Matteo Alvaro

EosFit7c and a Fortran module (library) for equation of state calculations

Abstract The relationship between linear elasticity theory of solids and their equations of state (EoS) is reviewed, along with the commonly-used types of isothermal EoS, thermal expansion models, and P-V-T EoS. A new console program, EosFit7c, is presented. It performs EoS calculations and fitting for both volume and linear isothermal data, isobaric data and P-T data. Linear data is handled by cubing the quantities and treating them as volumes in all EoS formulations. Least-squares fitting of EoS to data incorporates the option to weight the fit with the measurement uncertainties in P, V and T simultaneously. The EosFit7c program is built with a new library of subroutines for EoS calculations and manipulation, written in Fortran. The library has been incorporated as a module, cfml_eos, in the publicly-available CrysFML library. The module handles Murnaghan, Tait, Birch-Murnaghan, Vinet, and Natural Strain EoS. For P-V-T calculations any of these isothermal EoS can be combined with a variety of published thermal expansion models, including a model of thermal pressure. The entire library has been revalidated against other software and against an ab-initio re-derivation of the EoS, which identified a number of small errors in published formulae for some EoS.

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.

High-pressure phase transition of a natural pigeonite

Abstract High-pressure and room-temperature single-crystal X-ray diffraction (XRD) studies have been performed on crystals of a natural pigeonite sample with composition ca. Wo10En43Fs47 using diamondanvil cells. The unit-cell parameters were determined at 18 different pressures up to about 6 GPa. A first-order P21/c-C2/c phase transition was found between 3.5 and 3.6 GPa, associated with the disappearance of the b-type reflections (h + k = odd) and a strong discontinuity (about 1.7%) in the unit-cell volume. At the transition, a small hysteresis (~0.3 GPa) was observed. A third-order Birch-Murnaghan equation of state (BM3-EoS) fit to the 10 P-V data of the low-P phase yielded V0 = 431.93(2) Å3, KT0 = 96.8(8) GPa and K′ = 8.5(6). A second-order Birch-Murnaghan EoS fit to the 8 P-V data (between 3.6 and 6 GPa) of the C2/c high-P phase yielded V0 = 423.6(1) Å3 and KT0 = 112.4(8), indicating that the high-P C2/c phase is significantly stiffer than the low-P phase. In a separated experiment with crystals of the same sample, intensity data were collected and crystal structures were refined at 13 pressures up to 9.4 GPa. The M1-O and M2-O mean bond lengths of the low-P P21/c phase decrease by 0.7 and 2.1%, respectively. The two non-equivalent A and B tetrahedral chains become more kinked with pressure, with a reduction of their angle by 2.2 and 5.1%, respectively. At the transition the A-chain changes sense of rotation and both chains become equivalent and more kinked, with a further reduction of their angle by 2.5% up to 9.4 GPa. Strain calculations have been performed and the evolution of the spontaneous strain and the order parameter variation with pressure are discussed, considering geometrical parameters of the structure and comparing our results with the available data for other compositions.

Phase stability, elastic behavior, and pressure-induced structural evolution of kalsilite: A ceramic material and high-T/high-P mineral

Abstract The phase stability, elastic behavior, and pressure-induced structural evolution of a natural metamorphic kalsilite (ideal formula KAlSiO4) from Punalur (Kerala district in southern India), with P31c symmetry and a K/Na molar ratio of ~350, has been investigated by in situ X‑ray single-crystal diffraction up to ~7 GPa with a diamond-anvil cell under hydrostatic conditions. At high-pressure, a previously unreported iso-symmetric first-order phase transition occurs at ~3.5 GPa. The volume compression of the two phases is described by third-order Birch-Murnaghan equations-of-state: V0 = 201.02(1) Å3, KT0 = 59.7(5) GPa, K′ = 3.5(3) for the low-P polymorph, and V0 = 200.1(13) Å3, KT0 = 44(8) GPa, K′ = 6.4(20) for the high-P polymorph. The pressure-induced structural evolution in kalsilite up to 7 GPa appears to be completely reversible. The compression of both phases involves tetrahedral rotations around [0001], which close up the channels within the framework. In addition, compression of the low-pressure phase involves tilting of the tetrahedra. The major structural change at the phase transition is an increase in the tilting of the tetrahedra, but with a reversion of the tetrahedral rotations to the value found at ambient conditions. This behavior is in distinct contrast to that of nepheline, which has a tetrahedral framework of the same topology.

Tetragonal Almandine-Pyrope Phase, TAPP: finally a name for it, the new mineral jeffbenite

Abstract Jeffbenite, ideally Mg3Al2Si3O8, previously known as tetragonal-almandine-pyrope-phase (ʻTAPPʼ), has been characterized as a new mineral from an inclusion in an alluvial diamond from São Luiz river, Juina district of Mato Grosso, Brazil. Its density is 3.576 g/cm3 and its microhardness is ~7. Jeffbenite is uniaxial (-) with refractive indexes ω = 1.733(5) and ε = 1.721(5). The crystals are in general transparent emerald green. Its approximate chemical formula is (Mg2.62Fe2+0.27)(Al1.86Cr0.16)(Si2.82Al0.18)O12 with very minor amounts of Mn, Na and Ca. Laser ablation ICP-MS showed that jeffbenite has a very low concentration of trace elements. Jeffbenite is tetragonal with space group I4̄2d, cell edges being a = 6.5231(1) and c = 18.1756(3) Å. The main diffraction lines of the powder diagram are [d(in Å), intensity, hkl]: 2.647, 100, 2 0 4; 1.625, 44, 3 2 5; 2.881, 24, 2 1 1; 2.220, 19, 2 0 6; 1.390, 13, 4 2 4; 3.069, 11, 2 0 2; 2.056, 11, 2 2 4; 1.372, 11, 2 0 12. The structural formula of jeffbenite can be written as (M1)(M2)2(M3)2(T1)(T2)2O12 with M1 dominated by Mg, M2 dominated by Al, M3 dominated again by Mg and both T1 and T2 almost fully occupied by Si. The two tetrahedra do not share any oxygen with each other (i.e. jeffbenite is classified as an orthosilicate). Jeffbenite was approved as a new mineral by the IMA Commission on New Minerals and Mineral Names with the code IMA 2014-097. Its name is after Jeffrey W. Harris and Ben Harte, two world-leading scientists in diamond research. The petrological importance of jeffbenite is related to its very deep origin, which may allow its use as a pressure marker for detecting super-deep diamonds. Previous experimental work carried out on a Ti-rich jeffbenite establishes that it can be formed at 13 GPa and 1700 K as maximum P-T conditions.

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.

In-situ high-temperature emissivity spectra and thermal expansion of C2/c pyroxenes: Implications for the surface of Mercury

Abstract This work was carried out within the framework of the European Space Agency and Japanese Aerospace Exploration Agency BepiColombo space mission to Mercury and intends to provide valid tools for the interpretation of spectra acquired by the MErcury Radiometer and Thermal Infrared Spectrometer (MERTIS) on board of BepiColombo. Two C2/c augitic pyroxenes, with different Mg/Fe ratios and constant Ca contents, were investigated by in situ high-temperature thermal infrared spectroscopy and in situ high-temperature single-crystal X-ray diffraction up to temperatures of about 750 and 770 K, respectively. The emissivity spectra of the two samples show similar band center shifts of the main three bands toward lower wavenumbers with increasing temperature. In detail, with increasing temperature bands 1 and 2 of both samples show a much stronger shift with respect to band 3, which remains almost unchanged. Our results indicate that the center positions of bands 1 and 2 are strong functions of the temperature, whereas the center position of band 3 is a strong function of the Mg# [with Mg# = Mg/ (Mg + Fe2+) atomic ratio]. The analysis of the thermal behavior gives similar thermal expansion volume coefficients, αV, for the Mg-rich and Fe-rich samples, with αV = 2.72(8) and 2.72(7) × 10-5 K-1, respectively, using the Berman (1988) equation. This correspondence totally explains the band center shifts similarity between the two samples. Our data suggest that MERTIS spectra will be able to provide indications of C2/c augitic pyroxene Mg# and will allow a correct interpretation that is independent on the spectra acquisition temperature.

A new framework topology in the dehydrated form of zeolite levyne

Abstract The thermoelastic behavior and structural evolution of a natural levyne-Ca [(Ca7.8 Na2.2K1.1)Σ11.1 Al20.0Si34.2O108⋅51.5H2O; R3m; a = 13.377(4) Å, c = 22.870(1) Å, V = 3544.1(3) Å3] were studied by both T-resolved synchrotron X‑ray powder diffraction (SR-XRPD) between room temperature and 800 °C, and by conventional-source high-temperature single-crystal X‑ray diffraction (SC-XRD). Above 230 °C, water loss and reallocation of extraframework cations induce the straining and consequent breaking of T-O-T bridges in the D6R, with resulting migration of tetrahedral cations to new tetrahedral sites. The new tetrahedra share an edge with the previously occupied tetrahedra. This phenomenon gives rise to a new topology, which coexists to about 40%, with the original one. The new framework consists of a sequence of a novel zeolitic cage (described as a 20-hedron formed by fourteen 6mR and six 4mR) and two consecutive cancrinite cages along [0001]. This topology, which is reported in the database of the hypothetical zeolite structures as 166_2_293, belongs to the ABC-6 family and can be described by the following sequence of 6-rings: ABCBCACAB, to be compared with that of levyne AABCCABBC. In the new topology the extraframework cations are distributed over 3 new sites: one at the center of the 6mR ⊥ [0001] shared by the two cancrinite cages, one near the center of the 6mR ⊥ [0001] at the base of the new cage, and a last one in a 6mR window of the new cage. The 8mR bidimensional channel system originally present in levyne is therefore absent in the new topology and hence molecular diffusion is likely to be partially hindered in the dehydrated form. The phase transition is not completely reversible, at least in the short term, as only partial rehydration was demonstrated

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.