In a plasma, a Coulomb collision rarely results in a large deflection. The cumulative effect of the many small angle collisions, however, is often larger than the effect of the few large angle collisions that occur, so it is instructive to consider the collision dynamics in the limit of small deflections. We can consider an electron of charge -e and mass m_\text{e} passing a stationary ion of charge +Ze and much larger mass at a distance b with a speed v. The perpendicular force is Ze^2/(4\pi\varepsilon_0 b^2) at the closest approach and the duration of the encounter is about b/v. The product of these expressions divided by the mass is the change in perpendicular velocity: \Delta m_\text{e} v_\perp \approx \frac{Ze^2}{4\pi\varepsilon_0} \, \frac{1}{vb} Note that the deflection angle is proportional to 1/v^2. Fast particles are "slippery" and thus dominate many transport processes. The efficiency of velocity-matched interactions is also the reason that fusion products tend to heat the electrons rather than (as would be desirable) the ions. If an electric field is present, the faster electrons feel less drag and become even faster in a "run-away" process. In passing through a field of ions with density n, an electron will have many such encounters simultaneously, with various
impact parameters (distance to the ion) and directions. The cumulative effect can be described as a diffusion of the perpendicular momentum. The corresponding diffusion constant is found by integrating the squares of the individual changes in momentum. The rate of collisions with impact parameter between b and (b + db) is so the diffusion constant is given by \begin{align} D_{v\perp} &= \int \left(\frac{Ze^2}{4\pi\varepsilon_0} \frac{1}{vb}\right)^2 \, nv \left(2\pi b\, db\right) \\[1ex] &= \left(\frac{Ze^2}{4\pi\varepsilon_0}\right)^2 \, \frac{2\pi n}{v} \, \int \frac{db}{b} \end{align} Obviously the integral diverges toward both small and large impact parameters. The divergence at small impact parameters is clearly unphysical since under the assumptions used here, the final perpendicular momentum cannot take on a value higher than the initial momentum. Setting the above estimate for \Delta m_\text{e} v_\perp equal to mv, we find the lower cut-off to the impact parameter to be about b_0 = \frac{Ze^2}{4\pi\varepsilon_0} \, \frac{1}{m_\text{e} v^2} We can also use \pi b_0^2 as an estimate of the cross section for large-angle collisions. Under some conditions there is a more stringent lower limit due to
quantum mechanics, namely the
de Broglie wavelength of the electron, h/m_\text{e} v where h is the
Planck constant. At large impact parameters, the charge of the ion is
shielded by the tendency of electrons to cluster in the neighborhood of the ion and other ions to avoid it. The upper cut-off to the impact parameter should thus be approximately equal to the
Debye length: \lambda_\text{D} = \sqrt{\frac{\varepsilon_0 kT_\text{e}}{n_\text{e} e^2}} == Coulomb logarithm ==