Summary of Rb loss simulations The goal: assess different techniques for optimization density gradient at the ends of plasma cell.

Summary of Rb loss simulations

The goal: assess different techniques for optimization density gradient at

the ends of plasma cell

Parameters

• Temperature: T = 200⁰C = 473K• Density: n = 7x1020m-3

• Dimensions: inner diameter: 4cm; length: 10m• Knudsen number ~0.1 (rarefaction parameter ~10)• Mean free path ~3.6mm• Pressure: = 6.5Pa = 0.065mbar = 0.049torr• Thermal velocity: = 301m/s• Rubidium atom mass: 1.443x10-25kg• Rubidium atom diameter: 496-606pm

Fast valve, No orifice (iris)

Fast valve: 10msWhat is depletion length?

Shakhov EM, Non-stationary rarefied gas flow into vacuum from a circular pipe closed at one end

15cm at 0.67ms

50cm at 3-4msKersevan R, https://indico.cern.ch/event/328455/contribution/11/material/slides/1.pdf

𝑓 𝑛 (𝑥 , 𝑡 )=𝑛0

2 [1+erf (− 𝑥𝑡 √ 𝑚2𝑘𝑇 )]

𝑛𝑛0

=( 2𝛾 )

𝛾− 1( 2𝛾+1 √𝛾2 − 𝛾−1

𝛾+1 √ 𝑚2𝑘𝑇

𝑥𝑡 )

2 (𝛾−1 )50cm at 1-2ms

1D Theory (FM+C) + 1D DSMC Petrenko A, https://indico.cern.ch/event/357090/contribution/2/material/slides/2.pdf

2D DSMC Petrenko A, https://indico.cern.ch/event/357090/contribution/2/material/slides/2.pdf

50cm at 2-3ms

0

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3D DSMC Plyushchev G

50cm at 3ms

Electron trapping

Lotov KV, http://arxiv.org/pdf/1408.4448v1.pdf

=> Length of the transition region should not exceed 10-15cm

Very simple estimation of outflow through orifice

• Particles escapes from orifice (continuum regime) with speed of sound: = 275m/sec.

• Number of particles escaped per second: = 1.51x1019/sec => 2.18mg/sec

• In reality, it is half-density should be used => 1.09mg/sec

• Total mass of Rb in 10m@4cm = 1.27mg

Less simple estimation of outflow through orifice

• Mass outflow for infinitely large volume to vacuum (Sharipov F., Rarefied gas flow through a thin orifice):

• Rarefaction: 1.23 => W=1.15 => M=0.77mg/sec• For continuum: W=1.51 => M=1.01mg/sec• Total mass of Rb in 10m@4cm = 1.27mg

Accounts for rarefaction,for molecular regime = 1

Orifice radius

Pressure

Mass of Rb

Summary of leak rate values

• Molecular flow theory: 0.67mg/sec• Rarefied flow: 0.77mg/sec• Continuum flow: 1.01mg/sec• Simple continuum estimation: 1.09mg/sec

• Simulation: 0.52mg/sec

Possible explanation of error: we don’t have infinitely large volume

Analytical tails• When orifice small compared to inner diameter of plasma

cell, the physics, near orifice, is similar to gas flow through a thin orifice

• Gas density on axis of orifice (Danilatos G., Direct simulation Monte Carlo study of orifice flow):

• => ramp length is order of magnitude of orifice diameter.• => gas density on axis symmetric with respect to orifice

plane• This equation could be used for plasma wakefield simulation

to see the influence of this profile on electron trapping

3D DSMC simulation Double Orifice no Source: boundary conditions

The idea : To have large volume between both orifice to drive outflow to gain some time

Symmetry wall

Thermal wall

50cm

4cm 1cm 20cm

4cm

Fast valve

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3D DSMC simulation Double Orifice no Source: density profile (1e7 particles)

Density profile integrated over last 4ms (25.9-29.9ms) in order to increase statistics:

Red line: theory for infinitely large volumeBlue lines: orifice 1 and 2

3D DSMC simulation Double Orifice no Source: density profile (1e7 particles)

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0.00ms

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0.66ms

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1.33ms

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5.31ms

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9.96ms

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15.3ms

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19.9ms

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25.2ms

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29.9ms

Approximately 15cm at 30ms

3D DSMC simulation Double Orifice no Source: density profile (1e7 particles)

Time, sec

Den

sity

in p

lasm

a ce

ll, a

.u.

3D DSMC simulation Single Orifice with Source: boundary conditions

Symmetry wall

Thermal wall

Source (constant flux)

50cm

4cm 1cm

20cm

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2cm

DSMC: hard sphere model

3D DSMC simulation Single Orifice with Source: results

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Final data

Convergence reached at around 0.5sec (with plasma cell tube initially filled)

Simulation inflow (=outflow): 0.52mg/sec. This equivalent to 45g/day.If orifice will be open only 3 seconds each 30 seconds: 4.5g/day.

3D DSMC simulation Single Orifice with Source: density profile

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Density profile in the center of plasma cell (inside r=4mm)

3D DSMC simulation Double Orifice with Source: boundary conditions

Symmetry wall

Thermal wall

Source (constant flux)

50cm

4cm 1cm

10cm

2cm

2cm2cm

The idea of second orifice: 1. Prevent any possible vortex creation2. Both orifices are symmetrically placed with respect to

source tube => the symmetry simplifies the understanding of problem (in case with low collisions between particles, it could be considered as superposition of source and two orifices with plasma cell with orifice at the end

3D DSMC simulation Double Orifice with Source: results

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Final data

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Results are very similar to the simulation with single orifice: leak rate: 0.52mg/sec

convergence: 0.5sec

3D DSMC simulation Double Orifice with Source: Flow

3D DSMC simulation Double Orifice with Source (constant density): results

7x1020m-3 10x1020m-3

8x1020m-3

9x1020m-3

Conclusions: 1. In this case (stationary case) the density near the source is higher that the density in plasma cell.2. Thus the injection tube (between Rb source and plasma cell) should be very short and should have large diameter in order to have smaller density gradient difference.3. The injection tube should be as close as possible to orifice

Challenge 1

• For Rb vapor, pressure depends on temperature

• Pressure near the source should be higher, then in plasma cell.

• => source should be as close as possible to plasma cell (and to iris)

• Can our source provide this constant flux (~0.77mg/sec)?

Steck, D.A., Rubidium 85 D Line Data

Temperature, K

Den

sity

, 1020

m-3

Surf

ace,

m2

Flux = 0.77mg/sec

Evaporation rate

Pound G.M., Selected Values of Evaporation and Condensation Coefficients for Simple Substances

𝐽=𝛼𝑛𝑘𝑇 −𝑝𝑒𝑞

√2𝜋𝑚𝑘𝑇 𝑝𝑒𝑞=101325×10(4.312− 4040

𝑇 )

Possible practical application

Source (constant flux)

10m

Valve 1Valve 2

Questions: 1. If valve 1 and 2 close, what is the time to fill this volume with Rb source from one end? (Initially plasma cell is empty!) (for particular geometry it is > 5sec)2. If valve 2 is closed and valve 1 is open, what is the time to reach the equilibrium? (Initially plasma cell is filled with Rb!) (for particular geometry it is > 8sec, see next slide)

Total mass of Rb in 10m@4cm = 1.27mg. Flow through orifice = 0.52 - 0.77mg/sec

Possible closed loop Rb vapor system:

A valve which isnormally closedand opened tolet beams pass.It’s not necessaryfor this valve tobe leak tightand fast.

At 70 °C equilibrium vapor pressure of Rb is 2000 times lower than at 200 °C.

Rb in this system is in the closed loop because it’s either in a liquid or in a vapor form.

The amount of liquid Rb in the reservoir can be limited to ~10 cm3 (15 g). The main question is how much liquid Rb will stick to 70 °C walls before it starts to flow down to the reservoir? Let’s assume the 70 °C surface is ~ πR2 = 3.14*(10 cm)2 = 300 cm2 and Rb layer is 1 mm thick => V = 300 cm2 * 0.1 cm = 30 cm3 => The total mass of Rb is likely to be below 100 g.

Rb flow is 0.5 mg/sec = 100 g / 2 days =>Another option may be a cycled operation – 70 °C tank will be heated up once a day or so.

R ~ 10 cm

Rb70 °C

200 °C oil tank

190 °C

70 °C

Challenges: 2• If we going to have orifice system (or source

system) from both ends of plasma cell => the both sources should be perfectly aligned (to avoid density ramp)