Ryu Jisu, Yeola Sarang, Jonas David M
Department of Chemistry, University of Colorado, Boulder, CO 80309-0215.
Department of Physics, University of Colorado, Boulder, CO 80309-0390.
Proc Natl Acad Sci U S A. 2024 Sep 10;121(37):e2410280121. doi: 10.1073/pnas.2410280121. Epub 2024 Sep 3.
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, a predominantly 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.
我们给出了爱因斯坦系数谱，并基于自发发射和受激发射之间的联系，对它们之间的广义爱因斯坦关系进行了细致平衡推导。如果两个展宽的能级或能带在能量上重叠，它们之间的跃迁不一定纯粹是吸收或发射。因此，自发发射可以在两个跃迁方向上发生，四条爱因斯坦系数谱取代了一条谱线的三个爱因斯坦系数。在平衡状态下，这四条不同的谱线服从五个成对关系，并且一种线形可以生成所有四条谱线。这些关系与分子量子统计无关，并预测了与黑体辐射达到平衡所需的正向和反向跃迁之间的斯托克斯位移。对于玻尔兹曼统计，正向和反向跃迁的相对强度取决于初始能带和终态能带之间的形式化学势差，对于理想溶质，该形式化学势差成为标准化学势差。一个能带的形式化学势取代了量子能级的能量和简并度。与量子能级的能量一样，能带的形式化学势服从里德伯 - 里茨组合原理。每个受激发射的爱因斯坦系数谱给出一个频率相关的跃迁截面。跃迁截面服从因果律以及与自发发射的细致平衡条件，但不直接服从广义爱因斯坦关系。即使能量宽度远小于光子能量，一个能量宽度远大于热能的主要吸收性正向跃迁也可能有如此大的斯托克斯位移，以至于其反向跃迁截面变得主要是吸收性而非发射性。
