Analysis of the pair distribution function

Analysis of the pair distribution function, bond angle distribution and the collapsed unit cells of local structure at RT were based on the combined 18 RMC models, to improve the statistical accuracy. Initial analysis of the A site environment partial pair correlation functions showed good agreement with the long-range average structure derived from diffraction data (Table 3.2 and Figure 3.4). The A site bond lengths of the average structure along ⟨111⟩p results in three

long bonds (orange), three short (green) bonds to oxygen, and the small disordered off-axis displacement of Bi3+ forms a fourth short Bi−O bond (cyan). The five remaining Bi–O bonds fall into two groups (purple, dark blue) of intermediate distance. The local structure in the RMC model as fitted by 3 Gaussian functions, provide the three main group of bonds distances 2.237(1) Å, 2.543(8) Å, and 3.303(3) Å with ratio 3.2:4.2:4.6, which correspond to average short (green and cyan), intermediate (purple and dark blue) and long bond distances (orange) respectively. Thus, the partial pair correlation function gBi−O(r) showed a distribution of Bi−O distances similar to that observed in the averaged crystal structure.

(a) (b)

Figure 3.4. (a) 12-fold coordinated Bi3+ cation environment (purple sphere , centroid indicated as a black sphere and O coloured as red spheres) from the crystallographic R3c structure with dominant cation displacements parallel to the pseudocubic ⟨111⟩p polar axis (c direction), whereas a and b axes correspond

79 to⟨110⟩p and ⟨11 0⟩p directions respectively; The A site displacement results in three long bonds (orange), three short (green) and the small disordered off-axis displacement of Bi3+ forms a fourth short Bi−O bond (cyan). The five remaining Bi–O bonds fall into two groups (purple, dark blue) of intermediate distance; (b) partial pair distribution function gBi-O(r) in the RMC representing three group of

Bi-O bond distances.

Table 3.2. Comparison of Bi-O bond distances obtained from the long-range average rhombohedral R3c structure (bond distances split over the 3 sites, which results in the 12 unique bond lengths) and partial pair distribution function gBi-

O(r) in the RMC model. Distances for RMC configuration were obtained by fitting

Gaussian functions to the observed partial pair distributions function (Figure 3.4b). Average structure / Å Local structure (RMC model) / Å Bi-O 2.242(3) 2.282(2) 2.369(4) 2.440(5) 2.598(3) 2.604(3) 3.156(3) 3.161(2) 3.319(5) 3.401(3) 3.432(2) 3.495(3) 2.237(1) 2.543(8) 3.303(3) (3.2:4.2:4.6)*

*Approximate ratio of separations (summed to 12 oxygen coordinate system) obtained from the integrated areas of peaks in gBi−O(r).

The local bonding at the three B sites is also revealed by the partial pair distribution gB-O(r) and the instantaneous bond angle distributions gO-B-O(θ), gB-O- B(θ), derived from Gaussian function fits to the observed distributions. Comparison of the gB-O(r) distributions reveals differences between B site cations (Figure 3.5b). The gTi−O(r) has a sharp peak centered at 1.818(2) Å, with a broader feature at a higher r value of ~2.11 Å. The shorter distance of the sharp peak is more comparable to the shortest Ti−O distance in tetragonal BaTiO3 [8] (1.83 Å)

Furthermore gTi−O(r) has a breadth and shape of distribution similar to those observed for FR-PZT [10]. The longer distance indicated by the broad feature is

also comparable to the longer Ti−O distances in BaTiO3 (2.13 Å) and a number of

monoclinic phases of PZT (2.11−2.15 Å) [11, 12]. Thus, the modeled Ti−O distances agree well with expectations based on a range of polar perovskites containing titanium atoms. On the other hand, the partial gFe-O(r) and gMg-O(r) show single peaks centered at 2.021(2) and 2.060(1) Å respectively, consistent with more regular environments, where the displacement is almost exclusively along ⟨111⟩p to form three long and three short Fe/Mg−O bonds as observed in

average structure (2.111(3) and 1.960(2) Å) ( Table 3.3 and Figure 3.5a).

Table 3.3. Comparison of B site-O bond distances from the long-range average rhombohedral R3c structure and partial pair distribution function gBi-O(r)

in the RMC model.

Average structure / Å

Local structure (RMC model) / Å

B site – O 3x1.960(2) 3x2.111(3) Ti - O Fe – O Mg-O 1.818(2) 2.105(10) 2.021(2) 2.060(1) (a) (b)

81 crystallographic R3c structure with dominant cation displacements parallel to the pseudocubic ⟨111⟩p polar axis (c direction), whereas a and b axes correspond to⟨110⟩pand ⟨11 0⟩p directions respectively; (b) partial pair distribution function gB-O(r) in the RMC model.

The analysis of external bond angles B-O-B show comparable results for all Ti4+, Fe3+ and Mg2+ cations, whereas internal octahedral angles O-B-O (Table 3.4 and Figure 3.6a, b) reveals different tilting of Ti octahedra with broader distribution indicating more distorted behaviour, in comparison to Fe and Mg atoms. The average distribution of Fe and Mg bond angles are in the agreement with those derived from the average structure but not with the distribution of Ti angles, which is significantly broader.

Table 3.4. Comparison of bond angle distributions of B-O-B and O-B-O between local and average structures at RT.

Average structure angle / ° (RT) Local structure (RT) Angle / ° FWHM/ ° Ti-O-Ti Fe-O-Fe Mg-O-Mg Ti-O-Fe Ti-O-Mg Fe-O-Mg 156.1(1) 163.9(2) 157.14(5) 155.04(5) 160.29(7) 159.69(5) 156.26(3) 30.0(3) 18.6(1) 20.3(1) 24.4(2) 25.1(1) 19.9(1) O-Ti-O O-Fe-O O-Mg-O 3x80.5(1) 3x99.9(1), 89.6(1) 88.3(1), 166.1(1) 88.87(9), 180.5(5) 88.7(1), 177.6(7) 88.4(2), 178(1) 24.4(3), 34.7(9) 21.8(4), 31.0(1) 22.4(5), 31.0(1)

(a) (b)

Figure 3.6. Bond angle distributions of (a) O-B-O; (b) B-O-B for the local structure at RT.

The B site – B site cation pair correlation functions (Figure 3.7a) are consistent with gB-O(r) and bond angle distribution, where differences between Ti4+ and (Fe3+, Mg2+) cations are noticeable. The gTi-Ti(r) shows a much broader distribution of bond distances, demonstrating a wider variation in the separation of homopairs gTi-Ti(r) and heteropairs gTi-Fe(r), gTi-Mg(r) than between combinations of homopairs gFe-Fe(r), gMg-Mg(r).

(a) (b)

Figure 3.7. The partial pair distribution function of (a) gB site-B site(r) and (b) gBi- B site(r) in the RMC model.

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The analysis of partial pair correlation functions gBi−B(r) (Figure 3.7b) show comparable bond distance distribution for Fe3+ and Mg2+ cations, with defined high-r shoulders for peaks in the gBi−Fe(r) and gBi−Mg(r), in contrast to the single broad peaks observed for gBi−Ti(r). The gBi−Ti(r) does not have these features, and instead appear as broadened single peaks, which is consistent with its varied displacements. This behaviour of Ti is similar to those observed in the local structure of experimental and computational studies of PZT [13-15], where Ti4+ cations were found to be more active in off-center displacement than non- displaced zirconium B site cations.