A new theory of gas discharges based on experiments

1. A new theory of gas discharges based on experiments Francis F. Chen, University of California KAIST, Daejeon, S. Korea, April 2011

2. Two problems with gas discharges • Anomalous skin depth in ICPs • Electron diffusion across magnetic fields Problem 1: Density does not peak near the antenna (B = 0)

3. Problem 1: anomalous skin effect Data by John Evans In the plane of the antenna, the density peaks well outside the classical skin layer

4. FL First solution to skin depth problem Consider the nonlinear effect of the Lorentz force on the motion of an electron in an RF field Eq This force is in the radial direction, while E is in the q direction

5. An electron trajectory over four RF cycles with and without the Lorentz force FL

6. After 32,000 iterations

7. UCLA Density profile in four sectors of equal area Points are data from Slide 5

8. Problem 2: Diffusion across B Classical diffusion predicts slow electron diffusion across B Hence, one would expect the plasma to be negative at the center relative to the edge.

9. Density profiles are never hollow If ionization is near the boundary, the density should peak at the edge. This is never observed.

10. Helicons are always more positive at center

11. Resolution: discharges have finite length The Simon short-circuit effect Electrons are Maxwellian along each field line, but not across lines. A small adjustment of the sheath drop allows electrons to “cross the field”. This results in a Maxwellian even ACROSS field lines.

12. The E-field moves ions to the center This is why density is not peaked at the edge

13. Hence, the Boltzmann relation holds even across B As long as the electrons have a mechanism that allows them to reach their most probable distribution, they will be Maxwellian everywhere. This is our basic assumption. This radial electric field pushes ions from high density to low density, filling in hollow profiles so that, in equilibrium, the density is always peaked on axis.

14. An idealized model of a high-density discharge • B = 0 or B  0; it doesn’t matter. • The discharge is cylindrical, with endplates. • All quantities are uniform in z and θ(a 1-D problem). • The ions are unmagnetized, with large Larmor radii. • Ion temperature is negligible: Ti / Te = 0

15. The ion equations Motion: 1-D: Continuity: Pi is the ionization probability Pc is the collision probability We keep the two highlighted equations

16. We now have three equations Ion equation of motion: Ion equation of continuity: Electron Boltzmann relation: (which comes from) 3 equations for 3 unknowns: vr(r)(r)n(r)

17. Eliminate unknowns h(r) and n(r) This yields an ODE for the ion radial fluid velocity: Note that dv/dr  at v = cs (the Bohm condition, giving an automatic match to the sheath We next define dimensionless variables to obtain…

18. Reminder: Bohm sheath criterion

19. The universal equation Note that the coefficient of (1 + ku2) has the dimensions of 1/r, so we can define This yields Except for the nonlinear term ku2, this is a universal equation giving the n(r), Te(r), and(r) profiles for any discharge and satisfies the Bohm condition at the sheath edge automatically.

20. Solutions for different values of k = Pc / Pi We renormalize the curves, setting rain each case to r/a, where a is the discharge radius. Once v is known, all the other profiles can be calculated from the three equations. We then find that there’s a universal profile, the same for all discharges!

21. A universal profile for constant k This is independent of magnetic field!

22. A program EQM was devised by Curreli The universal profile is good for k = constant, where k is the ratio Pc/Pi. However, Pc depends on v(r) and Pi depends on Te. But we know v(r) from our solution, so we can calculate Pc(r) and use it. Te, however, depends on ionization balance and will be treated next. Ph.D. student Davide Curreli from the University of Padua, Italy, wrote a program to solve all the equations with radial profiles. These are density profiles for different values of constant Te, but with self-consistent Pc. It is no longer necessary to calculate a collisional presheath and match it to the collisionless sheath. It’s all included.

23. Ionization balance Consider the number of ions in a cylindrical shell dr of unit length input output dr Result: Previous equation (u = v/cs) Assume nn (or pressure po) to be uniform, and solve these equations simultaneously.

24. Ionization balance result For given tube radius a, there is only one Te that puts the sheath edge there. This gives the familiar result that Te varies inversely with po, But more accurately, since the profiles of v(r) and n(r) are taken into account. The p-Te relation depends on the tube radius a.

25. Neutral depletion What about nn(r)? This is more difficult, since the neutrals’ motion depends on the exact geometry of the discharge. Eqn. of continuity for neutrals We have to assume that the mfp’s are short, and the neutrals are collision-dominated, with a diffusion constant D. The come in and out uniformly at the surface. Inside the plasma they are ionized at a rate proportional to electron density n and ionization rate Pi(r).

26. Results for neutral depletion Neutral density profiles for various pressures in a 5-cm diam tube. The ne profiles (----) are peaked at 1012 cm-3. Same profiles as peak density is varied. Input pressure is 1 mTorr. The corresponding Te profiles (----) are shown, but these are not realistic because radiation losses have not yet been accounted for.

27. Energy balance: helicon discharges To implement energy balance requires specifying the type of discharge. The HELIC program for helicons and ICPs can calculate the power deposition Pin(r) for given n(r), Te(r) and nn(r) for various discharge lengths, antenna types, and gases. However, B(z) and n(z) must be uniform. The power lost is given by

28. Energy balance: the Vahedi curve The particle losses Wi and We are simple. The radiation loss Wr, however, has to be calculated from the Vahedi curve, known for argon. This gives the energy needed to create each ionization, including all the radiation that is lost before that happens. Energy balance gives us the data to calculate Te(r)

29. Iteration between EQM and HELIC The final step is the tedious process of iterating between EQM and HELIC. Starting with constant Te, EQM calculates n(r) and nn(r). These are put into HELIC to obtain Te(r). This is then put into EQM to get new n(r) and nn(r). The process is repeated until it converges. The right Te(r) is extremely important, since ionization varies exponentially with Te. Here are three cases, before iteration

30. Two cases with iteration

31. One case of absolute agreement The unfinished task is to predict the Big Blue Mode, which has complete ionization on axis. But that involves higher ionization states.

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