Trace element incorporation into pyrope–grossular solid solutions: an atomistic simulation study

 We have performed atomistic computer simulations on trace element incorporation into the divalent dodecahedral X-sites of pyrope (Py — Mg₃Al₂Si₃O₁₂) – grossular (Gr — Ca₃Al₂Si₃O₁₂) solid solutions. An ionic model and the Mott–Littleton two-region approach to defect energies were used to calculate the energetics of substitution by a range of divalent trace-elements and of charge-balanced substitution by trivalent ions in the static limit. Results are compared with experimental high-temperature, high-pressure garnet-melt trace element partitioning data obtained for the same garnet solid solution to refine our understanding of the factors controlling element partitioning into solid solutions. Defect energies (U _def,f), relaxation (lattice strain) energies (U _rel), and solution energies (U _sol) were derived using two different approaches. One approach assumes the presence of one type of hybrid X-site with properties intermediate between pure Mg and Ca sites, and the other assumes discrete Mg and Ca X-sites, and thus two distinct cation sublattices. The hybrid model is shown to be inadequate, since it averages out local distortions in the garnet structure. The discrete model results suggest trace elements are more soluble in Py₅₀Gy₅₀ than in either end-member compound. Physically this is due to small changes in size of the X-sites and the removal of unfavourable interactions between third nearest neighbours of the same size. Surprisingly, depending on the local order, large trace element cations may substitute for Mg²⁺ and small trace elements for Ca²⁺ in Py₅₀Gr₅₀. These solubilities provide an explanation for the anomalous trace-element partitioning behaviour along the pyrope–grossular join observed experimentally. Received: 27 January 2000 / Accepted: 14 February 2003     

Silicate solid solutions and geothermometry

Principal component analysis, using eigenvalues and eigenvectors, of the encountered variation in the chemistry of 153 garnets indicates the following: (1) In low grade metamorphic rocks, spessertite and almandine are distinctly isomorphous. The variability of pyrope content does not influence the binary Fe-Mn relationship. However there may be some influence of the grossularite content. (2) In high grade metamorphic rocks, pyrope and almandine are distinctly isomorphous. Variability of grossularite content does not affect the binary Mg-Fe relationship. However, variability in spessertite content may influence the linearity particularly when there is little pyrope.Similar statistical analysis of the chemical data on 119 samples of clinopyroxenes indicates that only significant changes in the concentrations of Al in octahedral and/or of Al in tetrahedral sites and Ti could cause some change in the FeMg ratio in the mineral.Principal component analysis of the data on coexisting garnet and clinopyroxene could be used to classify rocks into their petrogenetic types.Distribution coefficient calculated by assuming ideal binary solution of Mg and Fe members in pyroxene and garnet is useful to indicate the P-T of the formation of the rocks.     

Silicate solid solutions and geothermometry

The properties of the regular solution are described. A solid solution is assigned certain theoretical values of the heat of mixing and the results of the distribution of a component between two coexisting solutions are shown graphically on Roozeboom diagrams. Such representations may be useful in explaining the distribution of elements observed in natural mineral assemblages.     

Computer simulation of crystal structures

It is impossible to calculate from first principles the details of crystal structures. However, it is possible to rely on information which is certain, namely on interatomic distances, and calculate the details of a crystal structure based on these. Individual distances between atoms can be predicted accurately if one considers the coordination numbers of anions and cations, the extended electrostatic valence rule, and the effects of shared edges between different coordination polyhedra. The predicted interatomic distances are used as observations in a distance least squares refinement in which the positional parameters and the cell edges are adjusted until the calculated interatomic distances correspond as closely as possible to the predicted distances. The topology of the simulated structure has to be known or assumed. Appropriate weighting can facilitate proper modelling. Applications include: a) solution of pseudosymmetric structures; b) preliminary refinement of trial structures; c) geometric refinement without X-ray intensities; d) comparison of hypothetical structures with observed polymorphs of the same compositions; e) simulation of temperature dependence of structures; f) simulation of pressure dependence; g) calculation of structures isomorphous to known structures; h) calculation of thermal ellipsoids; i) calculation of local environments deviating from the overall symmetry of a structure; j) testing of hypotheses about the behavior of structures at varying conditions.     

Computer simulation of reef growth

Light is one of the major controls on reef growth and carbonate production. The growth of present reef builders depends largely upon the amount of light available for photosynthesis. As light decreases with water depth, so does reef growth. The computer model presented extends this principle by combining two functions, one for photosynthesis and the other for the extinction of light in water. The model is used to simulate the growth of Alacran Reef, Mexico, two reefs of the Great Barrier Reef and the reefs of the windward platform of St Croix, US Virgin Islands. The model also gives an accurate simulation of the growth of fore-reef walls in Belize, in agreement with the accretion hypothesis developed for this feature.     

Computer simulation of second order faults

SummaryComputer Simulation of Second Order Faults Faults of the second order can be simulated in the computer with an accuracy hitherto unknown, by using Malina's (1969, 1970) program version of the Finite Element Method. The computations show that stress rearrangements occur with shear movements along the principal fault and with propagation of the fault. These stress rearrangements contribute to the development of new secondary faults. The computed second order faults can be classified according to Chinnery (1966) as shear plane typesA, B, E, andF. Moreover, the computation shows two dominant shear planes which cannot be classified according to Chinnery.

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ZusammenfassungSimulation von Störungen zweiter Ordnung im Computer Störungen zweiter Ordnung lassen sich mit einer bisher noch nicht gekannten Genauigkeit mit Hilfe einer von Malina (1969, 1970) entwickelten Programmversion der Finite-Element-Methode im Computer simulieren. Die Rechnungen zeigen, daß mit einer Scherbewegung entlang der Hauptstörung und dem Weiterreißen der Störung Spannungsumlagerungen einhergehen, die mitbewirken, daß neue sekundäre Störungen entstehen. Die ermittelten Störungen zweiter Ordnung lassen sich nach der Klassifikation von Chinnery (1966) als die ScherflächentypenA, B, E undF einordnen. Darüberhinaus liefert die Rechnung zwei sehr häufig auftretende Scherflächen, die sich nach Chinnery nicht klassifizieren lassen.

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RésuméSimulation de failles de deuxième ordre par l'ordinateur Les failles de deuxième ordre peuvent être simulées avec une précision encore jamais atteinte grâce à l'utilisation d'une version d'une programme développé par Malina (1969, 1970) à partir de la méthode des éléments finis. Le calcul montre que, conjointement au cisaillement le long de la faille principale et à la propagation de cette faille, de nouvelles redistributions des contraintes apparaissent. Ces nouvelles redistributions contribuent essentiellement au développement de nouvelles surfaces de failles secondaires. Les failles de deuxième ordre peuvent être classées d'après Chinnery (1966) suivant les catégories de surfaces de cisaillementA, B, E, etF. De plus, le calcul indique fréquemment deux autres types de surfaces de cisaillement n'entrant pas dans la classification de Chinnery.

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With 11 Figures     

Thermodynamic properties of almandine-grossular garnet solid solutions

The mixing properties of Fe₃Al₂Si₃O₁₂-Ca₃Al₂Si₃O₁₂ garnet solid solutions have been studied in the temperature range 850–1100° C. The experimental method involves measuring the composition of garnet in equilibrium with an assemblage in which the activity of the Ca₃Al₂Si₃O₁₂ component is fixed. Experiments on the assemblage garnet solid solution, anorthite, Al₂SiO₅ polymorph and quartz at known pressure and temperature fix the activity of the Ca₃Al₂Si₃O₁₂ component through the equilibrium: 1 $$\begin{gathered} {\text{3CaAl}}_{\text{2}} {\text{Si}}_{\text{2}} {\text{O}}_{\text{8}} \rightleftarrows {\text{Ca}}_{\text{3}} {\text{Al}}_{\text{2}} {\text{Si}}_{\text{3}} {\text{O}}_{{\text{12}}} \hfill \\ {\text{Anorthite garnet}} \hfill \\ {\text{ + 2Al}}_{\text{2}} {\text{SiO}}_{\text{5}} {\text{ + SiO}}_{\text{2}} \hfill \\ {\text{ sillimanite/kyanite quartz}}{\text{.}} \hfill \\ \end{gathered}$$ This equilibrium, with either sillimanite or kyanite as the aluminosilicate mineral, was used to control \({\text{a}}_{{\text{Ca}}_{\text{3}} {\text{Al}}_{\text{2}} {\text{Si}}_{\text{3}} {\text{O}}_{{\text{12}}} }^{{\text{gt}}} \) . The compositions of the garnet solutions produced were determined by measurement of their unit cell edges. At 1 bar Fe₃Al₂Si₃O₁₂-Ca₃Al₂Si₃O₁₂ garnets exhibit negative deviations from ideality at the Fe-rich end of the series and positive deviations at the calcium end. With increasing pressure the activity coefficients for the Ca₃Al₂Si₃O₁₂ component increase because the partial molar volume of this component is greater than the molar volume of pure grossular. Previous studies indicate that the activity coefficients for the Ca₃Al₂Si₃O₁₂ component also increase with increasing (Mg/Mg+Fe) ratio of the garnet. The region of negative deviation from ideality implies a tendency towards formation of a stable Fe-Ca garnet component. Evidence in support of this conclusion has been found in a natural Fe-rich garnet which was found to contain two different garnet phases of distinctly different compositions.     

Activities of components in omphacitic solid solutions

Available experimental data on mixing of disordered C2/c clinopyroxenes in the system diopside-jadeite-hedenbergite-acmite are reviewed and evaluated. Because the methods used to determine jadeite activity suffer from severe uncertainty at high jadeite mol fractions, these data cannot be used to infer asymmetry in the jadeite-diopside or the jadeite-hedenbergite solid solutions. If the measurement uncertainties are taken into account, a single parameter (regular, or reciprocal energy) suffices to describe the mixing properties of these two solid solutions. It is argued that a two-site entropy of mixing satisfies the experiments and is consistent with the C2/c disordered nature of the solid solutions; the data in the range 600–1300° C are consistent with a temperature-independent interaction energy, implying no discernible excess entropy. The available experimental data imply W=26±2 kJ mol^–1 for jd-di, and W=25±3 kJ mol^–1 for jd-hd, solid-solutions. Landau theory for a tricritical phase transformation (C2/c-P2/n) is in good agreement with the calorimetrically determined disordering enthalpy, and may be used to derive a simple expression for the activities in ordered omphacite solid solutions. The derived activities of jadeite at 600° C in ordered omphacites are remarkably close to those reported previously for short-range ordered pyroxenes. A simple model is presented for determining the activities of end-members in the system jadeite-diopside-hedenbergite-acmite.     

The thermodynamic properties of reciprocal solid solutions

A multisite solid solution of the type (A, B) (X, Y) has the four possible components AX, AY, BX, BY. Taking the standard state to be the pure phase at the pressure and temperature of interest, the mixing of these components is shown not to be ideal unless the condition: $$\Delta G^0 = (\mu _{AX}^0 + \mu _{BY}^0 - \mu _{AY}^0 - \mu _{BX}^0 = 0$$ applies. Even for the case in which mixing on each of the individual sublattices is ideal, ΔG ⁰ contributes terms of the following form to the activity coefficients of the constituent components: $$RT\ln \gamma _{AX} = - X_{B_1 } X_{Y_2 } \Delta G^0$$ (X _Ji refers to the atomic fraction of J on sublattice i). The above equation, which assumes complete disorder on (A, B) sites and on (X, Y) sites is extended to the general n-component case. Methods of combining the “cross-site” or reciprocal terms with non-ideal terms for each of the individual sites are also described. The reciprocal terms appear to be significant in many geologically important solid solutions, and clinopyroxene, garnet and spinel solid solutions are all used as examples. Finally, it is shown that the assumption of complete disorder only applies under the condition: $$\Delta G^0 \ll zn_1 RT$$ where z is the number of nearest-neighbour (X, Y) sites around A and n ₁ is the number of (A, B) sites in the formula unit. If ΔG ⁰ is relatively large, then substantial short range oder must occur and the activity coefficient is given by (ignoring individual site terms): $$\gamma _{AX} = \left( {\frac{{1 - X'_{Y_2 } }}{{1 - X_{Y_2 } }}} \right)^{zn_1 }$$ where X′_Y2 is the equilibrium atomic fraction of Y atoms surrounding A atoms in the structure. The ordered model may be developed for multicomponent solutions and individual site interactions added, but numerical methods are needed to solve the simultaneous equations involved.     

Computer simulation of fracturing of layered rock

Fracturing of rocks is simulated using an electronic computer, on the assumption that when a stress acts on a model, the fracturing condition is determined according to Mohr's theory and that of Griffith. A rock body is regarded as an aggregate of a grid cell whose strength is expressed by a random number of Gaussian distribution functions and the layered fabric of the rock body is given by inserting layered parts with different strength and with various inclinations. The results displaying the fractures on the grid system suggest that some of the geologic structures such as the echelon or the zigzag pattern of fracture are ascribed to fracturing in a layered or heterogeneous body, and variation of strength and the presence of layered fabric in a rock have a significant influence on fracture patterns developed in the rock.     

Local structural heterogeneity in garnet solid solutions

Local heterogeneities in pyrope-almandine, almandine-grossular and pyrope-grossular solid solutions have been investigated using IR-powder absorption spectroscopy. Correlations of the wavenumber shifts and line broadening systematics with the thermodynamic mixing properties were found. Wavenumber shifts of the highest energy modes correlate closely with the Si-O bond distances and give an indirect view of the average distortions across the three solid solutions. They have a linear behaviour for Py-Alm, but show positive variations from linearity for Alm-Gr and Py-Gr systems. An effective line width (Δ_corr) of the absorption bands over a given wavenumber interval was obtained using the autocorrelation function. Line broadening is associated with local heterogeneities arising from cation substitution in the structure of samples at intermediate compositions. Non-linearities of the line broadening were found for Alm-Gr and Py-Gr and have a shape similar to the enthalpy of mixing, ΔH_mix. An empirical analysis was therefore carried out to compare ΔH_mix and Δ_corr quantitatively. Low-temperature far-IR spectra were recorded for the end-members pyrope, almandine and grossular and far-IR and mid-IR low-temperature spectra for Py₆₀Gr₄₀ in the temperature range 292–44 K. Softening of the lowest energy band with decreasing temperature was observed in the spectrum of pyrope and more enhanced in the spectrum of Py₆₀Gr₄₀. The same softening occurs by substitution of grossular component into pyrope. High energy modes of Py₆₀Gr₄₀ show the effect of saturation below 110–130 K, which correlates with the volume saturation at low temperature. This could provide an alternative explanation for the heat capacity anomaly found for Py-Gr solid solution at low-temperatures.     

断层几何形态反演的计算机模拟

构造反演是指利用构造变形形态反推使之变形的断层的几何参数。在拉张构造区域,从理论上解决构造反演问题是相当成功的。而对于挤压推覆构造区域,由于其逆断层因上盘上升过程,使得其推算的断层只能获取下盘断点以下的构造形态,所以在实际运用中,特别是对于复杂构造叠加区域,其反演过程更多的是依靠人工干预的方法来进行。鉴于此,利用较成熟的拉张构造区域的理论方法,自行提出了单剪变形的挤压构造反演方法来完善,并通过计算机模拟来实现,以此为正演平衡剖面计算,提供一个初始的断层几何模型,并为验证地震剖面解释的合理性提供辅助作用。     

基于GIS的固体矿产资源潜力评价

矿产资源潜力评价成果对提高矿产勘查工作的经济效益具有重要意义,从20世纪70年代起各国就相继开展不同规模的矿产资源潜力评价工作.矿产资源预测评价方法经过30多年的发展,形成了一套相对成熟的方法.基于GIS的矿产资源潜力评价方法,通过对各种地质信息的数字化,实现成矿信息的定量化识别与提取,最终实现对矿产资源位置的圈定、数...     

计算机模拟引擎及其在铀成矿过程模拟中的应用研究

本文介绍了计算机模拟引擎的设计与开发 ,铀成矿过程模拟的实施步骤 ,并对其在伊犁盆地南缘的应用作了探讨。     

Computer simulation of sedimentation in meandering streams

A dynamic mathematical model for simulation of sedimentation in meandering streams is briefly described. This is composed of component mathematical models which are formulated to predict the following aspects of the system for a given physical situation and a single time increment. (1) The characteristics of the plan form of the meander; (2) the movement of the meander in plan, and definition of cross-sections across the meander in which erosion and deposition are considered in detail; (3) the hydraulic properties of the channel in the bend and the erosional and depositional activity within the channel as defined in specific cross-sections; (4) the nature and occurrence of cut-off; (5) a relative measure of the discharge during a seasonal high water period, which is used in (3) and (4); (6) aggradation. The model, in the form of a FORTRAN IV computer program, has been used to simulate various aspects of sedimentation in meandering streams by performing a set of experiments with the program under different input conditions. The geometry of simulated point bar sediments, as controlled by channel migration over floodplains with variable sediment type, agrees broadly with the natural situation, however extensive sheets of point bar sediment cannot be simulated because large scale meander-belt movements are not accounted for. In the simulated sediments, successive surfaces of the point bar before falling stage deposition (lateral and vertical) may be picked out, and these delineate the epsilon cross-stratification of Allen (1963b). The epsilon unit thickness is that measured from bankfull stage down to the lowest channel position existing prior to deposition. The model records the characteristic fining upwards of grain sizes in the point bar, and the systematic distribution of sedimentary structures. Channel migration combined with seasonal scouring and filling across the channel section produces a characteristic relief in the basal scoured surfaces and facies boundaries (as defined by variation in grain size and sedimentary structure). A related lensing and inter-fingering of the facies may also be present. The model also records large-scale lateral changes in grain size and sedimentary structure associated with changes in the shape of developing meanders. When channel migration is combined with a constant aggradation rate the model predicts a general slope (relative to the land surface) of facies boundaries and scoured basal surfaces upward in the direction of channel movement. If aggradation sufficiently increases the thickness of fine-grained overbank material, there is a channel stabilization effect. It is shown that a complete sequence of point bar sediments capped by overbank sediments would rarely be preserved in the moving-phase situation. Such preservation only becomes likely when an aggrading section lies out of range of an eroding channel for a considerably longer time span than it takes a meander to move one half-wavelength downvalley. Deep channel scours have a higher preservation potential than contemporary shallower ones. Where appropriate field data exist the model can be used in the more accurate recognition of ancient fluviatile sediments. Inferences may be made about the erosion-deposition processes operating in the ancient channel system, and the geometry and hydraulics of the system can be alluded to. A representative application of the model to the quantitative interpretation of an ancient point bar deposit is illustrated. There is reasonable agreement between the natural and the simulated deposits, and a broad quantitative picture of the palaeoenvironment of sedimentation is obtained.