Categories: Physics, Plasma physics.

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.

- P.M. Bellan,
*Fundamentals of plasma physics*, 1st edition, Cambridge. - 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*Plasma physics: lecture notes*, 2021, unpublished.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.