From e28d2a982d0c65fcad9a2d2a4c20d06a9848fa8f Mon Sep 17 00:00:00 2001
From: Prefetch
Date: Tue, 5 Oct 2021 19:31:12 +0200
Subject: Expand knowledge base

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+---
+title: "Spitzer resistivity"
+firstLetter: "S"
+publishDate: 2021-10-05
+categories:
+- Physics
+- Plasma physics
+
+date: 2021-10-04T14:47:44+02:00
+draft: false
+markup: pandoc
+---
+
+# Spitzer resistivity
+
+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.
+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.
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