3 Non-linear Dynamics in Accelerators
101
3.8.2.2 Driving Terms
The treatment of the resonance map is still not fully understood and a standard
treatment using first order perturbation theory leads to a few wrong conclusions. In
particular it is believed that a resonance cannot be excited unless a driving term
for the resonance is explicitly present in the Hamiltonian. This implies that the
related map must contain the term for a resonance in leading order to reproduce
the resonance. This regularly leads to the conclusion that 3rd order resonances
are driven by sextupoles, 4th order are driven by octupoles etc. This is only a
consequence of the perturbation theory which is often not carried beyond leading
order, and e.g. a sextupole can potentially drive resonances of any order. Such a
treatment is valid only for special operational conditions such as resonant extraction
where strong resonant effects can be well described by a perturbation theory. A
detailed discussion of this misconception is given in [6]. A correct evaluation must
be carried out to the necessary orders and the tools presented here allow such a
treatment in an easier way.
3.8.3 Chromaticity and Chromaticity Correction
For reasons explained earlier, sextupoles are required to correct the chromaticities.
In large machines and in particular in colliders with insertions, these sextupoles
dominate over the non-linear effects of so-called linear elements.
3.8.4 Dynamic Aperture
Often in the context of the discussion of non-linear resonance phenomena the
concept of dynamic aperture in introduced. This is the maximum stable oscillation
amplitude in the transverse (x, y)-space due to non-linear fields. It must be
distinguished from the physical aperture of the vacuum chamber or other physical
restrictions such as collimators.
One of the most important tasks in the analysis of non-linear effects is to provide
answers to the questions:
• Determination of the dynamic aperture
• Maximising the dynamic aperture
The computation of the dynamic aperture is a very difficult task since no
mathematical methods are available to calculate it analytically except for the trivial
cases. Following the concepts described earlier, the theory is much more complete
from the simulation point of view. Therefore the standard approach to compute the
dynamic aperture is done by numerical tracking of particles.
101
3.8.2.2 Driving Terms
The treatment of the resonance map is still not fully understood and a standard
treatment using first order perturbation theory leads to a few wrong conclusions. In
particular it is believed that a resonance cannot be excited unless a driving term
for the resonance is explicitly present in the Hamiltonian. This implies that the
related map must contain the term for a resonance in leading order to reproduce
the resonance. This regularly leads to the conclusion that 3rd order resonances
are driven by sextupoles, 4th order are driven by octupoles etc. This is only a
consequence of the perturbation theory which is often not carried beyond leading
order, and e.g. a sextupole can potentially drive resonances of any order. Such a
treatment is valid only for special operational conditions such as resonant extraction
where strong resonant effects can be well described by a perturbation theory. A
detailed discussion of this misconception is given in [6]. A correct evaluation must
be carried out to the necessary orders and the tools presented here allow such a
treatment in an easier way.
3.8.3 Chromaticity and Chromaticity Correction
For reasons explained earlier, sextupoles are required to correct the chromaticities.
In large machines and in particular in colliders with insertions, these sextupoles
dominate over the non-linear effects of so-called linear elements.
3.8.4 Dynamic Aperture
Often in the context of the discussion of non-linear resonance phenomena the
concept of dynamic aperture in introduced. This is the maximum stable oscillation
amplitude in the transverse (x, y)-space due to non-linear fields. It must be
distinguished from the physical aperture of the vacuum chamber or other physical
restrictions such as collimators.
One of the most important tasks in the analysis of non-linear effects is to provide
answers to the questions:
• Determination of the dynamic aperture
• Maximising the dynamic aperture
The computation of the dynamic aperture is a very difficult task since no
mathematical methods are available to calculate it analytically except for the trivial
cases. Following the concepts described earlier, the theory is much more complete
from the simulation point of view. Therefore the standard approach to compute the
dynamic aperture is done by numerical tracking of particles.