1. HWR, SSR1 and SSR2 transverse kick summary Paolo Berrutti, Vyacheslav Yakovlev

2. Outline Study and re-optimization of SSR2 new geometry with four ports to allow cavity rotation in the cryostat. Estimation of the dipole kick introduced by a geometry misalignment for HWR and SSR1: spoke and beam pipe misalignments have been considered.

3. SSR2: cavity design with 4 ports SSR2 cavities has 4 ports to allow the rotation of two neighbor cavities of π/2, to compensate the transverse field asymmetry. Moving coupler and vacuum ports from the center to a π/4 angle means putting the ports where the magnetic field is significantly high: the magnetic enhancement factor needed to be re-optimized. The electric field area did not need to be modified, apart from the aperture in the spoke, which has been brought back to the original circular shape (racetrack previously). The new cavity looks like the old version, except from the ports.

4. SSR2: port optimization Figures below show the magnetic field on surface of the old cavity (right) and the new version (left), the rounding radius of the coupler was optimized, elliptical port was not successful. Rounding radius [mm] E_peak/E_accB_peak/E_acc [mT/(MV/m)] G [Ohm]R/Q [Ohm] 53.468.3112.63289.48 143.476.13112.96289.95 153.456.107112.98289.94 163.486.11112.99289.92

5. SSR2: transverse field asymmetry Figures below show the transverse field components, asymmetry parameter vs beta (whole energy range of usage) and transverse kick for particles as function of the angle with the X axis. The first row shows the two ports cavity design, the second the latest four ports design. The additional ports do not influence transverse field in the beam tube.

6. Geometry misalignments Spoke shifted of 1 mm for HWR, 0.5 mm for SSR1 and SSR2 Bilateral and unilateral BP shift, 1 mm for HWR, 0.5 mm for SSR1 and SSR2 The displacements considered involve transverse directions only, since z displacement influences just the acceleration. Synchronous phase has been taken into account for a more precise kick evaluation. The aim for SSR1 study is to evaluate the perturbation to beam dynamic we should expect for a real cavity, while for HWR is necessary to understand what the required alignment precision for the geometry is.

7. Dipole kick phase shift The transverse kick due to a geometry misalignment is usually shifted in phase from the accelerating kick of π/2, this is true for spoke and bilateral beam pipe misalignment: symmetric perturbation to the field. When only one of the beam pipe tubes is misaligned the field perturbation loses its symmetry, and so the phase shift condition changes.

8. Dipole kick calculation

9. Angle introduced by dipole

10. HWR spoke 1mm shift The maximum X and Y angle value is <1.6 mrad

11. HWR BP bilateral 1mm shift The maximum X and Y angle value is <3 mrad

12. HWR BP unilateral 1mm shift The maximum X and Y angle value is <1.4 mrad

13. HWR misalignments summary The total dipole kick can be estimated as the sum of the spoke and BP misalignments contribution, the same can be done with the angles. Total X dipole kick: Total Y dipole kick: Spoke 1mmBP 1 mmBP uni 1mmTotal ∆pc [MeV]0.10320.18290.1019< 0.3880 α [rad]0.0016 0.00290.0013<0.0058 Spoke 1mmBP 1 mmBP uni 1mmTotal ∆pc [MeV]0.09560.18090.1004< 0.3769 α [rad]0.00130.00290.0013< 0.0055

14. SSR1 spoke 0.5 mm shift The maximum X and Y angle value is <0.3 mrad

15. SSR1 BP 0.5 mm shift The maximum X and Y angle value is <0.4 mrad

16. SSR1 BP unilateral 0.5 mm shift The maximum X and Y angle value is <0.2 mrad

17. SSR1 misalignments summary The total dipole kick can be estimated as the sum of the spoke and BP misalignments contribution, the same can be done with the angles. Total X dipole kick: Total Y dipole kick: Spoke 0.5mmBP 0.5mmBP uni 0.5mmTotal ∆pc [MeV]0.04170.04650.0299< 0.118 α [rad]0.00029 0.000320.00018<0.00078 Spoke 0.5mmBP 0.5mmBP uni 0.5mmTotal ∆pc [MeV]0.03660.04710.0304< 0.1141 α [rad]0.00025 0.000330.00018<0.00076

18. SSR2 misalignments summary SSR2 study has been reported previously, here values are summarized in the tables. Total X dipole kick: Total Y dipole kick: Spoke 0.5mmBP 0.5mmBP uni 0.5mmTotal ∆pc [MeV]0.01920.01880.0126< 0.0506 α [rad]0.00007 0.00003<0.00017 Spoke 0.5mmBP 0.5mmBP uni 0.5mmTotal ∆pc [MeV]0.01230.01850.0127< 0.0435 α [rad]0.00005 0.000070.00003<0.00015

19. Conclusions and next steps HWR, SSR1 and SSR2 misalignment study is now complete. All the cavities analyzed showed that dipole kick is fully compensated by corrector satisfying the specification we already have. NEXT: SSR2 HOMs complete study SSR2 mechanical design