The average and local crystal structures of scheelite-type double molybdates and tungstates of formula NaRE(MO₄)₂ (RE = rare-earth, Y; M = Mo, W) are probed using a dual-space approach that combines Rietveld and PDF analyses of X-ray scattering data. The evolution of metal–oxygen distances as a function of the ionic radius of the rare-earth cation reveals that the crystal-chemical pictures derived from Rietveld and PDF analyses differ in their description of (1) the rigidity of M^VI–O bonds defining MO₄ tetrahedra and (2) the extent of Na/RE–O bond distance distortions in (Na/RE)O₈ dodecahedra. Analysis of the thermal ellipsoids of the oxygen atoms reveals that these differences stem from the presence of rotational disorder of MO₄ tetrahedra. Rietveld analysis averages random rotations of these tetrahedra yielding an abnormally broad range of Mo–O and W–O bond distances across the rare-earth series. By contrast, bond distances extracted from PDF analysis remain unchanged upon chemical substitution, in agreement with the well-known rigidity of M^VI–O bonds. In addition, PDF analysis shows that the rotational disorder of MO₄ tetrahedra translates into local distortions of Na/RE–O bond distances defining (Na/RE)O₈ dodecahedra. These results highlight the complementarity of Rietveld and PDF analyses and the significance of coupling both approaches to accurately probe local structural distortions through a single scattering measurement.

中文翻译：

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白钨矿型NaRE（MO ₄）_2中的旋转紊乱（RE =稀土，Y； M = Mo，W）

使用结合了Rietveld和PDF的双空间方法探查白钨矿型双钼酸盐和钨酸盐分子式NaRE（MO ₄）₂（RE =稀土，Y； M = Mo，W）的平均和局部晶体结构X射线散射数据分析。金属-氧气距离随稀土阳离子离子半径的变化而变化，表明从Rietveld和PDF分析得出的晶体化学图谱在以下方面有所不同：（1）M ^VI -O键的刚性定义MO ₄四面体和（2）（Na / RE）O ₈中Na / RE-O键距离畸变的程度十二面体。对氧原子的热椭球的分析表明，这些差异源于MO ₄四面体的旋转无序的存在。Rietveld分析将这些四面体的随机旋转平均化，从而在整个稀土系列中产生异常宽的Mo-O和W-O键距范围。相比之下，从PDF分析中提取的键距在化学取代后仍保持不变，这与众所周知的M ^VI –O键的刚性一致。另外，PDF分析表明，MO ₄四面体的旋转紊乱转化为Na / RE-O键距的局部畸变，从而定义了（Na / RE）O ₈十二面体。这些结果凸显了Rietveld和PDF分析的互补性，以及结合两种方法通过一次散射测量准确探测局部结构变形的重要性。