The evolutionary path taken by the interior of the Earth is strongly related to the behavior of the rock material from which it is made. Together, these represent the dominant phases of the Earth's mantle, now and in the distant past.
Introduction
These results are then compared to assess the effect of iron composition on the high temperature compression of silicate perovskite. Finally, we assess the plausibility of various competing models to explain the perovskite-rich lower mantle structures.
Methods
Cells were loaded with Ne pressure medium using the GSECARS gas loading system at the Advanced Photon Source (APS) at Argonne National Laboratory ( Rivers et al., 2008 ). The pressure for each spectrum was determined using the Ne pressure medium as a primary pressure marker, Dewaele et al. (2008), along with the offline ruby fluorescence measurements.
Analysis
Peak Fitting
It achieves this by approximately distributing the observed intensities between the diffraction peaks present in each part of the spectrum. This is usually done by assuming a reasonable basic model in terms of unit cell parameters for each spectrum and then locating the diffraction lines that appear to best match the predicted peak positions.
Determining P-T Conditions
In this study, we mainly rely on the diffraction lines of the neon pressure medium to determine the in situ pressures. Through this analysis, we obtain the final determinations of the pressure and its error, calculating the deviatoric stress.
Extracting Crystal Volumes and Cell Dimensions
We therefore use this model as a prior constraint on the deviatoric stress and refit the set of neon line positions to obtain the best constant for the deviatoric stress fraction for each data set, obtaining values of 0.5% and 0.6% for the Mg end-element and samples, containing Fe, according to the central peaks of the apparent deviatoric stress fraction histograms. By fitting two or three neon lines visible in each spectrum, we obtain unconstrained simultaneous estimates of axial deviatoric stress and neon unit cell size, which are converted to pressure estimates using the neon equation of state from Dewaela et al.
Obtaining the Equation of State from P-V-T Measurements
The prior simply expresses the probability distribution for the parameters prior to any particular step of the analysis. This finding is directly reflected in the best-fit estimates for the thermal parameters γ0 and q, which relate to the full scale of thermal expansion and its compression (or volume) dependence.
Discussion
Equation of State Comparison and Uncertainties
First, none of the previous investigations report the full covariance matrix (or equivalent correlation matrix) for their equation of state parameters. In the bottom panel of Figure 2.9, we show the linear evolution of the octahedral tilt angle with compression for the two samples.
Physical Property Profiles
Using these expressions, we determine the isothermal profiles of thermal expansion, bulk modulus and volume for the two perovskite samples. It should be noted that we obtain a significantly different thermal behavior for the Fe-free endmember sample than Fiquet et al. (2000).
Perovskite-Dominated Chemical Piles
In this exercise, use the mixture model described above to calculate the material properties of the mixture at the core-mantle interface. On the left and center of the figure, we show the percent contrast between the mixture model (as a function of composition and temperature) and the main mantle as represented by PREM (Dziewonski and Anderson, 1981). Using a simple linear expansion of the density and the bulk modulus difference at the core-mantle interface, we can obtain rough estimates for the height of the density crossover.
Conclusion
We can thus assess the relative plausibility of the passive pile and metastable dome explanations by calculating the fraction of the allowed parameter space occupied by material behavior consistent with the two theories—this is simply given by the relative area of the red-shaded region as com. - parried to the white area. This fraction is just below 10%, indicating that while metastable domes may be consistent with our current understanding of the perovskite equation of state, they impose very tight constraints on the temperature-dependent composition of the LLSVP material and are therefore not very likely. Since this intensity weighting map is independent of the collected data, it only needs to be calculated once for each detector configuration.
Introduction
Intro to Bayesian Statistics
As we will show later, the most well-known form of probability uses weighted least squares or. Since the prerequisite for any data analysis problem assumes that we can generate a model for any specific set of model parameters M~, we then use probability distributions for the data points {d} to determine the probability of the data given the model values. We can imagine that we would construct a local first-order Taylor expansion of the physical model in the region of the parameter space that best fits.
Bayesian Analysis of Powder Diffraction Data
Estimating Peak Positions and Uncertainties
We start the fitting process with an initial guess for the peak positions, based on the positions of a few of the most intense and easily identified lines (prior knowledge from previous studies is also useful to obtain a reasonable initial guess). This reflects the fact that the peak width directly controls the relative curvature of the spectrum in the peak region. This well-known result comes from the fact that the expected value of the error-weighted squared residuals is simply equal to one for a normal distribution, h(∆xi/σi)2i = 1.
Estimating Unit Cell Parameters and Accounting for Misidentified
On the left side of the figure are the results for a sample containing 13% Fe. In contrast, the modeling results for the 0% Fe sample are shown in the right panel of the figure. In doing so, we can estimate the expected linear trend of each of the normalized axial ratios.
Obtaining Unbiased Estimates and Uncertainties for Equation of
In the absence of zero-pressure measurements of the sample volume, the correct way to constrain the parameter values is to impose a reasonable composition-dependent assumption on V0. The one-dimensional marginal probability distributions for each parameter are shown on the edges of the figure. In addition, it is clear that the two different compositions have differences in their equations of state that are statistically discernible (though not necessarily all physically relevant, see discussion in Chapter 2), with little or no overlap in 2-D 95% confidence regions for any of the parameter pairs.
Conclusions
Second, and more important, is the fact that all previous investigations have determined a number of important equations of state parameters to improve apparent parameter constraints. We show how model parameters can be obtained by training on equation of state data for different crystal polymorphs that discretely sample the range of available coordination states. This method is applied to the MgO system and then used to calculate the coordination evolution and equation of state of liquid Mg over a wide range of pressures and temperatures.
Background
Ideal Mixing Models for Experimental Data
Given a measurement of fluid volumes and energies for a range of compositions at a particular P-T state, the ideal mixing model assumes that the volumes, entropies, and energies are additive and depend linearly on the fraction of each end-member composition. (Bottinga and Weill, 1970). Despite the widespread success, there are obvious shortcomings in the direct application of this approach, as explored by (Guo et al., 2013), in which they show evidence for large variations in the partial molar volume of the oxide component of of iron as a function of overall composition. Preliminary results from liquid shock wave experiments in the CaO−MgO−Al2O3−SiO2−FeO system show similar failures of the ideal mixture method (Thomas et al., submitted 2013), although it is not yet clear whether the practical approach of adding more end members that sample the range of dominant coordination states is sufficient to address the alignment deficiencies.
Atomic Simulation Using Molecular Dynamics
The major challenge with EPMD is related to the question of whether the parameter values for the empirical potential are applicable outside the conditions under which they were trained and thus the quality of the simulation results can vary widely depending on the details of the system. First-Principles Molecular Dynamics (FPMD) avoids many of these issues surrounding accuracy of the applied forces by determining them directly using first-principles quantum mechanics calculations. Although this is likely to be sufficient for comparing state properties of most systems, any physical effects that rely on length scales approaching the size of the simulation cell will not be captured by this approach, making the determination of dynamic material properties such as diffusivity and viscosity difficult. make. possibly much more uncertain.
Basic Hard Sphere Model
The initial investigations of hard sphere fluids represent some of the first computer simulations ever performed. For moderate packing fractions (low density liquid phase), particles frequently interact with each other revealing the importance of the excluded volume. In the low density case, the particle can access almost the entire free volume of the liquid.
Previous Application of Hard Spheres to Silicate Liquids
This was achieved by determining a single effective hard sphere size for each fluid that best matched the 1 bar density measurement at the reference temperature. Recently, Jing and Karato (2011) made significant progress in describing hard sphere silicate melts by fitting a set of hard sphere diameters to a collection of liquid density measurements in the CaO–MgO–Al2O3–SiO2–FeO system. Initially, fixed diameters of hard spheres were found to best fit the liquid data in a least-squares fashion.
Local Liquid Structure
From this equation, it is clear that the radiation is scattered with high intensity only in a limited set of preferred directions that are characteristic of the crystal structure. Without going into details, we briefly mention some of the available techniques and the general knowledge they provide about high-pressure silicate melts. For visualization purposes, we reproduce a figure from Karki et al. 2007) shown in Figure 4.4, which shows the evolution of different Si−O coordination polyhedra with increasing compression.
Coordinated HArd Sphere Model (CHASM)
A Qualitative Picture of Liquid Compression
Including the dependence of the internal energy on the packing fraction results in a perturbed hard sphere model, where the sphere sizes adjust with compression. At the top left of the figure we start with the orderly compression of solids, which is dominated by changes in bond length. In the third panel we show how a more realistic fluid image causes the size of the hard sphere particles to change as a function of pressure (or temperature).
Perturbing the Hard Sphere Mixture Model
Therefore, the microstates of the system are populated in such a way as to minimize the total free energy. Together, the ideal and hard-sphere terms represent parts of the standard hard-sphere mixing model. Finally we come to the last term in the free energy expression, which captures the effects of the internal degrees of freedom of each particle of the hard sphere.
Training Pair Potentials on Solid Structures
This dependence is only important as a higher order correction to the free energy in terms of the bond angle contribution. Panel (a) shows the variation of the equilibrium radius, (b) shows the energy well depth and (c) shows the width of the energy well. Panels b and c show the variation of the energy well depth and width with coordination number.
Determining Liquid Properties using CHASM
The inset figure shows a zoomed-in view of the minimum, which occurs for a coordination number close to 6. In the previous section, we developed a simple polynomial description of the pair potential terms as a function of coordination number, as shown in Figure 4.8. Also, in the introduction of the same figure, the change of the defined diameters of the solid spheres with compression is presented.
Conclusions
All of these combine to express the contribution to the free energy and entropy e. Stixrude (2006), First-principles calculations of the structural, dynamical, and electronic properties of liquid MgO, Phys. Meng (2008), Effect of Fe on the equation of state of mantle silicate perovskite above 1Mbar, Earth Physics and Planetary Interiors.
Cold compression data map for 13% Fe-bearing perovskite
Cold compression data map for Mg-perovskite
Measured spectral temperatures for 13% Fe-bearing perovskite
Measured spectral temperatures for Mg-perovskite
Apparent deviatoric stress experienced by perovskite samples as determined
Cold compression curves with equation of state confidence regions for 0%
Excess volumes due to thermal expansion with best-fit isothermal equation
Joint confidence regions for core-mantle boundary volumes and bulk moduli
Normalized axial ratios and octahedral tilt angles for 0% and 13% Fe-bearing
Isothermal profiles with confidence regions for the physical properties of 0%
Buoyant stability of a perovskite-dominated LLSVP determined for a range
Compression evolution of peak positions for Mg-perovskite sample
Compression evolution of peak positions for 13% Fe-bearing perovskite sample 76