Coulomb Logarithm Calculator

The Coulomb logarithm, ln Λ, is a number that accounts for the cumulative effect of many small-angle Coulomb collisions in a plasma, rather than a single large-angle collision. It arises from integrating the Coulomb scattering cross section between a minimum impact parameter (close encounters) and the Debye length (beyond which collisions are screened), and it appears directly in essentially every plasma collision-rate and transport calculation.

This calculator uses the standard NRL Plasma Formulary form for the classical, weakly-coupled regime: ln Λ = 24 − ln(√n_e[cm⁻³] / T_e[eV]), valid for electron temperatures above about 10 eV, the regime that covers most fusion and space plasmas.

Because it depends only logarithmically on density and temperature, even the enormous range of real plasma conditions (density spanning 20+ orders of magnitude from the solar wind to a tokamak core) only shifts ln Λ by a factor of about 2. This remarkable insensitivity is why plasma physicists often just use 'ln Λ ≈ 10' as a rough estimate when a precise value is not critical.

It sets the overall collision rate and resistivity of a plasma: the Spitzer resistivity, electron-ion collision frequency, and most other transport coefficients are directly proportional to ln Λ. A plasma physicist cannot compute resistivity, thermal conductivity, or viscosity without first knowing this number.

A charged particle passing through a plasma is nudged slightly by every other charged particle within a Debye sphere, and while each individual deflection is tiny, there are so many of them (recall N_D can be in the millions or billions) that their cumulative effect on the particle's trajectory outweighs the rare close, large-angle encounters. The Coulomb logarithm is essentially counting how many 'decades' of impact parameter contribute to this cumulative scattering.

The classical NRL formula used here assumes weakly-coupled, non-quantum Coulomb collisions, valid once the electron temperature is comfortably above about 10 eV. Below that, quantum diffraction effects on the minimum impact parameter become important and a different formula (with a quantum-corrected minimum impact parameter) applies instead.

Both quantities are built from the Debye length, and the Coulomb logarithm is approximately ln(9 N_D) for a simple hydrogen-like plasma, so a plasma with a very large Debye sphere particle count also has a somewhat larger Coulomb logarithm. The related <a href="/science/plasma-physics/debye-sphere-particle-count-calculator/">Debye Sphere Particle Count Calculator</a> computes N_D directly.

Only weakly, through the reduced mass and charge state in more detailed formulas; the basic electron-electron/electron-ion form used here depends only on electron density and temperature. More precise NRL formulary variants exist for electron-ion and ion-ion collisions with different ion charge states, but they typically give very similar numerical values.

A larger Coulomb logarithm would mean more effective collisional coupling (higher resistivity, faster thermalization between species), while a smaller one would mean a more 'collisionless' plasma. In practice its narrow real-world range (roughly 10 to 20) means it rarely dominates order-of-magnitude estimates the way density or temperature does.

In extremely dense, cold, strongly-coupled plasmas (like the interior of white dwarfs or laser-compressed matter), ln Λ can become small or the classical weakly-coupled picture can break down entirely, requiring strongly-coupled plasma physics instead of the simple formula used here. For essentially all fusion, space, and laboratory plasmas in the classical regime, ln Λ stays safely positive and in the 10-20 range.