3 Non-linear Dynamics in Accelerators
97
3.8 Beam Dynamics with Non-linearities
Following the overview of the evaluation and analysis tools, it is now possible to
analyse and classify the behaviour of particles in the presence of non-linearities.
The tools presented beforehand allow a better physical insight to the mechanisms
leading to the various phenomena, the most important ones being:
• Amplitude detuning
• Excitation of non-linear resonances
• Reduction of dynamic aperture and chaotic behaviour.
This list is necessarily incomplete but will serve to demonstrate the most important
aspects.
To demonstrate these aspects, we take a realistic case and show how the effects
emerge automatically.
3.8.1 Amplitude Detuning
It was discussed in Sect. 3.7.5 that the one-turn-map can be transformed into a
simpler map where the rotation is separated. A consequence of the non-linearities
was that the rotation frequency becomes amplitude dependent to perform this
transformation. Therefore the amplitude detuning is directly obtained from this
normal form transformation.
3.8.1.1 Amplitude Detuning due to Non-linearities in Machine Elements
Non-linear elements cause an amplitude dependent phase advance. The computational procedure to derive this detuning was demonstrated in the discussion on
normal for transformations in the case of an octupole Eqs. (3.125) and (3.129). This
formalism is valid for any non-linear element.
Numerous other examples can be found in [6] and [5].
3.8.1.2 Amplitude Detuning due to Beam–Beam Effects
For the demonstration we use the example of a beam-beam interaction because it is
a very complex non-linear problem and of large practical importance [7, 22].
In this simplest case of one beam-beam interaction we can factorize the machine
in a linear transfer map e :f 2 : and the beam-beam interaction e :F : , i.e.:
e : f 2 : · e : F : = e : h :
(3.166)
97
3.8 Beam Dynamics with Non-linearities
Following the overview of the evaluation and analysis tools, it is now possible to
analyse and classify the behaviour of particles in the presence of non-linearities.
The tools presented beforehand allow a better physical insight to the mechanisms
leading to the various phenomena, the most important ones being:
• Amplitude detuning
• Excitation of non-linear resonances
• Reduction of dynamic aperture and chaotic behaviour.
This list is necessarily incomplete but will serve to demonstrate the most important
aspects.
To demonstrate these aspects, we take a realistic case and show how the effects
emerge automatically.
3.8.1 Amplitude Detuning
It was discussed in Sect. 3.7.5 that the one-turn-map can be transformed into a
simpler map where the rotation is separated. A consequence of the non-linearities
was that the rotation frequency becomes amplitude dependent to perform this
transformation. Therefore the amplitude detuning is directly obtained from this
normal form transformation.
3.8.1.1 Amplitude Detuning due to Non-linearities in Machine Elements
Non-linear elements cause an amplitude dependent phase advance. The computational procedure to derive this detuning was demonstrated in the discussion on
normal for transformations in the case of an octupole Eqs. (3.125) and (3.129). This
formalism is valid for any non-linear element.
Numerous other examples can be found in [6] and [5].
3.8.1.2 Amplitude Detuning due to Beam–Beam Effects
For the demonstration we use the example of a beam-beam interaction because it is
a very complex non-linear problem and of large practical importance [7, 22].
In this simplest case of one beam-beam interaction we can factorize the machine
in a linear transfer map e :f 2 : and the beam-beam interaction e :F : , i.e.:
e : f 2 : · e : F : = e : h :
(3.166)