  Motivated by experiments, we performed a systematic study on the one
dimensional zone center phonons in $O_{h}$ space groups. All the one
dimensional phonon modes for different Wyckoff positions are tabulated. We show
that at least four (inequivalent) atoms (in one set of Wyckoff positions) are
needed to carry a single one dimensional phonon. A general restriction rule on
the number of atoms and the number of one dimensional phonons is obtained. The
same restriction applies to phonons of all cubic crystal systems ($T$, $T_{h}$,
$T_{d}$, $O$, $O_{h}$) and to magnons for crystals whose unitary symmetry
elements are in ($T$, $T_{h}$, $T_{d}$, $O$, $O_{h}$). Crystals with $A15$
structure are found to satisfy experimental requirements (to have a
$\Gamma_{2}^{+}$ phonon) while most crystals of $O_{h}$ space groups do not.
$A15$ is also used to demonstrate the rules we found regarding the phonon
structure in $O_{h}$ space groups.
