Vlasov description of the beam response to noise in the presence of wakefields in high-energy synchrotrons: beam transfer function, diffusion, and loss of Landau damping

Abstract

Noise can have severe impacts on particle beams in high-energy synchrotrons. In particular, it has recently been discovered that noise combined with wakefields can cause a diffusion that leads to a loss of Landau damping after a latency. Such instabilities have been observed in the Large Hadron Collider. This paper, therefore, studies the beam response to noise in the presence of wakefields, within the framework of the Vlasov equation. First, a wakefield beam eigenmode transfer function (MTF) is derived, quantifying the amplitude of a wakefield eigenmode when excited by noise. Then, the MTFs of all the wakefield eigenmodes are combined to derive the beam transfer function (BTF) including the impact of wakefields. It is found to agree excellently with multi-particle tracking simulations. Finally, the MTFs are also used to derive the single-particle diffusion driven by the wakefield eigenmodes. This new Vlasov-based theory for the diffusion driven by noise-excited wakefields is found to be superior to an existing theory by comparing to multi-particle tracking simulations. Through sophisticated simulations that self-consistently model the evolution of the distribution and the stability diagram, the diffusion is found to lead to a loss of Landau damping after a latency. The most important technique to extend the latency and thereby mitigate these instabilities is to operate the synchrotron with a stability margin in detuning strength relative to the amount of detuning required to barely stabilize the beam with its initial distribution.

Mitigation strategies for the instabilities induced by the fundamental mode of the HL-LHC Crab Cavities

“…In figure 6 we plot the octupole thresholds obtained for a range of chromaticities with and without crab cavities. With the RF feedback switched on, we obtain thresholds in the range 500 A to 600 A, when including the effect of noise (which is obtained, in good approximation, by multiplying the thresholds by a factor of two [11,12]). are higher than 500 A, hence potentially too high to ensure a good DA.…”

High intensity beam dynamics assessment and challenges for HL-LHC

“…Additional measurements of growth rates and octupole thresholds were performed in 2022 and are compared with -10 - Minimum DA ( ) simulations using the impedance model in figures 8 and 9, showing reasonable agreement as well. In the latter case, the effect of noise (detrimental to beam stability [67]) is removed by quickly decreasing the octupole current until it reaches the threshold, hence the lower threshold values compared to the operational case where latency effects have to be taken into account (see next section). Also, the behaviour at high chromaticy 𝑄 ′ is still under study -recently a good agreement with measurements was obtained, as visible in figure 9, by including the non-linearity of the RF bucket in the simulations.…”

“…This reduces the threshold by up to 20 % [79]. Also, note that latency effects [67,80] are taken into account. The octupole threshold with the RF comb filter at the most unstable chromaticity in the range 10 < 𝑄 ′ < 20 is 364 A and 443 A for relaxed and tight collimator settings, respectively.…”

Focus point on high-energy accelerators: advances, challenges, and applications

“…Finally, a selection of papers discussing future advances is given in [26][27][28][29]. The use of nonlinear beam dynamics to extend the domain of possible beam manipulations is discussed in [26,28], where resonances are used as a means to provide an alternative transition-crossing scheme [26]or to manipulate transverse emittances [28].…”

“…The use of nonlinear beam dynamics to extend the domain of possible beam manipulations is discussed in [26,28], where resonances are used as a means to provide an alternative transition-crossing scheme [26]or to manipulate transverse emittances [28]. The Vlasov description of collective effects in circular accelerators is described in detail in [27] and, last, a novel diffusive model to describe the beam halo dynamics is presented in [29] with the focus on the determination of effective techniques to reconstruct the functional form of the diffusion coefficient with the longer-term goal of estimating the beam lifetime.…”