Interaction of Plasmons

Abstract

The nonlinear interaction between plasma oscillation normal modes is studied in a hydrodynamic approximation. A modified quasilinear treatment is used, in which the normal modes are taken to affect each other through the slow nonlinear temporal and spatial changes they induce in the (equilibrium) background quantities. Plasmons, i.e., wave packets of plasma oscillations, are found in this approximation to have an interaction similar in form to the electromagnetic interaction. The plasmon interaction is describable in terms of scalar and vector potentials which satisfy wave equations of the form satisfied by their electromagnetic counterparts. Here, quantities proportional to the slow changes induced in the background number and current densities play the roles of the scalar and vector potentials, with the thermal velocity corresponding to the speed of light. The source terms in the potential wave equations are quadratic in the normal mode density perturbation. The normal mode density perturbation in turn satisfies a Klein-Gordon-like equation, so that, normalized, it would appear to play a role similar to the quantum-mechanical probability wave function Ψ. The nonlinear interaction leads to an instability with a growth rate proportional to the square root of the ratio of the plasma oscillation energy to the background thermal energy.