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

3. 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.

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

5. Consider a discharge of moderate length UCLA 1.Electrons are magnetized; ions are not. 2.Neglect axial gradients. 3.Assume T i << T e

6. Sheaths when there is no diffusion Sheath potential drop is same as floating potential on a probe. This is independent of density, so sheath drops are the same.

7. The Simon short-circuit effect Step 1: nanosecond time scale 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.

8. Sheath drops change, E-field develops Ions are driven inward fast by E-field The Simon short-circuit effect Step 2: 10s of msec time scale

9. The Simon short-circuit effect Step 3: Steady-state equilibrium Density must peak in center in order for potential to be high there to drive ions out radially. Ions cannot move fast axially because E z is small due to good conductivity along B.

10. 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.

11. We now have a simple equilibrium problem UCLA Ion fluid equation of motion ionizationconvection CX collisionsneglect Bneglect T i Ion equation of continuity where Result

12. The r-components of three equations Ion equation of motion: Ion equation of continuity: 3 equations for 3 unknowns: v r (r)  (r) n(r) Electron Boltzmann relation: (which comes from)

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

14. We obtain a simple equation Note that the coefficient of (1 + ku 2 ) has the dimensions of 1/r, so we can define This yields Except for the nonlinear term ku 2, this is a universal equation giving the n(r), T e (r), and  (r) profiles for any discharge and satisfies the Bohm condition at the sheath edge automatically.

15. Reminder: Bohm sheath criterion

16. Solutions for different values of k = P c / P i We renormalize the curves, setting  a in each case to r/a, where a is the discharge radius. No presheath assumption is needed. We find that the density profile is the same for all plasmas with the same k. Since k does not depend on pressure or discharge radius, the profile is “universal”.

17. A universal profile for constant k These are independent of magnetic field! k does not vary with pk varies with Te These samples are for uniform p and Te

18. Ionization balance and neutral depletion Ionization balance Neutral depletion Ion motion The EQM code (Curreli) solves these three equations simultaneously, with all quantities varying with radius. Three differential equations

19. 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 P in (r) for given n(r), T e (r) and n n (r) for various discharge lengths, antenna types, and gases. However, B(z) and n(z) must be uniform. The power lost is given by

20. Energy balance: the Vahedi curve This curve for radiative losses vs. T e gives us absolute values. Energy balance gives us the data to calculate T e (r)

21. Helicon profiles before iteration Trivelpiece-Gould deposition at edge Density profiles computed by EQM These curves were for uniform plasmas We have to use these curves to get better deposition profiles.

22. Sample of EQM-HELIC iteration UCLA It takes only 5-6 iterations before convergence. Note that the T e ’s are now more reasonable. Te’s larger than 5 eV reported by others are spurious; their RF compensation of the Langmuir probe was inadequate.

23. Comparison with experiment UCLA This is a permanent- magnet helicon source with the plasma tube in the external reverse field of a ring magnet. It is not possible to measure radial profiles inside the discharge. We can then dispense with the probe ex- tension and measure downstream. 2 inches

24. Probe at Port 1, 6.8 cm below tube UCLA 1.The density peaks on axis 2.T e shows Trivelpiece-Gould deposition at edge. 3.Vs(Maxw) is the space potential calc. from n(r) if Boltzmann.

25. Dip at high-B shows failure of model UCLA With two magnets, the B-field varies from 350 to 200G inside the source. The T-G mode is very strong at the edge, and plasma is lost axially on axis. The tube is not long enough for axial losses to be neglected.

26. Example of absolute agreement of n(0) UCLA The RF power deposition is not uniform axially, and the equivalent length L of uniform deposition is uncertain within the error curves.