The Coulomb Logarithm
The slowly varying factor that captures the dominance of many small-angle collisions in a plasma.
Why a logarithm appears
In a plasma, the cumulative effect of many distant, small-angle Coulomb encounters outweighs rare close ones. Summing their contributions produces an integral over impact parameter that diverges logarithmically at both ends and is cut off physically. The result is the Coulomb logarithm:
ln(Lambda) = ln( b_max / b_min ) = ln( lambda_D / b_90 )The upper cutoff b_max is the Debye length, beyond which the plasma screens the interaction; the lower cutoff b_min is the closest approach (or the de Broglie wavelength when quantum effects dominate). The ratio is large, so its logarithm is a slowly varying number, typically between 10 and 20 in fusion plasmas.
A convenient near-constant
Because ln(Lambda) changes so slowly with density and temperature, it is often treated as a constant around 15 to 17 for core fusion conditions. This is what lets collision rates, resistivity, and transport coefficients be written as clean power laws in n and T.
Where it enters
- The collision frequency and Spitzer resistivity both carry a factor of ln(Lambda)
- The Fokker-Planck friction and diffusion coefficients scale with it
- Fast-ion slowing-down times depend on it
Practical evaluation
Standard formulas (for example the NRL Plasma Formulary expressions) give ln(Lambda) for electron-electron, electron-ion, and ion-ion collisions in terms of density and temperature. Codes evaluate the appropriate form locally. Although a modest number, its logarithmic factor is carried consistently in collisional calculations for plasmas such as the Hyperion breeder core, to keep transport and current-drive estimates accurate.