The value of m is 1/2 for direct allowed, 2 for indirect allowed, 3/2 for direct forbidden, and 3 for indirect forbidden transitions [61].
For the crystalline semiconductors, p is theoretically equal to 1/2, 3/2, 2 or 3 for the direct allowed, direct forbidden, indirect allowed and indirect forbidden transitions, respectively.
The values of m = 1/2, 3/2, 2, and 3 are for direct allowed, direct forbidden, indirect allowed, and indirect forbidden transitions, respectively.
The value of n denotes the nature of the transition, n = 1/2 for direct allowed transitions, n = 3/2 for direct forbidden transitions, n = 2 for indirect allowed transitions, and n = 3 for indirect forbidden transition [39].
The optical band gap (E g) of the ZnO-NPs sample is estimated using the Tauc's equation which demonstrates a relationship between absorption coefficient and the incident photon energy (hν) as follows [27]: alpha hnu = beta (hnu - E_{text{g}} )^n, (4 where the exponent n is ½, 3/2, 2, 3 (for allowed direct, forbidden direct, allowed indirect and forbidden indirect transitions, respectively).
(3) Co3O4 shows different types of band gaps: direct allowed 2.06 and 1.44 eV, direct forbidden 1.38 and 1.26 eV, indirect allowed 1.10 eV (energy of phonon assisting indirect transition = 0.02) and indirect forbidden 0.75 eV (energy of phonon assisting indirect transition = 0.27) (Kabre 2011).
where α is the absorbance, h is the Planck constant, ν is the frequency, Egap is the optical band gap, and n is a constant associated to the different electronic transitions ( 2, or 3 for direct allowed, indirect allowed, direct forbidden, and indirect forbidden transitions, respectively).
where α is the linear absorption coefficient of the material, A is an energy-independent constant, Eg is the optical bandgap, and n is a constant which determines the type of optical transitions: for indirect allowed transition, n = 2; for indirect forbidden transition, m = 3; for direct allowed transition, n = 1/2; and for direct forbidden transition, m = 3/2.
Asymptotic series are given for the linear combinations of Airy functions and their derivatives needed in the theory of both direct allowed and direct forbidden transitions in the intermediate field case, and some examples of recent fits to experimental data in the literature are discussed.
where A is constant, Eg is the absorption band gap, α is the absorption coefficient, and n depends on the type of transition, n may assume the values 1/2, 2, 3/2 and 3 respectively corresponding to allowed direct, allowed indirect, forbidden direct and forbidden in direct transitions [19].
