Debye length

Remarks

It is the length for which the linearized equation $\nabla^2 \varphi = \varphi / \lambda_D^2$ holds, obtained from Poisson's equation when the carriers follow the Boltzmann distribution and the potential is weak. Example: a $1{:}1$ salt such as NaCl at concentration $c$ has two species with $q = \pm e$ and $n = c N_A$, where $N_A$ is Avogadro's number, so $\lambda_D = \sqrt{\varepsilon k_B T / (2 c N_A e^2)}$. In water at $25$ degrees Celsius this is about $0.304 / \sqrt{c}$ nanometres with $c$ in mol per litre, so about $0.96$ nm at $c = 0.1$. Non-example: a gas of neutral molecules has no free charges, so the sum in the denominator is $0$ and there is no screening length. For an electron plasma with immobile ions, the usual form is $\lambda_D = \sqrt{\varepsilon_0 k_B T_e / (n_e e^2)}$, where $T_e$ and $n_e$ are the electron temperature and density.