# Presentation: 698195

## Collider Ring—Nonlinear Compensations 

- Alex Bogacz

## Muon Collider Ring at 1.5 TeV CM

**Energy = 750 GeV ****	**
**Normalized emittance = 2 ********m rad**
**Relative momentum spread = ****±****0.005**
**Dipole bending field = 10 Tesla**
**Quadruople gradient = 250 Tesla/m**
**Number of  IRs  =  4**
**Betas at IR, *************x,y**** =  10 mm**
**Peak Luminosity/IP = 7********10****34**** s****-1**** cm-****2**** **
**(maximum allowed by the tune shift limit**

**Low Emittance MC Scenario:**

***New SBIR proposal with Muons Inc (submitted)**

## IR Collider Ring – Design Goals

**Low ************ = 1 cm**
**Small momentum compaction factor ********c**** ~ 10****-5**
**Small circumference ~ 4 km**
**Momentum acceptance  ±0.005**
**Dynamic aperture ( > 3******** at normalized emittance = 2 ********m rad)**

## Periodic FODO cell – 135 deg. phase advance

**Arc dipoles**:
- $B=100;  =>         10 Tesla
- $E0=750000;  =>      750 GeV
- $rho = 250 m
- $N=4*50;  =>          200
- $Lcell=1600;  =>        16 m
- $ang=360/$Ndip;  =>        0.45 deg.
- $Lb=$PI*$Hr*$ang/(180*$B);  =>  196.5 cm

**Arc quadrupoles:**
- L[cm] 		G[kG/cm]
- 260 	 	25.1
- 260		-25.1

- phase adv./cell (x= 1350, y=1350)

## Quadrant Arc – Momentum compaction

- 48 full cells

- 2 trans cells

- 2 trans cells

## Collider Ring layout

- Circumference = 3846 m

## IR Linear lattice 

**IR ********Linear lattice **

- x* = 10 mm
- y* = 10 mm

- xmax = 18,500 m
- ymax = 18,600 m

- 8 m

- 13 m

**Name ****	****L[cm] ****	****G[kG/cm]**
- DD1 	290 	-20.1
- FF 	510 	18.2
- DD2 	290 	-23.5

## IR Beam Envelopes (rms)

**IR ********Beam Envelopes (********rms****)**

- N = 2  m rad
-  = 0.3 nm rad

- 8 m

- 13 m

- xmax = 18,500 m
- ymax = 18,600 m

## IR - matching to the Ring 

- doublet

- FODO

**IR - matching to the Ring **

- x* = 10 mm
- y* = 10 mm

- xmax = 18,500 m
- ymax = 18,600 m

- FF

- IP

## IR – Dispersion wave (negative M56)  

- doublet

- FODO

**IR – Dispersion wave (negative M****56****)  **

- x* = 10 mm
- y* = 10 mm

- xmax = 18,500 m
- ymax = 18,600 m

- FF

- IP

## IR - beta chromaticity 

- x* = 5 mm
- y* = 5 mm

- xmax = 36,000 m
- ymax = 32,000 m

- p/p= ±0.005

## Beta Chromaticity 

- IR beta functions strongly vary for off momentum particles
- It is measured by the beta chromaticity functions:

- or by is the so called ‘envelope’ dispersion

- Typical values of the w-functions ~ 100, need to be ~10
- Could be corrected with sextupoles placed in the Matching region where dispersion is generated in a controlled fashion
- dipoles outside the IR so that *D *= 0 but  *D *‘ 0 at the IP

## Beta Chromaticity correction with sextupoles

- (- I)

- (- I) (- I) (- I)

## Beta Chromaticity correction with sextupoles

- sS1         	L[cm]=90         	S[kG/cm/cm)]=0.1	 Tilt[deg]=0
- sS2         	L[cm]=90         	S[kG/cm/cm)]=-0.1 	Tilt[deg]=0

## T126/T226 and T346/T446  correction with sextupoles

- ×4

**T****126****/****T226**** and T****346****/T****446****  correction with sextupoles**

- ELEGANT (Michael Borland) studies

## T126/T226 and T346/T446  correction with sextupoles

- initial

- 1 turn uncorrected

- 1 turn: S1=0.1 S2=-0.1

- x x’

- y y’

## Chromatic Aberrations and Mitigation schemes 

-  Chromatic aberrations
-  Beta chromaticity in the IR
-  Second order chromaticity and momentum compaction
-  Mitigation schemes
-  Localized Beta chromaticity correction with sextupoles in the IR-to-Arc matching sections
-  Chromaticity correction in the Arcs (two families of sextupoles)
-  Dynamic Aperture – octupoles in the IR quads

## Chromaticity Compensation with two families of Sextupoles

- y = 3 1800

## ×4

- ELEGANT (Michael Borland) studies

## Dynamic Aperture Considerations 

- Large cross-detuning makes the dynamic aperture small – octupole corrections may be necessary
- Due to larger dispersion at IR sextupoles the requires sextupole gradient is lower reducing** **adverse** **2nd order effects
- The 2nd order dispersion will be corrected with sextupoles in the matching section/Arcs

## Summary

- Proposed Optics design for the Collider Ring and IR  - Linear Lattice
- Periodic dispersion achromat arcs
- Beta chromaticity corrections with sextuoples outside the FF region
- Natural chromaticity compensation with 2 families of orthogonal sextupoles
- Compact matching from IR to the arcs
- Ring parameters:

- Energy = 750 GeV
- Total Length=3880 m
- Tunes: Qx=13.8043     Qy=10.2107
- Chromaticity: nuxp=-4011.85 	 nuyp=-2393.76
- Momentum compaction ~ 10-5

- p/p= 0.003