Beam dynamics aspects of crab cavities in the CERN Large Hadron Collider

Modern colliders bring into collision a large number of bunches to achieve a high luminosity. The long-range beam-beam effects arising from parasitic encounters at such colliders are mitigated by introducing a crossing angle. Under these conditions, crab cavities (CC) can be used to restore effective head-on collisions and thereby to increase the geometric luminosity. Such crab cavities have been proposed for both linear and circular colliders. The crab cavities are rf cavities operated in a transverse dipole mode, which imparts on the beam particles a transverse kick that varies with the longitudinal position along the bunch. The use of crab cavities in the Large Hadron Collider (LHC) may not only raise the luminosity, but it could also complicate the beam dynamics, e.g., crab cavities might not only cancel synchrobetatron resonances excited by the crossing angle but they could also excite new ones, they could reduce the dynamic aperture for off-momentum particles, they could influence the aperture and orbit, also degrade the collimation cleaning efficiency, and so on. In this paper, we explore the principal feasibility of LHC crab cavities from a beam dynamics point of view. The implications of the crab cavities for the LHC optics, analytical and numerical luminosity studies, dynamic aperture, aperture and beta beating, emittance growth, beam-beam tune shift, long-range collisions, and synchrobetatron resonances, crab dispersion, and collimation efficiency will be discussed.

Figure 1

Two local crab cavities in interaction region 5 (IR5) for LHC beam 1 which illustrates the local scheme (left) and one global crab cavity in IR4 for LHC beam 1 which illustrates the global scheme (right) (single global scenario).

Figure 2

Schematic for two crabbed beams at IP5, local scheme (with $\theta$ denoting the full crossing angle) (left); schematic of only one crabbed beam at LHC IP5 with a single global crab cavity, for beam 1 only (right).

Figure 3

Optics parameters for the local crab scheme in LHC IR5: nominal LHC collision optics (top) and lowbetamax collision optics (bottom). IP5 is at $s=0$.

Figure 4

Global crab optics at LHC IR4: nominal LHC collision optics (top) and lowbetamax collision optics (bottom). IP4 is at $s=0$.

Figure 5

Turn-by-turn $x\mathrm{\text{−}}z$ correlation of an off-momentum particle ($2.5{\sigma }_p$) at IP5 with 400-MHz and 800-MHz global crab cavity, for the nominal LHC optics (with rms bunch length ${\sigma }_z=75.5\mathrm{mm}$). The crab-cavity kick is linear within $1{\sigma }_z$, which recovers most of the geometric luminosity expected without crossing angle.

Figure 6

Minimum dynamic aperture over 60 error seeds for nominal LHC optics with or without crab cavity; the maximum decrease of $1\sigma$ due to crab cavity is acceptable. The dynamic-aperture tracking is performed for 100 000 turns.

Figure 7

Minimum dynamic aperture over 60 error seeds for lowbetamax collision optics with or without crab cavity; the maximum decrease of 1 or $2\sigma$ due to crab cavity is acceptable. The dynamic-aperture tracking is performed for 100 000 turns.

Figure 8

Minimum dynamic aperture over 60 error seeds for nominal LHC optics, without crab cavity (blue); with two local 800-MHz crab cavities and horizontal phase advance 0.518 (in units of $2\pi$) between them (red); with two local 800-MHz crab cavities and optimized horizontal phase advance 0.5 (in units of $2\pi$) between them (green).

Figure 9

Comparison of horizontal orbit change of a particle with $1{\sigma }_z$ offset for 800-MHz global crab-cavity case (red); of a particle with $1{\sigma }_z$ offset for 400-MHz global crab-cavity case (green); and for a $1{\sigma }_p$ off-momentum particle (blue), in the nominal LHC collision optics. The orbit change due to the crab cavity is comparable to, or smaller than, the dispersive orbit shift.

Figure 10

Comparison of horizontal orbit change of a particle with $1{\sigma }_z$ offset for 800-MHz global crab-cavity case (red); of a particle with $1{\sigma }_z$ offset for 400-MHz global crab-cavity case (green); and for a $1{\sigma }_p$ off-momentum particle (blue), in the lowbetamax collision optics. The orbit change due to the crab cavity is comparable to, or smaller than, the dispersive orbit shift.

Figure 11

Off-momentum beta beating for the nominal LHC optics with a relative momentum offset of 0.000 27 (top); $z$-dependent beta beating due to the global crab cavity (bottom), which is much smaller than the off-momentum beta beating.

Figure 12

Off-momentum beta beating for the lowbetamax collision optics with a relative momentum offset of 0.000 27 (top); $z$-dependent beta beating due to the global crab cavity (bottom), which is much smaller than the off-momentum beta beating.

Figure 13

Sketch of the coordinate system for the two colliding beams.

Figure 14

Comparison of normalized luminosity between analytical formulas (curves) and GUINEA-PIG simulations (dots), with two horizontally crabbed beams (each by a crabbing angle of $\theta /2$) at IP5 (left); and with only one horizontally crabbed beam (by a crabbing angle of $\theta /2$) at IP5 (right). Simulations and analytical expressions agree.

Figure 15

Relative luminosity gain (normalized to the case without crab cavity) versus crabbing angle at IP5, for the nominal LHC collision optics (left) and for the lowbetamax collision optics (right) with a single global crab cavity. A luminosity gain of 7% or 25% is achieved, respectively. For 20 MHz the optimum crab angle equals half the crossing angle as theoretically expected while this is no longer the case for the higher crab frequencies due to the rf curvature.

Figure 16

Relative luminosity gain (normalized with the corresponding crossing case and the emittance set to 3.75, 2.5, and $1\mu \mathrm{m}\mathrm{rad}$ for the three cases, respectively) for three cases with different beam emittance and bunch intensity (nominal LHC with ${\beta }^{*}=0.55\mathrm{m}$, $\theta =285–552\mu \mathrm{rad}$), from GUINEA-PIG simulations with a single global crab cavity. The luminosity gain is 10%–25% for lower emittance.

Figure 17

Check of crab cavity ramping by tracking reveals a ramp-up time of 5000 turns is sufficiently adiabatic with respect to the synchrotron motion.

Figure 18

Relative horizontal emittance growth for different ramping speed of one 800-MHz global crab cavity. A ramping period longer than 10 turns is required to avoid the emittance growth.

Figure 19

Relative horizontal emittance growth due to the crab cavity ramping with different frequency crab cavities (20, 400, and 800 MHz), without (left) and with (right) Landau octupoles, with 1 turn ramping speed. The lower the frequency is, the larger the emittance growth will be.

Figure 20

Relative horizontal emittance growth due to the crab cavity ramping with different frequency crab cavities (20, 400, and 800 MHz), longitudinal cut at $1\sigma$, with 1 turn ramping speed. The transverse emittance growth is due to particles in the longitudinal tails.

Figure 21

Crab dispersion (IP5 is at 13 329 m) due to one 800-MHz global crab cavity (red) and one 400-MHz global crab cavity (green); normal dispersion (blue).

Figure 22

$z$-dependent closed-orbit change induced by the global crab cavity in IR4. The maximum orbit change is about $1\sigma$.

Figure 23

Schematic of the impact parameter on the LHC primary collimator.

Figure 24

Average first-time impact parameter at specified turn for different cases. All cases have a $1\mu \mathrm{m}$ impact parameter for the first turn.

Figure 25

Local loss map for nominal LHC (top energy, ${\beta }_{\mathrm{IP}1,5}^{*}=0.55\mathrm{m}$), with one global crab cavity and horizontal beam halo (beam halo generated at $z=1{\sigma }_z$). A beam lifetime of 0.2 h is assumed to define the quench limit (red dashed line). This loss map is almost indistinguishable from the case without crab cavity.

Figure 26

Phase-space cut by various collimators in the presence of crab dispersion but zero momentum dispersion, with the hierarchy of primary (TCP), secondary (TCSG), tertiary (TCTH), beam dump (TCDQ) horizontal collimators, and shower absorbers (TCLA). $D_r(i_{\mathrm{coll}},\delta )=0$ in formula (24).

Figure 27

CC disturbed phase space for the circulating beam by applying formula (24) ($z=1{\sigma }_z$), with the hierarchy of primary (TCP), secondary (TCSG), tertiary (TCTH), beam dump (TCDQ) horizontal collimators, and shower absorbers (TCLA) maintained. The white region labeled in the middle is the phase space available to the circulating beam.

Figure 28

Horizontal tune shift (top) and vertical tune shift (bottom): crossing collision with one beam-beam element at IP5(H) and horizontal crossing (red); full crab collision with the same beam-beam element at IP5(H) and both beam crabbed (blue). The crab crossing tune shift is found to be the same as the one for the head-on collision case.

Figure 29

Frequency spectrum computed from the turn-by-turn horizontal beam position, without (left) and with (right) nonlinear magnet errors, for a particle launched with $1\sigma$ offset in $x$, $y$, and $z$. The second sideband synchrobetatron resonances introduced by the crossing collision are suppressed by the crab cavities (synchrotron tune $Q_s=0.00197$, and chromaticity $Q_{x,y}^{\prime }=2$). The spectrum with crossing angle and crab cavities resembles the head-on collision case.

Figure 30

Frequency spectrum computed from the turn-by-turn horizontal beam position, with nonlinear magnet errors, for a particle launched with $1\sigma$ offset in $x$, $y$, and $z$. The second sideband synchrobetatron resonances introduced by the crossing collision are partially suppressed by the single global 800-MHz crab cavity (synchrotron tune $Q_s=0.00197$, and chromaticity $Q_{x,y}^{\prime }=2$).

Figure 31

Minimum dynamic aperture over 60 error seeds for the nominal LHC optics with or without crab cavity; including all the head-on and long-range beam-beam interactions at IP1, IP2, IP5, and IP8. The dynamic-aperture tracking is performed for 100 000 turns.