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Muon Collider Design workshop, BNL, Upton NY December 3-7, 2007

FERMI NATIONAL ACCELERATOR LABORATORY US DEPARTMENT OF ENERGY. f. Muon Collider lattice design with chromatic correction in IR. Y.Alexahin & E.Gianfelice-Wendt. Muon Collider Design workshop, BNL, Upton NY December 3-7, 2007. Design Goals. Low  = 1 cm

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Muon Collider Design workshop, BNL, Upton NY December 3-7, 2007

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  1. FERMI NATIONAL ACCELERATOR LABORATORY US DEPARTMENT OF ENERGY f Muon Collider lattice design with chromatic correction in IR Y.Alexahin & E.Gianfelice-Wendt Muon Collider Design workshop, BNL, Upton NY December 3-7, 2007

  2. Design Goals • Low  = 1cm • Small momentum compaction factor c ~ 10-4 • Momentum acceptance ~ 1% • Dynamic aperture > 3 for N=25 m (HE option) • Small circumference • Difficulties to overcome: • Low + small momentum compactionhuge chromatic aberrations + large momentum acceptancevery strong sextupoles higher order effects small DA MC Lattice - Y. Alexahin, E. Gianfelice MCD workshop, BNL December 4, 2007

  3. Montague chromatic functions Chromatic functions definition: most important since it determines modulation of phase advance x,y Equations for chromatic functions (if started with (x, x’) set) ax,y is created first, and is converted into bx,y as phase advance x,y grows Equations look a bit different if started with (x, px) set as in MAD: MC Lattice - Y. Alexahin, E. Gianfelice MCD workshop, BNL December 4, 2007

  4. Chromatic correction with 1996 Oide’s design y x bx ax by ay dx/2 dx/2 -goes to - 4824.5 ! hor. CC ver. CC  * = 3mm, max = 901835m chromatic phase (arg a/b) 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 Lattice - Y. Alexahin, E. Gianfelice MCD workshop, BNL December 4, 2007

  5. New approach to chromatic correction The only way to killax,y before they convert into bx,y is to put sextupole correctors right into the IR, not in a separate CC section! For sextupoles to work the dispersion must be present in IR, it may be generated 1) by dipoles in the IR so that Dx= Dx´= 0 at the IP, 2) outside the IR so that Dx= 0 but Dx´ 0 at the IP (now pursued by Carol et al). We explore the first possibility (symmetric design) which has an apparent drawback – huge value of the dispersion invariant generated in the IR  being the dipole bend angle But we make a good use of it: we can earn a negative contribution to c while suppressing this huge Jx so that the rest of the lattice can be simple FODO cells which helps to reduce the circumference (FODO has the largest dipole packing factor). There is another drawback with this scheme: sextupoles do not constitute non-interleaved pairs -> large cross-detuning vs compactness of the ring MC Lattice - Y. Alexahin, E. Gianfelice MCD workshop, BNL December 4, 2007

  6. IR layout for *=1cm ! Interaction region IP: marker DR1: drift, L=6.5; ! Is it really necessary? Q1: quadrupole, L=2.6, k1=-.08; ! G=200T/m @ p=750 GeV/c DR2: drift, L=1; OCT1: multipole, knl:={ 0, 0, 0, kO1}; Q2: quadrupole, L=4, k1=.052; DP1: rbend, L=6, angle=0.018; ! B=7.5T OCT2: multipole, knl:={ 0, 0, 0, kO2}; DR3: drift, L=6; ! Elseparator? OCT3: multipole, knl:={ 0, 0, 0, kO3}; Q3: quadrupole, L=2, k1=-.05; SLB1: sextupole, L=4, k2=-0.15; DR4: drift, L=11.5; SLB2: sextupole, L=3, k2=0.30; Q4: quadrupole, L=1, k1=.06; DR5: drift, L=6; Q5: quadrupole, L=2, k1=-.067; DR6_1: drift, L=8; SLB3: sextupole, L=4, k2=-0.04; DR6_2: drift, L=6.5; Q6: quadrupole, L=2, k1=.047; DR7: drift, L=16; MC Lattice - Y. Alexahin, E. Gianfelice MCD workshop, BNL December 4, 2007

  7. Optics & chromatic functions x y Dx DDx/100 Wx Wy IR, negative dispersion and matching sections (sextupole polarity not indicated) MC Lattice - Y. Alexahin, E. Gianfelice MCD workshop, BNL December 4, 2007

  8. Choice of FODO arc cell parameters • Criteria: • contribution to momentum compaction factor • minimum length • Quadrupole and sextupole integrated strengths in a FODO lattice rather weakly depend on the phase advance per cell  = x= y As a consequence the total length occupied by quadrupoles and sextupoles is proportional to the number of cells and decreases with  increasing. And so does the inverse dipole packing factor We tried  as high as 108 (3/5) and 135 (3/4) per cell. MC Lattice - Y. Alexahin, E. Gianfelice MCD workshop, BNL December 4, 2007

  9. Choice of arc cells With  = 108 there is 120 arc cells total (including modified cells in matching sections). The regular cell layout: fodo: line=(QFH,DE1,DE2,DE3,BEND,DE3,SD,DE1, QD,DE1,DE2,DE3,BEND,DE1,SF,DE3,QFH); QFH: quadrupole, L=1, k1= 9.41993426E-02; ! half-quad QD: quadrupole, L=2, k1=-9.44619456E-02; ! G= 236 T/m SF: sextupole, L=0.5, k2= 7.57939220E-01; ! G2=1895 T/m^2 SD: sextupole, L=0.5, k2=-4.24293803E-01; DE1: drift, L=0.1; ! techno-gap DE2: drift, L=0.5; ! BPMs, correctors DE3: drift, L=0.1; ! techno-gap bend: rbend, L=5.8, angle=0.0226466; ! B=9.76 T The total machine circumference is C=3131.8m, The dipole packing factor for regular cells is 63%. MC Lattice - Y. Alexahin, E. Gianfelice MCD workshop, BNL December 4, 2007

  10. Tunes and momentum compaction w/o octupoles Qx Qy c No octupoles Second order chromaticity: Q1'' = 102511.04779854 Q2'' = 366.54867056 Normalized anharmonicities: dQ1/dE1 = 0.55557395E+08 dQ1/dE2 = 0.20800890E+09 dQ2/dE2 = 0.58845415E+08 Huge cross-detuning is the price to pay for not arranging sextupoles in non- interleaved pairs, it makes dynamic aperture virtually vanishing – octupoles necessary MC Lattice - Y. Alexahin, E. Gianfelice MCD workshop, BNL December 4, 2007

  11. Dynamic aperture with octupoles  CSIy [m]  CSIx [m] Second order chromaticity: Q1'' = 102517.98582532 Q2'' = 1127.89764247 Normalized anharmonicities: dQ1/dE1 = 0.65239168E+08 dQ1/dE2 = 0.47761742E+08 dQ2/dE2 = 0.37233974E+08 Dynamic aperture with ~ optimum octupole strength still is not sufficient for the high-emittance option: <1.5 for N=12.5 m (marginally O.K. for the low-emittance option) MC Lattice - Y. Alexahin, E. Gianfelice MCD workshop, BNL December 4, 2007

  12. Can we move 1st quad closer to IR? x1 y1 s - as Valeri Lebedev pointed out, not too much (even if the detector people agree): to provide enough focusing the integrated quad strength must increase ~1/d, but the bore radius cannot be reduced accordingly (it has to accommodate the shielding) Simplified problem:y=0 at quad exit x1 for *=0.5cm y1 for *=0.5cm x1 for *=1cm y1 for *=1cm d1 Parameters: B_tip = 10T, _liner = 3cm, r_beam_pipe = 4 for N = 25 m L1 for *=0.5cm L1 for *=1cm d1 No big gain for *=1cm MC Lattice - Y. Alexahin, E. Gianfelice MCD workshop, BNL December 4, 2007

  13. “Dipole First” IR Design Option x y Dx DDx/50 Wx Wy Dipole before the first quad creates larger dispersion in IR -> weaker sextupoles It may also help to protect the detector from backgrounds: decay electrons and Bethe-Heitler muons MC Lattice - Y. Alexahin, E. Gianfelice MCD workshop, BNL December 4, 2007

  14. “Dipole First” MC Lattice Properties Qx Second order chromaticity: Q1'' = 67698.83542578 Q2'' = 1860.74134081 Normalized anharmonicities: dQ1/dE1 = 0.43575747E+08 dQ1/dE2 = 0.16659793E+08 dQ2/dE2 = 0.14651033E+08 p Qy c Owing to larger dispersion in IR the required sextupole gradient became lower reducing 2nd order effects. Also, in this version 2nd order dispersion was corrected with sextupoles in the matching section. p Static momentum acceptance of ± 0.7% is O.K. for the high-emittance option (not for the low), however, the dynamic acceptance <0.45% due to change in c sign. MC Lattice - Y. Alexahin, E. Gianfelice MCD workshop, BNL December 4, 2007

  15. “Dipole First” MC Lattice Properties  CSIy [m]  CSIx [m] • The “dipole first” option gives a hope to obtain the required DA (by further optimization) with =1cm • It is not clear, however, if the synchrotron radiation from a dipole so close to the IP would be tolerable. • To proceed further to a realistic design a close collaboration with the detector, energy deposition and magnet technology groups is a must. • Nikolai estimates the time necessary for backgrounds evaluation and shielding design as ~ 0.5 FTE, but does not have a free person to tackle the issue. The 1024 turns DA is only marginally sufficient for the high-emittance option: ~3 for N=12.5 m (O.K. for the low) MC Lattice - Y. Alexahin, E. Gianfelice MCD workshop, BNL December 4, 2007

  16. Attempt to fix momentum compaction The simplest (but not interesting) way to avoid c =0 would be to increase positive contribution from the arcs. Another possibility is to make DDx0 (on average) in NDS by playing with IR sextupoles: k2l1= 0.1  0.2313k2l2=-0.9555 -1.1965 k2l3= 0.6689  0.5742 The increase in the strength of the first two sextupoles compromised the momentum acceptance due to a large increase in the horizontal 2nd order chromaticity Qx p Qy c p MC Lattice - Y. Alexahin, E. Gianfelice MCD workshop, BNL December 4, 2007

  17. 135 FODO arc cells (sneak peek) Qx  CSIy [m] Qy p c p  CSIx [m] Increase in the phase advance / cell from 108 to 135 (80 cells total) allows to increase the dipole packing factor from 63% to 71% and reduce the machine circumference from 3131.8m to 2811.8m – 10% gain in luminosity! However, the nonlinear effects become more pronounced: The 1024 turns DA is only slightly smaller than in the case of  = 108 - difficult, but not hopeless MC Lattice - Y. Alexahin, E. Gianfelice MCD workshop, BNL December 4, 2007

  18. Summary & Outlook • The “dipole first” option gives a hope to obtain the required DA for the HE option (by further optimization) with =1cm • Momentum compaction factor can be corrected in the momentum range ±0.7% but cannot be made smaller than c ~ 10-4 due to large 2nd derivative. • Circumference ~3km is possible with realistic magnet parameters • To proceed further to a realistic design a close collaboration with the detector, energy deposition and magnet technology groups is a must. MC Lattice - Y. Alexahin, E. Gianfelice MCD workshop, BNL December 4, 2007

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