High Luminosity LHC Robustness & tolerances Catia Milardi HL LHC Task 2.2, September 19 th 2012

Topics Tolerances to gradient errors in quadrupoles method results for quadrupoles in: IR5 Matching Section (MS) Dispersion Suppressor (DS) impact of the gradient errors on linear and nonlinear optics Next steps

Low-  triplet Matching Section quadrupoles Dispersion Suppressor quadrupoles: Q8, Q9, Q10, QT11 All the quads in MS and in DS are double bore magnet housing separated beam pipe for each ring Layout of half IR1 (IR5) IR1 and IR5 are equal in terms of quadrupole configuration

A  gradient error  k affects: linear optics in terms of:  -beating tune shift orbit and dispersion distortion chromaticity shift non-linear optics changing chromatic functions Linear optics variation can been evaluated analytically, under some approximations. In the present study the impact of gradient errors on the LHC optics are evaluated numerically by using the MAD-X model Where: Q is the betatron tune  is the phase advance between the quadrupole error and the observation point Gradient error  k

Optics: slhc/opt_0150_0150.madx (15 cm) Tolerance criteria: Parameter step: Parameter range Terms of the study (for triplet quadrupoles)

Method Optics is computed for each quadrupole and for each beam using a different k Output files return the following parameters: name, s,k1l, betx,bety,dx,dy,DDX …….. For each Quadrupole twiss_lhcb1.tfs twiss_lhcb1_kqx1l5_1p0001.tfs twiss_lhcb1_kqx1l5_1p0002.tfs twiss_lhcb1_kqx1l5_1p0003.tfs twiss_lhcb1_kqx1l5_1p0004.tfs twiss_lhcb1_kqx1l5_1p0005.tfs twiss_lhcb1_kqx1l5_1p0006.tfs twiss_lhcb1_kqx1l5_1p0008.tfs twiss_lhcb1_kqx1l5_1p0007.tfs twiss_lhcb1_kqx1l5_1p0009.tfs twiss_lhcb1_kqx1l5_p9991.tfs twiss_lhcb1_kqx1l5_1p001.tfs twiss_lhcb1_kqx1l5_p9993.tfs twiss_lhcb1_kqx1l5_p9992.tfs twiss_lhcb1_kqx1l5_p9994.tfs twiss_lhcb1_kqx1l5_p9996.tfs twiss_lhcb1_kqx1l5_p9995.tfs twiss_lhcb1_kqx1l5_p9997.tfs twiss_lhcb1_kqx1l5_p9998.tfs twiss_lhcb1_kqx1l5_p9999.tfs twiss_lhcb1_kqx1l5_p999.tfs twiss_lhcb2.tfs twiss_lhcb2_kqx1l5_1p0001.tfs twiss_lhcb2_kqx1l5_1p0002.tfs twiss_lhcb2_kqx1l5_1p0003.tfs twiss_lhcb2_kqx1l5_1p0005.tfs twiss_lhcb2_kqx1l5_1p0004.tfs twiss_lhcb2_kqx1l5_1p0006.tfs twiss_lhcb2_kqx1l5_1p0007.tfs twiss_lhcb2_kqx1l5_1p0008.tfs twiss_lhcb2_kqx1l5_1p0009.tfs twiss_lhcb2_kqx1l5_p9991.tfs twiss_lhcb2_kqx1l5_1p001.tfs twiss_lhcb2_kqx1l5_p9992.tfs twiss_lhcb2_kqx1l5_p9993.tfs twiss_lhcb2_kqx1l5_p9994.tfs twiss_lhcb2_kqx1l5_p9996.tfs twiss_lhcb2_kqx1l5_p9995.tfs twiss_lhcb2_kqx1l5_p9997.tfs twiss_lhcb2_kqx1l5_p9998.tfs twiss_lhcb2_kqx1l5_p9999.tfs twiss_lhcb2_kqx1l5_p999.tfs

SCRIPT ela.sh Deals with input-output files Runs two fortran programs F1 and F2 F1: compare the output file for each Dk/k with the reference one produce an summary file with:  q1  q2  d q1  d q2  k/k (  x /  x ) MAX (  y /  y ) MAX (  x /  x ) MAX (  ” x /  ” x ) MAX (  x /  x ) min (  y /  y ) min (  x /  x ) min (  ” x /  ” x ) min F2: Finds out which one among the parameters: (  x /  x ) MAX (  y /  y ) MAX (  x /  x ) min (  y /  y ) min violates the tolerance criteria Interpolate data to get the exact (  /  ) value corresponding to (  /  ) = ±0.1 Analysis

Tolerance for the quadrupoles in the IR5 triplet

Tolerances for the quadrupole strengths in IR5

Quadrupole Name (  k/k) min (  k/k) MAX kqx3_l kqx3_r kqx2b_r kqx2a_r kqx2b_l kqx2a_l kqx1_l kqx1_r Summary table for the IR5 triplet

Tune variation range

Chromaticity variation range

Second order dispersion variation range

Quad Name (  k/k) min (  k/k) MAX KQTL11.L5B KQTL11.R5B Quad Name (  k/k) min (  k/k) MAX KQTL11.L5B KQTL11.R5B Tolerances for the quadrupoles in MS and DS Beam 2 Beam 1

Tune variation range Beam 1

Chromaticity variation range Beam 1

Tune variation range Beam 2

Chromaticity variation range

Beam 2 Beam 1 Second order dispersion variation range

Quad Name (  k/k) min (  k/k) MAX kq4.l5b KQ4.R5B KQ5.L5B KQ5.R5B KQ6.L5B KQ6.R5B KQ7.L5B KQ7.R5B KQ8.L5B KQ8.R5B KQ9.L5B KQ9.R5B KQ10.L5B KQ10.R5B KQTL11.L5B KQTL11.R5B Quad Name (  k/k) min (  k/k) MAX KQ4.L5B KQ4.R5B KQ5.L5B KQ5.R5B KQ6.L5B KQ6.R5B KQ7.L5B KQ7.R5B KQ8.L5B KQ8.R5B KQ9.L5B KQ9.R5B KQ10.L5B KQ10.R5B KQTL11.L5B KQTL11.R5B Summary table for the quadrupoles in MS and DS Beam2Beam1

Conclusions and further studies Tolerances to gradient errors have been evaluated for quadrupoles in IR5, MS and DS and their values are quite reasonable As expected gradient errors have a relevant impact on linear and non-linear optics parameters Relying on the present results a further study is required to understand the optics flexibility in compensating such errors.