Reflection thermal diffuse x-ray scattering for quantitative determination of phonon dispersion relations

Synchrotron reflection x-ray thermal diffuse scattering (TDS) measurements, rather than previously reported transmission TDS, are carried out at room temperature and analyzed using a formalism based upon second-order interatomic force constants and long-range Coulomb interactions to obtain quantitative determinations of MgO phonon dispersion relations $\hslash {\omega }_j$(q), phonon densities of states $g$($\hslash \omega$), and isochoric temperature-dependent vibrational heat capacities $c_v\left(T\right)$. We use MgO as a model system for investigating reflection TDS due to its harmonic behavior as well as its mechanical and dynamic stability. Resulting phonon dispersion relations and densities of states are found to be in good agreement with independent reports from inelastic neutron and x-ray scattering experiments. Temperature-dependent isochoric heat capacities $c_v\left(T\right)$, computed within the harmonic approximation from $\hslash {\omega }_j$(q) values, increase with temperature from $0.4\times {10}^{-4}\mathrm{eV}/\mathrm{atom}\mathrm{K}$ at 100 K to $1.4\times {10}^{-4}\mathrm{eV}/\mathrm{atom}\mathrm{K}$ at 200 K and $1.9\times {10}^{-4}\mathrm{eV}/\mathrm{atom}\mathrm{K}$ at 300 K, in excellent agreement with isobaric heat capacity values $c_p\left(T\right)$ between 4 and 300 K. We anticipate that the experimental approach developed here will be valuable for determining vibrational properties of heteroepitaxial thin films since the use of grazing-incidence ($\theta \lesssim {\theta }_c$, where ${\theta }_c$ is the density-dependent critical angle) allows selective tuning of x-ray penetration depths to $\lesssim 10\mathrm{nm}$.

Figure 1

MgO(011) thermal-diffuse x-ray pole figures acquired with scattering vector $q$ equal to (a) 57.82, (b) 62.61, (c) 64.42, (d) and $68.58{\mathrm{nm}}^{-1}$. The pole figures are plotted as normalized logarithmic isointensity maps over azimuthal angles $\chi =0-{80}^{\circ }$ and polar angles $\phi =0-{360}^{\circ }$. The left side of each pole figure $(180<\phi <{360}^{\circ })$ is determined using synchrotron radiation; the right side $(0<\phi <{180}^{\circ })$ is computed from theoretical MgO phonon dispersion relations obtained by fitting experimentally measured thermal-diffuse intensities, including those shown on the left side of each pole figure, to a model based on second-nearest-neighbor interatomic forces and long-range Columbic interactions. (e) An illustration defining azimuthal $\chi$, rotational $\phi$, and Bragg $2\theta$ angles with respect to the MgO crystal and an area detector.

Figure 2

Typical MgO synchrotron thermal-diffuse x-ray intensity profiles (circles), plotted as a function of $\phi$ between 0 and ${360}^{\circ }$ with $\chi =20$, 50, and ${70}^{\circ }$, obtained from the pole figure shown in Fig. 1 acquired at scattering vector $q=68.58\mathrm{n}{\mathrm{m}}^{-1}$. Simulated diffracted intensities (solid lines) calculated using MgO phonon dispersion relations obtained by fitting the measured pole-figure intensities using a model based on second-nearest-neighbor interatomic forces and long-range Columbic interaction parameters.

Figure 3

Simulated individual contributions from each phonon branch to MgO(011) thermal-diffuse pole-figure intensities acquired with scattering vector $q=68.58\mathrm{n}{\mathrm{m}}^{-1}$. The panels (a)–(f) are in order of increasing phonon energy, with (a) corresponding to the lowest acoustic phonon branch, and (f) the highest optical branch. Plots are presented as logarithmic isointensity maps over azimuthal angles $\chi =0-{80}^{\circ }$ and rotational angles $\phi =0-{360}^{\circ }$. The sum of intensities in (a)–(f) yields the diffracted thermal-diffuse intensity pole-figure plot shown Fig. 1.

Figure 4

(a) MgO phonon dispersion relations (solid lines) obtained by fitting synchrotron reflection x-ray diffraction thermal-diffuse pole-figure intensities (Figs. 1, 2) using a model based upon second-order-interatomic forces and long-range Coulomb interactions. Points from INS experiments, Ref. [41], are plotted for comparison. (b) The first Brillouin zone of MgO showing high symmetry points corresponding to $\mathrm{\Gamma }\left[000\right],X\left[100\right], L\left[\frac{1}{2}\frac{1}{2}\frac{1}{2}\right], W\left[\frac{1}{2}10\right], U\left[\frac{1}{4}\frac{1}{4}\right]$, and $K\left[\frac{3}{4}\frac{3}{4}0\right]$. Indices are in units of $2\pi /a$ in which $a$ is the MgO lattice parameter. $b_1, b_2$, and $b_3$ are reciprocal lattice vectors.

Figure 5

MgO phonon density of states $g\left(\hslash \omega \right)$ (black line, lower TDS curve) obtained by fitting synchrotron x-ray diffraction thermal-diffuse pole-figure intensities (Figs. 1, 2) using a model based on second-order-interatomic forces and long-range Coulomb interactions. For comparison, $g\left(\hslash \omega \right)$ curves from inelastic neutron scattering experiments (green line, lower-middle INS curve) [41], inelastic x-ray scattering measurements (blue line, upper-middle INX curve) [42], and density functional theory calculations (red line, upper DFT curve) [33] are also shown.

Figure 6

Temperature-dependent MgO isochoric heat capacities $c_v\left(T\right)$ (solid line) obtained by fitting synchrotron x-ray diffraction thermal-diffuse pole-figure intensities (Figs. 1, 2), obtained at room-temperature, using a model based upon second-order interatomic forces and long-range Coulomb interactions. Squares and circles are isobaric heat capacities $c_p\left(T\right)$ from Refs. [43] and [44, 45, 46], respectively.