# Presentation: 492507

## f

**Muon Collider Lattice Design Issues**

**Y.Alexahin**
**(FNAL)**

-     	         **FERMI  NATIONAL  ACCELERATOR  LABORATORY**
**	****	****                      ****US  DEPARTMENT  OF  ENERGY**

## MC lattice requirements & challenges

**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**

## MC Designs & Designers

**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**

## IR Issues

**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?**

## IR design by K.Oide 

**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

## Approaches to chromatic correction

**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**

## “Local” chromatic correction with Oide’s design

**“****Local” chromatic correction with Oide’s design**

*** ** = 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

## New approach to chromatic correction 

**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!**

*** ** = 1cm, ** max = 32,772 m

**YA&EGW design**

## Arc cells with low momentum compaction

**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**

## KEK ATF arc cell with negative c

**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.**

## Finally: momentum acceptance

**Finally: momentum acceptance**

- Qx

- Qx

- Qy

- Qy

******c**

******c**

******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%**

## Dynamic Aperture with Oide’s design

**Dynamic Aperture with Oide’s design**

**Tracking performed with program SAD that automatically includes fringe fields for all elements**

## Dynamic Aperture computed with MAD

**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]**

****** ****CSIy [********m]**

****** ****CSIx [********m]**

## Dynamic Aperture with YA&EGW design

**Dynamic Aperture with YA&EGW design**

**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.**

## Sensitivity to errors

**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%. **

## Sensitivity to errors

**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**

## Sensitivity to errors (Carol’s designs)

**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**

## Summary

**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)**