Study of phonon dispersion relations in cuprous oxide by inelastic neutron scattering

Abstract

Phonon dispersion relations in ${\mathrm{Cu}}_{2}$O have been studied at 20\ifmmode^\circ\else\textdegree\fi{}C using inelastic neutron scattering. Seven acoustic branches and twelve optical branches have been studied in detail in the three symmetry directions [$00\ensuremath{\zeta}$], [$\ensuremath{\zeta}\ensuremath{\zeta}0$], and [$\ensuremath{\zeta}\ensuremath{\zeta}\ensuremath{\zeta}$] of the cubic lattice. Four of the six zone-center phonons have been observed and the assignments and energies are confirmed as ${\ensuremath{\Gamma}}_{25}=87\ifmmode\pm\else\textpm\fi{}2$ ${\mathrm{cm}}^{\ensuremath{-}1}$, $\ensuremath{\Gamma}_{12}^{}{}_{}{}^{\ensuremath{'}}=105\ifmmode\pm\else\textpm\fi{}3$ ${\mathrm{cm}}^{\ensuremath{-}1}$, ${\ensuremath{\Gamma}}_{15}=146\ifmmode\pm\else\textpm\fi{}1$ ${\mathrm{cm}}^{\ensuremath{-}1}$, and $\ensuremath{\Gamma}_{2}^{}{}_{}{}^{\ensuremath{'}}\ensuremath{\approx}347$ ${\mathrm{cm}}^{\ensuremath{-}1}$. The dispersion relations agree only qualitatively with the rigid-ion-model calculations. It is suggested that more detailed calculations may be performed in the light of the present measurements. The elastic constants obtained from the slopes of the acoustic branches are (units of ${10}^{11}$ dyn/${\mathrm{cm}}^{2}$) ${c}_{11}=12.61\ifmmode\pm\else\textpm\fi{}0.12$, ${c}_{44}=1.36\ifmmode\pm\else\textpm\fi{}0.04$, and ${c}_{12}=10.86\ifmmode\pm\else\textpm\fi{}1.14$, which are in close agreement with those obtained by the sound-echo technique.