# Unit conversion problem for the electron thermal conductivity

1. May 29, 2014

### goodphy

Hello.

In CGS unit electron thermal conductivity for plasma is expressed as $\frac{n_{e}T_{e}}{m_{e}\upsilon_{e}}\Gamma_{1}$ [1] where $\Gamma_{1}$ is the dimensionless transport coefficient. [2]
You can also find similar expression in http://farside.ph.utexas.edu/teaching/plasma/lectures/node35.html

$n_{e}$: electron number density in $cm^{-3}$.
$T_{e}$: electron temperature in erg.
$m_{e}$: electron mass in g (gram).
$\upsilon_{e}$: electron-ion collision frequency.

Experimentally, SI unit is useful and I've tried to convert unit of the formula to Si unit of $Wm^{-1}K^{-1}$ but failed.

I directly replaced erg by $gcm^{2}/s^{2}$(= erg) and arranged dimensions in the formula. The results is $cm^{-1}s^{-1}$.

This appears far from what is supposed to be in SI unit.

Could you help me to figure out what I was wrong in conversion?

Reference
• A. Esaulov, P. Sasorov, L. Soto, M. Zambra, and J. Sakai, "MHD simulation of a fast hollow cathode capillary discharge",
Plasma Phys. Control. Fusion 43, 571 (2001).
• E.M. Epperlein and M.G. Haines, "Plasma transport coefficients in a magnetic field by direct numerical solution of the Fokker-Pianck equation", Phys. Fluids 29, 1029 (1986).

2. May 30, 2014

### nasu

I think that in SI it should be
$\frac{n_{e} k_B^2 T_{e}}{m_{e}\upsilon_{e}}\Gamma_{1}$

Assuming that indeed gamma is dimensionless.

3. May 30, 2014

### goodphy

I just wonder how you get this expression? Could you provide me the reference with which you come up with that?

And is there any way to directly convert CGS unit to SI unit in this case? I at least think that the expression of the physics must be useful nomatter what unit system is used..

4. May 30, 2014

### UltrafastPED

The formula for thermal conductivity in a plasma is derived (in SI units) here:
http://www.pma.caltech.edu/Courses/ph136/yr2004/0419.1.K.pdf

See section 19.5.

Also see note 2, at the beginning, for how to convert from SI to gaussian units ... the opposite of your situation, but may be enlightening.

5. Jun 3, 2014

### goodphy

Oh, thanks everybody! I finally got the answer! I'm so happy!

The thermal conductivity in my question is actually the one used when the temperature in the heat flux equation (Fourier's law) $q=k\nabla T$ (q is local heat flux ($W/m^{2}$ unit)) is actually in energy, $k_{B}T$($T$ is temperature in Kelvin at this moment.).

When the equation is with true temperature, not energy, the conductivity in the equation becomes $k_{B}k$.

Dimension check confirmed.