Plasmon resonance broadening in small metal particles

A quantum mechanical method based on the Kramers-Heisenberg dispersion relation is used to evaluate the dielectric response of small metal particles, and thereby to determine the influence of particle size on the widths of the plasmon resonance line shapes. Several different particle shapes are considered (sphere, cylinder, rectangular prism, spherical shell, and cylindrical shell) and for each shape a free electron Schrödinger equation is used to determine conduction band energies and dipole matrix elements. The main emphasis in this work is on particle sizes large enough that only the first order deviations from the infinite size limit are important, and for such sizes we find that the size dependent contribution to the width can be expressed in terms of an effective length L_eff. This effective length is found to depend on the direction of the external field relative to the particle symmetry axes, and on the shape of the particle. For compact shapes, L_eff is accurately approximated by 0.65 L_av along each principal axis, where L_av is the ratio of particle volume to its projected area along the relevant axis. Comparison with previous classical and semiclassical calculations is considered, and for spherical particles, we find good agreement with the classical surface scattering model, differing by about 16%. More significant differences are found for other shapes, most notably because the classical theory ignores the dependence of resonance width on the orientation of the field relative to the particle.