Symmetry and Reststrahlen in Elemental Crystals

Lattice vibrations in elemental crystals can possess a first-order electric moment, and thus exhibit reststrahlen (symmetry-allowed one-phonon infrared absorption), by the mechanism of displacement-induced charge redistribution (dynamic charge). By a group-theoretical investigation of the relation between symmetry and reststrahlen, we show that a necessary and sufficient condition for the existence of reststrahlen in an elemental crystal is a structure with at least three atoms in the primitive unit cell, s≥3. Using n_νⁱʳ for the number of infrared-active phonon frequencies (reststrahlen bands), this minimum-complexity condition states: (a) n_νⁱʳ=0↔s=1 or 2; (b) n_νⁱʳ≥1↔s≥3. To derive (a) and (b), group-theoretical arguments are used to determine the number of infrared-active phonons and phonon frequencies, and thereby the form of the effective charge tensor (∂p∂u), in terms of crystal symmetry (group characters) and unit-cell structural complexity (structure factors specifying the number of sublattices invariant under factor-group symmetry operations). The proof of (a), s≥3 as a structural requirement for a reststrahlen-displaying elemental crystal, follows from these results and the observation that all $s=2$ elemental crystals possess an inversion operation which interchanges the two sublattices; (a) is equivalent to a generalization of the Lax-Burstein argument for the vanishing first-order moment in Ge. The demonstration of (b), s≥3 as a sufficient condition for a first-order moment, is obtained by developing an inequality relating n_νⁱʳ, s, and g (the order of the factor group), and by considering the highest-symmetry crystal classes in some detail. Other applications of his approach, to compounds as well as elemental crystals, are discussed.