How could I calculate the strength of an electromagnetic field sufficient enough to generate plasma in low pressure argon gas?

Question

For context, I wish to create a plasma toroid in a glass sphere of 25 Torr Argon gas like in the photo below; there are multiple examples of how to do this online using a class E oscillator oscillating around 10-15 MHz to sustain ionization of a noble gas.

I am confident I understand how the circuit works, however, I do not understand why the oscillation frequency was chosen to be 10-15 MHz. My best guess is that this is the frequency range that would create a magnetic field strong enough to ionize the gas. I wish to verify this through calculations.

How would I go about calculating the magnetic field strength necessary to ionize argon gas, assuming that some free electrons are already in the gas?

Answer 1

Actually calculating things specifically for argon is difficult, since it's orbitals are full and naively it looks like we can't use any single-electron approximations for which analytic calculations are possible. But hopefully we can guess the qualitative behavior correctly by thinking about single-electron physics.

At first glance, we can compute when the Zeeman shift of an electron would exceed argon's ionization energy of 15.7 eV

$E_\text{ionize} = \Delta E_\text{Zeeman} \propto B_z \mu_B \rightarrow B_{z, \text{ionize}} \propto 15.7~\text{eV} / \mu_B = 2.7 ~T$

Where the proportionality constant depends on the $m_s$ and $m_l$ quantum numbers of the orbital under consideration. See Wikipedia > Zeeman Effect. This is a huge magnetic field, well above what could realistically be sustained in a home-built device like pictured. It is also not necessarily a correct calculation. This is the magnetic field required to Zeeman-shift a static atom's orbital above the binding energy, but there is no guarantee that the atom will actually ionize under these conditions. The reason is that the Zeeman splitting can be both positive or negative*, and therefore some unoccupied energy levels will get pushed below that of the electron under consideration. So instead of ionizing, the electron can relax into one of these lower orbitals.

The full story is even more complicated, and to be completely honest I don't know what happens as you put Argon in increasingly stronger magnetic fields and if it will ever spontaneously ionize. But I do think we can be sure that these effects alone are not responsible for generating the plasma here. Quasistatic fields definitely can spontaneously ionize atoms, but again the electric fields involved are very large. Most likely the device uses electric fields to accelerate atoms until they collide with one another, and it is the accumulated kinetic energy released in these collisions that triggers ionization.

I realize that this answer only answers the atomic physics portion of your question, whereas you are probably interested in understanding how to calculate what actually does trigger the ionization in practice. One simple calculation I can imagine is to consider an argon atom freely accelerating under an applied electric field. If the mean free path at 25 Torr is long enough that atoms will gain more than 15.7 eV of kinetic energy before hitting another atom, then this could also lead to the formation of a plasma. But again I am not sure that this is the full story. Apologies for the partial answer. Will happily upvote a more thorough one.

*The Lorenz force can push the electron toward or away from the nucleus, depending on the direction of the orbit and the orientation of the magnetic field.