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). Problem 2: Diffusion across B. Classical diffusion predicts slow electron diffusion across B.
Problem 1: Density does not peak near the antenna (B = 0)
Classical diffusion predicts slow electron diffusion across B
Hence, one would expect the plasma to be negative at the center relative to the edge.
If ionization is near the boundary, the density should peak at the edge. This is never observed.
Sheath potential drop is same as floating potential on a probe.
This is independent of density, so sheath drops are the same.
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.
Step 2: 10s of msec time scale
Sheath drops change, E-field develops
Ions are driven inward fast by E-field
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 Ez is small
due to good conductivity along 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.
Ion fluid equation of motion
Ion equation of continuity
Ion equation of motion:
Ion equation of continuity:
(which comes from)
3 equations for 3 unknowns: vr(r)(r)n(r)
Eliminate h(r) and n(r) to get an equation for the ion vr
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
Note that the coefficient of (1 + ku2) has the dimensions of 1/r, so we can define
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.
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. 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”.
k does not vary with p
k varies with Te
These samples are for uniform p and Te
These are independent of magnetic field!
Three differential equations
The EQM code (Curreli) solves these three equations simultaneously, with all quantities varying with radius.
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
This curve for radiative losses vs. Te gives us absolute values.
Energy balance gives us the data to calculate Te(r)
Trivelpiece-Gould deposition at edge
Density profiles computed by EQM
We have to use these curves to get better deposition profiles.
These curves were for uniform plasmas
It takes only 5-6 iterations before convergence.
Note that the Te’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.
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.
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.
The RF power deposition is not uniform axially, and the equivalent length L of uniform deposition is uncertain within the error curves.