Kriging and Spatial Design Accelerated by Orders of Magnitude: Combining Low-Rank Covariance Approximations with FFT-Techniques

Computational power poses heavy limitations to the achievable problem size for Kriging. In separate research lines, Kriging algorithms based on FFT, the separability of certain covariance functions, and low-rank representations of covariance functions have been investigated, all three leading to drastic speedup factors. The current study combines these ideas, and so combines the individual speedup factors of all ideas. This way, we reduce the mathematics behind Kriging to a computational complexity of only $\mathcal{O}(dL^{*} \log L^{*})$ , where L ^? is the number of points along the longest edge of the involved lattice of estimation points, and d is the physical dimensionality of the lattice. For separable (factorized) covariance functions, the results are exact, and nonseparable covariance functions can be approximated well through sums of separable components. Only outputting the final estimate as an explicit map causes computational costs of $\mathcal{O}(n)$ , where n is the number of estimation points. In illustrative numerical test cases, we achieve speedup factors up to 10⁸ (eight orders of magnitude), and we can treat problem sizes of up to 15 trillion and two quadrillion estimation points for Kriging and spatial design, respectively, within seconds on a contemporary desktop computer. The current study assumes second-order stationarity and simple Kriging on a regular, equispaced lattice, without working with restricted neighborhoods. Extensions to many other cases are straightforward.     

Efficient Computation of Linearized Cross-Covariance and Auto-Covariance Matrices of Interdependent Quantities

In many geostatistical applications, spatially discretized unknowns are conditioned on observations that depend on the unknowns in a form that can be linearized. Conditioning takes several matrix–matrix multiplications to compute the cross-covariance matrix of the unknowns and the observations and the auto-covariance matrix of the observations. For large numbers n of discrete values of the unknown, the storage and computational costs for evaluating these matrices, proportional to n ², become strictly inhibiting. In this paper, we summarize and extend a collection of highly efficient spectral methods to compute these matrices, based on circulant embedding and the fast Fourier transform (FFT). These methods are applicable whenever the unknowns are a stationary random variable discretized on a regular equispaced grid, imposing an exploitable structure onto the auto-covariance matrix of the unknowns. Computational costs are reduced from O(n ²) to O(nlog₂ n) and storage requirements are reduced from O(n ²) to O(n).     

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.     

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.     

Ab-initio structure, energy and stable Fe isotope equilibrium fractionation of some geochemically relevant H-O-Fe complexes

The hexa-aqua complexes [Fe(H₂O)_6−m−n(OH)_n]⁽²⁻ⁿ⁾⁺n = 0 → 3, m = 0 → 6 − n; [Fe(H₂O)_6−m−n(OH)_n]⁽³⁻ⁿ⁾⁺n = 0 → 4, m = 0 → 6 − n were investigated by ab-initio methods with the aim of determining their ground-state geometries, total energies and vibrational properties by treating their inner solvation shell as part of their gaseous precursor¹ (or “hybrid approach”). After a gas-phase energy optimization within the Density Functional Theory (DFT), the molecules were surrounded by a dielectric representing the Reaction Field through an implicit Polarized Continuum Model (PCM). The exploration of several structural ligand arrangements allowed us to quantify the relative stabilities of the various ionic species and the role of the various forms of energy (solute-solvent electronic interaction, cavitation, dispersion, repulsion, liberation free energy) that contribute to stabilize the aqueous complexes. A comparison with experimental thermochemistries showed that ab-initio gas-phase + solvation energies are quite consistent with experimental evidence and allow the depiction of the most stable form in solution and the eventual configurational disorder of water/hydroxyl species around central cations. A vibrational analysis performed on the ⁵⁴Fe, ⁵⁶Fe, ⁵⁷Fe and ⁵⁸Fe isotopomers indicated important separative effects systematically affected by the extent of deprotonation. The role of the system’s redox state (fO₂) and acidity (pH) on the isotopic imprinting of the aqueous species in solution was investigated by coupling the separative effects with speciation calculations. The observed systematics provided a tool of general utility in the interpretation of the iron isotopic signature of natural waters. Applications to the interpretation of isotopic fractionation in solution dictated by redox equilibria and to the significance of the Fe-isotopic imprinting of Banded Iron Formations are given.     

Optical absorption and mössbauer spectra of purple and green yoderite,a kyanite-related mineral

Mössbauer and polarized optical absorption spectra of the kyanite-related mineral yoderite were recorded. Mössbauer spectra of the purple (PY) and green yoderite (GY) from Mautia Hill, Tanzania, show that the bulk of the iron is Fe³⁺ in both varieties, with Fe²⁺/(Fe²⁺+Fe³⁺) ratios near 0.05. Combining this result with new microprobe data for PY and with literature data for GY gives the crystallochemical formulae: $$\begin{gathered} ({\text{Mg}}_{{\text{1}}{\text{.95}}} {\text{Fe}}_{{\text{0}}{\text{.02}}}^{{\text{2 + }}} {\text{Mn}}_{{\text{0}}{\text{.01}}}^{{\text{2 + }}} {\text{Fe}}_{{\text{0}}{\text{.34}}}^{{\text{3 + }}} {\text{Mn}}_{{\text{0}}{\text{.07}}}^{{\text{3 + }}} {\text{Ti}}_{{\text{0}}{\text{.01}}} {\text{Al}}_{{\text{3}}{\text{.57}}} )_{5.97}^{[5,6]} \hfill \\ {\text{Al}}_{{\text{2}}{\text{.00}}}^{{\text{[5]}}} [({\text{Si}}_{{\text{3}}{\text{.98}}} {\text{P}}_{{\text{0}}{\text{.03}}} ){\text{O}}_{{\text{18}}{\text{.02}}} ({\text{OH)}}_{{\text{1}}{\text{.98}}} ] \hfill \\ \end{gathered}$$ and PY and $$\begin{gathered} ({\text{Mg}}_{{\text{1}}{\text{.98}}} {\text{Fe}}_{{\text{0}}{\text{.02}}}^{{\text{2 + }}} {\text{Mn}}_{{\text{< 0}}{\text{.001}}}^{{\text{2 + }}} {\text{Fe}}_{{\text{0}}{\text{.45}}}^{{\text{3 + }}} {\text{Ti}}_{{\text{0}}{\text{.01}}} {\text{Al}}_{{\text{3}}{\text{.56}}} )_{6.02}^{[5,6]} \hfill \\ {\text{Al}}_{{\text{2}}{\text{.00}}}^{{\text{[5]}}} [({\text{Si}}_{{\text{3}}{\text{.91}}} {\text{O}}_{{\text{17}}{\text{.73}}} {\text{(OH)}}_{{\text{2}}{\text{.27}}} ] \hfill \\ \end{gathered}$$ for GY. The Mössbauer spectra at room temperature contain one main doublet with isomer shifts and quadrupole splittings of 0.36 (PY), 0.38 (GY) and 1.00 (PY), 0.92 (GY) mm s^?1, respectively. These values correspond to Fe³⁺ in six or five-fold coordination. The doublet components have anomalously large half widths indicating either accomodation of Fe³⁺ in more than one position (e.g., octahedraA1 and five coordinatedA2) or the yet unresolved superstructure. Besides strong absorption in the ultraviolet (UV) starting from about 25,000 cm^?1, the polarized optical absorption spectra are dominated by strong bands around 16,500 and 21,000 cm^?1 (PY) and a medium strong band at around 13,800 cm^?1 (GY). Position and polarization of these bands, in combination with the UV absorption, explain the colour and pleochroism of the two varieties. The bands in question are assigned to homonuclear metal-to-metal charge transfer transitions: Mn²⁺(A1) Mn³⁺(A1′) ? Mn³⁺(A1) Mn²⁺(A1′) and Mn²⁺(A1) Mn³⁺(A2 ? Mn³⁺(A1) Mn²⁺(A2) in PY and Fe²⁺(A1) Fe³⁺(A1′) ? Fe³⁺(A1) Fe²⁺(A1′) in GY. The evidence for homonuclear Mn²⁺ Mn³⁺ charge transfer (CTF) is not quite clear and needs further study. Heteronuclear FeTi CTF does not contribute to the spectra. In PY, additional weak bands were resolved at energies around 17,700, 18,700, 21,000, and 21,900 cm^?1 and assigned to Mn³⁺ in two positions. Weak bands around 10,000 cm^?1 in both varieties are assigned to Fe²⁺ spin-alloweddd-transitions. Very weak and sharp bands, around 15,400, 16,400, 21,300, 22,100, 23,800, and 25,000 cm^?1 are identified in GY and assigned to Fe³⁺ spin-forbiddendd-transitions.     

Olivine/melt transition metal partitioning, melt composition, and melt structure—Melt polymerization and Q-speciation in alkaline earth silicate systems

The two most abundant network-modifying cations in magmatic liquids are Ca²⁺ and Mg²⁺. To evaluate the influence of melt structure on exchange of Ca²⁺ and Mg²⁺ with other geochemically important divalent cations (m-cations) between coexisting minerals and melts, high-temperature (1470-1650 °C), ambient-pressure (0.1 MPa) forsterite/melt partitioning experiments were carried out in the system Mg₂SiO₄-CaMgSi₂O₆-SiO₂ with ?1 wt% m-cations (Mn²⁺, Co²⁺, and Ni²⁺) substituting for Ca²⁺ and Mg²⁺. The bulk melt NBO/Si-range (NBO/Si: nonbridging oxygen per silicon) of melt in equilibrium with forsterite was between 1.89 and 2.74. In this NBO/Si-range, the NBO/Si(Ca) (fraction of nonbridging oxygens, NBO, that form bonds with Ca²⁺, Ca²⁺-NBO) is linearly related to NBO/Si, whereas fraction of Mg²⁺-NBO bonds is essentially independent of NBO/Si. For individual m-cations, rate of change of K_D(m−Mg) with NBO/Si(Ca) for the exchange equilibrium, m^melt + Mg^olivine ? m^olivine + Mg^melt, is linear. K_D(m−Mg) decreases as an exponential function of increasing ionic potential, Z/r² (Z: formal electrical charge, r: ionic radius—here calculated with oxygen in sixfold coordination around the divalent cations) of the m-cation. The enthalpy change of the exchange equilibrium, ΔH, decreases linearly with increasing Z/r² [ΔH = 261(9)-81(3)·Z/r² (Å⁻²)]. From existing information on (Ca,Mg)O-SiO₂ melt structure at ambient pressure, these relationships are understood by considering the exchange of divalent cations that form bonds with nonbridging oxygen in individual Qⁿ-species in the melts. The negative ∂K_D(m−Mg)/∂(Z/r²) and ∂(ΔH)/∂(Z/r²) is because increasing Z/r² is because the cations forming bonds with nonbridging oxygen in increasingly depolymerized Qⁿ-species where steric hindrance is decreasingly important. In other words, principles of ionic size/site mismatch commonly observed for trace and minor elements in crystals, also govern their solubility behavior in silicate melts.     

Direct Sequential Co-simulation with Joint Probability Distributions

The practice of stochastic simulation for different environmental and earth sciences applications creates new theoretical problems that motivate the improvement of existing algorithms. In this context, we present the implementation of a new version of the direct sequential co-simulation (Co-DSS) algorithm. This new approach, titled Co-DSS with joint probability distributions, intends to solve the problem of mismatch between co-simulation results and experimental data, i.e. when the final biplot of simulated values does not respect the experimental relation known for the original data values. This situation occurs mostly in the beginning of the simulation process. To solve this issue, the new co-simulation algorithm, applied to a pair of covariates Z ₁(x) and Z ₂(x), proposes to resample Z ₂(x) from the joint distribution F(z ₁,z ₂) or, more precisely, from the conditional distribution of Z ₂(x ₀), at a location x ₀, given the previously simulated value z₁^(l)(x₀)z_{1}^{(l)}(x_{0}) (F(Z₂|Z₁=z₁^(l)(x₀)F(Z_{2}|Z_{1}=z_{1}^{(l)}(x_{0}) ). The work developed demonstrates that Co-DSS with joint probability distributions reproduces the experimental bivariate cdf and, consequently, the conditional distributions, even when the correlation coefficient between the covariates is low.     

Complex formation of Cm(III) with formate studied by time-resolved laser fluorescence spectroscopy

Pore waters of natural clays, which are investigated as potential host rock formations for high-level nuclear waste, are known to contain large amounts of low-molecular weight organic compounds. These small organic ligands might impact the aqueous geochemistry of the stored radionuclides and, thus, their migration behavior. In the present work, the complexation of Cm(III) with formate in aqueous NaCl solution is investigated by time-resolved laser fluorescence spectroscopy (TRLFS) as a function of the ionic strength (0.5–3.0 mol/kg), the ligand concentration (0–0.2 mol/kg) and the temperature (20–90 °C). The Cm(III) speciation is determined by deconvolution of the emission spectra. The obtained distribution of Cm(III) species is used to calculate the conditional stability constants (log K′(T)) at a given temperature and ionic strength which are extrapolated to zero ionic strength by using the specific ion interaction theory (SIT). Thus, the thermodynamic log K⁰_n(T) values for the formation of [Cm(Form)_n]⁽³⁻ⁿ⁾⁺ (n = 1, 2) and the ion interaction coefficients (ε(i,k)) for [Cm(Form)_n]⁽³⁻ⁿ⁾⁺ (n = 1, 2) with Cl⁻ are obtained. The log K⁰₁(T) (2.11 (20 °C)–2.49 (90 °C)) and log K⁰₂(T) values (1.17 (30 °C–2.01 (90 °C)) increase continuously with increasing temperature. The log K⁰_n(T) values are used to derive the standard reaction enthalpies and entropies (Δ_rH⁰_m, Δ_rS⁰_m) of the respective complexation reactions according to the Van’t Hoff equation. In all cases, positive Δ_rH⁰_m and Δ_rS⁰_m values are obtained. Thus, both complexation steps are endothermic and entropy-driven.     

Mesoporous material Al-MCM-41 from natural halloysite

Aluminum-containing hexagonally ordered mesoporous silica (Al-MCM-41) with specific surface area of 509.4 m²/g was first synthesized using natural halloysite as source material by hydrothermal treatment, without addition of silica or aluminum regents. The samples were characterized by X-ray diffraction, transmission electron microscopy, N₂ adsorption–desorption measurements, and Fourier transform infrared spectra techniques. The results indicate that process parameters, including calcination temperature, pH value, n(SiO₂)/n(CTAB)/n(H₂O) ratio, and hydrothermal reaction time, show moderate effects on the preparation of Al-MCM-41. SiO₂/Al₂O₃ molar ratio could be effectively modulated by the calcination temperature for halloysite. Furthermore, we first clarified the structural evolution from natural halloysite to mesoporous material Al-MCM-41 at the atomic level.     

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.     

Estimating the Density and Compressibility of Seawater to High Temperatures Using the Pitzer Equations

The density and compressibility of seawater solutions from 0 to 95 °C have been examined using the Pitzer equations. The apparent molal volumes (X = V) and compressibilities (X = κ) are in the form $$ X_{\phi } = \bar{X}^{0} + A_{X} I/(1.2 \, m)\ln (1 + 1.2 \, I^{0.5} ) + \, 2{\text{RT }}m \, (\beta^{(0)X} + \beta^{(1)X} g(y) + C^{X} m) $$ where $ \bar{X}^{0} $ is the partial molal volume or compressibility, I is the ionic strength, m is the molality of sea salt, A_X is the Debye–Hückel slope for volume (X = V) or adiabatic compressibility (X = κ _s), and g(y) = (2/y ²)[1 ? (1 + y) exp(?y)] where y = 2I ^0.5. The values of the partial molal volume and compressibility ( $ \bar{X}^{0} $ ) and Pitzer parameters (β ^(0)X , β ^(1)X and C ^X ) are functions of temperature in the form $$ Y^{X} = \sum_{i} a_{i} (T-T_{\text{R}} )^{i} $$ where a _i are adjustable parameters, T is the absolute temperature in Kelvin, and T _R = 298.15 K is the reference temperature. The standard errors of the seawater fits for the specific volumes and adiabatic compressibilities are 5.35E?06 cm³ g^?1 and 1.0E?09 bar^?1, respectively. These equations can be combined with similar equations for the osmotic coefficient, enthalpy and heat capacity to define the thermodynamic properties of sea salt to high temperatures at one atm. The Pitzer equations for the major components of seawater have been used to estimate the density and compressibility of seawater to 95 °C. The results are in reasonable agreement with the measured values (0.010E?03 g cm^?3 for density and 0.050E?06 bar^?1 for compressibility) from 0 to 80 °C and salinities from 0 to 45 g kg^?1. The results make it possible to estimate the density and compressibility of all natural waters of known composition over a wide range of temperature and salinity.     

The distribution of sodium ions in aluminosilicate glasses: a high-field Na-23 MAS and 3Q MAS NMR study

The local configurations around sodium ions in silicate glasses and melts and their distributions have strong implications for the dynamic and static properties of melts and thus may play important roles in magmatic processes. The quantification of distributions among charge-balancing cations, including Na⁺ in aluminosilicate glasses and melts, however, remains a difficult problem that is relevant to high-temperature geochemistry as well as glass science.Here, we explore the local environment around Na⁺ in charge-balanced aluminosilicate glasses (the NaAlO₂-SiO₂ join) and its distribution using ²³Na magic-angle spinning (MAS) nuclear magnetic resonance (NMR) spectroscopy at varying magnetic fields of 9.4, 14.1, and 18.8 T, as well as triple-quantum (3Q)MAS NMR spectroscopy at 9.4 T, to achieve better understanding of the extent of disorder around this cation. We quantify the extent of this disorder in terms of changes in Na-O distance (d[Na-O]) distributions with composition and present a structural model favoring a somewhat ordered Na distribution, called a “perturbed” Na distribution model. The peak position in ²³Na MAS spectra of aluminosilicate glasses moves toward lower frequencies with increasing Si/Al ratios, implying that the average d(Na-O) increases with increasing R. The peak width is significantly reduced at higher fields (14.1 and 18.8 T) because of the reduced effect of second-order quadrupolar interaction, and ²³Na MAS NMR spectra thus provide relatively directly the Na chemical shift distribution and changes in atomic environment with composition. Chemical shift distributions obtained from ²³Na 3Q MAS spectra are consistent with MAS NMR data, in which deshielding decreases with R. The average distances between Na and the three types of bridging oxygens (BOs) (Na-{Al-O-Al}, Na-{Si-O-Al}, and Na-{Si-O-Si}) were obtained from the correlation between d(Na-O) and isotropic chemical shift. The calculated d(Na-{Al-O-Al}) of 2.52 Å is shorter than the d(Na-{Si-O-Si}) of 2.81 Å, and d(Na-{Al-O-Al}) shows a much narrower distribution than the other types of BOs. ²³Na chemical shifts in binary (Al-free) sodium silicate glasses are more deshielded and have ranges distinct from those of aluminosilicate glasses, implying that d(Na-NBO) (nonbridging oxygen) is shorter than d(Na-BO) and that d(Na-{Si-O-Si}) in binary silicates can be shorter than that in aluminosilicate glasses. The results given here demonstrate that high-field ²³Na NMR is an effective probe of the Na⁺ environment, providing not only average structural information but also chemically and topologically distinct chemical shift ranges (distributions) and their variation with composition and their effects on static and dynamic properties.     

Predicted model for hydrous modified olivine (HyM-α)

A possible structure for hydrous modified olivine (HyM-α) has been obtained by the subtraction of Mg₃SiO₅ from forsterite by crystallographic shear along a direction parallel to the [010] direction of olivine. The subtraction of Mg₃SiO₅ results in the subtraction of MgO from bulk chemistry (?Mg₃SiO₅=?Mg₂SiO₄?MgO). A possible structure for HyM-α thus obtained has the chemical formula Mg₉Si₅H₂O₂₀ (= 5 × Mg_1.8SiH_0.4O₄) with monoclinic unit cell a=4.754 Å, b=10.19 Å, c=29.90 Å, ρ=3.126 g cm^?3, and space group=Ac2m (no. 39). Since the X-ray powder diffraction pattern of HyM-α proposed in this study is very close to that of clinohumite, there is the possibility of this phase having been undiscovered. The humite group minerals and HyM-α proposed in this study make a homologous series as recombination structures: Mg_{(2 m + n )}Si _m H_{2 n} O_{2(2 m + n )} for the humite group and Mg_{(2 m + n )}Si_{( m + n )}H_{2 n} O_{4( m + n )} for HyM-α A characteristic feature is that Mg/Si > 2 for the humite group and Mg/Si < 2 for HyM-α. Forsterite specimens containing around 100 ppm H₂O reported in mantle xenoliths might be the disordered case with n=1 and m=1200 of the humite group or HyM-α.     

EPR study of chromium-doped forsterite crystals: Cr3+(M1) with associated trivalent ions Al3+ and Sc3+

Electron paramagnetic resonance (EPR) study of single crystals of forsterite co-doped with chromium and scandium has revealed, apart from the known paramagnetic centers Cr³⁺(M1) and Cr³⁺(M1)– $ V_{{{\text{Mg}}^{2 + } }} $ (M2) (Ryabov in Phys Chem Miner 38:177–184, 2011), a new center Cr³⁺(M1)– $ V_{{{\text{Mg}}^{2 + } }} $ (M2)–Sc³⁺ formed by a Cr³⁺ ion substituting for Mg²⁺ at the M1 structural position with a nearest-neighbor Mg²⁺ vacancy at the M2 position and a Sc³⁺ ion presumably at the nearest-neighbor M1 position. For this center, the conventional zero-field splitting parameters D and E and the principal g values have been determined as follows: D?=?33,172(29) MHz, E?=?8,482(13) MHz, g?=?[1.9808(2), 1.9778(2), 1.9739(2)]. The center has been compared with the known ion pair Cr³⁺(M1)–Al³⁺ (Bershov et al. in Phys Chem Miner 9:95–101, 1983), for which the refined EPR data have been obtained. Based on these data, the known sharp M1″ line at 13,967?cm^?1 (with the splitting of 1.8?cm^?1), observed in low-temperature luminescence spectra of chromium-doped forsterite crystals (Glynn et al. in J Lumin 48, 49:541–544, 1991), has been ascribed to the Cr³⁺(M1)–Al³⁺ center. It has been found that the concentration of the new center increases from 0 up to 4.4?×?10¹⁵?mg^?1, whereas that of the Cr³⁺(M1) and Cr³⁺(M1)– $ V_{{{\text{Mg}}^{2 + } }} $ (M2) centers quickly decreases from 7.4?×?10¹⁵?mg^?1 down to 3?×?10¹⁵?mg^?1 and from 2.7?×?10¹⁵?mg^?1 down to 0.5?×?10¹⁵?mg^?1, i.e., by a factor of 2.5 and 5.4, respectively, with an increase of the Sc content from 0 up to 0.22 wt?% (at the same Cr content 0.25 wt?%) in the melt. When the Sc content exceeds that of Cr, the concentration of the new center decreases most likely due to the formation of the Sc³⁺(M1)– $ V_{{{\text{Mg}}^{2 + } }} $ (M2)–Sc³⁺ complex instead of the Cr³⁺(M1)– $ V_{{{\text{Mg}}^{2 + } }} $ (M2)–Sc³⁺ center. The formation of such ordered neutral complex is in agreement with the experimental results, concerning the incorporation of Sc into olivine, recently obtained by Grant and Wood (Geochim Cosmochim Acta 74:2412–2428, 2010).     

Experimental constraints on hydrogen diffusion in garnet

The incorporation mechanisms and diffusional loss of hydrogen in garnet have been experimentally investigated. A suite of gem-quality hydrous spessartine- and grossular-rich garnets were analysed by Fourier transform infrared spectroscopy (FTIR) and by ion microprobe (SHRIMP-SI) to determine the calibration coefficients for quantification of FTIR data. The excellent agreement between measured absorption and OH/O indicates that the same molar extinction coefficient can be used for spessartine and grossular. The coefficient of 14400 l mol^??1 cm^??2 proposed by Maldener et al. (Phys Chem Miner 30:337–344, 2003) seems the most appropriate for both minerals. A grossular with 6.4% andradite and 1.6% almandine containing 834 ppm H₂O, and an almost pure spessartine with 282 ppm H₂O, were selected for diffusion experiments. 1.5-mm cubes of garnets were heated between 12 h and 10 days at 1 atm under various temperature (750–1050 °C) and oxygen fugacity (\({f_{{{\text{O}}_2}}}\)) conditions, (ΔQFM +?15.2 to ??3.0). Diffusion profiles were acquired from sections through the cubes using FTIR, with a deconvolution algorithm developed to assess peak-specific behaviour. Different families of peaks have been identified based on their diffusive behaviour, representing hydrogen incorporated in different H-bearing defects. A dominant, fast, strongly \({f_{{{\text{O}}_2}}}\)-dependent oxidation-related diffusion mechanism is proposed \(\left( {\{ {{\text{M}}^{2+}}+{{\text{H}}^+}\} +\frac{1}{4}{{\text{O}}_2}={{\text{M}}^{3+}}+\frac{1}{2}{{\text{H}}_2}{\text{O}}} \right)\) (M=Fe, Mn) with a relatively low activation energy (158?±?19 kJ mol^??1). This diffusion mechanism is likely restricted by availability of ferrous iron in grossular. At low oxygen fugacity, this diffusion mechanism is shut off and the diffusivity decreased by more than three orders of magnitude. A second, slower hydrogen diffusion mechanism has been observed in minor bands, where charge balance might be maintained by diffusion of cation vacancies, with much higher activation energy (≈?200–270 kJ mol^??1). Spessartine shows clear differences in peak retentivity suggesting that up to four different H sites might exist. This opens exciting opportunities to use hydrogen diffusion in garnet as speedometer. However, it is essential to constrain the main diffusion mechanisms and the oxygen fugacity in the rocks investigated to obtain timescales for metamorphic or igneous processes.     

Surface complexation model for multisite adsorption of copper(II) onto kaolinite

We measured the adsorption of Cu(II) onto kaolinite from pH 3-7 at constant ionic strength. EXAFS spectra show that Cu(II) adsorbs as (CuO₄H_n)ⁿ⁻⁶ and binuclear (Cu₂O₆H_n)ⁿ⁻⁸ inner-sphere complexes on variable-charge ≡AlOH sites and as Cu²⁺ on ion exchangeable ≡X--H⁺ sites. Sorption isotherms and EXAFS spectra show that surface precipitates have not formed at least up to pH 6.5. Inner-sphere complexes are bound to the kaolinite surface by corner-sharing with two or three edge-sharing Al(O,OH)₆ polyhedra. Our interpretation of the EXAFS data are supported by ab initio (density functional theory) geometries of analog clusters simulating Cu complexes on the {110} and {010} crystal edges and at the ditrigonal cavity sites on the {001}. Having identified the bidentate (≡AlOH)₂Cu(OH)₂⁰, tridentate (≡Al₃O(OH)₂)Cu₂(OH)₃⁰ and ≡X--Cu²⁺ surface complexes, the experimental copper(II) adsorption data can be fit to the reactions

     

Dissolution Kinetics of Calcite in NaCl–CaCl2–MgCl2 Brines at 25 °C and 1 bar pCO2

Calcite dissolution rates were measured as a function of saturation state in NaCl–CaCl₂–MgCl₂ solutions at 1 bar (0.1 MPa) pCO₂ and 25 °C. Rates measured in phosphate- and sulfate-free pseudo-seawater (Ca²⁺:Mg²⁺= 0.2, I= 0.7) were compared with those in synthetic brines. The brines were prepared by co-varying calcium and magnesium (Ca²⁺:Mg²⁺= 0.9; 2.0; 2.8; 3.1; 4.8; 5.8) along with ionic strength (I= 0.9; 1.1; 1.6; 2.1; 3.0; 3.7; 4.4 m) to yield solutions approximating those of subsurface formation waters. The rate data were modeled using the equation, R = k(1 ? Omega;) ⁿ , where k is the empirical rate constant, n describes the order of the reaction and ω is saturation state. For rates measured in the pseudo-seawater, n= 1.5 and k= 4.7 × 10^?2 mol m^?2 hr^?1. In general, rates were not significantly faster in the synthetic brines (n= 1.4 ± 0.2 and k= 5.0 ± 7 × 10^?2 mol m^?2 hr^?1). The rate coefficients agree within experimental error indicating that they are independent of ionic strength and Ca²⁺:Mg²⁺ over a broad range of brine compositions. These findings have important application to reaction-transport modeling because carbonate bearing saline reservoirs have been identified as potential repositories for CO₂ sequestration.     

A finite temperature chandrasekhar model: Determining the parameters and calculation of the characteristics of degenerate dwarfs

We suggest a generalization of the standard Chandrasekhar model for degenerate dwarfs. We apply an equation of state for a degenerate ideal electron gas in the form of a Sommerfeld expansion in the parameter k _B T(r)/m ₀ c ². The radial temperature distribution T(r) is modeled taking into account the presence of the isothermal core. The model has four dimensionless parameters, two microscopic (the relativistic parameter at the stellar center x ₀ and the chemical-composition parameter ?? _e = A/Z) and two macroscopic (the dimensionless temperature T ₀ ^* = k _B T _c /m ₀ c ² and dimensionless radius ?? ₀ = R _c /R of the core, where R _c and R are the radii of the core and dwarf). We found x ₀, ?? _e , and T ₀ ^* for about 3000 DA white dwarfs, based on their masses, radii, and effective temperatures from the Sloan Digital Sky Survey Data Release 4; ?? ₀ was treated as a free parameter. The influence of temperature effects on the macroscopic characteristics is analyzed, in particular, the minimum mass and maximum radii of the stars. Based on our computed energy-radius dependence, we suggest an interpretation of the observed radius distribution for these dwarfs.     

The heat capacity of two natural chlorite group minerals derived from differential scanning calorimetry

The heat capacity of natural chamosite (X_Fe=0.889) and clinochlore (X_Fe=0.116) were measured by differential scanning calorimetry (DSC). The samples were characterised by X-ray diffraction, microprobe analysis and Mössbauer spectroscopy. DSC measurements between 143 and 623?K were made following the procedure of Bosenick et?al. (1996). The fitted data for natural chamosite (CA) in J?mol^?1?K^?1 give: C _p,CA = 1224.3–10.685?×?10³?×?T ^??0.5???6.4389?× 10⁶?×T ^??2?+?8.0279?×?10⁸?×?T ^??3 and for the natural clinochlore (CE): C _p,CE = 1200.5–10.908?×?10³?×T ^??0.5?? 5.6941?×?10⁶?×?T ^??2?+?7.1166?×?10⁸?×?T ^??3. The corrected C _p-polynomial for pure end-member chamosite (Fe₅Al)[Si₃AlO₁₀](OH)₈ is C _p,CAcor = 1248.3–11.116?× 10³?×?T ^??0.5???5.1623?×?10⁶?×?T ^??2?+?7.1867?×?10⁸×T ^??3 and the corrected C _p-polynomial for pure end-member clinochlore (Mg₅Al)[Si₃AlO₁₀](OH)₈ is C _p,CEcor = 1191.3–10.665?×?10³?×?T ^??0.5???6.5136?×?10⁶?×?T ^??2?+ 7.7206?×?10⁸?×?T ^??3. The corrected C _p-polynomial for clinochlore is in excellent agreement with that in the internally consistent data sets of Berman (1988) and Holland and Powell (1998). The derived C _p-polynomial for chamosite (C _p,CAcor) leads to a 4.4% higher heat capacity, at 300?K, compared to that estimated by Holland and Powell (1998) based on a summation method. The corrected C _p-polynomial (C _p,CAcor) is, however, in excellent agreement with the computed C _p-polynomial given by Saccocia and Seyfried (1993), thus supporting the reliability of Berman and Brown's (1985) estimation method of heat capacities.