1. FCC-ee Lattice with Misalignments
Sergey Sinyatkin

2. Mission
Using ordinary tolerances for magnetic elements preserve small vertical emittance.
Transverse shift of elements is Gaussian with RMSx,y = 100 um, truncated at 2*Sigma.
Estimation of vertical emittance with misalignments of magnets in the area of extremely large betas (IR).
Estimation of vertical emittance after orbit and betatron coupling correction.

3. K. Oide

4. FCC-ee Twiss functions
FF:
βx_max = 500 m
βy_max = 10 km
GQD = -91 T/m
GQF = 84 T/m
CCY:
βx_max = 200 m
βy_max = 3-4 km
GQD = -8 T/m
CCX:
βx_max = 400 m
βy_max = 200 m
GQF = 11 T/m

5. Task (Quadrupole misalignment )
Quadrupoles are shifted randomly (Gauss) in transverse plain with Sigma_x,y = 100 um, truncated at 2*Sigma.
FF quads and those in V and H chromatic sections (extremely large beta) meanwhile are not shifted.
Q shift distorts CO, vertical dispersion and generates betatron coupling (COD at sextupoles) leads to vertical emittance growth.
I installed BPMs (4-5 per betatron wave) and correctors (3-4 per wave).
Skew-quads are installed in the FF quads, in the IR vertical chromatic section and in every hundredth arc quads.
Correctors and skew-quads correct COD, coupling and vertical dispersion. What vertical emittance can we achieve after correction?

6. Correction COD, coupling and Dy is corrected by MADX.
The process is time consuming.
All sextupoles and bends are perfectly aligned.
The results are preliminary

7. Q shift and CO
For the 100 um Q shift the closed orbit w/o correction is hardly can be found (but can)
For the 10 um Q shift w/o correction the probability for CO is ~ 50%.
For the 100 um Q shift after correction the CO exists for ~ 30-40% samples.
CO can not be found when the vertical tune moves to the half integer or integer resonance.

8. CO search by iterations
Correction of beam trajectory and betatron phase advance can be done step by step (BPM by BPM) like in beam line (not closed optics) starting from injection point. (very slowly with MADX)
Alternative is closed orbit correction iteratively with misalignments scaled from 0 um to 100 um.

9. CO search vertical tune limits the closed optics
For some seeds MADX failed with CO
E_y/Ex = 0.5-5x10-3
After skew quad correction. Q shift of 100 um (RMS).

10. COD after correction
COD around the ring (10 seeds) 100 um quadrupole shift produces average residue COD  70 um (x) and  50 um (y) after orbit correction
Steering magnet strength σkick= 6.5 urad

11. COD in the IR (after correction)
COD in the IR. Max horizontal COD  800 um

12. Optics distortion (9 samples) after correction
(βx_err/βx)max = 1.1 (βy_err/βy)max = 1.5
σDx= 1.5 cm σDy= 3.4 mm

13. Eigen mode oscillation (1 sample) after orbit correction
before Skew quads correction
εy/ εx = 1.6 %
after Skew quads correction
εy/ εx = 0.15 %

14. Vertical emittance excitation
CO correction
CO correction +
SQ in the IR +
SQ in the arcs
CO correction +
SQ in the IR
SQ: σK1sL= 80e-6 m-1

15. Q tilts Before correction After correction by SQ (εy/ εx ~0.5x10-3)
σK1sL= 76e-6 m-1
Quadrupoles rotate randomly (Gauss) with Sigma_tilt = 100 urad, truncated at 2Sigma.
FF and H/V chromatic sections quads are not rotated.

16. FF & CCS_XY quads misalignment
FF Quads are shifted randomly (Gauss) transversely with Sigma_x,y = 25,50 um, truncated at 2Sigma.
Misalignments of other elements are zero.
Q shift distorts CO, vertical dispersion  vertical emittance degrades.
Skew-quads are installed in the FF quads, IR vertical chromatic section and in every hundredth arc quads.
IR Steerers correct COD
Coupling and vertical dispersion is corrected by Skew-quads.

17. FF Quads shift and CO
There is not the closed orbit w/o correction for the 50 um Q shift.
For the 10 um Q shift w/o correction the probability for CO is ~ 50% (in many cases it is MADX problem to find CO).
For the 50 (100) um Q shift after correction the CO exists for all samples.

18. FF quads misalignment After orbit and vertical dispersion correction
<|ηy|>= 0.7 cm
εy / εx = 0.3 %
After closed orbit correction
<|ηy|>= 3 cm
εy / εx = 3 %
FF Quads: Sigma_x,y = 50 um, truncated at 2Sigma

19. Emittances ratio (FF Quads)
Δx,y = 50 um
After dispersion correction by SQ:
Tolerance 50 um
εy / εx ~ 2 %
Tolerance 25 um
εy / εx ~ 0.4 %
E = 175 GeV:
GQD = -90 T/m
GQF = 84 T/m
Δx,y = 25 um

20. FF quads misalignment
Acceptable value of FF quads tolerance < 50 um (expected 20 – 30 um).

21. CCS_XY quads misalignment Emittances ratio
CCS_XY Quads Tolerance 100 um
εy / εx ~ 0.07 % after dispersion correction by SQ
CO w/o correction for the 100 um CCS_XY Q shift is available.
CCS_X: βx_max = 400 m, βy_max = 200 m, GQF = 11 T/m (E=175 GeV)
CCS_Y: βx_max = 200 m, βy_max = 3-4 km, GQD =-8 T/m (E=175 GeV)

22. Task (Sextupoles misalignment )
All sextupoles are shifted randomly (Gauss) transversely with Sigma_x,y = 100 um, truncated at 2Sigma
The shift distorts CO, generates betatron coupling and vertical dispersion  vertical emittance degrades
BPMs (4-5 per betatron wave) and steerers (3-4 per wave) are installed
Skew-quads are installed in the FF quads, IR vertical chromatic section and in every fiftieth arc quads
Steerers and skew-quads correct COD, coupling and vertical dispersion. What vertical emittance can be achieved after correction?

23. Optics correction Average: y  0.3% x Average: y  10% x
A random sample before and after COD and coupling correction
(βx_err/βx)max = 1.7 (βy_err/βy)max = 2.3
σDx= 2.2 cm σDy= 5.2 cm
200 samples
Average: y  10% x
(βx_err/βx)max = 1.1 (βy_err/βy)max = 1.9
σDx= 1.2 cm σDy= 0.8 cm
10 samples
Average: y  0.3% x

24. Vertical emittance after correction
Arc sextupole misalignments 100 mkm
(E=175 GeV)
SSD = -360 T/m2
SSF = 240 T/m2
βx_max = 70 m
βy_max = 90 m
y/x ≈ 310-3
CCS_XY sextupole misalignments 100 mkm
(E=175 GeV)
SSD = T/m2
SSF = 1800 T/m2
βx_max = 400 m βy_max = 3-4 km

25. Dipoles misalignment Emittances ratio
Beam parameters are not distorted by 100 um XY misalignments of dipoles.
Vertical emittance is excited by 200 urad dipoles tilts.
εy / εx is reduced to 0.14 % after vertical dispersion correction by SQ.

26. BPM misalignment Misalignment of FF monitors: ΔX,Y = 25 um
y/x ≈ 210-3
Misalignment of CCS_XY monitors:
ΔX,Y = 50 um
y/x ≈ 710-3
Misalignment of ARC monitors:
ΔX,Y = 100 um
y/x ≈ 210-3

27. Misalignments of elements
Tolerance:
BPM:
FF ΔX,Y = 25 um
CCS ΔX,Y = 50 um
ARC ΔX,Y = 100 um
Dipole:
ΔTilt = 200 urad
Quadrupole
CCS ΔX,Y = 100 um
Sextupole
After CO correction
After CO&SQ correction

28. Misalignments of elements
(βx_err/βx)max = 1.3
(βy_err/βy)max = 2.2
σDx= 2.4 cm
σDy= 7 cm
y/x ≈ 11 %
σDx= 1.5 cm
σDy= 0.8 cm
y/x ≈ 0.37 %
After CO correction
After CO&SQ correction

29. Misalignments of elements Emittances ratio
BPM: FF ΔX,Y = 25 um, CCS ΔX,Y = 50 um, ARC ΔX,Y = 100 um;
Dipole: ΔTilt = 200 urad;
Quadrupole: FF ΔX,Y = 25 um, CCS ΔX,Y = 100 um, ARC ΔX,Y = 100 um;
Sextupole: CCS ΔX,Y = 100 um, ARC ΔX,Y = 100 um;
<εy / εx> is reduced from 15 % to 1.4 % after CO, betatron coupling and vertical dispersion correction.

30. Misalignments of elements (E =175 GeV; After CO & SQ correction)
dx,y,um
Tilt, mrad
before corr. CO
Dy, mm Qx Qy
εx, nm*rad
εy / εx, %
ARC BPM
100 +
5.5
0.003
0.041
1.40
0.66
FF BPM 25
3.2
0.000
0.002
1.39
0.20
CCS_XY BPM 50
7.8
0.043
0.72
Dipole -
0.2
2.6
0.14
Quads
3.3
0.063
1.46
0.12
FF Quads
4.0
0.38
CCS_XY Quad
1.8
0.024
0.07
Sextupole
3.1
0.005
0.009
1.42
0.26
CCS_XY Sext
0.004
0.082
0.28
Total *
7.0
0.011
0.048
1.44
* - all misalignments included.

31. Conclusion For Q shift w/o correction CO is unavailable.
After correcting CO and keeping the vertical tune far from resonance CO is available.
Vertical dispersion is strongly excited by FF quads misalignments.
Acceptable value of FF quads tolerance < 50 um (expected 20 – 30 um).
CO is strongly distored by BPM of FF and CCS areas after correction.
Misalignments between FF Quads and FF BPM must be small ( um).
In order to have better accuracy it is necessary more samples.