The first step in the process of investigating how first-principles calculations can assist the assignment of the NMR spectra of Y2SnxTi2–xO7
involved generating a series of structural models based on an approach detailed in previous studies,73,78 but using a more recent version of CASTEP,91
to take advantage of the various advances in DFT, including the development of new, more accurate and transferrable pseudopotentials, and
the implementation of ZORA for the calculation of NMR parameters.92,93
In this approach, the NNN environment for a single cation (Y or Sn) is systemically varied, by constructing a series of models, each with a different number and arrangement of Sn and Ti on the six NNN B sites, leading to 13 unique structural models. To generate a series of Y-centred models, the Y2Sn2O7 unit cell8 was used, with the environment of one Y atom varied, as
shown in Figure 5.10a. The Sn cations on the six NNN B sites surrounding the selected Y are then systematically substituted for Ti, leading to a set of structures with different numbers of Sn NNN and arrangements of these. This approach results in 13 unique structures based on Y2Sn2O7 as shown in
Figure 5.10b, (a series subsequently referred to as Y-centred, Sn-rich models). This process was repeated using the Y2Ti2O7 unit cell,90 leading to 13 Y-
centred Ti-rich models, and a total of 26 structural models.
An equivalent series of Sn-rich and Ti-rich Sn-centred structural models were generated, again starting from either Y2Sn2O7 or Y2Ti2O7, but now
systematically varying the NNN environment of a selected Sn cation. As Ti occupies all the B sites in Y2Ti2O7, the first step in constructing the Ti-rich
models, involved an initial substitution of one Ti for Sn, before the NNN environment of this Sn species was then systematically modified. In the Y- centred clusters the six NNN B sites are arranged in a hexagon around the Y (i.e., all lie in a similar structural plane), making it straightforward to represent the NNN arrangements using a two-dimensional projection, as shown in Figure 5.10b. As the six NNN B sites in the Sn-centred cluster form a triangular antiprism, with three cations above and three cations below the
Figure 5.10: (a) Schematic showing the modification of the NNN environment of a single Y species in Y2Sn2O7 to give a Y-centred cluster within the final unit cell. (b) Two-dimensional projection of the 13 unique ways of arranging Sn and Ti cations on the NNN B sites surrounding Y.
central Sn, each in a triangular arrangement (see Figure 5.2b), the Sn-centred cluster must be orientated to show the six B site cations in the same two- dimensional representation as seen for the Y-centred clusters. This involves aligning the Sn-centred cluster along the three-fold rotation axis. As shown in Figure 5.11, reorientation causes the six NNN B site cations to appear in a hexagonal arrangement around the central Sn, allowing the same numbering system, i.e., 1,2,3-, 1,2,4- or 1,3,5-Sn3Ti3, to be used for both Y-centred and Sn- centred cluster models. However, it is important to note that in the two- dimensional representation of the 1,3,5-Sn3Ti3 arrangement shown in Figure 5.11, the cations on adjacent NNN positions, e.g., position 1 and position 2, are actually on different faces of the triangular antiprism, with the cations on positions 3 and 5 on the same face as the cation on position 1. Figure 5.12 shows how the bond angle between two cations on the NNN B sites, joined through the central atom (Y or Sn) differs depending on whether a Y-centred or an Sn-centred cluster is being considered.
Figure 5.11: Schematic showing how an Sn-centred cluster is orientated such that the same NNN nomenclature for the Y-centred models can be used for the Sn-centred models.
Figure 5.12: The local environment of (a) a Y-centred and (b) a Sn-centred cluster model, with a two-dimensional representation of the NNN B site arrangement, showing the bond angles linking cations on positions 1 and 2, and positions 1 and 5 (same as the angle between positions 1 and 3) to one another in three-dimensions, i.e., through the central atom (note the central cation (Y or Sn) is not shown). In (b) the three cations on positions 2, 4 and 6 have been faded to represent them being arranged below the central Sn, whereas the three cations on positions 1, 3 and 5 are arranged above the central cation.
In summary, by manually modifying the Y2Sn2O7 and Y2Ti2O7 structures, a
series of structural models designed to investigate how the systematic change in n Sn NNN environment effects the calculated NMR parameters. These models consist of 13 Y-centred Sn-rich, 13 Y-centred Ti-rich, 13 Sn- centred Sn-rich and 13 Sn-centred Ti-rich structures, giving a total of 52 unique models, each of which can be investigated separately.
All first-principles calculations were performed using CASTEP 8.0, with an Ecut of 60 Ry, a k-point spacing of 0.04 2π Å–1 (see Appendix A1 for
convergence testing), the PBE EXC functional,94 default on-the-fly ultrasoft
pseudopotentials,95 a geom_energy_tol value of 1 × 10–4 eV / atom and an elec_energy_tol value of 1 × 10–5 eV / atom used (see Table 3.5 for a
definition of geom_energy_tol and elec_energy_tol, which are also
discussed in detail in Section 5.5.1.4). The CASTEP 8.0 convergence criteria used were 5 × 10–2
eV / Å, 1 × 10–3
Å and 1 × 10–1
GPa for the maximum forces, atomic displacement and stress, respectively. All structural models were geometry optimised to reduce the forces acting upon the atoms, with the subsequent calculations of NMR parameters carried out using ZORA correction. As described in Chapter 3, a reference shielding (σref) was used to
convert calculated isotropic shieldings (σiso) to isotropic shifts (δiso). For 89Y a
comparison of σiso and δiso for the two Y sites in Y2O3 gave a σref of 2705.64
ppm. For 119Sn a comparison of σ
iso and δiso for the single Sn site in Y2Sn2O7
gave a σref of 2601.93 ppm. However, in order to provide better agreement
with experimental NMR measurements, the calculated 89Y δ
iso and Ω values
were scaled by comparing the calculated and experimental NMR parameters for Y2Sn2O7 and Y2Ti2O7, i.e., the two compositional end-members of the 26 Y-
centred cluster models, as described in Appendix A2. Calculated NMR parameters were analysed using in-house Python scripts extending the CCP-
NC MagresPython module.96
The calculated 89
Y δiso for all Y cations in each Y-centred cluster models are
plotted as a function of n Sn NNN in Figure 5.13a, with the points coloured
according to model set, i.e., Sn-rich or Ti-rich structural models in Figure 5.13b. Figure 5.13 indicates that the incorporation of Ti, i.e., a decrease in n Sn NNN, is accompanied by a noticeable (~15 ppm) upfield shift. There is also little overlap between calculated 89
Y δiso for Y with different n Sn NNN, in
agreement with previous first-principles calculations73 and supporting the
initial assignment of the 89Y MAS NMR spectra of Y
2SnxTi2–xO7.71 While there
is a generally clear, systematic change in calculated 89Y δ
Figure 5.13: Plot of (a) calculated 89Y δ
iso as a function of n Sn NNN for the 26 Y-centred cluster models of Y2SnxTi2–xO7, and (b) with the data points coloured by the model type.
Figure 5.14: Plot of calculated 89Y δ
iso as a function of n Sn NNN for the 26 Y-centred cluster models of Y2SnxTi2–xO7, with the type of NNN environment indicated. For n Sn NNN = 4, 3 and 2, data points for the different arrangements have been vertically offset to facilitate comparison.
Figure 5.13 shows several outlying points, for Y with Sn3Ti3 or Sn2Ti4 NNN environments. Figure 5.14 shows calculated 89Y δ
iso plotted as a function of n
Sn NNN, with the data points coloured by n and shaped according to the specific NNN arrangements (see Figure 5.10b), with the outlying points corresponding to Y with either a 1,2,4-Sn3Ti3 or a 4,6-Sn2Ti4 arrangement.
Figure 5.15: Plot of calculated 89Y δ
iso as a function of (a) mean Y–O8a bond length, (b) mean Y–O48f bond length and (c) deviation of theO8a–Y–O8a bond angle from 180º, for the 26 Y- centred cluster models of Y2SnxTi2–xO7, with data points coloured by n Sn NNN.
Figure 5.15a shows that a systematic decrease in both calculated 89Y δ
iso and
mean Y–O8a bond length results from the successive substitution of Sn for Ti.
The data points for Y species with unexpectedly high 89Y δ
iso have very,
similar mean Y–O8a bond lengths to other Y with the same n Sn NNN
indicating that this is determined primarily by the nature of the species occupying the six NNN B sites. When the calculated 89Y δ
iso is plotted against
the corresponding mean Y–O48f bond length, as shown in Figure 5.15b, there
seems to be a general increase in mean bond length with increasing Ti incorporation. It also appears that the data points in Figure 5.15b can be split
Figure 5.16: Plot of (a) calculated 89Y Ω as a function of n Sn NNN for 26 Y-centred cluster models of Y2SnxTi2–xO7, with data points (b) coloured by model type and (c) shaped according to NNN cation arrangement. For n Sn NNN = 4, 3 and 2 in (c) the data points for the different NNN arrangements have been offset slightly to facilitate comparison.
Figure 5.17: Plot of calculated 89Y Ω against calculated 89Y δ
iso for the 26 Y-centred cluster models of Y2SnxTi2–xO7, with the data points coloured by n Sn NNN.
into two separate series, corresponding to the Sn-rich and Ti-rich structural models, respectively, with the latter exhibiting the shorter mean Y–O48f bond
lengths irrespective of the n Sn NNN. In addition, Y with unusually high 89Y
δiso have noticeably longer mean Y–O48f bonds compared to Y with the same n
Sn NNN. As shown in Figure 5.15c, the strongest correlation seems to be between calculated 89
Y δiso and the deviation in the O8a–Y–O8a bond angle
from 180º, the angle in an ideal pyrochlore structure. From Figure 5.16a, it is apparent that as seen for 89Y δ
iso, as the number of Sn
NNN decreases, 89Y Ω increases, with Sn-rich structural models yielding the
Figure 5.18: Plot of calculated 89Y Ω as a function of (a) mean Y–O8a bond length, (b) mean Y– O48f bond length and (c) deviation of the O8a–Y–O8a bond angle from 180º, for the 26 Y- centred cluster models of Y2SnxTi2–xO7, with the data points coloured by n Sn NNN.
largest values, particularly apparent for the Sn3Ti3 NNN environments, as
shown in Figure 5.16b. Environment with n Sn NNN have a reasonably well-
defined predicted range of 89Y Ω, with this parameter appearing to be a
reasonably sensitive probe of changes in the NNN environment. However, as shown in Figure 5.16c, calculated 89Y Ω for different arrangements of cations
for a particular value of n, i.e., 1,2-Sn4Ti2, 1,3-Sn4Ti2 and 1,4-Sn4Ti2, are too similar to allow them to be separated using this NMR parameter alone. As shown in Figure 5.17, when both the calculated 89Y δ
iso and Ω are considered
simultaneously, it can be seen that each environment with n Sn NNN
generally has a reasonably well-defined range for both NMR parameters, although several Y have noticeably high 89Y δ
iso, despite 89Y Ω being well
within the expected range for that particular n Sn NNN environment.
The relationship between calculated 89
Y Ω and local geometrical parameters
for the 26 Y-centred cluster models have also been investigated, as shown in Figure 5.18, with Figure 5.18a highlighting a very strong correlation between mean Y–O8a bond length and 89Y Ω. Figure 5.18b shows that irrespective of n
Sn NNN, Y in Sn-rich models have longer mean Y–O48f bond lengths,
resulting in an increase in 89Y Ω. In contrast to 89Y δ
iso, Figure 5.18c indicates
that there is no strong correlation between the O8a–Y–O8a bond angle
deviation from 180º and 89
Y Ω, with even significant angle variations of up to ~10º having little effect on Ω. This suggests that the magnitude of 89Y Ω is
dominated by n, with the deviation in O8a–Y–O8a bond angle having little
effect on this NMR parameter. Something evident in all three of the plots in Figure 5.18 is that as n decreases, the overlap in 89Y Ω for neighbouring
environments increases, with the largest range of 89
Y Ω seen for the Sn2Ti4, SnTi5 and Ti6 environments.
How the predicted 119
Sn NMR parameters for the 26 Sn-centred cluster models vary with local geometry has also been investigated. When calculated 119Sn δ
iso is plotted as a function of n Sn NNN (see Figure 5.19), it is
apparent that as NNN Ti content increases, the range of 119Sn δ
Figure 5.19: Plot of (a) calculated 119Sn δ
iso as a function of n Sn NNN for the 26 Sn-centred cluster models of Y2SnxTi2–xO7, with the data points (b) coloured by model type and (c) shaped by NNN arrangement. For n Sn NNN = 4, 3 and 2 in (c) the data points for the different arrangements have been offset slightly, to facilitate comparison.
different n begin to converge, with significant overlap seen for all but the n = 6 (Sn6) environment. This significant overlap in 119
Sn δiso for species with
different n Sn NNN is generally consistent with the appearance of the 119
Sn MAS NMR spectra of Y2SnxTi2–xO7, shown in Figure 5.8, which are dominated
by a maximum of three, very broad resonances. This perhaps indicates that in comparison to 89Y δ
iso, the 119Sn δiso is relatively more strongly affected by
the unit cell contraction caused by Ti incorporation, i.e., the upfield shift associated with the incorporation of subsequent Ti is roughly balanced by the downfield shift caused by the unit cell contraction. When, as shown in
Figure 5.19c, the data points are separated according to specific NNN arrangements, it is apparent that, in general, ranges of 119
Sn δiso for
environments that have the same n but different cation arrangements, i.e., 1,2-Sn4Ti2, 1,3-Sn4Ti2 and 1,4-Sn4Ti2, overlap significantly. The only exception to this seems to be is the 1,3,5-Sn3Ti3 arrangement, where the three NNN Sn cations occupy the same face of the Sn-centred antiprism, with the three Ti occupying the opposite face (see Figure 5.11), leading to an increase in 119Sn δ
iso, over that seen for 1,2,3- or 1,2,4-Sn3Ti3 environments. It should,
however, be noted that there are a relatively small number of structural models considered for some environment types, which may pose a particular problem when defining a typical range for a parameter. The extensive overlap in calculated 119Sn δ
iso for Sn-centred structural models indicates that
it will not be possible to unambiguously assign resonances in the 119Sn MAS
NMR spectra for many Y2SnxTi2–xO7 compositions.
The relationship between calculated 119
Sn δiso and the mean Sn–O48f bond
Figure 5.20: Plot of calculated 119Sn δiso as a function of mean Sn–O48f bond length for the 26 Sn-centred cluster models of Y2SnxTi2–xO7, with the data points coloured by (a) model type
Figure 5.21: Plot of calculated 119Sn Ω as a function of n Sn NNN for the 26 Sn-centred cluster models of Y2SnxTi2–xO7, with (b) the data points coloured by model type.
Figure 5.22: Plot of calculated 119Sn Ω against calculated 119Sn δ
iso for the 26 Sn-centred cluster models of Y2SnxTi2–xO7, with the data points coloured by n Sn NNN.
length has also been considered, with Figure 5.20a showing that the mean Sn–O48f bond length is significantly different in the Sn-rich and Ti-rich
structural models, with the latter having much shorter bond lengths. Figure
5.20b indicates that within each model type, the mean Sn–O48f bond lengths
are very similar for all environments with the same n Sn NNN. Figure 5.21
shows that there is an increase in 119
Sn Ω with decreasing n, but that (unlike for 89Y, as shown in Figure 5.16), the range of Ω values predicted also
environments with different values of n. Indeed, there appears to be more overlap in 119
Sn Ω than in 119
Sn δiso, as shown in Figure 5.19, meaning that this
anisotropic NMR parameter is less sensitive to small variations in local geometry than the isotropic component of the shielding tensor. As shown in Figure 5.22, considering the combination of calculated 119Sn δ
iso and Ω values
is not sufficient to separate Sn species according to specific n Sn NNN environments, supporting the assumption that the broad resonances seen in many of the 119Sn MAS NMR spectra of Y
2SnxTi2–xO7 (see Figure 5.8) result
from overlap between peaks from Sn species with different NNN B-site environments.
The Y- and Sn-centred cluster models provide good insight into the relationship between predicted NMR parameters and changes in local geometry, helping to explain many of the observations made from the 89Y
and 119Sn MAS NMR spectra of Y
2SnxTi2–xO7. For example, the calculated 89Y
δiso for the Y-centred structural models show a systematic decrease in shift
with increasing Ti content on the NNN B site. This confirms that the
additional resonances upfield of the peak seen in the 89Y MAS NMR
spectrum of Y2Sn2O7, which appear when Ti is introduced, can be attributed
to Y with different n Sn NNNs, generally supporting the initial assignment of the 89
Y MAS NMR spectra for Y2SnxTi2–xO7. 71
It was also shown that the small number of Y species that seemed to have anomalously high δiso, have
noticeably longer mean Y–O48f bond lengths and a much more distorted O8a–
Y–O8a bond angle than other Y with the same n Sn NNN. The decrease in n