Cation partitioning versus temperature in spinel: optimization of site occupants

 Pavese et al. (1999) examined cation partitioning vs. temperature in a synthetic spinel of composition (Mg_0.70 Fe_0.23 ³⁺) Al_1.97 O₄ using structure data obtained from in situ neutron powder diffraction. After imposing assumptions on the site assignment of vacancies and Fe³⁺, they assigned the remaining cations by applying least-squares minimization to chemical constraints on site-occupancy sums, site-scattering, chemical composition, and thermal expansion of the octahedral site. Their proposed site assignments exhibit a sharp discontinuity in occupancy fractions versus temperature, a necessary consequence of their assumptions on vacancy assignments. In this paper we reexamine the cation partitioning of the same spinel using the constrained least-squares formulation of OccQP (Wright et al. 2000), which optimizes site occupancies without ad hoc assumptions. We obtain strikingly different results, supporting the general view that spinel undergoes a lambda transition at ∼1000 K. For all observed parameters, the residuals obtained with the OccQP assignments are lower than those obtained with the Pavese et al. assignments, in some cases by more than 1 order of magnitude. Received: 05 April 2000 / Accepted: 19 October 2000     

Thermal expansion and crystal structure of FeSi between 4 and 1173 K determined by time-of-flight neutron powder diffraction

The thermal expansion and crystal structure of FeSi has been determined by neutron powder diffraction between 4 and 1173?K. No evidence was seen of any structural or magnetic transitions at low temperatures. The average volumetric thermal expansion coefficient above room temperature was found to be 4.85(5)?×?10^?5?K^?1. The cell volume was fitted over the complete temperature range using Grüneisen approximations to the zero pressure equation of state, with the internal energy calculated via a Debye model; a Grüneisen second-order approximation gave the following parameters: θ_D=445(11)?K, V ₀=89.596(8)?ų, K ₀′=4.4(4) and γ′=2.33(3), where θ_D is the Debye temperature, V ₀ is V at T=0?K, K ₀′ is the first derivative with respect to pressure of the incompressibility and γ′ is a Grüneisen parameter. The thermodynamic Grüneisen parameter, γ_th, has been calculated from experimental data in the range 4–400?K. The crystal structure was found to be almost invariant with temperature. The thermal vibrations of the Fe atoms are almost isotropic at all temperatures; those of the Si atoms become more anisotropic as the temperature increases.     

Phase transition of synthetic zinc ferrite spinel (ZnFe2O4) at high pressure, from synchrotron X-ray powder diffraction

 Synthetic Zn-ferrite (ideally ZnFe₂O₄; mineral name: franklinite) was studied up to 37 GPa, by X-ray powder diffraction at ESRF (Grenoble, France), on the ID9 beamline; high pressure was achieved by means of a DAC. The P-V equation of state of franklinite was investigated using the Birch-Murnaghan function, and the elastic properties thus inferred [K₀ = 166.4(±3.0) GPa K₀ ^′ = 9.3(±0.6) K₀ ^″ = −0.22 GPa⁻¹] are compared with earlier determinations for MgAl-spinel and magnetite. The structural behaviour of Zn-ferrite as a function of pressure was studied by Rietveld refinements, and interpreted in the light of a phase transition from spinel to either CaTi₂O₄- or MnFe₂O₄-like structure; this transformation occurs above 24 GPa. Received: 15 March 1999 / Accepted: 22 April 2000     

High temperature structural and thermoelastic behaviour of mantle orthopyroxene: an in situ neutron powder diffraction study

The temperature induced structural evolution and thermoelastic behaviour of a natural (Pbca) orthopyroxene (Opx), with chemical formula ^M2(Mg_0.856Ca_0.025Fe²⁺ _0.119) ^M1(Mg_0.957Fe²⁺ _0.011Fe³⁺ _0.016Cr_0.011Al_0.005)Al_0.032Si_1.968O₆, from a suite of high pressure ultramafic nodules of mantle origin, have been investigated by in-situ neutron powder diffraction at several temperatures starting from 1,200°C down to 150°C. Unit-cell parameter variations as a function of T show no phase transition within this temperature range. The volume thermal expansion coefficient, α = V ⁻¹(∂V/∂T) _P0, varies linearly with T. The axial thermal expansion coefficients, α_j = l _j⁻¹(∂l _j/∂T)_P0, increase non-linearly with T. The principal Lagrangian unit-strain coefficients (ɛ//a, ɛ//b, ɛ//c), increase continuously with T. However, the orientation of the unit-strain ellipsoid appears to change with T. With decreasing T, the values of the unit-strain coefficients along the b and c axes tend to converge. The orientation at ΔT = 1,080°C is maintained down to the lowest temperature (150°C). The two non-equivalent tetrahedral chains, TA _n OA_3n and TB _n OB_3n , are kinked differently. At room-T, the TB _n OB_3n chain is more strongly kinked by about 23° than the TA _n OA_3n chain. With increasing T, the difference decreases by 3° for the TB _n OB_3n chain. The intersite cation exchange reaction between M1 and M2 (Mg²⁺ and Fe²⁺) shows a slight residual order at 1,200°C followed by reordering with decreasing temperature although seemingly not with a definite progressive trend. At the lowest temperature reached (150°C), reordering has occurred with the same value of partitioning coefficient K _D as that before heating. The absence of the expected phase transition is most likely due to the presence of minor amounts of Fe³⁺, Al, Ca and Cr which must play a crucial role on the thermoelastic behaviour and phase stability fields in natural Opx, with consequent important petrologic and geological implications.     

Structural changes and oxidation of ferroan phlogopite with increasing temperature: in situ neutron powder diffraction and Fourier transform infrared spectroscopy

The thermal response of the natural ferroan phlogopite-1M, K₂(Mg_4.46Fe_0.83Al_{0. 34}Ti_0.22)(Si_5.51Al_{2. 49})O₂₀[OH_3.59F_0.41] from Quebec, Canada, was studied with an in situ neutron powder diffraction. The in situ temperature conditions were set up at ?263, 25, 100°C and thereafter at a 100°C intervals up to 900°C. The crystal structure was refined by the Rietveld method (R _p=2.35–2.78%, R _wp=3.01–3.52%). The orientation of the O–H vector of the sample was determined by the refinement of the diffraction pattern. With increasing temperature, the angle of the OH bond to the (001) plane decreased from 87.3 to 72.5°. At room temperature, a = 5.13 Å, b = 9.20 Å, c = 10.21 Å, β = 100.06° and V(volume) = 491.69 ų. The expansion rate of the unit cell dimensions varied discontinuously with a break at 500°C. The shape of the M-octahedron underwent some significant changes such as flattening at 500°C. At temperatures above 500°C, the octahedral thickness and mean distance was decreased, while the octahedral flattening angle increased. Those results were attributed to the Fe oxidation and dehydroxylation processes. The dehydroxylation mechanism of the ferroan phlogopite was studied by the Fourier transform infrared spectroscopy (FTIR) after heated at temperatures ranging from 25 to 800°C with an electric furnace in a vacuum. In the OH stretching region, the intensity of the OH band associated with Fe²⁺(N _B-band) begun to decrease outstandingly at 500°C. The changes of the IR spectra confirmed that dehydroxylation was closely related to the oxidation in the vacuum of the ferrous iron in the M-octahedron. The decrease in the angle of the OH bond to the (001) plane, with increasing temperature, might be related to the imbalance of charge in the M-octahedra due to Fe oxidation.     

Pressure–volume–temperature equation of state of andradite and grossular, by high-pressure and -temperature powder diffraction

 The thermoelastic parameters of natural andradite and grossular have been investigated by high-pressure and -temperature synchrotron X-ray powder diffraction, at ESRF, on the ID30 beamline. The P–V–T data have been fitted by Birch-Murnaghan-like EOSs, using both the approximated and the general form. We have obtained for andradite K ₀=158.0(±1.5) GPa, (dK/dT )₀=−0.020(3) GPa K⁻¹ and α₀=31.6(2) 10⁻⁶ K⁻¹, and for grossular K ₀=168.2(±1.7) GPa, (dK/dT)₀=−0.016(3) GPa K⁻¹ and α₀=27.8(2) 10⁻⁶ K⁻¹. Comparisons between the present issues and thermoelastic properties of garnets earlier determined are carried out. Received: 7 July 2000 / Accepted: 20 October 2000     

Structure and compressibility of synthetic ZnAl2O4 (gahnite) under high-pressure conditions, from synchrotron X-ray powder diffraction

 The structural behavior of synthetic gahnite (ZnAl₂O₄) has been investigated by X-ray powder diffraction at high pressure (0–43 GPa) and room temperature, on the ID9 beamline at ESRF. The equation of state of gahnite has been derived using the models of Birch–Murnaghan, Vinet and Poirier–Tarantola, and the results have been mutually compared (the elastic bulk modulus and its derivatives versus P determined by the third-order Birch–Murnaghan equation of state are K ₀=201.7(±0.9) GPa, K ^′ ₀=7.62(±0.09) and K ^″ ₀=−0.1022 GPa⁻¹ (implied value). The compressibilities of the tetrahedral and octahedral bond lengths [0.00188(8) and 0.00142(5) GPa⁻¹ at P=0, respectively], and the␣polyhedral volume compressibilities of the four-␣and␣sixfold coordination sites [0.0057(2) and 0.0041(2) GPa⁻¹ at P=0, respectively] are discussed. Received: 15 January 2001 / Accepted: 23 April 2001     

Crystal chemistry, cation ordering and thermoelastic behaviour of CoMgSiO4 olivine at high temperature as studied by in situ neutron powder diffraction

The thermal expansion, structural changes and the site partitioning of Co and Mg in synthetic CoMgSiO₄ olivine have been studied by in situ time-of-flight neutron powder diffraction as a function of temperature, between 25 and 1,000°C. Thermal expansion of the unit cell dimensions and volume are linear within this temperature range and give no indications of a phase transition, although the thermoelastic behaviour indicates a slight strain minimum around 700°C. Co²⁺ shows a strong preference for the M1 site throughout this temperature range with an oscillatory behaviour; it decreases slightly at about 300°C, climbing up to nearly its original value at around 800°C and then decreasing by about 30% at 1,000°C. This behaviour is in contrast with that of (Fe, Mg)₂SiO₄ olivine, in which the initial Fe²⁺ site preference for the M1 site switches to the M2 site beyond a cross-over temperature. The oscillatory site preference in (CoMg)-olivine as a function of temperature is reflected in the M–O polyhedral volume changes and M–O bond lengths, as well as, thermoelastic strain and atomic thermal displacement parameters. The imbalance between the increasing vibrational and decreasing configurational entropy contributions, together with covalent bonding effects rather than crystal field contributions, seem to drive the cation partitioning in (CoMg)-olivine.     

Determination of cation distribution in (Co,Ni,Zn)2SiO4 olivine by synchrotron X-ray diffraction

 The cation distribution of Co, Ni, and Zn between the M1 and M2 sites of a synthetic olivine was determined with a single-crystal diffraction method. The crystal data are (Co_0.377Ni_0.396Zn_0.227)₂SiO₄, M _r  = 212.692, orthorhombic, Pbnm, a = 475.64(3), b = 1022.83(8), and c = 596.96(6) pm, V = 0.2904(1) nm³, Z = 4, D _x  = 4.864 g cm⁻³, and F(0 0 0) = 408.62. Lattice, positional, and thermal parameters were determined with MoKα radiation; R = 0.025 for 1487 symmetry-independent reflections with F > 4σ(F). The site occupancies of Co, Ni, and Zn were determined with synchrotron radiation employing the anomalous dispersion effect of Co and Ni. The synchrotron radiation data include two sets of intensity data collected at 161.57 and 149.81 pm, which are about 1 pm longer than Co and Ni absorption edges, respectively. The R value was 0.022 for Co K edge data with 174 independent reflections, and 0.034 for Ni K edge data with 169 reflections. The occupancies are 0.334Co + 0.539Ni + 0.127Zn in the M1 sites, and 0.420Co + 0.253Ni + 0.327Zn in the M2 sites. The compilation of the cation distributions in olivines shows that the distributions depend on ionic radii and electronegativities of constituent cations, and that the partition coefficient can be estimated from the equation: ln [(A/B)_M1/(A/B)_M2] = −0.272 (IR _A -IR _B ) + 3.65 (EN _A −EN _B ), where IR (pm) and EN are ionic radius and electronegativity, respectively. Received: 8 April 1999 / Revised, accepted: 7 September 1999     

Kinetic study of the dehydroxylation of chrysotile asbestos with temperature by in situ XRPD

 The kinetics of the dehydroxylation of chrysotile was followed in situ at high temperature using real-time conventional and synchrotron powder diffraction (XRPD). This is the first time kinetics parameters have been calculated for the dehydroxylation of chrysotile. The value of the order of the reaction mechanism calculated using the Avrami model indicates that the rate-limiting step of the reaction is a one-dimensional diffusion with an instantaneous nucleation or a deceleratory rate of nucleation of the reaction product. Hence, the rate-limiting step is the one-dimensional diffusion of the water molecules formed in the interlayer region by direct condensation of two hydrogen atoms and an oxygen atom. The calculated apparent activation energy of the reaction in the temperature range 620–750 °C is 184 kJ mol⁻¹. The diffusion path is along the axis of the fibrils forming the fibers. The amorphous or short-range ordered dehydroxylate of chrysotile is extremely unstable because forsterite readily nucleates in the Mg-rich regions. Moreover, it is less stable than the dehydroxylate of kaolinite, the so-called metakaolinite, which forms mullite at about 950 °C. This difference is interpreted in terms of the different nature of the two ions Mg²⁺ and Al³⁺ and their function as glass modifier and glass-forming ion, respectively. Received: 10 April 2002 / Accepted: 7 January 2003 Acknowledgements This work is part of a COFIN project (04 Scienze della Terra, NR 17, 2000) supported by MURST. Dr Dapiaggi is kindly acknowledged for help during the data collection at the Dipartimento di Scienze della Terra, University of Milan.     

Hydrogen bonding interactions in phase A [Mg7Si2O8(OH)6] at ambient and high pressure

Neutron powder diffraction data of phase A (Mg₇Si₂O₈(OH)₆) were collected at ambient pressure and 3.2?GPa (calculated from the compressibility of phase A) from the deuterated compound, and the structure was refined using the Rietveld method. The derived crystal structure implies that hydrogen atoms occupy two distinct sites in phase A, both forming hydrogen bonds of different lengths with the same oxygen atom. This picture is supported by IR spectra, which exhibit two absorption bands at 3400 and 3513?cm^?1 corresponding to OH stretching vibrations, and proton NMR spectra, which display two peaks with equal intensities and isotropic chemical shifts of 3.7 and 5?ppm. The D-D distance [D(1)-D(2) distance] at ambient pressure was found to be 2.09?±?0.02?Å from the neutron diffraction data and 2.09?±?0.05?Å from the NMR spectra. At 3.2?GPa, there is no statistically significant increase in the O-D interatomic distance while the hydrogen bonding interaction D···O appears to increase for one of the hydrogen sites, D(1), which has the stronger hydrogen bonding interaction compared with the other hydrogen, D(2), at ambient pressure. The O-D bond valences, determined indirectly from the D···O distances were 0.86 and 0.91 at ambient pressure, and 0.83 and 0.90?at 3.2?GPa, for D(1) and D(2), respectively.     

High temperature dehydroxylation of muscovite-2M1: a kinetic study by in situ XRPD

Muscovite-2M₁ shows a major phase transition at about 800°C, which is generally attributed in the literature to the structural dehydroxylation process, although a number of structural models have been proposed for the dehydroxylated phase, and different transformation mechanisms have also been put forward. The observed first order transformation involves an increase in the cell volume, and it is not clear to date how the cell expansion is related to the loss of hydroxyl groups. The phase change has been re-investigated here by in situ high temperature powder diffraction, both in non-isothermal and isothermal modes, to combine for the first time the structural and the kinetic interpretation of the transformation. The results unequivocally confirm that the reaction taking place in the temperature range 700–1000°C is truly a dehydroxylation process, involving the nucleation and growth of the high temperature dehydroxylated phase, having Al in 5-fold coordination. Structural simulations of the basal peaks of the powder diffraction patterns indicate that the model originally proposed by Udagawa et al. (1974) for the dehydroxylated phase correctly describes the high temperature phase. The kinetic analysis of the isothermal data using an Avrami-type model yields values for the reaction order compatible with a reaction mechanism limited by a monodimensional diffusion step. Apparent activation energy of the process in vacuum is about 251 kJ/mol. Experiments carried out at temperatures much higher than the onset temperature of the reaction show that the dehydroxylation reaction overlaps with the reaction of formation of mullite, the final product in the reaction pathway. Received: 24 April 1998 / Revised, accepted: 12 October 1998     

High-pressure decomposition of MCr2O4 spinels (M=Mg, Mn, Zn) by ab initio methods

The high-pressure equation of state of the normal spinels MgCr₂O₄ (picrochromite), MnCr₂O₄ and ZnCr₂O₄, and their reaction of decomposition into Cr₂O₃ (eskolaite) and MO (rocksalt-type) component oxides, were investigated by periodic unrestricted Hartree-Fock calculations. All-electron basis sets, and an a posteriori correction for the electron correlation energy, based on Density-Functional-Theory, were employed. Interpolation of results by the P-V Murnaghan equation of state yielded the equilibrium volume and energy, and the bulk modulus and its pressure derivative, for each of the seven phases (three spinels, three rocksalt oxides and eskolaite) considered. The simulated behaviour of interatomic distances vs pressure shows similar compressibilities of M-O bonds in both octahedral and tetrahedral coordinations. Binding energies and formation enthalpies of spinels from oxides are also computed and compared to available experimental data. The predicted decomposition pressures of Mg, Mn and Zn chromium spinels are 19, 23 and 34 GPa, respectively. The greater stability of ZnCr₂O₄ is related to Zn²⁺ being better suited to tetrahedral coordination than the other M²⁺ cations. Such results are strongly supported by the excellent agreement previously obtained between simulated (11 GPa) and experimental (13 GPa) pressures of the decomposition of MgAl₂O₄ spinel into corundum and periclase. Received: 9 February 1998 / Accepted: 12 October 1998     

Study of the electron density in MgO, (Mg0.963Fe0.037)O and Cu2O by the maximum entropy method and multipole refinements: comparison between methods

 The electron densities determined by the maximum entropy method and by the multipole refinement approach are compared with each other, in terms of some topological properties according to the Bader formalism (Laplacian and eigenvalues of the Hessian matrix of the electron density at the critical points). The cases of MgO, (Mg,Fe)O and Cu₂O are examined. The best agreement is observed for the critical points along the Mg–O, (Mg,Fe)–O and Cu–O directions, whereas larger discrepancies occur at the other critical points. Plots of the electron densities generated by the maximum entropy method and the multipole formalism along the most representative crystallographic directions contribute to elucidating the comparison between approaches. Received: 3 July 2001 / Accepted: 7 March 2002     

Determination of structures of Ca2CoSi2O7, Ca2MgSi2O7, and Ca2(Mg0.55Fe0.45)Si2O7 in incommensurate and normal phases and observation of diffuse streaks at high temperature

 The structures of Ca₂CoSi₂O₇, Ca₂MgSi₂O₇, and Ca₂(Mg_0.55Fe_0.45)Si₂O₇ have been determined in the temperature range between 297 and 773 K with arbitrary intervals. The structures of the incommensurate phase of the three compounds are characterized by the presence of the six-, seven-, and eight-coordinated Ca–O polyhedra and of the bundles along the c-axes consisting of four arrays of the six-coordinated Ca–O polyhedra and an array of T¹O₄ (T¹: Co, Mg, or Mg–Fe) tetrahedra in the structures. The number of bundles in each material decreases at elevated temperatures. The incommensurate phase undergoes a phase transition into the normal phase at 493 K in Ca₂CoSi₂O₇, at 360 K in Ca₂MgSi₂O₇, and at 510 K in Ca₂(Mg_0.55Fe_0.45)Si₂O₇. The features of the structures of the normal phase are almost the same as those found in the basic structures (the averaged structures of the incommensurate structures), and this fact implies that the characteristics of the structures, such as the six-coordinated Ca–O polyhedra or fragments of the bundles, should be partially preserved at higher temperatures both in the incommensurate structures and also in the structures of the normal phase. Analyses of anisotropic displacement parameters clarified that disorder of the modulation waves is developed in the structures at higher temperatures. The evolution of a disorder in the structures was ascertained by observation of the circular diffuse streaks in the vicinity of the transition temperature between the incommensurate and normal phases. Received: 3 July 2000 / Accepted: 26 October 2000     

Variation of the oxygen content and point defects in olivines, (Fe x Mg 1−x )2SiO4, 0.2 ≤ x ≤ 1.0

 The variation of the oxygen content in olivines, (Fe _x Mg_{1− x} )₂SiO₄, with 0.2 ≤ x ≤ 1.0, was investigated by thermogravimetric measurements. Mass changes occurring upon oxygen activity changes were measured as a function of oxygen activity and cationic composition at 1130 and 1200 °C. During the measurements the samples were in direct contact with gases containing CO, CO₂ and N₂ and, at a few spots at the bottom of the sample stack, also with SiO₂. By fitting experimental data of mass changes to equations derived using point defect thermodynamics, it was shown for olivines with 0.2 ≤ x ≤ 1.0 at 1130 °C and 0.2 ≤ x ≤ 0.7 at 1200 °C within the oxygen activity ranges investigated that the observed variations in the oxygen contents are compatible with cation vacancies and Fe³⁺ ions on M sites and Fe³⁺ ions on silicon sites as majority defects if it is assumed that only three types of point defects occur as majority defects. Different cases were considered, closed systems, taking into account that ξ=[Si]/([Si]+[Fe]+[Mg]) is not necessarily equal to 1/3, and olivines in equilibrium with SiO₂ or pyroxenes. The oxygen content variations observed in this study are significantly smaller than those reported previously in the literature. It is proposed that these differences are related to the dissolution of Fe into noble metal containers used as sample holders in earlier studies and/or to the presence of secondary phases. Received: 1 November 1995 / Accepted: 15 September 2002 Acknowledgements This work was supported by the Cornell Center for Materials Research (CCMR), a Materials Research Science and Engineering Center of the National Science Foundation (DMR-0079992). The authors thank Mr. Daniel M. DiPasquo and Mr. Jason A. Schick for helping in experimental work.     

Crystal chemistry of ferric iron in (Mg,Fe)(Si,Al)O<Subscript>3</Subscript> majorite with implications for the transition zone

Fifteen samples of (Mg,Fe)SiO₃ majorite with varying Fe/Mg composition and one sample of (Mg,Fe)(Si,Al)O₃ majorite were synthesized at high pressure and temperature under different conditions of oxygen fugacity using a multianvil press, and examined ex situ using X-ray diffraction and Mössbauer and optical absorption spectroscopy. The relative concentration of Fe³⁺ increases both with total iron content and increasing oxygen fugacity, but not with Al concentration. Optical absorption spectra indicate the presence of Fe²⁺–Fe³⁺ charge transfer, where band intensity increases with increasing Fe³⁺ concentration. Mössbauer data were used in conjunction with electron microprobe analyses to determine the site distribution of all cations. Both Al and Fe³⁺ substitute on the octahedral site, and charge balance occurs through the removal of Si. The degree of Mg/Si ordering on the octahedral sites in (Mg,Fe)SiO₃ majorite, which affects both the c/a ratio and the unit cell volume, is influenced by the thermal history of the sample. The Fe³⁺ concentration of (Mg,Fe)(Si,Al)O₃ majorite in the mantle will reflect prevailing redox conditions, which are believed to be relatively reducing in the transition zone. Exchange of material across the transition boundary to (Mg,Fe) (Si,Al)O₃ perovskite would then require a mechanism to oxidize sufficient iron to satisfy crystal-chemical requirements of the lower-mantle perovskite phase.     

In situ high-temperature X-ray and neutron powder diffraction study of cation partitioning in synthetic Mg(Fe<Subscript>0.5</Subscript>Al<Subscript>0.5</Subscript>)<Subscript>2</Subscript>O<Subscript>4</Subscript> spinel

The intra-crystalline cation partitioning over T- and M-sites in a synthetic Mg(Fe,Al)₂O₄ spinel sample has been determined as a function of temperature by Rietveld structure refinements from powder diffraction data, combining in situ high-temperature neutron powder diffraction (NPD; POLARIS diffractometer, at ISIS, Rutherford Appleton Laboratory, UK), to determine the Mg and Al occupancy factors, with in situ high-temperature X-ray powder diffraction, to fix the Fe³⁺ distribution. The results obtained agree with a two-stage reaction, in which an initial exchange between Fe³⁺ and Mg, the former leaving and the latter entering tetrahedral sites, is successively followed by a rearrangement involving also Al. The measured cation distribution has then been compared and discussed with that calculated by the Maximum Configuration Entropy principle, for which only NPD patterns have been used. The cation partitioning has finally been interpreted in the light of the configuration model of O’Neill and Navrotsky.