3.1 X-ray powder diffraction

The X-ray diffraction pattern of the as-prepared material, Nd_1/3NbO₃, is shown in Fig. 2. Most of the reflections can be indexed in the orthorhombic system with the following cell parameters, a = 3.881 Å, b = 3.907 Å, c = 7.836 Å, close to the ones previously reported in the literature for this compound [9]; however, additional reflections can be noted (arrows in the figure). Further heat treatments at 1300 °C did not modify the X-ray diffraction pattern. Annealing at higher temperatures up to fusion (≈1400 °C) led to a decomposition into NdNbO₄ and Nb₂O₅. In order to understand the origin of these additional reflections, electron diffraction was then performed.

3.2 Electron diffraction

Typical electron diffraction patterns of Nd_1/3NbO₃ powder are shown in Fig. 3. Using the c* axis which can be easily identified (≈7.8 Å), Fig. 3a characterizes the [100] zone axis; tilting of about 45° around the c* axis leads to pattern (3b). To be fully interpreted, this pattern needs to double the a and b initial parameters. Obviously for instance, the reflection labeled (110) cannot be indexed without doubling a and b parameters. Thus, this pattern corresponds to [11¯0] zone axis. Starting from the pattern (3a) ([100] zone axis) and tilting of about 45° around the b* axis leads to pattern (3c) corresponding to the [101¯] zone axis. This pattern further confirms the superstructure 2a × 2b. Starting from the [11¯0] zone axis, any rotation of 45° around the c* axis gives patterns similar to Fig. 3d or a. Note that the a* and b* axes cannot be distinguished as they are metrically almost identical. The reciprocal lattice (Fig. 4) reconstructed from these patterns implies as for the X-ray data, a superstructure with the following cell parameters: a_C ≈ 7.76 Å, b_C ≈ 7.81 Å, c_C ≈ 7.84 Å. From the systematic condition of reflections h + k = 2n, which appears in the patterns, a C Bravais lattice can be deduced. Since we did not observe any additional conditions in the (0kl)* and (h0l)* reciprocal planes, the absence of glide planes orthogonal to a and b directions can be asserted.

Unfortunately it has not been possible to get information on a possible presence of a glide plane orthogonal to the c direction, as the (hk0)* reciprocal plane has not been observed.

In addition, one should stress that in most patterns, after long exposure, additional spots of weak or very weak intensity, more or less diffuse, can be evidenced, as shown in Fig. 5. The previous cell describes the main reflections but not the very weak additional ones which are observed around the main spots. These additional reflections are reproducible and present in various reciprocal planes. Due to their very weak intensity, it has not been possible to obtain additional information such as the exact location of these diffuse scattering or even their nature (spots, surface, cylinder, sphere…).

3.3 Rietveld structural refinements

On the basis of the electron diffraction study, the X-ray diffraction pattern of Nd_1/3NbO₃ was then refined using the Fullprof program. With respect to the work of Carillo et al. [15], the most symmetrical space group, i.e. Cmmm, was chosen with the initial cell parameters obtained from the previous electron diffraction study. In a first refinement, the initial atomic positions of Nd_1/3NbO₃ were deduced from those of La_1/3NbO₃ [15]. In the course of this work, Zhang et al. published the crystal structure of Nd_1/3NbO₃ based on neutron diffraction data [16]. A second refinement was therefore performed using the atomic positions (Cmmm space group). We also included a second phase, NdNbO₄, which could be identified in small amount. This resulted in a slightly improved refinement.

The peak shape was described by a pseudo-Voigt function, and the background level was fitted with linear interpolation between a set of 28 given points with refinable heights. The refined parameters include the scale factor, zero point correction, cell parameters, isotropic thermal and positional parameters for all atoms and three coefficients to describe the angular dependence of line width and asymmetry factors.

Data and details of the refinements are given in Table 2. The various reliability factors show that the refinement # 2, based on the atomic positions of Zhang et al., is improved, the final values being R_exp = 11.69; R_wp = 13.2; R_B = 4.72; χ² = 1.28.

Fig. 6 shows the excellent agreement between the calculated and observed diffraction profiles for Nd_1/3NbO₃. The atomic positions, isotropic thermal displacement and reliability factors (R_B and R_wp) of Nd_1/3NbO₃ are given in Table 3 and the interatomic distances in Table 4, only for the refinement based on Zhang atomic positions.

Additional efforts for positioning some amount of Nd in the empty plane and fitting with other space groups of lower symmetry were carried out without success. In all cases, the goodness of the fit did not improve. Finally, the amount of the impurity, NdNbO₄, was estimated to be around 1–2%.

3.4 Discussion: structural features

The structure of Nd_1/3NbO₃ is known to be of the deficient A-site perovskite type, with Nb⁵⁺ and Nd³⁺ ions located in octahedral and in dodecahedral sites, respectively. The refinement of the XRD pattern confirms the presence of empty Nd³⁺ sites in every second layer. The high-resolution image (Fig. 7) is consistent with this structure and evidences the cation ordering; the inset shows a simulated image calculated with the structural data obtained from the X-ray refinement that nicely confirms these results. This makes a framework of NbO₆ octahedra and partially occupied layers of Nd³⁺ cations at z = 0, alternating with Nd-vacant layers (z = 1/2), which causes the c-axis to be doubled (Fig. 8). At z = 0, Nd³⁺ ions occupy partially and randomly the dodecahedral sites in agreement with both refinements, the average composition being Nd_1/3NbO₃. In addition, Nb⁵⁺ atoms are slightly shifted from the octahedra center towards the empty Nd layer, which results in alternating long and short Nb–O bonds along the c-axis (cf. Nb–O₁ and Nb–O₂ in Table 4). Similarly, O₃, O₄ and O₅ atoms are slightly moved away from their ideal positions in the z = 1/4 plane due to electrostatic repulsions. Finally, the NbO₆ octahedra are somewhat distorted (see bond lengths and bond angles in Table 4), but the average Nb–O distance, 1.973 Å, is typical of Nb (V) oxides. The unit cell dimensions of a ≈ b ≈ 2a_p result from the tilting of NbO₆ octahedra around the b-axis. The superlattice reflections arising from this small rotation are very weak in the X-ray diffraction pattern, due to the much stronger scattering factors of Nb and Nd.

The very weak and more or less diffuse scattering observed in the electron diffraction patterns probably indicates some long-range ordering of neodymium atoms and vacancies. Such additional reflections have already been observed by Alario-Franco et al. [10] for the A-deficient perovskite Th_1/4NbO₃. These authors interpreted their observation by considering a modulation of the occupation of the Th crystallographic sites associated with a displacive modulation of these atoms and also of surrounding atoms. Preliminary observations indicate that the additional reflections observed in Nd_1/3NbO₃ by electron diffraction are not located at the same position as those observed for Th_1/4NbO₃, which likely results from different stoichiometry of the compounds, i.e. the cationic occupation in the central site is different and thus the modulation should be different. One should also note that, due to the weakness of these additional diffuse reflections appearing in the electron diffraction patterns, their presence in X-ray diffraction patterns is doubtful. A single crystal study should now be performed to provide a more complete description of the structure.