# Presentation: 492506

## f

**Muon Collider lattice design**
**with chromatic correction in IR**

**Y.Alexahin & E.Gianfelice-Wendt **

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

## Design Goals

**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 m****omentum acceptance**** ******** ****very strong sextupoles ******** ****higher order effects  ******** ****small DA**

## Montague chromatic functions

**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***

## Chromatic correction with 1996 Oide’s design

**Chromatic correction with 1996 Oide’s design**

*** ** = 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

## 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!**
**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**

## IR layout for  *=1cm 

**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;**

## Optics & chromatic functions

**Optics & chromatic functions**

**IR, negative dispersion and matching sections (sextupole polarity not indicated)**

## Choice of FODO arc cell parameters

**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.**

## Choice of arc cells

**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**

## Tunes and momentum compaction w/o octupoles

**Tunes and momentum compaction w/o octupoles**

**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**

## Dynamic aperture with octupoles

**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)**

## Can we move 1st quad closer to IR?

**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)**

**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

- L1 for *=1cm

- x1 for *=1cm

- d1

- d1

- L1 for *=0.5cm

**No big gain ****for *********=1cm**

## “Dipole First” IR Design Option

**“****Dipole First” IR Design Option**

**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**

## “Dipole First” MC Lattice Properties

**“****Dipole First” MC Lattice Properties**

- Qy

- Qx

******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**

## “Dipole First” MC Lattice Properties

**“****Dipole First” MC Lattice Properties**

** ****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)**

## Attempt to fix momentum compaction

**Attempt to fix momentum compaction**

- Qx

- Qy

******c**

******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**

## 135 FODO arc cells (sneak peek)

**135******** FODO arc cells (sneak peek)**

**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**

## Summary & Outlook

**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.**