Accuracy and feasibility of the ${\beta }^{*}$ measurement for LHC and High Luminosity LHC using $k$ modulation

The future regimes of operation of the LHC will require improved control of ${\beta }^{*}$ measurements to successfully level the luminosities in the interaction points. The method of $k$ modulation has been widely used in other machines to measure lattice beta functions. In the LHC, $k$ modulation of the closest quadrupoles to the interaction point (IP) is the most accurate method to measure ${\beta }^{*}$ at the IP. This paper highlights the challenge of high precision tune measurements (down to ${10}^{-5}$) for the correct determination of ${\beta }^{*}$ in the high luminosity LHC. Furthermore it presents a new analytical method for the calculation of ${\beta }^{*}$ using $k$ modulation.

Figure 1

Schematic representation of interaction region configuration. This corresponds to the default horizontal plane as defined in mad-x models of the LHC, with a focusing quadrupole on the right and a defocusing quadrupole on the left.

Figure 2

Schematic representation of the closest tune approach. The unperturbed tunes are labeled as $Q_x$ and $Q_y$, while the perturbed tunes are given by $Q_1$ and $Q_2$.

Figure 3

Fringe field as simulated [26] and modeled for the MQXF quadrupoles with an aperture of 150 mm. In black is the simulated field of the magnet, and in red is the modeled field using the fifth order Enge function. The modeled fringe field agrees very well with the simulated data and validates the used order of the Enge function.

Figure 4

Uncertainty on ${\beta }^{*}$ measurements for $\delta Q\pm {10}^{-5}$ using $k$ modulation for the LHC (top figure) and HL-LHC (bottom figure). The vertical axis shows the waist relative to the design ${\beta }^{*}$ (LHC: ${\beta }_{\text{design}}^{*}=0.55\mathrm{m}$, HL-LHC:${\beta }_{\text{design}}^{*}=0.15\mathrm{m}$).

Figure 5

Uncertainty $\mathrm{\Delta }{\beta }^{*}/{\beta }^{*}$ for the LHC (top figure) and HL-LHC (bottom figure) over the relevant tune shifts. The minimum error occurs at $\chi (0)$ for both the LHC and HL-LHC.

Figure 6

The calculated ${\beta }^{*}$ calc using the new analytical method in the LHC (top figure) and HL-LHC (bottom figure) including the effect of fringe fields on the last quadrupoles. Results are presented for the horizontal plane.

Figure 7

Calculated errors using the analytical method for different contributions and for the range of LHC (top figure) and HL-LHC (bottom figure) optics. In red are the errors found for simulations without fringe fields. Results as presented before are shown when including fringe fields are shown in blue, while errors found when including uncertainties in $K$ $(0.1\%)$ and $L^{*}$ (6 mm) are shown in green.

Figure 8

Error on ${\beta }^{*}$ measurements versus the tune accuracy for different optics. The error grows rapidly with increasing tune uncertainty. A tune resolution of $5\times {10}^{-5}$ will result in an 120% error on the ${\beta }^{*}$ measurement for the design optics of ${\beta }^{*}=0.15\mathrm{m}$.

Figure 9

Deviation of calculated ${\beta }^{*}$ for different values of coupling for the LHC (top figure) and HL-LHC (bottom figure). Results show an increased deviation for increasing coupling in the accelerator.

Figure 10

Deviations of calculated ${\beta }^{*}$ for the different optics of the LHC and HL-LHC, independently showing the effect of linear coupling with $C^{-}=5\times {10}^{-4}$ and a quadrupole tilt of 1 mrad. Though the effect is small for both, a quadratic increase of the deviations for decreasing ${\beta }^{*}$ is observed.