5Vdc from 48Vdc - Options?

I doubt it. You probably want enough to be useful. And I suspect that will be more than 1st order terms. First order terms certainly are those included by the derivative of Paschen's equation.

Speaking as someone largely ignorant, the basic idea has to do with the electric field that eventually reaches the point where the neutral gas molecules in within the field become ionized and form a plasma.

I don't know that much, but I remember something. For tiny values of pressure times distance, when the mean free path becomes on the order of the distance between the electrodes, then the majority of collisions are occurring more between the electrode surface and the gas and not within the length of the gas, itself. This inhibits the cascading effect that takes place when gas molecules ionize and makes it more difficult for the electric discharge to take place. So that means a higher electric field potential is required there. (This is more looked at by capacitor manufacturers, I suspect, who care about such things.)

For larger values of pressure times distance, there mean free path is very much smaller than the distance to the electrodes and the cascading effect then dominates.

There is a difference between electric field and the voltage. (I think so, anyway.) The voltage must provide sufficient acceleration to the secondary electrons that they can produce enough ionizing collisions before expending their energy. The field potential contributes to those collisions and adds energy once this process starts, but it's important I think to keep these isolated in mind as two distinct things. The voltage is contributing to the field, of course, but is also factors into the secondary electron energy leaving the electrode and the likelihood of creating sufficient ionizations once it leaves, while the field potential contributes as a force that continues to accelerate once that process begins. That's why, I think, there is a monotonic curve of field potential versus pressure times distance but instead a quasi-parabolic curve with voltage with a nice minimum that you can find with the derivative of Paschen's law.

At first, when the electron first leaves the electrode, there isn't enough inelastic collisions for you to see anything. So right near the electrode there is a kind of 'dark gap' of sorts. Not enough light coming from there. At low pressures, this gap widens. At first, I think, the electrons are moving straight towards the anode, but as the collisions take place they are scattered. If the anode gets really, really close then the electrons are too likely to reach the anode before causing an ionizing collision, which is why there is that hump (quasi-parabola) thing with the Paschen curve. I think. Anyway, if they make it across without ionizing, you don't get ionization. And you have to crank up the field potential to start seeing the ionization taking place, again. Something like that.

There is a kind of 'negative glow' region where ionization is taking place and where the electrons become more scattered. The length of this region is roughly speaking the mean free path, if I recall.

Anyway, I'm getting rapidly beyond my ken on this. But maybe it helps.

Well, as I mention above, capacitor folks are interested in an entirely different region of behavior -- in other words, where cascading ionization is NOT dominant. So please keep in mind that the dominant (first order, as you say) effect in one regime is quite different than the dominant effect in the other. It's only by keeping the all of it in mind that you can move smoothly from thinking about one domain to the other without completely messing up.

Jon

Why would you doubt my happiness? I just want to understand it. I don't plan to design any engines or spark plugs.

But that is the question. Is the ionization caused by the electric field that each molecule sees or does ionization happen when a free electron slams into an atom and creates more free ions? These are two separate mechanisms. I am thinking that the former is not as important as the second, but I don't know.

This could explain the low end where the resistance goes back up as the PxD drops so that the mean free path is approaching the electrode spacing. I would expect this increase to level off once the mean free path exceeds the electrode spacing.

The difference is the distance. Electric field is the voltage divided by the spacing.

That would be a way to distinguish between ionization from collisions vs. breakdown directly from the electric field.

I recall a field effect, which is usually associated with solids in practical situations, not gases, and is part of the process with electron beam guns. A thermal agitation effect, that is also used in electron beam guns along with the field effect, but also in vacuum tubes where thermal agitation is about all that is happening. And the ionization case you mentioned where an electron slams into an atom. Maybe more I'm not thinking about now. I don't think the field effect strips electrons off of the gas atoms, but instead off of the solid electrodes together perhaps with any thermal agitation present.

Jon

An interesting writeup and application

Looks consistent and provides a nice addition, I think. Thanks.

Jon