The Stefan-Boltzmann constant re-visited for photons thermally generated within matter

TL;DR

This paper revises the Stefan-Boltzmann law for photons generated within matter by incorporating material-specific photon mode densities, leading to new spectral density calculations and predictions of internal photon resonances.

Contribution

It introduces a material-dependent formulation of photon spectral densities inside matter, extending the classical Stefan-Boltzmann law to account for internal photon mode variations.

Findings

Spectral densities calculated for water, germanium, and silver.

Predicted internal photon resonances in dielectrics.

Verification possible through external spectral intensity measurements.

Abstract

The Stefan-Boltzmann constant arose from photon densities inside a cavity, but inside matter photon mode densities are material specific. Photon speeds are governed by the mode they occupy, so mode densities can be expressed in terms of speed. Cavity intensities at temperature \(T_{K}\) combined \(\left[\frac {8\pi k^4}{c^3h^3}\right] T_{K }^4\) with \(({\pi^4}/{15})\). A material dependent number from summation of internal photon spectral energy densities replaces \(\frac {\pi^4}{15}\). Spectral densities are presented for water, germanium and silver. Output intensity combines revised hemispherical emittance \(\epsilon_{Q,H}\) based on these densities, with universal factor \(\left[\frac {8\pi k^4}{c^3h^3}\right] T_{K }^4\). Emitted radiance after interface internal reflectance of directionally invariant internal radiance elements defines \(\epsilon_{Q,H}\). Predicted internal densities are verifiable using measured external spectral intensities, provided refraction upon exit is accounted for in emissivity, which the Kirchhoff rule neglects. Virtual bound state photon resonances are predicted in dielectrics and observed.