The technique of high resolution Brillouin spectroscopy has been used to determine the adiabatic elastic constants of single crystals of β-N2 and β-CO as a function of temperature, in an effort to study the α–β phase transition. For all elastic constants, there is an increase of approximately 1%/K on cooling and there is no evidence of further mode softening, even within 0.5 K of the phase transition. Three large single crystals of α-CO were grown. The orientations were determined, by Laue X-ray diffraction, and correlated to the orientations of the parent β-crystals. The β-phase hexagonal basal planes appear to transform to planes of the form {110} in the α-phase cubic structure.