2 nd Muon Collider Design workshop, JLab, Newport News VA December 8-12, 2008

1. FERMI NATIONAL ACCELERATOR LABORATORY US DEPARTMENT OF ENERGY f Update on the Muon Collider lattice design with chromatic correction in IR Y.Alexahin & E.Gianfelice-Wendt 2nd Muon Collider Design workshop, JLab, Newport News VA December 8-12, 2008

2. MC lattice requirements & challenges h z /  “Hour-glass factor” • Requirements: • low  ( 1cm) • small circumference (luminosity ~ 1/R) • momentum acceptance in % range and sufficient dynamic aperture • low momentum compaction (c ~ 10-4, better ~ 10-5) to obtain small z with moderate URF • absence of long straights (not to create "hot spots" of neutrino radiation) • protection of low beta quads from secondaries? ( may limit available field gradient) • tunability! (It's not worth while to have huge peak luminosity if the average ~0 due to difficulties in tuning) • Challenges: • chromatic effects - require strong sextupoles • dynamic aperture - suffers from strong sextupoles • sensitivity to errors - makes sophisticated scheme impractical • beam separation (in multibunch scheme) - headache in the presence of strong nonlinearities MC Lattice Update - Y. Alexahin 2nd MCD workshop, JLab, December 11, 2008

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 Update - Y. Alexahin 2nd MCD workshop, JLab, December 11, 2008

4. Approaches to chromatic correction Low-beta quads excite large chromatic -wave (described by functions ax,y and bx,yfrom the previous slide). There are different possibilities to suppress the chromatic -wave 1) with sextupole families in the arcs (classic method, “global” correction) 2) with sextupoles in special CC sections (“local” correction, but the locale is out of IR). Allows to organize the sextupoles into non-interleaved pairs with phase advances between the sextupoles in a pair =. This greatly improves DA. 3) local with sextupoles right in IR - saves space, less prone to errors but at the price of stronger higher-order effects MC Lattice Update - Y. Alexahin 2nd MCD workshop, JLab, December 11, 2008

5. “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 Update - Y. Alexahin 2nd MCD workshop, JLab, December 11, 2008

6. 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 Update - Y. Alexahin 2nd MCD workshop, JLab, December 11, 2008

7. “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 Update - Y. Alexahin 2nd MCD workshop, JLab, December 11, 2008

8. Longitudinal “kinetic energy” - asymmetry in c limits maximum synchrotron amplitude by p=0.55%, To improve dynamic momentum acceptance good control of is necessary. It can be achieved with the help of additional sextupoles in high Dx locations (beware of large DDx excitation!) MC Lattice Update - Y. Alexahin 2nd MCD workshop, JLab, December 11, 2008

9. “Dipole First” MC Lattice Dynamic Aperture  CSIy [m]  CSIx [m] • The 1024 turns DA is only marginally sufficient for the medium-emittance option: ~3 for N=12.5 m (O.K. for the low), • but NOT for the high-emittance option. • Before starting a principally new design, we tried to improve this one by: • rematching IR and the arcs to make c>0 • moving tunes closer to integer • introducing new multipole correctors (sextupoles and octupoles for now) to correct 1, DDx, second order chromaticity and detuning with amplitude CSIx,y = Courant-Snyder invariant. DA [sigma]=Sqrt(  CSIx / N) MC Lattice Update - Y. Alexahin 2nd MCD workshop, JLab, December 11, 2008

10. Upgrade Lattice Properties c Qx Qy p K p • Dynamic momentum acceptance is now 0.63%, • it can be further increased by • stronger octupoles to reduce Qx’’ • introducing decapoles to correct Qx’’’ p MC Lattice Update - Y. Alexahin 2nd MCD workshop, JLab, December 11, 2008

11. Upgrade Lattice Dynamic Aperture  CSIy [m]  CSIx [m] Second order chromaticity: Q1'' = 24074.96031867 Q2'' = 4020.58313978 Normalized anharmonicities: dQ1/dE1 = 0.25242152E+08 dQ1/dE2 = 0.19616977E+08 dQ2/dE2 = 0.18515914E+08 • The 1024 turns DA is now marginally sufficient for the high-emittance option: ~3 for N=25 m and is O.K. for low- and medium-emittance option. • It can be increased by compensating detuning with stronger octupoles • However: • fringe fields not included yet • optics errors will reduce DA • we need good LONG-TERM DA to work with protons One would expect It appeared twice larger in the ideal lattice: Q  0.9 MC Lattice Update - Y. Alexahin 2nd MCD workshop, JLab, December 11, 2008

12. Summary & Outlook “Dipole first” optics can provide necessary DA for the medium-emittance option. However, for the high-emittance option as well as for the long-term beam stability new ideas are necessary. Our next design - combination of the two approaches: chromatic correction in one plane with IR sextupoles and with a -I pair of sextupoles in the other plane further downstream from IP. Hopefully this will permit to avoid using dipole near IP MC Lattice Update - Y. Alexahin 2nd MCD workshop, JLab, December 11, 2008