1. Muon Collider Lattice Design Status Muon Collider Workshop, Telluride CO, June 27 – July 1 2011 Y. Alexahin (FNAL APC)  Lattice design - 1.5 TeV c.o.m Lattice - New 3 TeV c.o.m Lattice  Fringe Field and Multipole Errors  Strong-Strong Beam-Beam Simulations  Plans

2. Ring Lattice Requirements 2 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)0.012.30.322 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  very small momentum compaction factor - New ideas for IR magnets chromaticity correction needed! MC Design Status- Y. Alexahin MC workshop 06/30/2011

3. Chromatic Correction Basics 3 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 - horizontal correction requires 2 sextupoles 180  apart to cancel spherical aberrations 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 Design Status- Y. Alexahin MC workshop 06/30/2011

4. Magnet Requirements 4 MC Design Status- Y. Alexahin MC workshop 06/30/2011  Distance from IP to the 1st quad = 6 m  Bending field in the arcs = 10T, in large aperture IR dipoles 8T  Aperture diameter  > 10  max + 30 mm  Quad gradient < 10T/ (  /2)  Quad length < 2 m, dipole length < 6 m  Interconnects > (  1 +  2)/2 + 16 cm (typically + 2 cm added)  FF quads horizontally displaced (if possible) to provide a dipole component that: - generates additional dispersion for chromaticity correction - sweeps aside decay electrons

5.  *=1cm 1.5 TeV c.o.m. MC IR Optics MC Design Status- Y. Alexahin MC workshop 06/30/2011 5 essentially a focusing doublet chromaticity correction sextupoles

6. 6  *=1cm 1.5 TeV MC FF Quads Q3Q4Q5B1Q6 Q2 Q1 s(m)s(m) a(cm) 5x5x 5y5y ParameterUnitQ1Q2Q3 Coil aperturemm80110160 Nominal gradientT/m250187-130 Nominal currentkA16.6115.314.2 Quench gradient @ 4.5 KT/m281.5209.0146.0 Quench gradient @ 1.9 KT/m307.6228.4159.5 Coil quench field @ 4.5 KT 12.813.213.4 Coil quench field @ 1.9 KT 14.014.414.8 Magnetic lengthm1.51.7 Quads displaced horizontally by 0.1 aperture to create ~2T bending field MC Design Status- Y. Alexahin MC workshop 06/30/2011

7. 7  *=1cm 1.5 TeV MC Lattice Performance QxQx QyQy pp pp cc DA (  ) pp “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 Only muons at bunch center tracked ! Fractional parts of the tunes and momentum compaction factor vs. momentum deviation beam extent MC Design Status- Y. Alexahin MC workshop 06/30/2011

8. 8 Design Pros & Contras MC Design Status- Y. Alexahin MC workshop 06/30/2011 Pros:  Achieves all stated goals (momentum acceptance, DA, etc.)  Robust chromaticity correction scheme  Small horizontal beam size allows for close shielding to intercept secondaries  FF quads can be displaced horizontally to create a dipole field Contras:  Large  y_max  high sensitivity to magnet errors  Difficult to upgrade to higher energies: it may not be possible to retain 10T pole tip field in quads with apertures > 16 cm due to mechanical problems

9. Triplet vs Doublet FF 9 A simplified problem considered:  Point-to-parallel focusing   *=5mm,   N =25(  )mm  mrad, 1.5TeV/beam  First quad starts at 6m from IP  Continuously varying quad gradient G=8T / R_bore, R_bore= 5*Sqrt(  max*   N /  )+15 mm   ss In the case of triplet focusing  max is 3 times smaller! - effect of the gradient dependence on aperture MC Design Status- Y. Alexahin MC workshop 06/30/2011

10. 10 Q3 Q4Q5 B1 Q7 Q2 Q1 s(m)s(m) a(cm) 5x5x 5y5y Q8 Q6  *=5mm 3 TeV c.o.m. MC FF Quads (Preliminary!) Q1Q2Q3Q4-Q6Q7Q8 aperture (mm)80104130146 160 gradient (T/m)-250-192.3153.9136.5-136.7-121.4 length (m)1.851.751.952.051.752.6  Aperture requirement  >10  max +30 mm as in 1.5 TeV case  The number of different apertures increased to 5 to follow more closely the beam sizes  Length limit < 2 m not fulfilled for Q8, it can be cut in two pieces if necessary  No horizontal displacement due to large horizontal beam size M1 MC Design Status- Y. Alexahin MC workshop 06/30/2011

11. 11  *=5mm 3 TeV c.o.m. MC IR Optics (Preliminary!)  y (m)  x (m) 10*DDx (m) 20*Dx (m) s (m) Wy chromaticity correction sextupoles M2 s (m) Wx M1 MC Design Status- Y. Alexahin MC workshop 06/30/2011

12. 12  *=5mm 3 TeV MC Lattice Performance (w/o Arcs) Large Qx  = -1.65  10 5  octupole (and decapole) correctors at M2  DA 5   *(cm) y*y* pp x*x* QxQx QyQy pp Static momentum acceptance  0.5% and Dynamic Aperture ~ 5  seem feasible – the arc sextupoles are too weak to have any effect  CSIy [  m]  CSIx [  m] 55 1024 turns DAFractional parts of the tunes MC Design Status- Y. Alexahin MC workshop 06/30/2011

13. 13 3 TeV MC Arc Cell SY DDx(m)/2 Dx (m) SX SASY  x (m)  y (m)  Central quad and sextupole SA control the momentum compaction factor and its derivative (via Dx and DDx) w/o significant effect on 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~4.5 km with 10T dipole field) MC Design Status- Y. Alexahin MC workshop 06/30/2011

14. 14 Parameters of the Two Designs  s (TeV)1.53  * (cm) (bare lattice)10.5  _max (km)4894 Av. Luminosity / IP (10 34 /cm 2 /s) 1.254.4 Max. bending field (T)1010 Av. bending field in arcs (T)8.38.4 Circumference (km)2.5 (2.7)4.5 No. of IPs22 Repetition Rate (Hz)1512 Beam-beam parameter / IP0.0870.087 Beam size @ IP (  m)63 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 250 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 Design Status- Y. Alexahin MC workshop 06/30/2011

15. What’s Next? 15 MC Design Status- Y. Alexahin MC workshop 06/30/2011  Triplet FF solves the problem with large  y_max, but lacks some nice features of the doublet FF associated with small  x  With triplet FF the major concern is horizontal beam stability, whereas with doublet FF it is for the vertical plane  Is a compromise possible? For 3 TeV we must know what gradients can be realistically achieved in large aperture quads, G(A) curve is needed from magnet designers  For 1.5 TeV case we may try to optimize  y_max/  x_max ratio and reduce  *  Optimization should be performed with account of systematic and random magnet errors and their correction strategy - a lot of work to do!  Extra manpower is needed!

16. Fringe Field in IR quads (V.Kapin) 16 1024 turns DA for 1.5TeV lattice in units of initial coordinates at IP without (left) and with quadrupole fringe fields: center - embedded in MAD-X PTC hard-edge approximation, right - maps produced by COSY.  Only vertical motion suffers due to  y_max>>  x_max  PTC underestimates the effect y0 (m) x0 (m) MC Design Status- Y. Alexahin MC workshop 06/30/2011 y0 (m) x0 (m)

17. IR Open-Midplane Dipole Nonlinearities (V.Kapin) 17 MC Design Status- Y. Alexahin MC workshop 06/30/2011 Rref=40mm b1=10000 b3=-5.875 b5=-18.320 b7=-17.105 IR dipole coil cross-section and good field region Effect of multipole components on DA in 1.5TeV case: decapole is most detrimental

18. 18 MC Design Status- Y. Alexahin MC workshop 06/30/2011 DA in the plane of initial particle coordinates:. left - no multipole errors, center - sextupole error added, right - sextupole corrector placed at the 1 st  y maximum.  Effect of the sextupole error can also be compensated with octupole (Netepenko)  Sextupole error affects both x- and y-motion Correction of IR Dipole Nonlinearities (V.Kapin) SC1

19. Strong-Strong BB Simulations (K.Ohmi) 19 MC Design Status- Y. Alexahin MC workshop 06/30/2011  Very fast luminosity degradation (by 15%) observed, most likely due to initial mismatch  Dr. Ohmi will come at Fermilab in October to do more studies.

20. Plans 20  Lattice design: - complete 1.5TeV design with new tuning & collimation sections - finish the 3TeV design  Fringe fields & Multipoles: - include realistic long. profile (Enge function) in MAD-X (F.Schmidt, CERN) or borrow from COSY-Infinity (V.Kapin) - nonlinear corrector arrangement for fringe field and multipole error correction (V.Kapin, F.Schmidt)  Strong-Strong Beam-Beam Simulations: - K.Ohmi (KEK) will come at Fermilab in October - A.Valishev and E.Stern (FNAL) also promised to look  Self-Consistent Longitudinal Dynamics: - V.Balbekov & L.Vorobiev (FNAL GS) can address it (using ORBIT?) MC Design Status- Y. Alexahin MC workshop 06/30/2011