1. 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

2. 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.

3. 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

4. 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, (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.

5. 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

6. 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)

7. 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.

8. 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

9. 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

10. 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 % 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.

11. 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.

12. 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.

13. 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.

14. 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”.

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

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

17. 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”.

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

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

20. 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.

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

22. 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.

24. Back-up slides

25. 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

26. 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