A new method of material balance evaluation in metamorphic differentiated systems

The methods used by earlier workers for evaluating material balance in core-mantle-matrix type differentiated systems are examined in detail. It is demonstrated that these methods can be successfully employed only when the true core-mantle volume ratio is known. On geometric grounds, it is rarely possible to have a reliable estimate of this ratio from natural specimen. Consequently, the scope of balance evaluation by these methods is severely restricted. From theoretical consideration of mass transfer relations in differentiated systems, a new computational method is proposed that can be effectively employed for quantitative balance evaluation without any prior knowledge of the core-mantle volume ratio provided the chemical analyses of the core, mantle and matrix are available. This method involves the application of the following mass balance equation: $$m_1 x_1^i + m_2 x_2^i = m_0 x_0^i {\text{ (}}i = {\text{1,2}}...{\text{,}}n{\text{)}}$$ where m ₁ and m ₂ are the mass of the core and mantle respectively, m ₀ is the mass of the matrix involved in differentiation, and x ₁ ⁱ , x ₂ ⁱ , x ₀ ⁱ are the weight fractions of the component i in the core, mantle and matrix respectively. This method would also permit a quantitative estimation of the materials added to or removed from the system. Three differentiated systems previously investigated by Mehnert (1951, 1968), Loberg (1963) and Kretz (1966) are selected for balance evaluation by the proposed method and the results are compared with the published balance analyses.     

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.     

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.     

On the nonstoichiometry and point defects of olivine

This paper presents the point-defect thermodynamics for fayalite and olivine solid solutions (Fe _x Mg_1?x )₂SiO₄. By means of thermogravimetry, the metal-to-oxygen ratio of these silicates has been determined as a function of oxygen potential, compositionx and temperature. Experiments were performed in the range of 1,000° C≦T≦1,280° C and 0.2≦x≦1.0. It is found that V _Me ^″ , Fe _Me ^· and the associate {Fe′ _Si ^′ Fe _Me ^· } are the majority defects. With this knowledge it is possible to calculate the nonstoichiometry at given temperature as a function of \(p_{O_2 } \) and \(a_{SiO_2 } \) . The cation vacancy concentration shows a \(p_{O_2 }^{1/5} \) -dependence (forx≧0.2) and increases at givenT and \(p_{O_2 } \) almost exponentially with compositionx. In the composition range studied here, the silicates show an oxygen excess, and FeO is more soluble in the olivine than SiO₂.     

The system Fe-Si-O: Oxygen buffer calibrations to 1,500K

The five solid-phase oxygen buffers of the system Fe-Si-O, iron-wuestite (IW), wuestite-magnetite (WM), magnetite-hematite (MH), quartz-iron-fayalite (QIF) and fayalite-magnetite-quartz (FMQ) have been recalibrated at 1 atm pressure and temperatures from 800°–1,300° C, using a thermogravimetric gas mixing furnace. The oxygen fugacity, \(f_{{\text{O}}_{\text{2}} }\) was measured with a CaO-doped ZrO₂ electrode. Measurements were made also for wuestite solid solutions in order to determine the redox behavior of wuestites with O/Fe ratios varying from 1.05 to 1.17. For FMQ, additional determinations were carried out at 1 kb over a temperature range of 600° to 800° C, using a modified Shaw membrane. Results agree reasonably well with published data and extrapolations. The reaction parameters K, ΔG _r ^o , ΔH _r ^o , and ΔS _r ^o were calculated from the following log \(f_{{\text{O}}_{\text{2}} }\) /T relations (T in K): $$\begin{gathered} {\text{IW }}\log f_{{\text{O}}_{\text{2}} } = - 26,834.7/T + 6.471\left( { \pm 0.058} \right) \hfill \\ {\text{ }}\left( {{\text{800}} - 1,260{\text{ C}}} \right), \hfill \\ {\text{WM }}\log f_{{\text{O}}_{\text{2}} } = - 36,951.3/T + 16.092\left( { \pm 0.045} \right) \hfill \\ {\text{ }}\left( {{\text{1,000}} - 1,300{\text{ C}}} \right), \hfill \\ {\text{MH }}\log f_{{\text{O}}_{\text{2}} } = - 23,847.6/T + 13.480\left( { \pm 0.055} \right) \hfill \\ {\text{ }}\left( {{\text{1,040}} - 1,270{\text{ C}}} \right), \hfill \\ {\text{QIF }}\log f_{{\text{O}}_{\text{2}} } = - 27,517.5/T + 6.396\left( { \pm 0.049} \right) \hfill \\ {\text{ }}\left( {{\text{960}} - 1,140{\text{ C}}} \right), \hfill \\ {\text{FMQ }}\log f_{{\text{O}}_{\text{2}} } = - 24,441.9/T + 8.290\left( { \pm 0.167} \right) \hfill \\ {\text{ }}\left( {{\text{600}} - 1,140{\text{ C}}} \right). \hfill \\ \end{gathered}$$ These experimentally determined reaction parameters were combined with published 298 K data to determine the parameters G_f, H_f, and S_f for the phases wuestite, magnetite, hematite, and fayalite from 298 K to the temperatures of the experiments. The T? \(f_{{\text{O}}_{\text{2}} }\) data for wuestite solid solutions were used to obtain activities, excess free energies and Margules mixing parameters. The new data provide a more reliable, consistent and complete reference set for the interpretation of redox reactions at elevated temperatures in experiments and field settings encompassing the crust, mantle and core as well as extraterrestrial environments.     

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.     

An experimental study of Ti and Zr partitioning among zircon, rutile, and granitic melt

In order to evaluate the effect of trace and minor elements (e.g., P, Y, and the REEs) on the high-temperature solubility of Ti in zircon (zrc), we conducted 31 experiments on a series of synthetic and natural granitic compositions [enriched in TiO₂ and ZrO₂; Al/(Na + K) molar ~1.2] at a pressure of 10 kbar and temperatures of ~1,400 to 1,200 °C. Thirty of the experiments produced zircon-saturated glasses, of which 22 are also saturated in rutile (rt). In seven experiments, quenched glasses coexist with quartz (qtz). SiO₂ contents of the quenched liquids range from 68.5 to 82.3 wt% (volatile free), and water concentrations are 0.4–7.0 wt%. TiO₂ contents of the rutile-saturated quenched melts are positively correlated with run temperature. Glass ZrO₂ concentrations (0.2–1.2 wt%; volatile free) also show a broad positive correlation with run temperature and, at a given T, are strongly correlated with the parameter (Na + K + 2Ca)/(Si·Al) (all in cation fractions). Mole fraction of ZrO₂ in rutile $ \left( {\mathop X\nolimits_{{{\text{ZrO}}_{ 2} }}^{\text{rt}} } \right) $ in the quartz-saturated runs coupled with other 10-kbar qtz-saturated experimental data from the literature (total temperature range of ~1,400 to 675 °C) yields the following temperature-dependent expression: $ {\text{ln}}\left( {\mathop X\nolimits_{{{\text{ZrO}}_{ 2} }}^{\text{rt}} } \right) + {\text{ln}}\left( {a_{{{\text{SiO}}_{2} }} } \right) = 2.638(149) - 9969(190)/T({\text{K}}) $ , where silica activity $ a_{{{\text{SiO}}_{2} }} $ in either the coexisting silica polymorph or a silica-undersaturated melt is referenced to α-quartz at the P and T of each experiment and the best-fit coefficients and their uncertainties (values in parentheses) reflect uncertainties in T and $ \mathop X\nolimits_{{{\text{ZrO}}_{2} }}^{\text{rt}} $ . NanoSIMS measurements of Ti in zircon overgrowths in the experiments yield values of ~100 to 800 ppm; Ti concentrations in zircon are positively correlated with temperature. Coupled with values for $ a_{{{\text{SiO}}_{2} }} $ and $ a_{{{\text{TiO}}_{2} }} $ for each experiment, zircon Ti concentrations (ppm) can be related to temperature over the range of ~1,400 to 1,200 °C by the expression: $ \ln \left( {\text{Ti ppm}} \right)^{\text{zrc}} + \ln \left( {a_{{{\text{SiO}}_{2} }} } \right) - \ln \left( {a_{{{\text{TiO}}_{2} }} } \right) = 13.84\left( {71} \right) - 12590\left( {1124} \right)/T\left( {\text{K}} \right) $ . After accounting for differences in $ a_{{{\text{SiO}}_{2} }} $ and $ a_{{{\text{TiO}}_{2} }} $ , Ti contents of zircon from experiments run with bulk compositions based on the natural granite overlap with the concentrations measured on zircon from experiments using the synthetic bulk compositions. Coupled with data from the literature, this suggests that at T ≥ 1,100 °C, natural levels of minor and trace elements in “granitic” melts do not appear to influence the solubility of Ti in zircon. Whether this is true at magmatic temperatures of crustal hydrous silica-rich liquids (e.g., 800–700 °C) remains to be demonstrated. Finally, measured $ D_{\text{Ti}}^{{{\text{zrc}}/{\text{melt}}}} $ values (calculated on a weight basis) from the experiments presented here are 0.007–0.01, relatively independent of temperature, and broadly consistent with values determined from natural zircon and silica-rich glass pairs.     

Ephesite,Na(LiAl2) [Al2Si2O10] (OH)2: I. thermal stability and standard state thermodynamic properties

Ephesite, Na(LiAl₂) [Al₂Si₂O₁₀] (OH)₂, has been synthesized for the first time by hydrothermal treatment of a gel of requisite composition at 300≦T(° C)≦700 and \(P_{H_2 O}\) upto 35 kbar. At \(P_{H_2 O}\) between 7 and 35 kbar and above 500° C, only the 2M₁ polytype is obtained. At lower temperatures and pressures, the 1M polytype crystallizes first, which then inverts to the 2M₁ polytype with increasing run duration. The X-ray diffraction patterns of the 1M and 2M₁ poly types can be indexed unambiguously on the basis of the space groups C2 and Cc, respectively. At its upper thermal stability limit, 2M₁ ephesite decomposes according to the reaction (1) $$\begin{gathered} {\text{Na(LiAl}}_{\text{2}} {\text{) [Al}}_{\text{2}} {\text{Si}}_{\text{2}} {\text{O}}_{{\text{10}}} {\text{] (OH)}}_{\text{2}} \hfill \\ {\text{ephesite}} \hfill \\ {\text{ = Na[AlSiO}}_{\text{4}} {\text{] + LiAl[SiO}}_{\text{4}} {\text{] + }}\alpha {\text{ - Al}}_{\text{2}} {\text{O}}_{\text{3}} {\text{ + H}}_{\text{2}} {\text{O}} \hfill \\ {\text{nepheline }}\alpha {\text{ - eucryptite corundum}} \hfill \\ \end{gathered}$$ Five reversal brackets for (1) have been established experimentally in the temperature range 590–750° C, at \(P_{H_2 O}\) between 400 and 2500 bars. The equilibrium constant, K, for this reaction may be expressed as (2) $$log K{\text{ = }}log f_{{\text{H}}_{\text{2}} O}^* = 7.5217 - 4388/T + 0.0234 (P - 1)T$$ where \(f_{H_2 O}^* = f_{H_2 O} (P,T)/f_{H_2 O}^0\) (1,T), with T given in degrees K, and P in bars. Combining these experimental data with known thermodynamic properties of the decomposition products in (1), the following standard state (1 bar, 298.15 K) thermodynamic data for ephesite were calculated: H _f,298.15 ⁰ =-6237372 J/mol, S _298.15 ⁰ =300.455 J/K·mol, G _298.15 ⁰ =-5851994 J/mol, and V _298.15 ⁰ =13.1468 J/bar·mol.     

Atmospheric pressure changes due to volcanic eruptions and possible water level fluctuations

The data of Reed (1983) are analysed to produce the following empirical equations for the amplitude p ₀ (overall fluctuation) in Pascals of the air pressure wave associated with a volcanic eruption of volume V km³ or a nuclear explosion of strength M Mt: Here s is the distance from the source in km. $$\begin{gathered} \log _{10} p_0 = 4.44 + \log _{10} V - 0.84\log _{10} s \hfill \\ {\text{ }} = 3.44 + \log _{10} M - 0.84\log _{10} s. \hfill \\ \end{gathered} $$ Garrett's (1970) theory is examined on the generation of water level fluctuations by an air pressure wave crossing a water depth discontinuity such as a continental shelf. The total amplitude of the ocean wave is determined to be where c ₂ ¹ = gh ₁, c ₂ ² = gh ₂, g is acceleration of gravity, h ₁ and h ₂ are the water depths on the ocean and shore side of the depth discontinuity, c is the speed of propagation of the air pressure wave, and ? is the water density. $$B = \left[ {\frac{{c_2^2 }}{{c^2 - c_2^2 }} + \frac{{c^2 (c_1 - c_2 )}}{{(c - c_1 )(c^2 - c_2^2 )}}} \right]\frac{{p_0 }}{{g\varrho }}$$ It is calculated that a 10 km³ eruption at Mount St. Augustine would cause a 460 Pa air pressure wave and a discernible water level fluctuation at Vancouver Island of several cm amplitude.     

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.     

A possible relation between the masses of black holes in galactic nuclei and the parameters of the host galaxies

Data on about forty virialized galaxy clusters with bright central galaxies, for which both the galactic velocity dispersion (?? _gal) and the stellar velocity dispersion in the brightest galaxies (??*) are measured, have been used to obtain several approximate relations between ?? _gal, ??*, the absolute B magnitude of the brightest central galaxyM _B ^BCG , and the mass of the central massive black holeM _BH: $\begin{gathered} \log \sigma _* = (0.12 \pm 0.14)\log \sigma _{gal} + (2.1 \pm 0.4), \hfill \\ \log \sigma _* = - (0.15 \pm 0.02)M_B^{BCG} + (0.85 \pm 0.5), \hfill \\ \log M_{BH} = 0.51\log \sigma _{gal} + 7.28. \hfill \\ \end{gathered} $ . These relations can be used to derive crude estimates ofMBH in the nuclei of the brightest galaxies using the parameters of the both host galaxies and the host galaxy clusters. The last relation above confirms earlier suggestions of a quadratic relation between the masses of the coronas of the host systems and the masses their central objects: M _hg ^halo ?? M _cent ² . The relations obtained are consistent with the common evolution of subsystems with different scales and masses formed in the process of hierarchical clustering.     

Pyroxene-melt equilibria: an updated model

An updated model for pyroxene-melt equilibria at 1 atm has been developed and calibrated using new and existing experimental data in order to refine calculations of liquid lines of descent, which simulate the effect of igneous differentiation processes. We combine the Davidson and Lindsley (1985) model for activities of components in clinopyroxene and orthopyroxene solid solutions, a _i ^p , where i represents a quadrilateral endmember, with the Nielsen and Drake (1979) expressions for component activities in the melt, a _i ^L (two-lattice melt model). The chemical potential differences for pyroxene-melt equilibria are expressed in the form: $$\Delta \mu _{\iota } = 0 = In \left( {{{a_i^p } \mathord{\left/{\vphantom {{a_i^p } {a_i^L }}} \right.\kern-\nulldelimiterspace} {a_i^L }}} \right) + A_i + {{B_i } \mathord{\left/{\vphantom {{B_i } T}} \right.\kern-\nulldelimiterspace} T}$$ Pyroxene compositions were projected to quadrilateral compositions with the method of Lindsley and Anderson (1983). The regression constants A _i and B _i were calculated from experimental data that consists of 282 pyroxene-melt pairs, including 83 orthopyroxene-melt pairs. These experiments were all performed at 1 atm and represent compositions ranging from basalts (alkali to lunar) to dacites (42–66 wt% SiO₂). The model is calibrated for 1000相似文献   

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.     

Temperature gradient in the cold-seal pressure vessel

Temperature gradient curves are given for 0.5–2 kb and 300–600°C based on internal temperature measurements, and an empirical formula $$t_x = T + (a_0 + a_1 T + a_2 T^2 + a_3 T^3 + a_4 T^4 + a_5 TP + a_6 P)x$$ is established to calculate the temperature at any point,x, within the vessel. Uncertainty of 90 percent estimates by using this formula is less than 5 degrees. In most instances, real pressure within the vessel is only 60–80 percent of the designed pressure as based onP-V-T relations due to the existence of temperature gradient. This large pressure departure can be reduced to less than 30 bars by using $$\begin{array}{l} \bar t = 11.6 + 0.8973{\rm T} + (0.01 - 3.06 \cdot 10^{ - 5} T - 5.664 \cdot 10^8 T^2 \\ + 8.8667 \cdot 10^{ - 11} {\rm T}^3 ) (P - 1250) \\ \end{array}$$ on the assumption of uniform temperature within the vessel. All the temperature-dependent properties such as water fugacity, ionization constant, and the solubility of gases in fluid phase will vary with this temperature gradient. On the other hand, because the pressure within the vessel keeps uniform, there must be an opposite gradient with respect to water density. Therefore, changes will likewise be expected in all the properties which are functions of water density, such as dielectric constant and activity coefficient. These changes may be important for the interpretation and understanding of some experimental results.     

An experimental study of the partitioning of Fe and Mg between garnet and orthopyroxene

The partitioning of Fe and Mg between garnet and aluminous orthopyroxene has been experimentally investigated in the pressure-temperature range 5–30 kbar and 800–1,200° C in the FeO-MgO-Al₂O₃-SiO₂ (FMAS) and CaO-FeO-MgO-Al₂O₃-SiO₂ (CFMAS) systems. Within the errors of the experimental data, orthopyroxene can be regarded as macroscopically ideal. The effects of Calcium on Fe-Mg partitioning between garnet and orthopyroxene can be attributed to non-ideal Ca-Mg interactions in the garnet, described by the interaction term:W _CaMg ^ga -W _CaFe ^ga =1,400±500 cal/mol site. Reduction of the experimental data, combined with molar volume data for the end-member phases, permits the calibration of a geothermometer which is applicable to garnet peridotites and granulites: $$T(^\circ C) = \left\{ {\frac{{3,740 + 1,400X_{gr}^{ga} + 22.86P(kb)}}{{R\ln K_D + 1.96}}} \right\} - 273$$ with $$K_D = {{\left\{ {\frac{{Fe}}{{Mg}}} \right\}^{ga} } \mathord{\left/ {\vphantom {{\left\{ {\frac{{Fe}}{{Mg}}} \right\}^{ga} } {\left\{ {\frac{{Fe}}{{Mg}}} \right\}}}} \right. \kern-\nulldelimiterspace} {\left\{ {\frac{{Fe}}{{Mg}}} \right\}}}$$ and $$X_{gr}^{ga} = (Ca/Ca + Mg + Fe)^{ga} .$$ The accuracy and precision of this geothermometer are limited by largerelative errors in the experimental and natural-rock data and by the modest absolute variation inK _D with temperature. Nevertheless, the geothermometer is shown to yield reasonable temperature estimates for a variety of natural samples.     

Statistical mechanics of coupled solid solutions in the dilute limit

A statistical mechanical analysis of the limiting laws for coupled solid solutions shows that the random model, in which the configurational entropy is calculated as if atoms mix randomly on each crystallographic site, is correct as a first approximation. In coupled solid solutions, since atoms of different valence substitute on the same sites, significant short-range order which reduces the entropy can be expected. A first-order correction is rigorously obtained for the entropy in dilute binary short-range ordered coupled solid solutions: $$\bar S^{{\text{XS}}} {\text{/R = }}Q\left( {{\text{e}}^{--H_{\text{A}} /{\text{R}}T} \left( {\frac{{H_{\text{A}} }}{{{\text{R}}T}} + 1} \right) - 1} \right)N_2^a N_4^b ,$$ where Q is the number of positions an associated cation pair can assume per formula unit, H _A is the association energy per formula unit, and N ₂ ^a and N ₄ ^b are the site occupancy fractions for atoms 2 and 4 that are dilute on sites a and b. S ^XS is the configurational entropy minus the random model entropy. Aluminous pyroxenes on the joints diopside-jadeite and diopside-CaTs are examined as examples. A generalization for dilute multiple component solutions, including possible long-range ordering variations is given by: $$\frac{{\bar S^{{\text{XS}}} }}{{\text{R}}}{\text{ = }}\sum\limits_i {\sum\limits_j {\sum\limits_k {Q_i } } \left( {{\text{e}}^{--H_{\text{A}}^{j{\text{ }}k{\text{, }}i} /{\text{R}}T} \left( {\frac{{H_{\text{A}}^{j{\text{ }}k{\text{, }}i} }}{{{\text{R}}T}} + 1} \right) - 1} \right)N_j^l N_k^m ,} $$ where i labels each crystallographically distinct pair, j and k label atomic species, l and m label crystallographic sites, and the N's are site occupancy fractions for the solute atoms. A total association model is examined as well as the partial association and random models. Real solution behavior must lie between the total association model and the random model. Molecular models in which the ideal activity is proportional to a mole fraction, which in itself is not always unambiguously defined, do not lie in this range and furthermore have no physical justification.     

High temperature calorimetry of sulfide systems

The standard enthalpies of formation of FeS (troilite), FeS₂ (pyrite), Co_0.9342S, Co₃S₄ (linnaeite), Co₉S₈ (cobalt pentlandite), CoS₂ (cattierite), CuS (covellite), and Cu₂S (chalcocite) have been determined by high temperature direct reaction calorimetry at temperatures between 700 K and 1021 K. The following results are reported: $$\Delta {\rm H}_{f,FeS}^{tr} = - 102.59 \pm 0.20kJ mol^{ - 1} ,$$ $$\Delta {\rm H}_{f,FeS}^{py} = - 171.64 \pm 0.93kJ mol^{ - 1} ,$$ $$\Delta {\rm H}_{f,Co_{0.934} S} = - 99.42 \pm 1.52kJ mol^{ - 1} ,$$ $$\Delta {\rm H}_{f,Co_9 S_8 }^{ptl} = - 885.66 \pm 16.83kJ mol^{ - 1} ,$$ $$\Delta {\rm H}_{f,Co_3 S_4 }^{In} = - 347.47 \pm 7.27kJ mol^{ - 1} ,$$ $$\Delta {\rm H}_{f,CoS_2 }^{ct} = - 150.94 \pm 4.85kJ mol^{ - 1} ,$$ $$\Delta {\rm H}_{f,Cu_2 S}^{cc} = - 80.21 \pm 1.51kJ mol^{ - 1} ,$$ and $$\Delta {\rm H}_{f,CuS}^{cv} = - 53.14 \pm 2.28kJ mol^{ - 1} ,$$ The enthalpy of formation of CuFeS₂ (chalcopyrite) from (CuS+FeS) and from (Cu+FeS₂) was determined by solution calorimetry in a liquid Ni_0.60S_0.40 melt at 1100 K. The results of these measurements were combined with the standard enthalpies of formation of CuS, FeS, and FeS₂, to calculate the standard enthalpy of formation of CuFeS₂. We found \(\Delta {\rm H}_{f,CuFeS_2 }^{ccp} = - 194.93 \pm 4.84kJ mol^{ - 1}\) . Our results are compared with earlier data given in the literature; generally the agreement is good and our values agree with previous estimates within the uncertainties present in both.     

Physical conditions in faint gigahertz-peaked spectrum radio sources

Weak, compact radio sources (～100 mJy peak flux, L～1–10 pc) with their spectral peaks at about a gigahertz are studied, based on the complete sample of 46 radio sources of Snellen, drawn from high-sensitivity surveys, including the low-frequency Westerbork catalog. The physical parameters have been estimated for 14 sources: the magnetic field (H _⊥), the number density of relativistic particles (n _e), the energy of the magnetic field $(E_{H_ \bot } )$ , and the energy of relativistic particles (E _e). Ten sources have $E_{H_ \bot } \ll E_e $ , three have approximate equipartition of the energies $(E_{H_ \bot } \sim E_e )$ , and only one has $E_{H_ \bot } \gg E_e $ . The mean magnetic fields in quasars (10^?3 G) and galaxies (10^?2 G) have been estimated. The magnetic field appears to be related to the sizes of compact features as $H \sim 1/\sqrt L $ .     

Thermochemical and pressure-volume-temperature systematics of data on solids,examples: Tungsten and MgO

Data systematization using the constraints from the equation $$Cp = Cv + \alpha _P {}^2V_T K_T T$$ where C _p, C _v, α _p, K _T and V are respectively heat capacity at constant pressure, heat capacity at constant volume, isobaric thermal expansion, isothermal bulk modulus and molar volume, has been performed for tungsten and MgO. The data are $$K_T (W) = 1E - 5/(3.1575E - 12 + 1.6E - 16T + 3.1E - 20T^2 )$$ $$\alpha _P (W) = 9.386E - 6 + 5.51E - 9T$$ $$C_P (W) = 24.1 + 3.872E - 3T - 12.42E - 7T^2 + 63.96E - 11T^3 - 89000T^{ - 2} $$ $$K_T (MgO) = 1/(0.59506E - 6 + 0.82334E - 10T + 0.32639E - 13T^2 + 0.10179E - 17T^3 $$ $$\alpha _P (MgO) = 0.3754E - 4 + 0.7907E - 8T - 0.7836/T^2 + 0.9148/T^3 $$ $$C_P (MgO) = 43.65 + 0.54303E - 2T - 0.16692E7T^{ - 2} + 0.32903E4T^{ - 1} - 5.34791E - 8T^2 $$ For the calculation of pressure-volume-temperature relation, a high temperature form of the Birch-Murnaghan equation is proposed $$P = 3K_T (1 + 2f)^{5/2} (1 + 2\xi f)$$ Where $$K_T = 1/(b_0 + b_1 T + b_2 T^2 + b_3 T^3 )$$ $$f = (1/2)\{ [V(1,T)/V(P,T)]^{2/3} - 1\} $$ $$\xi = ({3 \mathord{\left/ {\vphantom {3 4}} \right. \kern-\nulldelimiterspace} 4})[K'_0 + K'_1 \ln ({T \mathord{\left/ {\vphantom {T {300}}} \right. \kern-\nulldelimiterspace} {300}}) - 4]$$ where in turn $$V(1,T) = V_0 [\exp (\int\limits_{300}^T {\alpha dT)]} $$ . The temperature dependence of the pressure derivative of the bulk modulus (K′₁) is estimated by using the shock-wave data. For tungsten the data are K′₀ = 3.5434, K′₁ = 0.032; for MgO K′₀ = 4.17 and K′₁ = 0.1667. For calculating the Gibbs free energy of a solid at high pressure and at temperatures beyond that of melting at 1 atmosphere, it is necessary to define a high-temperature reference state for the fictive solid.     

Thermal equation of state of synthetic orthoferrosilite at lunar pressures and temperatures

Iron-rich orthopyroxene plays an important role in models of the thermal and magmatic evolution of the Moon, but its density at high pressure and high temperature is not well-constrained. We present in situ measurements of the unit-cell volume of a synthetic polycrystalline end-member orthoferrosilite (FeSiO₃, fs) at simultaneous high pressures (3.4–4.8 GPa) and high temperatures (1,148–1,448 K), to improve constraints on the density of orthopyroxene in the lunar interior. Unit-cell volumes were determined through in situ energy-dispersive synchrotron X-ray diffraction in a multi-anvil press, using MgO as a pressure marker. Our volume data were fitted to a high-temperature Birch–Murnaghan equation of state (EoS). Experimental data are reproduced accurately, with a  $\varDelta P$ Δ P  standard deviation of 0.20 GPa. The resulting thermoelastic parameters of fs are: V ₀ = 875.8 ± 1.4 Å³, K ₀ = 74.4 ± 5.3 GPa, and $\frac{{\text d}K}{{\text d}T} = -0.032 \pm 0.005\,\hbox{GPa K}^{-1}$ d K d T = - 0.032 ± 0.005 GPa K - 1 , assuming ${K}^{\prime}_{0} = 10 $ K 0 ′ = 10 . We also determined the thermal equation of state of a natural Fe-rich orthopyroxene from Hidra (Norway) to assess the effect of magnesium on the EoS of iron-rich orthopyroxene. Comparison between our two data sets and literature studies shows good agreement for room-temperature, room-pressure unit-cell volumes. Preliminary thermodynamic analyses of orthoferrosilite, FeSiO₃, and orthopyroxene solid solutions, (Mg_1?x Fe _x ) SiO₃, using vibrational models show that our volume measurements in pressure–temperature space are consistent with previous heat capacity and one-bar volume–temperature measurements. The isothermal bulk modulus at ambient conditions derived from our measurements is smaller than values presented in the literature. This new simultaneous high-pressure, high-temperature data are specifically useful for calculations of the orthopyroxene density in the Moon.