Einstein coefficients and excited-state lifetimes

The first factor is associated with the electronic dipole transition probability between the electronic states the second factor is associated between vibrational levels of the lower state v and the excited state V, and is commonly known as the Franck-Condon factor, the third factor stems from the rotational levels involved in the transition, J and /, the rotational line-strength factor (often termed the Honl-London factor). In particular, the Franck-Condon information from the spectrum allows one to gain access to the relative equilibrium positions of the molecular energy potentials. Then, with a full set of the spectroscopic constants that are used to approximate the energy-level structure (see Equations (2.1) and (2.2)) and which can be extracted from the spectra, full potential energy curves can be constructed. 

Atoms and molecules in excited states can decay without the presence of an external light field, or photon interaction. The spontaneous decay rate dNj/dt or dNi/dt) is given by 

Since atoms in the upper (excited) level can decay both via spontaneous and stimulated emission, the total downward rate —ANjIdt or dNi/dt) is given by 

For a two-level system (lower level i and upper level j), the rate of an upward stimulated transition. 

It should be noted that, in spectroscopy, the transition intensity is frequently expressed in terms of the oscillator strength/, which is a dimensionless number and is useful when comparing different transitions (see Box 2.3 for a definition off). 

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