First $\beta$-beating measurement and optics analysis for the CERN Large Hadron Collider

Proton beams were successfully steered through the entire ring of the CERN Large Hadron Collider (LHC) on September the 10th of 2008. A reasonable lifetime was achieved for the counterclockwise beam, namely beam 2, after the radiofrequency capture of the particle bunch was established. This provided the unique opportunity of acquiring turn-by-turn betatron oscillations for a maximum of 90 turns right at injection. Transverse coupling was not corrected and chromaticity was estimated to be large. Despite this largely constrained scenario, reliable optics measurements have been accomplished. These measurements together with the application of new algorithms for the reconstruction of optics errors have led to the identification of a dominant error source.

Figure 1

The measurement of the horizontal (top) and vertical (bottom) $\beta$ functions for the LHC beam 2. The three different algorithms show consistent results. The locations of the LHC interaction regions (IRs) are marked on the top.

Figure 2

Comparison of the resulting measurement errors for the three algorithms used to reconstruct the $\beta$ functions. Histograms of the horizontal and vertical beta function relative measurement errors are shown on the top and bottom plots, respectively.

Figure 3

Measured $\beta$ functions (top two plots) and $\alpha$ functions (bottom two plots) for the LHC beam 2 together with the design values.

Figure 4

Measured horizontal (top) and vertical (bottom) beta beating versus longitudinal location together with tolerances.

Figure 5

Simulations (black and blue), observation (red), and tolerances (magenta) of the peak beta beating in the LHC, showing an unexpectedly large measured beta beating for beam 2.

Figure 6

Segment-by-segment approach in IR3. The top plot shows the gradient distribution versus longitudinal location. The two bottom plots show the horizontal and vertical beta functions from measurement and for two models (propagated by taking the initial $\alpha$ and $\beta$ as measured).

Figure 7

Comparison of measured beta functions (red) and the model (blue) after including all the findings from the segment by segment.

Figure 8

Comparison of model (blue) and measured beta functions (red) after five iterations of the model iterative correction (having started from the segment-by-segment results).

Figure 9

Integrated error distribution as obtained by the segment-by-segment and the iterative approaches.

Figure 10

Real and imaginary parts of the coupling resonance driving term $f_{1001}$ compared with a model reconstructed via the skew quadrupole response matrix.

Figure 11

Simulation of the performance of two different algorithms used to measure the $\beta$ functions. Histograms of the horizontal and vertical beta function relative measurement errors are shown on the top and bottom plots, respectively.

Figure 12

Relative deviation of measured and model beta functions as obtained from simulation.