Spitzer Resistivity

The collisional electrical resistivity of a fully ionized plasma, falling steeply as temperature rises.

The Formula

Spitzer resistivity is the electrical resistivity of a fully ionized plasma set by electron-ion Coulomb collisions. It scales as eta proportional to Z_eff ln(Lambda) / T_e^{3/2}, where Z_eff is the effective ion charge, ln(Lambda) the Coulomb logarithm, and T_e the electron temperature. The striking feature is the inverse three-halves temperature dependence: hotter plasmas conduct dramatically better.

Physical Origin

Resistivity arises because electrons carrying current are scattered by ions, transferring momentum and impeding the flow. Faster electrons in a hotter plasma have smaller Coulomb cross-sections (scattering falls with the fourth power of speed), so collisions become rare and resistivity plummets. The Coulomb logarithm accounts for the cumulative effect of many small-angle collisions, which dominate over rare large-angle ones.

Corrections

The basic Spitzer value is modified by trapped particles in a torus, which cannot carry current freely and raise the effective (neoclassical) resistivity, and by the effective charge Z_eff, which rises with impurity content. The parallel neoclassical resistivity used in transport codes multiplies Spitzer by a trapped-fraction correction factor, since trapped particles bounce rather than stream and cannot contribute to the parallel current. The Coulomb logarithm itself varies only weakly, so the temperature and charge dependences dominate the practical variation of resistivity across a discharge.

Relevance

Spitzer resistivity sets ohmic heating power (proportional to eta J^2), the resistive diffusion time for current-profile evolution, and the Lundquist number that governs reconnection. Its steep temperature dependence means ohmic heating becomes ineffective at reactor temperatures, requiring auxiliary heating, and that hot plasmas have enormous Lundquist numbers. For the Hyperion breeder concept these consequences frame the heating and current-drive strategy in design-stage modeling of a simulated device.