Categories: Physics, Plasma physics.

Boltzmann relation

In a plasma where the ions and electrons are in thermal equilibrium, in the absence of short-lived induced electromagnetic fields, the densities nin_i and nen_e can be predicted.

By definition, a charged particle in an electric field E=−∇ϕ\vb{E} = - \nabla \phi experiences a Lorentz force Fe\vb{F}_e. This corresponds to a force density fe\vb{f}_e, such that Fe=fedV\vb{F}_e = \vb{f}_e \dd{V}. For electrons:

fe=qeneE=−qene∇ϕ\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 pe=kBTenep_e = k_B T_e n_e, then the pressure pep_e leads to another force density fp\vb{f}_p:

fp=−∇pe=−kBTe∇ne\begin{aligned} \vb{f}_p = - \nabla p_e = - k_B T_e \nabla n_e \end{aligned}

At equilibrium, we demand that fe=−fp\vb{f}_e = - \vb{f}_p, and isolate this equation for ∇ne\nabla n_e, yielding:

kBTe∇ne=−qene∇ϕ  ⟹  ∇ne=−qe∇ϕkBTene=−∇(qeϕkBTe)ne\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 nen_e, known as the Boltzmann relation, due to its resemblance to the statistical Boltzmann distribution (see canonical ensemble):

ne(r)=ne0exp⁡ ⁣( ⁣− ⁣qeϕ(r)kBTe)\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 ne0n_{e0} represents the electron density for ϕ=0\phi = 0. We can do the same for ions instead of electrons, leading to the following ion density nin_i:

ni(r)=ni0exp⁡ ⁣( ⁣− ⁣qiϕ(r)kBTi)\begin{aligned} \boxed{ n_i(\vb{r}) = n_{i0} \exp\!\bigg( \!-\! \frac{q_i \phi(\vb{r})}{k_B T_i} \bigg) } \end{aligned}

But due to their large mass, ions respond much slower to fluctuations in the above equilibrium. Consequently, after a perturbation, the ions spend more time in a non-equilibrium state than the electrons, so this formula for nin_i is only valid if the perturbation is sufficiently slow, such that the ions can keep up. Usually, electrons do not suffer the same issue, thanks to their small mass and hence 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 M. Salewski, A.H. Nielsen, Plasma physics: lecture notes, 2021, unpublished.