Electron Interaction in Solids. General Formulation

A general Hamiltonian formalism is developed to treat from first principles the motion of electrons in solids, including their mutual Coulomb interaction. By a series of canonical transformations it is shown that under suitable circumstances (which obtain in nearly all solids) plasmons (a quantized collective plasma oscillation of the electron gas) represent a well-defined elementary excitation of the solid. The "existence criterion" for plasmons is found to be a high electronic polarizability. Where plasmons exist, we are able to give a satisfactory description of their properties when the majority of the individual electron oscillator strengths correspond to transitions in which the energy change is large or small with respect to the plasmon energy, $\hslash {\omega }_p=\hslash {\left(\frac{4\pi N{e}^2}{m}\right)}^{\frac{1}{2}}$. After the plasmon modes are separated out, the remaining electron interaction is found to be screened, with a range of the order of the interelectronic spacing. The usefulness of this effective Hamiltonian for the calculation of the electronic energy levels and cohesive energy in solids is discussed briefly.