Spitzer Resistivity
The collisional electrical resistivity of a fully ionized plasma, falling steeply as temperature rises.
Resistivity of an ionized gas
Spitzer and Harm derived the resistivity of a fully ionized plasma from electron-ion Coulomb collisions. The parallel resistivity scales as:
text
eta_parallel ~ Z ln(Lambda) / T_e^(3/2)where Z is the ion charge, ln(Lambda) the Coulomb logarithm, and T_e the electron temperature. The striking feature is the T_e^(-3/2) dependence: a hotter plasma is a much better conductor because fast electrons are deflected far less by Coulomb collisions.
Parallel versus perpendicular
Resistivity is anisotropic in a magnetized plasma. The perpendicular resistivity is about twice the parallel value in the Spitzer result, because the magnetic field does not change the collision physics along the field but modifies the effective transport across it.
Consequences
- Ohmic heating (eta j^2) becomes ineffective at high temperature, so auxiliary heating is required to reach fusion conditions
- The current-diffusion timescale (resistive time) grows with temperature, so hot plasmas hold their current profile for a long time
- Runaway electrons appear when the electric field exceeds the collisional drag, a direct consequence of the velocity dependence behind Spitzer resistivity
How it enters simulation
Resistive MHD and transport codes use the Spitzer value, corrected by a neoclassical factor that accounts for trapped particles (which raise the effective resistivity at low collisionality). The current-diffusion equation, a resistive-diffusion equation for the poloidal flux, uses this resistivity to evolve the q-profile.
In the Hyperion breeder, the low resistivity at fusion-relevant temperatures means the plasma is nearly a perfect conductor on fast timescales, justifying ideal MHD for stability while resistive effects govern slow current-profile evolution and tearing-mode onset.