Innovative method to measure the extent of the stable phase-space region of proton synchrotrons

The advent of circular accelerators based on superconducting magnets has revolutionized the field of beam dynamics, with particle motion turning from linear to nonlinear due to unavoidable high-order field errors generated by the ring magnets. Nonlinear dynamics was already well mastered, e.g., in the close field of celestial mechanics as similar problems had been considered and successfully tackled. Hence, several results were available to aid comprehension of the behavior of charged particle beams under the influence of nonlinear forces. Here, we discuss how concepts derived from the theory of dynamical systems, linked with the fundamental Kolmogorov–Arnold–Moser theory and Nekhoroshev theorem, can be successfully applied to the analysis of nonlinear motion of charged particles in a circular accelerator. Based on these ideas, an innovative method to measure the extent of the phase-space region within which bounded motion occurs is presented, which has been successfully tested for the first time at the CERN LHC.

Figure 1

Left: Dynamic aperture of a model of the LHC ring in normalized physical space. The red points represent initial conditions stable up to the maximum number of turns ($10^5$). The blue points represent unstable conditions and the marker’s size is proportional to the number of turns for which the motion is still bounded. Right: Time evolution of DA. The markers represent the numerical results, while the continuous line shows the fitted inverse logarithmic law, and the vertical dashed line represents $D_{\infty }$ (the fit parameters are $D_{\infty }=23.74\pm 0.01,b=0.439\pm 0.005,\kappa =1.40\pm 0.05$ [19]).

Figure 2

Left: Sketch of the principle of the standard method to measure DA. Right: Measured and simulated losses (from Ref. [38]).

Figure 3

Upper: Layout of the LHC (from Ref. [11]). The ring eight-fold symmetry is visible, together with the arcs and the long straight sections. Bottom: Layout of the LHC regular cell (from Ref. [11]). Six dipoles and two quadrupoles with the dipole, quadrupole, sextupole, and octupole magnets (for closed orbit, tune, chromaticity correction and beam stabilization, respectively) are shown. The spool pieces used to compensate the systematic $b_3$ component (MCS), $b_4$ and $b_5$ components (MCDO nested magnets) are also shown. The field imperfections of LHC magnets are given as $B_y+i{B}_x=B_{\mathrm{ref}}\sum _{n=1}^M(b_n+i{a}_n){(\frac{x+iy}{R_{\mathrm{r}}})}^{n-1}$ where $R_{\mathrm{r}}=17\mathrm{mm}$.

Figure 4

Sequence of measurements performed during the experimental session. The evolution of the beam intensity and of the MCO magnets’ strength is shown. The color is used to indicate the data selected for the analysis presented in the rest of the paper.

Figure 5

Relative beam intensity vs turns plotted for all octupolar configurations probed in the LHC DA experiment with Beam 1. The different intensity decay rate is visible.

Figure 6

DA estimates from simulations (lines) and measurements (lines with full markers) for three selected cases. The excellent agreement is visible, even if some difference in slope between measured and simulation results is visible for the case $\mathrm{\Sigma }{K}_{\mathrm{MCO}}=-50.3{\mathrm{m}}^{-4}$. The DA is not given in units of beam $\sigma$, but rather in units of the nominal $\sigma$ obtained with the nominal beam emittance of ${\epsilon }_{\mathrm{n}}=3.75\mu \mathrm{m}$.

Figure 7

Comparison between measured DA (lines with full markers) and extrapolation (lines) of the numerical estimate based on the scaling law (2) for four selected cases. The very good agreement is clearly visible. The upper right plot refers to the case with alternating signs for the strength of the MCOs. The DA is not given in units of beam $\sigma$, but rather in units of the nominal $\sigma$ obtained with the nominal beam emittance of ${\epsilon }_{\mathrm{n}}=3.75\mu \mathrm{m}$.

Figure 8

Summary of the measured and simulated DA values corresponding to $D(1\times {10}^6)$, plotted for the various configurations of the MCO strength. The error bars of measurement results are obtained by the spread of DA values for $10^6$ turns (see the experimental curves in Fig. 6). The error bars of simulation results are obtained from the DA distribution over the sixty realizations of the LHC lattice (both rms and min/max error bars are shown). The DA is not given in units of beam $\sigma$, but rather in units of the nominal $\sigma$ obtained with the nominal beam emittance of ${\epsilon }_{\mathrm{n}}=3.75\mu \mathrm{m}$.