Shapes of the noncentrosymmetric TeO₅E groups in tellurite glasses with P₂O₅ additions

1 Introduction

The cations A^k+ of several posttransitional main group elements such as Sn, Sb, and Te can have oxidation states two less than their group numbers. Two spin-paired valence electrons could occupy the lowest empty state, the outer s orbital. However, if the energy gap to the ligand’s p orbital is small, then these s orbitals participate in bonding while the antibonding electron density is pushed to the top of the valence band hybridizing on the empty cationic p orbitals [1]. In the case of oxygen ligands, the effect is almost always present. In the course of this, they form noncentrosymmetric cation environments. The antibonding electron density of the lone pair arranges eccentric to the cation’s core position while the strongest A−O bonds point toward the opposite side and are denoted primary bonds [2]. The A−O distances at the lone-pair side are the secondary bonds. Alcock [3] introduced the latter term for bonds that are longer than typical bonds but shorter than those based on only van der Waals interactions. Ligand pairs of short and long bonds in trans positions have been used to describe the noncentrosymmetric oxygen environments of Sb_(III) and Te_(IV) (Trömel 1980 [4]). At that time, the structural effects of the sterically active lone pair were explained by the repulsions from a negatively charged ligand.

Primarily, the noncentrosymmetric TeO _n E units are distinguished according to the number of their primary bonds where groups of n = 3, 4, and 5 are known. Earlier diffraction work on (TeO₂)_1−x(P₂O₅) _x glasses [5] detected Te−O coordination numbers, N_TeO, larger than four with mixtures of TeO₄E and TeO₅E units. Two TeO₅E units in crystal structures were known. Trömel [4] classified the first one [6] as a deformed pyramid while the other one [7] is a regular square pyramid with the five oxygens occupying the four base corners and the apex. A transition to the regular pyramid was expected for the glasses with increasing P₂O₅ content according to the behavior in the related crystals. However, the TeO₅E in all the glasses have bond lengths similar to those of the trigonal bipyramid TeO₄E (α-TeO₂ [8]) completed with a fifth bond a little longer with ∼0.240 nm [5]. These units were called TeO₄₊₁E in accordance with distorted TeO₄E units that are denoted TeO₃₊₁E at the transition from TeO₃E to TeO₄E [9]. The +1 denotes the long bond. Such longer bonds indicate a general problem. An exact definition of the TeO₅E is difficult. Zero minima do not occur beyond the first peaks of Te−O distances. On average, the bond lengths of the tellurite crystals show a minimum at ∼0.250 nm [10], which is close to the minima found in the Radial Distribution Functions, RDF_TeO(r), of the (TeO₂)_1−x(P₂O₅) _x glasses [5] and vitreous (v−)TeO₂ [11]. Molecular dynamics (MD) works combine this criterion with a bond angle limit [12, 13]. For crystal structures, it is recommended to use 0.245 nm to delimit typical bonds [14]. This length corresponds to ∼0.3 vu (units of bond valence). The Te−O bonds of 0.245 nm span angles of ∼90° to the lone pair’s direction. This angle describes the boundary between the hemispheres of the primary and secondary bonds.

Cut-offs of 0.230 or 0.250 nm were used to obtain the Te−O coordination numbers, N_TeO, of the (TeO₂)_1−x(P₂O₅) _x glasses [5, 15]. Both variants show increasing N_TeOs with increasing x with a difference of about 0.4. Clear fractions of TeO₅E exist. The increased N_TeO values are due to the effects of the isolated PO₄ units in the network. The PO₄ units show narrow P−O peaks at 0.154 nm [5]. Short P=O double bonds do not occur. Long bonds in P−O−P bridges can exist in small fractions only. The bonds of ∼1.25 vu of the isolated PO₄ need underbonded Te−O in the P−O−Te bridges. This is only possible with TeO₅E units or other units with a Te=O double bond, the latter is rather unlikely (see Discussion chapter). Supposing, all oxygens are forming Te−O−P or Te−O−Te bridges such as in the Te₂P₂O₉ crystal [16]. The corresponding model N_TeOs behave similarly to the N_TeOs resulting from the fits [15]. Why does the regular TeO₅E unit formed in the Te₂P₂O₉ crystal not exist in the glasses? This work’s goal is the determination of the TeO₅E shape. A mechanical analog is introduced to identify the TeO₅E groups that reproduce the RDF_TeO(r)s of the glasses [5]. The N_TeOs obtained with the cut-off of 0.250 nm give the best explanation of the structural behavior. The results of the glass series [5] can be extrapolated toward v-TeO₂ whose structural details are still in debate [11]. The results were presented at a conference [15]. Here, a thorough presentation follows.

Last but not least, it is needed to emphasize that the steric effects of lone pairs result from optimizations of bond energies [1, 2]. Te−O bonds, when using cationic s electrons, must bring the center of the lone pair’s antibonding electron density away from the Te⁴⁺ core. The gain in bond energy pulls the ligands toward one hemisphere of the Te⁴⁺ and stabilizes the lone-pair density in the other hemisphere. The ligands control the direction of the lone-pair shifts. For crystal structures, the specifics of the TeO _n E units are analyzed based on the coordinates of the Te and O atoms. Direct determinations of the center of the antibonding lone-pair density are not known, but the Density Functional Theory (DFT) [1] is a useful tool for its simulations. For glass structures, an analysis is more difficult. Only total numbers of oxygen neighbors and distributions of Te−O distances are available from diffraction methods. The TeO₃E units should be well-defined trigonal pyramids. An Sb_(III)-containing crystals analysis shows all SbO₃E units with three equal Sb−O distances [17]. The bond lengths of the SbO₄E units show variations that go beyond the lengths of a trigonal bipyramid. Square pyramids and transitional forms also exist. A regular TeO₅E square pyramid [7, 16] was considered the stable TeO₅E prototype [4, 14]. However, this unit is inappropriate to model the RDF_TeO(r)s of the (TeO₂)_1–x(P₂O₅) _x glasses [5]. The TeO₅E units are expected to show even more variations than the TeO₄E.

2 Experimental and modeling

2.1 Diffraction results

Five samples of (TeO₂)_1–x(P₂O₅) _x glasses were measured by neutron and X-ray diffraction with large upper limits of magnitudes of scattering vector, Q_max, of 400 and 300 nm⁻¹, respectively [5]. These values allow a good distance resolution. The contrast change of neutron and X-ray scattering helps to separate the RDF_TeO(r) functions from the P−O and O−O contributions that have also short-range distances. The Te−O distance distributions were fit with five Gaussian components whereby the broad peak at 0.280 nm is attributed to the secondary bonds and not considered part of the structural units. Secondary bonds have the maximum in this range [10]. In the case of glasses, the secondary bonds are neither recorded properly nor assigned to the structural units correctly. Figure 1 shows the RDF_TeO(r)s that are calculated with the peak parameters taken from [5]. These functions are compared with those calculated with the bond lengths of the crystals α-TeO₂ [8], c-Te₄P₂O₁₃ [18], c-Te₃P₂O₁₁ [19], and c-Te₂P₂O₉ [16]. The agreement is excellent for the sample tep02 and α-TeO₂ except for missing a few lengths at ∼0.24 nm. The behaviors for the sample tep19 and c-Te₄P₂O₁₃ are similar, at least. The curves of the samples tep27 and tep32 differ from those of the related crystal structures. These crystals have regular TeO₅E square pyramids (type 3), i.e., one short Te−O bond and four medium bond lengths as given in Table 1. This TeO₅E is suitable for linking to another TeO₅E and four PO₄ corners as would be excellent for glasses of 33 mol% P₂O₅. Two further species of TeO₅E units exist in other crystals. Their bond lengths are given in Table 1, as well. Type 2 was considered the Te₄₊₁E unit for the glassy networks [5].

Figure 1:

Te−O radial distributions in (TeO₂)_1−x(P₂O₅) _x glasses calculated with the parameters from peak fitting [5] (thick red solid lines). The P₂O₅ contents x are 0.02, 0.14, 0.19, 0.27, and 0.32. It is compared with curves (dash-dotted lines) that are calculated with the distances of the related crystal structures with P₂O₅ fractions of x = 0, 0.20, 0.25, 0.33, as appropriate. The final model curves (dotted lines) are calculated with distances of TeO₄E bipyramids and special TeO₅E groups (see Table 1). Their fractions follow the N_TeO values obtained with the distance cut-off of 0.250 nm.

Table 1:

Te−O bond lengths and bond valence sums (BVS) of the Te_(IV)O₅E units of various crystal structures and the TeO _n E units of (TeO₂)_1–x(P₂O₅) _x glasses. The asterisk on a crystal’s formula indicates the coexistence with Te_(VI) sites. Two asterisks denote secondary bonds that are close to the limit of primary bonds. The three different types of TeO₅E units are explained in the text.

Compound	Te site, type	Te−O bond lengths (in nm)					BVS
K2Te4O12* [6]	Te_5,    1	0.192	0.202	0.202	0.228	0.228	3.70 vu
NiTe2O5 [20]	Te_2,    1	0.189	0.200	0.200	0.225	0.225	3.99 vu
γ-TeO2 [21]	Te_1,    2	0.186	0.195	0.202	0.220	0.269**	3.87 vu
Te4P2O13 [18]	Te_1,    2	0.190	0.194	0.199	0.229	0.256**	3.80 vu
	Te_2,    2	0.190	0.192	0.203	0.220	0.240	3.96 vu
BaTe2O6* [7]	Te_2,    3	0.183	0.213	0.213	0.213	0.213	3.98 vu
Te2P2O9 [16]	Te_1,    3	0.191	0.205	0.212	0.213	0.223	3.75 vu
	Te_2,    3	0.192	0.207	0.207	0.210	0.212	3.97 vu
Glasses	TeO5E, 2	0.191	0.191	0.204	0.217	0.240	3.99 vu
	TeO4E	0.190	0.190	0.211	0.211	–	3.70 vu

2.2 Model constraints of the structural TeO _n E units

A model based on simple assumptions is presented that helps to understand the evolution of the TeO₅E units. (i) Close inspections of TeO _n E units of crystals reveal small O−Te−O angles (∼90°) between the primary bonds. This behavior is well described for a long time [4]. The competition for strong bonds causes the oxygens to push together to small O−O distances [1, 2]. These bonds are formed on the opposite side of the lone-pair hemisphere. An average minimum of the O−O distance of 0.28 nm is chosen for the model.

Christy & Mills [22] analyzed the polyhedral volumes of TeO₆ octahedra of numerous Te_(IV) sites in crystal structures. Also, the geometrical form of the oxygen positions was determined. (ii) The oxygen positions of the primary and secondary bonds are found on spherical surfaces whereby the oxygens are not evenly distributed on these surfaces. The average radius of these spheres is ∼0.265 nm. The centers of the spheres and the Te⁴⁺ cores are distant by 0.10 nm. Both parameters show distributions whereby a linear correlation was found between them [22]. Here, these distributions are neglected for the simplicity of the model. The sphere center can be considered the center of the antibonding electron density. Some authors performed MD simulations for v-TeO₂, where a core–shell potential simulates the polarizability of the Te⁴⁺ [12]. Coulombic forces emanate from the Te⁴⁺ core and the eccentrically arranged shell. The short-range repulsions are exerted from the spherical shell alone. A variable coupling of the core–shell distances corresponds to variable stereoactivities [12].

The larger the number n of the primary bonds, the more variations of the TeO _n E shapes exist as it was outlined in the Introduction. The number n depends on the glass compositions, i.e., on the character and amount of additional components. (iii) Local variations of the bond valences of the oxygen ligands and other network constraints influence the shape of the units. Glassy networks do not stabilize special symmetries of the TeO _n E units as would be possible from lattice symmetries in crystals. Second neighbors of the oxygen ligands and other network constraints modify the strengths and directions of the bonds in the TeO _n E groups.

The model described below is aimed at detecting a starting point to understand the evolution of the shape of the TeO₅E units in the glasses. For this purpose, the constraints (iii) emanating from the glassy networks are neglected for now and the structural unit of the minimum energy is obtained. This unit is stable if ligands of suitable bond valences and directions are available. Figure 2 illustrates the model for the trigonal TeO₃E pyramid and TeO₄E bipyramid to explain the different spheres. The surface of the middle sphere with the diameter Ø = 0.530 nm defines the oxygen positions as reported in [22]. Its center is about identical to the center of the lone pair’s density. The Te⁴⁺ core is fixed below this point distant by d = 0.10 nm. The oxygens (Ø = 0.280 nm) are not allowed to enter each other nor the inner sphere (Ø = 0.250 nm). The angle β describes the tilt between the Te−O bond and the backward extension of the lone pair’s direction (cf. Figure 2). The shortest possible bond of maximum energy in any direction is a function of this angle β whereby the oxygens touch the surface of the outer sphere (Ø = 0.810 nm) from the inside. The shortest and longest Te−O bonds (0.165 and 0.365 nm) are found at β = 0° and 180°, respectively. The bond lengths R_TeO are obtained by the cosine formula. R_TeO, d, and the radius of the sphere of oxygen positions (r = 0.265 nm) form a triangle that is shown in Figure 2 by thick solid lines. Figure 3a shows the change of R_TeO versus angle β. Thin straight lines delimit the range of typical primary bonds. Of course, an oxygen atom can form a less tight bond longer than the minimum length in a given direction. Then, it would intersect the outer sphere but that is not taken into account in the simple model.

Figure 2:

The layout of the balls-in-a-bowl model illustrated for the TeO₃E and TeO₄E units. The positions of the oxygen ligands are found on the surface of the sphere of middle diameter (green circle – Ø = 0.530 nm). Accordingly, the oxygens (Ø = 0.280 nm) contact the surface of the outer sphere from the inside (blue circle – Ø = 0.810 nm). They cannot enter the inner repulsive sphere (red circle). The side views show also some ligands in secondary bonds. An oxygen ligand of the TeO₄E is removed from the front so that the definition of the tilt angle β of the right oxygen is clear. The solid-lined triangle relates the bond length R_TeO to the angle β.

Figure 3:

The possible minimum bond lengths (a) of an oxygen ligand in bond orientations of the tilt angle β. The bond valences shown in part (b) correspond to the lengths in part (a). The thin lines delimit the range of the typical primary bonds (cf. Table 1).

2.3 Mechanical analog of the lone-pair effects

A mechanical analog is introduced that simulates the oxygen positions of the primary bonds. The oxygens are replaced by steel balls (Ø = 65 mm) that can move freely in a spherical bowl (Ø = 200 mm). The bowl corresponds to the outer sphere in Figure 2. The dimensions are chosen according to the ratios on the atomic length scale. Under the effect of their gravity, the balls compete for the lowest position in the bowl. The reduction of a ball’s height replaces the energy gain of a Te−O bond. Figure 3b shows the change of bond valences b_v with angle β. The value b_v can be related to the bond strength [2]. Value b_v changes with the bond length [2], and it is approximated with

(1) b v = exp R 0 − R i j B

The parameters R₀ and B depend on the atomic species of the bond partners [10, 22]. Here, the R₀ and B values from [23] with 1.96 Å and 0.389 Å are used where R _ij is the Te−O distance. The values b_v decrease uniformly for 20° < β < 80° and have reduced slopes outside this range. The bond valences give a criterion to check the bond lengths formed of all the different atomic sites. The bond valence sums, BVS, of O²⁻ (2.0 vu) and Te⁴⁺ (4.0 vu) sum up the valences of all the bonds of a given atomic site and should agree with the expected value. The outer right column of Table 1 lists the BVS of the Te sites in the related crystal structures. Figure 3b shows that a Te=O double bond (b_v = 2 vu) is possible but that requires small angles β < ∼25°.

For the Te−O bond energy and the gravitational potential of a ball, the dependencies on the angle β should not differ too much to guarantee the validity of the model. For ionic bonding, a relationship based on 1/R_TeO approximates the bond energy with the bond lengths taken from Figure 3a. The energies are set to zero for β = 0° and, by a factor, normalized to unity for β = 180°. The angular dependencies are given in Figure 4. The bond energies increase uniformly in the range 40° < β < 110° and have smaller slopes outside this range. The gravitational potential of a steel ball depends on the ball’s height relative to the Te⁴⁺ core. It follows the value –cosβ · R_TeO. Using equivalent boundary conditions, the angular dependency of the ball’s height is similar to that of the bond energies but with a visible shift. The gradient of gravitation appears reduced for β < 60° if compared with the bond energies. Regardless of that, the mechanical analog is sufficient for qualitative discussions on the TeO₅E behavior in the glasses. The full line of the ball’s height ends at 0.5 (cross marker in Figure 4). Higher balls can no longer stick to the bowl’s wall but they would fall inward. This deficit concerns only the secondary bonds, which are not considered by the model.

Figure 4:

The angular dependency of the bond energy (ionic bonding) obtained from the lengths given in Figure 3a is compared with the gravitational energy of the steel balls they suffer in a spherical bowl. The bond energy is set to zero for β = 0°. The value at β = 180° is normalized to unity by a factor. The mechanical analog is only applicable for balls up to half height (0.5), which corresponds to a tilt angle β of ∼110°.

3 Results

The mechanical analog gives clear results. Starting from any configurations, stable TeO _n E units with n = 3, 4, and 5 balls are obtained as shown in Figure 5. The TeO₃E pyramid with three equivalent O sites is found as expected. The change from the TeO₄E square pyramid to a trigonal bipyramid (Figure 5b) is weakly driven, which explains the large variety of TeO₄E units in the different crystals and glasses. The stable TeO₅E unit (Figure 5c) has only two-fold symmetry. Figure 6 shows a photograph of the mechanical analog with this TeO₅E unit. This unit is not known of any Te phosphate crystal. However, it is found in some other crystalline compounds [6, 20] and is denoted type 1 in Table 1. The regular TeO₅E unit is a square pyramid of four-fold symmetry (one oxygen on the bottom and four on the sides – type 3 in Table 1). This TeO₅E reveals highly unstable in the bowl and needs stabilization by additional spacers.

Figure 5:

The resulting arrangements obtained with three, four, and five balls. The TeO₃E and TeO₄E units are known as trigonal pyramids or trigonal bipyramids. The TeO₅E unit is a highly distorted pyramid with a single symmetry plane. The side views show arbitrarily chosen ligands in secondary bonds.

Figure 6:

The TeO₅E unit as obtained from the balls-in-a-bowl model. The five steel balls (Ø = 65 mm) simulate the oxygen ligands. The Te⁴⁺ core is not shown, but it sits 24 mm below the center of the spherical bowl (Ø = 200 mm).

The bond lengths of the type 1 unit (Figure 5c) are in the first peak’s range of the RDF_TeO(r)s (Figure 1). However, the one short bond of ∼0.190 nm is not sufficient for well-fitted distance peaks. The best TeO₅E group for approximation is given by the Te_2 (type 2) of the Te₄P₂O₁₃ crystal [18]. A mixture of TeO₄E and TeO₅E units with the bond lengths given in Table 1 is used for fitting the RDF_TeO(r)s (Figure 1). Only small variations of the distances are needed. Excellent agreement is obtained up to distances of 0.240 nm (dotted curves in Figure 1). The fractions used for the TeO₄E and TeO₅E units are taken from the N_TeOs obtained with a cut-off of 0.250 nm [5]. The values are N_TeO = 4.35, 4.66, 4.86, 4.96, and 5.00 for the samples of P₂O₅ contents of 2, 14, 19, 27, and 32 mol%, respectively (for more details, see the Discussion chapter).

To clarify the relationship between the TeO₅E types 1 and 2, these units are shown in Figure 7. The Te_2 and Te_1 sites of the Te₄P₂O₁₃ crystal [18], both type 2 as shown in (b) and (c), are compared with the type 1 of two-fold symmetry in (a). Starting from type 1 (a), the oxygens of type 2 shift away from the center of the bowl, for (c) a little more than for (b). Interestingly, the Te site of γ-TeO₂ [21] shows a similar sequence of bond lengths (Table 1) whereby the fifth bond of 0.269 nm is only a little longer than that of the Te_1 of the Te₄P₂O₁₃ crystal [18]. The shift from type 1 toward type 2 is simulated easily with the five balls in the spherical bowl if one applies a weak lateral force.

Figure 7:

Variations of the TeO₅E units. The TeO₅E unit in (a) is found by the balls-in-a-bowl model and it is called type 1. The TeO₅E units (b) and (c) are distorted variants (type 2) of (a). They exist in the Te₄P₂O₁₃ crystal [18] and are also suggested for the glasses. The TeO₅E unit (e) is found in the Te₂P₂O₉ crystal [16] (type 3). It is obtained from the unit (d) by lowering the positions of the five oxygen ligands. Therefore, the central O intersects the outer sphere. The definitions of the particles are the same as in Figure 5.

Above it was stated that the TeO₅E units must show even more variations than the TeO₄E. Accordingly, the bond lengths obtained in the RDF_TeO(r) fits as given in Table 1 are the average of differently distorted TeO₅E units. Both, types 1 and 2, are suggested to exist in the glasses. Note, the distortions of type 1 in the opposite direction of that shown in Figure 7 bear the same changes in the bond lengths. Hence, most TeO₅E units appear rather a little distorted (type 2 in Figure 7b) than undistorted (type 1 in Figure 7a). The question may arise whether the lone-pair orientation of the TeO₅E type 2 remains vertical as defined in the mechanical analog. If one would consider only the first neighbors, the orientation of the lone pair and the center of mass of the ligands appear decoupled. The oxygen ligands of type 2 are pushed outward from the bowl’s center. The origin of this shift is due to second neighbors acting unevenly from different sides. Hence, it can be stated that the lone-pair orientation arises from the equilibrium with the overall cationic environment and cannot be inferred from the ligand positions alone.

The next question concerns the existence of TeO₅E type 3 units in the glasses. Figure 7d shows an example of this group. The oxygen in the apex could form a Te=O double bond with a short length of ∼0.170 nm (cf. Figure 3a). The adjacent O−Te−O angles are expected close to 80° [4, 16] which means four Te−O bonds of ∼0.230 nm. This length corresponds to a bond valence of ∼0.4 vu and the BVS of this Te site would reach 3.6 vu only. The known TeO₅E square pyramids (cf. Table 1) possess longer middle Te−O bonds of 0.183 nm [7] or 0.192 nm [16]. This bond lengthening causes the oxygen to intersect the outer sphere, which is illustrated in Figure 7e. The lengthened Te−O allows the four neighboring oxygens to shorten their bonds at smaller angles β. The four shortened Te−O bonds overcompensate for the loss of bond valence of the one longer bond and the BVS of this Te reaches ∼4.0 vu (Te_2 of Te₂P₂O₉ in Table 1). This unit is well suited to link with other TeO₅E via the apex and with PO₄ units via the four base corners. Possibly, a Te=O double bond causes problems for the TeO₅E unit to reach the full bond valence. However, it is more significant that the change from the unit in Figure 7d to that in 7e needs stabilization from the outside, e.g., from the four-fold symmetry of a crystal lattice. The uniform pressure of all four outer ligands keeps the central ligand at an enlarged distance. The disordered networks of glasses do not stabilize such structural units.

4 Discussion

The networks of tellurite glasses are formed of noncentrosymmetric TeO _n E units. The corner with the lone pair E occupies one hemisphere while n = 3, 4, or 5 primary bonds point into the opposite hemisphere. The tellurite glasses with network modifiers such as alkali oxides show transitions from TeO₄E to TeO₃E units with increasing modifier content [24] where a distorted TeO₄E unit, the TeO₃₊₁E, can occur. The RDF(r)s from neutron diffraction on vanadium tellurite glasses were analyzed with two visible Te−O distances [25]. Assuming the TeO₃E as a trigonal pyramid with three equally short bonds and the TeO₄E as a trigonal bipyramid with two short and two long bonds, the process of modifier additions becomes obvious in the total N_TeOs and in the corresponding fractions of two bond lengths. The extrapolation toward v-TeO₂ suggests the existence of a continuous network of corner-connected TeO₄E units. However, this view of the structural evolution revealed too simple. Neutron diffraction on v-TeO₂ finds a rather broad distribution of bond lengths with a total N_TeO of ∼3.7 [26]. ¹²⁵Te NMR indicates that 11% of the Te sites belong to other units than TeO₄E, probably TeO₃E [27]. Possibly existing TeO₃E units in v-TeO₂ raise the question of the existence of Te=O double bonds in these TeO₃E, a next point of debate [28].

High-energy X-ray diffraction is excellent to resolve the Te−O and Te−Te correlations [5]. The huge peak centered at 0.36 nm was attributed to the Te−Te distances of corner-connected units. Edge-connected units can exist only in marginal numbers. They would form Te−Te distances <0.32 nm. The combination with neutron diffraction introduces an excellent consideration of the O−O distances, which allows analyzing the RDF_TeO(r)s up to lengths of ∼0.30 nm [5, 11]. It reveals that the first peak of Te−O distances with its shoulder is not separate from that of the secondary bonds. Assuming a cut-off length of 0.236 nm [11] gives an N_TeO of ∼4.0 for v-TeO₂, well knowing that a few more bonds exist beyond. An extrapolation of the results of the (TeO₂)_1−x(P₂O₅) _x glasses [5] toward v-TeO₂ supports the results given in [11]. Structural interpretations of v-TeO₂ [11, 13] often compare with the TeO₄E units of γ-TeO₂ [21]. Different from the trigonal bipyramid TeO₄E of α-TeO₂ the unit known from γ-TeO₂ is a distorted pyramid with a tendency toward a TeO₅E type 2. Table 1 compares the first five Te−O distances of this unit with those of the type 2 units of the Te₄P₂O₁₃ crystal. The Te_2 site of the latter crystal was essential to fit the RDF_TeO(r) peaks of the (TeO₂)_1−x(P₂O₅) _x glasses (Figure 1). A fraction of ∼30% of such TeO₅E was found for the sample tep02 of only 2 mol% P₂O₅, which makes it likely that there is a significant fraction of TeO₅E type 2 in v-TeO₂, as well. An exact answer to this problem is difficult due to several uncertainties. The cut-off of bond lengths of the structural unit is somewhat arbitrary (cf. Introduction). Ab initio MD simulations of v-TeO₂ [13] used a procedure based on the obtained electronic structure that allows the definition of bonds. The obtained cut-off was 0.246 nm for v-TeO₂ with an N_TeO of 4.12. This is comparable to N_TeO = 4.22 with a cut-off of 0.236 nm from X-ray diffraction [11] because the ab initio MD had a small shift of 0.005 nm in the bond lengths. The authors [11, 13] stand by the N_TeO result of ∼4.0 though there is no reason to exclude the larger N_TeO values, thus, a small fraction of distorted TeO₅E units.

Any TeO₃E unit existing in v-TeO₂ needs a Te=O double bond to balance the bond valences as predicted for experiments with N_TeOs < 4 [26, 27], whereas TeO₅E units in v-TeO₂ would need three-coordinated oxygens. Assuming all oxygens of the (TeO₂)_1−x(P₂O₅) _x glasses in Te−O−P and Te−O−Te bridges yields

which was the working hypothesis [5, 15]. Figure 8 compares this model function with the values N_TeO used above (Results chapter) and with values obtained with the recommended sharp cut-off at 0.245 nm [14]. Even the latter limit yields values larger than those of the model, which means that three-coordinated oxygens exist in the glasses. Such oxygens contradict the traditional concepts of glass structures (Zachariasen [29]). However, the distributions of bond lengths of the TeO₅E units obtained from fitting the RDF_TeO(r)s show a significant contribution at 0.240 nm (Table 1). The bond valence corresponding to this length is only 0.32 vu, which is too small to be part of a real oxygen bridge. Hence, three-coordinated oxygens must exist. The disorder of a glassy network causes strong distortions of the TeO₅E units. The regular TeO₅E square pyramid (type 3) and the TeO₅E of two-fold symmetry (type 1), the latter determined as that of minimum energy from the mechanical analog, have only bonds shorter than 0.240 nm. However, these units are not stable against the tensions from the network disorder. The bond lengths of the TeO₅E unit type 2 obtained for the glasses are the average of a greater variety of pyramids. It is not justified to assume that all TeO₅E of the glasses have exactly the geometry of the Te_2 of the Te₄P₂O₁₃ crystal. The bond lengths of the TeO₄E of α-TeO₂, γ-TeO₂, and the TeO₅E types 1 and 2 (Table 1) indicate that these units can exist in all transitional forms. The idea of clearly definable structural units has to be abandoned for these glasses. Te−O bonds of 0.240 nm and three-coordinated oxygens are not to avoid in the (TeO₂)_1−x(P₂O₅) _x system and, probably, also in v-TeO₂.

Figure 8:

The evolution of the N_TeOs of the (TeO₂)_1−x(P₂O₅) _x glasses versus the P₂O₅ content x. The values N_TeO of limit 0.250 nm correspond to the area of the dotted-lined peaks in Figure 1. The second series of N_TeOs is obtained with a sharp cut-off at 0.245 nm as recommended by the crystallographers [14]. The model N_TeOs follow from assuming all oxygens form either Te−O−Te or Te−O−P bridges.

Commonly, more variations of structural groups than in reality are obtained by computer-aided structural simulations such as conventional MD or ab initio MD work on v-TeO₂ [12, 13]. Proportions of three-, four-, and five-coordinated Te sites are listed, a smaller part of the units with a terminal Te−O bond. Ab initio MD allows the localization of the lone pairs. Distributions of the angles θ_W-Te-O are given, which means the angles between the lone pair’s directions and the Te−O bonds [13]. The angle θ_W-Te-O is equal to 180° – β as we used above. Unfortunately, the angular distributions are not differentiated for the TeO _n E of the different n. The correlations between bond lengths and angles θ_W-Te-O would be interesting such as it is shown in Figure 3. The TeO₅E unit was not the focus, and the shapes of the TeO _n E are not reported [13].

The O−Te coordination number from the MD work [13] is nearly two but with ∼10% terminal Te−O and ∼8% three-fold coordinated oxygen. The latter oxygen species got little interest so far, but the Te=O double bonds in TeO _n E units are a subject of ongoing discussions [26–28]. Very few crystal structures have Te−O bonds shorter than 0.180 nm [10] that are Te=O bonds belonging to TeO₅E square pyramids. Such units are not significant in the (TeO₂)_1−x(P₂O₅) _x glasses. Crystal structures with TeO₃E and TeO₄E units, that contain Te=O bonds [30, 31], are reported [14]. However, close inspection of the crystal structures reveals additional bonds at ∼0.240 nm. According to the balls-in-a-bowl model, the Te=O bond is only possible with β < ∼25°. Adjacent bonds must have angles β > 70°, which means their bond valences are less than 0.5 vu. Two or three such bonds together with the Te=O are not sufficient that the bond valence sum (BVS) of that Te reaches the needed 4 vu.

The increase of the Te−O coordination number with P₂O₅ additions in the glasses studied is the reaction of the TeO _n E units on overbonded P−O bonds (cf. Introduction). Sometimes, the P=O double bond is mistakenly assumed to be a stable feature of a PO₄ unit. However, such bonds are restricted to glasses close to the pure P₂O₅ composition. If possible, then already the oxygen of the P=O corner in the PO₄ branching group (three P−O−P bridges) coordinates a modifier cation [32]. The P=O bonds of other PO₄ units (chain, end, or isolated groups) are distributed to all nonbridging PO₄ corners by bond resonance [33]. Additionally, the P−O−P bridges are broken as much as possible using the oxygen of the other oxides.

At first, the effects were discussed for phosphate glasses with typical network modifiers [32]. The concept proved to be valid also in the presence of network-forming oxides. The value N_GeO of the GeO₂–P₂O₅ glasses increases with P₂O₅ additions [34]. N_GeOs close to 5.5 were obtained in Na₂O–GeO₂–P₂O₅ glasses [35]. Molecular orbital calculations were performed on Te₅PO₁₈H₁₁ clusters to simulate the Raman spectra of a 90TeO₂–10P₂O₅ glass [36]. Both the agreement of the model spectra and the lower energy favor the cluster with the PO₄ with four equivalent bonds and one TeO₄E unit that is changed to a TeO₄₊₁E. It was compared to a cluster with the PO₄ with a P=O corner and all TeO₄ groups being unchanged.

In the evolution of the Te−O coordination numbers caused by the effects of the PO₄, the possible TeO _n E units affect the range of glass formation. TeO₆E units are not known and the regular TeO₅E (type 3) was found unstable in glasses. The N_TeOs in Figure 8 do not exceed value five. The steady rise of N_TeO weakens at x = ∼0.2 (Te₄P₂O₁₃ composition). Beyond this, regular TeO₅E (type 3) units should be formed but they do not appear. The phase diagram reports an ultimate limit x < 0.33 [37] and above immiscibility becomes effective. As the second phase in addition to Te₂P₂O₉, a Te pyrophosphate was suggested. The first limitations appear already in the range 0.2 < x < 0.33. A neutron diffraction work of poor real-space resolution on TeO₂–P₂O₅ glasses reported yellowish but transparent samples up to x = 0.20 while the sample of x = 0.26 is dark and poorly transparent [38]. Similarly, our two samples of x > 0.2 changed to dark brownish color, which can be explained by traces of reduced Te_(II) [5]. An appropriate TeO _n E unit of a P₂O₅-rich phase with Te_(IV) is not known. The Te_(II) could form TeO₃E units, which contribute to constant N_TeOs for x = ∼0.30. A similar change of the oxidation state with P₂O₅ additions is known in binary V₂O₅–P₂O₅ glasses where vanadium reduces with increasing P₂O₅ content [39]. The VO₅ square pyramid turned out to be the unit of the largest possible N_VO [40]. The occurrence of Te_(II) sites may be accompanied by dimers of PO₄ groups. The peak widths of the P−O distances indicate the dominance of isolated PO₄ where, however, small numbers of P−O−P linkages cannot be excluded [5]. Bearing these findings in mind, the TeO₂ resists any change to a network modifier.

Binary TeO₂–P₂O₅ glasses are not suitable for practical use due to a metastable immiscibility gap [37]. Differently, studies of ternary glasses are promising, such as that on the ZnO–TeO₂–P₂O₅ system [41]. Fractions of TeO₅E units are not reported though related crystal structures show this feature (e.g., K₂TeOP₂O₇ [42]). The noncentrosymmetric TeO _n E units are essential when considering the optical properties of tellurite glasses. These materials show enormous magnitudes of nonlinear effects that are related to their high polarizabilities [43]. Structural details are needed to elaborate realistic concepts to explain these properties.

5 Conclusions

The nonbonding lone pairs of electrons for Te_(IV) make the oxide TeO₂ a conditional glass former with important applications. TeO _n E groups of noncentrosymmetric shapes form the disordered tellurite networks. The rare case of TeO₅E units exists for (TeO₂)_1–x(P₂O₅) _x glasses. Their shapes were determined based on the distance distributions of the Te−O bonds from diffraction methods and using a simple mechanical analog. The competition of the five oxygen ligands for maximum bond energy in the limits of their mutual repulsions and the repulsions by the antibonding lone-pair density are taken into account. Five balls (oxygens) move freely in a spherical bowl and arrange such as the oxygen ligands. The bowl simulates the repulsions by the antibonding states. They act through a spherically shaped potential that is eccentric to the Te⁴⁺ core. The decrease in the ball’s height in the bowl simulates the gain in bond energy.

The TeO₅E shape detected by the model is that with two-fold symmetry of the three known types of TeO₅E in the crystals. This unit and the distorted pyramids of the Te₄P₂O₁₃ crystal are suggested to exist in the glasses while the regular TeO₅E square pyramid needs stabilization from a crystal lattice.

The bond lengths obtained from diffraction experiments are the average distribution of large varieties of distorted TeO₅E in the glasses. The available bond valences and directions of the oxygen ligands of the disordered networks cause numerous TeO₅E shapes. The shape of the TeO₄E of γ-TeO₂ suggests the existence of transitional forms between the TeO₄E and TeO₅E units. A well-definable difference does not exist.

The bond lengths (and bond valences) of the model change with the tilt angle between the Te−O bond and the lone-pair direction. If a Te=O double bond would exist, the four neighboring oxygens cannot reach sufficient bond valence for the Te⁴⁺ site. Units with n < 5 are even more unlikely for double bonds. On average, the distorted TeO₅E units have one bond of 0.240 nm with a bond valence of only 0.3 vu. The corresponding oxygen must be three-coordinate to reach the full bond valence.

The special TeO₅E of the Te₄P₂O₁₃ crystal (x = 0.2) is related to the compositional limit of transparent glasses. The TeO₅E square pyramid of the other Te₂P₂O₉ crystal is unstable in glasses. The brownish color of glasses for x > 0.2 indicates traces of Te_(II) that appears as the reaction on the larger P₂O₅ additions before immiscibility becomes effective.