Interatomic Forces and Phonon Anomalies in bcc3dTransition Metals

An expression is given for the interatomic force constants in terms of two-, three-, and four-body contributions within the tight-binding approximation. These contributions, which depend on the induced bond charge, allow a microscopic real-space interpretation of the phonon anomalies in the bcc $3d$ transition metals V, Cr, and Fe. The real-space formalism is complementary to the $\overrightarrow{q}$-space approach of Varma and Weber, but has the advantage of being directly applicable to problems such as defects which lack translational symmetry.