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In physics, a forbidden mechanism or forbidden line is a spectral line emitted by atoms undergoing nominally "forbidden" energy transitions not normally allowed by the selection rules of quantum mechanics. In formal physics, this means that the process cannot proceed via the most efficient (electric dipole) route. Although the transitions are nominally "forbidden", there is a small probability of their spontaneous occurrence, should an atom or molecule be raised to an excited state. More precisely, there is a certain probability that such an excited atom will make a forbidden transition to a lower energy state per unit time; by definition this probability is much lower than that for any transition permitted by the selection rules. Therefore, if a state can de-excite via a permitted transition (or otherwise, e.g. via collisions) it will almost certainly do so rather than choosing the forbidden route. Nevertheless, "forbidden" transitions are only relatively unlikely: states that can only decay in this way (so-called meta-stable states) usually have lifetimes of order milliseconds to seconds, compared to less than a microsecond for decay via permitted transitions.
In solid-state physics 
Forbidden transitions in rare earth atoms such as erbium and neodymium make them useful as dopants for solid-state lasing media. In such media the atoms are held in a matrix which keeps them from de-exciting by collision, and the long half life of their excited states makes them easy to "optically pump" to create a large population of excited atoms. Neodymium doped glass derives its unusual coloration from "forbidden" f-f transitions within the neodymium atom, and is used in extremely high power solid state lasers.
In astrophysics 
Forbidden emission lines have only been observed in extremely low-density gases and plasmas, either in outer space or in the extreme upper atmosphere of the Earth. Even the hardest laboratory vacuum on Earth is still too dense for forbidden line emission to occur before atoms are collisionally de-excited. However, in space environments, densities may be only a few atoms per cubic centimetre, making atomic collisions unlikely. Under such conditions, once an atom or molecule has been excited for any reason into a meta-stable state, then it is almost certain to decay by emitting a forbidden-line photon. Since meta-stable states are rather common, forbidden transitions account for a significant percentage of the photons emitted by the ultra-low density gas in space.
Forbidden lines of nitrogen ([N II] at 654.8 and 658.4 nm), sulfur ([S II] at 671.6 and 673.1 nm), and oxygen ([O II] at 372.7 nm, and [O III] at 495.9 and 500.7 nm) are commonly observed in astrophysical plasmas. These lines are important to the energy balance of such things as planetary nebulae and H II regions. The forbidden 21-cm hydrogen line is particularly important for radio astronomy as it allows very cold neutral hydrogen gas to be seen.
Forbidden line transitions are noted by placing square brackets around the atomic or molecular species in question, e.g. [O III] or [S II].