Distorted Oxygen Environments of Titanium in Phosphate Crystals and Glasses – A Discussion of Scattering Results

1. Introduction

Some titano-phosphates—such as the potassium titano-phosphate crystal—have attracted considerable interest for optical applications due to their large nonlinear refractive indices [1]. This property was assigned to the distortions of the oxygen environments of Ti. In the KTiO(PO₄) crystal, some Ti–O–Ti bridges are out of balance and consist of a short and a long bond [2]. This difference was interpreted using the second-order Jahn-Teller (SOJT) theorem [1]. More generally, the distortions of the MO₆ octahedra (M – transition metal with a d⁰ configuration) lowering its symmetry were described as out-of-center shifts of the M sites [3]. The empty d orbitals of M mix with occupied p orbitals of the oxygen ligands, and the shifts of M avoid the nearly degenerate states. The smaller the energy gap between the orbitals, the stronger the tendency to the out-of-center shift of M. V⁵⁺ and Mo⁶⁺ were classified as strong distorters, which never occur in undistorted MO₆ octahedra [3,4]. The moderate distorters Ti⁴⁺, Nb⁵⁺, and Ta⁵⁺, and W⁶⁺ can form undistorted MO₆ octahedra, but unbalanced outer forces cause an out-of-center shift of M against only slight resistance. The shift of an M is directed to octahedral corners, edges, or faces, accompanied by characteristic changes in the M–O bond lengths (see Figure S1). For example, the oxygens of the TiO₆ in crystalline PbTiO₃ [5] form a nearly regular octahedron, but the Ti⁴⁺ is shifted by 0.033 nm toward one of the corners.

It is known that Ti⁴⁺ cations form TiO₄ or TiO₅ units in addition to the TiO₆ units, for example, in silicate glasses [6]. The ratios of the ionic radii for a given atom A surrounded by spherical oxygen ligands determine which polyhedra are preferably formed [7]. Ratios r_A/r_O of 0.414 or 0.225 correspond to exactly filling the gaps within octahedra or tetrahedra. The corresponding ratios for Ti⁴⁺ and O^2- are 0.45 and 0.31, respectively, as calculated from the ionic radii [8], indicating a preference for octahedral units. The slightly smaller Ge⁴⁺ has ratios of 0.39 and 0.29. GeO₄ tetrahedra are preferred, and vitreous GeO₂ consists of a continuous network of such units, but GeO₆ octahedra are also known in several crystals and glasses. The Ti-centered oxygen polyhedra could differ in silicate and phosphate environments. Si–O bonds in SiO₄ units possess bond valences of about 1.0 v.u. (valence units), while the P–O bonds in bridges with neighboring M have larger bond valences [9], which vary with the connectivity of the phosphate network and local constraints (see Figure S1). Hence, Ti–O bonds of bond valences <1.0 v.u. occur and favor TiO₅ or TiO₆ units.

If large ranges of scattering vector magnitude (Q) are used in scattering experiments on glasses, then details about different bond lengths, in addition to the mean coordination numbers, are obtained. Such experiments on TiO₂-containing glasses were performed on silicate and phosphate systems [10,11,12,13,14,15]. The scattering contrast of Ti atoms is changed by isotopic substitution in neutron scattering experiments [10,11,12]. Another profitable approach is the contrast change when combining X-ray and neutron scattering data [14,15].

The study of the Ti environments in glasses is restricted to distributions of Ti–O pair distances. Detailed geometries of the Ti-centered oxygen polyhedra, as available for crystals, are not accessible for glass structures. The total Ti–O coordination number gives the average of the frequencies of the TiO₄, TiO₅, and TiO₆ units. The known Ti–O distances of the polyhedra of related crystals support the analysis. Considering the crystals' coordination polyhedra indicates a problem. Well-defined bond lengths for the structural groups are not observed due to the strong TiO₆ distortions. The empirical relationship between bond valence and bond length, as introduced by Brown & Altermatt [16], is a helpful tool. Here, more recent parameters for the Ti–O distances are used [17].

In this work, our previous scattering results for TiO₂-P₂O₅ glasses with minor Al₂O₃ impurities [15] are re-examined and partially modified. This is necessary for the subsequent comparison with recent results on binary TiO₂-P₂O₅ glasses [14]. Although slight interactions between the TiOₓ and AlO₄ units for the glasses in [15] cannot be ruled out, it is useful to consider these data together. They complement each other in the TiO₂/P₂O₅ ratio. Small Al₂O₃ contents extend the range of glass formation. Furthermore, the out-of-center shifts of the Ti positions within their oxygen octahedra, as predicted for this d⁰ transition element by SOJT, were not discussed for glass structures at that time [15]. The comparison with the GeO₂-P₂O₅ glasses [18], Ge is a main-group element with fully occupied d orbitals, highlights the role of the SOJT effects.

2. Materials and Methods

The experiments have been described in [15]. Here, only the essential parameters of the experiments are repeated. The quasi-binary titano-phosphate glasses have compositions 54TiO₂-40P₂O₅-6Al₂O₃ and 61TiO₂-35P₂O₅-4Al₂O₃. The samples contain sufficient oxygen to avoid P–O–P bridges. AlO₄ tetrahedra are expected in the presence of isolated PO₄. To a good approximation, both samples are quasi-binary glasses with TiO₂ contents of 61.4 and 66.3 mol%, respectively, after subtracting a suitable AlPO₄ content. The samples are labelled TP60 and TP65 as in [15]. Atomic number densities ρ_T of 74.1 and 74.4 atoms/nm³ were calculated from the mass densities. X-ray diffraction was performed at the BW5 beamline of the DORIS III synchrotron at the German Electron Synchrotron (Hamburg), and neutron diffraction at the GEM instrument of the ISIS spallation source at the Rutherford Appleton Laboratory (Chilton, UK). The upper limits of the magnitude of scattering vector, Q_max, are 280 nm^–1 (X-rays) and 500 nm^–1 (neutrons), where Q is calculated from the scattering angle, 2θ, and the radiation wavelength, λ by $Q=(4\pi /\lambda )\sin \theta$.

3. Results

The static structure factors S(Q) were obtained from the normalized scattering intensities as shown in [15]. The short-range order parameters of the glasses are determined from the real-space correlation functions T(r), which are obtained from S(Q) by Fourier transformation with

$$T\left(r\right)=4\mathsf{\pi }r{\rho }_{\mathrm{T}}+2/\mathsf{\pi }{\int }_0^{Q_{\mathrm{m}\mathrm{a}\mathrm{x}}}Q\left[S\left(Q\right)-1\right]\sin \left(Qr\right)\mathrm{d}Q$$

(1)

Different from [15] and in accordance with [14], damping is not used. The values of Q_max used for the X-ray and neutron scattering data are 250 and 360 nm^–1, respectively, to exclude noisy scattering intensities at high Q-ranges. The T(r) curves are shown in Figure 1.

The Q_max limits in the Fourier integral (Equation 1) cause a peak broadening and termination ripples. These effects are taken into account during parameter fitting according to the convolution method of Mozzi & Warren [19,20]. Fits were made simultaneously to the X-ray and neutron T(r) functions, which helps separate the Al–O, Ti–O, and O–O correlations in the distance range 0.17 < r < 0.25 nm. The Ti–O weighting factors change strongly for the X-ray and neutron correlations. The peak of the Ti–O distances appears as a minimum (cf. Figure 1b) due to the negative neutron scattering length of Ti in natural isotopic abundance. Compared to X-ray scattering, the O–O correlation is significantly enhanced in neutron scattering.

Gaussian functions approximate the distance peaks. Their parameters are the coordination number N_ij, the distance r_ij, and the full width at half maximum (fwhm) Δr_ij. The Δr_ij of a Gaussian is related to the root of the mean square deviation σ_ij by Δr_ij = 2.335 σ_ij. A single Gaussian function is used for the P–O peak, corresponding to the four equivalent bonds in isolated PO₄ tetrahedra. Only these units have been detected by ³¹P-NMR spectroscopy [21]. A next small single Gaussian approximates the Al–O peak. Three Gaussians are needed to model the Ti–O first-neighbor distances. The shortest O–O distances at ~0.251 nm can be assigned to the edges of the PO₄ units. The edge lengths of the AlO₄ and TiO₆ units are only slightly longer than those of the PO₄. Thus, the next Gaussian in the O–O correlation overlaps with the first one. Different from [15], the N_OO value of the first Gaussian is fixed to 3.1, as reported for the X-ray data in [14]. The use of identical N_OO parameters in the T(r) model peaks for the X-ray and neutron scattering data requires the assumption of a small fraction of Ti–O distances near 0.24 nm. The Gaussians at ~0.192 and ~0.214 nm describe the main part of the Ti–O peak. The parameters obtained from the fits are listed in Table 1. The large peak in the X-ray T(r) function at ~0.33 nm is approximated by four P–Ti and three Ti–Ti pairs.

The TiO₂ and Al₂O₃ components of both samples provide sufficient oxygen for all PO₄ to exist as isolated units. The parameters of the Al–O peaks are close to those known for crystalline AlPO₄ with Al–O distances of 0.174 nm and N_AlO = 4 [22]. There, all oxygens are in Al–O–P bridges. The parameters of the Ti–O peaks can be compared with those of the related crystals Ti₅O₄(PO₄)₄ [23], Ti₂O(PO₄)₂ [24], Ti(P₂O₇) [25], and TiO₂ [26] (Brookite). The Ti₅O₄(PO₄)₄ structure shows the most strongly distorted TeO₆ units with Ti–O bond lengths ranging from 0.174 to 0.26 nm. The total N_TiO values of the glasses are 6.0 ± 0.2 (TP60) and 5.7 ± 0.2 (TP65). The first O–O peak at 0.252 nm with N_OO = 3.1 exceeds the calculated coordination numbers of 2.94 (TP60) and 2.72 (TP65), assuming that only the oxygen edges of the PO₄ units contribute to this peak. Thus, the edges of other oxygen polyhedra are involved in this first Gaussian component of the O–O correlation.

4. Discussion

4.1. Total Ti–O Coordination Number and Density Behavior

In this chapter, the Ti phosphate glasses with small Al₂O₃ impurities [15] are considered quasi-binary glasses, then with TiO₂ fractions of 61.4 (TP60) and 66.3 (TP65) mol%. The Al₂O₃ is assumed to be bound in AlPO₄-like structures. Regarding the P/O ratio, AlPO₄ corresponds to a 60TiO₂-40P₂O₅ glass, which is close to that of sample TP60. Figure 2a compares the total Ti–O coordination numbers. To ensure comparability with the N_TiO values from [14], distances ≥0.24 nm are also disregarded for our values (cf. Table 1). The latter values of TP60 and TP65 are given as small circles. The N_GeO values for binary GeO₂-P₂O₅ glasses [18], the GeP₂O₇ [27], and Ge₅O(PO₄)₆ [28] crystals are added for comparison. They exemplify the behavior of a main-group A⁴⁺ cation of similar size.

The coordination behavior of Ti and Ge differs significantly, although single Ti–O bonds (0.182 nm) are only slightly longer than single Ge–O bonds (0.175 nm) [13,17,18]. The N_GeO values follow a curve called M_TO, given by $M_{\mathrm{T}\mathrm{O}}=2\left(x+1\right)/x$, until the range of glass formation ends at x = ~0.8. A change in N_GeO according to M_TO indicates a network structure with exclusively doubly coordinated oxygens. This behavior is also realized in the crystals TiP₂O₇ [24] and Ti₂O(PO₄)₂ [25]. For x < 0.7, the N_TiO values of the TiO₂-P₂O₅ glasses show a similar trend, but with a significant upward offset. Finally, the N_TiO values start to increase for x > 0.7. The relation N_TiO > M_TO indicates fractions of triply coordinated oxygen, thus, oxygen with either three Ti neighbors or one P and two Ti neighbors. A model for the evolution of the corresponding oxygen fractions is detailed in [14]. It is noteworthy that the number of short Ti–O distances in the glass sample TP60 (5.2 at 0.192 nm) corresponds well with the respective M_TO value (circle with hatched fill in Figure 2a). In other words, the oxygens in the glassy network of TP60 seem almost doubly coordinated if only the strong bonds are considered. Is Zachariasen’s rule [29] on the exclusion of triply coordinated oxygen in oxide glass networks violated only by elongated bonds? It is unclear whether this also applies to the glasses with higher TiO₂ contents. Analyzing this effect is difficult for the other samples. For sample TP65, the number of short Ti–O bonds is lower than the M_TO value. However, there is a significant overlap with the second distance component, as is also the case for the Ti–O peaks of the samples reported in [14].

Figure 2b shows the evolution of the number densities of the titano-phosphate materials. The number densities of atoms in crystals with constant N_TiO = 6 increase with increasing x, which is attributed to an increasing fraction of triply coordinated oxygen and exchange of tetrahedral PO₄ for octahedral TiO₆ units. On the other hand, if the networks avoid triply coordinated oxygen, then TiO₆ units change to TiO₄, and the number densities decrease. Apart from a definite offset, the ρ_T values of the glasses follow the density trend of those crystals, which are accompanied by fractions of triply coordinated oxygen. Evidently, the number of terminal oxygen atoms (M_TO) per Ti does not play a decisive role in the structure of the TiO₂-P₂O₅ glasses. The opposite was established for the GeO₂-P₂O₅ glasses (cf. Figure 2a) [18].

4.2. Details of the Ti–O Bond Lengths

The scattering data are of sufficient quality to discuss detailed shapes of the Ti–O distance peaks. The Ti–O peaks in Figure 3a are calculated using the parameters obtained from the fits (Table 1 in this work and Table VII in [14]). In these comparisons, the radial distribution function (RDF) is used with RDF(r) = r∙T(r). Then, the peak area is related to the number of atomic neighbors. For the crystal structures in Figure 3b, Δr_TiO values are chosen similar to those of the glasses. Two effects are visible. The mixture of TiO₄ and TiO₆ units in Ti₂O(PO₄)₂ [24] causes shorter bond lengths than in TiP₂O₇ [25], where all Ti are exclusively located in TiO₆ units. Both crystals possess only doubly coordinated oxygen. The other two crystal structures show significant components of lengths r_TiO > 0.210 nm. The brookite [26] shows only a small effect. All oxygens are triply coordinated. However, the TiO₆ octahedra are connected asymmetrically along their edges, causing the Ti position to shift slightly away from the TiO₆ center [3]. On the other hand, only one of the five Ti positions in the Ti₅O₄(PO₄)₄ crystal [23] has six equal Ti–O bond lengths. The other four TiO₆ units exhibit strong 1+4+1 deformations (Ti shifts toward a corner - one short, four medium, and one long bond). Three of these Ti sites have an occupancy of only 90%. The remaining Ti atoms occupy even more distorted octahedral sites. Since the latter sites cause only a marginal change, they have been neglected when calculating the Ti–O peak.

The origin of the strong TiO₆ distortions in crystalline Ti₅O₄(PO₄)₄ lies in a mixture of 70% doubly coordinated and 30% triply coordinated oxygens, as emphasized in [14]. As calculated by the sophisticated structural model for TiO₂-P₂O₅ glasses [14], the fraction of triply coordinated oxygens increases with the TiO₂ content at the expense of doubly coordinated sites. In this process, the number of triply coordinated Ti–O–Ti₂ sites increases at the expense of P–O–Ti₂. This change results in a decrease in the first peak component, accompanied by an increase in the number of longer distances, as is evident in Figure 3a.

At this point, a closer look at the Ti₅O₄(PO₄)₄ crystal [23] is useful, in which a relationship between bond length and bond valence (bv) is employed. It is given by the formula $r_{\mathrm{T}\mathrm{i}\mathrm{O}}=R_0-\mathrm{l}\mathrm{n}\left(bv\right)*B$, with R₀ = 0.1819 nm and B = 0.0342 nm [17]. The Ti–O bonds exist in four different configurations. The oxygen atoms occupy bridging Ti–O–Ti and P–O–Ti sites or triply linked Ti–O–Ti₂ and P–O–Ti₂ sites. For simplicity, all P–O bond valences of isolated PO₄ could be fixed to 1.25 v.u. However, the valence of the P–O bond in P–O–Ti₂ has 1.18 v.u. as calculated from the distances. This value is compensated with 1.273 v.u. in the P–O–Ti bridges. For each O site, the remaining bond valence is assumed to be distributed equally among the Ti–O bonds. Figure S2 shows the dependence of bond valences on the length of the Ti–O bond. The four types of Ti–O bonds (in the order as listed above) have bond valences of 1.0, 0.73, 0.67, and 0.41 v.u., which leads to Ti–O distances of 0.182, 0.192, 0.196, and 0.212 nm, respectively. A Ti–O peak is calculated using these lengths and the respective proportions of the crystal’s bond types. In Figure 3b, this peak (dotted line) is compared with the broader peak (solid line) calculated from the detailed distances in the crystal. Evidently, knowledge of the nature of the neighboring oxygen sites is insufficient to predict the real distribution of Ti–O distances.

Hence, additional distortions are effective in the Ti₅O₄(PO₄)₄ crystal [23]. Strong variations in bond lengths from 0.173 to 0.240 nm occur at oxygen atoms with three Ti neighbors. The variations are large in Ti–O–Ti bridges, while only varying within the limits of the widths Δr_TiO used to calculate the Ti–O distances to corners of PO₄ units. The Ti⁴⁺ cations in TiO₂-P₂O₅ glasses have heterogeneous environments with doubly and triply coordinated oxygens, whereby a P⁵⁺ as a second neighbor causes much stronger repulsions than a Ti⁴⁺. As pointed out in the Introduction chapter, the SOJT effect is of crucial importance [1,3]. Due to external forces, a Ti⁴⁺ cation can be easily displaced away from the center of its oxygen octahedron against only slight resistance. As a result, the distribution of bond lengths broadens. Due to the lower repulsions, bonds to O with only Ti as a second neighbor are most affected. Ti–O distances near 0.175 nm are absent in the glasses, whereas lengths near 0.24 nm are observed (cf. Figure 3a and Table 1). In distorted TiO₆ octahedra, all short Ti–O bonds (even two or three) are clearly directed toward one side of the octahedron. Figure 4a shows an example of a TiO₆ octahedron with two short bonds to one side, while the longest bonds point in opposite directions. An analogous behavior of V⁵⁺ in its oxygen environments was demonstrated for vanadium-phosphate glasses using ab initio molecular dynamics [30]. V⁵⁺ is likewise a d⁰ transition element.

4.3. The Ti–O Units in Different Environments and a Comparison with the Ge–O Units

It appears helpful to compare the behavior of Ti⁴⁺ and Ge⁴⁺ cations. The Ge⁴⁺ cation, similar in size, is free from SOJT effects. Isostructural crystals KAO(PO₄) are known with A = Ti [2] and Ge [31], both in octahedral sites, where Ti–O distances range from 0.172 to 0.216 nm, while Ge–O bonds range from 0.179 to 0.202 nm. The corresponding distance peaks are compared in Figure 5. The TiO₆ units show a 1+4+1 deformation, as already discussed for the Ti₅O₄(PO₄)₄ crystal [23]. The GeO₆ unit in KGeO(PO₄) also has significantly varying bond lengths, albeit less pronounced than in the TiO₆ unit. In contrast to the highly asymmetric Ti–O–Ti bridges in KTiO(PO₄) with bond lengths of 0.173 and 0.213 nm [1,2], the bonds in the Ge–O–Ge bridges in KGeO(PO₄) have nearly equal lengths (0.179 and 0.181 nm) [31].

The A–O peaks of the glasses are shifted to slightly smaller distances. Due to N_AO < 6 for both glasses, there are small fractions of AO₄ and AO₅ units with shorter bonds. The peaks have equal widths. In contrast to the Ge–O peak, the Ti–O peak shows a significant tail toward long distances. The deformation type of the TiO₆ units in the glasses cannot be specified. A 1+4+1 deformation with a single short bond, such as in the crystal, is not detected in the glasses.

The TiO₅ square pyramid, as in the Na₂TiOSiO₄ crystal [33], is a variant of the 1+4+1 deformation, in which the sixth oxygen atom is displaced to a greater distance [3]. The close similarity to the scattering results [10,11] for the K₂O-TiO₂-2SiO₂ glasses confirms the existence of such units in silicate glasses (cf. Figure S4), which have also been found in silicate glasses of other similar compositions [6]. A 100KTiPO₅-15P₂O₅-5SiO₂ (KTP) glass was studied [15] to find a distorted TiO₆ unit similar to that of the KTiO(PO₄) crystal [2]. This proved unsuccessful, and a nearly symmetrical Ti–O peak from TiO₆ units was obtained (cf. Figure S4).

The SiO₂-rich Ti-silicate glasses provide a suitable environment for Ti⁴⁺ to form TiO₄ tetrahedra with four single bonds in Si–O–Ti bridges [6,13]. In this case, a TiO₆ octahedron would require triply coordinated oxygen atoms in Si–O–Ti₂ configurations—a feature that appears problematic. This behavior with TiO₄ is not transferable to phosphate glasses. In that case, Ti⁴⁺ is always confronted with over-bonded P–O bonds (bv > 1 v.u., see Figure S1), which then force under-bonded Ti–O linkages resulting in TiO₆ units.

Despite only minor differences in the cationic sizes of A = Ge and Ti — the lengths of their single A–O bonds are 0.175 and 0.182 nm, respectively — their A₂O-P₂O₅ systems exhibit distinctly different tendencies regarding crystal structures and glass formation. The flexibility of distorted TiO₆ octahedra—compared to rather rigid GeO₆ octahedra (cf. Figure 4)—exerts an enormous influence on the structures observed. Potentially strong TiO₆ distortions enable the incorporation of triply coordinated oxygen atoms within the octahedron, resulting in N_TiO > M_TO and dense packing fractions. Thus, there is a pronounced tendency to octahedral structural units. Figure 4a shows an example of a distorted TiO₆ unit with two short bonds. At the same time, the preferred TiO₆ units, accompanied by an increasing fraction of triply coordinated oxygen, limit glass formation at a TiO₂ content x > 0.75 [14]. To explain the values N_TiO < 6 for the TiO₂-P₂O₅ glasses discussed, some TiO₅ square pyramids must exist. The TiO₅ are similar to the GeO₅ pyramids shown in Figure 4b, which could form in the presence of isolated PO₄ units with bond valences of 1.25 v.u. Triply coordinated oxygens are not needed to participate in the TiO₅.

In contrast to TiO₂-P₂O₅ glasses, GeO₂-P₂O₅ glasses are obtained up to vitreous GeO₂ but are limited to x > ~0.8 [18] (cf. Figure 2a). The addition of P₂O₅ in the form of isolated PO₄ units is accompanied by an increase in N_GeO due to over-bonded P–O bonds (1.25 v.u.) in P–O–Ge bridges. A GeO₅ square pyramid is possible with these PO₄ units, as shown in Figure 4b. What is happening concerning a GeO₆ octahedron? It could use six P–O–Ge linkages with P–O bond valences of 1.33 v.u. (cf. Figure 4c - such as in the GeP₂O₇ crystal [27]) or triply coordinated oxygen neighbors. Otherwise, the sum of bond valences of the Ge would exceed 4.0 v.u. Bond valences of 1.33 v.u. are typical for PO₄ end groups (see Figure S1) [9]. Due to strong repulsions in P–O–Ge₂ configurations, triply coordinated oxygens are not possible in GeO₆ units, which results in N_GeO = M_TO. Due to the dominance of isolated PO₄ in glasses with x > 0.6, rigid GeO₆ units are difficult to insert into the networks. This problem is solved in the crystals Ti₂O(PO₄)₂ [24] and Ge₅O(PO₄)₆ [28] by strong distortions of the isolated PO₄ units. These distortions compensate for the different bond valences required for the AO₄ and AO₆ units, implying significant internal stresses within the PO₄ units—a scenario that is unlikely in glasses. Consequently, GeO₂-P₂O₅ glasses form only within a narrow compositional range close to vitreous GeO₂.

Glasses approaching the composition 50AO₂-50P₂O₅—whether with A = Ti or Ge—are difficult to obtain due to the strong tendency to crystallize to AP₂O₇ forms [25,27] that contain highly symmetric AO₆ octahedra surrounded by PO₄ end groups (cf. Figure 4c). This tendency toward crystallization is suppressed in ternary Na₂O-GeO₂-P₂O₅ glasses [32]. These glasses, whose compositions predominantly contain PO₄ end groups, are excellently suited for incorporating Ge into GeO₆ octahedra, as illustrated in Figure 4c. A maximum Ge–O coordination number of 5.6 ± 0.4 was observed.

5. Conclusions

Recently, it was shown that the range of glass formation in the (TiO₂)_x(P₂O₅)_1–x system is limited to 0.71 < x < 0.75, whereby triply coordinated oxygens are involved in the glassy networks. Here, the analysis of our scattering data of similar glasses with x = 0.61 and 0.66 – they have small Al₂O₃ impurities – has been repeated, and all results are discussed together.

To achieve agreement between the results of X-ray and neutron scattering, it proved necessary to allow for small proportions of Ti–O distances at ~0.24 nm. Mixtures of TiO₅ and TiO₆ units are observed, whereby the large Ti–O distances belong to highly distorted TiO₆ octahedra. The Ti–O coordination numbers indicate networks of dominantly doubly coordinated oxygens with significant fractions of triply coordinated oxygens in TiO₆ units. Triply coordinated oxygen enforces shorter cation-cation distances than oxygen bridges. The associated repulsion drives distortions. They are then facilitated by the SOJT effects of the d⁰ transition element Ti⁴⁺_, resulting in displacements of Ti⁴⁺ cations away from the octahedral centers. This is accompanied by asymmetric distributions of bond lengths at oxygens with two or three Ti bonding neighbors.

Given the potential for strong octahedral distortions, Ti favors octahedral oxygen environments but requires specific support to occupy tetrahedral positions. In contrast, the main-group element Ge—which is only slightly smaller than Ti—prefers relatively undistorted octahedral environments that do not accommodate triply coordinated oxygen. Consequently, it forms glass structures containing GeO₆ octahedra only to a limited extent, favoring GeO₄ units instead. The preferences for different oxygen polyhedra in the TiO₂-P₂O₅ and GeO₂-P₂O₅ systems define distinct regions of glass formation.