Three-branch normal mode dispersion relation for metals: The case of large momentum transfer

In N. Barberán, J. da Providência, Jr., Z. Phys. B 101 (1996) 3–12, a variational procedure appropriate for the description of the collective modes of valence electrons in a metal was proposed. This approximation, which is based on the quantum mechanical action principle and uses a Slater determinant as the wave function of the cloud of valence electrons, is here generalized. In the present note we use the semiclassical approximation, considering now that the generator of the fluctuations S ˆ = ∑ i N S ( r i , − i ħ ∇ i , t ) of the Slater determinant ( | ϕ 〉 = exp ( i S ˆ / ħ ) | ϕ 0 〉 ) is a polynomial of order five in the single particle momentum, introducing in this way six collective coordinates (each one as a coefficient of the corresponding power of the single particle momentum operator). As a result we obtain a branch with higher energy and two branches with lower energy, which are important for describing the experimental data obtained for large momentum transfer. Comparison with experimental data is presented for aluminum, beryllium and lithium.