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[ O III ]   ,  [ S II ] ,   [ N II ] etc...

The brackets are called forbidden lines. This means that the atoms are undergoing forbidden transitions in the low-density nebulae. These transitions are not possible in the higher density laboratory situations. What does it mean by forbidden?

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Forbidden lines are not really forbidden - just very very unlikely. Usually the atoms will be jostled into another transition by collisions with other atoms before they can undergo these transitions. Therefore forbidden is a bad phrase, like so many in astronomy, but it's here to stay. Unlikely is a better term!

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So that's why these forms ( like O III or S II ) are more likely to form and survive in low density spaces than in higher ones since there is less collision at lower densities.

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Atomic nuclei are are surrounded by electons in discrete orbitals or energy levels.

Electrons can be promoted to higher, unoccupied energy levels by absorbing discreet quanta or packets of energy in the form of EM radiation. These are the 'allowed' transitions and are specific for any given element.

When electrons are promoted in such a way their spin direction 'flips'. Electrons can be promoted to discreet energy levels which doesn't involve flipping their spin direction - these are 'forbidden' transitions in the sense that they cannot be achieved by absorbing energy in the form of EM radiation. They have to gain the energy in some other way (eg. by colliding with other atoms) and as JulianO says this is much less likely.

When these electrons return to their ground states from these forbidden levels (eg. O-III) they emit a specific wavelength of light. The reason why that is not possible in higher density laboratory situations is that the atom is more likely to lose its energy through collision than through emitting the energy as radiation.

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forbidden lines,  in astronomical spectroscopy, bright emission lights in the spectra of certain nebulae (HII region), not observed in the laboratory spectra of the same gases, because on Earth the gases cannot be rarefied sufficiently. The term forbidden is misleading; a more accurate description would be “highly improbable.” The emissions result from electrons in long-lived orbits within the radiating atoms—i.e., the transition from an upper energy level to a lower energy level that produces the emissions requires a long time to take place. As a result, emission lines corresponding to such atomic transitions are extremely weak compared with other lines.

Hope this is something of a help

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we like to confuse! Usually its "forbidden" in a dipole transition. however, in higher-pole transitions (eg quadrapole or octapole) they become allowed. Also may occur as two-photon transitions etc. Probibility of the transition varies greatly. The usual reading of "forbidden" is that the dipole transition is not allowed. It does not meann that the probibility of the transition is zero though.   I said we like to confuse!! :)  

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