Configurational thermodynamics of aluminous pyroxenes: A generalized pair approximation

A pair approximation is used to estimate the effects of short-range order on the thermodynamic properties of aluminous clinopyroxenes on the joins diopside (CaMg-Si₂O₆)-jadeite (NaAlSi₂O₆) and diopside-CaTs (CaAl₂SiO₆). The generalized pair approximation is the simplest model for concentrated solutions which includes short-range order. Short-range order is expected to be especially significant in coupled solid solutions, such as aluminous pyroxenes, since atoms of different valence substitute for each other. The calculations show that the random model, in which the configurational entropy is calculated as if atoms on each crystallographic site mix randomly, is appropriate as a first approximation. The excess entropy relative to the random model behaves regularly, is always negative, and becomes more negative as temperature decreases or the ordering energies increase. The excess entropy relative to the random model can be modeled reasonably well with a simple power series, or Margules-type, formulation. In contrast, the excess entropy relative to a molecular model, in which the ideal activity is assumed to be equal to some mole fraction, is irregular, can be positive or negative, and even changes in sign with variations in temperature and composition. The configurational enthalpy is positive at high temperatures, and becomes negative with decreasing temperature or increasing ordering energy. The mixing enthalpy can have non-configurational contributions, in addition to the effective short-range configurational contributions considered explicitly. The pair approximation predicts an ordering transition from C2/c to P2₁/n for CaTs and diopside-CaTs solutions at moderate to low temperatures, respectively. A field where C2/c orders to C2 is also found. A higher order approximation, different relative ordering energies, or quantitative consideration of strain contributions is required to account for the C2/c to P2/n transition in omphacites. There is no justification for molecular models, in which the configurational entropy is calculated as if endmember “molecules” were mixing in the crystal, in either concentrated or dilute solutions. Molecular models do not represent limiting ordered states for coupled solid solutions.     

A computational study of Al/Si ordering in cordierite

The ordering of Al and Si in Mg cordierite Mg₂Al₄Si₅O₁₈ is considered using computer simulation. First the enthalpy of interaction J _ij between sites is derived by computer modelling 101 different Al/Si configurations and analysing their energies. They are compared with similar results for three other minerals and with ab initio calculations to assess the whole approach. Secondly the ordering process is studied using Monte Carlo simulation applied to the J _ij . The ordering phase transition temperature T _c is found as 1800°C in reasonable agreement with the experimental estimate of 1450° C. These are much lower than the estimate T _c(ABW)≈7600°C obtained from Bragg-Williams theory. Strong short-range order sets in below T _c(ABW), and the reasons for much lower temperature T _c of long-range ordering are discussed. Strong short-range also sets in very rapidly in a simulated anneal, in agreement with experiment. Thirdly an attempt is made to compare our calculated enthalpies directly with the results of NMR and calorimetry experiments, not completely successfully. A free energy ΔG≈4.6 eV for the activation barrier for ordering is suggested.     

Computer simulations of the structural and physical properties of the olivine and spinel polymorphs of Mg2SiO4

The aim of the work presented is to develop a computer simulation technique which will predict the structure and physical properties of forsterite and ringwoodite, the major mantle-forming polymorphs of Mg₂SiO₄. The technique is based upon energy minimization, in which all structural parameters are varied until the configuration with the lowest energy is achieved. The lattice energy and physical properties (e.g. elasticity and dielectric constants) are calculated from interatomic potentials, which generally include electrostatic and short-range terms. We investigate several types of traditional potential models, and present a new type of model which includes partial ionic charges and a Morse potential to describe the effect of covalency on the Si-O bond. This new form of potential model is highly successful, and not only reproduces the zero-pressure structural, elastic and dielectric properties of forsterite and ringwoodite, but also accurately describes their pressure dependence.     

Energy gap and density in SiO2 polymorphs

A relationship between the energy gap (E _G) and density (ρ) for pure SiO₂ polymorphs is derived from atomic weights and first ionization potentials of free silicon and oxygen atoms. Theoretical considerations are based on the Lorentz electron theory of solids. The eigenfrequency v₀ of elementary electron oscillators, in energy units h v₀, is identified with the energy gap of a solid. The numerical relation is expressed as \(E_G = \sqrt {139.24 - 13.8327\rho } \) is in eV. For low-quartz with a density of 2.65 g/cm³ and also for stishovite with a density of 4.28 g/cm³, the energy gap E _G=10.1 eV and 8.9 eV, respectively. From laboratory measurements for low-quartz E _G=10.2 eV. The energy gap-density relation suggests a critical density value of ρ_x ≈ 10.1 g/cm³ for an SiO₂ phase when the energy gap vanishes (E _G=0), which is consistent with estimated densities for a high pressure silica polymorph with the fluorite structure.     

Structural states of Mg cordierite III: Infrared spectroscopy and the nature of the hexagonal-modulated transition

The time evolution of the Al, Si ordering and the ferroelastic distortion of the Mg-cordierite structure are quantified on a local length scale by Hard Mode Infrared Spectroscopy (HMIS). The line profiles of various absorption peaks were measured at room temperature and at 80 K. Their integrated intensities, frequencies and half width are correlated with the interacting order parameters Q _od (Al, Si ordering), Q (displacive orthorhombic distortion) and their equivalent short-range analogs. It is shown that the phase transition between hexagonal and modulated cordierite is stepwise, as predicted earlier. The local structural state of quenched, modulated cordierite is essentially equivalent to that of the orthorhombic phase. A general concept is outlined which allows, in general, the independent determination of various interacting order parameters using HMIS.     

Pressure and temperature dependence of cation distribution in Mg-Mn olivine

Synthetic (Mg_0.51, Mn_0.49)₂SiO₄ olivine samples are heat-treated at three different pressures; 0, 8 and 12 GPa, all at the same temperature (～500° C). X-ray structure analyses on these single crystals are made in order to see the pressure effect on cation distribution. The intersite distribution coefficient of Mg and Mn in M1 and M2 sites, K _D = (Mn/Mg) _M1/(Mn/Mg) _M2, of these samples are 0.192 (0 GPa), 0.246 (8 GPa) and 0.281 (12 GPa), indicating cationic disordering with pressure. The small differences of cell dimensions between these samples are determined by powder X-ray diffraction. Cell dimensions b and c decrease, whereas a increases with pressure of equilibration. Cell volume decreases with pressure as a result of a large contraction of the b cell dimension. The effect of pressure on the free energy of the cation exchange reaction is evaluated by the observed relation between the cell volume and the site occupancy numbers. The magnitude of the pressure effect on cation distribution is only a fifth of that predicted from the observed change in volume combined with thermodynamic theory. This phenomenon is attributed to nonideality in this solid solution, and nonideal parameters are required to describe cation distribution determined in the present and previous experiments. We use a five-parameter equation to specify the cationic equilibrium on the basic of thermodynamic theory. It includes one energy parameter of ideal mixing, two parameters for nonideal effects, one volume parameter, and one thermal parameter originated from the lattice vibrational energy. The present data combined with some of the existing data are used to determine the five parameters, and the cation distribution in Mg-Mn olivine is described as a function of temperature, pressure, and composition. The basic framework of describing the cationic behavior in olivine-type mineral is worked out, although the result is preliminary: each of the determined parameters is not accurate enough to enable us to make a reliable prediction.     

Kinetics of Fe<Superscript>2+</Superscript>–Mg order–disorder in orthopyroxene: experimental studies and applications to cooling rates of rocks

We determined the forward rate constant (K⁺) for the Fe²⁺–Mg order–disorder between the M2 and M1 sites of orthopyroxene (OPx), which is described by the homogeneous reaction Fe²⁺ (M2) + Mg(M1) ↔ Mg(M2) + Fe²⁺ (M1), by both ordering and disordering experiments at isothermal condition and also by continuous cooling experiments. The rate constant was determined as a function of temperature in the range of 550–750°C, oxygen fugacity between quartz–fayalite–iron and Ni–NiO buffers, and at compositions of 16 and 50 mol% ferrosilite component. The K⁺ value derived from disordering experiment was found to be larger than that derived from ordering experiment at 550°C, while at T>580°C, these two values are essentially the same. The fO₂ dependence of the rate constant can be described by the relation K⁺ α (fO₂) ⁿ with n=5.5–6.5, which is compatible with the theoretically expected relation. The Arrhenius relation at the WI buffer condition is given by

Dislocation strain energy in the Al2SiO5 polymorphs

Molar elastic strain energy arising from dislocations in andalusite and sillimanite were calculated using equations derived from a non-core, linear elasticity model. For perfect (unit) c screw dislocations in these polymorphs, minimum dislocation densities of about 10¹⁰/cm² are necessary to significantly perturb the andalusite=sillimanite equilibrium boundary in P-T space. Compared to unit c dislocations, smaller energy perturbations arise from dissociated c screw dislocations, which are commonly observed in kyanite and sillimanite. A low computed value of stacking fault energy (～30 ergs/cm²) in these polymorphs is compatible with the large separations of dissociated dislocations in these phases. Dislocation densities in naturally occurring Al₂SiO₅ polymorphs are typically <10⁸/cm². Assuming that these densities are representative of those existing during metamorphism, as is supported by the lack of microtextures indicative of strong recovery, it is concluded that molar strain energies corresponding to observed dislocation densities (<10⁸/cm²) result in insignificant perturbation of P-T phase equilibrium boundaries of the Al₂SiO₅ polymorphs.     

Polytypism and the factors determining the stability of spinelloid structures

The relative stabilities of spinelloid polytypic modifications are analysed in terms of the relative magnitudes of the interaction energies between first, second and third neighbour structural modules. Structures which exhibit minimum interaction energies are deduced, and it is found that of all the possible polytypic modifications considered, only five spinelloid structures can possess a minimum value for the interaction energy between modules. It is shown that the spinelloid structures adopted in the system Ni₂SiO₄-NiAl₂O₄ can be satisfactorily analysed in terms of these interaction energies, and it is suggested that the dominant factors influencing the energy of interaction between structural units in this system are electrostatic charge imbalance effects. Strain energy effects, associated with cation size mismatch, however, appear to play a significant role in determining the behaviour of the analogous iron and cobalt systems.     

Electrical conductivity measurements on olivines and pyroxenes under defined thermodynamic activities as a function of temperature and pressure

Electrical conductivities of Ni₂SiO₄, Fe₂SiO₄, and MgSiO₃ were measured on synthetic powders in the temperature range 340° to 1,100° C and at pressures up to 20 kbars. For ternary compounds such as olivines and pyroxenes the control of two further variables, like the chemical activities of two components are needed, besides temperature and pressure. The activities of the corresponding binary oxides were controlled by equilibrating the samples with their neighbour-phases. Control of the oxygen partial pressure was achieved by buffer techniques. From the slopes of the lg σ vs. 1/T lines the activation energies were calculated for 10 kbar: 0.56 eV and 2.7 eV for Ni₂SiO₄ in equilibrium with SiO₂ and Ni/NiO-buffer for the temperature range 500°–800°C and 800°–1,000°C resp. 0.52 eV for Fe₂SiO₄ in equilibrium with SiO₂ and metallic iron, and 0.38 eV in equilibrium with SiO₂ and magnetite; 1.11 eV for MgSiO₃ in equilibrium with SiO₂, and 1.25 eV in equilibrium with Mg₂SiO₄.     

Crystal structures of Ni2SiO4 and Fe2SiO4 as a function of temperature and heating duration

X-ray structure refinements of Ni₂SiO₄ and Fe₂SiO₄ spinels have been made as a function of temperature and heating duration by intensity measurements at high temperatures and room pressure. The lattice parameters of Ni₂SiO₄ spinel linearly increased with temperature up to 1,000° C. However, Fe₂SiO₄ spinel exhibited a nonlinear thermal expansion and was converted to a polycrystalline mixture of spinel and olivine by heating of less than one-hour at 800° C. The ratios between the octahedral and tetrahedral bond lengths D _oct/D _tetr and between the shared and unshared edge distances (O-O)_sh/(O-O)_unsh in Fe₂SiO₄ spinel were both much larger than those in Ni₂SiO₄. These ratios increase with temperature. The Fe₂SiO₄ spinel more readily approached a activation state which facilitated the transition to the olivine structure than the Ni₂SiO₄ spinel. The lattice parameter of Ni₂SiO₄ spinel decreased with heating period at constant temperatures of 700° C and 800° C. The parameter of the quenched sample after heating for 52 h at 700° C was smaller than that of the nonheated sample. The refinements of the site occupancies at each heating duration indicated an increase in the cation deficiency in both tetrahedral and octahedral sites. Electron microprobe analysis, however, proved no significant difference in the chemical compositions between the quenched and nonheated samples. Si and Ni atoms displaced from normally occupied spinel lattice sites are assumed to settle in vacant sites defined by the cubic close packed oxygen sublattice in a manner which preserves the electric neutrality of the bulk crystal.     

A computer simulation study of Al/Si ordering in gehlenite and the paradox of the low transition temperature

The ordering of Al and Si in gehlenite is considered using computer simulation. The enthalpy of ordering ΔH per 2Al+2Si atoms is found to be 0.52 eV. It is dominated by the nearest neighbour interaction, but the analysis is carried out to fifth neighbours. The nearest neighbour interaction differs significantly from that for other materials. The structure does not have a connected network of ordering sites, which mainly accounts for the unobservably low transition temperature for Al/Si ordering. Two alternatives are given for the likely ordering pattern.     

Magnetic structure and cation distribution in (Fe,Mn)2SiO4 (olivine) by neutron diffraction

The cation distribution in the synthetic samples of olivine-type structure with composition (Fe _x Mn_1?x )₂SiO₄ was determined at room temperature and confirms previous Mössbauer results. At low temperature an antiferromagnetic ordering is observed. The magnetic structures can be described in the crystallographic cell (i.e. k=0). They are interpreted on the basis of the irreducible representations (modes) of the symmetry groups which are compatible with Pnma. The dominant modes observed for all compounds, including Fe₂SiO₄ and Mn₂SiO₄, only differ in their direction. The main direction of magnetization is dominated by the Fe²⁺ single-ion anisotropy. At 4.2K, for x=0.29, it is parallel to the c-axis, whereas for x=0.76 the direction is parallel to the b-axis. The anisotropy of the M1-sites dominates in the first case, whereas M2-anisotropy dominates in the second case. The influence of temperature is demonstrated for x=0.50 where c is the main direction at 4.2K, when it is b at 38K.     

Lattice dynamics and inelastic neutron scattering from forsterite,Mg2SiO4: Phonon dispersion relation,density of states and specific heat

Magnesium-rich olivine (Mg_0.9Fe_0.1)₂SiO₄ is considered to be a major constituent of the Earth's upper mantle. Because of its major geophysical importance, the temperature and pressure dependence of its crystal structure, elastic and dielectric constants, long-wavelength phonon modes and specific heat have been measured using a variety of experimental techniques. Theoretical study of lattice dynamics provides a means of analyzing and understanding a host of such experimental data in a unified manner. A detailed study of the lattice dynamics of forsterite, Mg₂SiO₄, has been made using a crystal potential function consisting of Coulombic and short-range terms. Quasiharmonic lattice dynamical calculations based on a rigid molecular-ion model have provided theoretical estimates of elastic constants, long-wavelength modes, phonon dispersion relation for external modes along the three high symmetry directions in the Brillouin zone, total and partial density of states and inelastic neutron scattering cross-sections. The neutron cross-sections were used as guides for the coherent inelastic neutron scattering experiment on a large single crystal using a triple axis spectrometer in the constant Q mode. The observed and predicted phonon dispersion relation show excellent agreement. The inelastically scattered neutron spectra from a powder sample have been analyzed on the basis of a phonon density of states calculated from a rigid-ion model, which includes both external and internal modes. The experimental data from a powder sample show good agreement with the calculated spectra, which include a multiphonon contribution in the incoherent approximation. The computed phonon densities of states are used to calculate the specific heat as a function of temperature using both the rigid molecular-ion and rigid ion models. These results are in very good agreement with the calorimetric measurement of the specific heat. The interatomic potential developed here can be used with some confidence to study physical properties of forsterite as a function of pressure and temperature.     

Determination of the mechanism of cation ordering in magnesioferrite (MgFe2O4) from the time- and temperature-dependence of magnetic susceptibility

The kinetics of non-convergent cation ordering in MgFe₂O₄ have been studied by measuring the Curie temperature (T _c) of synthetic samples as a function of isothermal annealing time. The starting material was a synthetic sample of near-stoichiometric MgFe₂O₄, synthesised from the oxides in air and quenched from 900 °C in water. Ordering experiments were performed using small chips of this material and annealing them at temperatures between 450 °C and 600 °C. The chips were periodically removed from the furnace, and their Curie temperatures were determined from measurements of alternating-field magnetic susceptibility (χ) as a function of temperature (T) to 400 °C. The Curie temperature of MgFe₂O₄ is very sensitive to the intracrystalline distribution of Fe³⁺ and Mg cations between tetrahedral and octahedral sites of the spinel crystal structure, and hence provides a very sensitive probe of the cation ordering process. The χ-T curve for the starting material displays a single sharp magnetic transition at a temperature of 303 °C. During isothermal annealing, the χ-T curve develops two distinct magnetic transitions; the first at a temperature corresponding to T _c for the disordered starting material and the second at a higher temperature corresponding to T _c for the equilibrium ordered phase. The size of the magnetic signal from the ordered phase increases smoothly as a function of time, until equilibrium is approached and the shape of the χ-T curve corresponds to a single sharp magnetic transition for the homogeneous ordered phase. These observations demonstrate that cation ordering in MgFe₂O₄ proceeds via a heterogeneous mechanism, involving the nucleation and growth of fine-scale domains of the ordered phase within a matrix of disordered material. Disordering experiments were performed by taking material equilibrated at 558 °C and annealing it at 695 °C. The mechanism of isothermal disordering is shown to involve nucleation and growth of disordered domains within an ordered matrix, combined with continuous disordering of the ordered matrix. This mixed mechanism of disordering may provide an explanation for the difference between the rates of ordering and disordering observed in MgFe₂O₄ using X-ray diffraction. The origin of the heterogeneous ordering/disordering mechanism is discussed in terms of the Ginzburg-Landau rate law. It is argued that heterogeneous mechanisms are likely to occur in kinetic experiments performed far from equilibrium, whereas a homogeneous mechanism may operate under slow equilibrium cooling. The implications of these observations for geospeedometry are discussed. Received: 12 May 1998 / Accepted: 25 June 1998     

Thermodynamics of open networks: Ordering and entropy in NaAlSiO4 glass,liquid, and polymorphs

The thermodynamic properties of carnegieite and NaAlSiO₄ glass and liquid have been investigated through C _p determinations from 10 to 1800 K and solution-calorimetry measurements. The relative entropies S ₂₉₈-S₀ of carnegieite and NaAlSiO₄ glass are 118.7 and 124.8 J/mol K, respectively. The low-high carnegieite transition has been observed at 966 K with an enthalpy of transition of 8.1±0.3 kJ/mol, and the enthalpy of fusion of carnegieite at the congruent melting point of 1799 K is 21.7±3 kJ/mol. These results are consistent with the reported temperature of the nepheline-carnegieite transition and available thermodynamic data for nepheline. The entropy of quenched NaAlSiO₄ glass at 0 K is 9.7±2 J/mol K and indicates considerable ordering among AlO₄ and SiO₄ tetrahedra. In the liquid state, progressive, temperature-induced Si, Al disordering could account for the high configurational heat capacity. Finally, the differences between the entropies and heat capacities of nepheline and carnegieite do not seem to conform to current polyhedral modeling of these properties     

Elasticity of the olivine and spinel polymorphs of Ni2SiO4

The single-crystal elastic moduli, c _ij x, of the olivine (α) and spinel (γ) polymorphs of nickel orthosilicate have been measured at atmospheric pressure and 20° C by Brillouin spectroscopy. The results are (Mbar), Ni₂SiO₄ olivine: c ₁₁=3.40(2), c ₂₂=2.38(2), c ₃₃=2.53(2), c ₄₄=0.71(1), c ₅₅=0.87(1), c ₆₆=0.78(1), c ₁₂=1.09(2), c ₁₃=1.10(4), c ₂₃=1.13(3), Ni₂SiO₄ spinel: c ₁₁=3.66(3), c ₄₄=1.06(1), c ₁₂=1.55(3). In comparing these results with extant elasticity data for olivine- and spinel-type compounds we find distinctive elastic characteristics related to crystal structure, and systematic trends due only to compositional variation. For silicate olivines, the longitudinal moduli decrease in the order c ₁₁>c ₃₃>c ₂₂, regardless of composition. The moduli c ₅₅ and c ₆₆ are approximately equal, and greater than c ₄₄. The former relationship is related to differences in polyhedral linkages along the crystallographic axes, whereas the latter may result from rotational freedom of SiO₄ tetrahedra in response to different directions of shear. Composition affects elasticity most directly through the relative magnitudes of \(\bar c_{12} > \; = (c_{12} + c_{13} + c_{23} )/3\) and \(\bar c_{44} = (c_{44} + c_{55} + c_{66} )/3\) . When transition-metal cations are six-coordinated by oxygen \(\bar c_{12} > \bar c_{44}\) , and when alkaline-earth cations are six-coordinated \(\bar c_{44} > \bar c_{12}\) . The longitudinal moduli along and normal to the close-packed directions of spinels are similar, reflecting the framework-like arrangement of octahedra. These longitudinal moduli exhibit little compositional dependence upon tetrahedral cations but vary dramatically with octahedral substitution. Our data indicate that tetrahedral cations affect elastic properties more as the oxygen positional parameter, u, decreases. The u parameter is also directly related to elastic anisotropy. While γ-Ni₂SiO₄ (u=0.244) is elastically isotropic, anisotropy increases rapidly as u approaches a limiting value near 0.27, and may be related to mechanical stability of the spinel structure. The longitudinal wave velocities along close-packed directions in α and γ Ni₂SiO₄ are equal. Thus, for an α-γ polymorphic pair, the assumptions of elastic isotropy of the γ phase and equal velocities in close-packed directions of α and γ allows the c _ij's and shear modulus of a spinel-structure silicate to be estimated from c ₁₁ of the corresponding α phase and the bulk modulus of the γ phase.     

The ordering behaviour of reedmergnerite,NaBSi3O8

Reedmergnerite (NaBSi₃O₈) has been synthesised hydrothermally from gels containing 10 wt.% excess SiO₂. The degree of B, Si order increases with time at constant temperature and pressure. Complete order is achieved in 250 h at 400° C, P _fluid=1 kbar and in 700 h at 300° C, 1 kbar. Lower pressures and/ or low water contents greatly reduce the rate of ordering. The ordering behaviour of reedmergnerite is insensitive to the composition of the co-existing fluid and this contrasts with the behaviour of albite. It is suggested that ordering takes place by a process of solution and re-precipitation.     

A MEG study of the olivine and spinel forms of Mg2SiO4

In this paper we present a theoretical investigation of the structures and relative stability of the olivine and spinel phases of Mg₂SiO₄. We use both a purely ionic model, based on the Modified Electron Gas (MEG) model of intermolecular forces, and a bond polarization model, developed for low pressure silica phases, to investigate the role of covalency in these compounds. The standard MEG ionic model gives adequate structural results for the two phases but incorrectly predicts the spinel phase to be more stable at zero pressure. This is mainly because the ionic modeling of Mg₂SiO₄ only accounts for 95 percent of the lattice energy. The remainder can be attributed to covalency and many-body effects. An extension of the MEG ionic model using “many-body” pair potentials corrects the phase stability error, but predicts structures which are in poorer agreement with experiment than the standard ionic approach. In addition, calculations using these many-body pair potentials can only account for 10 percent of the missing lattice energy. This model predicts an olivine-spinel phase transition of 8 GPa, below the experimental value of 20 GPa. Therefore, in order to understand more fully the stability of these structures we must consider polarization. A two-shell bond polarization model enhances the stability of both structures, with the olivine structure being stabilized more. This model predicts a phase transition at about 80 GPa, well above the observed value. Also, the olivine and spinel structures calculated with this approach are in poorer agreement with experiment than the ionic model. Therefore, based on our investigations, to properly model covalency in Mg₂SiO₄, a treatment more sophisticated than the two-shell model is needed.     

The reaction forsterite+cordierite=aluminous orthopyroxene+spinel in the system MgO-Al2O3-SiO2

Reversal experiments at 1,150–1,300°C on the reaction forsterite+cordierite=aluminous orthopyroxene+spinel in the system MgO-Al₂O₃-SiO₂ show the equilibrium to have a negativedT/dP. The slope andT-P location of this equilibrium have been modelled using available heat capacity data and various structural models which explore the configurational entropy contributions to the totalΔS. The experimental data are consistent with the aluminous orthopyroxene model of Ganguly and Ghose (1979) where limited Al disorder occurs between theM1 andM2 sites, Al-Si mixing occurs on the tetrahedralB site with the ‘aluminum avoidance’ principle maintained, and Mg-Al disorder occurs in spinel with an interchange enthalpy of 9–12 kcal mol^?1. Additionally, Al-Si disordering which occurs in the indialite structure of cordierite is inconsistent with the experimental data and all pyroxene and spinel mixing models; consequently, Si and Al in anhydrous cordierites to 1,300°C in the system MgO-Al₂O₃-SiO₂ must be largely ordered.