Atoms or ions have different electronic energy levels, and transitions between such levels often involve the emission or absorption of light (photons). An absorbed photon can deliver the energy for an atom or ion to get into a higher-lying energy level, whereas spontaneous or stimulated emission releases energy which was previously stored in the atom or ion. Such transitions are used e.g. as laser transitions in laser gain media.
The likelihood of such transitions depends on the electronic levels involved. Strong transitions are those where certain selection rules are satisfied. For example, dipole transitions can occur only between energy levels with the angular momentum parameter ℓ differing by one. Therefore, dipole transitions between energy levels with same parity are not allowed, i.e. they are forbidden. Some “less strongly forbidden” transitions are those which would be forbidden if the approximation of LS coupling were exact.
Dipole-forbidden transitions between energy levels may nevertheless occur based on other mechanisms such as quadrupole transitions. Also, for ions embedded in a crystal lattice or in a glass, internal electric and magnetic fields can break certain symmetries, so that e.g. originally dipole-forbidden transitions become possible by mixing of states with different parity. Such processes, however, are usually much less likely, i.e., they exhibit a small oscillator strength. The resulting transitions are sometimes called weakly allowed transitions rather than strictly forbidden transitions because there are mechanisms for such transitions, although not very strong ones. Whereas typical upper-state lifetimes are of the order of only a few nanoseconds in the case of allowed transitions for spontaneous emission, forbidden transitions of isolated atoms or ions can have upper-state lifetimes of milliseconds or even many seconds, and for ions in crystals or glasses typically between microseconds and milliseconds. Such long-lived levels are called metastable states.
Transitions in Solid-state Laser Gain Media
Essentially all the laser transitions in doped-insulator solid-state lasers (but not in semiconductor lasers and color center lasers) are weakly allowed transitions which are enabled by internal electric fields. The low transition rates lead to long upper-state lifetimes, allowing significant energy storage, which is the basis of pulse generation by Q switching. The combination of long upper-state lifetimes and low transition cross-sections also causes a tendency for spiking phenomena and pronounced relaxation oscillations for such lasers.
Note that the achievable gain on forbidden transitions is not necessarily lower than for allowed transitions because spontaneous emission is also weak, so that one can more easily maintain a high excited state population. In other words, the <$\sigma -\tau$> product can be large despite the small emission cross-section <$\sigma$> because the weak transitions allow for a high upper-state lifetime <$\tau$>.
Transitions for Optical Clocks
Forbidden transitions of isolated atoms or ions are used for optical clocks (clock transitions). Here, the long upper-state lifetime is important because it leads to an extremely narrow linewidth of the transition, so that the transition frequency is very well defined. Unfortunately, the low transition rates also make it more challenging to probe such transitions.
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