Metallic Absorptivity at Normal Incidence above Far-infrared

The absorptivity of monochromatic radiation at normal incidence on metals is described by the Hagen-Rubens (H-R) relation, provided the wave energy is not higher than similar to 100 meV (wavelength not shorter than similar to 10 mu m), that is, it is restricted to the far-infrared and downwards. The H-R relation is a consequence of Maxwell's equations together with Ohm's law. This work proposes an extension of the H-R relation above the far-infrared by taking into account inertial effects due to charge carriers. The influence of inertia is described by introducing a relaxation time for the current density in the framework of a generalized Ohm's law. In the extended approach, it is found that the unique characteristic length scale of the problem, the penetration depth of the wave into the conductor, splits in two such that two distinct regimes for the skin effect can be identified: at low frequencies, the fields exhibit the usual attenuated oscillations; at high frequencies, they become purely attenuated in space. A transition frequency between the regimes is also identified. It is shown as well that the relaxation time can be fully determined from the metallic permittivity and angular frequency. The extended formulation is applied to aluminum and it is found that its relaxation time similar to 6.91 fs at room temperature. It is also shown that the absorptivity by aluminum is in excellent agreement with the extended theory up to the near-infrared (similar to 200 meV).