# Radiative Lifetime

Author: the photonics expert Dr. Rüdiger Paschotta

Definition: lifetime of an electronic state in the (hypothetical) situation where only radiative processes depopulate that level

Category: physical foundations

Units: s

Formula symbol: <$\tau_\textrm{rad}$>

DOI: 10.61835/ian Cite the article: BibTex plain textHTML Link to this page

The radiative lifetime of an excited electronic state e.g. in a laser gain medium is the lifetime which would be obtained if radiative decay via the unavoidable spontaneous emission were the only mechanism for depopulating this state. It is given by the equation

$$\frac{1}{{{\tau _{{\rm{rad}}}}}} = \frac{{8\pi \;{n^2}}}{{{c^2}}}\;\int {{\nu ^2}{\sigma _{{\rm{em}}}}(\nu )\;{\rm{d}}\nu } = 8\pi \;{n^2}c\;\int {\frac{{{\sigma _{{\rm{em}}}}(\lambda )}}{{{\lambda ^4}}}\;{\rm{d}}\lambda } $$which shows that high emission cross-sections and a large emission bandwidth inevitably lead to a low radiative lifetime. This is because the transition cross-sections determine not only the strength of stimulated emission but also that of spontaneous emission. The derivation of that equation is based on an equation for the mode density of free space, as is also used e.g. for the derivation of Planck's law for the power spectral density of thermal radiation. This means that the equation does *not* hold in microcavities (as often used in experiments on quantum electrodynamics) because such cavities can substantially modify the mode density.

Note also the influence of the refractive index via the mode density. If fluorescence lifetime measurements are done using a powder with a grain size well below the wavelength of light, the refractive index of the ambient medium (rather than that of the powder grains) becomes relevant. For example, the upper-state lifetime measured for powder in air can be longer compared with that for solid crystals. Such observations should not be misinterpreted as evidence for quenching effects in crystals.

Another important aspect is that a shorter mean wavelength of the emission implies a shorter radiative lifetime. This results from the increased mode density of the radiation field. A consequence is that ultraviolet lasers tend to have a higher threshold pump power than e.g. infrared lasers.

As the gain efficiency of a laser medium is (in simple cases) proportional to the product of the maximum emission cross-section and the upper-state lifetime (the *<$\sigma -\tau$> product*), lasers based on broadband gain media have a higher threshold pump power.

The actual lifetime of an electronic level can be lower than the radiative lifetime, if non-radiative quenching processes also significantly depopulate the level. This means that the quantum efficiency of the transition is below unity. The radiative and non-radiative transition rates, being independent of each other, simply add up to result in a total transition rate, the inverse of which is the actual level lifetime.

If the quantum efficiency is known to be close to unity, the above equation can be used for obtaining the absolute scaling of emission cross-sections, the wavelength dependence of which is already known from the shape of the emission spectrum (→ *Füchtbauer–Ladenburg equation*). In other cases, where the scaling of emission cross-sections is known (e.g. obtained from absorption cross-sections via the reciprocity method), the quantum efficiency of the fluorescence can be obtained by comparing the calculated radiative lifetime with the upper-state lifetime.

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