Categories: Physics, Plasma physics.

Spitzer resistivity

If an electric field with magnitude \(E\) is applied to the plasma, the electrons experience a 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 \(\ln\!(\Lambda)\), 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 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, 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.

© Marcus R.A. Newman, a.k.a. "Prefetch". Available under CC BY-SA 4.0.
uses