Categories: Physics, Plasma physics.

Boltzmann relation

In a plasma where the ions and electrons are both in thermal equilibrium, and in the absence of short-lived induced electromagnetic fields, their densities \(n_i\) and \(n_e\) can be predicted.

By definition, a particle in an electric field \(\vb{E}\) experiences a Lorentz force \(\vb{F}_e\). This corresponds to a force density \(\vb{f}_e\), such that \(\vb{F}_e = \vb{f}_e \dd{V}\). For the electrons, we thus have:

\[\begin{aligned} \vb{f}_e = q_e n_e \vb{E} = - q_e n_e \nabla \phi \end{aligned}\]

Meanwhile, if we treat the electrons as a gas obeying the ideal gas law \(p_e = k_B T_e n_e\), then the pressure \(p_e\) leads to another force density \(\vb{f}_p\):

\[\begin{aligned} \vb{f}_p = - \nabla p_e = - k_B T_e \nabla n_e \end{aligned}\]

At equilibrium, we demand that \(\vb{f}_e = - \vb{f}_p\), and isolate this equation for \(\nabla n_e\), yielding:

\[\begin{aligned} k_B T_e \nabla n_e = - q_e n_e \nabla \phi \quad \implies \quad \nabla n_e = - \frac{q_e \nabla \phi}{k_B T_e} n_e = - \nabla \bigg( \frac{q_e \phi}{k_B T_e} \bigg) n_e \end{aligned}\]

This equation is straightforward to integrate, leading to the following expression for \(n_e\), known as the Boltzmann relation, due to its resemblance to the statistical Boltzmann distribution (see canonical ensemble):

\[\begin{aligned} \boxed{ n_e(\vb{r}) = n_{e0} \exp\!\bigg( \!-\! \frac{q_e \phi(\vb{r})}{k_B T_e} \bigg) } \end{aligned}\]

Where the linearity factor \(n_{e0}\) represents the electron density for \(\phi = 0\). We can do the same for ions instead of electrons, leading to the following ion density \(n_i\):

\[\begin{aligned} \boxed{ n_i(\vb{r}) = n_{i0} \exp\!\bigg( \!-\! \frac{q_i \phi(\vb{r})}{k_B T_i} \bigg) } \end{aligned}\]

However, due to their larger mass, ions are much slower to respond to fluctuations in the above equilibrium. Consequently, after a perturbation, the ions spend much more time in a transient non-equilibrium state than the electrons, so this formula for \(n_i\) is only valid if the perturbation is sufficiently slow, allowing the ions to keep up. Usually, electrons do not suffer the same issue, thanks to their small mass and fast response.

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 M. Salewski, A.H. Nielsen, Plasma physics: lecture notes, 2021, unpublished.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.
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