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

Ring Lattice Requirements 2 What we would like to achieve compared to other machines: MCTevatronLHC Beam energy (TeV)  * (cm) Momentum spread (%)0.1< Bunch length (cm)15015 Momentum compaction factor (10^-3) Geometric r.m.s. emittance (nm) Particles / bunch (10^11) Beam-beam parameter,  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

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

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)/ 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

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

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 aperturemm Nominal gradientT/m Nominal currentkA Quench 4.5 KT/m Quench 1.9 KT/m Coil quench 4.5 KT Coil quench 1.9 KT Magnetic lengthm Quads displaced horizontally by 0.1 aperture to create ~2T bending field MC Design Status- Y. Alexahin MC workshop 06/30/2011

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 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

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 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) gradient (T/m) length (m)  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  *=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  *=5mm 3 TeV MC Lattice Performance (w/o Arcs) Large Qx  =  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 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 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) Max. bending field (T)1010 Av. bending field in arcs (T) Circumference (km)2.5 (2.7)4.5 No. of IPs22 Repetition Rate (Hz)1512 Beam-beam parameter / IP Beam IP (  m)63 Bunch length (cm)10.5 No. bunches / beam11 No. muons/bunch (10 12 )22 Norm. Trans. Emit. (  m)2525 Energy spread (%) Norm. long. Emit. (m) Total RF voltage (MV) at 800MHz 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

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!

Fringe Field in IR quads (V.Kapin) 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)

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

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

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.

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