What are helicons? Helicons are partially ionized RF discharges in a magnetic field. They are basically whistler modes confined to a cylinder. They are much different than in free space; they have E-fields. OLD NEW Long cylinderPermanent magnet

Helicons pose unending problems UCLA Why does the amplitude oscillate along the cylinder? Why is a right-helical antenna better than a left one? What causes the high ionization efficiency? Why does an endplate near the antenna increase n? Why is the ion temperature so high? Why is a half-wavelength antenna better than a full? Why is the density peaked at the center? Most discharge theorists treat only collision cross sections and ion distribution functions.

The Trivelpiece-Gould mode: edge ionization UCLA An electron cyclotron wave near the edge deposits most of the RF energy

Edge ionization should give a hollow profile UCLA But density is almost always peaked at center, even in KTe is peaked at the edge.

Previous attempt for an ICP UCLA

Let’s take the simplest realistic problem UCLA Eliminate all unnecessary features, and not  length! Treat a 1D problem in radius r

The problem is how to treat the ends UCLA The sheath drop is normally independent of density

Ion diffusion upsets the balance UCLA The short-circuit effect “moves” electrons across B. Sheaths change to preserve neutrality. Electrons can now follow the Boltzmann relation. This happens in nanoseconds.

Sheath drops interchange, creating E r UCLA

In equilibrium, n is peaked on center UCLA E r and diffusion must be outward if axial flow is slow. n(r) is flat in the limit of all ionization at edge.

Three equations in 3 unknowns: v, n, and  UCLA Ion equation of motion: Ion equation of continuity: Use the Boltzmann relation: Simplify the collision terms:

Reduce to one dimension in r UCLA Eliminate n and  to get an equation for v(r): Non-dimensionalize: This is an ordinary differential equation for all the plasma profiles.

Rescale r to see structure of the equation UCLA We had: Rescale r: Finally: k contains the plasma information:

Solutions for uniform pressure and KT e UCLA Solutions for three values of k Rescale  so that  a  1 in each case This profile is independent of pressure, size, and magnetic field. It depends on KT e, but is always peaked at the center.

This profile IS modified: UCLA When T e is changed or varies with r When n n varies with r (neutral depletion, treated later) When k varies with r But the central peaking remains

Ionization balance restricts KT e for real r UCLA Our previous dimensional equation Solved simultaneously

Improved T e – p 0 relation UCLA Old, radially averaged data: M.A. Lieberman and A.J. Lichtenberg, Principles of Plasma Discharges and Materials Processing, 2nd ed. (Wiley-Interscience, Hoboken, NJ, 2005). F. F. Chen and J.P. Chang, Principles of Plasma Processing (Kluwer/Plenum, New York, 2002),

The EQM program solves simultaneously: UCLA Ion motion Neutral depletion Ionization balance

Last step: iteration with HELIC UCLA

Another layer off the onion! UCLA

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