Collision frequency

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The collision frequency is the number of 90 degree scatterings that a particle goes through in one unit of time. This frequency changes drastically in different regimes. In the classical regime, the basic electron collision frequency is:

\nu_e=\frac{4\sqrt{2\pi} n \lambda e^4}{3\sqrt{m_e}\left(kT_e\right)^{3/2}}

where λ is the coulomb logarithm . For ions:

\nu_i=\frac{4\sqrt{\pi} n \lambda e^4}{3\sqrt{m_i}\left(kT_i\right)^{3/2}}

When deriving the distance of closest approach , we find that the angle executed by one particle scattering around another is given by:

\tan\frac{\theta}{2}=\frac{b_0}{b}

This means that the perpendicular veolicty change will be given by:

\left(\Delta v\right)_\perp^2 = v^2 \sin^2 \theta = \frac{4 v^2 \left(b/b_0\right)^2}{\left(1+\left(b/b_0\right)^2\right)^2}

So we can integrate over all of the impact parameters. Consider a particle moving distance dx forward. From radius b to b+db there are 2\pi n b\,db\,dx particles. We can integrate the effects of all particles from 0 to λD, where the coulomb force is no longer important. This gives us:

\left\langle \left(\Delta v \right)_\perp^2\right\rangle= \int_0^{\lambda_D}{\frac{8\pi n b v^2 \left(b/b_0\right)^2}{\left(1+\left (b/b_0\right)^2\right)^2}db\, dx}

Performing the integral in the limit \lambda_D\gg b_0:

\left\langle \left(\Delta v \right)_\perp^2\right\rangle= 8\pi n v^2 b_0^2 \ln \frac{\lambda_D}{b_0}\, dx

Remembering that the distance of closest approach b0 = e2 / kT, we can note that:

\frac{\lambda_D}{b_0}=\sqrt{\frac{kT}{4\pi n e^2}}\frac{kT}{e^2}=4\pi n\lambda_D^3 \approx \Lambda

To write:

\left\langle \left(\Delta v \right)_\perp^2\right\rangle= 8\pi n v^2 b_0^2 \ln \Lambda \, dx

So to get a 90 degree collision, we go through distance v / νe and change the velocity by v:

v^2 =\left(v/\nu_e\right) 8\pi n v^2 b_0^2 \ln \Lambda

Giving the frequency:

\nu_e=8 \pi n v b_0^2 \ln \Lambda=\frac{8 \pi n v e^4 \ln \Lambda}{k^2T^2}

This is off by a small numerical factor from the actual result. We need a full derivation involving the collision operator to recover that value.

This page was recovered in October 2009 from the Plasmagicians page on Collision_frequency dated 22:28, 3 March 2007.

Retrieved from "https://qed.princeton.edu/main/PlasmaWiki/Collision_frequency"
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