then the absorption process dominates, and there is a net attenuation of photons. If the populations of the two states are the same (N1 = N2), the rate of absorption of light exactly balances the rate of emission; the medium is then said to be optically transparent. If the higher energy state has a greater population than the lower energy state (N1 < N2), then the emission process dominates, and light in the system undergoes a net increase in intensity. It is thus clear that to produce a faster rate of stimulated emissions than absorptions, it is required that the ratio of the populations of the two states is such that N2/N1 > 1; In other words, a population inversion is required for laser operation. Selection rules Many transitions involving electromagnetic radiation are strictly forbidden under quantum mechanics. The allowed transitions are described by so-called selection rules, which describe the conditions under which a radiative transition is allowed. For instance, transitions are only allowed if ΔS = 0, S being the total spin angular momentum of the system. In real materials, other effects, such as interactions with the crystal lattice, intervene to circumvent the formal rules by providing alternate mechanisms. In these systems, the forbidden transitions can occur, but usually at slower rates than allowed transitions. A classic example is phosphorescence where a material has a ground state with S = 0, an excited state with S = 0, and an intermediate state with S = 1. The transition from the intermediate state to the ground state by emission of light is slow because of the selection rules. Thus emission may continue after the external illumination is removed. In contrast fluorescence in materials is characterized by emission which ceases when the external illumination is removed. Transitions that do not involve the absorption or emission of radiation are not affected by selection rules. The radiationless transition between levels, such as between the excited S = 0 and S = 1 states, may proceed quickly enough to siphon off a portion of the S = 0 population before it spontaneously returns to the ground state. The existence of intermediate states in materials is essential to the technique of optical pumping of lasers (see below). Creating a population inversion As described above, a population inversion is required for laser operation, but cannot be achieved in our theoretical group of atoms with two energy-levels