1. Muon Collider Design workshop, BNL, Upton NY December 3-7, 2007 Muon Collider Lattice Design Issues Y.Alexahin (FNAL) FERMI NATIONAL ACCELERATOR LABORATORY US DEPARTMENT OF ENERGY f

2. MC design issues - Y. Alexahin MCD workshop, BNL December 4, 2007 MC lattice requirements & challenges 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 U RF 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 h  z /   “Hour-glass factor”

3. MC design issues - Y. Alexahin MCD workshop, BNL December 4, 2007 MC Designs & Designers 1996  =3mm designs N. Gelfand (very naive) A. Garren (lattice file not available) K. Oide (complete lattice with remarkable properties) Present day  =1cm designs Y.A. & Eliana Gianfelice-Wendt C. Johnstone, M. Berz, P. Snopok A. Bogacz (lattice file not available) Topics to discuss  IR design issue  Chromatic correction  Arc cell choice  Momentum acceptance, dynamic aperture  Sensitivity to errors

4. MC design issues - Y. Alexahin MCD workshop, BNL December 4, 2007 IR Issues How many IRs? Is 2 detectors with luminosity L each better than 1 detector with 2L? How close to IP we may put magnets? – 6.5m stay clear requirement was obtained for 4TeV collider, not for the 1.5TeV case. Will deflection of secondaries from the center of the detector with “dipole first” be helpful? Is LB quad protection from inside necessary? (e.g. K.Oide made no provision) What degree of stability in current and position of the magnets is achievable?

5. MC design issues - Y. Alexahin MCD workshop, BNL December 4, 2007 IR design by K.Oide Is this design practical? - huge  max ~ 1000 km - small quad aperture For beam energy 2TeV the first quad gradient is 214T/m, the aperture radius can be increased to ~5cm, but is this enough to accommodate the shielding? In the 750GeV case there will be no problem with the first quads, but there are issues with other magnets to be discussed later. QC1-QC4 have small octupole component

6. MC design issues - Y. Alexahin MCD workshop, BNL December 4, 2007 Approaches to chromatic correction Low-beta quads excite large chromatic  -wave (described by functions a x,y and b x,y in the previous report). 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

7. MC design issues - Y. Alexahin MCD workshop, BNL December 4, 2007 “Local” chromatic correction with Oide’s design  y  x axax ayay d  x /2  - goes to - 4824.5 ! bxbx byby d  x /2   * = 3mm,  max = 901,835 m 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 x,y / b x,y ) advances by 2  at locations where respective beta- functions are low

8. MC design issues - Y. Alexahin 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!  x  y Dx DDx/50 Wx Wy  * = 1cm,  max = 32,772 m YA&EGW design

9. MC design issues - Y. Alexahin MCD workshop, BNL December 4, 2007 Arc cells with low momentum compaction 1) KEK ATF arc cells (negative  c  problems with vertical chromaticity) 2) FODO - used in YA & EGW (+ neg. dispersion section) and Alex’ designs The quadrupole and sextupole integrated strengths in a FODO lattice rather weakly depend on the phase advance per cell  =  x =  y, therefore in order to obtain large dipole packing factor (and minimize the machine circumference C) we should choose as high as possible: We tried  as high as 108  (3  /5) and 135  (3  /4) per cell. But let us start with KEK ATF arc cells incorporated in Oide’s design The number of regular FODO cells required to obtain the desired  c is

10. MC design issues - Y. Alexahin MCD workshop, BNL December 4, 2007 KEK ATF arc cell with negative  c 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.

11. MC design issues - Y. Alexahin MCD workshop, BNL December 4, 2007 Finally: momentum acceptance Qx Qy cc cc pp pp pp pp The two designs: Oide’s with  * = 3mm at 1 IP and YA&EGW with  * = 1cm at 2 IPs have about the same static momentum acceptance and similar problems with  c limiting dynamic acceptance by 0.4%

12. MC design issues - Y. Alexahin MCD workshop, BNL December 4, 2007 Dynamic Aperture with Oide’s design Tracking performed with program SAD that automatically includes fringe fields for all elements

13. MC design issues - Y. Alexahin MCD workshop, BNL December 4, 2007 Dynamic Aperture computed with MAD On the diagonal it is meager 7  m instead of 35nm  7098 (  ) = 250  m ! Katsunobu suggested that since there is no fringe fields the multipole correctors produce more harm than good. But switching them off has little effect. We should learn Japanese (SAD I mean). Octupoles adjusted to minimize vertical detuning with amplitude All octupole and decapole correctors off MAD8 does not support fringe fields for elements other than dipole  CSIx [  m]  CSIy [  m]  CSIx [  m]

14. MC design issues - Y. Alexahin MCD workshop, BNL December 4, 2007 Dynamic Aperture with YA&EGW design  CSIy [  m]  CSIx [  m] The 1024 turns DA is only marginally sufficient for the high-emittance option: ~3  for   N =12.5  m The “dipole first” option gives a hope to obtain with  =1cm the DA required for   N =25  m - by further optimization and possibly employing higher-order multipole correctors (up to dodecapole). Whether the fringe fields are important with this optics remains to be seen.

15. MC design issues - Y. Alexahin MCD workshop, BNL December 4, 2007 Sensitivity to errors Sensitivity to errors shows: 1) whether the machine operation is at all possible with techically achievable stability of parameters, 2) how difficult it will be to find the stability window: tuning may take more time than data- taking devaluating the high peak luminosity we are after! The most important are tolerances on quadrupole errors and misalignments (which also contribute to quadrupole errors through feeddown effect in sextupole magnets). These errors can make it impossible to obtain a circulating beam. The sextupole errors are less dangerous since they may affect only intensity of the circulating beam (via momentum acceptance and DA). Still if the tolerances are too tight it will be difficult to operate the machine. I looked (with the help of MAD8) at the effect of quadrupole and sextupole random field errors (Gaussian distribution truncated at 2.5  ) on linear optics stability for: 1) Oide’s lattice (1 IP,  * = 3mm,  max = 901,835 m), 2) YA&EGW lattice (2 IPs,  * = 1cm,  max = 32,772 m), 3) lattice designs provided by Carol (1 IP,  * = 3mm,  max = 145,928 m and  * = 1cm,  max = 43,212 m) For each error magnitude 100 “seeds” were taken. With quadrupole errors the lattice was considered unstable if there was no solution for |  p |  0.3%. (Quite often there were no solution at  p =0 whereas with  p  0 the lattice was stable!) With sextupole errors the stability was required for |  p |  0.5%.

16. MC design issues - Y. Alexahin MCD workshop, BNL December 4, 2007 Sensitivity to errors % unstable relative sextupole field error Oide YA&EGW lattice has 2 IRs. With 1 IR the probability of losing stability will be lower. Still it will be difficult to find the stability window % unstable relative quadrupole field error not a single case with YA&EGW lattice YA&EGW Oide Some unstable cases are MAD artefacts - sometimes it cannot find optics for linearly stable lattice

17. MC design issues - Y. Alexahin MCD workshop, BNL December 4, 2007 Sensitivity to errors (Carol’s designs) % unstable relative quadrupole field error  * = 3mm,  max = 145,928 m  * = 1cm,  max = 43,212 m Since the  * = 3mm design had no nonlinear chromaticity correction the required stability range was lowered to |  p |  0.01%. The  * = 1cm did not have even linear chromaticity correction so the required range was |  p |  0.001%. Probably these lattices can be made more robust by adjusting the tunes (with  * = 3mm design Qx = 27.519, with  * = 1cm Qy = 24.029

18. MC design issues - Y. Alexahin MCD workshop, BNL December 4, 2007 Summary  designs with  as low as 3mm do not seem practical due to high sensitivity to errors of all kinds  chromatic correction in special CC sections is - sensitive to both quadrupole and sextupole errors - noticeably increases the machine circumference thus lowering luminosity  multipole correctors (probably up to dodecapole) are necessary for both improving DA chromatic correction With the present level of understanding (I mean myself of course) it seems possible:   =1cm   c ~ 10 -4  momentum acceptance ±0.7%  Dynamic aperture > 3  for   N =25  m (HE option)  Circumference ~3km (all at the same time)