On the accuracy of the Debye shielding model

Abstract

The expression for the Debye shielding in plasma physics is usually derived under the assumptions that the plasma particles are weakly coupled, so that their total kinetic energy is much greater than their electrostatic interaction energy, and that the velocity distributions of the plasma species are Maxwellian. The first assumption also establishes that the number of particles within a sphere with a Debye radius, known as the plasma parameter $N_D$, should be significantly greater than 1, and determines the difference between weakly and strongly coupled plasmas. Under such assumptions, Poisson's equation can be linearized, and a simple analytic expression is obtained for the electrostatic potential. However, textbooks rarely discuss the accuracy of this approximation. In this work we compare the linearized solution with a more precise numerical (or ``exact'') solution, and show that the linearization, which underestimates the ``exact'' solution, is reasonably good even for $N_D \sim 40$. We give quantitative criteria to set the limit of the approximation when the number of particles is very small, or the distance to the test charge too short.