# The mean free path of electrons in high vacuum is 26 billion kilometres?

I used this formula to calculate the mean free path length of an electron in high vacuum.

$$\lambda = \frac{k T}{\sqrt{2}\cdot4\pi r^2 \cdot p}$$

where k is the Boltzmann konstant, T the temperature in Kelvin, r the radius of the particle in question, in m and p the pressure in pascal.

I used the following numbers:

$$k = 1.38*10^{-23}, T = 300, r \approx3*10^{-15}, p = 10^{-6}$$

Am I crazy for getting a result of

$$\lambda \approx 2.6*10^{13}m$$

I mean am I missing something? Maybe the radius of the electron changes when it moves? (Actually it moves with 50 keV but I have neglected that so far.)

I am sorry but this result seems just so....unreal. And I have to be sure because I have an exam about this stuff soon!

• You better use the size of the atoms, which is bigger, not the electron which is much smaller, so $10^{-10}$ instead of $3 ~ 10^{-15}$. – Kostas Oct 5 '19 at 20:32
• The formula you are using appears to be one that might apply to the molecules of a gas. When you speak of a high vacuum, are you referring to a low density of molecules (as in outer space) or a plasma of electrons? In either case you need a collision cross section based on the probability that the electron wave packet will interact with the other particles in your system. – R.W. Bird Oct 5 '19 at 23:36

The presence of $$r$$ in the formula you used implies that it is based a contact scattering assumption.