Line shape of the low-energy tail of exciton absorption in molecular crystals

The influence of crystal defects and phonons on the line shape of the low-energy tail of exciton absorption is calculated for molecular crystals with the lower limit of their exciton band at $\overrightarrow{\mathrm{k}}=0\mathrm{}$. The exciton Hamiltonian for a crystal with lattice defects is derived from that for a perfect crystal with lattice vibrations. Calculation of the line shape is done assuming that the individual lattice defects have independent effects on the unperturbed exciton band. Three examples of the molecular exciton energy bands are considered: (1) one-dimensional tight-binding exciton band, (2) the exciton bands with Hubbard's model of density of states, and (3) exciton bands with density of states independent of the exciton energies. At very low temperatures where the influence of the defects is dominant, we have found the asymmetric low-energy tail due to defects, acting as shallow traps, in all the three examples. The observed asymmetric line shape in 1,2-dibromonaphthalene single crystals is, therefore, attributed to such shallow traps and analogous to the Urbach-Martienssen tail observed in alkali halides. As the temperature increases and phonons become more active, the line shape becomes symmetric due to the dominant influence of thermal broadening.