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

2. 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

4. 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

5. 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

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

7. 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

8. 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

9. Tolerance for the quadrupoles in the IR5 triplet

10. Tolerances for the quadrupole strengths in IR5

11. Quadrupole Name (  k/k) min (  k/k) MAX kqx3_l5 -0.000203330.00018697 kqx3_r5 -0.000203330.00018697 kqx2b_r5 -0.000232320.00021381 kqx2a_r5 -0.000254740.00023409 kqx2b_l5 -0.000383840.00035410 kqx2a_l5 -0.000344280.00031755 kqx1_l5 -0.000612960.00066475 kqx1_r5 -0.000612960.00066478 Summary table for the IR5 triplet

12. Tune variation range

13. Chromaticity variation range

14. Second order dispersion variation range

15. Quad Name (  k/k) min (  k/k) MAX KQTL11.L5B1-0.359190.38942 KQTL11.R5B1-0.678270.72733 Quad Name (  k/k) min (  k/k) MAX KQTL11.L5B2-0.288540.30940 KQTL11.R5B2-0.629520.68251 Tolerances for the quadrupoles in MS and DS Beam 2 Beam 1

16. Tune variation range Beam 1

17. Chromaticity variation range Beam 1

18. Tune variation range Beam 2

19. Chromaticity variation range

20. Beam 2 Beam 1 Second order dispersion variation range

21. Quad Name (  k/k) min (  k/k) MAX kq4.l5b1-0.0109490.010209 KQ4.R5B1-0.0107170.0098841 KQ5.L5B1-0.0226190.024253 KQ5.R5B1-0.0219010.023742 KQ6.L5B1-0.0474950.044304 KQ6.R5B1-0.0377780.034854 KQ7.L5B1-0.00781510.0072148 KQ7.R5B1-0.00847850.0079143 KQ8.L5B1-0.0839390.078648 KQ8.R5B1-0.129600.12034 KQ9.L5B1-0.0638190.059148 KQ9.R5B1-0.0842200.080542 KQ10.L5B1-0.0115480.010774 KQ10.R5B1-0.0113840.010498 KQTL11.L5B1-0.359190.38942 KQTL11.R5B1-0.678270.72733 Quad Name (  k/k) min (  k/k) MAX KQ4.L5B2-0.0112850.010408 KQ4.R5B2-0.0115290.010750 KQ5.L5B2-0.0244210.026475 KQ5.R5B2-0.0252220.027045 KQ6.L5B2-0.0351080.032390 KQ6.R5B2-0.0468580.043708 KQ7.L5B2-0.00822210.0076722 KQ7.R5B2-0.00811960.0074961 KQ8.L5B2-0.0809640.075036 KQ8.R5B2-0.0871920.081707 KQ9.L5B2-0.0640340.059996 KQ9.R5B2-0.0805980.074786 KQ10.L5B2-0.0131990.012179 KQ10.R5B2-0.0112500.010491 KQTL11.L5B2-0.288540.30940 KQTL11.R5B2-0.629520.68251 Summary table for the quadrupoles in MS and DS Beam2Beam1

22. 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.