Plasmons propagating in periodic lattices and their dispersion

Starting from the equation of motion for the electron density in terms of the momentum flux density, a non-local r space theory is developed for describing the propagation of plasmons in a periodic lattice. The theory is valid for long wavelengths, and is shown to yield correct results for the dispersion relation $\omega $(k) for a uniform electron gas in the high density limit, to order $k^{2}$, k being the wavenumber. By using a local density approximation, corrected by higher order terms involving gradients of the density, the non-local equation for the density change can be expressed explicitly in terms of the periodic lattice density. The way in which the dispersion relation $\omega $(k) can be obtained around k = 0 in the various plasmon branches is discussed, a method paralleling the Wigner-Seitz theory in electronic energy band structure being proposed. Numerical calculations are planned within such a spherical cell model. The connexion with Bloch's hydrodynamic model of collective oscillations is finally discussed.