Short-Range Order and Dynamics in Crystalline α ... - ACS Publications

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Short-Range Order and Dynamics in Crystalline α-TeO2 E. R. Barney,†,‡ A. C. Hannon,*,† and D. Holland§ †

ISIS Facility, Rutherford Appleton Laboratory, Chilton, Didcot OX11 0QX, United Kingdom Novel Photonic Glasses Research Group, Electrical Systems & Optics Research Division, Faculty of Engineering, University of Nottingham, Nottingham NG7 2RD, United Kingdom § Physics Department, University of Warwick, Coventry CV4 7AL, United Kingdom ‡

ABSTRACT: The short-range order and dynamics in crystalline α-TeO2 have been investigated by neutron and X-ray total scattering and by Rietveld refinement of neutron diffraction data. The true lengths of the two bonds in a Te
O
Te bridge are 1.882(1) and 2.117(1) Å, and the high valence, 1.293, of the strong, short bond is balanced by the low valence, 0.686, of the weak, long bond. The root-mean-square (rms) thermal variation, 0.083(1) Å, in the long bond length is nearly twice the rms thermal variation, 0.048(1) Å, in the short bond length because the largest motion of both Te and O atoms is perpendicular to the short bonds. A bond-valence model for the thermal variation in bond lengths, in which both the average and the instantaneous positions of the atoms conform to bond-valence requirements, accounts closely for the observed distribution of Te
O distances in α-TeO2. This has important implications for the interpretation of diffraction experiments on tellurite glasses.

1. INTRODUCTION In this Article, we report a short-range order study of the thermal vibrations in crystalline α-TeO2 (paratellurite), with the primary aim of enabling a better interpretation of experimental structural studies of the short-range order in tellurite glasses. Materials containing TeO2 have received much attention for their potential use in optical devices, due to the unusual properties of Te4+. Tellurite glasses, in particular, are of great interest due to their high nonlinear susceptibility, and the resultant nonlinear optical (NLO) properties that they exhibit. Several studies of the NLO properties of glasses have linked this behavior with the structure of the glasses and, in particular, with the presence of the lone pair (LP) of electrons on the Te4+ ion.1,2 The LP exerts a strong steric effect on the tellurium environment, producing an asymmetric distribution of oxygen atoms around tellurium atoms (see, for example, Figure 1) and forming voids in the structure. It has been demonstrated, in a series of studies,3
13 that the tellurium environment in a glass is altered by the addition of modifiers, such as an alkali oxide (e.g., K2O). This alters the NLO response, which is highest in tellurium-rich glasses comprised of [TeO4] units and decreases rapidly with the addition of a modifier and the associated formation of [TeO3] units.1,2 To allow the unusual properties of tellurite glasses to be utilized effectively, an understanding of the relationship between tellurium environment, glass composition, and NLO properties is necessary. However, a clear description of the structure of tellurite glasses is yet to emerge, due to the relatively complex nature of the bonding between the tellurium and oxygen atoms. The method of total scattering was originally developed mainly for structural studies of noncrystalline materials,14 but is r 2011 American Chemical Society

increasingly being used to study the short-range order in crystalline materials too.15,16 However, the correlation functions measured for tellurite glasses are much more difficult to interpret than is usually the case for oxide glasses. Most oxide glasses are based on a glass former such as SiO2 for which the distribution of nearest neighbor distances between the glass former cations and oxygen (e.g., Si
O) is narrow and well-defined, and furthermore is well separated from other interatomic distances, making it possible to determine reliable coordination numbers (e.g., nSiO) from the measured correlation function. In contrast, for tellurite glasses, there is a wide, asymmetric distribution of nearest neighbor Te
O distances that overlaps significantly with the distribution of other interatomic distances in the glass; this makes it much more challenging to determine a reliable Te
O coordination number, and a clear description of the Te
O coordination polyhedra. Instead of a narrow peak in the correlation function, the nearest neighbor Te
O distribution is broad and asymmetric, with a sharp leading edge and a broader tail extending to relatively long distances. Figure 2 shows the neutron correlation function measured for a vanadium tellurite glass containing 4 mol % V2O5,12 which has been chosen because the very small coherent neutron scattering length of vanadium, together with the low concentration present in this sample, results in a correlation function that may be expected to be very close to that for pure amorphous TeO2 (it is very difficult to make pure TeO2 in an amorphous form17). The fit shown in Figure 2 Received: August 16, 2011 Revised: December 8, 2011 Published: December 14, 2011 3707

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Figure 2. The correlation function, TN(r), for a vanadium tellurite glass with 4 mol % V2O5 (continuous black line), together with the three peaks used to fit the Te
O distribution (dashed lines) and their sum (continuous gray line).12 Figure 1. Two connected [TeO4] units from the structure of α-TeO2. The long Te
O bonds are shown dashed. Smaller (gold) spheres represent Te atoms, and larger (red) spheres represent oxygens.

demonstrates that the Te
O peak can only be fitted with a symmetric peak if several components are used, and variations in the method used to fit the distribution lead to uncertainty in the extracted tellurium coordination number.7,9 Furthermore, the Te
O peak in the correlation function of a tellurite glass covers a very large range in interatomic distance, comparable to the Te
O bond length distribution reported for crystalline K2Te4O9, which ranges from 1.816 to 2.489 Å.18 Consequently, the long distance part of the Te
O peak in the correlation function cannot be fully separated from the O
O peak, which arises from distances between oxygen atoms in the [TeO4] and [TeO3] units. In the extensive literature on tellurite glasses (for example, see refs 3, 7, and 12, and references therein), the complex Te
O peak in the correlation function has always been interpreted solely in terms of static disorder, that is, a complex distribution of bond lengths. There is, however, another possibility that has not been considered; this is that thermal vibrations also play a significant role in determining the complex distribution of nearest neighbor Te
O distances. Crystalline paratellurite, α-TeO2, has a simple structure19 with no static disorder, and so it provides an excellent opportunity to investigate this possibility. Each Te atom in α-TeO2 is bonded to four oxygen atoms, forming a [TeO4] unit, and these units are connected together by sharing a bridging oxygen between two units (see Figure 1). Each Te atom has two oxygen neighbors at a distance of about 1.88 Å, two more oxygen neighbors at a distance of about 2.12 Å, and then a further two oxygen neighbors at a distance of about 2.86 Å.19 The average ^ e
O angle for adjacent pairs of oxygen atoms is 89.9
(with O
T values ranging from 70.8
to 115.5
), which is close to the ideal value of 90
for an octahedron, and Brown20 has interpreted this geometry in terms of a distorted TeO6 octahedron, with pairs of “strong”, “intermediate”, and “weak” Te
O bonds, respectively. However, Brown’s description of the bonding is not consistent with the standard model for Te4+ according to hybridization theory, whereby the coordination of the Te atom involves a TeO4E pseudo trigonal bipyramid (TBP), based on dsp3 hybridization,21 where E represents a LP of electrons. The LP is located between the Te atom and the two oxygen atoms which are at about 2.86 Å, and thus it is incorrect to identify these oxygen

atoms as being bonded to the Te atom. As predicted by the ^ e
O Valence Shell Electron Pair Repulsion Model,22 the O
T angle for the two equatorial oxygen atoms in the TBP is less than the ideal value of 120
, and the axial Te
O bonds are longer than the equatorial Te
O bonds; with only two different bond lengths, α-TeO2 has essentially no static disorder, making it ideally suited for an investigation of the influence of thermal vibrations on the distribution of nearest neighbor Te
O distances. Each oxygen bridge involves one short equatorial Te
O bond and one long axial Te
O bond. Bridging oxygen is a key concept for structural models of glasses,23 but most oxygen bridges involve two bonds of very similar length; thus, the large difference in bond lengths for the oxygen bridge in α-TeO2 is very unusual and is of particular interest from a glass science perspective.

2. THEORY 2.1. Total Scattering. A total scattering experiment measures the total (X-ray or neutron) scattering from the sample, I(Q), where Q is the magnitude of the scattering vector for elastic scattering.14 The total scattering is the sum of the self-scattering, IS(Q) (the interference between waves scattered from the same atom), and the distinct scattering, i(Q) (the interference between waves scattered from different atoms), which is expressed simply for neutron diffraction as a differential cross-section:

dσ ¼ I N ðQ Þ ¼ I S ðQ Þ þ iN ðQ Þ dΩ

ð1Þ

For both X-rays and neutrons, the self-scattering can be calculated within an approximation for a sample of known composition and is subtracted from the corrected experimental data to give the distinct scattering. For neutron diffraction, iN(Q) can then be Fourier transformed to give a real-space correlation function: T N ðrÞ ¼ 4πF0 rð þ

2Z π

∑i cl b̅ l Þ2

Qmax 0

QiN ðQ ÞMðQ Þ sinðrQ Þ dQ

ð2Þ

where M(Q) is a modification function, which is used to reduce termination ripples in the Fourier transform. F0 is the atomic 3708

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number density, and cl and bl are, respectively, the atomic fraction and coherent neutron scattering length for element l. The resolution in real-space depends on the modification function, and principally on the maximum momentum transfer, Qmax, of the experimental data. The correlation function obtained in this way is essentially a measure of the frequency of occurrence of interatomic distances in the sample. In the case of X-ray diffraction, the situation is more complex because X-rays scatter from the electron cloud, yielding a relatively broad “electron
electron” correlation function. To account for this, the sharpened distinct scattering is obtained by first dividing by Æf(Q)æ2 (where Æf(Q)æ is the mean X-ray form factor for the sample),24,25 to approximate scattering from point sources: iX ðQ Þ ¼

I X ðQ Þ
Æf ðQ Þ2 æ Æf ðQ Þæ2

ð3Þ

The X-ray correlation function is then obtained according to T X ðrÞ ¼ 4πF0 r þ

2 Z Qmax X Qi ðQ ÞMðQ Þ sinðrQ Þ dQ π 0 N

ð4Þ

X

The total correlation functions T (r) and T (r) are the weighted sums of partial correlation functions, tll0 (r) (neglecting the effect of form factors on X-ray peak shapes): T N ðrÞ ¼

∑l, l cl b̅ l b̅ l tll ðrÞ 0

0

T X ðrÞ ¼

and

0

∑l, l 0

cl Zl Zl0 tll0 ðrÞ Z2 ð5Þ

The summation is over all of the pairwise combinations of elements in the sample. For a peak in TN(r) due to a particular pair of elements l and l0 , the coordination number, nll0 , can be calculated from the area, All0 , and position, rll0 , of the peak and the coefficient for tll0 (r), according to nll0 ¼

rll0 All0 ð2
δll0 Þcl b̅ l b̅ l0

ð6Þ

where δll0 is the Kronecker delta. 2.2. Rietveld Analysis of Neutron Diffraction Data. The diffraction pattern of a crystalline powder may be analyzed using the Rietveld profile refinement technique,26 in which the intensity, Ihkl, of the (hkl) Bragg reflection is proportional to the square modulus of the structure factor:27 Fhkl ¼

∑α b̅ α e½2πiðhx

α

þ kyα þ lzα Þ ½
Q T Æuα uα T æQ =2

e

ð7Þ

where h, k, and l are the Miller indices, and the summation is over the atoms, α, in the unit cell, which have coordinates xα, yα, and zα, and atomic displacement uα. In general, the mean square displacement of an atom is then expressed by the anisotropic temperature factor tensor: 0 1 Æu21 æ Æu1 u2 æ Æu1 u3 æ B C 2 C Uij ¼ Æui uj T æ ¼ B ð8Þ @ Æu1 u2 æ Æu2 æ Æu2 u3 æ A 2 Æu1 u3 æ Æu2 u3 æ Æu3 æ where i and j denote displacement along specific coordinate directions. Using the anisotropic thermal parameters, the mean square displacement of an atom in a particular direction, ^s (for example, in the direction of a bond), can be calculated from the

Table 1. Measured and Calculated Density of α-TeO2 density (g cm
3) measured by helium pycnometry

6.025(3)

calculated from previous crystallographic study19

6.024(1)

calculated from neutron Rietveld refinement

6.0231(3)

anisotropic tensor as27 Æu2s æ ¼ ^sT U^s

ð9Þ

2.3. Thermal Parameters. The peaks in the correlation function, T(r), are broadened due to thermal vibrations of the atoms. It has been shown experimentally28
31 that the peak widths are narrower at short distances, similar to the nearest neighbor distance, but that the width rapidly rises at longer distances to become equal to the width predicted from crystallographic thermal parameters. The reason for the small thermal width at short distances is that T(r) depends on the instantaneous distances between pairs of atoms. Therefore, a peak in T(r) is only broadened by a thermal vibration to the extent that it causes a change in the interatomic distance. Consequently, the correlated motions (principally the acoustic phonons) of atoms that are bonded (or very close together) have a smaller broadening effect on the related peak in T(r), and hence there is relatively little thermal broadening of the first few peaks in T(r). For example, a hypothetical vibrational mode in which two bonded atoms move in phase and with the same amplitude produces no broadening of the associated peak in T(r). A crystallographic analysis of diffraction data, such as Rietveld refinement, yields the time-averaged coordinates of individual atoms within the unit cell. The crystallographic thermal parameters then describe the motions of the atoms as isolated oscillators. The reason for this difference from total scattering is that Bragg scattering is necessarily elastic,32 whereas the total scattering approach uses all contributions to the scattering, elastic and inelastic. If anisotropic temperature factors have been determined, then the isotropic mean square displacement of an atom is given by diagonalizing the U matrix and then calculating:

Æu2 æiso ¼

1 traceUdiag 3

ð10Þ

If two atoms, l and l0 , are far apart in the crystal, then there is little correlation between their motions, so that on average (for the associated peak in T(r), any effect of anisotropy will be averaged out) the mean square variation in their separation is given by Æu2ll0 æav ¼ Æu2l æiso þ Æu2l0 æiso

ð11Þ

Thus, the thermal width for longer distance peaks in T(r) is given by adding, in quadrature, the isotropic mean square displacements of the two atoms concerned.

3. EXPERIMENTAL DETAILS A neutron diffraction study of α-TeO2 was carried out on commercial material (Alfa Aesar, 99.99 mol % TeO2), the density of which was measured using a Quantachrome pycnometer with helium as a working fluid (see Table 1). The neutron diffraction data were measured at room temperature using the GEM time-of-flight (t-o-f) neutron diffractometer33 at the ISIS pulsed neutron source, Rutherford Appleton Laboratory. The powder sample was contained in an 8.3 mm 3709

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Figure 3. The measured distinct scattering data: (a) i (Q) for α-TeO2, showing the high Q region in the inset, and (b) iX(Q) for α-TeO2. N

diameter can made of thin, 25 μm, vanadium foil so as to minimize the container background and absorption. Data were also collected for the empty container, an 8.34 mm diameter vanadium rod, and the empty diffractometer so that a full set of experimental corrections could be performed using the GudrunN program,34 and the ATLAS data analysis software.35 GudrunN corrects for background, absorption, inelastic, and multiple scattering effects. The data used for Rietveld refinement, and for extraction of the correlation function, were processed using exactly the same corrections procedure to ensure that the thermal factors from the two analysis methods are directly comparable. The corrected diffraction data, in both reciprocal- and real-space, are available from the ISIS disordered materials database.36 X-ray diffraction data were measured on a Panalytical X’pertPro diffractometer using Ag Kα radiation (λ = 0.560885 Å) without a monochromator, but with a rhodium filter. The sample was held in a 0.3 mm diameter silica capillary, and data were collected over 24 h to give a good signal-to-noise ratio. Background measurements of the empty diffractometer and the empty capillary were also made. Data were measured up to Qmax = 21.5 Å
1 and were corrected using the program GudrunX,34,37 which follows the same procedure as GudrunN, detailed above, but also corrects for fluorescence and Bremsstrahlung effects.

4. RESULTS 4.1. Neutron Total Scattering. Figure 3a shows the corrected distinct scattering, iN(Q), obtained by a combination of data from detector banks 3, 4, 5, 6, and 7 (at mean scattering angles of 34.26
, 61.78
, 91.80
, 147.23
, and 161.09
, respectively) on the GEM diffractometer.33 123Te has a resonant absorption of neutrons at 2.334 eV,38 and a composite iN(Q), covering the

Figure 4. The measured total correlation functions. (a) TN(r) for αTeO2, and (b) TX(r) for α-TeO2, showing experimental data (continuous line) and, where appropriate, the sum (gray) and components (dashed line) of the fit. The inset shows the fit in greater detail.

widest achievable range in Q, was obtained by combining data from the given detector banks, while carefully avoiding the region for each bank which is adversely affected by the resonance. Before carrying out a Fourier transform of the data, the low Q region was extrapolated to zero with a horizontal line. The iN(Q) data were Fourier transformed using the Lorch modification function,39 and Qmax = 40 Å
1 (beyond which oscillations in the data were not observed). The measured density agrees very closely with the value calculated from the lattice parameters reported by Thomas,19 and a final normalization of the data was achieved with reference to the first term in eq 2, so that TN(r) oscillates about zero prior to the first peak. Figure 4a shows TN(r) for α-TeO2 together with a fit to the first two Te
O peaks. The peaks were fitted using a Gaussian convoluted with the experimental real-space resolution,40 and the parameters of the fit are given in Table 2. The Te
O bond lengths (1.882 ( 0.001 and 2.117 ( 0.001 Å) and associated coordination numbers (1.95 ( 0.02 and 2.00 ( 0.02; sum 3.95 ( 0.03) are very close to the values from the previous crystallographic study.19 The close similarity of the fitted coordination numbers to the ideal values for this crystal (2 and 2; sum 4) is evidence of the success of the approach used to achieve the final normalization. 4.2. X-ray Total Scattering. Figure 3b shows the corrected distinct scattering, iX(Q), measured using X-rays emitted by a silver tube. The tellurium K edge is at a higher energy (31.8138 keV41) than characteristic silver Kα1 X-rays (21.16292 keV42), and so ideally a TeO2 sample should have low absorption and no fluorescence. However, for our setup, the X-ray beam is filtered but not monochromated, and hence the source emits a significant distribution of Bremsstrahlung X-rays, and they give rise to fluorescence from tellurium.34 The effect of fluorescence was 3710

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modeled by subtracting a constant value from the data. This is a crude approximation, neglecting the increased absorption of fluorescence at low angles, and results in an iX(Q) that is only reliable up to a Qmax of 15 Å
1. Figure 4b shows TX(r) obtained using this Qmax and a Lorch modification function.39 The weak intensity of the first peak makes fitting the Te
O distribution in TX(r) unreliable, and it has not been attempted here. However, the X-ray correlation function does yield accurate Te
Te distances that can be compared to the crystal structure as determined by Rietveld refinement. 4.3. Rietveld Refinement of Neutron Data. Rietveld refinement with anisotropic thermal parameters was performed using neutron diffraction data measured with Bank5, the high resolution (Δd/d ≈ 0.51%) 90
bank on GEM,33 because data measured at a scattering angle of 90
can usually be corrected most successfully. The program TF12LS43 was used for the Rietveld refinement, starting with numerical values from the previous crystal structure report19 and refining the parameters to fit the data. The refinement was performed over a neutron Table 2. Parameters from Fitting the First Two Te
O Peaks in TN(r) for α-TeO2 Te
O short Te
O long bond length from previous

1.878(1)

2.122(1)

crystallographic study19 (Å) bond length from current Rietveld

1.882(1)

2.117(1)

sum

refinement (Å) fitted bond length, rTeO (Å)

1.882(1)

2.117(1)

fitted thermal width, Æu2TeOæ1/2 (Å)

0.048(1)

0.083(1)

fitted coordination number, nTeO

1.95(2)

2.00(2)

3.95(3)

valence of Te
O bond

1.293(2)

0.686(2)

1.979(3)

valence  Æu2TeOæ1/2

0.062(1)

0.057(1)

t-o-f range from 3000 to 17 500 μs, which corresponds to a d-spacing range from 0.450 to 2.622 Å, or a Q-range from 2.396 to 13.977 Å
1. Figure 5 shows both the experimental data and the final refinement, for which a weighted profile R factor of 10.85 was achieved. Usually the aim in a Rietveld refinement is to achieve a value of R close to the ideal of one. However, as a consequence of the need to Fourier transform the whole diffraction pattern, the diffraction data for a total scattering study are generally collected for a much longer counting time than is typical for a conventional powder diffraction study. Hence, Rietveld refinement of a total scattering data set often has a relatively poor agreement factor, even for apparently good fits with small residuals. Long counting times lead to small statistical errors in iN(Q), and any imperfection in the description of the peak shape used in the refinement, or other sampledependent effects, will result in deviations of the fit from the data that exceed the error bars, and hence a value for the R factor that is greater than one.44 The refinement was performed in space group P41212, and the lattice parameters for α-TeO2 were found to be a = b = 4.8088(1) Å and c = 7.6112 (1) Å. The statistical errors on our lattice parameter values are smaller than those for the previous single crystal X-ray diffraction study (a = 4.8082(3) Å and c = 7.612(1) Å), 19 probably as a consequence of the high resolution that arises from the use of the t-o-f neutron diffraction technique, and the agreement with the previous values is good. Also, the density calculated from our lattice parameter values agrees well with the measured density (Table 1). The two Te
O bond lengths derived from the Rietveld refinement are the same as those determined by fitting T N(r) (Table 2). The unit cell of α-TeO2 contains one unique atom of each type, and the atomic positions and anisotropic thermal parameters are listed in Table 3.

Figure 5. Final fitted profiles (crosses, observed; line, calculated) from Rietveld refinement for α-TeO2 shown in the upper frame. Vertical tick lines indicate the positions of the allowed reflections. The lower frame shows the residuals (Iobs
Icalc)/esd for the fit. 3711

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Table 3. Atomic Coordinates and Thermal Parameters for α-TeO2 from Rietveld Refinement of the Neutron Diffraction Dataa x

atom

a

y

z

U11

U22

U33

U23

U13

U12

Te (4a)

0.0266(1)

0.0266(1)

0.0

0.0090(3)

0.0090(3)

0.0058(4)


0.0004(3)

0.0004(3)


0.0026(4)

O (8b)

0.1397(1)

0.2578(2)

0.1862(1)

0.0130(4)

0.0120(4)

0.0084(3)


0.0020(3)


0.0010(3)


0.0035(3)

Space group P41212, a = b = 4.8088(1) Å, c = 7.6112(1) Å.

5. DISCUSSION 5.1. Total Scattering. The coherent neutron scattering lengths of Te and O are virtually identical (bTe = 5.80 fm, bO = 5.803 fm45), and hence the three different partial correlation functions (see eq 5) all have significant contributions to the neutron correlation function, although those involving oxygen are larger than the Te
Te function:

T N ðrÞ ¼ 0:112tTeTe ðrÞ þ 0:224tTeO ðrÞ þ 0:224tOO ðrÞ ð12Þ However, due to the large atomic number difference (ZTe = 52, ZO = 8), the weighting factors for the X-ray correlation function are very different: T X ðrÞ ¼ 1:754tTeTe ðrÞ þ 0:540tTeO ðrÞ þ 0:083tOO ðrÞ ð13Þ The X-ray correlation function is strongly dominated by tellurium, and especially the Te
Te contribution, with the result that, while X-rays can be used to determine the position of Te atoms well, the oxygen positions can be determined much more accurately using neutrons; thus, the first two Te
O peaks are much more significant in TN(r) than in TX(r) (see Figure 4a and b). Furthermore, for our data the higher value of Qmax for the neutron measurement leads to a narrower real-space resolution, so that the first two Te
O peaks are well resolved in TN(r), but not in TX(r). The fit to these two peaks yields coordination numbers very close to the ideal value of two (see Table 2), demonstrating the reliability of the fit. The thermal width of the first peak, Æu2TeOæ1/2 = 0.048(1) Å, is significantly narrower than that of the second peak, 0.083(1) Å (see Table 2). As there is no static disorder present in the crystal (there is only one Te and one O site), this is clear evidence for a significant difference in the thermal variation in the lengths of the two bonds. The difference in thermal widths of the two bonds indicates that the motions of the two atoms connected by a short bond are more highly correlated than for two atoms connected by a long bond; a short Te
O bond is thus stronger than a long Te
O bond. The bond-valence approach can be of great use in considering the bond lengths in solids.46,47 According to this approach, the valence of an atom, i, may be expressed in the form:   Rij
dij Vi ¼ vij ¼ exp ð14Þ b j j

∑

∑

where the summation is performed over its neighbors, j. dij and vij are, respectively, the length and the valence of the bond between atoms i and j. b is an empirical constant (=0.37 Å), and Rij is the bond-valence parameter for the atom pair (i,j); the bond-valence parameter for Te
O bonds has the value RTeO = 1.977 Å.46 The valences of the two bonds given in Table 2 were calculated using eq 14.

Figure 6. The valence of a Te
O bond as a function of bond length, dTeO.46 The bond length corresponding to a valence of one is indicated by the dot-dashed line; this bond length is used to define the difference between short and long bonds.

The short and long Te
O bonds in α-TeO2 are, respectively, equatorial and axial bonds in the TeO4E pseudo TBP (Figure 1), but the bond-valence method can provide a useful insight into how the lengths of these two bonds arise as a consequence of a balance between their respective valences, and this provides the basis for our approach to the thermal variation in bond lengths, which is discussed below. According to Pauling’s charge balance principle,48 the sum of the valences of the two bonds to a bridging oxygen must equal the formal valence (two) of oxygen, and the value given in Table 2, 1.979(3), satisfies this requirement very well. Figure 6 shows the variation of the valence of a Te
O bond as a function of its bond length, dTeO. If an oxygen atom that forms a bridge between two tellurium atoms were equidistant from these two tellurium atoms, then both bonds would have a valence of one and a length equal to the bond-valence parameter, 1.977 Å. Hence, we define a Te
O bond to be “short” or “long” depending whether its length is shorter or longer than RTeO. The length of each of the two Te
O bonds can then be seen as arising from the need to balance the other bond in the Te
O
Te bridge, so that the valence sum for the oxygen is two. It is also of note that the sum of the valences of the two long bonds in a [TeO4] unit is 1.372, similar to the valence of one short bond, so that the total valence along any orthogonal axis of the [TeO4] unit (see Figure 1) is almost the same, with a value close to 4/3. This observation suggests a bond-valence interpretation of the large difference in the strengths of the long, weak axial and short, strong equatorial bonds: If there is a requirement for approximately equal valences in each of the three orthogonal axes of the [TeO4] unit, then the presence of two long bonds on the same axis is what leads to them having half the bond strength of the short bonds. Correspondingly, the other two axes have solely one short bond, due to the presence of the LP of electrons, 3712

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Table 4. Crystallographic Results for the Mean Square Displacement of the Te and O Atoms in the Directions of the Two Different Te
O Bonds, and the Summed Root-Mean-Square Displacement along Each Bond, Together with the Total Scattering Result for the Root-Mean-Square Variation in Bond Length previous study19

this work

Æu2Teæ in direction of bond (Å) Æu2Oæ1/2 in direction of bond (Å) Æu2TeOæ1/2 unc in direction of bond (Å) Æu2TeOæ1/2 from TN(r) fit (Å)

short Te
O bond

long Te
O bond

short Te
O bond

long Te
O bond

0.0061(4)

0.0115(5)

0.0074(1)

0.0083(1)

0.0066(3)

0.0082(4)

0.0053(6)

0.0059(7)

0.113(2)

0.140(2)

0.113(3)

0.119(3)

0.048(1)

0.083(1)

and hence the short bonds have twice the bond strength of the long bonds. In this way, the large difference in the bond lengths in an oxygen bridge is related to the presence of the LP. The values in Table 2 show that the product of valence and Æu2æ1/2 is roughly constant. This shows that the thermal variation in length of the Te
O bond increases as the bond strength (i.e., the valence) becomes weaker; the rms (root-mean-square) variation in Te
O bond length is approximately inversely proportional to its valence. 5.2. Rietveld Refinement of Neutron Data. The total scattering approach for a powder sample used here treats all directions in the crystal as the same, but crystallographic methods do not, and they allow us to investigate the anisotropy in the atomic motions. The anisotropic thermal parameters from crystallographic refinement were used in eq 9 to calculate the mean displacement of each atom in the directions of the two different Te
O bonds, as given in Table 4. We have also calculated the combined uncorrelated effect of these two dis2 2 1/2 placements as Æu2TeOæ1/2 to enable comparunc = (ÆuTeæ + ÆuOæ) ison between the two different directions, although it should be emphasized that this quantity is not the rms variation in Te
O bond length. Our results indicate that, according to this approach, the atomic displacements in the direction of the long bond are significantly larger than the atomic displacements in the direction of the short bond. The rms variation in the two bond lengths determined by fitting the first two peaks in TN(r) also shows that there is larger motion in the direction of the long bond than in the direction of the short bond. The actual values determined from TN(r) are markedly smaller than those from crystallographic analysis because of the effects of correlated motion, as discussed in section 2.3. Note that it is the value of Æu2TeOæ1/2 from the correlation function that gives the true rms variation in the Te
O bond length, whereas the value from crystallographic analysis, Æu2TeOæ1/2 unc, is appropriate to a pair of atoms whose motions are uncorrelated, that is, a pair of atoms with a relatively large separation. The anisotropic thermal factors from the previous X-ray diffraction study19 give values for the individual atomic displacements, Æu2Teæ and Æu2Oæ, along the two bonds that do not agree closely with the values obtained from our neutron diffraction study (see Table 4). In particular, the atomic displacements calculated from the results of the previous structural study show very little difference in the two directions and are thus inconsistent with the Te
O peak widths from TN(r). An advantage of neutron diffraction, which may lead to more accurate anisotropic thermal factors, is that both elements contribute significantly to the diffraction pattern, whereas X-ray diffraction from TeO2 is dominated by Te (see eqs 12 and 13). In addition, the superior

Figure 7. The thermal ellipsoids for a pair of connected [TeO4] units in α-TeO2, determined by Rietveld refinement (see Figure 1 for key).

Q-range for neutron diffraction is an advantage; as indicated by eq 7, the effect of thermal motion on the intensity of a Bragg peak is given by a Debye
Waller factor, e
2W,40,49,50 where 2W ¼

Q 2 Æu2 æ 2

ð15Þ

The previous study19 used X-ray diffraction with a maximum Q of 11.365 Å
1, whereas our neutron Rietveld refinement was performed up to 13.977 Å
1, and this higher maximum Q, together with the absence of the X-ray form factor problem, may mean that the anisotropic thermal factors from neutron diffraction are more reliable. The values of the anisotropic thermal parameters lead to the same conclusion as drawn from TN(r), that there is more thermal motion along the long bonds than there is along the short bonds. However, the crystallographic results have the advantage that it is possible to examine the anisotropy of the motion. Figure 7 shows the crystallographically determined thermal ellipsoids, from which it is apparent that the largest displacements of both Te and O are in a direction that is approximately perpendicular to ^ the short Te
O bond. The Te
O
Te bond angle is 138.809
, and hence an oxygen atom moving perpendicular to the short bond has a significant displacement along the direction of the ^ e
O bond angles are similar to 90
(or long bond. The O
T 180
), and hence a tellurium atom moving perpendicular to the two short bonds in a [TeO4] unit is moving in the direction of the two long bonds. Thus, both atoms have larger displacements in the direction of the long bond. 5.3. Simulations of T(r). The atomic coordinates and lattice parameters determined by Rietveld refinement were used to simulate the total correlation functions (both neutron and X-ray) using the XTAL program.51 For these simulations, the ideal 3713

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Table 5. Mean Square Displacements of Each Atom along the Three Principal Axes of Its Thermal Ellipsoid, and the Average “Isotropic” Mean Square Displacement atom

Æu21æ (Å2)

Æu22æ(Å2)

Æu23æ (Å2)

Æu2æiso (Å2)

Te

0.0057(3)

0.0064(3)

0.0117(3)

0.0079(2)

O

0.0065(3)

0.0108(3)

0.0161(3)

0.0111(2)

Table 6. Isotropically Averaged Root-Mean-Square Variation in Interatomic Distance, Æu2ll0 æ1/2 av , for Each Atom Pair, Calculated from the Average Mean Square Displacements of the Atoms atom pair

Æu2ll0 æ1/2 av (Å)

Te
Te

0.126(3)

Te
O

0.138(3)

O
O

0.149(3)

partial correlation functions were broadened for the effects both of experimental real-space resolution (arising from the finite Qmax of the experiment) and of thermal motion. For longer distances, the isotropically averaged rms variation in interatomic distance, Æu2ll0 æ1/2 av , was calculated from the anisotropic thermal parameters (determined by Rietveld refinement) using eqs 10 and 11, as shown in Tables 5 and 6. For short distances, however, the rms variation in interatomic distance has a smaller value, due to the effect of correlated motion (as discussed in section 2.3); the values used for the thermal widths of the Te
O peaks were taken directly from the peak fits to TN(r) (Table 2). In addition, the peaks arising from the distance between two oxygen atoms in a [TeO4] unit, and the distance between two Te atoms connected via a Te
O
Te bridge, also have smaller widths due to correlated motions. The thermal widths of the first O 3 3 3 O and Te 3 3 3 Te peaks were estimated and then varied to optimize the agreement between experiment and simulation, as shown in Figure 8, and the final values are given in Table 7. The potential benefit of using both neutron and X-ray data to determine shortrange structure is clearly demonstrated by the simulated correlation functions shown in Figure 8; the peaks in TN(r) arise predominantly from Te
O and O
O peaks, while TX(r) is dominated by Te
Te correlations. Although our results have been determined from neutron diffraction data, the close agreement between the simulated and experimental TX(r) shown in Figure 8b is strong confirmation of the validity of the results. Figure 9 shows a comparison between the simulated and experimental neutron correlation function, TN(r), at short distance, demonstrating the remarkably close agreement between the bond lengths determined by the Rietveld refinement and by fitting the correlation function (see Table 2). The interatomic distances determined from Rietveld refinement results represent the distance between the time-averaged positions of the atoms, whereas the interatomic distances determined from the correlation function represent the average, instantaneous distance between two atoms and are a measure of the true bond lengths. Hence, the apparent bond length determined by Rietveld refinement is always shorter than the true bond length, due to the effect of atomic vibrations.27,52,53 For phases of SiO2, the bond lengths determined by the Rietveld method have been found to be significantly shorter (for β-cristobalite the bond lengths from Rietveld analysis are of order 0.07 Å shorter54) due

Figure 8. Simulations of the total correlation function for (a) neutrons and (b) X-rays using the structure determined by Rietveld refinement of the neutron data and thermal displacements from Table 7. The most significant partial contributions are also shown: Te
O and O
O for the neutron simulation, and Te
Te for the X-ray simulation.

Table 7. Thermal Broadening Parameters Used To Simulate the Neutron and X-ray Correlation Functions atom pair

interatomic distance (Å)

Æu2ll0 æ1/2 (Å)

Te
O short bond

1.882

0.048

Te
O long bond

2.117

0.083

Te
O not bonded

>2.117

O
O (O
Te
O link)

2.690
2.947

0.138 0.100

O
O not bonded Te
Te (Te
O
Te link)

>2.947 3.744

0.149 0.090

Te
Te (not bonded)

>3.744

0.126

to rigid unit modes of the SiO4 tetrahedra playing a major role in atomic vibrations.55 The close agreement between the Te
O bond lengths determined from Rietveld and T(r) analysis (Table 2) is probably an indication that rigid unit modes do not play an important role in α-TeO2, and that the [TeO4] units do not vibrate as rigid entities. 5.4. A Bond-Valence Model for the Thermal Variation in Bond Lengths. The width of a peak in the correlation function, T(r), arises from three different causes: static disorder, thermal disorder, and real-space resolution. Each specific pair of atoms has an average separation, and static disorder is present if there is a distribution of the average separations of the pairs of atoms giving rise to the peak. Thermal disorder involves the variation in the instantaneous separations between atoms, which arises as a result of their thermal motions. Real-space resolution-broadening 3714

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Figure 9. A comparison between the simulated and measured neutron correlation function, TN(r), for short distances.

occurs as a result of the finite value of Qmax, and the choice of modification function, M(Q), and its effect can be calculated (see eqs 2 and 4). For α-TeO2, there are two different Te
O bond lengths, short and long, and because it is an ordered crystal this is the only source of static disorder in the region of the first two Te
O peaks in the correlation function. The widths of these two peaks then arise from thermal atomic motion and real-space resolution. The effect of real-space resolution can be taken into account analytically, and the challenge is then to account for the thermal widths of the two Te
O peaks. As shown above, the Te
O bond lengths observed in α-TeO2 are in very close agreement with predictions based on the Te
O bond-valence parameter, and thus we have attempted to develop a bond-valence model to account for the different shapes and widths of the first two Te
O contributions to the correlation function. The bond-valence approach46 was developed entirely on the basis of the static positions of atoms in crystals, as determined by crystallographic studies, and so it may seem surprising that we have used a bond-valence approach to develop a model for dynamical properties. However, both the static and the dynamic positions of the atoms are determined by the same interatomic potentials, and this is why it may be reasonable to account for the effect of the atomic displacements using a bondvalence approach. Our bond-valence model considers an oxygen atom that forms a bridge between two tellurium atoms (as shown in Figure 1) and attempts to account for the thermal variations in the lengths of the bonds to the two tellurium atoms. The assumptions made by our bond-valence model for the thermal variation in bond lengths are as follows: (1) The equilibrium positions of the atoms satisfy bondvalence requirements exactly. As discussed above, our results show that the static atomic positions of α-TeO2 satisfy bond-valence requirements closely. (2) The instantaneous positions of the atoms are such that bond-valence requirements are exactly satisfied. This means that, at any point in time, all atoms are thermally displaced a small distance from their equilibrium positions in such a way that their positions satisfy bond-valence requirements exactly. (Assumption 2 is made so that a tractable model can be developed. However, it should be noted that the atomic coordinates of crystal structures do not satisfy bond-valence requirements exactly, but are

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consistent within a small margin of error. Therefore, it is more likely that in reality the instantaneous atomic positions satisfy bond-valence requirements within a small margin of error too.) (3) The instantaneous Te
O bond lengths for one of the bonds to oxygen (either the short bond or the long bond) are distributed according to a Gaussian. We have used the Gaussian form for the distribution not for a fundamental reason, but because it is simple to calculate, and it appears to have a form similar to the experimental results. A bridging oxygen has one short Te
O bond and one long Te
O bond, and according to assumption 2 of our model, their instantaneous valences and bond lengths, vi and ri, satisfy (see eq 14):     RTeO
rs RTeO
rl vs ¼ exp and vl ¼ exp b b ð16Þ 48

According to Pauling’s charge balance principle, the sum of these two valences must equal the formal valence (two) of the oxygen, so that vs þ vl ¼ 2

ð17Þ

Now let ns(rs) be the distribution of short bond lengths, so that ns(rs) drs is the probability that the instantaneous length of a short bond is in the range (rs, rs + drs), and let nl(rl) be the corresponding distribution for the long bond lengths. (Note that the n(r) functions are essentially proportional to the radial distribution function, rT(r), and not to the correlation function.) If the lengths of the short and long Te
O bonds to a bridging oxygen obey eqs 16 and 17, then these two distributions are not independent, but are related by a Jacobian: vl ð18Þ nl ðrl Þ ¼ ns ðrs Þ vs This equation shows that if one of the distributions is symmetric (e.g., a Gaussian), then the other distribution must be asymmetric. We have attempted to apply our bond-valence model to simulate the first two peaks of the neutron correlation function of α-TeO2 using two different approaches. Our first approach was to assume that the short bond length distribution, ns(rs), is a Gaussian, because the short bond region of the correlation function of tellurite glasses is more symmetric than the long bond region (as in Figure 2).3,7,12,13,56 The long bond distribution, nl(rl), which corresponds to this Gaussian, was then calculated according to eqs 16
18. The two contributions were broadened to take into account the effect of real-space resolution and were added appropriately to yield a simulated correlation function, Tsim(r). The position and standard deviation of the short bond Gaussian, ns(rs), were 1.882 and 0.048 Å, respectively (the same as the fit parameters in Table 2), giving good agreement between Tsim(r) and TN(r) in the region of the short bond peak, as shown in Figure 10a. Figure 10b shows the two contributions to Tsim(r), without broadening for the effect of real-space resolution. The symmetric short bond distribution gives rise to a long bond distribution with a marked asymmetry. Figure 10a shows that this leads to a simulation, Tsim(r), which has a width roughly similar to the experiment, TN(r), in the region of the long bond peak, but has a significantly different shape. Note that by definition the model has Te
O coordination numbers of two for the short and long bond peaks, and hence the 3715

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Figure 10. The simulated correlation function according to the bondvalence model of the thermal variation in bond lengths, with a Gaussian distribution of short Te
O bonds, as compared to experiment. (a) The experimental TN(r) (thin line) together with the resolution-broadened simulation, Tsim(r) (thick line), and its residual (dotted line). (b) The experimental TN(r) (continuous line) together with the two component Te
O peaks (short dashed line for short bonds, long dashed line for long bonds) without resolution-broadening.

area under the simulated peaks necessarily agrees well with experiment; it is not a success of the model that the simulation has area similar to the experiment. Although our first approach involves an asymmetric long bond distribution and a symmetric short bond distribution, it is perhaps more likely that the short bond thermal distribution is asymmetric. Thermal motion that brings the oxygen and tellurium atoms closer together will be resisted more strongly because of the hard sphere part of the Te
O potential. Also, the variation of valence with bond length according to bond-valence (see Figure 6) indicates a strong resistance to shorter Te
O bonds. On the other hand, there is less reason for a marked asymmetry of the thermal distribution of long bonds, which experiences a less rapidly changing part of the Te
O potential and is in a region where the valence of the bond is changing less rapidly (see Figure 6). Therefore, our second approach was to simulate the long bond distribution with a Gaussian. The position and standard deviation of the long bond Gaussian, nl(rl), were 2.117 and 0.083 Å, respectively, giving good agreement between Tsim(r) and TN(r) in the region of the long bond peak, as shown in Figure 11a. The two contributions to Tsim(r), are shown in Figure 11b without broadening for the effect of real-space resolution, and it is apparent that the symmetric long bond distribution gives rise

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Figure 11. The simulated correlation function according to the bondvalence model of the thermal variation in bond lengths, with a Gaussian distribution of long Te
O bonds, as compared to experiment. The detailed key for the parts of this figure is the same as for Figure 10.

to an asymmetric short bond distribution. Figure 11a shows that this leads to a simulation, Tsim(r), which agrees fairly well with experiment, TN(r). It is notable that both the width and the shape of the simulation reproduce the experimental function quite closely in the region of the short bond peak. The agreement shown in Figure 11a is not perfect, but the form of the experimental correlation function suggests that there should be a slight asymmetry in the long bond distribution, which would improve the agreement further. The asymmetry of the short bond distribution is not readily apparent in Tsim(r), first because of the broadening by the symmetric resolution function, and second because the asymmetric tail at higher r is masked by the overlap with the contribution from the long bond distribution. The agreement between simulation and experiment shown in Figure 11a indicates that the experimental data are reasonably consistent with a model in which the instantaneous positions of the atoms are such that bond-valence requirements are satisfied. Potentially, this model provides a new approach to the consideration of thermal atomic displacements in solids; rather than considering individual vibrational modes, it considers the instantaneous positions of the atoms, as determined by the contributions of all of the thermally activated modes. The model gives a justification for the experimental observation that the thermal width of the long bond peak is significantly larger than the thermal width of the short bond peak. This result needs to be taken into account when interpreting measured correlation functions for tellurite glasses; the observation of a long bond contribution to a glass correlation function that is broader than the short bond contribution 3716

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The Journal of Physical Chemistry C is not necessarily evidence for a broader distribution of static bond lengths. On the other hand, our model indicates that the thermal broadening for the long bond peak is symmetric, or only slightly asymmetric; a significant asymmetry in the long bond contribution to a glass correlation function is thus evidence of an asymmetric distribution of static bond lengths. 5.5. Implications for Studies of Tellurite Glasses. Put simply, the bond-valence analysis of our structural results on α-TeO2 shows that the long bond in an oxygen bridge is significantly weaker than the short bond. Consequently, a narrow distribution of short bonds must be balanced by a broader distribution of long bonds. There is no evidence for static disorder in α-TeO2, and hence this difference in the widths of the two distributions is solely a dynamic effect. Unlike crystalline α-TeO2, glasses have static disorder (i.e., a distribution of bond lengths) as well as thermal disorder: For the first component of the peak fit (i.e., the short bond contribution at 1.890 Å) to the correlation function for vanadium tellurite glass (shown in Figure 2), the rms variation in Te
O distance is Æu2TeOæ1/2 = 0.068 Å,12 and this value is typical for high real-space resolution (i.e., t-o-f neutron diffraction) studies of the structure of tellurite glasses.3,7,12,13,56 This value is significantly higher than the thermal width, 0.048 Å (see Table 2), of the short bond distribution in α-TeO2 and is evidence that the glass has static disorder. If the glass has the same thermal variation in short Te
O bond length as for α-TeO2, then the rms variation in static short Te
O bond length in the glass is (0.0682
0.0482)1/2 = 0.048 Å. Thus, the static and thermal widths for the short Te
O bonds in the glass are equal in magnitude. The long bond region (∼2.0
2.3 Å) of the correlation function measured for glasses (e.g., see Figure 2) is much broader than the peak at ∼1.89 Å corresponding to short bonds, and also the long bond distribution shows a marked asymmetry. Our experimental data and simulations for α-TeO2 show that thermal vibrations are a factor causing the long bond distribution to be broader than the short bond distribution. However, our results do not provide any evidence that the asymmetry of the long bond distribution is due to thermal vibrations. Instead, we propose that the observed asymmetry in the long Te
O bond distribution in glasses (see Figure 2a) arises as a consequence of the static distribution of short bonds. Our first bond-valence approach at modeling the thermal variation in bond lengths illustrated in Figure 10 shows that a symmetric distribution of short bonds (as may arise from static disorder in the glass) leads to an asymmetric long bond distribution. A model of this type may thus be the basis for an understanding of the complex distribution of Te
O bond lengths in tellurite glasses, but with the additional need to also consider the effect of other cations (e.g., V5+ in vanadium tellurite glasses) on the oxygen bonding.

6. CONCLUSIONS Crystalline α-TeO2 has two different Te
O bonds, a short bond and a long bond, of lengths 1.882(1) and 2.117(1) Å, respectively. A measurement of the neutron correlation function shows that the rms (root-mean-square) thermal variation, 0.083 Å, in the length of the long bonds is nearly twice the rms thermal variation, 0.048 Å, in the length of the short bonds. This is an important observation for studies of tellurite glass structure, because it indicates that thermal vibrations are an important factor contributing to a relatively broad distribution of long Te
O bonds in glasses. A bond-valence model for the thermal

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variation in bond lengths has been developed, which is able to account closely for the observed distribution of Te
O distances in α-TeO2, suggesting that the instantaneous positions of the atoms, as well as the average positions, may conform to bondvalence requirements. A comparison with results for glasses indicates that there is a significant amount of static disorder in the lengths of the Te
O bonds in glasses, and it is proposed that the observed asymmetry in the distribution of long Te
O bonds in glasses arises as a consequence of the static disorder in the lengths of the short bonds.

’ AUTHOR INFORMATION Corresponding Author

*Tel.: +44-1235-445358. Fax: +44-1235-445720. E-mail: alex. [email protected].

’ ACKNOWLEDGMENT We are most grateful to Dr. Uwe Hoppe of Rostock University for providing glass diffraction data. The preliminary stage of this work was partly funded by the CLRC Centre for Materials Physics and Chemistry, Grant no. CMPC04108. Prof. Pam Thomas and Dr. Dean Keeble of Warwick University are thanked for crystallography discussions. ’ REFERENCES (1) Fargin, E.; Berthereau, A.; Cardinal, T.; LeFlem, G.; Ducasse, L.; Canioni, L.; Segonds, P.; Sarger, L.; Ducasse, A. J. Non-Cryst. Solids 1996, 203, 96. (2) Jeansannetas, B.; Blanchandin, S.; Thomas, P.; Marchet, P.; Champarnaud-Mesjard, J. C.; Merle-Mejean, T.; Frit, B.; Nazabal, V.; Fargin, E.; Le Flem, G.; Martin, M. O.; Bousquet, B.; Canioni, L.; Le Boiteux, S.; Segonds, P.; Sarger, L. J. Solid State Chem. 1999, 146, 329. (3) Johnson, P. A. V.; Wright, A. C.; Yarker, C. A.; Sinclair, R. N. J. Non-Cryst. Solids 1986, 81, 163. (4) Sekiya, T.; Mochida, N.; Ohtsuka, A.; Tonokawa, M. J. NonCryst. Solids 1992, 144, 128. (5) Shimizugawa, Y.; Maeseto, T.; Suehara, S.; Inoue, S.; Nukui, A. J. Mater. Res. 1995, 10, 405. (6) Shimizugawa, Y.; Maeseto, T.; Inoue, S.; Nukui, A. Phys. Chem. Glasses 1997, 38, 201. (7) Sinclair, R. N.; Wright, A. C.; Bachra, B.; Dimitriev, Y. B.; Dimitrov, V. V.; Arnaudov, M. G. J. Non-Cryst. Solids 1998, 232, 38. (8) Akagi, R.; Handa, K.; Ohtori, N.; Hannon, A. C.; Tatsumisago, M.; Umesaki, N. Jpn. J. Appl. Phys. 1999, 38, 160. (9) Iwadate, Y.; Mori, T.; Hattori, T.; Nishiyama, S.; Fukushima, K.; Umesaki, N.; Akagi, R.; Handa, K.; Ohtori, N.; Nakazawa, T.; Iwamoto, A. J. Alloys Compd. 2000, 311, 153. (10) McLaughlin, J. C.; Tagg, S. L.; Zwanziger, J. W.; Haeffner, D. R.; Shastri, S. D. J. Non-Cryst. Solids 2000, 274, 1. (11) McLaughlin, J. C.; Tagg, S. L.; Zwanziger, J. W. J. Phys. Chem. B 2001, 105, 67. (12) Hoppe, U.; Yousef, E.; Russel, C.; Neuefeind, J.; Hannon, A. C. Solid State Commun. 2002, 123, 273. (13) Barney, E. R. The structural role of lone pair ions in novel glasses. Ph.D. Thesis, University of Warwick, 2008. (14) Wright, A. C. J. Non-Cryst. Solids 1989, 112, 33. (15) Hibble, S. J.; Hannon, A. C.; Fawcett, I. D. J. Phys.: Condens. Matter 1999, 11, 9203. (16) Hibble, S. J.; Hannon, A. C. In From Semiconductors to Proteins: Beyond the Average Structure; Billinge, S. J. L., Thorpe, M. F., Eds.; Kluwer Academic/Plenum Publishers: New York, 2002; p 129. (17) Lambson, E. F.; Saunders, G. A.; Bridge, B.; El-Mallawany, R. A. J. Non-Cryst. Solids 1984, 69, 117. 3717

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The Journal of Physical Chemistry C

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(18) Becker, C. R.; Tagg, S. L.; Huffman, J. C.; Zwanziger, J. W. Inorg. Chem. 1997, 36, 5559. (19) Thomas, P. A. J. Phys. C 1988, 21, 4611. (20) Brown, I. D. J. Solid State Chem. 1974, 11, 214. (21) Ziolo, R. F.; Troup, J. M. J. Am. Chem. Soc. 1983, 105, 229. (22) Gillespie, R. J. Chem. Soc. Rev. 1992, 21, 59. (23) Shelby, J. E. Introduction to Glass Science and Technology; The Royal Society of Chemistry: Letchworth, 1997. (24) Patterson, A. L. Z. Kristallogr. 1935, 90, 517. (25) Soper, A. K. J. Phys.: Condens. Matter 2007, 19, 335206. (26) Rietveld, H. J. Appl. Crystallogr. 1969, 2, 65. (27) Willis, B. T. M.; Pryor, A. W. Thermal Vibrations in Crystallography; Cambridge University Press: New York, 1975. (28) Kaplow, R.; Averbach, B. L.; Strong, S. L. J. Phys. Chem. Solids 1964, 25, 1195. (29) Fessler, R. R.; Kaplow, R.; Averbach, B. L. Phys. Rev. 1966, 150, 34. (30) Lagneborg, R.; Kaplow, R. Acta Metall. 1967, 15, 13. (31) Jeong, I. K.; Heffner, R. H.; Graf, M. J.; Billinge, S. J. L. Phys. Rev. B 2003, 67, 104301. (32) Wright, A. C.; Sinclair, R. N. J. Non-Cryst. Solids 1985, 76, 351. (33) Hannon, A. C. Nucl. Instrum. Methods Phys. Res., Sect. A 2005, 551, 88. (34) Soper, A. K. Rutherford Appleton Laboratory Technical Report, RAL-TR-2011-013, 2011. (35) Hannon, A. C.; Howells, W. S.; Soper, A. K. IOP Conf. Ser. 1990, 107, 193. (36) Hannon, A. C. ISIS Disordered Materials Database; http:// www.isis.rl.ac.uk/disordered/database. (37) Soper, A. K.; Barney, E. R. J. Appl. Crystallogr. 2011, 44, 714. (38) Mughabghab, S. F.; Divadeenam, M.; Holden, N. E. Neutron Cross Sections. Vol. 1. Neutron Resonance Parameters and Thermal Cross Sections. Part A, Z=1
60; Academic Press: New York, 1981. (39) Lorch, E. J. Phys. C 1969, 2, 229. (40) Hannon, A. C.; Grimley, D. I.; Hulme, R. A.; Wright, A. C.; Sinclair, R. N. J. Non-Cryst. Solids 1994, 177, 299. (41) Bearden, J. A.; Burr, A. F. Rev. Mod. Phys. 1967, 39, 125. (42) Bearden, J. A. Rev. Mod. Phys. 1967, 39, 78. (43) David, W. I. F.; Ibberson, R. M.; Matthewman, J. C. Rutherford Appleton Laboratory Report, RAL-92-032, 1992. (44) Hill, R. J.; Madsen, I. C. J. Appl. Crystallogr. 1984, 17, 297. (45) Sears, V. F. Neutron News 1992, 3, 26. (46) Brese, N. E.; O’Keeffe, M. Acta Crystallogr., Sect. B 1991, 47, 192. (47) Hannon, A. C.; Parker, J. M. Phys. Chem. Glasses 2002, 43C, 6. (48) Pauling, L. J. Am. Chem. Soc. 1929, 51, 1010. (49) Wright, A. C.; Leadbetter, A. J. Phys. Chem. Glasses 1976, 17, 122. (50) Lonsdale, K. Acta Crystallogr. 1948, 1, 142. (51) Hannon, A. C. Rutherford Appleton Laboratory Report, RAL-93-063, 1993. (52) Busing, W. R.; Levy, H. A. Acta Crystallogr. 1964, 17, 142. (53) Barrera, G. D.; Bruno, J. A. O.; Barron, T. H. K.; Allan, N. L. J. Phys.: Condens. Matter 2005, 17, R217. (54) Dove, M. T.; Keen, D.; Hannon, A. C.; Swainson, I. P. Phys. Chem. Miner. 1997, 24, 311. (55) Tucker, M. G.; Squires, M. P.; Dove, M. T.; Keen, D. A. J. Phys.: Condens. Matter 2001, 13, 403. (56) Hoppe, U.; Gugov, I.; B€urger, H.; Jovari, P.; Hannon, A. C. J. Phys.: Condens. Matter 2005, 17, 2365.

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