In determining the optical band gap from absorption coefficient spectra, one can determine whether the band gap is direct or indirect. Depending on the exponent used, it can be allowed or forbidden (in)direct semiconductor. So what does it mean allowed and forbidden transitions in this case?
It implies on parametric and non-parametric nonlinear phenomenas i.e. the transition which the atomic states return back to its primary stable states and the other the transition which the atomic states return back to some unstable states
While considering the transitions, one important concept is "selection rules". one of the selection rule states that "the transitions which involve a change in spin quantum number of an electron, during the transition are not allowed to take place. These are called as Forbidden transitions".
In a crystal, for an optical transition, the stronger contribution is the dipolar term. The selection rules are calculated considering a dipolar interaction. Using the Wigner-Eckart theorem, we say that the transition is "allowed" when the matrix element between initial and final states (wavefunctions) and the dipole operator P is nonzero, and forbidden when this is zero. Symmetry arguments will be enough to determine this. The form of the absorption and the exponent in each case comes from the functional dependence on energy of the density of states curve close to the band extrema that participate in the transition.
The optical selection rules result from the conservation of angular momentum, energy and linear momentum. Since a photon carries spin, momentum and energy, all these quantities must be absorbed by the induced transition. If the energy difference between the initial and final state corresponds to the photon energy, the transition is called resonant. If the angular momentum of the initial and final state differs by \pm 1 the dipole matrix element is nonzero and the transition is called dipole allowed. And last but not least, the photon momentum is absorbed by the center of mass momentum of the induced transition. Since the photon momentum is negligible, this means that only direct transitions, i.e. between equal k-states are allowed. Only transitions that fulfill all these criteria simultaneously can be induced by a single photon process (allowed). Other transitions require interactions with additional particles (photons or phonons) involved and are much more unlikely (forbidden).
Well, it may be important to note that the selection criterion may also be a function of the characteristic wavelengths and their corresponding intensities. For lower transitions and/or longer wavelengths, dipolar transitions dominate.
The selection rules for electromagnetic radiation have been identified by Oskar Laporte and others in the 1920s. Consequently they have been described in the literature and text books since almost a century. Electrodynamics (Jackson's text book and many others) describes a "multipole expansion of the radiation field". The lowest order of radiation is the electric dipole radiation (E1) which connects levels of different parity and carries one unit of angular momentum. The next higher order of the expansion consists of magnetic dipole (M1) and electric quadrupole (E2) radiation (no change of parity), then magnetic quadrupole (M2) and electric octupole (E3) (change of parity), and so on.
Usually the low-order radiation is strongest and the high order contribution minor. However, in some situations (governed by parity and total angular momentum), E1 radiation cannot occur (is "forbidden"), and then the next higher order multipole radiation is the dominant one (and so on). The various types of radiation scale differently with the properties of the emitting atom (nuclear charge Z, transition energy) and have different spatial emission characteristics. "Forbidden" transitions usually mean "E1-forbidden". The photons do not carry distinguishing marks whether they originated as E1 transitions or M1 etc. Emission and absorption of photons by atoms (or other entities) are described by transition rates. These are usually much higher for (E1) "allowed" than for (E1) "forbidden" transitions, and the process is symmetric for emission and absorption.
Of course, sloppy lab jargon abounds and often denotes something diffusely as "forbidden" that is less frequent than something else that is "allowed", without necessarily relating to the clearly defined atomic physics situation.
Optical selection rule distinguishes allowed (nonzero dipole matrix element at k0) and forbidden (zero dipole matrix element at k0), where k0 is the band extrema close to which we consider absorption. Absorption occurs in both cases. However, for forbidden transitions, one has to account k-dependence of the matrix element in the vicinity of k0 that results in different frequency dependence of the absorption coefficient as compared to the one caused by allowed transitions.