1. The HiLumi LHC Design Study is included in the High Luminosity LHC project and is partly funded by the European Commission within the Framework Programme 7 Capacities Specific Programme, Grant Agreement 284404. Work supported by the US LHC Accelerator Research Program (LARP) through US Department of Energy contracts DE-AC02-07CH11359, DE-AC02-98CH10886, DE-AC02-05CH11231, and DE- AC02-76SF00515. Field Quality Requirements for Separation Dipoles and Matching Quadrupoles at Collision Energy Based on Dynamic Aperture Yuri Nosochkov (SLAC) Y. Cai, M.-H. Wang (SLAC) S. Fartoukh, M. Giovannozzi, R. de Maria, E. McIntosh (CERN) 3 rd Joint HiLumi LHC—LARP Meeting 11—15 November 2013, Daresbury, UK

2. Introduction New large aperture magnets are planned for the HL-LHC lattice: superconducting 150 mm D1 and 105 mm D2 separation dipoles, 90 mm Q4 and 70 mm Q5 matching quadrupoles near IP1 and IP5. High beta functions in these magnets enhance sensitivity to their field errors causing reduction of dynamic aperture (DA). Field quality in these magnets needs to be evaluated and optimized to satisfy two conflicting requirements: the field errors must be small enough to provide a sufficient DA (~10  ), but large enough to be realistically achievable. Estimates of field quality obtained from measured data and magnetic field calculations are used as a starting point for evaluation and optimization. Impact of the field errors on DA is determined in tracking simulations using SixTrack. Lattice: SLHCV3.1b with  *=15/15 cm at IP1 and IP5, SC IT quadrupoles with 150 mm coil diameter and 150 T/m gradient, 7 TeV beam energy.

3. Beta functions High  -functions in the D1, D2, Q4, Q5 enhance beam sensitivity to their field errors. Field correctors for the IT also compensate the low order D1 field errors (n=3-6) since the two beams share the D1 aperture. 2-in-1 D2 and Q4, Q5 magnets do not have local correctors. D1 D2Q4Q5D1D2Q4Q5 ≈180°

4. IT field quality specifications at r 0 = 50 mm (“IT_errortable_v66”) These IT specifications is the result of previous optimization studies. In this study, the IT specification errors are always included. skewmeanuncertaintyrandom normalmeanuncertaintyrandom a300.800 b300.820 a400.650 b400.570 a500.430 b500.420 a600.310 b60.8000.550 a700.1520.095b700.095 a800.0880.055b800.065 a900.0640.040b900.035 a1000.0400.032b100.0750.100 a1100.0260.0208b1100.0208 a1200.014 b1200.0144 a1300.010 b1300.0072 a1400.005 b14-0.0200.0115

5. Q4 field errors at r 0 = 30 mm (“Q4_errortable_v1”) skewmeanuncertaintyrandom normalmeanuncertaintyrandom a300.6821.227b301.2821.500 a400.4280.893b400.4830.465 a500.1770.406b500.2030.431 a600.4840.277b605.1871.487 a700.0940.189b700.0940.189 a800.1930.257b800.1930.257 a900.088 b900.088 a1000.120 b1003.5870.956 a1100.3260.489b1100.3260.489 a120 0.4450.222b1200.4450.222 a1300.6060.303b1300.6060.303 a1400.8270.413b1402.0670.413 a1501.1270.564b1501.1270.564 Estimate is based on scaling from the measured field of existing MQY quadrupole with 70 mm aperture and applied to 90 mm Q4. This estimate is expected to be updated.

6. Q5 expected field quality at r 0 = 17 mm (“Q5_errortable_v0”) skewmeanuncertaintyrandom normalmeanuncertaintyrandom a300.5000.900b300.9401.100 a400.2300.480b400.2600.250 a500.0700.160b500.0800.170 a600.1400.080b601.5000.430 a700.0200.040b700.0200.040 a800.0300.040b800.0300.040 a900.010 b900.010 a1000.010 b1000.3000.080 a1100.0200.030b1100.0200.030 a1200.0200.010b1200.0200.010 a1300.0200.010b1300.0200.010 a1400.0200.010b1400.0500.010 a1500.0200.010b1500.0200.010 Estimate is based on the measured field of existing MQY quadrupole with 70 mm aperture which is of the same type as Q5.

7. D1 expected field quality at r 0 = 50 mm (“D1_errortable_v1”) skewmeanuncertaintyrandom normalmeanuncertaintyrandom a200.679 b200.200 a300.282 b3-0.90.727 a400.444 b400.126 a500.152 b500.365 a600.176 b600.060 a700.057 b70.40.165 a800.061 b800.027 a900.020 b9-0.590.065 a1000.025 b1000.008 a1100.007 b110.470.019 a1200.008 b1200.003 a1300.002 b1300.006 a1400.003 b1400.001 a1500.001 b15-0.040.002 Estimate is based on magnetic field calculations for 160 mm aperture D1 magnet (T. Nakamoto, E. Todesco, CERN-ACC-2013-002).

8. D2 field errors at r 0 = 35 mm (“D2_errortable_v3”) skewmeanuncertaintyrandom normalmeanuncertaintyrandom a200.6790.6790b2±653.0 a300.2820.2820b3-305.0 a400.4440.4440b4±251.0 a500.152 b5-4.01.0 a600.176 b600.060 a700.057 b7-0.20.165 a800.061 b800.027 a900.020 b90.090.065 a1000.025 b1000.008 a1100.007 b110.030.019 a1200.008 b1200.003 a1300.002 b1300.006 a1400.003 b1400.001 a1500.001 b1500.002 Estimate is based on magnetic field calculations for 2-in-1 D2 dipole (E. Todesco 01-Jan-2013). These values were obtained for a shorter magnet, therefore they may be potentially reduced for the longer D2. The large values of b2, b3, b4, b5 terms are due to field saturation.

9. Latest D2 field estimate at r 0 = 35 mm (“D2_errortable_v4”) skewmeanuncertaintyrandom normalmeanuncertaintyrandom a200.6790.6790b2±65→±253.0→2.5 a300.2820.2820b3-30→3.05.0→1.5 a400.4440.4440b4±25→±2.01.0→0.2 a500.152 b5-4.0→-1.01.0→0.5 a600.176 b600.060 a700.057 b7-0.20.165 a800.061 b800.027 a900.020 b90.090.065 a1000.025 b1000.008 a1100.007 b110.030.019 a1200.008 b1200.003 a1300.002 b1300.006 a1400.003 b1400.001 a1500.001 b1500.002 The recent optimization of iron geometry and coil in D2 (E. Todesco) resulted in significant reduction of b2, b3, b4, b5 terms at collision energy (D2_errortable_v4). It also significantly reduced the mean values of b3 (95.8→3.8) and b5 (15→3.0) at injection energy. However, for most of this study, the D2_errortable_v3 was used as a reference table.

10. Typical set-up for SixTrack tracking 100,000 turns 60 random error seeds 30 particle pairs per amplitude step (2   11 angles 7 TeV beam energy Initial  p/p = 2.7e-4 Tune = 62.31, 60.32 Normalized emittance = 3.75  m-rad Arc errors and correction are included IT local correctors to compensate an, bn errors of order n=3-6 in IT quads and D1 dipoles are included

11. DA without D1, D2, Q4, Q5 errors This is our starting DA with only IT and arc errors included. The goal is to optimize D1, D2, Q4, Q5 errors in order to keep minimum DA near 10 .

12. Kicks due to an, bn terms in D1, D2 at x=10  x (and similar for y’ from a n ) Largest kicks are produced by b2m, b3m, b4m in D2_v3 table. They are further amplified by ≈180° phase between left and right side D2 magnets around IP, and by the fact that mean bn terms of even order are of opposite sign in the left and right D2. These kicks are substantially reduced in the D2_v4 error table.

13. Impact of large b2 in D2_v3 on DA Large b2 terms in D2_v3 table affect beta functions by increasing  * and reducing peak  in the IT. This reduces impact of IT, D1, D2 errors resulting in a larger DA. For comparison,  KL focusing strength from b2m=65 in one D2 is equivalent to ¼ of a regular arc quad or 7% of Q4. Beta perturbation from left side D2 is amplified by the right side D2. To maintain luminosity, this perturbation will be compensated in operation. To simulate such compensation and avoid too optimistic DA, we set b2=0 in D2 in all simulations.

14. Example of beta perturbation due to b2 in D2_v3 b2=0 b2×1/8 b2×2/8 b2×3/8 b2×2/8 b2×1/8 b2=0 IT peak beta IP beta X X Y Y

15. Feed-down due to orbit in D1, D2 Beam trajectory in D1, D2 is, on average, horizontally offset relative to magnet axis resulting in lower order feed-down bn, an terms. Example for D2_v3 table: b4m=25 creates an average feed-down term =3.6 (12% of main b3m), and b5m=-4 creates feed-down =-0.77 (3% of main b4m). Feed-down effect on DA will be verified in tracking.

16. Relative impact of Q4_v1, Q5_v0, D1_v1, D2_v3 errors Q4, Q5 errors have minimal effect on DA, hence are acceptable. D2 errors without b2, b3, b4, b5 are acceptable. D1 errors should be optimized. D2_v3

17. Impact of bn in D1_v1 with D2 errors off Large DA reduction due to b7m, b9m. Consider b7m, b9m reduction by a factor of 2. D1 errors for n ≤ 6 are compensated using local IT correctors. 7 th order resonances 3 rd, 6 th, 9 th order resonances

18. Scan of b5 in D2_v3 Effect of b5 in D2_v3 on DA is small. Minor effect from feed-down.

19. Scan of b4 in D2_v3 Strong impact on DA requires a factor of 10 reduction of b4 relative to D2_v3 table. Feed-down has minor impact at reduced b4.

20. Scan of b3 in D2_v3 Strong DA reduction requires a factor of 10 reduction of b3 relative to D2_v3 table. Feed-down effect is small.

21. Sensitivity to b3, b4 uncertainty and random terms b3u,r should be reduced at least a factor of 2 relative to D2_v3 table. b4u,r impact is small.

22. Scan of D2 b3m and b4m with other errors close to D2_v4 table Consider reducing b3m by a factor of 20 relative to D2_v3 (i.e. factor of 2 relative to D2_v4). D2_v4 value

23. Recommended target for D1 field quality skewmeanuncertaintyrandom normalmeanuncertaintyrandom a200.6790.6790b200.200 a300.2820.2820b3-0.90.727 a400.4440.4440b400.126 a500.152 b500.365 a600.176 b600.060 a700.057 b70.4→0.20.165 a800.061 b800.027 a900.020 b9-0.59→-0.2950.065 a1000.025 b1000.008 a1100.007 b110.470.019 a1200.008 b1200.003 a1300.002 b1300.006 a1400.003 b1400.001 a1500.001 b15-0.040.002 Reduce b7m and b9m a factor of 2 relative to D1_errortable_v1.

24. Recommended target for D2 field quality skewmeanuncertaintyrandom normalmeanuncertaintyrandom a200.6790.6790b2±252.5 a300.2820.2820b33.0→1.51.5 a400.4440.4440b4±2.00.2 a500.152 b50.5 a600.176 b600.060 a700.057 b7-0.20.165 a800.061 b800.027 a900.020 b90.090.065 a1000.025 b1000.008 a1100.007 b110.030.019 a1200.008 b1200.003 a1300.002 b1300.006 a1400.003 b1400.001 a1500.001 b1500.002 1) Use D2_errortable_v4 and further reduce b3m a factor of 2. 2) Minimize the b2 term or compensate its impact on beta function. Correction options are not yet decided, but may include adjustment of Q4 gradient or D2 spool-piece correctors.

25. DA with Q4_v1, Q5_v0 and target D1, D2 errors Minimum DA is 9.9  dominated by vertical DA.

26. DA with target D1, D2 errors versus original D1_v1, D2_v4 Most improvement is near the vertical plane: +0.5  for DAmin and +0.7  for DAave. Further increase of vertical DA may be possible by optimizing vertical phase advance between IP1 and IP5 (in SLHCV3.1b it is integer x 2  causing amplification of systematic error effects in vertical plane in the IP1 and IP5 magnets).

27. Summary Impact on DA from estimated field errors in the large aperture D1, D2 separation dipoles and Q4, Q5 matching quadrupoles in the SLHCV3.1b lattice has been evaluated. The Q4, Q5 field errors were found to be acceptable. Field error terms – b7m, b9m in D1, and b3m in D2 – were identified as having the most impact on the DA. In order to obtain the minimum DA near 10 , these three terms were reduced a factor of 2 relative to D1_errortable_v1 and D2_errortable_v4, respectively. It is also critical to minimize the large D2 term b2 or compensate its impact on beta function. Correction options are not yet decided, but may include adjustment of Q4 strength or implementation of D2 spool-piece correctors. The feed-down effect due to offset orbit in the D1, D2 was found to be small. In summary, the evolution of minimum DA in optimization is as follows:   With the IT errors and without D1, D2, Q4, Q5 errors the starting DAmin is 10.38 .   With the IT, Q4, Q5 errors and D1_v1, D2_v3 error tables the DA would be extremely small.   With the IT, Q4, Q5 errors and D1_v1, D2_v4 error tables before optimization DAmin is 9.36 .   With the IT, Q4, Q5 errors and optimized D1, D2 errors DAmin is 9.90 . Further improvement of the minimum DA may be possible by optimizing vertical phase advance between IP1 and IP5.