In a plasma, particles often appear to collide, although actually it is caused by Coulomb forces, i.e. the “collision” is in fact Rutherford scattering. In any case, the particles’ paths are deflected, and it would be nice to know whether those deflections are usually large or small.
Let us choose as an example of a large deflection angle. Then Rutherford predicts:
Isolating this for the impact parameter then yields an effective radius of a particle:
Therefore, the collision cross-section for large deflections can be roughly estimated as the area of a disc with radius :
Next, we want to find the cross-section for small deflections. For sufficiently small angles , we can Taylor-expand the Rutherford scattering formula to first order:
Clearly, is inversely proportional to . Intuitively, we know that a given particle in a uniform plasma always has more “distant” neighbours than “close” neighbours, so we expect that small deflections (large ) are more common than large deflections.
That said, many small deflections can add up to a large total. They can also add up to zero, so we should use random walk statistics. We now ask: how many small deflections are needed to get a large total of, say, radian?
Traditionally, is chosen instead of for convenience. We are only making rough estimates, so those two angles are close enough for our purposes. Furthermore, the end result will turn out to be logarithmic, and is thus barely affected by this inconsistency.
You can easily convince yourself that the average time between “collisions” is related like so to the cross-section , the total density of charged particles, and the relative velocity :
Therefore, in a given time interval , the expected number of collision for impact parameters between and (imagine a ring with these inner and outer radii) is given by:
In this time interval , we can thus turn our earlier sum into an integral of over :
Using the formula , we thus define as the effective cross-section needed to get a large deflection (of radian), with an average period :
Where we have replaced with our earlier Taylor expansion. Here, we recognize :
But what are the integration limits? We know that the deflection grows for smaller , so it would be reasonable to choose as the lower limit. For very large , the plasma shields the particles from each other, thereby nullifying the deflection, so as upper limit we choose the Debye length , i.e. the plasma’s self-shielding length. We thus find:
Here, is known as the Coulomb logarithm, with the plasma parameter defined below, equal to times the number of particles in a sphere with radius :
The above relation between and gives us an estimate of how much more often small deflections occur, compared to large ones. In a typical plasma, is between 6 and 25, such that is 2-3 orders of magnitude larger than .
Note that is now fixed as the period for small deflections to add up to radian. In more useful words, it is the time scale for significant energy transfer between partices:
Where we have used that , for some temperature . Consequently, in hotter plasmas, there is less energy transfer, meaning that a hot plasma is hard to heat up further.
- P.M. Bellan, Fundamentals of plasma physics, 1st edition, Cambridge.
- M. Salewski, A.H. Nielsen, Plasma physics: lecture notes, 2021, unpublished.