Generalized Einstein Relations between Absorption and Emission Spectra at Thermodynamic Equilibrium

TL;DR

This paper derives generalized Einstein relations for absorption and emission spectra at thermodynamic equilibrium, accounting for overlapping energy levels and broadening effects, and introduces a formal chemical potential concept for bands.

Contribution

It presents a detailed-balance derivation of generalized Einstein relations involving four spectra, extending classical relations to overlapping and broadened energy levels, independent of quantum statistics.

Findings

Four Einstein coefficient spectra replace the classical three for overlapping levels.

The relations predict the Stokes' shift and depend on a formal chemical potential for bands.

Transition cross sections obey causality and detailed balance, with broadening affecting emission and absorption characteristics.

Abstract

We present Einstein coefficient spectra and a detailed-balance derivation of generalized Einstein relations between them that is based on the connection between spontaneous and stimulated emission. If two broadened levels or bands overlap in energy, transitions between them need not be purely absorptive or emissive. Consequently, spontaneous emission can occur in both transition directions, and four Einstein coefficient spectra replace the three Einstein coefficients for a line. At equilibrium, the four different spectra obey five pairwise relationships and one lineshape generates all four. These relationships are independent of molecular quantum statistics and predict the Stokes' shift between forward and reverse transitions required by equilibrium with blackbody radiation. For Boltzmann statistics, the relative strengths of forward and reverse transitions depend on the formal chemical potential difference between the initial and final bands, which becomes the standard chemical potential difference for ideal solutes. The formal chemical potential of a band replaces both the energy and degeneracy of a quantum level. Like the energies of quantum levels, the formal chemical potentials of bands obey the Rydberg-Ritz combination principle. Each stimulated Einstein coefficient spectrum gives a frequency-dependent transition cross section. Transition cross sections obey causality and a detailed-balance condition with spontaneous emission, but do not directly obey generalized Einstein relations. Even with an energetic width much less than the photon energy, an absorptive forward transition with an energetic width much greater than the thermal energy can have such an extreme Stokes' shift that its reverse transition cross section becomes predominantly absorptive rather than emissive.