where m κ is the mass of atom κ, Q qν is a phonon coordinate indicating the extent of the mode's excitation, and R 0 lk is the equilibrium position of the atom. The eigenvector components W qνκα and the phase factor e iq•R 0 lk are in general complex. The requirement that atomic displacements are real-valued imposes a condition on the phonon coordinates (i.e.