Optimization of Triplet Field Quality in Collision Yuri Nosochkov (SLAC) Y. Cai, M-H. Wang (SLAC) S. Fartoukh, M. Giovannozzi, R. de Maria, E. McIntosh (CERN) 2nd Joint HiLumi LHC-LARP Annual Meeting 14—16 November 2012 INFN Frascati, Italy

Introduction High beta functions in the inner triplet (IT) of the HL-LHC lattice in collision increase the effects of IT field errors on dynamic aperture (DA). To maintain sufficient DA, the required IT field quality must be re-evaluated. The first part of the study assumed 120 mm aperture IT quadrupoles. Field components were projected from nominal MQXB quadrupole to the new quadrupole based on scaling laws. Then DA sensitivity to IT field errors was determined in tracking, and a self-consistent set of field errors was constructed producing DAmin = 12.3s. This result was reported at IPAC12. As a next step, these errors were compared with realistically expected field quality in such magnets and found too tight. Therefore, the IT field error tolerances must be relaxed as much as possible towards the achievable field quality. The second part of study is performed for IT quadrupoles with 150 mm coil aperture. Strategy to relax the tolerances included improvement of IT multipole field correction, setting a tighter level for acceptable minimum DA, and optimization of high order field coefficients. Lattice: HL-LHC V3.01, collision option “4444” with b*=15/15 cm at IP1 and IP5, SC IT quadrupoles with 150 mm coil diameter and 120 T/m gradient, 7 TeV beam energy. Tracking code SixTrack.

HL-LHC triplet b-functions High IT b-functions increase the non-linear effects of triplet field errors on DA. 150 mm IT quadrupoles will provide the necessary aperture for larger beam in the triplet and help relaxing the field tolerances. Nominal LHC b* = 55/55 cm HL-LHC V3.01 option “4444” b* = 15/15 cm bmax~ 4.5km bmax~ 21.5km

Multipole field scaling in a SC quadrupole where n=2 is for a quadrupole, etc. BQ is the main quadrupole field at r0 The bn and an coefficients represent a relative value of n-th order “normal” and “skew” fields compared to the main quad field at a given reference radius r0 (in 10-4 units). In LHC specifications, the an and bn are split in two components: the “uncertainty” terms anu, bnu (deviation from systematic) and the “random” terms anr, bnr. Their values correspond to sigma of a Gaussian distribution. Scaling with reference radius r0 does not affect dynamic aperture. Nominal MQXB quad r0 = 17 mm → new quad r0 = 50 mm. Scaling with coil diameter dc in a SC quad (B. Bellesia, et al., Phys. Rev. ST-AB 10, 062401 (2007)). Larger aperture reduces bn, an at fixed r0. Nominal MQXB dc = 70 mm → new dc = 150 mm. Scaling with peak IT beta function bmax to keep contribution of the IT field non-linear resonance driving terms constant (S. Fartoukh, SLHC Project Report 0038). Nominal bmax = 4.5 km → new bmax = 21.5 km.

Triplet field correctors The A3, B3, A4, B4, B6 correctors compensate the corresponding IT field errors thus relaxing the tolerances. The A5, B5, A6 correctors are also planned to be implemented, however they were not included in this study. A3, B3, A4, B4, B6 correctors IP

Set-up for SixTrack tracking 100,000 turns 60 random error seeds 30 particle pairs per amplitude step (2s) 11 angles 7 TeV beam energy Initial Dp/p = 2.7e-4 Tune = 62.31, 60.32 Normalized emittance = 3.75 mm-rad IT field correctors to compensate A3, B3, A4, B4, B6 terms are included For simulations, the IT quadrupole reference radius is set to 40 mm Arc errors and correction are included No field errors in D1, D2 separation dipoles and Q4 quadrupoles (future study)

Expected to achieve field quality in 150 mm aperture IT quadrupole at r0 = 50 mm ``REVIEW OF ESTIMATES OF RANDOM COMPONENTS IN THE INNER TRIPLET’’ E. Todesco, Hi-Lumi and LARP Collaboration Meeting, CERN, June 7, 2012 This table will be further referred as error table “target4”. For SixTrack simulations, the coefficients are scaled to r0 = 40 mm.

DA sensitivity to IT field errors DA sensitivity to individual An, Bn terms was initially studied for IT quads with dc=120 mm. The An, Bn terms were obtained by scaling from measured field of the nominal MQXB quadrupole. Conclusions for dc=120 mm: The impact of high-order terms is not small except the Anu terms. Triplet correctors for n=3,4 terms work very well. The B6 correction showed some residual impact on DA, possibly due to effects of uncorrected feed-down terms caused by IR orbit. Anu Anr (An, Bn in normalized units) Bnu Bnr

Error table providing DAmin = 12 Error table providing DAmin = 12.3s based on the initial sensitivity study (normalized to “target4” values) This table is based on approximately equal impact of each coefficient on DA. Most values are tight relative to the “target4” except for high order Anu coefficients. skew uncertainty rms normal a3 0.40 0.49 b3 0.32 0.61 a4 0.78 0.68 b4 0.27 a5 0.36 0.08 b5 0.25 0.19 a6 0.43 0.10 b6 0.37 0.38 a7 0.66 0.15 b7 0.06 0.07 a8 0.82 b8 0.22 0.04 a9 2.48 0.16 b9 0.14 0.13 a10 1.95 b10 0.39 a11 2.42 b11 0.26 0.31 a12 2.59 0.17 b12 0.21 a13 2.63 b13 0.28 a14 2.17 0.42 b14 0.70

Strategy for relaxing the tolerances The desired goal is to approach the values of the error table “target4”. There is an indication that with some effort the achievable field quality could be further improved by as much as 50%. Therefore, our target range is 50-100% values of the table “target4”. Relaxing the tolerances requires an improvement of the IT field correction. Therefore, we take into account that A5, B5, A6 correctors are already planned to be included (although they are not explicitly included in the studied lattice). To simulate their effect, we allow the A5, B5, A6 terms to be smaller assuming that they represent residual errors after correction. Secondly, the looser tolerances require a compromise on the acceptable minimum DA. In this study, we set the acceptable DA limit to 10.6s for the design 3.75 mm-rad emittance. Note that this corresponds to 13s if the HL-LHC emittance will be 2.5 mm-rad as predicted. A smaller acceptable DA reduces the relative importance of highest order field terms thus helping to relax the other terms as well. Final optimization of the tolerances require fine tuning of the field coefficients using scans. This process should start from scanning and relaxing the high order terms since their tolerances are likely more difficult to control.

Dynamic aperture at full and half values of table “target4” ”target4”, DAmin=6.79s Half “target4”, DAmin=8.69s DA is not sufficient even if all the errors are set to half-values of table “target4”. Clearly, as a first step, improvement of IT field correction needs to be considered.

Scan of A5, B5, A6 As a first step, the A5, B5, A6 terms are considered to be partially corrected with the A5, B5, A6 correctors. This scan indicates that for DA >10.6s, the residual A5, B5, A6 errors after correction should be near 0.2 level relative to “target4” values. The least sensitive terms n=3,4 and Anu (n=7-14) can be set at “target4” values.

Scan of high order Anr, Bnu, Bnr (n=7-14) when A5, B5, A6 are at 0 Scan of high order Anr, Bnu, Bnr (n=7-14) when A5, B5, A6 are at 0.2 relative to “target4” The case in the circle (table “target424”) satisfies our minimal requirement , where DA>10.6s, and the terms are at least half of the “target4” values, except A5, B5, A6 which are set at 0.2 (assumed to be after correction). The non-sensitive terms n=3,4 and Anu (n=7-14) are set to full “target4” values.

Scan of high order Bn (n=9-14) Starting from table “target424”, the high-order Bn (n=9-14) can be relaxed from 0.5 to 0.7 (relative to “target4” values) while keeping the DAmin >10.6s. This corresponds to table “target427”.

Scan of B8 Starting from table “target427”, the B8 can be relaxed to 0.6 (relative to “target4” value) corresponding to table “target431”.

Scan of B7 Starting from table “target431”, this scan shows that B7 should be kept at 0.5 relative to “target4” value.

Scan of high order Anr (n=9-14) Starting from table “target431”, the high-order Anr (n=9-14) can be relaxed to 0.7 (relative to “target4” values) corresponding to table “target468”.

Scan of A8r Starting from table “target468”, this scan shows that A8r should be kept at 0.5 relative to “target4” value.

Scan of A7r Starting from table “target468”, this scan shows that A7r should be kept at 0.5 relative to “target4” value.

The currently best error tolerance table “target468” (normalized to “target4” values) skew uncertainty rms normal a3 1.0 b3 a4 b4 a5* 0.2 b5* a6* b6 0.5 a7 b7 a8 b8 0.6 a9 0.7 b9 a10 b10 a11 b11 a12 b12 a13 b13 a14 b14 * A5, B5, A6 terms are assumed to be residual errors after A5, B5, A6 correction.

Dynamic aperture for error table “target468” DAmin=10.69s

Summary Field error tolerances for 150 mm aperture IT quadrupoles in the HL-LHC V3.01 lattice have been optimized. The resultant error table (“target468”) is reasonably close to the expected achievable field quality (table “target4”), while providing DAmin > 10.6s at 3.75 mm-rad normalized emittance. Most of the un-corrected terms exceed half-values of table “target4”, with the high order terms set to at least 0.7 level of “target4”. The least sensitive terms n=3,4 (corrected) and Anu (n=7-14) can be set to full values of “target4”. It was identified that A5, B5, A6 correctors need to be included, and that they need to reduce the residual A5, B5, A6 errors to the level of 0.2 relative to “target4”. It will be important to verify the level of this correction in tracking when these correctors are included.

Back-up slides

Error table “target468” at r0 = 50 mm skew uncertainty rms normal a3 0.712 b3 a4 0.512 b4 a5* 0.074 b5* a6* 0.192 0.053 b6 0.720 a7 0.168 0.084 b7 a8 0.128 0.064 b8 0.077 a9 0.045 b9 a10 0.048 0.033 b10 a11 0.032 0.022 b11 a12 0.021 0.015 b12 a13 0.014 0.010 b13 a14 0.009 0.006 b14

Error table “target4” at r0 = 40 mm skew uncertainty rms normal a3 0.5696 B3 a4 0.3277 b4 a5 0.1884 b5 a6 0.3932 0.1081 b6 0.5898 0.4194 a7 0.0551 b7 a8 0.0336 b8 a9 0.0134 b9 a10 0.0081 b10 a11 0.0043 b11 a12 0.0023 b12 a13 0.0012 b13 a14 0.0006 b14