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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.

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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

Hence, one would expect the plasma to be negative at the center relative to the edge.


Density profiles are almost never hollow

If ionization is near the boundary, the density should peak at the edge. This is never observed.


Consider a discharge of moderate length

  • Electrons are magnetized; ions are not.
  • Neglect axial gradients.
  • Assume Ti << Te



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.


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.


The Simon short-circuit effect

Step 2: 10s of msec time scale

Sheath drops change, E-field develops

Ions are driven inward fast by E-field


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 Ez is small

due to good conductivity along B.


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.


We now have a simple equilibrium problem

Ion fluid equation of motion



CX collisions

neglect B

neglect Ti

Ion equation of continuity





The r-components of three equations

Ion equation of motion:

Ion equation of continuity:


Boltzmann relation:

(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

to obtain…


We obtain a simple 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.


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”.


A universal profile for constant k

k does not vary with p

k varies with Te

These samples are for uniform p and Te

These are independent of magnetic field!


Ionization balance and neutral depletion

Ionization balance

Neutral depletion

Ion motion

Three differential equations

The EQM code (Curreli) solves these three equations simultaneously, with all quantities varying with radius.


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


Energy balance: the Vahedi curve

This curve for radiative losses vs. Te gives us absolute values.

Energy balance gives us the data to calculate Te(r)


Helicon profiles before iteration

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


Sample of EQM-HELIC iteration

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.



Comparison with experiment

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



Probe at Port 1, 6.8 cm below tube

  • The density peaks on axis
  • Te shows Trivelpiece-Gould deposition at edge.
  • Vs(Maxw) is the space potential calc. from n(r) if Boltzmann.



Dip at high-B shows failure of model

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.



Example of absolute agreement of n(0)

The RF power deposition is not uniform axially, and the equivalent length L of uniform deposition is uncertain within the error curves.