Phonon Modes of Molecular Crystals

Abstract

In order to account for the phonon modes of solids, the Bravais space cell is used by molecular spectroscopists to obtain the irreducible representation for the lattice vibrations. The crystallographic unit cell may be identical with the Bravais cell or it may be larger by a multiple of two, three or four. The relationship between the Bravais cell and ciystallographic cell can be obtained from the Hermann-Mauguin x-ray symbol which is used to designate the crystal symmetry. For all crystal structures designated by a symbol P (primitive), the ciystallographic unit cell and the Bravais unit cell are identical. Crystal structures designated with capital letters A, B, C, or I are doubly primitive and, therefore, the ciystallographic unit cells contain two Bravais cells. Crystal structures designated with the letters R or F are triply and quadruply primitive, respectively, and the ciystallographic unit cells contain three and four Bravais cells, respectively. Therefore, to obtain the number of molecules per Bravais space cell from the crystallographic unit cell, one simply divides the number of molecules in the crystallographic unit cell by the number of lattice points. In summary, the number of molecules in the Bravais space cell, Z’, is the number of molecules in the crystallographic unit cell (Z) divided by the number of lattice points (LP): Z’ = Z/LP, and this reduction is illustrated with the data in Table 1.