Thermodynamics of plagioclase II: Temperature evolution of the spontaneous strain at the I\bar 1 \leftrightarrow P\bar 1 phase transition in anorthite

The temperature dependence of the lattice parameters of pure anorthite with high Al/Si order reveals the predicted tricritical behaviour of the \(I\bar 1 \leftrightarrow P\bar 1\) phase transition at T _c ^* =510 K. The spontaneous strain couples to the order parameter Q° as x _i∝S _xQ ⁱ Q°² with S _xQ ¹ =4.166×10^?3, S _xQ ² =0.771×10^?3, S _xQ ³ =?7.223×10^?3 for the diagonal elements. The temperature dependence of Q° is $$Q^{\text{o}} = \left( {1 - \frac{T}{{510}}} \right)^\beta ,{\text{ }}\beta = \tfrac{{\text{1}}}{{\text{4}}}$$ A strong dependence of T _c ^* , S _xQ ⁱ and β is predicted for Al/Si disordered anorthite.     

The effect of Al/Si disorder on the I \bar 1-P \bar 1Co-Elastic phase transition in Ca-rich plagioclase

The influence of Al/Si disorder in the anorthite tetrahedral framework upon the I \(\bar 1\) -P \(\bar 1\) displacive transition of that framework has been investigted at high-temperature by powder X-ray diffraction. The temperature-dependence of the order parameter in a heat-treated (disordered) anorthite and a Ca-rich plagioclase has been determined from spontaneous strain measurements. Both samples show appreciable disorder, with Q _od = 0.88 in both cases. In each, the critical exponent β appears to be intermediate between values for classical tricritical (1/4) and second-order (1/2) phase transitions. This critical behaviour is consistent with a Landau potential in which the coefficient of the quartic term is positive but smaller than the coefficient of the sixth order term, corresponding to a second-order phase transition close to a tricritical point. There does not appear to be any defect strain tail near T _c and inhomogeneities in Q _od appear to be on rather a short length scale in these samples. The role of changing Q _od appears to be more important than that of changing composition (albite component). The data are interpreted using a model of a homogeneous field due to changing Q _od which renormalizes the transition temperature, T _c ^* , and the fourth order coefficient, B _eff, in the Landau expansion. The results are consistent with classical Landau behaviour, and demonstrate the care which must be taken in interpreting apparently non-classical critical exponents for phase transitions close to a tricritical point.     

The R{\overline{3}} c \to R{\overline{3}} m transition in nitratine, NaNO3, and implications for calcite, CaCO3

The temperature dependences of the crystal structure and superstructure intensities in sodium nitrate, mineral name nitratine, NaNO₃, were studied using Rietveld structure refinements based on synchrotron powder X-ray diffraction. Nitratine transforms from $R{\overline{3}} c\;\hbox{to}\;R{\overline{3}} m$ at T _c = 552(1) K. A NO₃ group occupies, statistically, two positions with equal frequency in the disordered $R{\overline{3}} m$ phase, but with unequal frequency in the partially ordered $R{\overline{3}} c$ phase. One position for the NO₃ group is rotated by 60° or 180° with respect to the other. The occupancy of the two orientations in the $R{\overline{3}} c$ phase is obtained from the occupancy factor, x, for the O1 site and gives rise to the order parameter, S = 2x ? 1, where S is 0 at T _c and 1 at 0 K. The NO₃ groups rotate in a rapid process from about 541 to T _c, where the a axis contracts. Using a modified Bragg–Williams model, a good fit was obtained for the normalized intensities (that is, normalized, NI^1/2) for the (113) and (211) reflections in $R{\overline{3}} c\hbox {\,NaNO}_{3},$ and indicates a second-order transition. Using the same model, a reasonable fit was obtained for the order parameter, S, and also supports a second-order transition.     

The thermodynamics of the I \overline{1} –P \overline{1} phase transition in Ca-rich plagioclase from an assessment of the spontaneous strain

In-situ powder diffraction measurements between 90 and 935?K on four anorthite-rich plagioclase samples (An₁₀₀, An₉₆Ab₄, An₈₉Ab₁₁ and An₇₈Ab₂₂) were used to determine the detailed evolution of these samples through the $I \overline{1} $ – $P \overline{1} $ phase transition. The c-type reflections indicative of $P \overline{1} $ symmetry were detected only in An₁₀₀, An₉₆Ab₄, whereas deviations in the evolution of the unit-cell parameters with temperature were observed in all samples, most prominently in the β unit-cell angle. The c-type reflections disappear at ~510 and ~425?K in An₁₀₀ and An₉₆Ab₄ respectively, and their intensity decreases according to a tricritical trend $ I^{2} \propto \left( {T - T_{\text{c}} } \right) $ . The cell parameter changes were used to determine the spontaneous strains arising from the transition which were modelled with Landau theory, allowing for low-temperature quantum saturation, in order to determine the thermodynamic behaviour. In An₁₀₀ tricritical behaviour was observed [T _c?=?512.7(4)?K; θ_s?=?394(4)] in good agreement with previous studies, and the c-type superlattice reflections indicative of $P \overline{1} $ symmetry persist up to the T _c determined from the spontaneous strain, and then disappear. The evolution of the spontaneous strain in An₉₆Ab₄ is tricritical at low temperatures [T _c?=?459(1) K, θ_s?=?396(5)] up to the temperature of disappearance of c-type reflections, but becomes second order beyond ~440?K. In An₈₉Ab₁₁ the strain displays second-order behaviour throughout [T _c?=?500(1) and θ_s?=?212(5)], and the c-type reflections are not detected in the powder diffraction patterns at any temperature. The apparent discrepancy between the absence of c-type reflections in temperature ranges where the cell parameters display significant spontaneous strain is resolved through consideration of the sizes of the anti-phase domains within the crystals. It is deduced that the tricritical phase transition occurs in well-ordered crystals with large domains in which the behavior of individual domains is dominant (i.e. in pure anorthite) or where the $P \overline{1} $ distortions within the domains are large enough to dominate the structural coherency strains between the domains. When both the magnitude of the $P \overline{1} $ pattern of displacements of the tetrahedral framework become smaller and the influence of the structural coherency between anti-phase domains becomes significant, the thermodynamic behavior becomes 2nd-order in character, the c-type reflections disappear, and the orientation of the spontaneous strain changes.     

Constraints on the thermodynamic mixing properties of plagioclase feldspars

Taking account of the Cˉ1/Iˉ1 (Al/Si order/disorder) transformation at high temperatures in the albite-anorthite solid solution leads to a simple model for the mixing properties of the high structural state plagioclase feldspars. The disordered (Cˉ1) solid solution can be treated as ideal (constant activity coefficient) and, for anorthite-rich compositions, deviations from ideality can be ascribed to cation ordering. Values of the activity coefficient for anorthite in the Cˉ1 solid solution (γ _An ^Cˉ1 ) are then controlled by the free energy difference between Cˉ1 and Iˉ1 anorthite at the temperature (T) of interest according to the relation: ΔˉG _ord ^Iˉ1 ⇌Cˉ1 =RT ln γ _An ^Cˉ1 . If the Iˉ1⇌Cˉ1 transformation in pure anorthite is treated, to a first approximation, as first order and the enthalpy and entropy of ordering are taken as 3.7±0.6 kcal/mole (extrapolated from calorimetric data) and 1.4–2.2 cal/mole (using an equilibrium order/disorder temperature for An₁₀₀ of 2,000–2,250 K), a crude estimate of γ _An ^Cˉ1 for all temperatures can be made. The activity coefficient of albite in the Cˉ1 solid solution (γ _Ab ^Cˉ1 ) can be taken as 1.0. The possible importance of this model lies in its identification of the principal constraints on the mixing properties rather than in the actual values of γ _An ^Cˉ1 and γ _Ab ^Cˉ1 obtained. In particular it is recognised that γ _An ^Cˉ1 depends critically on ordering in anorthite as well as, at lower temperatures, any ordering in the Cˉ1 solid solution. A brief review of activity-composition data, from published experiments involving ranges of plagioclase compositions and from the combined heats of mixing plus Al-avoidance entropy model (Newton et al. 1980), reveals some inconsistencies. The values of γ _An ^Cˉ1 calculated using the approach of Newton et al. (1980), although consistent with Orville's (1972) ion exchange data, are slightly lower than values derived from experiments by Windom and Boettcher (1976) and Goldsmith (1982) or from ion-exchange experiments of Kotel'nikov et al. (1981). Based on the Cˉ1/Iˉ1 transformation model, values of γ _An ^Cˉ1 <1.0 are unlikely. Discrepancies between the experimental data sets are attributed to incomplete (non-equilibrium) Al/Si order attained during the experiments. It is suggested that the choice of activity coefficients remains somewhat subjective. The development of accurate mixing models would be greatly assisted by better thermodynamic data for ordering in pure anorthite and by more thorough characterisation of the state of order in plagioclase crystals used for phase equilibrium experiments.     

Equilibrium thermodynamics of Al/Si ordering in anorthite

Variations in the equilibrium degree of Al/Si order in anorthite have been investigated experimentally over the temperature range 800-1535° C. Spontaneous strain measurements give the temperature dependence of the macroscopic order parameter, Q, defined with respect to the \(C\bar 1 \rightleftharpoons I\bar 1\) phase transition, while high temperature solution calorimetric data allow the relationship between Q and excess enthalpy, H, to be determined. The thermodynamic behaviour can be described by a Landau expansion in one order parameter if the transition is first order in character, with an equilibrium transition temperature, T _tr, of ～2595 K and a jump in Q from 0 to ～0.65 at T_tr. The coefficients in this Landau expansion have been allowed to vary with composition, using Q=1 at 0 K for pure anorthite as a reference point for the order parameter. Published data for H and Q at different compositions allow the calibration of the additional parameters such that the free energy due to the \(C\bar 1 \rightleftharpoons I\bar 1\) transition in anorthite-rich plagioclase feldspars may be expressed (in cal. mole^-1) as: \(\begin{gathered}G = \tfrac{1}{2} \cdot 9(T - 2283 + 2525X_{Ab} )Q^2 \\ {\text{ + }}\tfrac{1}{4}( - 26642 + 121100X_{Ab} )Q^4 \\ {\text{ + }}\tfrac{1}{6}(47395 - 98663X_{Ab} )Q^6 \\ \end{gathered}\) where X _Ab is the mole fraction of albite component. The nature of the transition changes from first order in pure anorthite through tricritical at ～An₇₈ to second order, with increasing albite content. The magnitude of the free energy of \()\) ordering reduces markedly as X _Ab increases. At ～700° C incommensurate ordering in crystals with compositions ～An₅₀–An₇₀ needs to have an associated free energy reduction of only a few hundred calories to provide a more stable structure. These results, together with a simple mixing model for the disordered ( \()\) ) solid solution, an assumed tricritical model for the incommensurate ordering and published data for ordering in albite have been used to calculate a set of possible free energy relations for the plagioclase system. The incommensurate structure should appear on the equilibrium phase diagram, but its apparent stability with respect to the assemblage albite plus anorthite at low temperatures depends on the values assigned to the mixing parameters of the $$$$ solid solution.     

A dynamical model for the I\bar 1 - P\bar 1 phase transition in anorthite,CaAl2Si2O8

The non-ferroic triclinic to triclinic \(I\bar 1 - P\bar 1\) phase transition in anorthite is described in terms of the spontaneous onset of an order parameter η. A triclinic to triclinic phase transition can be driven by order parameters (representations) arising from the Γ, Z, X, U, V, R, Y, and T points of symmetry of the Brillouin zone. Each point leads to a set of two inequivalent representations and thus there is a total of sixteen inequivalent order parameters. However, only the R ₁ ⁺ representation is consistent with the change from the body-centered to primitive cell (increase of primitive cell size of two) and also with the origin of the two space groups (inversion center) being at the same position. The R ₁ ⁺ order parameter of the high symmetry triclinic phase \(P\bar 1_0\) (or equivalently \(I\bar 1\) ) causes a reciprocal lattice change and, in terms of the lower symmetry reciprocal lattice, the order parameter corresponds to the b^* point. This is consistent with experimentally observed x-ray diffuse scattering. Using induced representation theory, microscopic distortions compatible with the R ₁ ⁺ order parameter are obtained. Assuming a distortion in an arbitrary direction at the general 2(i) Wyckoff position (x₀,y₀,z₀) of \(P\bar 1_0\) (the higher symmetry phase) induced representation theory demands an opposite displacement at the position (x₀, y₀, z₀), an opposite displacement at (x₀+1,y₀+1,z₀+1), and the same displacement at ( \(\bar x\) ₀+1, \(\bar y\) ₀+1, \(\bar z\) ₀+1) of \(P\bar 1_0\) . This is also consistent with experiment. The presence of the weak c-type reflections above the transition is attributed to the fluctuating lower symmetry antiphase domains related by the translation (1/2, 1/2, 1/2).     

Interdiffusion of NaSi—CaAl in peristerite

The ‘average’ interdiffusion coefficient ( \(\bar D\) ) for NaSi—CaAl exchange in plagioclase for the interval from An₀ to An₂₆ was estimated from experimentally determined homogenization times for peristerite exsolution lamellae. The average spacing between adjacent (unlike) lamellae is 554±77 Å. Dry heating in air at 1,100°C for 98 days produced no change in the exsolution microstructure; thus \(\bar D\) (dry)<10^?17 cm²/s. This limit is consistent with the recently reported ‘average’ \(\bar D\) (dry) values for the Huttenlocher interval (An_70–90) at this temperature. At 1.5 GPa with about 0.2 weight percent water added the ‘average’ diffusion coefficient from 1,100°C to 900°C is given by: \(\bar D\) (wet)=18 _?15 ⁺¹⁰⁸ (cm²/s) exp (?97±5 (kcal/mol)/RT), where R is the gas constant, and T is °K. This \(\bar D\) (wet) at 1,100°C is more than three orders of magnitude greater than \(\bar D\) (dry) for Na- and Ca-rich plagioclases.     

Experimental delineation of the “e” ⇌ I\bar 1 and “e” ⇌ C\bar 1 transformations in intermediate plagioclase feldspars

Approximately 125 hydrothermal annealing experiments have been carried out in an attempt to bracket the stability fields of different ordered structures within the plagioclase feldspar solid solution. Natural crystals were used for the experiments and were subjected to temperatures of ～650°C to ～1,000°C for times of up to 370 days at \(P_{{\text{H}}_{\text{2}} {\text{O}}} \) =600 bars, or \(P_{{\text{H}}_{\text{2}} {\text{O}}} \) =1,200 bars. The structural states of both parent and product materials were characterised by electron diffraction, with special attention being paid to the nature of type e and type b reflections (at h+k=(2n+1), l=(2n+1) positions). Structural changes of the type C \(\bar 1\) → I \(\bar 1\) , C \(\bar 1\) → “e” structure, I \(\bar 1\) → “e” and “e” structure → I \(\bar 1\) have been followed. There are marked differences between the ordering behaviour of crystals with compositions on either side of the C \(\bar 1\) ? I \(\bar 1\) transition line. In the composition range ～ An₅₀ to ～ An₇₀ the e structure appears to have a true field of stability relative to I \(\bar 1\) ordering, and a transformation of the type I \(\bar 1\) ? e has been reversed. It is suggested that the e structure is the more stable ordered state at temperatures of ～ 800°C and below. For compositions more albite-rich than ～ An₅₀ the upper temperature limit for long range e ordering is lower than ～ 750°C, and there is no evidence for any I \(\bar 1\) ordering. The evidence for a true stability field for “e” plagioclase, which is also consistent with calorimetric data, necessitates reanalysis both of the ordering behaviour of plagioclase crystals in nature and of the equilibrium phase diagram for the albite-anorthite system. Igneous crystals with compositions of ～ An₆₅, for example, probably follow a sequence of structural states C \(\bar 1\) → I \(\bar 1\) → e during cooling. The peristerite, Bøggild and Huttenlocher miscibility gaps are clearly associated with breaks in the albite, e and I \(\bar 1\) ordering behaviour but their exact topologies will depend on the thermodynamic character of the order/disorder transformations.     

X-ray diffraction study of the orientational order/disorder transition in NaNO3: Evidence for order parameter coupling

The orientational ordering transition R \(R\bar 3m - R\bar 3c\) in NaNO₃ near 552 K has been investigated using x-ray diffraction techniques. NaNO₃ is a model system for CaCO₃ and other minerals with orientational disorder of triangular molecules in a simple NaCl-type matrix. The temperature evolution of the integrated intensities of the superlattice reflection \(\bar 1\) 23 and the fundamental reflection 110 are discussed in terms of Landau theory of two coupled order parameters. It is shown that the known phenomenological critical exponent (Poon and Salje 1988) and the anomalous thermal expansion at T > T _tr (Reeder et al. 1988) can be understood as the result of a Z point instability which mainly describes the NO ₃ ^- disorder, and a second order parameter linked with the spontaneous strain of this phase transition.     

Fitting matrix-valued variogram models by simultaneous diagonalization (Part I: Theory)

Suppose that ¯(x₁),...,¯Z(x_n). are observations of vector-valued random function ¯(x). In the isotropic situation, the sample variogram γ^*(h) for a given lag h is $$\bar \gamma ^ * (h) = \frac{1}{{2N(h)}}\mathop \sum \limits_{s(h)} (\overline Z (x_1 ) - \overline Z (x_1 )) \overline {(Z} (x_1 ) - \overline Z (x_1 ))^T $$ where s(h) is a set of paired points with distance h and N(h) is the number of pairs in s(h).. For a selection of lags h₁, h₂, .... h_k such that N (h₁) > O. we obtain a ktuple of (semi) positive definite matrices $\bar \gamma ^ * (h_{ 1} ),. . . ., \bar \gamma ^ * (h_{ k} )$ . We want to determine an orthonormal matrix B which simultaneously diagonalizes the $\bar \gamma ^ * (h_{ 1} ),. . . ., \bar \gamma ^ * (h_{ k} )$ or nearly diagonalizes them in the sense that the sum of squares of offdiagonal elements is small compared to the sum of squares of diagonal elements. If such a B exists, we linearly transform $\overline Z (x)$ by $\overline Y (x) = B\overline Z (x)$ . Then, the resulting vector function $\overline Y (x)$ has less spatial correlation among its components than $\overline Z (x)$ does. The components of $\overline Y (x)$ with little contribution to the variogram structure may be dropped, and small crossvariograms fitted by straightlines. Variogram models obtained by this scheme preserve the negative definiteness property of variograms (in the matrix-valued function sense). A simplified analysis and computation in cokriging can be carried out. The principles of this scheme arc presented in this paper.     

Kinetic rate laws derived from order parameter theory IV: Kinetics of Al,Si disordering in Na-feldspars

The kinetic rate laws of Al-Si disordering under dry conditions (T = 1353K, 1253 K, 1223 K, 1183 K) and in the presence of water (p = 1 kbar, T = 1023 K, 1073 K, 1103 K) were studied both experimentally and theoretically. A gradual change of the degree of order was found under dry conditions. For intermediate degrees of order broad distributions of the order parameter Q _od occur. The variations of Q _od are correlated with structural modulations as observed in the transmission electron microscope. The time evolution of the mean value of Q _od can be well described by the rate law: $$\frac{{dQ_{od} }}{{dt}} = - \frac{\gamma }{{RT}}\exp \sum\limits_{i = 1}^n {X_i^2 } \left[ {\frac{{ - (G_a^0 + \varepsilon (\Delta Q_{od} )^2 )}}{{RT}}} \right]\frac{{dG}}{{dQ_{od} }}$$ with the excess Gibbs energy G and G _a ⁰ = 433.8 kJ/mol, ?= -27.4 kJ/mol, γ = 1.687 · 10¹⁴ h ^?1. Under wet conditions, two processes were found which occur simultaneously. Firstly, some material renucleated with the equilibrium degree of order. Secondly, the bulk of the material transformed following the same rate law as under dry conditions but with the reduced activation energy G _a ⁰ = 332.0 kJ/mol and ? = -43.0 kJ/ mol, γ = 1.047 · 10¹³ h^?1. The applicability of the kinetic theory is discussed and some ideas for the analysis of geological observations are evolved.     

Deformation mechanisms in experimentally deformed single crystals of pyrrhotite,Fe1−xS

Single crystals of hexagonal and monoclinic pyrrhotite, Fe_1?xS, have been experimentally deformed by uniaxial compression at 300 MPa confining pressure, and at a strain rate of 1 × 10^?5 s^?1 in the temperature range from 200° C to 400° C. Very high anisotropy characterizes the mechanical behaviour of the crystal structure. During compression parallel to thec-axis, when no slip system may be activated, the maximum strength is observed. One or two degrees of non-parallelism between [c] and σ₁ results in slip on the basal plane, illustrating the very low resistance of the lattice against shear in this plane. At σ₁ Λ(0001)=45°, i.e. when maximum resolved shear stress is attained on the basal plane, the strength reaches a minimum. Thecritical resolved shear stress (CRSS) increases from less than 4.7 MPa at 400° C to 52 MPa at 200° C. A new slip system, \((10\overline 1 0)\parallel \left\langle {1\overline 2 10} \right\rangle \) prism slip, is described. It is activated only at high angles (>70°) between σ₁ and [c]. The CRSS of the prism slip ranges from 7 MPa (400° C) to 115 MPa (200° C). Twinning on \((10\overline 1 2)[(10\overline 1 2):(1\overline 2 10)]\) , earlier reported by several authors, has been produced only at the highest temperature either as secondary feature during pressure release (compression ‖[c]) or in heterogeneously strained areas (compression ⊥[c]). As twinning and prism slip attain their maximum values of the Schmidt factor under nearly equal stress conditions it is postulated that the former of the two deformation modes has the higher shear resistance.     

Gold Solubility in SiO2—HCl—H2O System at 200℃：a Preliminary Assessment of the Implications of Silicification with Regard to Gold Mineralization

The complexation between gold and silica was experimentally, confirmed and calibrated at 200 °C: $$\begin{gathered} Au^ + + H_3 SiO_4^ - \rightleftharpoons AuH_3 SiO_4^0 \hfill \\ \log K_{(200^\circ C)} = 19.26 \pm 0.4 \hfill \\ \end{gathered} $$ Thermodynamic calculations show that AuH₃SiO ₄ ⁰ would be far more abundant than AuCl ₂ ^? under physicochemical conditions of geological interest, suggesting that silica is much more important than chloride as ligands for gold transport. In systems containing both sulfur and silica, AuH₃SiO ₄ ⁰ would be increasingly more important than Au (HS) ₂ ^? as the proportion of SiO₂ in the system increases. The dissolution of gold in aqueous SiO₂ solutions can be described by the reaction: $$\begin{gathered} Au + 1/4O_2 + H_4 SiO_4^0 \rightleftharpoons AuH_3 SiO_4^0 + 1/2H_2 O \hfill \\ log K_{(200^\circ C)} = 6.23 \hfill \\ \end{gathered} $$ which indicates that SiO₂ precipitation is an effective mechanism governing gold deposition, and thus explains the close association of silicification and gold mineralization.     

The P–V–T equation of state of CaPtO3 post-perovskite

Orthorhombic post-perovskite CaPtO₃ is isostructural with post-perovskite MgSiO₃, a deep-Earth phase stable only above 100 GPa. Energy-dispersive X-ray diffraction data (to 9.4 GPa and 1,024 K) for CaPtO₃ have been combined with published isothermal and isobaric measurements to determine its P–V–T equation of state (EoS). A third-order Birch–Murnaghan EoS was used, with the volumetric thermal expansion coefficient (at atmospheric pressure) represented by α(T) = α₀ + α₁(T). The fitted parameters had values: isothermal incompressibility, $ K_{{T_{0} }} $  = 168.4(3) GPa; $ K_{{T_{0} }}^{\prime } $  = 4.48(3) (both at 298 K); $ \partial K_{{T_{0} }} /\partial T $  = ?0.032(3) GPa K^?1; α₀ = 2.32(2) × 10^?5 K^?1; α₁ = 5.7(4) × 10^?9 K^?2. The volumetric isothermal Anderson–Grüneisen parameter, δ _T , is 7.6(7) at 298 K. $ \partial K_{{T_{0} }} /\partial T $ for CaPtO₃ is similar to that recently reported for CaIrO₃, differing significantly from values found at high pressure for MgSiO₃ post-perovskite (?0.0085(11) to ?0.024 GPa K^?1). We also report axial P–V–T EoS of similar form, the first for any post-perovskite. Fitted to the cubes of the axes, these gave $ \partial K_{{aT_{0} }} /\partial T $  = ?0.038(4) GPa K^?1; $ \partial K_{{bT_{0} }} /\partial T $  = ?0.021(2) GPa K^?1; $ \partial K_{{cT_{0} }} /\partial T $  = ?0.026(5) GPa K^?1, with δ _T  = 8.9(9), 7.4(7) and 4.6(9) for a, b and c, respectively. Although $ K_{{T_{0} }} $ is lowest for the b-axis, its incompressibility is the least temperature dependent.     

On the least-squares fit of small and great circles to spherically projected orientation data

Given the direction cosines a _i ^′ = (a ₁ ⁱ , a ₂ ⁱ , a ₃ ⁱ )corresponding to a set of pspherically projected fabric poles, an initial estimate x′ = (x₁, x₂, x₃, x₄)for the angular radius x₄,and direction cosines of the center of the least-squares small circle which minimizes the sum of the squares of the angular residuals $$r = \sum\limits_p {\left[ {x_4 - \cos ^{ - 1} \left( {a_1^i x_1 + a_2^i x_2 + a_3^i x_3 } \right)} \right]} ^2 $$ can be iteratively improved by taking x_j+1 = x_j + Δxwhere x_j is the value of xat the jth iteration and $$\Delta x = - H_j^{ - 1} \left[ {q_j + x_j \left( {x'_j H_j^{ - 1} x_j } \right)\left( {q_j - x'_j H_j^{ - 1} q_j } \right)} \right],$$ where As an initial approximation for xwe have found it convenient to ignore the fact that the data are constrained to lie on the surface of the reference sphere and to use the parameters of a least-squares plane through the given poles. Generalization of this approach to fitting variously constrained great and small circles is easily made. The relative merits of differently constrained fits to the same data can be tested approximately if it is assumed that the errors in the location of the poles are isotropic and normally distributed. It is thus possible to statistically assess the relative significance of conflicting structural models which predict different geometrical patterns of fabric elements.     

Carbon Dioxide in igneous petrogenesis: I

A number of experimental CO₂ solubility data for silicate and aluminosilicate melts at a variety of P- T conditions are consistent with solution of CO₂ in the melt by polymer condensation reactions such as SiO _4(m ^4? +CO_2(v)+Si _n O _3n+1(m) ⁽²ⁿ⁺¹⁾ ?Si _n+1O _3n+4(m) ^(2n+4)? +CO _3(m ^)2? . For various metalsilicate systems the relative solubility of CO₂ should depend markedly on the relative Gibbs free change of reaction. Experimental solubility data for the systems Li₂O-SiO₂, Na₂O-SiO₂, K₂O-SiO₂, CaO-SiO₂, MgO-SiO₂ and other aluminosilicate melts are in complete accord with predictions based on Gibbs Free energies of model polycondesation reactions. A rigorous thermodynamic treatment of published P- T-wt.% CO₂ solubility data for a number of mineral and natural melts suggests that for the reaction CO_2(m) ? CO_2(v)

1. CO₂-melt mixing may be considered ideal (i.e., { \(a_{{\text{CO}}_{\text{2}} }^m = X_{{\text{CO}}_{\text{2}} }^m \) );
2. \(\bar V_{{\text{CO}}_{\text{2}} }^m \) , the partial molal volume of CO₂ in the melt, is approximately equal to 30 cm³ mole^?1 and independent of P and T;
3. Δ C _p ⁰ is approximately equal to zero in the T range 1,400° to 1,650 °C and
4. enthalpies and entropies of the dissolution reaction depend on the ratio of network modifiers to network builders in the melt. Analytic expressions which relate the CO₂ content of a melt to P, T, and \(f_{{\text{CO}}_{\text{2}} } \) for andesite, tholeiite and olivine melilite melts of the form

$$\ln X_{{\text{CO}}_{\text{2}} }^m = \ln f_{{\text{CO}}_{\text{2}} } - \frac{A}{T} - B - \frac{C}{T}(P - 1)$$ have been determined. Regression parameters are (A, B, C): andesite (3.419, 11.164, 0.408), tholeiite (14.040, 5.440,0.393), melilite (9.226, 7.860, 0.352). The solubility equations are believed to be accurate in the range 3<P<30 kbar and 1,100°<T<1,650 °C. A series of CO₂ isopleth diagrams for a wide range of T and P are drawn for andesitic, tholeiitic and alkalic melts.     

Perrierite-(La), (La,Ce,Ca)4(Fe2+,Mn)(Ti,Fe3+,Al)4(Si2O7)2O8, a new mineral species from the Eifel volcanic district, Germany

Non-metamict perrierite-(La) discovered in the Dellen pumice quarry, near Mendig, in the Eifel volcanic district, Rheinland-Pfalz, Germany has been approved as a new mineral species (IMA no. 2010-089). The mineral was found in the late assemblage of sanidine, phlogopite, pyrophanite, zirconolite, members of the jacobsite-magnetite series, fluorcalciopyrochlore, and zircon. Perrierite-(La) occurs as isolated prismatic crystals up to 0.5 × 1 mm in size within cavities in sanidinite. The new mineral is black with brown streak; it is brittle, with the Mohs hardness of 6 and distinct cleavage parallel to (001). The calculated density is 4.791 g/cm³. The IR spectrum does not contain absorption bands that correspond to H₂O and OH groups. Perrierite-(La) is biaxial (-), α = 1.94(1), β = 2.020(15), γ = 2.040(15), 2V _meas = 50(10)°, 2V _calc = 51°. The chemical composition (electron microprobe, average of seven point analyses, the Fe²⁺/Fe³⁺ ratio determined from the X-ray structural data, wt %) is as follows: 3.26 CaO, 22.92 La₂O₃, 19.64 Ce₂O₃, 0.83 Pr₂O₂, 2.09 Nd₂O₃, 0.25 MgO, 2.25 MnO, 3.16 FeO, 5.28 Fe₂O₃, 2.59 Al₂O₃, 16.13 TiO₂, 0.75 Nb₂O₅, and 20.06 SiO₂, total is 99.21. The empirical formula is (La_1.70Ce_1.45Nd_0.15Pr_0.06Ca_0.70)_Σ4.06(Fe _0.53 ²⁺ Mn_0.38Mg_0.08)_Σ0.99(Ti_2.44Fe _0.80 ³⁺ Al_0.62Nb_0.07)_Σ3.93Si_4.04O₂₂. The simplified formula is (La,Ce,Ca)₄(Fe²⁺,Mn)(Ti,Fe³⁺,Al)₄(Si₂O₇)₂O₈. The crystal structure was determined by a single crystal. Perrierite-(La) is monoclinic, space group P2₁/a, and the unit-cell dimensions are as follows: a =13.668(1), b = 5.6601(6), c = 11.743(1) Å, β = 113.64(1)°; V = 832.2(2) ų, Z = 2. The strong reflections in the X-ray powder diffraction pattern are [d, Å (I, %) (hkl)]: 5.19 (40) (110), 3.53 (40) ( $\overline 3 $ 11), 2.96 (100) ( $\overline 3 $ 13, 311), 2.80 (50) (020), 2.14 (50) ( $\overline 4 $ 22, $\overline 3 $ 15, 313), 1.947 (50) (024, 223), 1.657 (40) ( $\overline 4 $ 07, $\overline 4 $ 33, 331). The holotype specimen of perrierite-(La) is deposited at the Fersman Mineralogical Museum, Russian Academy of Sciences, Moscow, Russia, with the registration number 4059/1.     

Kinetic rate laws as derived from order parameter theory II: Interpretation of experimental data by laplace-transformation,the relaxation spectrum,and kinetic gradient coupling between two order parameters

A unifying theory of kinetic rate laws, based on order parameter theory, is presented. The time evolution of the average order parameter is described by $$\langle Q\rangle \propto \smallint P(x)e^{^{^{^{^{^{^{ - xt} } } } } } } dx = L(P)$$ where t is the time, x is the effective inverse susceptibility, and L indicates the Laplace transformation. The probability function P(x) can be determined from experimental data by inverse Laplace transformation. Five models are presented:

1. Polynomial distributions of P(x) lead to Taylor expansions of 〈Q〉 as $$\langle Q\rangle = \frac{{\rho _1 }}{t} + \frac{{\rho _2 }}{{t^2 }} + ...$$
2. Gaussian distributions (e.g. due to defects) lead to a rate law $$\langle Q\rangle = e^{ - x_0 t} e^{^{^{^{^{\frac{1}{2}\Gamma t^2 } } } } } erfc\left( {\sqrt {\frac{\Gamma }{2}} t} \right)$$ where x ₀ is the most probable inverse time constant, Γ is the Gaussian line width and erfc is the complement error integral.
3. Maxwell distributions of P are equivalent to the rate law 〈Q〉∝e^?k√t .
4. Pseudo spin glasses possess a logarithmic rate law 〈Q〉∝lnt.
5. Power laws with P(x)=x ^a lead to a rate law: ln〈Q〉=-(α + 1) ln t.

The power spectra of Q are shown for Gaussian distributions and pseudo spin glasses. The mechanism of kinetic gradient coupling between two order parameters is evaluated.