Debye shielding

From Physics wiki

Within the plasma the electric potential $\Phi(\mathbf{r})\,$ may fluctuate, in response to, and producing, fluctuations in the charge density $\rho_i\,$ of each species. The potential energy of a single particle of each species is given by $q_i \Phi(\mathbf{r})\,$ (which is independent of its momentum), so that the density of each species is given by Maxwell-Boltzmann statistics

$n_i(\mathbf{r}) = n_{i,0} e^{-q_i \Phi(\mathbf{r})/k_B T_i}\,$ ,

and the charge density is

$\rho(\mathbf{r}) = \sum_i q_i n_{i,0} e^{-q_i \Phi(\mathbf{r})/k_B T_i}\,$ .

If the plasma is very hot, i.e., $k_B T_i \gg |q_i \Phi(\mathbf{r})|\,$ , we can write

$n_i(\mathbf{r}) \approx n_{i,0} \left( 1 -\frac{q_i \Phi(\mathbf{r})}{k_B T_i} \right)\,$ .

Poisson's equation reads

$\nabla^2 \Phi(\mathbf{r}) = \frac{\rho(\mathbf{r})}{\epsilon_0}\,$ ,

$\nabla^2 \Phi(\mathbf{r}) \approx \sum_i q_i n_{i,0} -\sum_i \frac{q_i^2 n_{i,0} }{\epsilon_0 k_B T_i} \Phi(\mathbf{r})\,$ .

Defining the Debye length $\lambda_D\,$ via

$\frac{1}{\lambda_D^2} = \sum_i \frac{q_i^2 n_{i,0} }{\epsilon_0 k_B T_i}\,$ ,

we obtain the Helmholtz equation

$\nabla^2 \Phi(\mathbf{r}) + \frac{1}{\lambda^2} \Phi(\mathbf{r}) = \sum_i q_i n_{i,0}\,$ .

As we shall see, the free species have the effect of shielding or screening the usual Coulomb potential, an effect known as Debye shielding or Debye screening. For a quasineutral plasma consisting of two species with charge $+e\,$ and $-e\,$ respectively, at the same temperature $T\,$ , and both with equilibrium number density $n_0\,$ ,