Octahedral oxide glass network in ambient pressure neodymium titanate

Rare-earth titanates form very fragile liquids that can be made into glasses with useful optical properties. We investigate the atomic structure of 83TiO2-17Nd2O3 glass using pair distribution function (PDF) analysis of X-ray and neutron diffraction with double isotope substitutions for both Ti and Nd. Six total structure factors are analyzed (5 neutron + 1 X-ray) to obtain complementary sensitivities to O and Ti/Nd scattering, and an empirical potential structure refinement (EPSR) provides a structural model consistent with the experimental measurements. Glass density is estimated as 4.72(13) g cm−3, consistent with direct measurements. The EPSR model indicates nearest neighbor interactions for Ti-O at \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\overline{r}_{TiO}$$\end{document}r¯TiO = 1.984(11) Å with coordination of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n_{TiO}$$\end{document}nTiO = 5.72(6) and for Nd-O at \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\overline{r}_{NdO}$$\end{document}r¯NdO = 2.598(22) Å with coordination of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n_{NdO}$$\end{document}nNdO = 7.70(26), in reasonable agreement with neutron first order difference functions for Ti and Nd. The titanate glass network comprises a mixture of distorted Ti-O5 and Ti-O6 polyhedra connected via 71% corner-sharing and 23% edge-sharing. The O-Ti coordination environments include 15% nonbridging O-Ti1, 51% bridging O-Ti2, and 32% tricluster O-Ti3. This structure is highly unusual for oxide glasses melt-quenched at ambient pressure, as it consists of Ti-Ox predominantly in octahedral (with nearly no tetrahedral) coordination.

where is the atomic number density and is the sample volume. (Effects of sample absorption were checked and confirmed to be negligible, so all expressions here are given without absorption corrections.) The vanadium scattering, , can be expressed similarly while noting the absence of magnetic scattering and that incoherent (i.e., self) scattering is the dominant term: where , is vanadium's incoherent scattering length.
The goal of data reduction is to isolate the sample's distinct scattering, from which the total scattering structure factor can be obtained (e.g., Eqn. 15 in the main text). By defining the ratio of sample-to-vanadium scattering, and combining with Eqns. S1-S2, the sample distinct scattering is given by: Self scattering is equal to the composition-averaged square of the incoherent scattering lengths: Inelastic scattering is handled by a Placzek correction 1 , and the paramagnetic scattering is given in terms of a magnetic form factor for Nd 3+ 2,3 . However, given the small sample sizes (e.g., < 0.08), it was difficult to fully correct the weak sample scattering using these expressions. Instead, an empirical fit using a pseudo-Voigt function, , is used to account for the combination of self, inelastic, and magnetic scattering. Eqn. S4 can then be simplified to: where is a lumped, sample-dependent ( -independent) constant. The experimentally measured and empirically fitted are shown for the five ND samples in Fig. S7. Finally, the normalized total scattering structure factor, ( ), is given by: where ⟨ ⟩ is the composition-average coherent scattering length of the sample. The neutron differential PDF is then obtained using Eqn. 12, which combined with Eqn. S7 yields: is a sample-dependent constant. As described in the main text for HEXRD analysis, ( ) is a Lorch modification function and the Gudrun top hat convolution was used with ( ) to remove residual, long-wavelength background in -space. ND total PDFs were calculated according to Eqn. 13.
The normalization of ND data is now equivalent to determining the appropriate value of for each of the five ND samples. This normalization was optimized by using the theoretical limiting behavior of ( ) = 0 at low-(i.e., at real-space distances shorter than any bonds). Even with the top hat convolution, some residual background in -space results in oscillations at low-, so the most robust determination of is obtained by using as large a range of as possible. Toward this goal, for each sample, the ( ) obtained from HEXRD and ND were weighted so as to eliminate the first peak (from the partial) in an HEXRD/ND difference function. Because the HEXRD weighting factors are -dependent, this difference must be performed in -space prior to the Fourier transformation. Beginning with Eqns. 12-13 and dividing by the Ti-O weighting factor, , of each measurement, the difference function is defined as: This expression for ∆ − in principle should be zero at low-. With the Ti-O peak now eliminated, the value of (contained in the expression for in Eqn. S10) was determined by minimizing the sum-square of ∆ − over = 0-2.0 Å.

Neutron Diffraction First Order Difference Functions
For the purpose of explanation, the Ti first order difference will be described, though the same methodology applies for the Nd first order difference. The total PDF for a neutron diffraction (ND) sample can be expressed as a summation of the atomic partial pair correlations, , as given in Eqn. 14. Because the weighting factors, , are -independent for ND, Eqns. 2 and 14 can be rearranged to give: For a pair of samples that are identical in composition except for substitution of Ti isotopes, all terms in the summation of Eqn. S11 not containing or = Ti will be equivalent between samples. Thus, the difference function has only atomic-pairs containing Ti: where "1" and "2" subscripts represent the 46 Ti-nat Nd and 48 Ti-nat Nd samples, respectively. Since we are most interested in using the difference function to investigate Ti-O coordination, the pertinent function is defined as: for which the first peak will correspond to the unweighted partial . For a given atomic partial pair correlation, , the average coordination number is given by: where the integral bounds are selected to bracket the peak in the PDF. Since the and correlations do not overlap with the first peak of , the function ∆ can be used to integrate the first peak and obtain the Ti-O coordination: The same process was used to obtain the function ∆ ( ) from the nat Ti-144 Nd and nat Ti-145 Nd samples, and was determined according to Eqn. S15 and integrating over = 2.00-3.40 Å. Fig. 4 provides a plot of ∆ ( ) and ∆ ( ).

Estimating Uncertainty in Atomic Coordination Numbers
Five sources of uncertainty were evaluated during the calculation of atomic coordination numbers: diffraction measurement uncertainty, samples' composition uncertainty, 145 Nd scattering length, sample density, and ND normalization.
The measurement uncertainty arises from counting statistics (i.e., Poisson noise), which was propagated through the data reduction to obtain ( ) and through the Fourier transform to obtain the PDFs 4,5 . The contribution of measurement uncertainty was small relative to effects of density and ND normalization, and thus was not considered in the final analysis.
Uncertainty in the samples' chemical compositions leads to uncertainty in the composition-average scattering lengths: 〈 〉 for ND or 〈 〉 for HEXRD. Based on the energy dispersive spectroscopy reported in the main text, the samples' compositions were off-nominal by 1.2 mol. % Nd2O3. Assuming this value provides an upper bound for the uncertainty among different samples, the compositional variation leads to 1-2% changes in 〈 〉 2 or 〈 〉 2 , which contributes less uncertainty to the final analysis than the effects of density and ND normalization. Thus, compositional uncertainty was not considered in the final analysis.
The coherent neutron scattering length for 145 Nd has never been directly measured, and the value provided by Sears ( = 14(2) fm) 6 was deduced from the scattering lengths known for nat Nd and some of the Nd isotopes at the time, assuming that 143 Nd and 145 Nd would have the same scattering length. To probe the effect of this uncertainty, ND difference functions and EPSR were carried out using = 14 ± 2 fm for 145 Nd. The uncertainty propagated to the results (e.g., coordination numbers) was smaller than that arising from density and ND normalization, so it was not considered further. Given the scattering length uncertainty of 2/14 ≈ 14%, this at first seems surprising. However, because Nd has the lowest atomic concentration in the glass, the Nd-containing weighting factors are already quite small, which explains the small effect of 145 Nd scattering length uncertainty.
HEXRD was identified as the most reliable method for estimating density, as discussed in the main text. Density and its uncertainty were defined as the weighted mean and standard deviation of values extracted from HEXRD measurements of the six samples. This uncertainty was propagated through the analysis (e.g., ND difference functions and EPSR) to obtain the corresponding uncertainty in atomic coordination numbers.
For the ND normalization procedure described earlier in the SI, the HEXRD/ND difference function (Eqn. S10) was calculated with each of the six HEXRD measurements, yielding six values for the normalization constant, , for each sample. The effect of ND normalization uncertainty on atomic coordination numbers was determined by considering the extreme cases of different possible values. For example, the Ti ND difference function was first calculated using the mean values of for 46 Ti-nat Nd and 48 Ti-nat Nd (resulting in = 5.43). Then the ND difference was recalculated using the maximum for 46 Ti-nat Nd and the minimum for 48 Ti-nat Nd (resulting in = 5.42), and recalculated once more using the minimum for 46 Ti-nat Nd and the maximum for 48 Ti-nat Nd ( = 5.48). This approach yielded the full range of coordination numbers that might result from the ND normalization uncertainty.
ND normalization depends on the sample density, so the overall uncertainty in coordination number was defined as the maximum of the two contributions (density and ND normalization). For , the density effect was larger (uncertainty of ± 0.15 in ND difference calculation), and for the ND normalization effect was larger (uncertainty of ± 0.35 in ND difference calculation).

Atomic-Pair Partial Structure Factors from Weights Matrix Inversion
The solution to Eqn. 3 is obtained by inverting the weights matrix, which must be completed for each discrete value of the structure factors because the weighting factors for HEXRD are -dependent. To provide an example of the relative magnitudes of terms in the inverted matrix, the solution to Eqn. 3  (S16) The partial structure factors (Fig. S3) and partial PDFs (Fig. S4) were calculated using the fullydependent solution to Eqn. 3. The partial also corresponds to the Ti second order neutron difference function, which is a linear combination of the 46 Ti-nat Nd ( 1 ), 48 Ti-nat Nd ( 2 ), and null Ti-nat Nd ( 5 ) ND structure factors.      Figure S6. Comparison of (a) the structure factor and (b) the differential PDF for the nat Ti-144 Nd sample before and after the top hat convolution 7 in Gudrun 8 . The top hat convolution alters only the nonphysical oscillations at low-in the PDF. Dashed black line indicates the initial slope given by −4 . Figure S7. The ratio of ND sample scattering to scattering of the vanadium standard (blue curve, in Eqn. S6), and the empirically fitted baseline (dashed green, in Eqn. S6) that includes self, inelastic, and Nd 3+ paramagnetic scattering.   6 , and the elemental scattering lengths shown here are calculated using certificates of analysis for the isotope-enriched powders.