Existence of an Infinity in the Frequency Distributiong(ν)of Monatomic Body-Centered Cubic Crystals

It is shown that if a certain inequality involving interatomic force constants is satisfied, then there exists a logarithmic infinity in the frequency distribution function $g\left(\nu \right)$ of monatomic body-centered cubic crystals. Analysis of data derived from inelastic coherent scattering of slow neutrons shows that this infinity occurs for sodium, potassium, chromium, iron, and tungsten, while niobium and tantalum do not show it. Inclusion of anharmonic effects will reduce this infinity to a finite peak, the width of which is directly associated with the phonon lifetime. Possible experimental ways to observe this singularity are discussed, and it is suggested that the Mössbauer technique might provide the necessary high resolution required for such a measurement. It is conjectured that this infinity might also exist in more complex cases, such as the vibrational spectra of polyatomic crystals and the electron density of states.