# Presentation: 660650

## ‘Dogbone’ RLA  Error Tolerances and Tracking

- Jefferson Lab

-   Symmetric 5 GeV ‘Dogbone’ RLA  – Linear Optics
-   Front-to-End Multi-particle Tracking - 30 mm rad Acceptance (normalized)
-  Magnet Misalignment Errors – DIMAD Monte Carlo Simulation
-   Focusing Errors Tolerance – Betatron Mismatch Sensitivity and Tunability
-   Magnet Field Quality Specs – Emittance Dilution due to Nonlinearities

**‘****Dogbone’ RLA ******** Error Tolerances and Tracking**

## Symmetric Muon Acceleration Complex 

**Linear pre-accelerator (273 MeV/c – 1.5 GeV) **
**Symmetric ‘Dogbone’ RLA (allowing to accelerate both ********+**** and ********-**** species), 3.5-pass (1.5 – 5 GeV)**

## Linac Optics – Arcs, multi-pass linac 

- Multi-pass linac optics additionally constrained by the mirror symmetry of the droplet arcs
-  at the exit/entrance from/to the previous/next linac: the betas are equal and the alphas are of the opposite sign
- Optimized 'bisected’ linac was chosen as follows:
- 900 phase advance/cell is set for the  'half pass’  linac (1.5-2GeV).
- as a consequence linac phase advance/cell in the first part of 1-pass drops to about 450.
- to avoid large 'beta beating' one chooses to keep 450 phase advance/cell throughout the second part of the linac (Bob Palmer).
- the phase advance at the end of 2-pass linac drops by another factor of two (22.50).
- the 'beta beating' is rather small on higher passes (2 and 3)

## Initial beam emittance/acceptance after cooling at 273 MeV/c 

| Study IIb |  | rms | A = (2.5)2  |
| --- | --- | --- | --- |
| normalized emittance: x/y  | mmrad  | 4.8 | 30 |
| longitudinal emittance: l 
lp z/mc)
     momentum spread: p/p 
     bunch length: z | mm 

mm | 27

0.07
176 | 150

0.17
442  |

## Pre-accelerator  different style cryo-modules

|  | Short | Medium | Long |
| --- | --- | --- | --- |
| Number of periods | 12 | 18 | 22 |
| Total length of one period | 3 m | 5 m | 8 m |
| Number of cavities per period | 1 | 1 | 2 |
| Number of cells per cavity | 1 | 2 | 2 |
| Cavity accelerating gradient  | 15 MV/m | 15 MV/m | 15 MV/m |
| Real-estate gradient  | 3.72 MV/m | 4.47 MV/m | 5.59 MV/m |
| Aperture in cavities (2a) | 460 mm | 460 mm | 460 mm |
| Aperture in solenoids (2a) | 460 mm | 460 mm | 460 mm |
| Solenoid length | 1 m | 1 m | 1 m |
| Solenoid maximum field | 1.5 T | 1.9 T | 3.9 T |

## Linear Pre-accelerator – Twiss functions and beam envelope (2.5 )

## Introduction of synchrotron motion in the initial part of the linac

## Injection Chicane  both  and  

## Chicane  ‘dogleg’

## Linac-Arc1-Linac Matching

- ‘half pass’  (1.5-2GeV)

- 1-pass (2-3GeV)

- (out = in ,and out = -in , matched to the linacs)

**Arc1 (2GeV)**

## Linac-Arc2-Linac Matching

- 2-pass (3-4GeV)

- (out = in ,and out = -in , matched to the linacs)

**Arc2 (3GeV)**

- 1-pass (2-3GeV)

## Linac-Arc3-Linac Matching

- (out = in ,and out = -in , matched to the linacs)

**Arc1 (3GeV)**

- 3-pass (4-5GeV)

- 2-pass (3-4GeV)

## Arc 1 – Mirror-symmetric Optics 

- Arc 1 – Mirror-symmetric Optics** **

- dipoles (2 per cell)
- $Lb=150;  =>         150 cm
- $ang0=10.3283 deg
- $ang=(90+$ang0)/($Nin-2*$Nout);  =>  8.36 deg
- $B=$PI*$Hr*$ang/(180*$Lb);  =>  6.537 kGauss

- quadrupoles (triplet):
- L[cm] 		G[kG/cm]
- 68 		-0.326
- 125 		0.328
- 68 		-0.326

- 16 cells

- 2 cells

- 2 cells

- (out = in ,and out = -in , matched to the linacs)

- $Nout=2, need minimum of 3 triplets to match 6 Twiss parameters

## Arc 2 – Mirror-symmetric Optics 

- dipoles:
- $Lb=150;  =>         150 cm
- $E=2920.75;  =>     2920.75 MeV
- $ang0=10.3283;  =>     10.33 deg.
- $B0=-$PI*$Hr*$ang0/(180*$Lb);  => -12.12 kGauss
- $ang=8.3607 deg.
- $B=$PI*$Hr*$ang/(180*$Lb);  =>  9.81 kGauss

- quadrupoles (triplet):
- L[cm] 		G[kG/cm]
- 68 		-0.490
- 125 		0.492
- 68 		-0.490

- (out = in ,and out = -in , matched to the linacs)

## Arc 3 – Mirror-symmetric Optics 

- dipoles:
- $E=3929.86 MeV
- $B0=-8.0755 kGauss
- $ang0= 5.1577 deg
- $BP=$PI*$Hr*$ang/(180*$Lb);  =>  10.64 kGauss
- $ang=(90+$ang0)/($Nin-2*$Nout);  =>  6.797 deg
- $Ang_out=$ang0+2*$Nout*$ang;  =>  45.94 deg
- $Ang_in=2*$Nin*$ang;  =>  271.88 deg

- quadrupoles (triplet):
- L[cm] 		G[kG/cm]
- 68 		-0.6537
- 125 		0.6565
- 68 		-0.6537

- (out = in ,and out = -in , matched to the linacs)

## Magnet Misalignment Errors 

- Lattice sensitivity to random misalignment errors was studied via DIMAD Monte-Carlo assuming:

- Gaussian distribution was chosen for individual quad misalignments
- Resulting reference orbit distortion (uncorrected) for Arc 2 is illustrated below

- Similar level of dipole misalignment errors had virtually no effect on random steering

## Arc 2 – Magnet Misalignment Errors 

- Same  level of orbit drifts due to quad misalignments for other ‘Dogbone’ segments (Arc 1, 3 and  linacs)
- Orbit drifts at the level of ~3 cm can easily  be corrected by pairs of hor/vert correctors (2000 Gauss cm each) placed at every triplet girder

**RMS Orbit Displacement [m]:  **
- X:	0.9486e-02
- y: 	0.7003e-02

**Extr. Orbit Displacement [m]:**
- Xmax:	0.2538E-01
- Xmin:	-0.2782E-01
- ymax:	0.1434E-01
- ymin:	-0.1697E-01

## Initial beam emittance/acceptance after cooling at 273 MeV/c 

| Study IIb |  | rms | A = (2.5)2  |
| --- | --- | --- | --- |
| normalized emittance: x/y  | mmrad  | 4.8 | 30 |
| longitudinal emittance: l 
lp z/mc)
     momentum spread: p/p 
     bunch length: z | mm 

mm | 27

0.07
176 | 150

0.17
442  |

## Longitudinal Beam Dynamics – Tracking 

## Large Momentum Compaction for a ‘droplet’ arc

## Cumulative Focusing Errors – Magnet Tolerances

- Focusing ‘point’ error perturbs the betatron motion leading to the Courant-Snyder invariant change:

- where,  m =1 quadrupole, m =2 sextupole, m=3 octupole, etc

- Each source of field error (magnet) contributes the following Courant-Snyder variation

## Cumulative mismatch/emittance increase along the lattice (N sources):

- Standard deviation of the Courant-Snyder invariant is given by:

- Assuming uncorrelated errors  at each source the following averaging (over the betatron phase) can by applied:

- Cumulative Focusing Errors – Magnet Tolerances

## Some useful integrals .... :

- will reduce the coherent contribution to the C-S variance as follows:

- Including the first five multipoles yields:

- Cumulative Focusing Errors – Magnet Tolerances

## Cumulative Focusing Errors – Magnet Tolerances

- One can define a ‘good fileld radius’  for a given type of magnet as:

- Assuming the same multipole content for all magnets in the class one gets:

- The first factor  purely depends on the beamline optics (focusing), while the second one describes field tolerance (nonlinearities) of the magnets:

## The higher, m > 1, multipoles will contribute to the emittance dilution – ‘limited’ by design via a separate allowance per each segment (Arc, linac)  (e.g. 1%) 

- Field Error Tolerances – Magnet Specs

- The linear errors, m =1, cause the betatron mismatch – invariant ellipse distortion from the design ellipse without changing its area – no emittance increase.
- By design, one can tolerate some level (e.g. 10%) of Arc-to-Arc betatron mismatch due to the focusing errors,  (quad gradient errors and dipole body gradient) to be compensated by the dedicated matching quads

## Arc1-Linac1:    

- Arc1-Linac1:

- 			Fmin = 1 m (GdL=41 kG)

## Arc2-Linac2:    

- Arc2-Linac2:

- 			Fmin = 1.5 m (GdL=65 kG)

## Arc1-Linac3:    

- Arc1-Linac3:

- 			Fmin = 1.46 m (GdL=92 kG)

## Summary

- Symmetric ‘Dogbone’ RLA (allowing to accelerate both + and - species), 3.5-pass (1.5 – 5 GeV) scheme – Complete linear Optics
- multi-pass linac optics – optimized focusing profile (tolerable phase ‘slippage’ )
- mirror-symmetric droplet’ Arc optics based on constant phase advance/cell (900)
- Front-to-End Multi-particle Tracking – **30 mm rad** normalized acceptance, 5 % particle loss
- Magnet misalignment error analysis (DIMAD Monte Carlo on the above lattice) shows quite manageable level of orbit distortion for  ~1 mm  level of magnet misalignment error.
- Great focusing errors tolerance for the presented lattice  10% of Arc-to-Arc betatron mismatch limit  sets the  quadrupole field spec at 0.1%