The LHC relies on Landau damping for longitudinal stability. To avoid decreasing the stability margin at high energy, the longitudinal emittance must be continuously increased during the acceleration ramp. Longitudinal blow-up provides the required emittance growth. The method was implemented through the summer of 2010. We inject band-limited RF phase-noise in the main accelerating cavities during the whole ramp of about 11 minutes. Synchrotron frequencies change along the energy ramp, but the digitally created noise tracks the frequency change. The position of the noise-band, relative to the nominal synchrotron frequency, and the bandwidth of the spectrum are set by pre-defined constants, making the diffusion stop at the edges of the demanded distribution. The noise amplitude is controlled by feedback using the measurement of the average bunch length. This algorithm reproducibly achieves the programmed bunch length of about 1.2 ns (4 ) at flat top with low bunch-to-bunch scatter and provides a stable beam for physics coast.

/pdf/longitudinal-emittance-blow-up-in-the-lhc-1ovnjfaxgr.pdf

Longitudinal Emittance Blow-Up in the LHC

The HL-LHC is being designed to deliver an integrated luminosity of at least 250fb−1/year in each of the two LHC General-Purpose Experiments ATLAS and CMS for proton operation [1], while operating the other two experiments, ALICE and LHCb, at very low and moderate instantaneous luminosity, of about 10 and 1–2×1033 cm−2s−1, respectively [2, 3]. The ambitious performance target for ATLAS and CMS cannot be met without pushing both the optics to their extreme, namely β∗, and the nominal parameters of the LHC beam. It relies as well on a number of very challenging new equipment and key innovative technologies, such as:

/pdf/the-hl-lhc-accelerator-physics-challenges-4dthynq1vh.pdf

The HL-LHC accelerator physics challenges

Longitudinal collective modes of a bunched beam with a repulsive inductive impedance (the space charge below transition or the chamber inductance above it) are analytically described by means of reduction of the linearized Vlasov equation to a parameterless integral equation. For any multipolarity, the discrete part of the spectrum is found to consist of infinite number of modes with real tunes, which limit point is the incoherent zero-amplitude frequency. In other words, notwithstanding the rf bucket nonlinearity and potential well distortion, the Landau damping is lost. Hence, even a tiny coupled-bunch interaction makes the beam unstable; such growth rates for all the modes are analytically obtained for arbitrary multipolarity. In practice, however, the finite threshold of this loss of Landau damping is set either by the high-frequency impedance roll-off or intrabeam scattering. Above the threshold, growth of the leading collective mode should result in persistent nonlinear oscillations.

Longitudinal modes of bunched beams with weak space charge

As the second stage of the CEPC-SPPC project, SPPC (Super Proton-Proton Collider) aims at exploring new physics beyond the Standard Model. The key design goal for the SPPC accelerator complex is to reach 75 TeV in center of mass energy with a circumference of 100 km for the collider and an injector chain of four accelerators in cascade to support the collider. As an important part of the SPPC conceptual study, the longitudinal beam dynamics was studied systematically, which includes the dynamics in the collider and its complex injector chain. First, the bunch filling scheme of the SPPC complex was designed on the basis of various constraints such as the technical challenges of the kicker magnets, limited extraction energy per injection and so on. Next, the study on the longitudinal dynamics in the collider focused on the RF scheme to meet the requirements for luminosity and mitigate relevant instabilities. A higher harmonic RF system (800 MHz) together with the basic RF system (400 MHz) to form a dual-harmonic RF system was employed to mitigate collective instabilities and increase the luminosity by producing shorter bunches. In addition, the longitudinal matchings between the bunches in the different accelerator stages were studied, with special attention to the space charge effects and the beam loading effect in the two lower energy rings (p-RCS and MSS), which result in an optimization of the RF schemes. A set of self-consistent beam and RF parameters for the SPPC complex was obtained. The collider and the three proton synchrotrons of the injector chain have unprecedented features, thus this study demonstrates how a future proton-proton collider complex looks like.

Optimization of design parameters for SPPC longitudinal dynamics

Landau damping mechanism plays a crucial role in providing single-bunch stability in LHC, High-Luminosity LHC, other existing as well as previous and future (like FCC) circular hadron accelerators. In this paper, the thresholds for the loss of Landau damping (LLD) in the longitudinal plane are derived analytically using the Lebedev matrix equation (1968) and the concept of the emerged van Kampen modes (1983). We have found that for the commonly-used particle distribution functions from a binomial family, the LLD threshold vanishes in the presence of the constant inductive impedance Im$Z/k$ above transition energy. Thus, the effect of the cutoff frequency or the resonant frequency of a broad-band impedance on beam dynamics is studied in detail. The findings are confirmed by direct numerical solutions of the Lebedev equation as well as using the Oide-Yokoya method (1990). Moreover, the characteristics, which are important for beam operation, as the amplitude of residual oscillations and the damping time after a kick (or injection errors) are considered both above and below the threshold. Dependence of the threshold on particle distribution in the longitudinal phase space is also analyzed, including some special cases with a non-zero threshold for Im$Z/k = const$. All main results are confirmed by macro-particle simulations and consistent with available beam measurements in the LHC.

Thresholds for loss of Landau damping in longitudinal plane.

Loss of Landau damping leading to a single bunch longitudinal instability has been observed in the LHC during the ramp and on the 3.5 TeV flat top for small injected longitudinal emittances. The first measurements are in reasonable agreement with the threshold calculated for the expected longitudinal reactive impedance budget of the LHC as well as with the threshold dependence on beam energy. The cure is a controlled longitudinal emittance blow-up during the ramp which for a constant threshold through the cycle should provide an emittance proportional to the square root of energy.

/pdf/loss-of-landau-damping-in-the-lhc-sayzbacshf.pdf

Loss of Landau Damping in the LHC

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Loss of Landau Damping in the LHC