1. Rb flow in AWAKE Gennady PLYUSHCHEV (CERN - EPFL)

2. AWAKE plasma cell: overview Ø4cm, 10m tube Rb, 200°C, 7x10 14 cm -3 Rarefied regime Goal: sharp density gradient. Fast valves are too slow: (10ms x 300m/s = 3m) Solution: orifices + continuous flow

3. Theory

4. Density profile

5. Flows in AWAKE Evaporation area (m 2 ) required to provide the mass flow rate of 1.0mg/s as a function of density and temperature of source: Calculated flows in order to have 10% density gradient in plasma cell:

6. Source temperatures in AWAKE Evaporation area: 10cm 2 T1T1 T2T2 T1T1 T2 - T1 To set the density gradient with 0.5% accuracy, the temperature of sources should be set with 0.14°C accuracy

7. COMSOL simulation -Qualitative analysis -Continuum flow -What is the structure of density near orifice inside plasma cell? Is there density maximum near orifice? n xx n

8. COMSOL results: Pressure

9. COMSOL results: Pressure Zoom There is density overshoot (1-2%) near orifice in front of source

10. Ends of plasma cell -At each end of plasma cell there is Volume for Rb to Expand (Expansion Volume). -The volume has cold wall in order to condensate the Rb and reduce the density -Thus sharp density gradient created through the orifice -Simulation to study the Rb flow beyond the plasma cell (Volume for Rb to Expand) Goal: 1. calculate density on axis 2. calculate Rb deposition Grant estimation: 0.5m available for Expansion Volume

11. Condensation rate: Maxwell Con Condensation Evaporation Thus, according to this theory, directed flux on surface will condensate immediately.

12. Density in infinitely large volume

13. Saturation pressure / density Saturation density of to 1.75x10 10 cm -3 corresponds to 28°C. Thus it is logical to keep the Expansion Volume under this temperature. In this case, the pressure in Expansion Volume will be determined mainly by the theoretical limit for infinite volume and not by the saturation density due to evaporation from the Expansion Volume surface.

14. Results of simulation: 27°C The simulation shows good agreement with the theoretical curve for infinite volume (for the case with T wall < 28°C and L<0.5m): -Cylindrical volume r = 0.1m; L = 0.2m -Base of the cylinder with orifice at 200°C -Another base and tube is at low temperature 27°C -Side of the cylinder is also at low temperature 27°C

15. Results of simulation: 27°C

16. In order to have density profile close to the theoretical limit for infinite volume: Rb flow should not hit the surface with temperature higher than 28°C (for Expansion Volume of 50cm). Thus all Rb which hits walls will be condensed. The transition from hot to cold temperature should be on lateral wall parallel to Rb flow: Rb Cold walls (< 28°C) Plasma Cell Transition from hot to cold temperature should be on this surface (parallel to Rb flow). Guidelines for Expansion Volume The shape and lateral size of Expansion Volume is not crucial for density profile on axis.

17. Rb deposition for T wall < 28°C Orifice Plasma Cell Expansion Volume Θ ϕ Expansion Volume Wall r Normalization for cosine distribution For cylindrical Expansion Volume (r = 0.2m; L = 0.5m) the Rb layer per 2weeks:

18. Perfect Expansion Volume shape Rb Plasma Cell

19. Conclusions Plasma Cell: -The density profiles and flows inside the plasma cell was calculated analytically, using the long tube approximation; -The on axis density has overshoot (~2%) near the orifice in front of the source; -The temperature of Rb sources should be controlled with better than 0.1°C precision; Expansion Volume: -Even in infinitely large volume the minimum of density is limited by vacuum flow propagation; -For the 50cm long Expansion Volume, the temperature of walls should be less than 28°C; -The transition from hot (200°C) to low (28°C) temperature should be on lateral wall of Expansion Volume which is parallel to Rb flow; -The Expansion Volume should be long enough to capture most of the Rb; -The shape and lateral size of Expansion Volume is not crucial for density profile on axis; -For homogeneous Rb deposition the Expansion Volume should have the spherical shape.

20. Expansion Volume with cone shape

21. z axis (along cell axis, starts at the end of orifice) r axis is perpendicular to orifice ξ along the conical wall z r ξ Cone shape: coordinates

22. The temperature of conical surface as a function of ξ is calculated according the heat equation (the walls are perfectly isolated; the temperature is fixed at the ends). 1 Cone shape: temperature profile At ξ = 0 the wall temperature is equal to 200°C, at ξ = 0.15m the wall temperature is equal to 28°C r 0 = 0.06m α = 60°

23. The density on axis increases up to 15% due to the conical surface Additional density Cone shape: Density on axis