Categories: Physics, Plasma physics.

Coulomb logarithm

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 π/2\pi/2 as an example of a large deflection angle. Then Rutherford predicts:

q1q24πε0v2μblarge=tan ⁣(π4)=1\begin{aligned} \frac{q_1 q_2}{4 \pi \varepsilon_0 |\vb{v}|^2 \mu b_\mathrm{large}} = \tan\!\Big( \frac{\pi}{4} \Big) = 1 \end{aligned}

Isolating this for the impact parameter blargeb_\mathrm{large} then yields an effective radius of a particle:

blarge=q1q24πε0v2μ\begin{aligned} b_\mathrm{large} = \frac{q_1 q_2}{4 \pi \varepsilon_0 |\vb{v}|^2 \mu} \end{aligned}

Therefore, the collision cross-section σlarge\sigma_\mathrm{large} for large deflections can be roughly estimated as the area of a disc with radius blargeb_\mathrm{large}:

σlarge=πblarge2=q12q2216πε02v4μ2\begin{aligned} \sigma_\mathrm{large} = \pi b_\mathrm{large}^2 = \frac{q_1^2 q_2^2}{16 \pi \varepsilon_0^2 |\vb{v}|^4 \mu^2} \end{aligned}

Next, we want to find the cross-section for small deflections. For sufficiently small angles θ\theta, we can Taylor-expand the Rutherford scattering formula to first order:

q1q24πε0v2μb=tan ⁣(θ2)θ2    θq1q22πε0v2μb\begin{aligned} \frac{q_1 q_2}{4 \pi \varepsilon_0 |\vb{v}|^2 \mu b} = \tan\!\Big( \frac{\theta}{2} \Big) \approx \frac{\theta}{2} \quad \implies \quad \theta \approx \frac{q_1 q_2}{2 \pi \varepsilon_0 |\vb{v}|^2 \mu b} \end{aligned}

Clearly, θ\theta is inversely proportional to bb. 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 bb) 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 NN small deflections θn\theta_n are needed to get a large total of, say, 11 radian?

n=1Nθn21\begin{aligned} \sum_{n = 1}^N \theta_n^2 \approx 1 \end{aligned}

Traditionally, 11 is chosen instead of π/2\pi/2 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 τ\tau between “collisions” is related like so to the cross-section σ\sigma, the total density nn of charged particles, and the relative velocity v|\vb{v}|:

1τ=nvσ    1=nvτσ\begin{aligned} \frac{1}{\tau} = n |\vb{v}| \sigma \qquad \implies \qquad 1 = n |\vb{v}| \tau \sigma \end{aligned}

Therefore, in a given time interval tt, the expected number of collision NbN_b for impact parameters between bb and b ⁣+ ⁣dbb\!+\!\dd{b} (imagine a ring with these inner and outer radii) is given by:

Nb=nvtσb=nvt(2πbdb)\begin{aligned} N_b = n |\vb{v}| t \: \sigma_b = n |\vb{v}| t \:(2 \pi b \dd{b}) \end{aligned}

In this time interval tt, we can thus turn our earlier sum into an integral of NbN_b over bb:

1n=1Nθn2=Nbθ2db=nvt2πθ2bdb\begin{aligned} 1 \approx \sum_{n = 1}^N \theta_n^2 = \int N_b \:\theta^2 \dd{b} = n |\vb{v}| t \int 2 \pi \theta^2 b \dd{b} \end{aligned}

Using the formula nvτσ=1n |\vb{v}| \tau \sigma = 1, we thus define σsmall\sigma_{small} as the effective cross-section needed to get a large deflection (of 11 radian), with an average period tt:

σsmall=2πθ2bdb=2πq12q224π2ε02v4μ2b2bdb\begin{aligned} \sigma_\mathrm{small} = \int 2 \pi \theta^2 b \dd{b} = \int \frac{2 \pi q_1^2 q_2^2}{4 \pi^2 \varepsilon_0^2 |\vb{v}|^4 \mu^2 b^2} b \dd{b} \end{aligned}

Where we have replaced θ\theta with our earlier Taylor expansion. Here, we recognize σlarge\sigma_\mathrm{large}:

σsmall=q12q222πε02v4μ21bdb=8σlarge1bdb\begin{aligned} \sigma_\mathrm{small} = \frac{q_1^2 q_2^2}{2 \pi \varepsilon_0^2 |\vb{v}|^4 \mu^2} \int \frac{1}{b} \dd{b} = 8 \sigma_\mathrm{large} \int \frac{1}{b} \dd{b} \end{aligned}

But what are the integration limits? We know that the deflection grows for smaller bb, so it would be reasonable to choose blargeb_\mathrm{large} as the lower limit. For very large bb, the plasma shields the particles from each other, thereby nullifying the deflection, so as upper limit we choose the Debye length λD\lambda_D, i.e. the plasma’s self-shielding length. We thus find:

σsmall=8σlargeln(Λ)=q12q22ln(Λ)2πε02v4μ2\begin{aligned} \boxed{ \sigma_\mathrm{small} = 8 \sigma_\mathrm{large} \ln(\Lambda) = \frac{q_1^2 q_2^2 \ln(\Lambda)}{2 \pi \varepsilon_0^2 |\vb{v}|^4 \mu^2} } \end{aligned}

Here, ln(Λ)\ln(\Lambda) is known as the Coulomb logarithm, with the plasma parameter Λ\Lambda defined below, equal to 9/29/2 times the number of particles in a sphere with radius λD\lambda_D:

ΛλDblarge=6πnλD3\begin{aligned} \boxed{ \Lambda \equiv \frac{\lambda_D}{b_\mathrm{large}} = 6 \pi n \lambda_D^3 } \end{aligned}

The above relation between σsmall\sigma_\mathrm{small} and σlarge\sigma_\mathrm{large} gives us an estimate of how much more often small deflections occur, compared to large ones. In a typical plasma, ln(Λ)\ln(\Lambda) is between 6 and 25, such that σsmall\sigma_\mathrm{small} is 2-3 orders of magnitude larger than σlarge\sigma_\mathrm{large}.

Note that tt is now fixed as the period for small deflections to add up to 11 radian. In more useful words, it is the time scale for significant energy transfer between partices:

1t=nvσsmall=q12q22ln(Λ)n2πε02μ2v3nT3/2\begin{aligned} \frac{1}{t} = n |\vb{v}| \sigma_\mathrm{small} = \frac{q_1^2 q_2^2 \ln(\Lambda) \: n}{2 \pi \varepsilon_0^2 \mu^2 |\vb{v}|^3} \sim \frac{n}{T^{3/2}} \end{aligned}

Where we have used that vT|\vb{v}| \propto \sqrt{T}, for some temperature TT. Consequently, in hotter plasmas, there is less energy transfer, meaning that a hot plasma is hard to heat up further.


  1. P.M. Bellan, Fundamentals of plasma physics, 1st edition, Cambridge.
  2. M. Salewski, A.H. Nielsen, Plasma physics: lecture notes, 2021, unpublished.