Muon Collider Lattice Design

FERMI NATIONAL ACCELERATOR LABORATORY US DEPARTMENT OF ENERGY

fYuri Alexahin, Eliana Gianfelice-Wendt(Accelerator Physics Center)Workshop on Dynalic Aperture for Ultimate Storage Rings, IU, Bloomington November 11-12, 2010

MC Lattie Design Workshop on DA for Ultimate Storage Rings, IU November 11, 2010The Road to High Luminosity2The 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 ?)P average muon beam power (limited by the P-driver power )

MC Lattice Design Challenges3 What we would like to achieve compared to other machines:MCTevatronLHCBeam energy (TeV)0.750.987* (cm) 12855Momentum spread (%)>0.1

Various MC lattice designs studied 1996 by Carol J., A. Garren 1996 by K.Oide Dipole first (2007) ~ satisfy the requirements Elianas synthetic (2009) Asymmetric dispersion Flat top Three sextupoles scheme the final solution based on Elianas preliminary design.________________ 1996 designs used special -I chromatic correction sections, had very high sensitivity to field errors. Oides 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.4MC Lattie Design Workshop on DA for Ultimate Storage Rings, IU November 11, 2010

Chromatic Correction Basics5Montague chromatic functions :Ax,y are created first, and then converted into Bx,y as phase advances x,y growK1 , K2 are normalized quadrupole and sextupole gradients, Dx is dispersion function: Dx = dxc.o./dpThe mantra:Kill As before they transform into Bs !- difficult to achieve in both planes- requires dispersion generated close to IP or large Dx0 at IPBx,y are most important since they determine modulation of phase advance x,yx,y = -x,y /2 , x,y are Twiss lattice functions, p is relative momentum deviation.Equations for chromatic functionsMC Lattie Design Workshop on DA for Ultimate Storage Rings, IU November 11, 2010

IR design by K.Oide *=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~5cmbut is this enough to accommodate the shielding?QC1-QC4 have small octupole componentMC Lattie Design Workshop on DA for Ultimate Storage Rings, IU November 11, 20106

Local chromatic correction with Oides design * = 3mm, max = 901,835 mhor. CCver. CCwith chromatic correction sections separated from IR there inevitably are places with large chromatic modulation of betatron phase advances potential for a troublechromatic phase (arg ax,y/bx,y) advances by 2 at locations where respective beta-functions are lowMC Lattie Design Workshop on DA for Ultimate Storage Rings, IU November 11, 20107

KEK ATF arc cell with negative cThe 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, 20108

Momentum acceptance with Oide designStatic 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, 20109

Dynamic Aperture with Oides designTracking performed with program SAD that automatically includes fringe fields for all elementsMC Lattie Design Workshop on DA for Ultimate Storage Rings, IU November 11, 201010

MC design issues - Y. Alexahin MCD workshop, BNL December 4, 2007Sensitivity to errorsDipole 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 windowSome unstable cases are MAD artefacts - sometimes it cannot find optics for linearly stable latticeMAD8 simulations of the random field errors (Gaussian distribution truncated at 2.5) effect on linear optics stability for: 1) Oides 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

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 sectionto 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 4.5K 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, 201012

3 Sextupoles Chromatic Correction Scheme MC Lattie Design Workshop on DA for Ultimate Storage Rings, IU November 11, 201013

One More Innovation: the Arc Cell 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, 201014

Momentum AcceptanceFractional 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 = -1.4510-5, can be made even smaller but we dont 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 DAMC Lattie Design Workshop on DA for Ultimate Storage Rings, IU November 11, 201015

Dynamic ApertureDA= ( CSI / N )1/2= 4.7 for N=25 m1024 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=0Delta(p)/p: 0.00000000 M o d e 1 M o d e 2Fractional tunes: Q1 = 0.55999422 Q2 = 0.57999563First order chromaticity: Q1' = 0.32480253 Q2' = -1.98911112Second order chromaticity: Q1'' = -862.88043160 Q2'' = 2077.02609602Normalized anharmonicities: dQ1/dE1 = -0.24797555E+06 dQ1/dE2 = 0.14326980E+06 dQ2/dE2 = 0.73894849E+05MAD8 STATIC command output - very imprecise! CSIx [m] CSIy [m]MC Lattie Design Workshop on DA for Ultimate Storage Rings, IU November 11, 201016

Dynamic Aperture with Beam-Beam InteractionDiagonal 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 slicesOnly muons at bunch center tracked !MC Lattie Design Workshop on DA for Ultimate Storage Rings, IU November 11, 201017

Muon Collider Parameterss (TeV)1.53Av. Luminosity / IP (1034/cm2/s) 1.25*5Max. bending field (T)1014Av. bending field in arcs (T)8.312Circumference (km)2.54No. of IPs22Repetition Rate (Hz)1512Beam-beam parameter / IP0.0870.087* (cm)10.5Bunch length (cm)10.5No. bunches / beam11No. muons/bunch (1012)22Norm. Trans. Emit. (m)2525Energy spread (%)0.10.1Norm. long. Emit. (m)0.070.07Total RF voltage (MV) at 800MHz20 230----------------------------------------------------------------------- *) With increase by the beam-beam effectP average muon beam power (~ ) C collider circumference (~ if B=const) muon lifetime (~ )* beta-function at IP beam-beam parameterMC Lattie Design Workshop on DA for Ultimate Storage Rings, IU November 11, 201018

Lesson - Symplecticity matters!MC Lattie Design Workshop on DA for Ultimate Storage Rings, IU November 11, 2010191000 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?)

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