Measurement of nonlinear observables in the Large Hadron Collider using kicked beams

The nonlinear dynamics of a circular accelerator such as the Large Hadron Collider (LHC) may significantly impact its performance. As the LHC progresses to more challenging regimes of operation it is to be expected that the nonlinear single particle dynamics in the transverse planes will play an increasing role in limiting the reach of the accelerator. As such it is vital that the nonlinear sources are well understood. The nonlinear fields of a circular accelerator may be probed through measurement of the amplitude detuning: the variation of tune with single particle emittance. This quantity may be assessed experimentally by exciting the beam to large amplitudes with kicks, and obtaining the tunes and actions from turn-by-turn data at Beam Position Monitors. The large amplitude excitations inherent to such a measurement also facilitate measurement of the dynamic aperture from an analysis of beam losses following the kicks. In 2012 these measurements were performed on the LHC Beam 2 at injection energy (450 GeV) with the nominal magnetic configuration. Nonlinear coupling was also observed. A second set of measurements were performed following the application of corrections for $b_4$ and $b_5$ errors. Analysis of the experimental results, and a comparison to simulation are presented herein.

Figure 1

Measurement of nonlinear chromaticity in the horizontal (top) and vertical (bottom) planes of the LHC at 450 GeV in 2011, before and after correction with octupolar and decapolar correctors [7, 8].

Figure 2

Illustration of the method used to measure the dynamic aperture. The value of $(\mathrm{DA}-N_{\sigma \mathrm{KICK}})$ is determined from the beam loss over many turns.

Figure 3

Decoherence of ($2{\sigma }_x,2{\sigma }_y$) kicks at injection. The decoherence of kicks at the three different optics studied are shown: nominal injection settings settings (red), with Landau octupoles powered to zero and the MCO residual field zeroed (green), and the corrected configuration with $Q^{″}$ and $Q^{‴}$ corrections applied on top of the preceding state (blue).

Figure 4

Detuning with amplitude of the LHC at nominal injection settings (red), and with Landau octupoles set to zero and additional corrections for $Q^{″}$ and $Q^{‴}$ applied (blue). The tune is plotted against the horizontal and vertical single particle emittance of the kicks. Points shown in black represent kicks where there was substantial coupling of the kick into the opposite plane.

Figure 5

Turn-by-turn (TbT) and phase space data for a ($8.3{\sigma }_x,0.6{\sigma }_y$) kick with nominal LHC injection settings in the vertical plane at BPM.31L1.B2 (top), and the horizontal plane at BPM.26L1.B2 (bottom).

Figure 6

Histogram showing the distribution of first order detuning with amplitude of $\sim 2000$ random instances of the LHC, with $|f_{1001}|$ defined on an even distribution within the band defined by measurements of local fluctuations of the RDT around the LHC ring, and a uniform distribution in the phase of $f_{1001}$ at the matching location. The WISE seeds were also varied randomly on a uniform distribution.

Figure 7

Modeled and measured variation in the tune-split with vertical kick amplitude at nominal injection optics.

Figure 8

${\epsilon }_{\max }$ calculated using Eq. (4) for the horizontal and vertical kicks applied during amplitude detuning measurements at nominal injection optics.

Figure 9

${\epsilon }_{\max }$ calculated using Eq. (4) for the horizontal and vertical kicks applied during amplitude detuning measurements with Landau octupoles depowered to zero and $Q^{″}$ and $Q^{‴}$ corrections applied.

Figure 10

Comparison of the modeled and measured ${\epsilon }_{\mathrm{MAX}}$ of kicks in the horizontal or vertical planes at nominal injection optics.

Figure 11

Measured TbT data from a large amplitude vertical kick at nominal injection optics.

Figure 12

Variation of the simulated tune split ($|Q_y-Q_x|$) with the action of vertical kicks at nominal injection optics. This is shown for the nominal tune split with unkicked beams, and for an increased initial separation of the tunes.

Figure 13

Surviving beam intensity versus horizontal (top) and vertical (bottom) kick amplitude in nominal sigma, 30 s after the kick was applied.

Figure 14

Beam intensity recorded by the LHC Beam Current Monitors following the highest amplitude kicks performed in the horizontal, vertical, and diagonal planes at the corrected optics configuration.

Figure 15

Comparison of the modeled and measured dynamic aperture at nominal injection optics in the LHC. Simulations with $|C^{-}|=0.002$ and $|C^{-}|=0.004$ have been considered in order to characterize the uncertainty in the DA arising from the reproduction of $|f_{1001}|$ in the model. Similarly, four phases ($0.0\pi$, $0.5\pi$, $1.0\pi$ and $1.5\pi$) of $f_{1001}$ at the matching point were considered for each amplitude examined. The area shown for the simulated amplitudes corresponds to the maximum and minimum DA defined by the four phases.

Figure 16

Comparison of the modeled and measured dynamic aperture, following depowering of the MO and the application of corrections for $Q^{″}$ and $Q^{‴}$. Simulations with $|C^{-}|=0.002$ and $|C^{-}|=0.004$ have been considered in order to characterize the uncertainty in the DA arising from the reproduction of $|f_{1001}|$ in the model. Similarly, four phases ($0.0\pi$, $0.5\pi$, $1.0\pi$, and $1.5\pi$) of $f_{1001}$ at the matching point were considered for each amplitude examined. The area shown for the simulated amplitudes corresponds to the maximum and minimum DA defined by the four phases. Collimator settings are indicated by orange lines, where the patterned orange area indicates the uncertainty in collimator aperture defined by an assumed maximum beta-beat of 5%.