Neutron diffraction measurements and first-principles study of thermal motion of atoms in selectMn+1AXnand binaryM

Herein, we compare the thermal vibrations of atoms in select ternary carbides with the formula ${M}_{n+1}A{X}_{n}$ (``MAX phases,'' $M$ $=$ Ti, Cr; $A$ $=$ Al, Si, Ge; $X$ $=$ C, N) as determined from first-principles phonon calculations to those obtained from high-temperature neutron powder diffraction studies. The transition metal carbides TiC, TaC, and WC are also studied to test our methodology on simpler carbides. Good qualitative and quantitative agreement is found between predicted and experimental values for the binary carbides. For all the MAX phases studied---Ti${}_{3}$SiC${}_{2,}$ Ti${}_{3}$GeC${}_{2}$, Ti${}_{2}$AlN, Cr${}_{2}$GeC and Ti${}_{4}$AlN${}_{3}$---density functional theory calculations predict that the $A$ element vibrates with the highest amplitude and does so anisotropically with a higher amplitude within the basal plane, which is in line with earlier results from high-temperature neutron diffraction studies. In some cases, there are quantitative differences in the absolute values between the theoretical and experimental atomic displacement parameters (ADPs), such as reversal of anisotropy or a systematic offset of temperature-dependent ADPs. The mode-dependent Gr\"uneisen parameters are also computed to explore the anharmonicity in the system.