---
title: "Spitzer resistivity"
date: 2021-10-05
categories:
- Physics
- Plasma physics
layout: "concept"
---

If an [electric field](/know/concept/electric-field/)
with magnitude $E$ is applied to the plasma, the electrons experience
a [Lorentz force](/know/concept/lorentz-force/) $q_e E$
(we neglect the ions due to their mass),
where $q_e$ is the electron charge.

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

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

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

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

This is Ohm's law, where $\eta$ is the resistivity.
From our derivation of the [Coulomb logarithm](/know/concept/coulomb-logarithm/) $\ln(\Lambda)$,
we estimate $f_{ei}$ to be as follows,
where $n_i$ is the ion density,
$\sigma$ is the collision cross-section,
and $\mu$ is the [reduced mass](/know/concept/reduced-mass/)
of the electron-ion system:

$$\begin{aligned}
    f_{ei}
    = n_i \sigma v_e
    = \frac{1}{2 \pi} \Big( \frac{q_e q_i}{\varepsilon_0 \mu} \Big)^2 \frac{n_i}{v_e^3} \ln(\Lambda)
    \approx \frac{1}{2 \pi} \frac{Z q_e^4}{\varepsilon_0^2 m_e^2} \frac{n_e}{v_e^3} \ln(\Lambda)
\end{aligned}$$

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

Inserting this expression for $f_{ei}$ into
the so-called **Spitzer resistivity** $\eta$ then yields:

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

A reasonable estimate for the typical velocity $v_e$
at thermal equilibrium is as follows,
where $k_B$ is Boltzmann's constant,
and $T_e$ is the electron temperature:

$$\begin{aligned}
    \frac{1}{2} m_e v_e^2
    = \frac{3}{2} k_B T_e
    \quad \implies \quad
    v_e
    = \sqrt{\frac{3 k_B T_e}{m_e}}
\end{aligned}$$

Other choices exist,
see e.g. the [Maxwell-Boltzmann distribution](/know/concept/maxwell-boltzmann-distribution/),
but always $v_e \propto \sqrt{T_e/m_e}$.
Inserting this $v_e$ into $\eta$ then gives:

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



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