Categories: Physics, Plasma physics.

Spitzer resistivity

If an electric field with magnitude EE is applied to the plasma, the electrons experience a Lorentz force qeEq_e E (we neglect the ions due to their mass), where qeq_e is the electron charge.

However, collisions slow them down while they travel through the plasma., This can be modelled as a drag force feimevef_{ei} m_e v_e, where feif_{ei} is the electron-ion collision frequency (we neglect feef_{ee} since all electrons are moving together), mem_e is their mass, and vev_e their typical velocity relative to the ions in the background. Balancing the two forces yields the following relation:

qeE=feimeve\begin{aligned} q_e E = f_{ei} m_e v_e \end{aligned}

Using that the current density J=qeneveJ = q_e n_e v_e, we can rearrange this like so:

E=feimeJneqe2=mefeineqe2J=ηJ\begin{aligned} E = f_{ei} m_e \frac{J}{n_e q_e^2} = \frac{m_e f_{ei}}{n_e q_e^2} J = \eta J \end{aligned}

This is Ohm’s law, where η\eta is the resistivity. From our derivation of the Coulomb logarithm ln(Λ)\ln(\Lambda), we estimate feif_{ei} to be as follows, where nin_i is the ion density, σ\sigma is the collision cross-section, and μ\mu is the reduced mass of the electron-ion system:

fei=niσve=12π(qeqiε0μ)2nive3ln(Λ)12πZqe4ε02me2neve3ln(Λ)\begin{aligned} f_{ei} = n_i \sigma v_e = \frac{1}{2 \pi} \Big( \frac{q_e q_i}{\varepsilon_0 \mu} \Big)^2 \frac{n_i}{v_e^3} \ln(\Lambda) \approx \frac{1}{2 \pi} \frac{Z q_e^4}{\varepsilon_0^2 m_e^2} \frac{n_e}{v_e^3} \ln(\Lambda) \end{aligned}

Where we used that μme\mu \approx m_e, and qi=Zqeq_i = -Z q_e for some ionization ZZ, and as a result neZnin_e \approx Z n_i due to the plasma’s quasi-neutrality. Beware: authors disagree about the constant factors in feif_{ei}; recall that it was derived from fairly rough estimates. This article follows Bellan.

Inserting this expression for feif_{ei} into the so-called Spitzer resistivity η\eta then yields:

η=mefeineqe2=12πZqe2ε02me1ve3ln(Λ)\begin{aligned} \boxed{ \eta = \frac{m_e f_{ei}}{n_e q_e^2} = \frac{1}{2 \pi} \frac{Z q_e^2}{\varepsilon_0^2 m_e} \frac{1}{v_e^3} \ln(\Lambda) } \end{aligned}

A reasonable estimate for the typical velocity vev_e at thermal equilibrium is as follows, where kBk_B is Boltzmann’s constant, and TeT_e is the electron temperature:

12meve2=32kBTe    ve=3kBTeme\begin{aligned} \frac{1}{2} m_e v_e^2 = \frac{3}{2} k_B T_e \quad \implies \quad v_e = \sqrt{\frac{3 k_B T_e}{m_e}} \end{aligned}

Other choices exist, see e.g. the Maxwell-Boltzmann distribution, but always veTe/mev_e \propto \sqrt{T_e/m_e}. Inserting this vev_e into η\eta then gives:

η=16π3Zqe2meε02(kBTe)3/2ln(Λ)\begin{aligned} \eta = \frac{1}{6 \pi \sqrt{3}} \frac{Z q_e^2 \sqrt{m_e}}{\varepsilon_0^2 (k_B T_e)^{3/2}} \ln(\Lambda) \end{aligned}


  1. P.M. Bellan, Fundamentals of plasma physics, 1st edition, Cambridge.
  2. M. Salewski, A.H. Nielsen, Plasma physics: lecture notes, 2021, unpublished.