Evaluation of Lattice Sums in the Calculation of Crystal Spectra

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Abstract
Transition dipole—transition dipole exciton interaction lattice sums are shown to be amenable to evaluation by the general procedure of Nijboer and de Wette [Physica 23, 309 (1957)]. The practical evaluation of lattice sums in certain cubic and tetragonal crystals is then discussed, with emphasis upon the importance of the manner in which the spectrum of a molecular solid is obtained. The particle size of a polycrystalline sample is shown to strongly affect the value of the lattice sum. For single crystals, the orientation of the crystal and its transition moment directions with respect to the electric vector of the exciting field is of importance. Specific formulas relating crystal field splittings to dipole derivatives (in vibrational spectra) and to oscillator strengths (in electronic spectra) are given and tested with a variety of vibrational transitions of various cubic crystals as well as with the deformation mode of N3— in the tetragonal alkali azides. The oscilator strength of the 1∏←1Σ+ transition of CO is calculated to be 0.148, using some recent data on this spectrum obtained by Dressler. Splittings in the related transition in solid N2 are also discussed. The results indicate that several conclusions in earlier calculations based upon dipole coupling models have been in error due to inadequate summation procedures. The results also indicate that ``exciton packets'' extend to the boundaries of the crystallites of which a polycrystalline sample is composed. Limitations to the general applicability of the summation procedures presented here are also discussed.