Spitzer Resistivity

Spitzer resistivity (whose inverse is Spitzer conductivity) is the classical collisional electrical resistivity η of a fully ionized plasma arising from cumulative small-angle Rutherford Coulomb scattering between current-carrying electrons and positively charged ions. Derived through exact numerical integration of the Fokker-Planck kinetic collision equation by American astrophysicists Lyman Spitzer Jr. and Richard Härm in 1953 (following earlier Lorentz-gas approximations by Spitzer, Cohen, and Routly in 1950), Spitzer resistivity demonstrates that the electrical resistivity of an ionized gas is virtually independent of electron number density n_e and decreases rapidly with electron temperature as η ∝ Z ln Λ T_e^−3/2. Within astronomy, solar physics, and controlled fusion plasma physics, Spitzer resistivity governs the magnetic diffusivity η_m = c²η / (4π) (in CGS units) or η / μ₀ (in SI units), setting the global Ohmic decay timescale of magnetic fields in stellar interiors, accretion disks, and galaxy clusters while establishing the resistive dissipation baseline for Sweet-Parker magnetic reconnection and Petschek magnetic reconnection.

Lorentz Gas Approximation and Momentum-Transfer Frequency

When an electric field E is applied to a fully ionized plasma of electron number density n_e and ion number density n_i (with charge state Z such that quasi-neutrality holds, n_e = Z n_i), electrons of mass m_e and charge −e accelerate opposite to E much more rapidly than ions of mass m_i ≫ m_e. Steady-state electrical resistance arises because drifting electrons transfer directed momentum to the nearly stationary ions via electrostatic Coulomb collisions.

In a classical 90°-deflection binary encounter at thermal speed v_th,e = (k_BT_e / m_e)^1/2, the distance of closest approach (the Landau length b₀) occurs where the electrostatic potential energy equals the electron thermal kinetic energy:

b₀ = (Z e²) / (4π ε₀ m_e v_th,e²) = (Z e²) / (4π ε₀ k_B T_e) (SI units)

However, because the bare Coulomb potential Φ(r) ∝ 1/r is a long-range interaction screened only beyond the Debye length:

λ_D = (ε₀ k_B T_e / (n_e e²))^1/2

an electron traversing a weakly coupled plasma (where the plasma parameter N_D = (4π/3)n_eλ_D³ ≫ 1) experiences vastly more distant, small-angle deflections (b₀ < b < λ_D) than single large-angle scatterings. Integrating the Rutherford differential cross section over impact parameters b from b_min = b₀ (or the quantum thermal de Broglie wavelength at high temperatures) to b_max = λ_D multiplies the large-angle cross section πb₀² by 8 ln Λ, where ln Λ = ln(λ_D / b₀) is the Coulomb logarithm (typically ln Λ ≈ 10 to 25 across astrophysical and laboratory plasmas). The resulting electron–ion momentum-transfer collision frequency ν_ei is:

ν_ei = (4 √(2π) n_e Z e⁴ ln Λ) / [ 3 (4π ε₀)² m_e^1/2 (k_B T_e)^3/2 ]

Origin of the Density Independence and T_e^−3/2 Scaling

In the simple Lorentz gas model—which treats ions as fixed scattering centers and neglects electron–electron collisions— balancing the electric force −*e*E against collisional drag m_e ν_ei u_e for the current density J = −n_e*e*u_e = E / η_L yields the Lorentz resistivity:

η_L = (m_e ν_ei) / (n_e e²) = [ π Z e² m_e^1/2 ln Λ ] / [ (4π ε₀)² (2 k_B T_e)^3/2 ]

Two remarkable physical properties emerge directly from this formula:

1. Cancellation of Electron Density (n_e): Although doubling n_e doubles the number of charge carriers carrying the current J, it simultaneously doubles the ion density n_i = n_e/Z and thus doubles the collision frequency ν_ei. The two factors of n_e cancel in the ratio ν_ei / n_e, leaving only a logarithmic dependence on density inside ln Λ.
2. Inverse Three-Halves Temperature Scaling (T_e^−3/2): Because faster thermal electrons (v_th,e ∝ T_e^1/2) spend less time in the electrostatic field of each passing ion, the Rutherford cross section drops as b₀² ∝ v_th,e⁻⁴ ∝ T_e⁻², so the collision rate ν_ei ≈ n_i σ v_th,e scales as T_e^−3/2. Consequently, a hot plasma is an exceptionally good electrical conductor: a 10⁷ K solar flare loop or tokamak core has an electrical conductivity comparable to or exceeding metallic copper at room temperature.

In a real plasma, electrons collide not only with ions but also with other electrons at a comparable rate ν_ee ~ ν_ei / Z. Although electron–electron collisions conserve total electron momentum and therefore cannot generate resistivity by themselves, they redistribute velocity-space current: because ν_ei(v) ∝ v⁻³, high-velocity tail electrons in the Maxwellian distribution carry a disproportionate share of the current in a Lorentz gas. Electron–electron collisions scatter these fast runaway-like tail electrons back into the slower thermal bulk, where ions scatter them more effectively, thereby increasing the parallel resistivity above the Lorentz value η_L by a charge-dependent factor 1 / γ_E(Z).

In their landmark 1953 paper, Spitzer and Härm solved the coupled Fokker-Planck equation for the perturbed electron velocity distribution function f₁(v) across ionic charges Z = 1, 2, 4, 16, and ∞ (Lorentz limit), establishing the exact parallel Spitzer resistivity η_∥ (for currents flowing parallel to a magnetic field B, or in an unmagnetized plasma):

η_∥ = η_L / γ_E(Z) ≈ 5.26 × 10⁻⁵ (Z ln Λ) / [ γ_E(Z) (T_e / 1 eV)^3/2 ] Ω·m

where for a pure hydrogen plasma (Z = 1), γ_E(1) = 0.5816 (so 1 / γ_E(1) ≈ 1.72, giving η_∥ = 0.5064 m_e ν_ei / (n_ee²) in Braginskii's definition of ν_ei).

Parallel vs. Perpendicular Spitzer Resistivity (η_⊥ / η_∥ ≈ 1.96)

When a strong magnetic field B is present such that the electron gyrofrequency ω_ce = eB / m_e considerably exceeds the collision frequency ν_ei (ω_ce τ_ei ≫ 1), electrons executing tight Larmor gyrations perpendicular to B cannot freely stream along the high-velocity tail of the distribution in the direction of J_⊥. As shown by Spitzer (1956) and Stanislav I. Braginskii (1965), this magnetic suppression of the high-velocity current tail eliminates the γ_E enhancement along the perpendicular direction, causing the perpendicular Spitzer resistivity η_⊥ (for currents J_⊥ ⊥ B) in a Z = 1 hydrogen plasma to be 1.96 times larger than the parallel resistivity:

η_⊥ = 1.96 η_∥ (for Z = 1 and ω_ce τ_ei ≫ 1)

In resistive magnetohydrodynamics (MHD), combining Faraday's law of induction with Ohm's law E + v × B = η J and Ampère's law yields the magnetic induction equation:

∂B / ∂t = ∇ × (v × B) + η_m ∇²B

where η_m = η / μ₀ (SI) = c²η / (4π) (CGS) is the magnetic diffusivity (with dimensions of length² / time):

η_m ≈ 10⁹ ln Λ T_e^−3/2 cm² s⁻¹ = 10⁵ ln Λ T_e^−3/2 m² s⁻¹ (with T_e in Kelvin, Z = 1)

Because astronomical length scales L are immense while η_m is small in hot plasmas, the global Ohmic diffusion timescale τ_Ohm = L² / η_m and the dimensionless Lundquist number S = L v_A / η_m (where v_A is the Alfvén speed) are huge:

• Stellar Magnetic Fossil Fields: For the radiative interior of the Sun or an Ap/Bp star (L ≈ 5 × 10⁸ m, T_e ≈ 5 × 10⁶ K, η_m ≈ 0.2 m² s⁻¹), the Ohmic decay timescale is τ_Ohm ≈ 4 × 10¹⁰ years—longer than the current age of the universe (Cowling's 1945 theorem).
• Solar Coronal Loops: For a coronal arcade of scale L = 10⁷ m and T_e = 2 × 10⁶ K (η_m ≈ 5 m² s⁻¹, v_A ≈ 10⁶ m s⁻¹), the diffusive timescale is τ_Ohm ≈ 6 × 10⁵ years and the Lundquist number is S ≈ 2 × 10¹². This tiny classical Spitzer diffusivity is the fundamental reason why Sweet-Parker magnetic reconnection (M_A = S^−1/2 ~ 10⁻⁶) cannot explain impulsive 100-second solar flares without invoking localized Petschek magnetic reconnection, plasmoid instabilities, or collisionless Hall/kinetic dissipation.

Breakdown: Dreicer Runaway Fields and Anomalous Turbulence

Classical Spitzer resistivity breaks down in three distinct physical regimes:

1. Partially Ionized Cold Gas: In cold protoplanetary disks and molecular clouds (T < 1,000 K, ionization fraction x_e < 10⁻⁴), electron–neutral collisions dominate over Coulomb scattering, and non-ideal MHD is governed instead by neutral-electron Ohmic resistivity, Hall drift, and ambipolar diffusion.
2. Dreicer Electric Field and Runaway Electrons: Because the Coulomb drag force on a suprathermal electron decreases with speed as v⁻², if the applied electric field exceeds the Dreicer field E_D = (n_e e³ ln Λ) / (4π ε₀² k_B T_e), thermal electrons accelerate continuously without reaching a terminal drift velocity, generating relativistic runaway electron beams observed in solar flares and tokamak disruptions.
3. Anomalous (Wave-Particle) Resistivity: When the electron drift velocity u_e = J / (n_ee) exceeds the ion-acoustic speed c_s = (k_BT_e/m_i)^1/2 or the electron thermal speed v_th,e inside a thin current sheet, micro-instabilities (such as the ion-acoustic, Buneman, or lower-hybrid drift instabilities) excite electrostatic plasma turbulence. Electrons scatter off collective wave electric field fluctuations at an effective frequency ν_eff ≫ ν_ei, elevating the effective anomalous resistivity by 4 to 8 orders of magnitude above the classical Spitzer value.

Why does Spitzer resistivity decrease as a plasma gets hotter, whereas the resistivity of a metal wire increases with temperature?

In a solid metallic conductor like copper, conduction electrons move through a periodic crystal lattice of ions; raising the temperature increases the amplitude of thermal lattice vibrations (phonons), increasing the scattering cross section and raising resistivity. In a fully ionized plasma, there is no crystal lattice—electrons scatter via electrostatic Coulomb attraction to bare ions. As electron temperature T_e increases, thermal electrons move much faster (v ∝ T_e^1/2) and spend far less time near each ion during a flyby, causing the Rutherford deflection cross section to plummet as T_e⁻² and the net resistivity to drop as T_e^−3/2.

Why is Spitzer resistivity almost independent of plasma density?

Doubling the electron number density n_e in a quasi-neutral plasma (n_e = Z n_i) provides twice as many free electrons to carry the electric current, which by itself would halve the resistivity. However, it simultaneously doubles the number density of positively charged ions n_i that scatter those electrons, doubling the collision frequency ν_ei. The two linear factors of density cancel in η ∝ ν_ei / n_e, leaving only a weak logarithmic dependence inside the Coulomb logarithm ln Λ.

Why does ohmic heating become ineffective in ultra-hot fusion plasmas above a few keV?

Because Spitzer resistivity scales as η ∝ T_e^−3/2, the volumetric Ohmic heating power P_Ohm = η J² produced by a fixed plasma current density J drops rapidly as the plasma heats up. Above T_e ≈ 3–5 keV (~3.5–6 × 10⁷ K) in a tokamak, Spitzer resistivity becomes so small that Ohmic heating alone can no longer overcome radiative and transport losses to reach the 10–20 keV ignition threshold of deuterium-tritium fusion, requiring auxiliary heating via neutral beam injection (NBI) or ion/electron cyclotron resonance heating.

1. Spitzer, L., & Härm, R. (1953). "Transport phenomena in a completely ionized gas." Physical Review, 89(5), 977–981. https://doi.org/10.1103/PhysRev.89.977
2. Cohen, R. S., Spitzer, L., & Routly, P. M. (1950). "The electrical conductivity of an ionized gas." Physical Review, 80(2), 230–238. https://doi.org/10.1103/PhysRev.80.230
3. Spitzer, L. (1962). Physics of Fully Ionized Gases (2nd ed.). Interscience Publishers / Wiley. https://books.google.com/books?id=T34ZAQAAIAAJ
4. Braginskii, S. I. (1965). "Transport processes in a plasma." Reviews of Plasma Physics, 1, 205–311. https://ui.adsabs.harvard.edu/abs/1965RvPP....1..205B