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Coulomb Collisions & Rutherford Scattering

Source lecture(s): PC368 Lec 4

Intuition

"Collision" in a plasma almost never means a hit. The Coulomb force has infinite range, so a particle is always being deflected slightly by everything inside its Debye sphere. The surprise is the outcome of adding up those nudges: the accumulation of many tiny deflections dominates the rare large ones by a factor of \(\ln\Lambda \approx 10\)–\(20\). A plasma randomises its velocities by a million gentle shoves, not by billiard-ball impacts.

Rutherford scattering

For a particle of charge \(q_1\), mass \(m\), speed \(v\) passing a stationary charge \(q_2\) at impact parameter \(b\), the deflection angle satisfies

\[\tan\frac{\theta}{2} = \frac{b_{90}}{b}, \qquad b_{90} = \frac{q_1q_2}{4\pi\epsilon_0 m v^2}\]

where \(b_{90}\) is the impact parameter giving a 90° turn. The differential cross-section is Rutherford's:

\[\frac{d\sigma}{d\Omega} = \left(\frac{q_1q_2}{16\pi\epsilon_0 E}\right)^2\frac{1}{\sin^4(\theta/2)}\]

The \(\sin^{-4}\) divergence at small angles is the mathematical statement of "long-range force": distant, glancing encounters are overwhelmingly the most common.

Why small angles win

Compare the two contributions to velocity randomisation.

Large-angle (single) scattering: cross-section \(\sigma_{90} \approx \pi b_{90}^2\).

Small-angle (cumulative) scattering: each encounter at impact parameter \(b\) gives a random transverse kick \(\Delta v_\perp \propto 1/b\). Kicks add randomly, so the mean square accumulates. Integrating \(\langle\Delta v_\perp^2\rangle\) over all impact parameters with \(2\pi b\,db\) weighting gives \(\int db/b\) — a logarithm:

\[\sigma_{\rm eff} \approx 8\pi b_{90}^2 \ln\Lambda, \qquad \Lambda = \frac{\lambda_D}{b_{90}} \approx 9N_D\]

The integral needs cutting off at both ends, and the physics supplies both cutoffs: at large \(b\), Debye shielding kills the interaction beyond \(\lambda_D\); at small \(b\), the expansion fails once the deflection is order-unity, at \(b_{90}\). The Coulomb logarithm \(\ln\Lambda\) is typically 10–20 — so cumulative small-angle scattering beats large-angle scattering by an order of magnitude.

That \(\ln\Lambda\) is only ever approximately known, and every transport coefficient in plasma physics inherits that factor-of-two vagueness. Plasma physicists have made peace with it.

Collision frequency and resistivity

\[\nu_{ei} \approx \frac{n e^4 \ln\Lambda}{4\pi\epsilon_0^2 m_e^2 v_{te}^3} \;\propto\; \frac{n}{T_e^{3/2}}\]

The \(T^{-3/2}\) is the single most consequential scaling in the subject. It gives the Spitzer resistivity

\[\eta \propto \frac{\ln\Lambda}{T_e^{3/2}}\]

with two spectacular consequences:

  • Hot plasmas are superb conductors. A 10 keV tokamak plasma conducts better than copper. This is what makes ideal MHD and the frozen-in theorem such good approximations, and why reconnection is so hard to make fast enough.
  • Ohmic heating switches itself off. Drive current through a cold plasma and it heats; as it heats, \(\eta\) falls, and the heating dies away. Ohmic heating alone plateaus around 1–2 keV, which is why tokamaks need neutral beams and RF.

Common mistakes

  • Treating plasma collisions like neutral-gas collisions. There is no mean free path between hard spheres; there is a diffusion in velocity space driven by everything at once.
  • Ignoring the \(\ln\Lambda\) cutoffs. Without both of them the integral diverges logarithmically at each end. The cutoffs are not a fudge — they are Debye shielding and the breakdown of the small-angle expansion.
  • Assuming faster particles collide more. The opposite: \(\nu \propto v^{-3}\). Fast particles are nearly collisionless, which is why runaway electrons run away.

Knowledge graph position

Prerequisites: Debye shielding, plasma. Leads to: Vlasov equation, transport theory, Spitzer resistivity, MHD.

Quiz

Q1 (conceptual). Why do small-angle collisions dominate, and what is \(\ln\Lambda\)?

Answer

Because the Coulomb force is long-ranged, encounters at large impact parameter are far more numerous (\(2\pi b\,db\) weighting) even though each is weak. Their mean-square velocity kicks accumulate as \(\int db/b\), giving \(\ln\Lambda = \ln(\lambda_D/b_{90})\). Numerically 10–20, so cumulative small-angle scattering beats large-angle by that factor.

Q2 (computational). Ohmic heating raises a tokamak plasma from 100 eV to 1 keV. By what factor does the resistivity change?

Answer

\(\eta \propto T^{-3/2}\), so a tenfold temperature rise cuts \(\eta\) by \(10^{3/2} \approx 32\). The ohmic heating power \(\eta j^2\) falls by the same factor — which is exactly why ohmic heating alone cannot reach fusion temperatures.

Q3 (MCQ). Runaway electrons occur because the collision frequency scales as:

  • (a) \(v^{3}\) — fast electrons collide more and are held back
  • (b) \(v^{-3}\) — fast electrons collide less, so an electric field accelerates them without limit
  • (c) independent of \(v\)
  • (d) \(v^{-1}\), and runaways are impossible
Answer

(b). Drag falls as \(v^{-3}\) while the electric force is constant, so above a critical speed acceleration always beats friction. Runaways in a disrupting tokamak can reach tens of MeV and punch holes in the wall.