Does anyone know about the cross section of electrical conductivity in plasma state of these elements that are being used for HID?

Mercury = ?
Sodium = ?

What about noble gases as well?

https://en.wikipedia.org/wiki/Electrical_resistivity

Varying somewhere between 1e-12 till 1e+12...

Well, that first line was halfway meant as a joke, but really only half way.
You just can never state any fixed number, as the conductivity of a plasma is a directly proportional to the ionization (more atoms ionized means more free charge carriers to carry the electricity) density and the free path length (longer the free distance, faster, so easier they travel, so higher conductivity).
The concentration depends on the ionization vs recombination balance, the free path length depends on the equivalent size and density of the neutral particles.
Both are more or less directly a function of the gas pressure, current fed into the discharge and the external irradiation and temperature (so the surrounding plasma).

Consequently for a given pressure, the arc voltage drop (so electrical field) uses to be rather independent on the current, mainly because higher voltage means there are more ionization than recombination and vice versa; the actual voltage then depends on the geometry and pressure.

Of course, if you approach a state, where all (or majority) of the gas material get ionized, the conductivity likely becomes about constant (I'm just guessing), but that requires either high vacuum (then we do not talk about a high pressure discharge; there the plasma conductivity measurement is usually influenced by the free electron current) or conditions found only in nuclear explosions or inside living stars, no way attainable within an artificial sustained electrical discharge (so not a high pressure discharge lamp either)...

Thanks, Medved.  I guess I needed to do more research with this.

You may calculate back the actual plasma conductivity of a given lamp operated at given condition (so at given moment in the mains period, as the conductivity changes more than an order of magnitude), e.g. the instant of the maximum current with a 80W MV lamp on series reactor ballast (assume EU installation):
The arc voltage is constant, so 110V. From that you have to subtract the cathode fall, which is about 15V. So the plasma drops then about 85V.
The current is nearly a sinewave, so assume a sine wave. The 0.8Arms means 0.8*1.414A, so 1.13A during that peak.
So the plasma forms a resistor of 85V/1.13A=75Ohm, conductance is then 1/75=13.3mS.
The length is about 30mm, arc diameter about 2mm (both dimensions I'm guessing now for the sake of this example, exact dimensions differ according to the given particular lamp design), so we have a 30mm long, 3.14mm^2 cross section conductor having 75Ohm resistance.
So the (average, as it is not constant within the arc) conductivity is COnductance*Length/CrossSection = 0.0133 * 0.03 * 3.14e-6 S/m = 127 S/m.
But the arc is not a homogenous conductor, it's conductivity varies across it's cross section (maximumj in the middle, geting reduced towards the arc perimeter). The maximum conductivity within  the arc center I would guess to be about double, in this example it would be 254 S/m.
But for a better accuracy you would need really a good view about the distribution of the conductivity within the arc body, otherwise you may just guess the order of magnitude, mainly the "distribution" related guess is very inaccurate figure.

That is the answer I was looking at

It really depends upon the amount of Mercury content, amount of load (Volts times Ampere) and the length of conductivity.

But that answer doesn't show any fixed value, I used only one example with one particular hypothetical lamp and at just one moment in the mains cycle.
Of course, if something conduct electricity, you may calculate something, what would correspond to resistance or so, but the question is, what use is for such value, when it is valid only for the exact arrangement and condition, so when you alter any parameter of the setup, the conductivity always changes, e.g.:
If we shift the time 4.5ms (assume 50Hz), the conductivity of the plasma, in exactly the same lamp, spot in the arc and on the same gear, will be about 6x lower. If we shift 0.4ms further (so 4.9ms from the maximum), the conductivity will be 30x lower than in the maximum, just because it follows the arc current (the dynamic of the plasma generation is way faster than the mains sinewave, so the plasma does maintain the equilibrium quite well).
If you change the lamp design by e.g. adding a tiny amount of an arc fattening element (used with MH's to reach fatter, less swirling arc), you will end up with about the same voltage drop, just the arc diameter will increase. The conductivity will be reduced a lot, because the arc will have larger cross sectional area, but the voltage drop remains the same, no change on the lamp electrical behavior. So the figure you get becomes completely different, but no externally visible change at all.

The material properties like conductivity or resistivity were introduced to allow to predict the behavior of something new by some simple calculation without any need to carry any measurement of the new arrangement, but the usability of these is limited with one condition: The property defined that way must be really constant, or dependent on only very limited number of well defined external parameters like temperature (e.g. the resistivity of the pure copper being 16uOhm.m at 25degC with given temperature coefficient you use to estimate the resistance, so power losses of e.g. a new transformer winding). Otherwise it is useless, so it does not make any real sense to extract any exact value (What will be the use of "Conductivity of a mercury plasma in a 30mm long, 2mm diameter arc running at 0.8Arms 550Hz sinewave is at it's maximum 254S/m", othher than an information "That arctube has an arc voltage drop of85V on the anode column"; actually the later describes way broader operating range...).

For the discharge is way more practical to calculate with free path length and ionization energies and so on - it is way complex math, but these parameters are about constant for a given material, so when the math is done, you are able to really predict the behavior of e.g. a new lamp design. But never with the "Conductivity in that lamp is 254S/m", well except when you want to exactly replicate that lamp. But then what is the point of any calculation, if you have a working example of that design...


So yes, you may calculate any parameter you define out of any arrangement or use any definition to calculate any material, but don't expect that to be written in some tables or so, when the usability is virtually zero...

I guess we have work to cut out for the plasma research