# Presentation: 667871

## ‘Dogbone’ RLA Lattice Optimization Including Errors 

- Jefferson Lab

-   Symmetric 5 GeV ‘Dogbone’ RLA  – Linear Optics
-  Optimized multi-pass linac focusing (1 GeV per pass)
-  Mirror-symmetric ‘Droplet’ Arcs (2, 3 and 4 Gev)  lattices with geometric ‘closure’
-  Arc-to-Linacs betatron matching
-   Magnet Misalignment Errors – DIMAD Monte Carlo Simulation
-   Focusing Errors Tolerance – Betatron Mismatch Sensitivity

**‘****Dogbone’ RLA Lattice Optimization Including Errors**** **

## 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)**
- 5 – 10 GeV FFAG
- 10 – 20 GeV FFAG

## Symmetric ‘Dogbone’ RLA (3.5-pass) Scheme 

- Main Linac (1 GeV/pass) - triplet focusing
- 3 ‘droplet’ Arcs based on periodic triplet cells (900 betatron phase advance per cell)

## Linac Optics - Beam Transport Choices 

- Multi-pass linac focusing scheme guarantees  (by design) 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)

## Multi-pass Linac Optics

- ‘half pass’ (n = 0), 1.5-2GeV

- 1-pass (n = 1), 2-3GeV

**mirror symmetry cond. **(outn = inn+1,and outn = -inn+1, n = 0,1 pass index)

- uniform phase adv/cell (x=0.25, y=0.25)

- last cell phase adv. (x=0.11, y=0.16)

## Multi-pass Linac Optics

- last cell phase adv. (x=0.07, y=0.14)

- 2-pass (n = 2), 3-4GeV

- 3-pass (n = 3), 4-5GeV

**mirror symmetry cond. **(outn = inn+1,and outn = -inn+1, n = 2, 3 pass index)

## Arc Optics - Beam Transport Choices 

- There is great advantage to have 900 phase advance/cell in the arcs – cancellation of chromatic effects.
- Phase advance 'mismatch' from/to the linacs is not detrimental - it induces larger 'beta beating' in Spr/Rec regions, but we match betas in this region anyway
- Dipole (horizontal) separation of multi-pass beams in RLA – a pair of dipoles (linac ends)
- No need to maintain achromatic Spreaders/Recombiners
- Compact Spreaders/Recombiners – minimized uniform focusing breakdown
- Arc1 and Arc 2 - scaled optics:
- Keep the same number of cells in each arc (Spr + 2 out + 16 in  + 2 out + Rec )
- Scale Arc1 (bends and quads) by factor of 3/2 to get Arc2
- Arc 3 optics
- Increase number of cells (Spr + 3 out + 20 in  + 3 out + Rec )
- Maintain the same 900 phase advance /cell.

## Arc 1  Layout 

**Arc dipoles**
- $Lb=150;  =>         150 cm
- $ang0=10.3283;  =>     10.328 deg
- $Nin=16;  =>          16
- $Nout=2;  =>           2
- $ang=(90+$ang0)/($Nin-2*$Nout);  =>  8.36 deg.
- #
- $Ang_out=$ang0+2*$Nout*$ang;  =>  43.77 deg.
- $Ang_in=2*$Nin*$ang;  =>  267.54 deg.
- $BP=$PI*$Hr*$ang/(180*$Lb);  =>  6.537 kGauss
- $Lring=227.3 m

## Periodic Triplet Cell - Arc1 Optics

- ‘outward’ cell’ at 2GeV (T = 1911.64 MeV)

- $P= 2014.529
- $Hr=$P/$c*1e11;  =>  6719.745
- dipoles (2 per cell)
- $Lb=150;  =>         150 cm
- $ang=(90+$ang0)/($Nin-2*$Nout);  =>  8.36 deg
- $BP=$PI*$Hr*$ang/(180*$Lb);  =>  6.537 kGauss

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

- phase adv/cell (x=0.25, y=0.25)

## Droplet Arc – Optics ‘Building Blocks’ 

- 900 phase advance/cell: inward and outward cells, missing dipole, empty cells

## 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

## Linac-Arc1-Linac Matching

- ‘half pass’  (1.5-2GeV)

- 1-pass (2-3GeV)

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

- Arc1 (2GeV)

## 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

## Focusing Error Tolerances – Quadrupole Field Spec 

- Standard deviation of Courant-Snyder invariant:

- Cumulative Arc-to-Arc Optics mismatch as measured by Courant-Snyder invariant change:

## Focusing Error Tolerances – Quadrupole Field Spec

- By design, one can tolerate Arc-to-Arc  mismatch at the level of **1%** (to be compensated by the dedicated matching quads).

- For any given Arc and the following Linac one can evaluate:    Fmin ≈1 m    and
- Thanks to well balanced, tight focusing in the Arcs and compact Spr/Rec optics the last number, 50 m, is factor of **6** smaller than the corresponding quantity for a typical CEBAF Arc-Linac segment.

- This yields  the required design specification for quadrupoles  of  **0.2%****:**

## 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’ in the higher pass linacs
- mirror-symmetric Arc optics based on constant phase advance/cell (900) compact lattice architecture for Spr/Rec/Trans
- geometric ‘closure’ of the ‘droplet’ Arcs
- 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  1% of Arc-to-Arc betatron mismatch limit  sets the  quadrupole field spec at 0.2%