Spitzer Resistivity and Conductivity Calculator

Spitzer resistivity is the electrical resistivity of a fully ionized plasma, derived by Lyman Spitzer from the physics of electron-ion Coulomb collisions. Unlike a metal, where resistivity comes from electrons scattering off a fixed lattice, plasma resistivity comes from electrons scattering off moving, thermally-agitated ions.

η = 1.03×10⁻⁴ × Z × lnΛ / Te[eV]^(3/2) ohm-metres, a standard closed-form result from the NRL Plasma Formulary, where Z is the ion charge state, lnΛ is the Coulomb logarithm, and Te is the electron temperature in electronvolts.

Faster electrons (higher temperature) spend less time near each ion as they fly past, so each Coulomb collision deflects them less, resistivity scales as Te^(-3/2). This is the opposite of a metal, where resistivity typically increases with temperature due to more lattice vibrations.

Copper's resistivity at room temperature is about 1.7×10⁻⁸ ohm-metres. A 10 keV tokamak plasma has a Spitzer resistivity around 1.8×10⁻⁹ ohm-metres, roughly ten times lower (a better conductor) than copper, and this gap widens further at higher temperatures.

Higher-charge ions (like fully-stripped carbon or oxygen instead of hydrogen) scatter electrons more strongly per collision because the Coulomb force scales with the ion's charge, so resistivity increases linearly with Z. This is one reason fusion reactors try to minimize high-Z impurity contamination, it directly raises plasma resistivity and radiated power.

Conductivity σ is simply the reciprocal of resistivity, σ = 1/η, and both describe the same underlying physics, resistivity emphasizes how much a plasma resists current flow, while conductivity emphasizes how easily current flows through it.

The Coulomb logarithm is a required ingredient of the Spitzer formula, and since it only depends on the same density and temperature already entered here, this calculator computes it automatically using the same standard NRL formula as the dedicated <a href="/science/plasma-physics/coulomb-logarithm-calculator/">Coulomb Logarithm Calculator</a>, so no separate lookup is needed.

Ohmic heating drives a current through the resistive plasma to heat it directly, but since resistivity falls as Te^(-3/2), the heating power (which scales as current squared times resistivity) drops off rapidly as the plasma gets hotter. Beyond a few keV, ohmic heating alone becomes too weak, so tokamaks switch to auxiliary heating methods like neutral beam injection or radiofrequency waves.

No, this calculator computes the parallel (along the magnetic field) resistivity, which is the lowest of the two. Perpendicular resistivity (across field lines) is generally higher because gyrating electrons cannot move freely across the field the way they can along it, though the parallel value is the one most often quoted as 'the' Spitzer resistivity.

The classical Spitzer formula assumes a fully ionized, weakly-coupled, collisional plasma and works well for describing the bulk resistive behavior of most laboratory and fusion plasmas, though real tokamaks show additional 'anomalous' resistivity from turbulence beyond this simple collisional picture, particularly during current disruptions or in the plasma edge.