Accurate crab cavity modeling for the high luminosity Large Hadron Collider

As part of the Large Hadron Collider high luminosity upgrade it is proposed to include crab cavities in the lattice in order to enhance the luminosity. For one proposed cavity design the dynamics of the cavity is considered in terms of its impact upon the dynamic aperture of the machine. Taylor maps of the cavity are created and used to perform this analysis with a full assessment of their validity. Furthermore from these Taylor maps, symplectic methods are developed further, guided by the knowledge gained in the study of the physics contained in them.

Figure 1

Local crab crossing scheme showing creation of “crab bump.”

Figure 2

Four rod cavity design (not to scale) with wavelength $\lambda$. The kick of the cavity shown is orientated in the $x$ plane, i.e., a horizontal crab cavity, projected in the (a) $x$-$y$ and (b) $z$-$x$ planes.

Figure 3

$E$ fields with varying $s$ at $x=0.01\mathrm{m}$ and $y=0.00\mathrm{m}$. (a) $E_x$, (b) $E_y$, and (c) $E_z$. The Interpolated mesh fields and Fitted and Taylor analytical representations are shown. (d) $E_z$ transverse variation in $x$ and $y$ away from variation at $y=0\mathrm{m}$ from Taylor field.

Figure 4

$E$ fitting error with varying $x$. Difference between interpolated and Taylor fitted fields; (a) $E_x$ and (b) $E_z$.

Figure 5

Convergence of integration. Final $p_x$ error produced with WFR integration with varying step size compared with integration with step size 0.5 mm. Initial conditions $(x,p_x,y,p_y,\delta )=0$, $z=0.01$.

Figure 6

The computation steps taken to pass through one crab cavity in SixTrack to allow for a Taylor map to work in a thin tracking code.

Figure 7

Impact of different crab cavity models on the minimum dynamic aperture after $10^5$ turns, comparing the 500 coefficient Taylor map with the rf-multipole kick model (using multipoles calculated using the Helmholtz and PW decompositions) and the simple kick model.

Figure 8

Symplectic error, $E$, absolute matrix terms for initial values $(x,y,z)=0.01$ and $(p_x,p_y,\delta )=0.0001$ from eighth order in Taylor series map. Voltage normalized to 4.155 MV.