1. Muon Collider Lattice Design FERMI NATIONAL ACCELERATOR LABORATORY US DEPARTMENT OF ENERGY f Yuri Alexahin, Eliana Gianfelice-Wendt (Accelerator Physics Center) Workshop on Dynalic Aperture for Ultimate Storage Rings, IU, Bloomington November 11-12, 2010

2. MC Lattie Design Workshop on DA for Ultimate Storage Rings, IU November 11, 2010 The Road to High Luminosity 2 The recipe:  Pack as many muons per bunch as the beam-beam effect allows (in practice this means 1 bunch / beam)  Make beams round to maximize the beam-beam limit  Invent new chromatic correction scheme to reduce  *  Do not leave free spaces to reduce C (also necessary to spread neutrino radiation) C – collider circumference (limited from below by available B-field)  – muon lifetime  * – beta-function at IP (limited from below by chromaticity of final focusing and aperture restrictions in IR magnets), small  * requires small  z  large  p / p =  || /  z – beam-beam parameter (limited by particle stability,  < 0.1/IP ?) h  z /   “Hour-glass factor” P  – average muon beam power (limited by the P-driver power )

3. MC Lattice Design Challenges 3 What we would like to achieve compared to other machines: MCTevatronLHC Beam energy (TeV)0.750.987  * (cm) 12855 Momentum spread (%)>0.1<0.010.0113 Bunch length (cm)15015 Momentum compaction factor (10^-3)0.052.30.322 Geometric r.m.s. emittance (nm)3.530.5 Particles / bunch (10^11)202.71.15 Beam-beam parameter,  0.10.0250.01 Muon collider is by far more challenging:  much larger momentum acceptance with much smaller  *  ~ as large Dynamic Aperture (DA) with much stronger beam-beam effect - New ideas for IR magnets chromaticity correction needed! MC Lattie Design Workshop on DA for Ultimate Storage Rings, IU November 11, 2010

4. Various MC lattice designs studied  1996 by Carol J., A. Garren  1996 by K.Oide  “ Dipole first ” (2007) ~ satisfy the requirements  Eliana ’ s “ synthetic ” (2009)  Asymmetric dispersion  “ Flat top ”  “Three sextupoles” scheme – the final solution based on Eliana’s preliminary design. ________________  1996 designs used special -I chromatic correction sections, had very high sensitivity to field errors.  Oide’s lattice: sextupoles arranged in non- interleaved pairs.  “ Dipole first ” - our first exercise defying conventional wisdom: all sextupoles were interleaved. Decent momentum acceptance and DA, but high sensitivity to beam-beam effect. 4 MC Lattie Design Workshop on DA for Ultimate Storage Rings, IU November 11, 2010

5. Chromatic Correction Basics 5 Montague chromatic functions : A x,y are created first, and then converted into B x,y as phase advances  x,y grow K 1, K 2 are normalized quadrupole and sextupole gradients, D x is dispersion function: D x = dx c.o. /d  p The mantra: Kill A’s before they transform into B’s ! - difficult to achieve in both planes - requires dispersion generated close to IP or large D x  0 at IP B x,y are most important since they determine modulation of phase advance  x,y  x,y = -  x,y /2,  x,y are Twiss lattice functions,  p is relative momentum deviation. Equations for chromatic functions MC Lattie Design Workshop on DA for Ultimate Storage Rings, IU November 11, 2010

6. IR design by K.Oide  *=3mm, E beam =2TeV Is this design practical? - huge  max ~ 1000 km - too small quad aperture? First quad G=214T/m (NbTi), with Nb 3 Sn B0=10T  a~5cm but is this enough to accommodate the shielding? QC1-QC4 have small octupole component MC Lattie Design Workshop on DA for Ultimate Storage Rings, IU November 11, 2010 6

7. “Local” chromatic correction with Oide’s design  y  x axax ayay d  x /2  - goes to - 4824.5 ! bxbx byby d  x /2   * = 3mm,  max = 901,835 m hor. CC ver. CC with chromatic correction sections separated from IR there inevitably are places with large chromatic modulation of betatron phase advances – potential for a trouble chromatic phase (arg a x,y / b x,y ) advances by 2  at locations where respective beta- functions are low MC Lattie Design Workshop on DA for Ultimate Storage Rings, IU November 11, 2010 7

8. KEK ATF arc cell with negative  c The defocusing sextupoles in the arc cells have gradients up to 67000 T/m^2 at the beam energy 2TeV. In the 750GeV case, if we reduce the geometrical sizes ~E, the sextupole gradient will actually rise as 1/E^2. - Again, not very practical solution. MC Lattie Design Workshop on DA for Ultimate Storage Rings, IU November 11, 2010 8

9. Momentum acceptance with Oide design Qx Qy cc pp pp Static momentum acceptance is ~0.55%, but change in  c sign limits dynamic acceptance to ~0.4% MC Lattie Design Workshop on DA for Ultimate Storage Rings, IU November 11, 2010 9

10. Dynamic Aperture with Oide’s design Tracking performed with program SAD that automatically includes fringe fields for all elements MC Lattie Design Workshop on DA for Ultimate Storage Rings, IU November 11, 2010 10

11. MC design issues - Y. Alexahin MCD workshop, BNL December 4, 2007 Sensitivity to errors “Dipole First” lattice had 2 IPs. With 1 IP the probability of losing stability would be lower. Still it would be difficult to find the stability window % unstable relative sextupole field error Oide % unstable relative quadrupole field error not a single case with “dipole first” lattice “dipole first” Oide Some unstable cases are MAD artefacts - sometimes it cannot find optics for linearly stable lattice MAD8 simulations of the random field errors (Gaussian distribution truncated at 2.5  ) effect on linear optics stability for: 1) Oide’s lattice (1 IP,  * = 3mm,  max = 901.8 km), 2) “Dipole first” lattice (2 IPs,  * = 1cm,  max = 32.8 km),  With quadrupole errors stability was required for |  p |  0.3%.  With sextupole errors stability was required for |  p |  0.5%. 11

12. New paradigm  Chromaticity of the larger  -function should be corrected first (before  is allowed to change) – and in one kick to reduce sensitivity to errors!  To avoid spherical aberrations it must be  y  then small  x will kill all detuning coefficients and RDTs (this will not happen if  y   x )  Chromaticity of  x should be corrected with a pair of sextupoles separated by -I section to control DDx (smallness of  y is welcome but not sufficient)  Placing sextupoles in the focal points of the other  -function separated from IP by  =  integer reduces sensitivity to the beam-beam interaction. These considerations almost uniquely determine the IR layout. Requirements adopted for the latest version:  full aperture A = 10sigma_max + 2cm (A.Zlobin adds 1cm on top of that)  maximum quad gradient 12% below quench limit at 4.5°K as calculated by A.Zlobin  bending field 8T in large-aperture open-midplane magnets, 10T in the arcs  IR quad length < 2m (split in parts if necessary!) – no shielding from inside  Sufficient space for magnet interconnects (typically 30-40cm) MC Lattie Design Workshop on DA for Ultimate Storage Rings, IU November 11, 2010 12

13. 3 Sextupoles Chromatic Correction Scheme  y  x Chrom. Correction Block Wx Wy correctors sextupoles bends Dx (m) quads RF multipoles for higher order chrom. correction MC Lattie Design Workshop on DA for Ultimate Storage Rings, IU November 11, 2010 13

14. One More Innovation: the Arc Cell SY DDx/5 Dx (m) SX SASY xx yy  Central quad and sextupole SA control the momentum compaction factor and its derivative (via Dx and DDx) w/o significant effect on tunes and chromaticity  Large  -functions ratios at SX and SY sextupole locations simplify chromaticity correction  Phase advance 300  / cell  spherical aberrations cancelled in groups of 6 cells  Large dipole packing factor  small circumference (C=2.5 km with 10T dipole field) MC Lattie Design Workshop on DA for Ultimate Storage Rings, IU November 11, 2010 14

15. Momentum Acceptance Fractional parts of the tunes  Static momentum acceptance =  1.2%, while the baseline scheme calls for only  0.3%  Central value of the momentum compaction factor = -1.45  10 -5, can be made even smaller but we don’t know yet what is practical With 2 IPs the central tunes are 18.56, 16.58 - good (!) for beam-beam effect - good for the orbit stability and DA pp cc x*x* y*y* pp QxQx QyQy pp MC Lattie Design Workshop on DA for Ultimate Storage Rings, IU November 11, 2010 15

16. Dynamic Aperture DA= (  CSI /   N ) 1/2 = 4.7  for   N =25  m 1024 turns DA (Lie3 tracking method): blue – Courant-Snyder Invariants of stable muons, red – lost muons  Dynamic Aperture is marginally sufficient for   N =50  m  DA can be further increased with vertical nonlinear correctors N  =0 Delta(p)/p: 0.00000000 M o d e 1 M o d e 2 Fractional tunes: Q1 = 0.55999422 Q2 = 0.57999563 First order chromaticity: Q1' = 0.32480253 Q2' = -1.98911112 Second order chromaticity: Q1'' = -862.88043160 Q2'' = 2077.02609602 Normalized anharmonicities: dQ1/dE1 = -0.24797555E+06 dQ1/dE2 = 0.14326980E+06 dQ2/dE2 = 0.73894849E+05 MAD8 STATIC command output - very imprecise!  CSIx [  m]  CSIy [  m] MC Lattie Design Workshop on DA for Ultimate Storage Rings, IU November 11, 2010 16

17. Dynamic Aperture with Beam-Beam Interaction “Diagonal” Dynamic Aperture (Ax=Ay) vs. (constant) momentum deviation in the presence of beam-beam effect (  = 0.09/IP) for normalised emittance   N =25  m. Opposing bunch is represented by 12 slices Only muons at bunch center tracked ! pp DA (  ) beam extent MC Lattie Design Workshop on DA for Ultimate Storage Rings, IU November 11, 2010 17

18. Muon Collider Parameters  s (TeV)1.53 Av. Luminosity / IP (10 34 /cm 2 /s) 1.25*5 Max. bending field (T)1014 Av. bending field in arcs (T)8.312 Circumference (km)2.54 No. of IPs22 Repetition Rate (Hz)1512 Beam-beam parameter / IP0.0870.087  * (cm)10.5 Bunch length (cm)10.5 No. bunches / beam11 No. muons/bunch (10 12 )22 Norm. Trans. Emit. (  m)2525 Energy spread (%)0.10.1 Norm. long. Emit. (m)0.070.07 Total RF voltage (MV) at 800MHz20 230 ----------------------------------------------------------------------- *) With increase by the beam-beam effect P  – average muon beam power (~  ) C – collider circumference (~  if B=const)  – muon lifetime (~  )  * – beta-function at IP – beam-beam parameter h  z /   “Hour-glass factor” MC Lattie Design Workshop on DA for Ultimate Storage Rings, IU November 11, 2010 18

19. Lesson - Symplecticity matters! MC Lattie Design Workshop on DA for Ultimate Storage Rings, IU November 11, 2010 19 1000 turns DA for one of the earlier MC versions with 1 IP calculated with MAD8 using different tracking methods. With METHOD=LIE4 not a single particle has survived! Almost all were lost in ~30 turns. Probable cause for discrepancy: kinematic nonlinearities in long IR dipoles (or roundoff errors?)

20. Conclusions  Arrangement of sextupoles in non-interleaved -I pairs helps to reduce detuning and improve DA, but is not a “must”.  Vertical chromaticity correction sextupole may be leaved w/o a pair if at its location  x <<<  y  New arc cell design allows for efficient (and independent!) control of  c and d  c /d  p and attainment of very low |  c |. It turned out that a similar design was considered by Yuri Senichev for CERN PS2 project (but not accepted).  Fringe-fields are of utmost importance, especially in the case of large  ’s at strong large-aperture magnets.  Symplecticity of tracking algorithm may be necessary for as little as 100 turns!  It seems that MAD (at least MAD8) can not properly handle non-linear effects in lattices with very large  ’s. MC Lattie Design Workshop on DA for Ultimate Storage Rings, IU November 11, 2010 20