4 Impedance and Collective Effects
135
Fig. 4.18 Required
normalised (to the
synchrotron tune Q s ) tune
spread q to stabilise the
bunch in both cases of
instabilities without and with
Transverse Damper (TD),
corresponding to the cases of
Fig. 4.14 [94]. The
normalised parameter x is
proportional to the bunch
intensity [63]
0.0
0.2
0.4
0.6
0.8
1.0
1.2
0.0
0.5
1.0
1.5
2.0
x
Δq
1-mode approach (with TD)
2-mode approach (with TD)
2-mode approach (without TD)
Fig. 4.19 Reduction of the
tune footprint (and associated
projections on the transverse
tunes axes, responsible for
Landau damping) vs. linear
coupling (described here by
the “closest tune approach”
|C − |) [98]. Courtesy of L.R.
Carver
believed to be the case with the Batman instability of the HERA proton ring
[97]: due to the features discussed in Ref. [97], the name “coupled head–tail
instability” was suggested for this instability in the HERA proton ring). Recently,
linear coupling was also observed to be detrimental in the CERN LHC [98], as
revealed by both measurements and macroparticle simulations (see Fig. 4.19). This
required a careful measurement and correction of linear coupling all along the LHC
cycle to avoid to use much more Landau octupoles current than foreseen. One has
also to remember that linear coupling modifies also the transverse emittances [99,
100].
In the case of additional space-charge nonlinearities, the stability diagram will be
shifted and beam stability can be obtained or lost, depending on the coherent tune.
The influence of space-charge nonlinearities on the Landau damping mechanism of
transverse coherent instabilities has first been studied by Möhl and Schönauer for
coasting and rigid bunched beams [101, 102]. It was studied in detail in the past
years for higher-order head–tail modes from both theory [103–106] and numerical
simulations [107].
135
Fig. 4.18 Required
normalised (to the
synchrotron tune Q s ) tune
spread q to stabilise the
bunch in both cases of
instabilities without and with
Transverse Damper (TD),
corresponding to the cases of
Fig. 4.14 [94]. The
normalised parameter x is
proportional to the bunch
intensity [63]
0.0
0.2
0.4
0.6
0.8
1.0
1.2
0.0
0.5
1.0
1.5
2.0
x
Δq
1-mode approach (with TD)
2-mode approach (with TD)
2-mode approach (without TD)
Fig. 4.19 Reduction of the
tune footprint (and associated
projections on the transverse
tunes axes, responsible for
Landau damping) vs. linear
coupling (described here by
the “closest tune approach”
|C − |) [98]. Courtesy of L.R.
Carver
believed to be the case with the Batman instability of the HERA proton ring
[97]: due to the features discussed in Ref. [97], the name “coupled head–tail
instability” was suggested for this instability in the HERA proton ring). Recently,
linear coupling was also observed to be detrimental in the CERN LHC [98], as
revealed by both measurements and macroparticle simulations (see Fig. 4.19). This
required a careful measurement and correction of linear coupling all along the LHC
cycle to avoid to use much more Landau octupoles current than foreseen. One has
also to remember that linear coupling modifies also the transverse emittances [99,
100].
In the case of additional space-charge nonlinearities, the stability diagram will be
shifted and beam stability can be obtained or lost, depending on the coherent tune.
The influence of space-charge nonlinearities on the Landau damping mechanism of
transverse coherent instabilities has first been studied by Möhl and Schönauer for
coasting and rigid bunched beams [101, 102]. It was studied in detail in the past
years for higher-order head–tail modes from both theory [103–106] and numerical
simulations [107].