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

2. MC Lattice - Y. Alexahin, E. Gianfelice MCD workshop, BNL December 4, 2007 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

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

4. MC Lattice - Y. Alexahin, E. Gianfelice MCD workshop, BNL December 4, 2007 Chromatic correction with 1996 Oide’s design  y  x axax ayay d  x /2  - goes to - 4824.5 ! bxbx byby d  x /2   * = 3mm,  max = 901835m hor. CC ver. CC with chromatic correction sections separated from IR there inevitably are places with large chromatic modulation of betatron phase advances – potential for a trouble chromatic phase (arg a / b ) advances by 2  at locations where respective beta- functions are low

5. MC Lattice - Y. Alexahin, E. Gianfelice MCD workshop, BNL December 4, 2007 New approach to chromatic correction The only way to kill a x,y before they convert into b x,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 D x = D x ´= 0 at the IP, 2) outside the IR so that D x = 0 but D x ´  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 But we make a good use of it: we can earn a negative contribution to  c while suppressing this huge J x 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  being the dipole bend angle

6. MC Lattice - Y. Alexahin, E. Gianfelice MCD workshop, BNL December 4, 2007 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;

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

8. MC Lattice - Y. Alexahin, E. Gianfelice MCD workshop, BNL December 4, 2007 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.

9. MC Lattice - Y. Alexahin, E. Gianfelice MCD workshop, BNL December 4, 2007 Choice of arc cells With  = 108  there is 120 arc cells total (including modified cells in matching sections). The regular cell layout: The total machine circumference is C=3131.8m, The dipole packing factor for regular cells is 63%. 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

10. MC Lattice - Y. Alexahin, E. Gianfelice MCD workshop, BNL December 4, 2007 Tunes and momentum compaction w/o octupoles Qy Qx c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

11. MC Lattice - Y. Alexahin, E. Gianfelice MCD workshop, BNL December 4, 2007 Dynamic aperture with octupoles 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)  CSIx [  m]  CSIy [  m]

12. MC Lattice - Y. Alexahin, E. Gianfelice MCD workshop, BNL December 4, 2007 Can we move 1 st quad closer to IR? - 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)  y1  x1 s Simplified problem:  y =0 at quad exit Parameters: B_tip = 10T,  _liner = 3cm, r_beam_pipe = 4  for   N = 25  m  y1 for  *=1cm  x1 for  *=0.5cm  y1 for  *=0.5cm L 1 for  *=1cm  x1 for  *=1cm d1d1 d1d1 L 1 for  *=0.5cm No big gain for  *=1cm

13. MC Lattice - Y. Alexahin, E. Gianfelice MCD workshop, BNL December 4, 2007 “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

14. MC Lattice - Y. Alexahin, E. Gianfelice MCD workshop, BNL December 4, 2007 “Dipole First” MC Lattice Properties Qy Qx pp pp Owing to larger dispersion in IR the required sextupole gradient became lower reducing 2 nd order effects. Also, in this version 2 nd order dispersion was corrected with sextupoles in the matching section. 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 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. cc

15. MC Lattice - Y. Alexahin, E. Gianfelice MCD workshop, BNL December 4, 2007 “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)

16. MC Lattice - Y. Alexahin, E. Gianfelice MCD workshop, BNL December 4, 2007 Attempt to fix momentum compaction Qx Qy cc pp pp 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  0.2313 k2l2=-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 2 nd order chromaticity

17. MC Lattice - Y. Alexahin, E. Gianfelice MCD workshop, BNL December 4, 2007 135  FODO arc cells (sneak peek) Qx Qy cc pp pp  CSIy [  m]  CSIx [  m] The 1024 turns DA is only slightly smaller than in the case of  = 108  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: - difficult, but not hopeless

18. MC Lattice - Y. Alexahin, E. Gianfelice MCD workshop, BNL December 4, 2007 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.