Ross J. Angel

EosFit7-GUI: A new graphical user interface for equation of state calculations, analyses and teaching

EosFit7-GUI is a full graphical user interface designed to simplify the analysis of thermal expansion and equations of state (EoSs). The software allows users to easily perform least-squares fitting of EoS parameters to diffraction data collected as a function of varying pressure, temperature or both. It has been especially designed to allow rapid graphical evaluation of both parametric data and the EoS fitted to the data, making it useful both for data analysis and for teaching.

Geobarometry from host-inclusion systems: The role of elastic relaxation

Abstract Minerals trapped as inclusions within other host minerals can develop residual stresses on exhumation as a result of the differences between the thermo-elastic properties of the host and inclusion phases. The determination of possible entrapment pressures and temperatures from this residual stress requires the mutual elastic relaxation of the host and inclusion to be determined. Previous estimates of this relaxation have relied on the assumption of linear elasticity theory. We present a new formulation of the problem that avoids this assumption. We show that for soft inclusions such as quartz in relatively stiff host materials such as garnet, the previous analysis yields entrapment pressures in error by the order of 0.1 GPa. The error is larger for hosts that have smaller shear moduli than garnet.

Olivine with diamond-imposed morphology included in diamonds. Syngenesis or protogenesis?

The identification of syngenetic inclusions in diamond (i.e. inclusions of minerals that crystallized at the same time and by the same genesis as their host) has long been of paramount importance in diamond studies. However, the widespread assumption that many or most inclusions in diamonds are syngenetic is based on qualitative morphological criteria and few direct measurements. In order to provide statistically significant information on inclusion–host genetic relations for at least one kimberlite, we have determined the crystallographic orientations of 43 olivine inclusions with diamond-imposed morphology, a feature generally interpreted to indicate syngenesis, in 20 diamonds from the Udachnaya kimberlite (Siberia). Our unprecedented large data set indicates no overall preferred orientation of these olivines in diamond. However, multiple inclusions within a single diamond frequently exhibit similar orientations, implying that they were derived from original single monocrystals. Therefore, regardless of the possible chemical re-equilibration during diamond-forming processes, at least some of the olivines may have existed prior to the diamond (i.e. they are protogenetic). Our results imply that a diamond-imposed morphology alone cannot be considered as unequivocal proof of syngenicity of mineral inclusions in diamonds.

A century of mineral structures: How well do we know them?

Abstract This decade marks the centenary of the discovery of X-ray diffraction. The development of mineralogy as a scientific discipline in which the properties of minerals are understood in terms of their atomic-scale structures has paralleled the development of diffraction crystallography. As diffraction crystallography revealed more precise details of mineral structures, more subtle questions about mineral properties could be addressed and a deeper understanding of the relationship between the two could be attained. We review the developments in X‑ray single‑crystal diffraction crystallography over the last century and show how its power to provide fundamental information about the structures of minerals has evolved with the improvements in data quality and the increased technological capacity to handle the data. We show that modern laboratory X‑ray diffraction data are of the quality such that mineralogical results are no longer limited by the data quality, but by the physical validity of the refinement models used to interpret the data.

Structural controls on the anisotropy of tetrahedral frameworks: the example of monoclinic feldspars

The structural variation of monoclinic C2/m alkali feldspars has been analysed, based on 50 previously-determined structures and 10 new structure refinements including three determinations of the structure of an Or90 orthoclase at temperatures of 600 K, 900 K and 1075 K. The influence of temperature, composition and state of Al,Si order on the overall average tetrahedral bond lengths of these structures is statistically insignificant. Analysis of the structures in terms of the tilts of the tetrahedra shows that the tilts evolve uniformly with unit-cell volume irrespective of whether the volume is changed by temperature or exchange of the extra-framework cation. There is a small but systematic decrease in the distortion of the T1 tetrahedron with increasing unit-cell volume. Comparisons of the refined structures with the results of geometric modelling show that volume changes are driven by exchange of the extra-framework cation, or by an increase in its thermal vibrations upon heating. The extreme anisotropy of the changes in the unit-cell parameters of monoclinic feldspars is not due to anisotropic interaction of the extra-framework cation with the anions of the framework, but due to the tilting of the tetrahedra controlled in part by O-O interactions. The anisotropy and tilting is not significantly modified by either Al,Si ordering or the distortions of the tetrahedra provided that the latter remain constant. The monoclinic feldspars show a slightly reduced anisotropy of strains as the unit-cell volume increases as a result of a decrease in the angular distortion of the T1 tetrahedron. The general conclusion is drawn that the pattern of anisotropy of the elastic properties (thermal expansion, compressibility, elastic compliances) and the cell-parameter changes of a tetrahedral framework structure with changing extra-framework species are intrinsic to the topology of the framework. The relative insensitivity of the anisotropy of the strains induced by volume changes to distortions of the tetrahedra also means that framework models which incorporate regular tetrahedra can be safely used to predict anisotropy, provided that the tetrahedral distortions do not change. If, as in the monoclinic feldspars, such a model does not reproduce exactly the observed anisotropy of the real structures, then this immediately indicates that there is a significant change in the distortion of the framework tetrahedra.

The elastic tensor of monoclinic alkali feldspars

Abstract The full elastic tensors of two K-rich monoclinic alkali feldspars, Or83Ab15 sanidine and Or93Ab7 orthoclase, have been determined by using the Impulse Stimulated Light Scattering technique to measure surface acoustic wave velocities. The new data confirm that alkali feldspars exhibit extreme elastic anisotropy, so the bounds of their isotropic average properties span a wide range. The measured adiabatic moduli are, for Or83Ab15 and Or93Ab7, respectively, KReuss = 54.7(7), 54.5(5) GPa; KVoigt = 62.9(1.1), 64.4(0.6) GPa; GReuss = 24.1(1), 24.5(1) GPa; and GVoigt = 36.1(5), 36.1(7) GPa. The small differences in moduli between the samples suggests that variations in composition and in state of Al, Si order only have minor effects on the average elastic properties of K-rich feldspars. The new measurements confirm that the earliest determinations of elastic wave velocities of alkali feldspars, widely used to calculate wave velocities in rocks, resulted in velocities systematically and significantly too slow by 10% or more.

Re-investigation of the crystal structure of enstatite under high-pressure conditions

Abstract A synthetic single crystal of pure orthoenstatite (MgSiO3, space group Pbca) has been investigated at high pressure for structural determinations by in situ single-crystal X-ray diffraction using a diamond-anvil cell. Ten complete intensity data collections were performed up to 9.36 GPa. This study significantly improved the accuracy of structural parameters in comparison to a previous high-pressure structural study, allowing a more detailed examination of structural behavior of orthoenstatite at high pressures and a comparison to other more recent structural studies performed on orthopyroxenes with different compositions. The structural evolution determined in this work confirms the high-pressure evolution found previously for other orthopyroxenes and removes some ambiguities originating from the less accurate published data on the MgSiO3 structure at high pressure. The structural compression is mostly governed by significant volume decrease of the Mg1 and Mg2 octahedra, affecting in turn the kink of the tetrahedral chains, especially the TB chain of larger SiO4 tetrahedra. The Mg2 polyhedron undergoes the largest volume variation, 8.7%, due especially to the strong contraction of the longest bond distance (Mg2-O3B), whereas Mg1 polyhedral volume decreases by about 7.4%. The compressional behavior of the tetrahedral sites is quite different from previously published data. The TA and TB tetrahedral volumes decrease by about 2.8 and 1.8%, respectively, and no discontinuities can be observed in the pressure range investigated. Using the data on the pure orthoenstatite as reference, we can confirm the basic influences of element substitutions on the evolution of the crystal structure with pressure.

A simple and generalised P–T–V EoS for continuous phase transitions, implemented in EosFit and applied to quartz

Continuous phase transitions in minerals, such as the α–β transition in quartz, can give rise to very large non-linear variations in their volume and density with temperature and pressure. The extension of the Landau model in a fully self-consistent form to characterize the effects of pressure on phase transitions is challenging because of non-linear elasticity and associated finite strains, and the expected variation of coupling terms with pressure. Further difficulties arise because of the need to integrate the resulting elastic terms over pressure to achieve a description of the P–T–V equation of state. We present a fully self-consistent simplified description of the equation of state of minerals with continuous phase transitions based on a purely phenomenological adaptation of Landau theory. The resulting P–T–V EoS includes the description of the elastic softening occurring in both phases with the minimum number of parameters. By coupling the volume and elastic behaviour of the mineral, this approach allows the EoS parameters to be determined by using both volume and elastic data, and avoids the need to use data at simultaneous P and T. The transition model has been incorporated in to the EosFit7c program, which allows the parameters to be determined by simultaneous fitting of both volume and elastic data, and all types of equation of state calculations to be performed. Quartz is used as an example, and the parameters to describe the full P–T–V EoS of both α- and β-quartz are determined.

EosFit-Pinc: A simple GUI for host-inclusion elastic thermobarometry

Abstract Elastic geothermobarometry is a method of determining metamorphic conditions from the excess pressures exhibited by mineral inclusions trapped inside host minerals. An exact solution to the problem of combining non-linear Equations of State (EoS) with the elastic relaxation problem for elastically isotropic spherical host-inclusion systems without any approximations of linear elasticity is presented. The solution is encoded into a Windows GUI program EosFit-Pinc. The program performs host-inclusion calculations for spherical inclusions in elastically isotropic systems with full P-V-T EoS for both phases, with a wide variety of EoS types. The EoS values of any minerals can be loaded into the program for calculations. EosFit-Pinc calculates the isomeke of possible entrapment conditions from the pressure of an inclusion measured when the host is at any external pressure and temperature (including room conditions), and it can calculate final inclusion pressures from known entrapment conditions. It also calculates isomekes and isochors of the two phases.

Chemically induced renormalization phenomena in Pb-based relaxor ferroelectrics under high pressure

The pressure-induced phase transition sequence in PbSc(0.5)Ta(0.5)O(3) (PST) and PbSc(0.5)Nb(0.5)O(3) (PSN) heavily doped with homo- and heterovalent cations on the A- or B-site of the perovskite-type structure (ABO(3)) was analysed by in situ synchrotron x-ray diffraction and Raman spectroscopy up to pressures of 25 GPa. We focused on the structural phenomena occurring above the first pressure-induced phase transition at p(c1) from a relaxor state to a non-polar rhombohedral phase with antiphase tilting of the BO(6) octahedra. The samples studied were PST doped with Nb(5+) and Sn(4+) on the B-site, PST doped with Ba(2+) and La(3+) on the A-site and PSN doped with Sr(2+) and La(3+) on the A-site. All of them exhibit a second pressure-induced phase transition at p(c2), similar to pure PST and PSN. The second transition involves the development of either order of antiparallel Pb(2+) displacements and complementary a(+)b(-)b(-) octahedral tilts, or a(-)b(-)b(-) (0 ≤ a < b) tilting alone. As in pure PST and PSN, the second phase transition is preceded by the occurrence of unequal octahedral tilts on the local scale. The substitution of Nb(5+) for Ta(5+) as well as the coupled substitution of Sn(4+) for Sc(3+) + Ta(5+) on the octahedral B sites increases the second critical pressure. The doping by Nb(5+) also reduces the length of coherence of antipolar Pb(2+) order developed at p(c2). The isovalent substitution of the larger Ba(2+) for Pb(2+) on the A-site suppresses the antipolar Pb(2+) order due to the induced local elastic stresses and thus significantly increases p(c2). The substitution of smaller cations for Pb(2+) on the A-site generally favours the development of long-range order of antiparallel Pb(2+) displacements because of the chemically enhanced a(-)a(-)a(-) octahedral tilts. However, this ordering is less when the dopant is aliovalent, due to the charge imbalance on the A-site. For all of the relaxors studied here, the dynamic compressibility estimated from the pressure derivative of the wavenumber of the soft mode associated with the first phase transition is larger in the pressure interval between p(c1) and p(c2) than above p(c2). The dynamic compressibility of the phase above p(c2) decreases if the antipolar Pb(2+) order is disturbed.