1. Transverse kick of HWR and SSR2 Paolo Berrutti, Vyacheslav Yakovlev

2. Outline HWR donut: evaluation of the kick and of the quadrupole effect in the whole beta domain: comparison with the previous HWR version Kick dependence on RF phase in the beta domain Modified SSR2 geometry based on HWR experience: SSR2 donut Electric and magnetic components analysis of SSR2 transverse kick (comparison with HWR) SSR2 modified design to have better quadrupole compensation in the whole beta domain Conclusions

3. HWR donut geometry Old (on the left) and new (right) HWR geometry. The central part of the electrode now is donut shaped.

4. HWR donut Transverse kick and Q β=0.11HWR Ell_BPHWR Donut Δpx [keV]2.272e+032.535e+03 Δpy [keV]2.385e+032.521e+03 Q-0.04880.0054

5. HWR Electric and Magnetic Q Q due to electric field is represented in figs below. β range covers the whole energy range in which SSR2 cavity is used (2.1-10.8 MeV). The electric quadrupole component is reduced by the donut shape, and it is the major part of the total Q of HWR. ElectricMagnetic

6. HWR Electric kick component Δpc due to electric field is represented in figs below. β range covers the whole energy range in which HWR cavity is used (2.1-10.8 MeV ). HWR donut presents more symmetry than HWR elliptical beam pipe. HWR elliptical BP HWR donut

7. HWR Magnetic kick component Δpc due to magnetic field is represented in figs below. β range covers the whole energy range in which HWR cavity is used (2.1-10.8 MeV ). HWR donut and HWR elliptical beam pipe present almost the same magnetic field contribution to transverse kick. HWR elliptical BP HWR donut

8. HWR kick X and Y component Δpc along X and Y axis is represented in figs below. β range 2.1-10.8 MeV. HWR donut has a better behavior in the whole beta range because of the symmetry of the two main (electric) components, since the magnetic contribution is small. HWR elliptical BPHWR donut

9. Quadrupole parameter vs RF phase Particle energy= 2.1 MeV, HWR Donut cavity Δpc X and Y components are synchronous in phase, Q parameter does not depend on RF phase.

10. Quadrupole parameter vs RF phase Particle beta= 0.11, HWR donut cavity Δpc X and Y components are synchronous in phase, Q parameter does not depend on RF phase.

11. Quadrupole parameter vs RF phase Particle energy= 10.8 MeV, HWR donut cavity Δpc X and Y components are synchronous in phase, Q parameter does not depend on RF phase.

12. SSR2 donut Transverse kick and Q β=0.47HWR Ell_BPHWR Donut Δpx [keV]5.518e+026.714e+02 Δpy [keV]5.512e+024.573e+02 Q0.00120.3793

13. SSR2 Electric and Magnetic Q Q due to electric field is represented in figs below. β range covers the whole energy range in which SSR2 cavity is used (35-165 MeV). Donut shape helps reducing the electric contribution, but it makes tha magnetic even worse. ElectricMagnetic

14. SSR2 Electric kick component Δpc due to electric field is represented in figs below. β range covers the whole energy range in which SSR2 cavity is used (35-165 MeV). SSR2 donut electric kick presents more symmetry than SSR2 racetrack aperture. SSR2 racetrack apSSR2 donut

15. SSR2 Magnetic kick component Δpc due to magnetic field is represented in figs below. β range covers the whole energy range in which SSR2 cavity is used (35-165 MeV). SSR2 donut and SSR2 racetrack aperture show similar magnetic kick in the whole beta range. The shape of the central part of the spoke affects mostly the electric field. SSR2 racetrack apSSR2 donut

16. SSR2 kick X and Y component Δpc X and Y components are represented in figs below. β range covers the whole energy range in which SSR2 cavity is used (35-165 MeV). The kick X and Y component are largely affected by magnetic field. As β increases the magnetic effect gets higher and the X component of magnetic field can not be reduced because of the shape of the field lines around the spoke. SSR2 racetrack apSSR2 donut

17. Electric field SSR2 3D 3D electric field inside the spoke resonator is reported in figures below, X-Y view is on the left, Y-Z on the right. X and Y components of the electric field can be modified by changing the shape of the aperture and beam pipe areas.

18. Magnetic field SSR2 3D 3D magnetic field inside the spoke resonator is reported in figures below, Y-Z view is on the right, X-Y on the right. It is clear that the magnetic field line are circles around the spoke electrode. The magnetic field X component is bigger than the Y one in the aperture region.

19. SSR2 new version Since the donut shape is not effective in reducing the quadrupole component the old version has been modified: the aperture is circular instead of racetrack beam pipe profile has changed from circular to elliptical. The figs below show Q vs beta of the new SSR2 design (right) and the comparison with RT aperture and donut (left).

20. Quadrupole parameter vs RF phase Particle energy= 35 MeV, SSR2 new version Δpc X and Y components are synchronous in phase, Q parameter does not depend on RF phase.

21. Quadrupole parameter vs RF phase Particle β=0.47, SSR2 new version Δpc X and Y components are synchronous in phase, Q parameter does not depend on RF phase.

22. Quadrupole parameter vs RF phase Particle energy= 165 MeV, SSR2 new version Δpc X and Y components are synchronous in phase, Q parameter does not depend on RF phase.

23. Conclusions Donut shape does help to reduce the quadrupole effect in HWR cavity because: The main kick component is due to electric field (low beta) Donut shape symmetrizes X and Y electric component, and so it symmetrizes the transverse kick Magnetic field contribution to Δpc transverse is not affected by the donut but it is very small. SSR2 quadrupole component is not correctable using the donut shape: Donut shape symmetrizes only electric field components and the magnetic ones are almost unmodified Hy is bigger than Hx in the aperture region, this is due to the spoke cavity magnetic field lines The ratio between magnetic and electric contribution to Δpc gets higher as beta gets higher The design of SSR2 showing the best quadrupole component vs beta has elliptical BP and circular aperture. HWR and SSR2 quadrupole effect does not depend on RF phase for any of the betas analyzed in the range of interest.