects
109
Therefore, there is an intensity-dependent step in the equilibrium bunch length
at transition, which leads to a longitudinal mismatch and subsequent quadrupolar
oscillations. If these bunch shape oscillations are not damped they will eventually
result in filamentation and longitudinal emittance blow-up. It’s worth mentioning
that in presence of significant space charge, the minimum of bunch length is not
reached right at transition anymore, but after about one nonadiabatic time T c , i.e.
after ~2 ms in the present case [11, 12]. The same kind of mechanism appears
with the inductive part of the longitudinal machine impedance (see Sect. 4.2). The
only difference is that in this case, the equilibrium bunch length is shorter below
transition and longer above transition.
If transition crossing cannot be avoided, the γ t jump is the only (known) method
to overcome all the intensity limitations. It consists in an artificial increase of the
transition crossing speed by means of fast pulsed quadrupoles. The idea is that
quadrupoles at nonzero dispersion locations can be used to adjust the momentum
compaction factor. The change in momentum compaction (called γ t jump) depends
on the unperturbed and perturbed dispersion functions at the kick-quadrupole
locations. These schemes were pioneered by the CERN PS group [13–16]. Such a γ t
jump scheme makes it possible to keep the beam at a safe distance from transition,
except for the very short time during which the transition region is crossed at a speed
increased by one or two orders of magnitude. Looking at Fig. 4.1(left) clearly reveals
why an asymmetric jump was proposed in the past [14] to damp the longitudinal
quadrupolar oscillations arising from the space charge induced mismatch: the idea
is to jump rapidly from an equilibrium bunch length below transition to the same
value above. The amplitude of the jump is defined by the time needed to go to the
same equilibrium bunch length above transition. The minimum amplitude of the
jump corresponds to the case represented with the dashed blue line starting right at
transition. However, in this case the initial longitudinal phase space ellipse is tilted
(see Fig. 4.1(right)), while the final one is almost not, which is not ideal. One might
want therefore to start the jump earlier, when the longitudinal phase space is almost
not tilted, for instance at x ≈ −2, which requires a larger jump (see the dashed
orange line in Fig. 4.1(left)).
Coming back to the transverse space charge, in the case of an elliptical beam
(instead of a round one), one has to replace 2σ x
2 by σ x (σ x + σ y ) and 2σ y
2 by
σ y (σ x + σ y ). Furthermore, due to the nonlinear nature of the space charge, the
tune shifts of the different particles will not be the same, which will lead to a tune
spread: plotted in the tune diagram it is called a tune footprint. The latter has to
be accommodated in the tune diagram, without crossing harmful resonance lines,
which might lead to emittance growth and/or beam losses. The exact tune footprint
depends on the distribution and to illustrate this effect we consider in the following
a round beam with quasi-parabolic distribution function, whose particle density
extends up to ~3.2σ [17, 18]. The corresponding horizontal 2D (i.e. neglecting
the longitudinal distribution) space charge force is plotted in Fig. 4.2(left), and the
tune footprint in Fig. 4.2(right). The unperturbed (low-intensity) working point is
in the top right corner, the small-amplitude particles have the largest tune shifts
while the largest amplitude particles have the smallest tune shifts. If the longitudinal
109
Therefore, there is an intensity-dependent step in the equilibrium bunch length
at transition, which leads to a longitudinal mismatch and subsequent quadrupolar
oscillations. If these bunch shape oscillations are not damped they will eventually
result in filamentation and longitudinal emittance blow-up. It’s worth mentioning
that in presence of significant space charge, the minimum of bunch length is not
reached right at transition anymore, but after about one nonadiabatic time T c , i.e.
after ~2 ms in the present case [11, 12]. The same kind of mechanism appears
with the inductive part of the longitudinal machine impedance (see Sect. 4.2). The
only difference is that in this case, the equilibrium bunch length is shorter below
transition and longer above transition.
If transition crossing cannot be avoided, the γ t jump is the only (known) method
to overcome all the intensity limitations. It consists in an artificial increase of the
transition crossing speed by means of fast pulsed quadrupoles. The idea is that
quadrupoles at nonzero dispersion locations can be used to adjust the momentum
compaction factor. The change in momentum compaction (called γ t jump) depends
on the unperturbed and perturbed dispersion functions at the kick-quadrupole
locations. These schemes were pioneered by the CERN PS group [13–16]. Such a γ t
jump scheme makes it possible to keep the beam at a safe distance from transition,
except for the very short time during which the transition region is crossed at a speed
increased by one or two orders of magnitude. Looking at Fig. 4.1(left) clearly reveals
why an asymmetric jump was proposed in the past [14] to damp the longitudinal
quadrupolar oscillations arising from the space charge induced mismatch: the idea
is to jump rapidly from an equilibrium bunch length below transition to the same
value above. The amplitude of the jump is defined by the time needed to go to the
same equilibrium bunch length above transition. The minimum amplitude of the
jump corresponds to the case represented with the dashed blue line starting right at
transition. However, in this case the initial longitudinal phase space ellipse is tilted
(see Fig. 4.1(right)), while the final one is almost not, which is not ideal. One might
want therefore to start the jump earlier, when the longitudinal phase space is almost
not tilted, for instance at x ≈ −2, which requires a larger jump (see the dashed
orange line in Fig. 4.1(left)).
Coming back to the transverse space charge, in the case of an elliptical beam
(instead of a round one), one has to replace 2σ x
2 by σ x (σ x + σ y ) and 2σ y
2 by
σ y (σ x + σ y ). Furthermore, due to the nonlinear nature of the space charge, the
tune shifts of the different particles will not be the same, which will lead to a tune
spread: plotted in the tune diagram it is called a tune footprint. The latter has to
be accommodated in the tune diagram, without crossing harmful resonance lines,
which might lead to emittance growth and/or beam losses. The exact tune footprint
depends on the distribution and to illustrate this effect we consider in the following
a round beam with quasi-parabolic distribution function, whose particle density
extends up to ~3.2σ [17, 18]. The corresponding horizontal 2D (i.e. neglecting
the longitudinal distribution) space charge force is plotted in Fig. 4.2(left), and the
tune footprint in Fig. 4.2(right). The unperturbed (low-intensity) working point is
in the top right corner, the small-amplitude particles have the largest tune shifts
while the largest amplitude particles have the smallest tune shifts. If the longitudinal