2. ICPs show anomalous skin depth
Plasma density is peaked on axis even when antenna is on periphery.
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3. Helicon discharges are also peaked on axis
Typical RF deposition profiles are peaked at edge due to the TG mode
Typical center-peaked density profile
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4. Consider a discharge of moderate length
Electrons are magnetized; ions are not.
Neglect axial gradients.
Assume Ti << Te
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5. We now have a simple equilibrium problem
Ion fluid equation of motion
ionization
convection
CX collisions
neglect B
neglect Ti
Ion equation of continuity
where
Result
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6. Reduce to 1D and normalize
ion motion
Result
ion continuity
All radial dependences are kept so far.
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7. Electrons: short-circuit effect
Assume ionization is higher on the outside (tube 1)
Ions will diffuse inwards, to tube 2
Electrons can’t follow, but the sheath drop in 2 can increase to trap more electrons to neutralize the ions.
Electrons can effectively “move” across B by the sheath adjustment
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8. The electrons can then follow Boltzmann
The short-circuit effect allows electrons to thermalize across B in nsecs.
Electrons then fall into their most probable distribution: Maxwellian
They then follow the Boltzmann relation everywhere.
This is our basic assumption
It has amazing consequences!
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9. 3 equations for 3 unknowns: v, n, 
ion motion
ion continuity
electron Boltzmann
Eliminating n and , we have an ODE for v :
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10. Matching to Debye sheath is automatic
This “plasma solution” has dv/dr   as v  cs at the sheath edge.
All radial variations can be taken into account in this equation.
Introduce dimensionless variables
This equation yields the profiles of v, n, and Vs in equilibrium
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11. Properties of this “universal” equation
Plasma properties enter only in k (r ) in the nonlinear u 2 term.
The equilibrium profiles are the same if k is the same.
The profiles are unchanged if the quantities in  are changed.
The B-field does not appear. B can be strong or zero.
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12. Solutions for constant k
Let nn and Te be uniform, so k is a constant.
Self-similar profiles for 3 values of k
Re-scale so a = r / a
All profiles become identical!
And n (r ) is always peaked on axis.
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13. How to treat non-uniform nn, Te
Local ionization balance gives inverse relation between nn and Te.
Equation of continuity for neutrals yields neutral depletion profile.
A code EQM was developed to solve these equations for given Te(r).
Te(r) depends on the specific discharge. We did this for helicons.
The HELIC code gives Prf(r) for given n(r), nn(r), and Te(r).
Iteration of EQM with HELIC gives absolute-value profiles.
One example: n(r) and Prf(r)
Te(r) and neutral depletion
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14. UCLA

15. Title here
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