Analysis of transverse beam stabilization with radio frequency quadrupoles

A radio frequency (rf) quadrupole has been considered as a potential alternative device for Landau damping in circular hadron colliders. The objective of this study is to benchmark and confirm its stabilizing effect predicted by stability diagram theory by means of numerical tracking simulations. To that end, two complementary models of the device are implemented in pyheadtail, a 6D macroparticle tracking code designed to study the formation and mitigation of collective instabilities. The rf quadrupole model is applied to a slow head-tail instability observed experimentally in the Large Hadron Collider to show that such a device can in principle provide beam stability similarly to magnetic octupoles. Thereafter, alternative usage schemes of rf quadrupoles also in combination with magnetic octupoles are proposed, discussed, and benchmarked with simulations.

Figure 1

Qualitative description of the quadrupolar fields leading to a transverse kick on a charged particle (cross-section). The direction of the momentum change $\mathrm{\Delta }{\mathbf{p}}_{\perp }$ is indicated for a positively charged particle. Left: Elliptical cavity type operating in a TM mode. Right: Four-vane cavity operating in a TE mode.

Figure 2

Normalized electric (left) and magnetic (right) fields for an octant of the elliptical (top) and for half of the four-vane (bottom) cavity respectively.

Figure 3

Illustration of the detuning that a particle (red) experiences as it passes through an rf quadrupole (orange) turn after turn. Given enough time, the longitudinal position of the particle will be evenly distributed over the interval $[-{\widehat{z}}_i,{\widehat{z}}_i]$ due to the synchrotron motion. It will hence experience a changing detuning every time it passes through (black dashed curve). If the rf quadrupole phase is ${\phi }_0=0$ (top), the average detuning will not vanish and there is a net incoherent tune spread. If ${\phi }_0=-\pi /2$ there will be no effective tune spread.

Figure 4

Comparison of incoherent betatron tune distributions in the $(Q_x,Q_y)$-space introduced by magnetic octupoles (blue) powered in a two-family scheme with negative (positive) polarity in the focusing (defocusing) family, and a single rf quadrupole (red). The lines and the cross in green mark the bare machine tunes $(Q_{x,0},Q_{y,0})$. Histograms on the top and on the side show the projections of the tune distributions onto the $Q_x$- and the $Q_y$-axis respectively.

Figure 5

Dependence of the RMS tune spread on the beam energy for both LHC magnetic octupoles at maximum strength (blue) and for an rf quadrupole (red), designed in such a way as to produce the same RMS tune spread at 7 TeV (green dashed line).

Figure 6

Top: Incoherent tune distributions obtained analytically for a Gaussian beam with a single rf quadrupole with a phase of ${\phi }_0=0$ assuming ${\beta }_x={\beta }_y$. Bottom: Corresponding stability diagrams determined by solving Eq. (9) numerically for an azimuthal mode $m=0$ head-tail instability.

Figure 7

Comparison of the incoherent betatron tune distributions introduced by an rf quadrupole for the detuning (red) and the localized kick (grey) model implemented in pyheadtail. Both models give consistent results. The detuning obtained from pyheadtail for particles with respectively $J_z={\sigma }_z^2/2{\beta }_z$ (green) and $J_z=0$ is in good agreement with analytical calculations (blue).

Figure 8

Left: Horizontal bunch centroid evolution over time with an exponential fit of the rise time (red). Middle: Frequency spectra of the bunch centroid at different times during the simulation (see dashed lines in the first plot). The most unstable mode is an azimuthal mode $m=-1$ in agreement with the experiments. Right: Horizontal slice centroid vs. longitudinal position for 80 consecutive turns. The head-tail motion is characterized by one node in the center of the bunch.

Figure 9

pyheadtail results of the bunch centroid motion over $3\times {10}^5$ turns in the LHC for different values of the currents in the Landau octupole magnets.

Figure 10

The dashed lines are stability diagrams obtained with the pyssd code for two different Landau octupole currents $I_{\mathrm{oct},f}=-15\pm 5\mathrm{A}$ (green) and $I_{\mathrm{oct},f}=-17.5\pm 1\mathrm{A}$ (orange). The colored areas are the corresponding uncertainties. The former corresponds to the values obtained experimentally, while the latter is determined from the unperturbed complex coherent tune shifts (red circle: bimbim, blue cross: pyheadtail).

Figure 11

pyheadtail results of the bunch centroid motion over $3\times {10}^5$ turns in the LHC for different values of the rf quadrupole strength $b^{(2)}$.

Figure 12

The dashed line is the stability diagram obtained by numerically solving the approximate dispersion relation in Eq. (9) for an rf quadrupole with $b^{(2)}=0.0235\pm 0.0010\mathrm{Tm}/\mathrm{m}$. The colored area is the corresponding resolution of the scan in $b^{(2)}$. The unperturbed complex coherent tune shifts from bimbim (red circle) and from pyheadtail (blue cross) are shown as well.

Figure 13

pyheadtail tracking simulations illustrating the stabilizing efficiency in the horizontal (red) and the vertical (blue) planes with an rf quadrupole with ${\phi }_0=0$ (top) and ${\phi }_0=\pi$ (bottom) respectively.

Figure 14

Evolution of the unperturbed coherent tune shift in the complex tune space as a function of intensity. Stability diagrams for the horizontal (solid lines) and the vertical (dashed lines) planes are overlaid for two different strengths for an rf quadrupole operating with ${\phi }_0=0$.

Figure 15

pyheadtail tracking simulations illustrating the stabilizing efficiency in the horizontal (red) and the vertical (blue) planes with the two-family rf quadrupole scheme.

Figure 16

pyheadtail tracking simulations showing the combined stabilizing effect from LHC Landau octupoles and an rf quadrupole in the HL-LHC.