Coulomb Logarithm

Coulomb Logarithm

The Coulomb logarithm (\ln \Lambda) appears throughout plasma physics whenever we calculate collision rates, transport coefficients, or energy exchange rates. It accounts for the cumulative effect of many small-angle Coulomb collisions.

Why a Logarithm?

In a Coulomb interaction, the scattering cross-section diverges at large impact parameters (weak, distant encounters). In a plasma, Debye shielding cuts off this divergence at the Debye length (\lambda_D). The ratio of the maximum to minimum impact parameter gives the logarithm.

The 90° Deflection Parameter

The minimum impact parameter is the 90° deflection parameter (b_{90}) — the distance at which the electron's kinetic energy equals the potential energy:

$b_{90} = \frac{e^2}{4\pi\varepsilon_0 m_e v^2}$

Using the thermal velocity (v = \sqrt{k_B T / m_e}), we get (b_{90} \propto 1/T).

Coulomb Logarithm Formula

$\ln \Lambda = \ln\left(\frac{\lambda_D}{b_{90}}\right) \approx \ln\left(12\pi n \lambda_D^3\right)$

where the Debye length is:

$\lambda_D = \sqrt{\frac{\varepsilon_0 k_B T}{n e^2}}$

Typical values range from (\ln \Lambda \approx 10) to (20) in laboratory and space plasmas.

Electron-Ion Collision Frequency

Using a simplified SI expression, the electron-ion collision frequency is:

$\nu_{ei} = \frac{2.91 \times 10^{-12} \cdot n \cdot \ln \Lambda}{T^{3/2}}$

where (n) is in m(^{-3}) and (T) is in Kelvin. The (T^{-3/2}) dependence means hotter plasmas are more collisionless — a fundamental and counterintuitive property of plasmas.

Physical Significance

• (\ln \Lambda \sim 10): Typical fusion plasma
• (\ln \Lambda \sim 20): Solar corona
• (\ln \Lambda < 2): Strongly coupled plasma (solid-like behavior)

The Coulomb logarithm enters into resistivity, thermal conductivity, viscosity, and energy equilibration times — making it one of the most important parameters in plasma transport theory.