Nanometer-range atomic order directly recovered from resonant diffuse scattering

The method for three-dimensional imaging with an atomic resolution, based on the measurement of resonant scattering of x rays, is presented and tested on a nanoscale-range occupational ordering of niobium and magnesium ions in the lead magnesium niobate (${\mathrm{PbMg}}_{1/3}{\mathrm{Nb}}_{2/3}{\mathrm{O}}_3$) single crystal. X-ray diffuse scattering experiments performed at two wavelengths close to the absorption edge of niobium allowed us to record two $1024\times 1024\times 1024$ data sets of scattering intensities covering densely a large volume of the reciprocal space (up to $Q_{\max }=8.5{\AA }^{-1}$, with steps smaller than $\delta Q=0.05{\AA }^{-1}$). It is demonstrated that the anomalous part of the scattering intensity, including both discrete diffraction spots and diffuse scattering, can be employed to reconstruct the local atomic environment around the niobium cation up to the distance of several nanometers.

Figure 1

The intensity map of x-ray diffuse scattering from the PMN single crystal in the (0 0 $0.25a^{*}$) plane of the reciprocal space. (a) Data collected at the photon energy of 18.500 keV. (b) The anomalous portion of the x-ray diffuse scattering obtained as difference between the data taken at the photon energies of 18.500 keV and 18.971 keV. The insets show the enlarged part of the scattering intensity from the marked regions, $a^{*}=2\pi /a$ refers to the lattice parameter of PMN ($a=0.405$ nm).

Figure 2

The intensity map of x-ray diffuse scattering in the (0 0 $2.5a^{*}$) plane of the reciprocal space collected at the photon energy of 18.500 keV (a) and the anomalous portion of it obtained as difference between the data taken at the photon energies of 18.500 and 18.971 keV (b).

Figure 3

The real-space image of the average surroundings of niobium ions reconstructed from the measured $I_{\mathrm{a}}\left(\mathbf{Q}\right)$. The coordinate system is related to the reference niobium ion at $\mathbf{r}=(0,0,0)$. Top (a), the reconstructed 3D image. For clarity, lead and oxygen ions are suppressed. Middle (b), the atomic plane parallel to the (001) crystallographic plane at $z=0$. Bottom (c), reconstructed average electron density function $f\left(\mathbf{r}\right)$ along the (100) direction. The horizontal line in (c) stands for the average B-site electron density $\widehat{f_{\mathrm{B}}}\doteq 34 e{\AA }^{-3}$ obtained from the Bragg reflections only. Left side of (c) gives the color code for the data displayed in (a) and (b).

Figure 4

Absolute value of the normalized oscillatory part of the electron density at perovskite B sites around the reference niobium ion as a function of the interatomic distance. Point symbols are data obtained from the $f\left(\mathbf{r}\right)$ of Fig. 3, the continuous line is a two-exponential fit described in the text.

Figure 5

Plausible distributions of niobium and magnesium ions within a (100)-oriented atomic plane. Top (a), atomic distribution with a priori designed $\beta$-CORs (dark-shaded areas) in a stoichiometric disordered matrix. Middle (b), identical distribution but decorated with $\gamma$-CORs, determined according to the algorithm described in the text. Bottom (c), random atomic distribution generated under requirements of stoichiometry and strict absence of magnesium ion pairs at the nearest neighbor sites. The $\gamma$-CORs are determined according to the algorithm described in the text.