Phonon profiles in superconducting ${\mathrm{YNi}}_2{\mathrm{B}}_2\mathrm{C}$ and ${\mathrm{LuNi}}_2{\mathrm{B}}_2\mathrm{C}$

We performed a systematic inelastic neutron scattering, between 4 and 620 K, of the profiles of the anomalous acoustic and optic phonon observed in both compounds in the vicinity of ${\overrightarrow{\xi }}_m\cong \left(0.5\mathrm{}\hspace{0.5em}0\hspace{0.5em}0\right).$ Above $T_c$ we find that, in both compounds, the frequencies of the acoustic modes decrease with decreasing temperature and there is intensity transfer from the upper (optic) to the lower (acoustic) mode. The frequencies of the optic modes of the Y compound, on the other hand, decrease slower than those of the Lu compound as the temperature decreases towards $T_c$ and, as a result, the two branches approach each other in the Lu compound, whereas they separate in the Y compound. In both compounds the observed phonon profiles above $T_c$ can be described satisfactorily in terms of a simple couple-mode model. Comparison of the phonon spectra of the two compounds below $T_c$ demonstrates that the sharp peak at approximately 4 meV with a broad shoulder at the higher-energy side, observed in both compounds, arises from the change in the acoustic phonon profile upon entering the superconducting state.