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Muon Collider Lattice Design

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

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Muon Collider Lattice Design

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

  2. The Road to High Luminosity 2 h z /  “Hour-glass factor” P – average muon beam power (limited by the P-driver power ) – beam-beam parameter (limited by particle stability,  < 0.1/IP ?) • 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 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) MC Lattie Design Workshop on DA for Ultimate Storage Rings, IU November 11, 2010

  3. MC Lattice Design Challenges 3 What we would like to achieve compared to other machines: MC Tevatron LHC Beam energy (TeV) 0.75 0.98 7 * (cm) 1 28 55 Momentum spread (%) >0.1 <0.01 0.0113 Bunch length (cm) 1 50 15 Momentum compaction factor (10^-3) 0.05 2.3 0.322 Geometric r.m.s. emittance (nm) 3.5 3 0.5 Particles / bunch (10^11) 20 2.7 1.15 Beam-beam parameter ,  0.1 0.025 0.01 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 4  1996byCarol 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. MC Lattie Design Workshop on DA for Ultimate Storage Rings, IU November 11, 2010

  5. Chromatic Correction Basics 5 Montague chromatic functions : Bx,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 Ax,y are created first, and then converted into Bx,y as phase advances x,ygrow K1 , K2 are normalized quadrupole and sextupole gradients, Dx is dispersion function: Dx= dxc.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 Dx0 at IP MC Lattie Design Workshop on DA for Ultimate Storage Rings, IU November 11, 2010

  6. IR design by K.Oide 6 • *=3mm, Ebeam=2TeV • Is this design practical? • - huge max ~1000 km • too small quad aperture? • First quad G=214T/m (NbTi), with Nb3Sn 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

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

  8. KEK ATF arc cell with negative c 8 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

  9. Momentum acceptance with Oide design 9 Qx Qy p c 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

  10. Dynamic Aperture with Oide’s design 10 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

  11. Sensitivity to errors 11 % unstable 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%. Oide “dipole first” relative quadrupole field error % unstable Oide “Dipole First” lattice had 2 IPs. With 1 IP the probability of losing stability would be lower. Still it would bedifficult to find the stability window not a single case with “dipole first” lattice Some unstable cases are MAD artefacts - sometimes it cannot find optics for linearly stable lattice relative sextupole field error MC design issues - Y. Alexahin MCD workshop, BNL December 4, 2007

  12. New paradigm 12  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 xshould 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

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

  14. One More Innovation: the Arc Cell 14 Dx (m) SY SA SY SX DDx/5 SX x 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

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

  16. Dynamic Aperture 16 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! N=0 1024 turns DA (Lie3 tracking method): blue – Courant-Snyder Invariants of stable muons, red – lost muons  CSIy [m] DA= (  CSI / N )1/2= 4.7 for N=25 m  CSIx [m]  Dynamic Aperture is marginally sufficient for N=50 m  DA can be further increased with vertical nonlinear correctors MC Lattie Design Workshop on DA for Ultimate Storage Rings, IU November 11, 2010

  17. Dynamic Aperture with Beam-Beam Interaction 17 DA () beam extent 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. Opposing bunch is represented by 12 slices Only muons at bunch center tracked ! MC Lattie Design Workshop on DA for Ultimate Storage Rings, IU November 11, 2010

  18. Muon Collider Parameters 18 h z /  “Hour-glass factor” s (TeV) 1.5 3 Av. Luminosity / IP (1034/cm2/s) 1.25* 5 Max. bending field (T) 10 14 Av. bending field in arcs (T) 8.3 12 Circumference (km) 2.5 4 No. of IPs 2 2 Repetition Rate (Hz) 15 12 Beam-beam parameter / IP 0.087 0.087 * (cm) 1 0.5 Bunch length (cm) 1 0.5 No. bunches / beam 1 1 No. muons/bunch (1012) 2 2 Norm. Trans. Emit. (m) 25 25 Energy spread (%) 0.1 0.1 Norm. long. Emit. (m) 0.07 0.07 Total RF voltage (MV) at 800MHz 20 230 ----------------------------------------------------------------------- *) With increase by the beam-beam effect P – average muon beam power (~  ) – beam-beam parameter • C – collider circumference (~  if B=const) • – muon lifetime (~ ) • * – beta-function at IP MC Lattie Design Workshop on DA for Ultimate Storage Rings, IU November 11, 2010

  19. Lesson - Symplecticity matters! 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?) MC Lattie Design Workshop on DA for Ultimate Storage Rings, IU November 11, 2010

  20. Conclusions 20  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

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