Structure of diopside, enstatite, and magnesium aluminosilicate glasses: A joint approach using neutron and x-ray diffraction and solid-state NMR

Magnesium is an enigmatic element within network-forming oxide glasses. It can adopt a variety of different local environments, where Mg–O coordination numbers in the range from four to six or more are found in the structural chemistry of its crystalline oxides,^1–6 and there is debate about the structural role played by four-coordinated magnesium as it is sometimes regarded as a network-forming species.⁷ Definitive experimental information on the coordination environment of Mg²⁺ and its dependence on the glass composition is therefore desirable in order to clarify the structural role of magnesium in disordered network structures. Such information is, however, scarce because of (i) the chameleon-like nature of the magnesium coordination environment, (ii) limitations in x-ray scattering experiments that originate from the low atomic number of magnesium, and (iii) difficulty in interpreting the results obtained from ²⁵Mg solid-state nuclear magnetic resonance (NMR) experiments.⁸ This isotope of magnesium has the only NMR active nucleus, but its natural abundance is small (10%) and the nucleus of this isotope has a low gyromagnetic ratio and significant quadrupolar moment (nuclear spin I = 5/2).⁹ 

In this paper, we investigate the structure of the diopside and enstatite composition glasses (CaO)_0.25(MgO)_0.25(SiO₂)_0.5 and (MgO)_0.5(SiO₂)_0.5, respectively, along with four magnesium aluminosilicate (MgAS) glasses (MgO)_x(Al₂O₃)_y(SiO₂)_1−x−y, with x = 0.375 or 0.25 along the 50 mol. % silica tie-line (1 − x − y = 0.5), or with x = 0.3 or 0.2 along the 60 mol. % silica tie-line (1 − x − y = 0.6). The latter correspond to glass compositions for which the ratio R = x/y is either three or unity, respectively. These glasses were chosen for investigation because they cover a range of compositions where the role of Mg²⁺ is expected to change from a network modifying (R → ∞) to a predominantly charge-compensating (R = 1) role. They also serve as model systems for technological materials such as commercial display glass^10,11 and magmatic materials of interest in the geosciences.^12–15 

The glass structure was probed by combining neutron diffraction with magnesium isotope substitution,¹⁶ high energy x-ray diffraction, and ²⁹Si, ²⁷Al, and ²⁵Mg solid-state NMR spectroscopy. The diffraction and NMR experiments were performed on the same set of samples. The neutron diffraction method was selected because it is designed to deliver unambiguous site-specific structural information.¹⁷ The experiments pointed, however, to a systematic error in the published value for the bound coherent neutron scattering length of the isotope ²⁵Mg.¹⁸ This parameter was therefore remeasured in neutron powder diffraction experiments on isotopically enriched crystalline ²⁵MgO. The requirement in the neutron diffraction work of isotopically enriched samples helped to motivate a reassessment of solid-state ²⁵Mg NMR for delivering unambiguous information on the coordination environment of magnesium in glass.

This paper is organized as follows. The essential diffraction theory is given in Sec. II and the experimental methods are described in Sec. III. The results are presented in Sec. IV and are discussed in Sec. V. Conclusions are drawn in Sec. VI.

In a diffraction experiment on a powdered crystalline sample, kinematic theory shows that the intensity of elastically scattered neutrons is given by¹⁹ 

where 2θ is the scattering angle, F_hkl is the structure factor for the Miller indices h, k, l, the fractional coordinates x_j, y_j, z_j give the site of atom j within a unit cell, b_j is the coherent neutron scattering length of atom j, and SOF_j is the site occupation factor. For crystalline MgO, which has the NaCl structure type, the fractional coordinates for Mg (0, 0, 0) and O (1/2, 1/2, 1/2) are known from its space group Fm$3̄$m, and SOF_j = 1 for both sites because of the well-defined 1:1 stoichiometry of the oxide.² Hence, if b_O is known accurately but there is uncertainty associated with the value for magnesium $bMgtrial$⁠, a Rietveld refinement of the measured diffraction pattern with fixed scattering lengths will deliver a fitted value $SOFMgfit≠1$⁠. The correct value for the magnesium scattering length follows from the expression $b̃Mg=bMgtrial×SOFMgfit$⁠, where the tilde indicates that the value will be sensitive to the isotopic enrichment.

In a neutron diffraction experiment, the measured total structure factor is given by¹⁷ 

where k is the magnitude of the scattering vector, c_α is the atomic fraction of chemical species α, b_α is the bound coherent neutron scattering length of chemical species α, and S_αβ(k) is the Faber–Ziman partial structure factor for the chemical species α and β. In an x-ray diffraction experiment, the scattering lengths b_α in Eq. (2) are replaced by the k-dependent form factors f_α(k) and the total structure factor is usually rewritten as

where the mean value $f(k)=∑αcαfα(k)$⁠. Neutral atom form factors were used in the x-ray diffraction data analysis.²⁰ 

Let neutron diffraction experiments be performed on two glasses that are identical in every respect, except that one contains magnesium of natural isotopic abundance, ^natMg, and the other is isotopically enriched with ²⁵Mg. If the measured total structure factors are denoted by $FNnat(k)$ and $FN25(k)$⁠, respectively, then the difference function

where $ΔbMg=bnatMg−b25Mg$⁠, eliminates all those pair-correlation functions that do not involve magnesium. The Mg-α correlations with α ≠ Mg can also be eliminated by forming the difference function given by

The difference functions ΔF_Mg(k) and ΔF(k) therefore simplify the complexity of correlations associated with a single total structure factor.

The real-space functions corresponding to F_N(k), ΔF_Mg(k), and ΔF(k) are obtained by Fourier transformation and are denoted by D_N(r), ΔD_Mg(r), and ΔD(r), respectively, where r is a distance in real space. For example, the total pair-distribution function is given by

where the normalization factor $b2=(∑αcαbα)2$ is obtained by taking the modulus of Eq. (2) after setting all the S_αβ(k) functions to zero and ⊗ is the one-dimensional convolution operator. M(k) is a window function given by M(k) = 1 for k ≤ k_max and M(k) = 0 for k > k_max, where k_max is the maximum k-value and M(r) is the real-space manifestation of M(k). If k_max is sufficiently large that M(k) does not truncate oscillations in F_N(k), Eq. (6) will deliver the unmodified total pair-distribution function

where ρ is the atomic number density and g_αβ(r) is the partial pair-distribution function for the chemical species α and β. Similarly,

where

and the normalization factor B is obtained by taking the modulus of Eq. (4) after setting all the S_αβ(k) functions to zero. It follows that, if the first peak in ΔD_Mg(r) is attributable to Mg–O correlations alone, then the Mg–O coordination number can be obtained from

where r₁ and r₂ give the lower and upper boundaries of the peak, respectively. In addition

where

and the normalization factor C is obtained by taking the modulus of Eq. (5) after setting all the S_αβ(k) functions to zero.

At r-values below the distance of closest approach between two atoms of chemical species α and β, the partial pair-distribution function g_αβ(r) = 0. It follows that, at these small r-values, D_N(r) = ΔD_Mg(r) = ΔD(r) = −4πρr, i.e., all three functions follow the so-called density line. Provided the datasets have been correctly normalized, the application of M(r) and the finite counting statistics will generate oscillations around the density line in the measured functions. If the low-r oscillations in one of these r-space functions are set to the values given by the density line and the function is Fourier transformed to k-space, this back Fourier transform should be in agreement with the measured dataset at all k-values.²¹ The level of agreement between a measured dataset and the back Fourier transform therefore gives a measure of the accuracy of the diffraction results. In the initial analysis of the datasets using the published value for the scattering length of ²⁵Mg,¹⁸ this self-consistency check was not obeyed, which pointed to the need for a remeasurement of the scattering length for this isotope.

To accommodate the effect of the window function, each peak i in rg_αβ(r) was represented by the Gaussian function

where $rαβi$⁠, $σαβi$⁠, and $n̄αβ(i)$ are the peak position, standard deviation, and coordination number of chemical species β around α, respectively. In the neutron diffraction work, a measured r-space function was fitted to a suitable sum of these Gaussian peaks, convoluted with M(r), using the procedure described in Ref. 22. The goodness-of-fit was assessed by the parameter R_χ.²³ 

In the x-ray diffraction work, the total pair-distribution function $DX′(r)$ was obtained from Eq. (6) after replacing $FN(k)/b2$ by $SX(k)−1=FX(k)/f(k)2$⁠. The contribution of each Gaussian peak $pαβi(r)$ to S_X(k) − 1 was calculated and then Fourier transformed to real space using the same M(k) function as used for the experimental data.²² A least squares procedure was used to fit an appropriate sum of these Fourier transforms to the function $TX′(r)=TX(r)⊗M(r)$ using the program PXFIT,⁸⁰ where $TX(r)≡DX(r)+TX0(r)$ and $TX0(r)=4πρr$⁠. The fitted functions are presented as $DX′(r)=TX(r)−TX0(r)⊗M(r)=TX′(r)−4πρr$ for ease of comparison with the neutron diffraction results. The goodness-of-fit was also assessed by the parameter R_χ²³ as calculated for the $DX′(r)$ representation of the x-ray datasets.

The enstatite samples were prepared using the method described in Ref. 16 from either ^natMgO (⁠$>$99%) or isotopically enriched ²⁵MgO and SiO₂ (⁠$>$99%). The isotopic enrichment of ²⁵Mg was measured to be 1.00(2)% ²⁴Mg, 98.49(3)% ²⁵Mg, and 0.51(1)% ²⁶Mg using Inductively Coupled Plasma Mass Spectrometry (ICP-MS).

The isotopically enriched diopside glass was made from powders of ²⁵MgO (0.81% ²⁴Mg, 98.79% ²⁵Mg, 0.40% ²⁶Mg), CaCO₃ (⁠$>$99%), and SiO₂ (⁠$>$99%) using the method described in Ref. 24. The powders were calcined overnight at 800 °C and were then melted in air at a processing temperature T_proc ∼1492 °C in a Pt/10%Rh crucible. The melt was quenched by immersing the bottom of the crucible in water. The melt and quench process was repeated to ensure a homogeneous glass. The diopside glass of natural isotopic abundance was prepared from powders of ^natMgO (Aldrich, ≥99.99%), CaCO₃ (Aldrich, ≥99.999%), and SiO₂ (Alfa Aesar, 99.9%). The powders were calcined overnight at 800 °C and mixed by shaking. A batch of mass ∼3 g was transferred to a Pt/10%Rh crucible, melted by heating from room temperature to T_proc = 1550 °C in about 1 h, and held at this temperature for 1 h. The melt was quenched by placing the bottom of the crucible onto a liquid-nitrogen cooled copper block. The sample was then ground, remelted in a furnace at 1550 °C for 1 h, and quenched again. The mass loss was 0.08% on the first melt and 0.15% on the second melt, which was attributed to the loss of water re-adsorbed during the preparation procedure.

The aluminosilicate samples are designated by MgAS_x_(100-x-y) with x and y expressed in mol. %. They were prepared from either ^natMgO (Aldrich, ≥99.99%) or isotopically enriched ²⁵MgO (Isoflex, 0.32% ²⁴Mg, 99.38% ²⁵Mg, 0.30% ²⁶Mg), Al₂O₃ (Sigma-Aldrich, 99.998%), and SiO₂ (Alfa Aesar, 99.9%). The powders were calcined at 1000 °C for 2 h in a Pt or Pt/10%Rh crucible.

For the MgAS_37.5_50 and MgAS_25_50 samples, an appropriate mass of each powder was mixed by shaking. The mixture was then transferred to a Pt/10%Rh crucible and dried at 1000 °C for 10 min to remove water adsorbed during the weighing and mixing process. Each batch of mass ∼3 g was melted by heating from room temperature to a temperature T_proc of either 1550 °C (MgAS_37.5_50) or 1650 °C (MgAS_25_50) in about 1 or 1.25 h, respectively, and was then held at this temperature for 1 h. The melt was quenched by placing the bottom of the crucible onto a liquid-nitrogen cooled copper block. The mass loss on melting was $<$0.3%, which was attributed to the loss of re-adsorbed water.

The MgAS_30_60 and MgAS_20_60 glasses were made by adding SiO₂ to either MgAS_37.5_50 or MgAS_25_50, respectively, after the latter had been finely ground and dried at 200 °C for 1 h. The powders were mixed by shaking. Each batch of mass ∼2 g was melted in a Pt/10%Rh crucible by putting that crucible into a furnace at 1650 °C for 1 h. The melt was quenched by placing the bottom of the crucible onto a liquid-nitrogen cooled copper block. Each sample was ground before the melting and quench process was repeated. The mass loss was ≤0.46% on the first melt and ≤0.29% on the second melt, which was attributed to the loss of re-adsorbed water.

The stoichiometry of the enstatite glasses was measured using electron microprobe analysis and the density was measured using Archimedes’s principle with toluene as the immersion medium.¹⁶ The stoichiometry of the other glasses was taken from the batch composition and their density was measured by He pycnometry (Table I).

For each glass composition, the electronic structure will not depend on isotope enrichment, so the x-ray diffraction patterns measured for the ^natMg and ²⁵Mg samples should be the same within statistical error. Figure 1 shows this is the case to a good level of approximation for the investigated MgAS materials. The agreement for the diopside glass is less satisfactory, which likely originates from differences in the sample preparation procedure for the ^natMg and ²⁵Mg samples.

Crystalline samples of ^natMgO and ²⁵MgO, of mass 195.6 and 312.3 mg, respectively, were prepared from the enstatite glasses. The silicate was placed in a polytetrafluoroethylene (PTFE) beaker and dissolved in 40% hydrofluoric acid. The resulting solution was evaporated, several milliliters of concentrated sulfuric acid were then added, and the solution was re-evaporated. The resulting MgSO₄ was transferred to a corundum crucible and fired at 1200 °C for 3 h to obtain pure MgO. The isotopic abundance of ^natMg was measured to be 74.6(2)% ²⁴Mg, 11.0(1)% ²⁵Mg, and 14.4(1)% ²⁶Mg using ICP-MS. In comparison, the standard isotopic abundance of ^natMg, as used in the preparation of standard tables of neutron scattering lengths,^25,26 is 78.992(25)% ²⁴Mg, 10.003(9)% ²⁵Mg, and 11.005(19)% ²⁶Mg.²⁷ 

The instrument D4c at the Institut Laue-Langevin²⁸ was used to perform diffraction experiments on the powdered crystalline samples because of the high flux of incident neutrons and its ability to deliver diffraction patterns with excellent reproducibility and count-rate stability.²⁹ The incident neutron wavelength λ was either 0.4971(5) or 0.6983(4) Å, as found from a Rietveld refinement of the diffraction pattern measured for a powdered nickel standard, and the λ/2 scattering from the monochromator was removed using either a Rh or Ir filter, respectively. In each experiment, powder diffraction patterns were measured for the ²⁵MgO or ^natMgO sample in a cylindrical vanadium container of inner diameter 4.78 mm and wall thickness 0.3 mm, the empty container, the empty instrument, and a cylindrical vanadium rod of diameter 6.08 mm for normalization purposes. In these experiments, the longer wavelength leads to improved resolution of the Bragg peaks in a powder diffraction pattern at the expense of a reduction in k_max.

For each wavelength, the datasets were processed in two different ways. In the first procedure, the diffraction pattern for the sample in its container was corrected for background and container scattering by subtracting a linear combination of the diffraction patterns for the empty instrument and empty container. Corrections for sample attenuation and multiple scattering were then applied. In the second procedure, the total structure factor F_N(k) was obtained using the procedure described in Ref. 21, which takes into account sample and container attenuation, multiple scattering from both the sample and container, and inelasticity corrections. Finally, the variable was changed from k to the scattering angle 2θ by using the expression k = (4π/λ) sin(θ).¹⁷ 

A Rietveld refinement was performed on each dataset, represented as a function of 2θ, using both the FullProf (version 7.60)^30,31 and GSAS-II (version 5241)³² packages. In each refinement, the scattering length b_O = 5.805(4) fm^25,33 and a rock-salt structural model was employed. The zero-point shift and overall scale factor were initially refined, followed by the cell parameter a and thermal displacement parameters for the Mg and O atoms. The background originating from diffuse scattering was fitted using a twelfth-order Chebyshev polynomial. In FullProf, the peak shapes were fitted using the pseudo-Voigt function $pV(x)=η̃L(x)+(1−η̃)G(x)$⁠, which has both Lorentzian L(x) and Gaussian G(x) contributions, with the variable x = θ − θ_h, where θ_h is the Bragg peak position and $η̃$ $(0≤η̃≤1)$ is a weighting factor. The full-width at half-maximum (FWHM) of the Gaussian function is given by the Caglioti relation FWHM² = U tan² θ + V tan θ + W.^34,35 The parameters U, V, W, and $η̃$ were obtained by refinement along with two asymmetry parameters.³¹ In GSAS, the peak shapes were fitted using an axial-divergence-broadened pseudo-Voigt function with the FWHM of the Gaussian function given by the Caglioti relation. The parameters U, V, W, and $η̃$ were refined along with the axial-divergence asymmetry parameter SH/L.

In a Rietveld refinement, the scattering length of magnesium cannot be obtained directly (Sec. II A). Instead, it was found by refining the occupation factor for the Mg atom after the magnesium scattering length is set to the published value of $b25Mg$ = 3.62(14) fm¹⁸ for the ²⁵MgO sample or $bnatMg$ = 5.375(4) fm³⁶ for the ^natMgO sample. The set magnesium scattering length is then multiplied by the refined occupation factor to give a corrected value, which is denoted by $b̃25Mg$ or $b̃natMg$⁠, respectively, and corresponds to the isotopic enrichment of magnesium in the measured sample.

In total, for each of the ²⁵MgO and ^natMgO samples, eight values of the magnesium scattering length were generated. The values did not show any obvious dependence on (i) the λ value chosen for the diffraction experiment, (ii) the procedure used to process the diffraction patterns, or (iii) the software package used for the Rietveld refinement. The mean value of the scattering length for a given sample was therefore calculated using all the available results and the error was found from the standard deviation about the mean.

The ²⁹Si magic angle spinning (MAS) NMR spectra were recorded at room temperature at a magnetic flux density of 5.64 or 14.1 T using an Agilent DD2 or Bruker Avance Neo 600 spectrometer, respectively, equipped with commercial MAS NMR probes. The investigated samples were isotopically enriched with ²⁵Mg. The spectra for the diopside and enstatite glasses were measured at 5.64 T using 4.0 mm rotors spun at 5 kHz, 90° pulses of 4 µs length, and a relaxation delay of 900 s.⁸ The spectra for the MgAS glasses were measured at 14.1 T using 2.5 mm rotors spun at 10 kHz, 90° pulses of 2.7 µs length, and a relaxation delay of 900 s.

The ²⁷Al MAS NMR spectra were recorded at room temperature at a magnetic flux density of 14.1 T on the Bruker Avance Neo 600 spectrometer, using 2.5 mm rotors, spinning rates between 15 and 25 kHz, 90° pulses of 1.2 µs length, and a relaxation delay of 0.5 s. The samples were isotopically enriched with ²⁵Mg or contained magnesium of natural isotopic abundance. The ²⁷Al triple-quantum magic angle spinning (TQMAS) experiments³⁷ utilized the standard three-pulse zero-filtering sequence, with pulse lengths of 3.9 and 1.7 µs for the creation and conversion of triple-quantum coherences, respectively, with a power level of 120 W. The soft detection pulse applied for the zero-to-single quantum conversion had a length of 10.2 µs with a power level of 1.5 W. The relaxation delay was 0.1 s. A series of free induction decay (FID) signals in the TQMAS measurements was Fourier transformed and sheared using the software TopSpin to obtain the separate isotropic (F1) and anisotropic (F2) dimensions. The isotropic chemical shift δ_iso and quadrupolar product P_Q were derived from the sheared two-dimensional spectra using the expressions³⁸ 

and

Here, δ₁ and δ₂ are the chemical shifts in the F1 and F2 dimensions obtained from integration over the regions of interest, respectively, C_Q and η (0 ≤ η ≤ 1) are parameters that describe the magnitude and symmetry of the electric field gradient tensor interacting with the nuclear electric quadrupole moment, respectively, ν₀ is the Zeeman frequency, and S is the spin quantum number of the nucleus.

The MAS NMR spectra were simulated using the DMfit program.³⁹ Gaussian/Lorentzian line shapes were assumed for (i) the spin-1/2 ²⁹Si nucleus and (ii) the spinning sidebands associated with the |m| = 1/2 ↔ |m| = 3/2 satellite transitions for the ²⁷Al nucleus (for spin-5/2 nuclei, the second-order quadrupolar broadening effects are negligible for these transitions⁴⁰). In the above, m represents the magnetic quantum number. The ²⁷Al central-transition line shapes were simulated with the Czjzek model,⁴¹ which is based on the assumption of a wide distribution of quadrupolar coupling strengths.

The ²⁹Si and ²⁷Al chemical shifts are reported relative to tetra-methylsilane (TMS) and 1M Al(NO₃)₃ aqueous solution, respectively, using solid kaolinite [δ_iso(²⁹Si) = −91.5 ppm] and AlF₃ [δ_iso(²⁷Al) = −16 ppm] as secondary references, respectively.

The ²⁵Mg MAS NMR spectra were measured at room temperature at a magnetic flux density of 14.1 T (36.8 MHz) in the form of rotor-synchronized Hahn spin echoes on the Bruker Avance Neo 600 spectrometer using 3.2 mm rotors spun at 20.0 kHz. Experiments were performed at two different power levels corresponding to nonselective nutation frequencies ν₁ of 25 and 80 kHz (measured on cubic MgO). The effective 90° pulse lengths were optimized by maximizing the echo intensity in a t_p − τ − 2t_p − τ acquisition scheme, with τ = 50 µs and t_p values of 9 and 3.5 µs, respectively, for the two nutation frequencies employed. The relaxation delay was 0.5 s.

The static ²⁵Mg NMR spectra were measured using a Wideband Uniform Rate Smooth Truncation (WURST) pulse sequence⁴² combined with the Carr–Purcell–Meiboom–Gill (CPMG) echo train acquisition scheme.^43,44 To cover a wide-frequency range, the signal was acquired at multiple distinct base frequencies, corresponding to resonance offsets from the center of the m = 1/2 ↔ m = − 1/2 transition of ±5000, ±10 000 ppm and up to ±30 000 ppm. These spectra were recorded with the objective of testing for the presence of noncentral (|m| = 1/2 ↔ |m| = 3/2 and |m| = 3/2 ↔ |m| = 5/2) satellite transitions, which are affected to different extents by the anisotropy of nuclear electric quadrupolar interactions. The WURST experiments were performed using excitation and refocusing pulses with an identical length of 50.0 µs, an excitation bandwidth of 1 MHz, and a recycle delay of 0.2 s. The pulse spacing was set to give a spikelet separation of 6.7 kHz upon Fourier transformation and the number of Meiboom–Gill loops was set to 256.

The simulations of the ²⁵Mg MAS line shape used the Czjzek model and were performed using the ssNake⁴⁵ software. The simulations of the ²⁵Mg static line shapes also used the Czjzek model and were performed using in-house Matlab^® code with SIMPSON⁴⁶ as the kernel. Details of the simulation procedures are described elsewhere.⁸ 

The ²⁵Mg chemical shifts are reported relative to aqueous MgCl₂ solution using solid ^natMgO [δ_iso(²⁵Mg) = 27 ppm] as a secondary reference.

The neutron diffraction with isotope substitution work on the amorphous materials employed the diffractometer D4c,²⁸ chosen because of the attributes described in Sec. III B and its ability to access a wide k-range.

The incident neutron wavelength was λ = 0.4958(1) Å for the experiments on the diopside, MgAS_25_50 and MgAS_37.5_50 compositions, λ = 0.4955(1) Å for the experiment on the MgAS_20_60 composition, and λ = 0.4978(1) Å for the experiment on the MgAS_30_60 composition. In each experiment, the coarsely ground samples were held in a cylindrical vanadium container of inner diameter 4.8 mm (diopside or MgAS_30_60) or 6.8 mm (all other samples) and wall thickness 0.1 mm. The same container was used for both samples of a given glass composition in order to reduce the occurrence of systematic errors.²¹ Diffraction patterns were measured at room temperature (≃298 K) for each of the samples in its container, the empty container, the empty instrument, and a cylindrical vanadium rod of diameter 6.08 mm for normalization purposes. The relative counting times for the sample-in-container and empty container measurements were optimized in order to minimize the statistical error on the container corrected intensity.⁴⁷ Data analysis followed the procedure described in Ref. 21.

The diffraction experiments on the enstatite glasses also used D4c with λ = 0.497 Å.¹⁶ The raw datasets were reanalyzed using the new scattering length for ²⁵Mg (Sec. IV A) following the same procedure used for the other glass samples.²¹ 

The x-ray diffraction experiments were performed at room temperature using beamline 6-ID-D at the Advanced Photon Source with an incident photon energy of 100.233 keV. A Varex 4343CT amorphous silicon flat panel detector was employed with the sample to detector distance set at 311 mm as found from the diffraction pattern measured for crystalline CeO₂. Powdered glass samples were held in cylindrical Kapton polyimide tubes of 1.80(1) mm internal diameter and 0.051(6) mm wall thickness. Diffraction patterns were measured for each sample in its container, an empty container, and the empty instrument. The data were converted to one-dimensional diffraction patterns using FIT2D.⁴⁸ The background scattering, beam polarization, attenuation and Compton scattering corrections were performed using PDFgetX2.⁴⁹ 

Figure 2 shows the powder diffraction patterns for the crystalline ²⁵MgO and ^natMgO samples measured at both neutron wavelengths, where the raw datasets were processed by using the first correction procedure described in Sec. III B. The diffraction patterns were refined using both the FullProf (see Fig. 2) and GSAS-II packages. As a consistency check, the raw datasets were also processed using the second correction procedure described in Sec. III B, and the corrected diffraction patterns were refined using both Rietveld refinement packages. For a given sample, the results from all these measurements were the same, within the experimental error.

Both the ²⁵MgO and ^natMgO samples contain finite amounts of all three of the stable magnesium isotopes ²⁴Mg, ²⁵Mg, and ²⁶Mg. The system of linear equations for extracting all three of the scattering length values is therefore under-determined, so we adopted an iterative approach to obtain a consistent set of values.

For the ²⁵MgO sample, an average value $b̃25Mg$ = 3.745(10) fm was obtained from the Rietveld refinements of all the diffraction patterns, where the tilde indicates a contribution from small amounts of the ²⁴Mg and ²⁶Mg isotopes. To correct for this contribution, the scattering lengths $b24Mg$ = 5.66(3) fm²⁶ and $b26Mg$ = 4.89(15) fm¹⁸ were initially employed, along with the isotopic abundance of our ²⁵Mg sample as measured by ICP-MS (Sec. III A). Here, the reported value for $b24Mg$ was calculated from the value $bnatMg$ = 5.375(4) fm measured by neutron interferometry³⁶ using the values $b25Mg$ = 3.62(14) fm and $b26Mg$ = 4.89(15) fm found in previous neutron powder diffraction work¹⁸ and the standard isotopic abundance of ^natMg reported in Ref. 27, where $b24Mg$ is the dominant isotope (Sec. III A). Hence, a new value for $b25Mg$ was obtained, leading to a new value for $b24Mg$ and, in turn, a new value for $b25Mg$⁠. This iterative procedure converged rapidly and the high isotopic enrichment of the ²⁵MgO sample led to a small correction of the order 0.7%. The final scattering length values are listed in Table II.

For the ^natMgO sample, an average value $b̃natMg$ = 5.304(35) fm was obtained from the Rietveld refinements of all the diffraction patterns, where the tilde indicates the isotopic abundance of our ^natMg sample as measured by ICP-MS (Sec. III A). In comparison, the value $b̃natMg$ = 5.328(31) fm is obtained from the scattering lengths for the magnesium isotopes listed in Table II and the isotopic abundance of our ^natMg sample. The discrepancy between these scattering lengths is 0.45%, which is within the precision of the measurements.

For completeness, the value $bnatMg$ = 5.373(29) fm is obtained from the scattering lengths for the magnesium isotopes listed in Table II and the standard isotopic abundance of ^natMg reported in Ref. 27 (Sec. III A), which is in agreement with the value $bnatMg$ = 5.375(4) fm measured using neutron interferometry.³⁶ The scattering lengths listed in Table II are therefore self-consistent, although the values for ²⁴Mg and ²⁶Mg should be remeasured.

Finally, the F_N(k) function measured for the ²⁵MgO sample using λ = 0.4976 Å, and corrected using the measured value $b̃25Mg$ = 3.745(10) fm that takes into account the isotopic enrichment, is shown in Fig. 3. Here, the wavelength was found from the diffraction pattern measured for the powdered nickel standard by fitting the five Bragg peaks that occur in the range 12° ≤ 2θ $≤30$° using a procedure in which the wavelength and zero-angle 2θ offset are varied, keeping the lattice parameter determining the Bragg peak positions fixed and allowing the parameters describing the intensity and shape of the Lorentzian–Gaussian peak profiles to vary. The 0.1% discrepancy with the value found from the Rietveld refinement of the same nickel diffraction data is well within the precision of the measurements. For comparison with the glass structures, the corresponding $DN′(r)$ function is shown in Fig. 4 and was fitted by constraining the coordination numbers of the fitted Gaussian peaks to the values expected for the measured crystal structure.² The first peak in $DN′(r)$ originates from Mg–O correlations and is symmetrical, corresponding to a bond distance of 2.102(2) Å and coordination number $n̄MgO$ = 6.00.

Figure 5 shows the ²⁹Si MAS NMR spectra for the isotopically enriched samples. The signal-to-noise ratio is low because of the small sample size. The resonances are poorly resolved and, for the MgAS glasses, no sensible peak deconvolution was possible. Such poor resolution is expected from the large variety of possible silicon local coordination environments generated by the interaction of silicon with both the network modifier and aluminum species. The resonance positions are consistent with four-coordinated Si atoms.⁵¹ 

In alumina free glasses, the introduction of a network modifier oxide to silica leads to the breakage of Si–O–Si linkages and the formation of non-bridging oxygen (NBO) atoms. The tetrahedral [SiØ_4/2]⁰ units of pure silica glass, where Ø denotes a bridging oxygen (BO) atom, are thereby converted to tetrahedral [SiØ_3/2O]⁻, [SiØ_2/2O₂]²⁻, [SiØ_1/2O₃]³⁻, and [SiO₄]⁴⁻ units. These silicon species are designated by Qⁿ, where n is the number of Si-BO bonds within a tetrahedron, and can be identified and quantified from a deconvolution of the measured ²⁹Si MAS NMR spectra.⁵¹ The results for the diopside and enstatite glasses are reported in Ref. 8.

The spectra for vitreous aluminosilicates are, however, more complicated.^52,53 In the standard model for (MO)_x(Al₂O₃)_y(SiO₂)_1−x−y glasses containing divalent modifier cations M²⁺ with R = x/y ≥ 1, which comprises the glass compositions relevant to this study, all the aluminum is present in the form of [AlØ_4/2]⁻ units. These units can form Si–O–Al linkages and their formal negative charge is compensated by the positive charge associated with M²⁺ ions. The M²⁺ species that fulfill this role are referred to as charge-compensating cations. All the NBO atoms are associated with the silicon atoms such that the probability of an Si-NBO bond is given by p_Si-NBO = N_NBO/4N_Si, where N_NBO and N_Si are the numbers of NBO and Si atoms, respectively.

Both the ligation of Si to NBO atoms and its ligation to [AlØ_4/2]⁻ units are expected to shift the ²⁹Si resonance toward higher (less negative) frequencies.⁵⁴ Along a tie-line with fixed mol. % silica in the regime where R ≥ 1, the NBO to silicon atom ratio N_NBO/N_Si will decrease as MO is replaced by Al₂O₃, thus shifting the ²⁹Si resonance toward lower (more negative) frequencies. In contrast, as the mole fraction of alumina increases there is a greater propensity for Si–O–Al linkages, thus shifting the ²⁹Si resonance in the opposite direction, i.e., toward higher (less negative) frequencies. In consequence, there are two competing effects on the shift in position of the ²⁹Si resonance.

The composition dependence of N_NBO/N_Si can be calculated from the glass composition.⁵⁰ The number of Si–O–Al linkages per silicon atom N_Si–O–Al/N_Si can be estimated by first assuming a binomial distribution of Si-BO and Si-NBO bonds such that the fraction of Qⁿ species is given by

A binomial distribution of Si–O–Si and Si–O–Al linkages, in which the probability of an Si–O–Al linkage is given by p_Si–O–Al = N_Al/(N_Si + N_Al) where N_Al is the number of Al atoms,⁵⁵ can then be used to find the fraction of Qⁿ species that form m Si–O–Al bonds (m ≤ n). These species are denoted by $QmAln$ and their fraction is given by

In this way, the overall fraction of the Si-centered tetrahedral units that form $QmAln$ species can be deduced and the ratio N_Si–O–Al/N_Si can be calculated.

Figure 6 compares the R dependence of the ratios N_NBO/N_Si, N_Si–O–Al/N_Si, and N_NBO/N_Si–O–Al calculated for the standard model along both the 50 and 60 mol. % silica tie-lines. The ratio N_NBO/N_Si → 0 as R → 1 and the ratio N_NBO/N_Si–O–Al varies linearly (50 mol. % silica) or almost linearly (60 mol. % silica) with R. The results confirm that the observed shift of the ²⁹Si resonance toward smaller (more negative) chemical shifts in the investigated MgAS glasses as R → 1 (Table III) is associated with a decrease in the N_NBO/N_Si–O–Al ratio. This shift is smaller for the 60 mol. % silica tie-line where, for the investigated R-range, the fractional change in N_NBO/N_Si–O–Al is concomitantly smaller.

The above considerations are, nevertheless, approximate because the high cation field strength of Mg²⁺ will promote the formation of a fraction of higher-coordinated aluminum species when R ≥ 1⁵⁰ and because NBO atoms associated with Al species are neglected. The influence of these changes in speciation on the ²⁹Si chemical shift is not well documented. Nevertheless, its effect on the R dependence of N_NBO/N_Si can be predicted by the so-called GYZAS model of Ref. 50, in which the fraction of Al(IV) species is set by an empirical parameter p found from ²⁷Al MAS NMR experiments: f_Al(IV) = (1 + p)/2 when R ≥ 1. As compared to the standard model, the ratio N_NBO/N_Si is no longer zero at R = 1 (Fig. 6), i.e., the shift in the ²⁹Si resonance toward lower frequencies originating from the loss of NBO atoms will be comparatively smaller.

Figure 7 shows the central transition in the measured ²⁷Al MAS NMR spectra and the line shape fits based on the parameters listed in Table IV. The asymmetrical shape of each spectrum is characteristic of a wide distribution of quadrupolar coupling parameters. Most of the datasets could be fitted by assuming two distinct aluminum sites, with Czjzek line shape components having isotropic chemical shifts near 63 and 38 ppm. According to these chemical shifts, the first site is assigned to four-coordinated aluminum and the second site suggests the formation of five-coordinated aluminum.^52,53 For the ²⁵MgAS_25_50 sample, there is also evidence for a small fraction of six-coordinated aluminum.

To confirm the presence of five-coordinated aluminum, Fig. 8 shows representative SATellite TRansition Spectra (SATRAS) for the isotopically enriched samples. As previously discussed,⁵⁷ the allowed m = ±1/2 ↔ m = ±3/2 Zeeman transitions of I = 5/2 nuclei are broadened by the anisotropy of the nuclear electric quadrupolar interactions. However, second-order effects on these Zeeman transitions are rather weak, such that magic angle spinning transforms the inhomogeneously broadened line shape into a manifold of sharp spinning sidebands separated from each other by multiples of the spinning frequency. Thus, owing to the absence of second-order quadrupolar broadening effects, the spectroscopic resolution for these satellite-MAS-peak patterns is superior to the resolution for the central-transition line shape, as observed in Fig. 8.

The ²⁷Al MAS NMR results therefore support the presence of five-coordinated Al species in all the MgAS glasses of the present study, with a contribution to the area of a spectrum that varies between 16% and 25%. The results for the MgAS_20_60 glass are in good agreement with those previously obtained.^50,58 The average coordination number $n̄AlO$ can be deduced from the fractional areas found from the fitted spectra and are listed in Table IV. This information was used in fitting the neutron and x-ray diffraction data (Sec. IV C).

Figure 9 shows the complementary spectra measured by ²⁷Al TQMAS NMR. They confirm the presence of both four- and five-coordinated Al, and the application of Eqs. (14) and (15) yields the average isotropic chemical shifts $δiso$ and second-order quadrupolar effects $PQ$ listed in Table IV. For four-coordinated Al, the $δiso$ and $PQ$ values obtained from TQMAS NMR are in reasonable agreement. The $PQ$ values extracted from TQMAS NMR tend to be smaller than the $|CQ|$ values extracted from the MAS-Czjzek fits because of the lower triple-quantum excitation efficiency observed for the Al species at the upper end of the quadrupole coupling constant distribution. For this reason, the $|CQ|$ values from the MAS-Czjzek fits are considered more representative. For five-coordinated Al, the $PQ$ values are considerably underestimated because of the low signal-to-noise ratio. For the isotopically enriched sample ²⁵MgAS_25_50, the small peak at δ₂ ≃ 5 ppm in the TQMAS spectrum indicates the presence of some six-coordinated Al, which is confirmed by our analysis of the MAS NMR spectrum. The amount of Al(VI) listed in Table IV may, however, be an overestimate because the presence of Al(VI) is not clear from the SATRAS data for this sample (Fig. 8). In general, for a given composition, the interaction parameters extracted from the ²⁷Al NMR spectra measured for the isotopically enriched ²⁵Mg vs ^natMg glasses are the same, within experimental errors.

Figure 10 shows the solid-state ²⁵Mg MAS NMR spectra, acquired with a Hahn-echo sequence. The MAS NMR line shape for all the glasses is similar and is dominated by a wide chemical shift dispersion due to a distribution of quadrupolar coupling strengths whose effects upon the NMR line shape can be treated in the regime of second-order perturbation theory. The entire line shape could be successfully modeled using a Czjzek distribution, implemented within the ssNake program.⁴⁵ The simulations were performed assuming the finite MAS rate regime, where 16 spinning sidebands were simulated by the Carousel averaging approach implemented in the software. An unconstrained fitting approach was used to determine the best-fit parameters. A Gaussian line-broadening (LB) parameter was employed to account for a distribution of isotropic chemical shifts.

The simulation parameters are listed in Table V, where the mean chemical shift takes a value in the range from 12 to 28 ppm. There is imprecision in the value of this parameter that originates from the large linewidth. In crystalline compounds, a chemical shift between zero and 30 ppm relative to aqueous MgCl₂ solution is typical of magnesium in a six-coordinated environment Mg(VI), whereas a chemical shift of 40–80 ppm is typical of magnesium in a four-coordinated environment Mg(IV).⁹ The results therefore suggest that the Mg²⁺ ions in the different glasses are predominantly six-coordinated. As discussed in Sec. V, this interpretation is in stark contrast to the results found in the diffraction work.

We have also recorded the static ²⁵Mg NMR spectrum for the isotopically enriched sample ²⁵MgAS_30_60 using the WURST-CPMG pulse sequence with a stepwise acquisition, with the carrier frequency varied in steps of 360 kHz (10 000 ppm). The spectrum is shown in Fig. 11 on two different scales, showing the central-transition line (left column) and the spikelet patterns visible on the high-frequency side of the central resonance (right column), which can be attributed to the satellite transition manifold. The line shape (spikelet envelope) is broadly consistent with a simulation based on the parameters obtained from the fit to the MAS NMR spectrum.

The scattering lengths used in the analysis of the neutron diffraction data for the glassy materials were b_Al = 3.449(5) fm, b_Si = 4.1491(10) fm, b_O = 5.803(4) fm,²⁶ $bnatMg$ = 5.375(4) fm (Table II), and taking into account the isotopic enrichment, $b25Mg$ = 3.739(12) fm for diopside, $b25Mg$ = 3.745(10) fm for enstatite, and $b25Mg$ = 3.729(12) fm for the MgAS glasses. The measured neutron total structure factors are shown in Fig. 12. The contrast in shape with the x-ray total structure factors (Fig. 1) reflects a difference in weighting of the S_αβ(k) functions.

The measured ΔF_Mg(k) functions are illustrated in Fig. 13 and show contrast between the total structure factors measured for each glass. The first peak in the corresponding $ΔDMg′(r)$ functions (Fig. 14) is attributed to nearest-neighbor Mg–O correlations and encompasses typical Mg–O nearest-neighbor distances. For example, the Mg–O bond length is 1.923(1) Å for the MgO₄ polyhedra of crystalline MgAl₂O₄,¹ is in the range 2.09(9)–2.31(38) Å for the MgO₆ polyhedra of crystalline MgO,² MgSiO₃,³ β-Mg₂SiO₄,⁴ and Mg_0.5AlSiO₄,⁵ and is 2.27(8) Å for the MgO₈ polyhedra of crystalline Mg₃Al₂Si₃O₁₂.⁶ Unlike crystalline MgO (Fig. 4), the first Mg–O peak is broad and asymmetric and was therefore represented by two Gaussian functions. A third Gaussian function was used in the fitting procedure to constrain the high-r side of these Mg–O peaks.

The fitted $ΔDMg′(r)$ functions are shown in Fig. 14. The fitted parameters are listed in Table VI along with the weighted mean bond distance given by

where g_MgO(r) was obtained from the sum of the Gaussian functions fitted to the first peak and r₁ and r₂ define the overall extent of this peak. For each glass, the overall Mg–O coordination number $n̄MgO$(sum) obtained from the fitting procedure is in agreement with the value found directly from $ΔDMg′(r)$ by using Eq. (10) to integrate over the first peak after the application of a Lorch modification function.^59,60 The value of $n̄MgO$(sum) = 4.46(4) for enstatite glass is in accord with the value of 4.50(2) measured previously using neutron diffraction with magnesium isotope substitution.¹⁶ 

The measured ΔF(k) functions for the enstatite, diopside, and MgAS glasses are shown in Fig. 15 and the corresponding ΔD′(r) functions are shown in Fig. 16.

For enstatite glass, the first peak in ΔD′(r) originates solely from Si–O correlations and was fitted to a single Gaussian function. A second Gaussian function was used to fit the O–O correlations, which are expected at a mean distance of $r̄OO=8/3r̄SiO≃$ 2.65 Å for a tetrahedral SiO₄ unit, where $r̄SiO$ is the mean intra-tetrahedral Si–O bond distance. The fitted function is shown in Fig. 16 and the fitted parameters are listed in Table VII. The diffraction results are consistent with $n̄SiO$ = 4, as found from ²⁹Si MAS NMR experiments on enstatite glass.^8,61,62

For the MgAS glasses, there is also a contribution to the first peak in ΔD′(r) from the Al–O correlations, which causes this peak to become more asymmetric as the alumina content of the glass increases (Fig. 16). These correlations were represented by an additional Gaussian function with $n̄AlO$ set to the value found from the ²⁷Al MAS NMR experiments (Table IV). There is little change to aluminum speciation with the glass composition, so there is little change to the mean Al–O distance $r̄AlO$ (Table VII).

For diopside glass, in addition to the contributions to ΔD′(r) from nearest-neighbor Si–O and O–O correlations, there is also a contribution from nearest-neighbor Ca–O correlations. To help assess their effect, the $DX′(r)$ functions were fitted using (i) a single Gaussian function for the nearest-neighbor Si–O correlations with the coordination number set at the value $n̄SiO$ = 4 found from ²⁹Si MAS NMR experiments,^8,63 (ii) two Gaussian functions for the Mg–O correlations with the coordination numbers set at the values found from fitting $ΔDMg′(r)$ (Table VI), and (iii) single Gaussian functions for both the Ca–O and O–O correlations. The x-ray datasets were used because of their enhanced sensitivity to the calcium pair-correlation functions: The atomic number of Ca (Z = 20) is larger than for the other chemical species present in the glass. The fitted function is shown in Fig. 17 and the fitted parameters are listed in Tables VIII and IX. A Ca–O coordination number of 5.01–5.06 was obtained with an associated bond distance of 2.353(5)–2.357(5) Å. The neutron ΔD′(r) function (Fig. 16) was then fitted following the procedure adopted for enstatite, but with an additional Gaussian function for the Ca–O correlations and $n̄CaO$ constrained to the range of values found in the x-ray diffraction work. The results give $n̄CaO≃5$ with $r̄CaO$ = 2.374(5) Å (Table VII), which compare to $n̄CaO=8$ for crystalline diopside with Ca–O distances in the range 2.352–2.717 Å.⁶⁴ The $n̄CaO$ value for the glass may, however, be larger than indicated if the distribution of nearest neighbors is asymmetric, as found for the coordination environment of magnesium in the same material.