Abstract
This chapter applies the coherent mode framework of Chap. 5 to periodic focusing, where coherent space charge eigenmodes can be subject to parametrically driven, resonant instability as in the case of the well-known envelope instability. Their main characteristic is that they follow a half-integer resonance condition between a coherent eigenmode of oscillation and the periodic focussing – in contrast with the (also half-integer) single particle parametric resonances discussed in the mismatch context. The subject will be discussed here from various points of view: as Vlasov perturbation theory; in smooth approximation; using the nonlinear rms envelope equations; and finally with multiparticle simulation comparing 2D and 3D models. This chapter is particularly relevant to periodic lattices in linear high intensity accelerators, but in a variety of aspects also to circular accelerators.
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Notes
- 1.
Note that half-integer is relative to a linac focussing or a ring structure period; also integer cases cannot be excluded, but they are much weaker.
- 2.
It is equally absent in the coherent betatron resonances of Chap. 8.
- 3.
In [2] these modes where simply called space-charge induced instabilities, and the notion parametric was not used. The term parametric more precisely describes their nature; moreover it distinguishes them from the so-called “Gluckstern mode” instabilities (see Sect. 7.3.2.3) also found in [2], which are not parametric, but intrinsic to the non-monotonic KV-distribution.
- 4.
The presence of such lower order terms does not mean lower order modes are included.
- 5.
Following [2] this stopband is actually composed of several bands with gaps in between.
- 6.
Precisely speaking, each branch splits further into two adjacent modes.
- 7.
A historical side note: this instability – frequently called “Gluckstern mode” – was of major concern in the pioneering time of space charge studies for linacs in the 1970s, until it was later recognized as irrelevant for realistic beams.
- 8.
Mentioned already in early simulations of a coasting waterbag beam in [17].
- 9.
Noting that in a symmetric FODO-channel the slow and fast envelope modes are actually not separate modes.
- 10.
With reference to linacs; long bunches as in synchrotrons or storage rings require different approaches in the third dimension.
- 11.
Note that in this sum mode section the resonance condition is written using tunes without space charge, and the space charge shifts are absorbed in the correspondingly modified Δν coh, s of Eq. 7.5.
- 12.
Here it is assumed that transverse tunes and emittances are identical, which is roughly the case in linacs.
- 13.
Noting that in circular accelerators lattices a 90∘ phase advance condition is usually avoided for reasons of structural resonances.
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Hofmann, I. (2017). Coherent Parametric Instabilities. In: Space Charge Physics for Particle Accelerators. Particle Acceleration and Detection. Springer, Cham. https://doi.org/10.1007/978-3-319-62157-9_7
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