1. Electric field calculation for the neutron EDM SNS experiment. Septimiu Balascuta Ricardo Alarcon ASU, 02/07/2008

2. The requirements for the electric field in the neutron EDM experiment: Electric field in the cell > 50 kV/cm The electric field uniformity should be less then 0.1% (optimum performance). The minimum performance is 1%. The reverse accuracy <1% Spark rate: 0.1/year. Eddy current heating should be < 1 mW.

3. Summary: Electric field uniformity inside measurement cell for three geometries of the cell and electrode. The effect of the optical guides on the electric field uniformity. A method to decrease the power loss (due to eddy currents) in the film on the surface of the electrodes in the presence of and external RF magnetic field.

4. The 1 st configuration of the electrodes and cell. All the cell walls are made from Lucite and are 1.3 cm thick. High Voltage electrode dimensions: height = 29.8 cm, radius=5 cm. Grounded electrode: height=24.8 cm, radius=2.5 cm. Cell boundaries: 5 cm <X< 12.6 cm, -5.08 cm<Y<5.08 cm, -25 cm<Z<25 cm Y X 5.08 1.3 1.62 1.9 2.5 1.3 6.38 1.9 5 3.7 5 1.62 1.3 Fig.1A Fig.1B

5. 1C 1D 1E 1F

6. Fig. 1G A zone map of the electric field in the vertical plane XOY through the middle of the cell.

7. High voltage electrode height = 32 cm, radius=5 cm; Grounded electrodes height 24 cm<H (g) <29 cm, radius=2.5 cm; Lateral walls of the cell are conductive and 1.3 cm thick; Top/bottom walls of the cell are not conductive and 1.3 cm thick. Cell boundaries: 5 cm <X< 12.6 cm, -5.08 cm<Y<5.08 cm, -25 cm<Z<25 cm. The 2 st configuration of the electrodes and cell with conductive lateral walls. Y X 5.08 0.72 0.47 2.5 1.3 6.38 16 5 3.7 5 7.6 7.47 H(g) 1.3 Fig.2A Fig.2B

8. 2C2D 2E2F

9. 2G A line map of the electric field in the vertical plane XOY. Fig.2G

10. The 3 rd configuration of the electrodes and cell with conductive walls. High voltage electrode height=11cm +R(HV) where R(HV) (the radius of the High Voltage electrode) was varied 5 cm< R(HV)< 5.75 cm. Grounded electrodes height 27.5 cm <H(g)<30 cm and radius equal with 2.8 cm; Lateral walls of the cell are conductive and 1.3 cm thick; Top/bottom walls of the cell are insulator (Lucite): 1.3 cm thick. Cell boundaries: 5 cm <X< 12.6 cm, -5.08 cm<Y<5.08 cm, -25 cm<Z<25 cm. Fig.3A

11. 3B3C 3D3E

12. Fig. 3F A zone map of the electric field in the vertical plane XOY.

13. The effect of the optical guides on the electric field uniformity inside the measurement cell. Model A: The lateral optical guides are attached to the external surface of the back and front cell walls. High Voltage electrode has radius 5.35 cm and height 32.7 cm (the 2 nd configuration presented above). The grounded electrode has the radius 2.8 cm and height 28 cm. Lateral guides are rectangular sheets: 0.99 cm (close to H.V. electrode) and 0.86 cm thick (close to 0 V electrode). Fig.4A

14. The Electric field uniformity close to the back wall for: A: No optical guides B: Optical guides with dielectric constant 2 and 3.5. Fig.4B Fig.4C Fig.4D Fig.4E

15. Vertical cross section (at 4 cm upstream) front the front wall of the cell (Z=29 cm). Fig.4F

16. Model B: The lateral optical guides (1.3 cm thick) pass through the recess inside the electrodes. The surface of the guide is covered with a conductive film. The top and bottom Lucite guides are 1.2 cm thick and 7.6 cm long in the OX direction) are not conductive. The height of the lateral optical guides is 29.68 cm. 5A Figure 5B: The lateral optical guides pass through the recess of the electrodes. Figure 5A: Making the surface of the Optical Guides (O.G.) not conductive increases the relative electric field variation up to 2%. 5B

17. Fig.5C: A cross section in a vertical plane (perpendicular to the OZ axis) and passing through the center of the cell (Z=0 cm). The conductive lateral walls of the cell are marked in red. Fig.5D: A cross section in a vertical plane (perpendicular to the OZ axis) and passing through the front wall of the cell (Z=25 cm). Fig. 5C Fig. 5D

18. 5E 5F Figure 5E: A cross section in a vertical plane at 1 cm distance upstream from the front wall of the cell. Figure 5F: A 3D picture of the Cell + Electrodes + Optical guides. The optical guides are made from Lucite. The lateral guides pass through the recess and come out from the front and back of the electrodes.

19. Figure (A,B). The relative electric field variation is calculated along directions parallel with the horizontal OX axis on the vertical surface of the back face of the cell) and in the Z=0 vertical plane passing through the middle of the cell. The electric field in the center of the cell (X=8.8 cm, Y=Z=0 cm) is E0=52.63 kV/cm for a 400 kV voltage applied on the High Voltage electrode. The thickness of the top and bottom Lucite walls is 1.92 cm. 5G 5H

20. Conclusion: A uniformity of the electric field less then 0.1% can be obtained inside the measurement cells for the 2 nd and 3 rd configurations presented above. However only for the 2 nd configuration the maximum electric field is less then 1.2 times the electric field in the center of the cell. The electric field uniformity can be less then 0.1% if the lateral optical guides are located inside the recess volume of the electrodes.

21. A method to decrease the power P loss (due to the eddy currents) in the conductive film on the surface of the electrodes in an external RF magnetic field. The thickness of the conductive film will be about 200 micrometers thick. “P” depends linearly on the conductivity of the film. “P” decreases if there are gaps (maxim1 mm wide) in vertical planes (parallel with XOY) on the surface of the film. The power loss in the (1 mm) film on the surface of the electrodes is plotted versus the conductivity of the film (Sigma_LC) for a film with no gaps, and for a film with 2, 8, 10 and 12 gaps. Gaps located at Z= ± 5cm, ± 10 cm, ±15 cm, ±20 cm, ±25 cm and ±35 cm along the OZ axis. B A

22. Fig A,B,C,D: The electric field uniformity calculated near the Z=5 cm gap on the HV electrode (at X=5.0 ) and on the grounded electrode (X=12.6 cm) for a film with no gaps (A) and a film with (eight) gaps 0.5 cm wide (fig. B) 0.2 cm wide (fig. C) and 0.1 cm wide (fig. D). AB C D