A new method to calculate end-member thermodynamic properties of minerals from their constituent polyhedra II: heat capacity, compressibility and thermal expansion

Thermodynamic calculations in petrology are generally performed at pressures and temperatures beyond the standard state conditions. Accurate prediction of mineral equilibria therefore requires knowledge of the heat capacity, thermal expansion and compressibility for the minerals involved. Unfortunately, such data are not always available. In this contribution we present a data set to estimate the heat capacity, thermal expansion and compressibility of mineral end‐members from their constituent polyhedra, based on the premise that the thermodynamic properties of minerals can be described by a linear combination of the fractional properties of their constituents. As such, only the crystallography of the phase of interest needs to be known. This approach is especially powerful for hypothetical mineral end‐members and for minerals, for which the experimental determination of their thermodynamic properties is difficult. The data set consists of the properties for 35 polyhedra in the system K–Na–Ca–Li–Be–Mg–Mn–Fe–Co–Ni–Zn–Al–Ti–Si–H, determined by multiple linear regression analysis on a data set of 111 published end‐member thermodynamic properties. The large number of polyhedra determined allows calculation of a much larger variety of phases than was previously possible, and the choice of constituents together with the large number of thermodynamic input data results in estimates with associated uncertainty of generally <5%. The quality of the data appears to be sufficiently accurate for thermodynamic modelling as demonstrated by modelling the stability of margarite in the CASH system and the position of the talc–staurolite–chloritoid–pyrope absent invariant point in the KMASH system. In both cases, our results overlap within error with published equivalents.     

Comparative compressibility and equation of state of orthorhombic and tetragonal edingtonite

The high-pressure (HP) behaviour of a natural orthorhombic and tetragonal edingtonite from Ice River, Canada, has been investigated using in situ single-crystal X-ray diffraction. The two isothermal equations of state up to 6.74(5) GPa were determined. V₀, K_T0 and K refined with a third-order Birch–Murnaghan equation of state (BM-EoS) are: V₀ = 598.70(7) ų, K_T0 = 59(1) GPa and K=3.9(4) for orthorhombic edingtonite and V₀ = 600.9(2) ų, K_T0 = 59(1) GPa and K=4.2(5) for tetragonal edingtonite. The experiments were conducted with nominally hydrous pressure penetrating transmitting medium. No overhydration effect was observed within the pressure range investigated. At high-pressures the main deformation mechanism is represented by cooperative rotation of the secondary building unit (SBU).Si/Al distribution slightly influences the elastic behaviour of the tetrahedral framework: the SBU bulk moduli are 125(8) GPa and 111(4) GPa for orthorhombic and tetragonal edingtonite, respectively. Extra-framework contents of both zeolites show an interesting behaviour under HP conditions: the split Ba2 site at P >2.85 GPa is completely empty; only the position Ba1 is occupied. Electronic Supplementary Material. Supplementary material to this paper (Observed and calculated structure factors) is available in electronic form at Electronic Supplementary Material Supplementary material is available in the online version of this article at     

Vibrational spectroscopy of end-member silicate garnets

Infrared reflectance (IR) and Raman spectra were collected on small (ca. 500 micron) single crystals of 5 natural garnets with nearly end-member compositions: pyrope (98% Mg₃Al₂Si₃O₁₂), almandine (83% Fe₃Al₂Si₃O₁₂), spessartine (98% Mn₃Al₂Si₃O₁₂), grossular (97% Ca₃Al₂Si₃O₁₂), and andradite (99% Ca₃Fe₂Si₃O₁₂). Frequencies and symmetry assignments were determined for all 17 IR modes and all 25 Raman modes. By using factor group analysis and by correlating the bands by their intensities, bands were assigned to either one of the SiO₄ internal motions, as a rotation, or to a type of translation. The assignments are supported by (1) the distinct trends of frequencies with cell size and cation masses for each of the different types of motion, (2) the similarity of garnet energies for each of the different types of motion to those of olivine with the same cation, and (3) the closeness of the T ₁ u IR frequencies to the T ₂ g Raman frequencies. Mode mixing appears to be weak. Correlations between frequencies and structural parameters suggests a direct dependence of force constants on lattice parameter. This relationship arises from bond lengths in the garnet structure being constrained by the size and compressibility of adjacent polyhedra through edge-sharing. Comparison of our endmember data with previous powder IR studies of intermediate garnets indicates that dodecahedral (X) and octahedral (Y) sites alone exhibit two-mode behavior for those solid solutions involving two ions with considerably different masses. However, for solid solutions involving cations of much different ionic radii, two-mode behavior is found for the translations of SiO₄ groups. This is the first report of two-mode behavior that is unrelated to mass, and instead is due to significantly different force constants in the pyralspites compared to the ugrandites.Anomalies in mixing volumes are linked to two-mode behavior of the SiO₄ translations, which leads to the suggestion that the mixing volume behavior is caused by the resistance of the Si-O bond to expansion and compression, as well as to changes in the dodecahedral site. Crystal-field effects may also play an important role within the ugrandite series. Deviation of molar volume dependence on composition from a linear to a asymmetric, non-linear (sometimes sigmoidal) dependence can be linked to solid solutions that possess slightly non-equivalent cation sites.     

Exsolution in ternary feldspars

It is shown that the microstructures of the antiperthites of the South Norwegian anorthosites strongly suggest an exsolution origin. In addition to the normal exsolution reactions responsible for most antiperthites, exsolution of potash feldspar in plagioclase may occur by a discontinuous mode of precipitation.     

On recasting analyses of garnet into end-member molecules

Most published analyses of garnet deviate from structural ideality. Consequently, compositions expressed as molecular percentages of end-member molecules may differ if different re-calculation sequences have been used. A suitable standard calculation procedure is presented, and is demonstrated to be satisfactory by its application to 69 published analyses of garnets both common and rare.It is seldom necessary to use molecules other than pyrope, almandine, spessartine, grossular, andradite, uvarovite and hydrogrossular, and most analyses can be recast into four or less molecules which exceed 3% of the garnet. This means that most analyses can be visually displayed in a composition tetrahedron.It is suggested, that the percentage number of cations which can be allocated to garnet molecules is a figure useful for assessment of analytical quality. More than 95% of the cations can be so allocated in the majority of the analyses considered.Full details of the proposed scheme are appended together with a worked example which demonstrates the abbreviated procedure which applies to most common garnets. A compilation is given of the common physical properties which have been measured for synthetic end-member garnets of the types used in the calculation scheme.S.A. UMP Publication No. 5.     

Exsolution in ternary feldspars

The equilibrium precipitate formed by exsolution in alkali feldspar containing calcium in solid solution is a two-phase intergrowth of plagioclase and quartz. It is the kinetics of the precipitation, however, which decides whether or not this reaction will take place. It is shown that lamellar growth of plagioclase and quartz is only accomplished by the advancement of an incoherent boundary originating at an alkali feldspar — alkali feldspar boundary or at an interface between alkali feldspar and plagioclase.     

Activity-composition relations in feldspars

Seck's (1971a) compositional data on coexisting feldspars in the Or-Ab-An ternary at 650° C and 1 kb were used to calculate the activity-composition relations in binary alkali feldspar and binary plagioclase. The energy constants in Guggenheim's expression for excess free energy of mixing are A ₀=3920 and A ₁=657 cal/mole for alkali feldspar, in excellent agreement with values obtained by Thompson and Waldbaum (1969), and 1320 and 373 cal/ mole for plagioclase. Using Orville's (1972) data from ion-exchange experiments between plagioclase and Na—Ca chloride solutions at 700° C and 2 kb, we obtained 967 cal/mole for A ₀ and 715 cal/mole for A ₁ in the plagioclase crystalline solution.Activity-composition relations for plagioclase are interpreted in terms of a continuous, random substitution of CaAl for NaSi across the high structural state plagioclase series. This interpretation is consistent with that obtained from a consideration of lattice parameters.     

Radiation defects in feldspars

Various radiation defects were characterized and analyzed by electron paramagnetic resonance (EPR) in 15 feldspars of different compositions after X-ray irradiation. A hole center on oxygen adjacent to two aluminum ions is formed in most feldspars, except those with very high An content. Since the hole is not localized at room temperature, clusters of more than two Al must be present in all feldspars in amounts of at least 100 ppm. Less frequent radiation defects are trivalent titanium and holes on oxygen ions adjacent to a small divalent ion of a yet unidentified nature on a T site with Si and in some cases also Pb as further neighbors. The directions of the magnetic axes for these centers allowed their assignment to specific sites in the feldspar structure. Characteristic absorption and thermolu-minescence emission bands could also be assigned to these centers. Their properties are remarkably independent of composition and Al, Si disorder of the feldspars.     

Thermal expansion of plagioclase feldspars

The volume thermal expansion coefficient and the anisotropy of thermal expansion were determined for nine natural feldspars with compositions, in terms of albite (NaAlSi₃O₈, Ab) and anorthite (CaAl₂Si₂O₈, An), of Ab₁₀₀, An₂₇Ab₇₃, An₃₅Ab₆₅, An₄₆Ab₅₄, An₆₀Ab₄₀, An₇₈Ab₂₂, An₈₉Ab₁₁, An₉₆Ab₄ and An₁₀₀ by high resolution powder diffraction with a synchrotron radiation source. Unit-cell parameters were determined from 124 powder patterns of each sample, collected over the temperature range 298–935 K. The volume thermal expansion coefficient of the samples determined by a linear fit of V/V ₀ = α(T − T ₀) varies with composition (X _An in mol %) as:

The superstructure of plagioclase feldspars

A modulation function representing the position and density of (Na, Ca) atoms in the superstructure of the e-plagioclase has been derived from the average structures of different plagioclase and a general modulation theory. Based on this function the superstructure of bytownite (An₇₃) has been studied with the single crystal X-ray method. The cell dimensions by Megaw's axes are a=7.946(3)A, b=67.09(2)A, c=12.236(4)A, α=39.03(1)°, β=45.63(1)° and γ=59.63(1)°. Z=18(Na, Ca) Al(Al, Si)Si₂O₈. The initial phase factor of the modulation function for bytownite has been obtained from the intensity data of the satellite reflections. This modulation function indicates a coherent small-scale alternation of the Na-rich and Ca-rich bands in the superstructure. This superstructure has been refined by applying the albite and anorthite structures to the Na-rich and Ca-rich bands, respectively. The change of the superstructure of the e-plagioclase due to the compositional change has been described based on the movements of the satellites in reciprocal space. The direction of the coherent small-scale intergrowth of the anorthite-like and albite-like bands is perpendicular to the t vector. The thickness of the intergrowth is 1/|t|. Both direction and thickness change regularly from An₇₅ to An₂₅.     

Myrmekite-like intergrowths in larvikite feldspars

Electron microprobe analyses of feldspars and nepheline from a number of larvikite specimens are presented. Nepheline occurs as primary grains, as vermicular grains in mrrmekite-like intergrowths with potash feldspar, as small blebs in potash feldspar lamellae and as larger grains surrounded by irregular, K-rich feldspar domains. It is believed that the different forms of nepheline associated with potash feldspar are formed by exsolution in the solid state.     

High-temperature thermodynamic properties of alkali feldspars

Recent hydrofluoric acid solution calorimetric data are used to derive standard enthalpies and Gibbs free energies of formation of low-albite, high-albite, NaAlSi₃O₈ glass, microcline, sanidine, and KAlSi₃O₈ glass. The data are presented as high-temperature functions from 298.15 to 1400° K.     

The magma mixing origin of mantled feldspars

The key to mantled feldspar genesis is epitaxial nucleation of plagioclase on K-feldspar or K-feldspar on plagioclase. Once this nucleation takes place there is a relatively straightforward process of crystal growth yielding rapakivi and antirapikivi textures. The most common mantling is plagioclase on K-feldspar which occurs in both volcanic and plutonic environments. In the volcanic environment the morphology of the plagioclase overgrowth typically is dendritic, though in subvolcanic and shallow plutonic environments dendritic growth is followed by a more or less continuous non-cellular shell of plagioclase. In the plutonic environment, early stages of plagioclase overgrowth also tend to be dendritic, although with coarser-grained characteristics. Dendritic morphology is thus a common denominator in rapakivi genesis. Since growth of dendritic plagioclase is clearly related to marked undercooling in silicate melt systems its occurrence in many volcanic rocks is to be expected. Equivalent quenching in the plutonic environment requires a cooling mechanism independent of conductive heat transfer to wallrock and also independent of effective cooling related to sudden loss of volatile phases that could only occur late in the crystallization of most magmas and therefore after much dendritic plagioclase had already formed. Internal quenching of portions of magma systems must occur if mafic magma is abruptly mixed with felsic magma. Such magma mixing yields a heterogeneous system at first, one that is in a drastic state of disequilibrium and tending to force nucleation of one feldspar type on the surface of another resulting in epitaxial crystallization of dendritic plagioclase on K-feldspar. Mantling of one feldspar type by another during magma mixing is paralleled by dendritic growth zones in coexisting plagioclase crystals.Mantling textures occur in hybrid rocks of magma mixing origin. Some of the hybrid rocks are fine-grained, mafic-rich, and may contain phenocrysts of quartz, plagioclase, and K-feldspar. They occur as rounded inclusions in calc-alkaline granites and granodiorites. The host plutons themselves commonly have mantled feldspars or at least plagioclase with the unusual zoning characteristics commonly accompanying rapakivi texture. Magma-mixing tends to occur in batches so that hybrid crystal-melt systems, the calc-alkaline granitic plutons, become intrusive into earlier hybrid crystal-melt systems, represented by the mafic-rich inclusions.     

On the constitution of the alkali feldspars

Summary The crystalline variants of the alkali feldspars differ from each other in the pattern of Al/Si disorder. The interrelations of the several variants are graphically shown in Figs. 1, 2, and 3. In high albite thekind of disorder is different from that in any other alkali feldspar; therefore the transition high albitemonalbite is diffusive; but the transition monalbiteanalbite is displacive. See Fig. 10.The phase diagram of the system NaAlSi₃O₈–KAlSi₃O₈ is presented in Fig. 4. The volume relations of the mixed crystals (Figs. 5 and 8) indicate that albite exhibiting monoclinic symmetry at room temperature has a defect lattice (defect monalbite) with vacant Na sites. A vacant cation site behaves as if a large cation (for example K) were present; therefore the defect monalbite behaves like a mixed crystal with some K replacing Na.

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Zusammenfassung Die verschiedenen kristallinen Modifikationen der Alkalifeldspate unterscheiden sich bezüglich der Art und Weise ihrer Al/Si-Verteilung. Die Interrelationen sind in den Abb. 1, 2 und 3 dargestellt. Im Hochalbit ist die Al/Si-Verteilung andersartig als in irgendeinem anderen Alkalifeldspat. Die meisten Alkalifeldspate haben eine zwifache Unendlichkeit von Variationsmöglichkeiten der Al/Si-Verteilung (entsprechend der in Abb. 2 eingezeichneten Ebene), Hochalbit aber steht einzig da und weist eine dreifache Unendlichkeit von Möglichkeiten auf. Infolgedessen entspricht dem Übergang HochalbitMonalbit ein diffuser Umwandlungsmechanismus, dem Übergang MonalbitAnalbit aber ein displaciver (Abb. 10).Das Phasendiagramm des Systems NaAlSi₃O₈–KAlSi₃O₈ ist in Abb. 4 dargestellt. Die Volumverhältnisse der Mischkristalle (Abb. 5 und 8) deuten darauf hin, daß der bei gewöhnlicher Temperatur monokline Albit ein defektes Gitter hat (= defekter Monalbit mit teilweise unbesetzten Na-Punktlagen). Die Wirkung einer unbesetzten Kation-Punktlage im Gitter ist der Wirkung eines großen Kations (z. B. Kalium) gleich; hierdurch erklärt sich die Tatsache, daß der defekte Monalbit die Eigenschaften eines etwas K-haltigen Mischkristalls nachahmt.

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With 10 Figures

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Dedicated to ProfessorF. Machatschki on the occasion of his 70th birthday.     

Structure energies of the alkali feldspars

Structural energetics of the alkali feldspars have been studied using a “lattice” or structure energy model. Electrostatic energies, U _e,for 20 well-refined, non-intergrown alkali feldspars were calculated using Bertaut's (1952) summation procedure and average about ?13,400 kcal/mol; the repulsive energies of the alkali site in each structure (～15 kcal/mol) were calculated using repulsive parameters for K-O and Na-O interactions estimated from bulk modulus data for NaF and KF and the exponential form of the repulsive potential. Using a procedure in which the position of the alkali cation was varied while the oxygen cage was kept fixed, structure energy gradients for the alkali sites of high albite and a hypersolvus Ab₄₂Or₅₈ structure were computed. In both cases, a broad structure energy well, elongated approximately parallel to c and subparallel to the observed split Na positions, was found. In both structures there is a single energy minimum corresponding closely with the observed single alkali positions. Comparison of U _e values for the alkali feldspars with different K/Na ratios shows that intermediate compositions are predicted to be less “stable” than either endmember and that the potassic end-member is predicted to be less “stable” than the sodic one, assuming that all other factors contributiong to the free energies of each phase are approximately the same. Comparison of U _e values for the high albite and low sanidine structures with different Al/Si distributions and a fixed tetrahedral framework indicates that the ordered charge distributions are 63.0 and 54.8 kcal/mol, respectively, more “stable” than the disordered distributions. Smaller, more realistic energy differences were obtained by using U _evalues averaged from four separate calculations with a +3 charge on a different T site in each and with +4 charges on the other T sites. If, in addition, the charges on cations and oxygen are reduced to half their nominal formal charges, in agreement with Pauling's electroneutrality principle and the results of recent molecular orbital calculations on silicates, the predicted electrostatic energy differences are reduced to 3.6 and 1.6 kcal/mol, respectively. These calculations also indicate that the T₁O site in the high albite structure energetically favors Al and that the Al/Si distribution determines the Na position within the alkali site.     

Structural state of detrital alkali feldspars

The structural state of an alkali feldspar is determined by the nature of the distribution of Al and Si in the tetrahedral sites of the feldspar structure. This state is a function of a number of genetic controls including temperature, cooling rate, deformation, crystal size, and several chemical factors. Together these controls constitute a genetic regime. Identification of the structures of detrital feldspars may enable recognition of genetic regimes and be useful in provenance interpretation and mineral province definition. A 2-peak method of X-ray diffraction (XRD) determination as a variant of an earlier proposed 3-peak method of structural state determination is used in this study. Total analytical time for each determination is about 50–55 min per grain. Results of structural state identification of 126 detrital feldspars from Holocene stream sands derived from volcanic, plutonic and metamorphic rocks are presented in order to illustrate the application and potential of the technique. Detrital feldspars from the metamorphic rocks are all maximum microcline; those from the plutons range from orthoclase to microcline depending on the age of the pluton, whereas those from the volcanic rocks are sanidines.     

Non-equilibrium mixing in natural alkali feldspars

Chemical-optical and X-ray analysis of some perthite groups, from cristalline rock massifs, came into evidence that in perthites with the same order degree of Si/Al distribution, the observed variations of 2V_X are strictly related to the Na content in the potassic phase; i.e. 2V_X decreases when Ab mixed in the K-phase increases.This fact suggests that for any Si/Al, distribution, not in equilibrium, exists a certain range of non equilibrium mixing between Or and Ab.     

Flow laws of multiphase materials and rocks from end-member flow laws

Rheological properties of polyphase rocks play an important role in the dynamics of the lithosphere and asthenosphere. However, flow laws for large portions of the polyphase rocks in the Earth's crust and mantle have not been well determined. An analysis based on the theory of mixtures has been made to calculate the general flow laws of coarse (≥5 μm), nearly equant-grained (aspect-ratio ≤3), and massive polyphase rocks and materials for which only the flow laws and volume fractions of the constituents are taken into consideration in the modeling of the bulk rheology and effects of microstructure could be ignored. The theoretical analysis is based on three assumptions: (1) the polyphase composite and the monophase aggregates of its constituents obey the same kind of flow laws (linear, power or exponential), (2) there is no change in the operative deformation mechanism of each phase when it is in the composite as compared to when it is in a monophase aggregate, and (3) neither chemical (metamorphic) reactions take place among the constituent phases nor eutectic melting occurs due to the phase mixing. The proposed iterative process allows to predict, to the first approximation, the flow laws for a large number of polyphase rocks in terms of the experimentally determined flow laws of a relatively small number of monomineralic aggregates. Applications of this approach to typical polyphase rocks such as granite, diorite, diabase, aplite and websterite as well as to synthetic two-phase materials such as forsterite–enstatite mixtures and water ice–ammonia dehydrate aggregates yield quite accurate approximations to the experimental values.