Rolf Heinemann

Time and temperature variation of the intracrystalline Fe2+,Mg fractionation in Johnstown meteoritic orthopyroxene

The partitioning of Fe 2+ and Mg on the M1 and M2 sites of orthopyroxenes from the Johnstown meteoritic diogenite has been equilibrated between 1000°C and 700°C in ordering and disordering runs. The method of bivariate high order truncation analysis (Kroll et al., 1997) has been employed to refine the site occupancies from conventional X-ray intensity data. The Fe 2+ ,Mg distribution coefficient varies according to \[\mathrm{In\ K_{D}\ =\ 0.417\ (121)\ {-}\ 2540(136)/T[K].}\] From isothermal kinetic ordering and disordering experiments an exceptionally large activation energy was derived. Combining our data with those of Zema et al. (1997a) results in the Arrhenius equation for the rate constant: \[\mathrm{In\ k_{dis}[min^{{-}1}]\ =\ 41.4({\pm}0.9)\ {-}\ 97.8({\pm}1.9)[kcal/mol]\ /\ RT.}\] For the first time, non-linear continuous cooling experiments were performed in which the crystals were cooled from 850°C to 250°C at an average rate of 10°C/day. The Fe 2+ ,Mg distributions were determined after the crystals had reached 650°C, 550°C, 450°C, 350°C, and 250°C, respectively. Using the Mueller rate equation (Ganguly, 1982) and employing the temperature dependencies of K D and k dis as given above, the experimentally delineated ordering path is closely reproduced by the calculated path. However, due to the large activation energy, cooling rates calculated for the untreated crystals turn out to be physically unreasonable, i.e. some 10 −5 K/My. By contrast, Arrhenius parameters determined in the literature on orthopyroxenes with compositions similar to the Johnstown crystals produce physically reasonable rates of some hundred K/My. TEM studies do not show a significant difference between the microtextures of untreated and annealed samples. All orthopyroxenes studied contain clinopyroxene exsolution lamellae and abundant Guinier-Preston zones. At present, we can neither prove nor disprove the concept that the large activation energy of the Johnstown orthopyroxenes is related to their intricate exsolution microtexture.

Temperature dependence of Fe,Mg partitioning in Acapulco olivine

Abstract The temperature dependence of the intracrystalline Fe,Mg partitioning (KD) in two olivine crystals (Fa11) separated from the Acapulco meteorite was determined by single-crystal X-ray structure analysis. The independent atom model (IAM) was compared with a “bond model” which accounts for bond-induced charge accumulations on the Si-O bonds. Outliers in the set of structure amplitudes observed when using the IAM disappeared upon introducing the bond model. The crystals were equilibrated at 750, 650, and 500 °C. The refined site occupancies yield the relation ln(KD) = 0.345(60) - 204(53)/T, where T is in K, which is in qualitative agreement with earlier work. Comparison of these data with the higher temperature data of Artioli et al. (1995) suggests an unusual temperature variation of the Fe,Mg distribution within two temperature regimes. Below 880 °C, Fe tends to order onto the M1 site with increasing temperature whereas it concentrates on the M2 site above 880 °C. In principle, olivine may serve as a geospeedometer similar to orthopyroxene. However, at present its usefulness is restricted because (1) the relatively weak dependence of ln(KD) on temperature needs to be even more tightly constrained than presented here and (2) low-temperature extrapolation of the rate constants for the Fe,Mg site exchange, derived from interdiffusion coefficients, is uncertain.

Order and anti-order in olivine I: Structural response to temperature

The crystal structure of Fe0.48Mg0.52[SiO4] olivine from the Boseti volcano, Ethiopia, has been investigated by single-crystal X-ray diffractometry at temperatures between 20 °C and 900 °C. For temperatures up to 600 °C, data were collected on crystals equilibrated at 600 °C. These data can therefore be assumed to reflect structural changes that are exclusively caused by thermal effects, whereas data collected between 600 °C and 900 °C carry additional information about the Fe2+,Mg equilibrium distribution. The in situ experiments at elevated temperatures were complemented by ambient temperature data collections on quenched crystals in order to check for possible Fe2+ and Mg redistributions during quenching. Such effects were found absent in crystals quenched from 800 °C or below. The derived temperature dependence of the Fe2+,Mg site distribution is \batchmode \documentclass[fleqn,10pt,legalpaper]{article} \usepackage{amssymb} \usepackage{amsfonts} \usepackage{amsmath} \pagestyle{empty} \begin{document} \[lnK_{D}\ =\ 0.4422({\pm}0.0070)\ {-}\ 140.0({\pm}6.5)/T\ (K)\] \end{document} according to which Fe2+ progressively anti-orders into the M1 “octahedral” site as temperature is raised. A reverse ordering reaction at ≈650 °C leading to a strong segregation of Fe2+ into the other “octahedral” site, M2, as reported by Redfern et al. (2000), could not be detected. Both the and mean bond distances continuously increase with temperature, exhibiting, however, a change in the increase rate at about 600 °C which conforms with an enrichment of the larger Fe2+ cation on the M1 site and its concomitant depletion on M2. In terms of bond lengths, the octahedral distortion of the M2 site is larger than that of M1. The opposite is true for the distortion defined in terms of the angles subtended at the cation site. Similar to the distances, the behaviour of the distortion parameters both of which increase above 600 °C reflects the Fe2+,Mg anti-order. The relative magnitudes as well as the variation with temperature of both bond length and angular distortions can be rationalized considering the different geometrical environments of the M1 and M2 sites. With respect to isotropic displacement parameters, U(M1)equiv is found larger than U(M2)equiv at all temperatures, also at variance with Redfern et al. (2000).

Order and anti-order in olivine III: Variation of the cation distribution in the Fe,Mg olivine solid solution series with temperature and composition

The structures of four Fe,Mg olivine crystals Fa11.6, Fa22.3, Fa27.6 and Fa27.8 have been investigated by ex situ and in situ experiments using single-crystal X-ray diffractometry. For the ex situ experiments, the crystals were quenched after equilibration at temperatures between 500 °C and 900 °C. Resetting effects were only observed for samples quenched from 900 °C. The in situ experiments were performed at temperatures up to 750 °C. With increasing equilibration temperature, Fe 2+ was found to progressively segregate into the octahedral M1 site, i.e. the degree of anti-order increases with rising temperature. Thus, contrary to the results of Rinaldi et al. (2000) and Redfern et al. (2000), no indication of an ordering reversal at high temperatures is found. Incorporation of the results of Heinemann et al. (2006) on olivine Fa47.9 into the present study allows for the first time to systematically investigate the compositional variation of the temperature dependence of the Fe 2+ ,Mg site occupancies. The formulation of Thompson (1969, 1970) for solid solutions undergoing non-convergent disordering processes, (1) \[{-}\ RTln\ K_{D}\ =\ [{\Delta}H_{exch}^{0}\ {-}\ (L_{M1}^{H}\ {-}\ L_{M2}^{H})X]\ {-}\ T[{\Delta}S_{exch}^{0}\ {-}\ (L_{M1}^{S}\ {-}\ L_{M2}^{S})X],\] yielded \begin{eqnarray*}&&{\Delta}H_{exch}^{0}\ =\ 1153\ ({\pm}67)\ J/mol{\ }{\ }{\Delta}S_{exch}^{0}\ =\ 3.743({\pm}0.072)\ J/molK\\&&L_{M1}^{H}\ {-}\ L_{M2}^{H}\ =\ 973\ ({\pm}211)\ J/mol{\ }{\ }L_{M1}^{S}\ {-}\ L_{M2}^{S}\ =\ {-}\ 0.89\ ({\pm}0.24)\ J/molK.\end{eqnarray*} ΔH exch 0 and ΔS exch 0 are the exchange enthalpy and entropy, respectively, related to the Fe 2+ ,Mg site exchange, L M1,M2 H and L M1,M2 S denote enthalpic and entropic intrasite interaction parameters. The compositional parameter X varies between −1 for forsterite (Fa0) and +1 for fayalite (Fa100). The magnitudes and signs of the four refined quantities can be rationalized in terms of thermodynamic and crystal-chemical considerations. (L M1 H − L M2 H ) > 0 indicates that the interactions between the M1 sites are stronger than those between the M2 sites, conforming with the M1-M1 distances being shorter than the M2-M2 distances. (L M1 S − L M2 S ) exch 0 increases towards the Mg endmember while ΔS exch 0 decreases. Since ΔH exch 0 > 0 stabilizes the ordered state whereas ΔS exch 0 >0 stabilizes the anti-ordered state, it follows that the ordered state is progressively favoured with increasing Mg content. Consequently, ordered site distributions should occur only in slowly cooled Mg-rich olivines, whereas olivines with Fa > 25 mol% should be found in their anti-ordered states frozen during cooling. This conclusion is supported by structure refinements and ordering path simulations for a metamorphic olivine Fa12.4 as well as two volcanic olivines Fa25.6 and Fa27.8.

Order and anti-order in olivine. II: Thermodynamic analysis and crystal-chemical modelling

The equilibrium order/anti-order behaviour in olivine Fe 0.48 Mg 0.52 [SiO 4 ] is analysed in terms of the Thompson (1969, 1970) model for the Gibbs energy due to ordering, G ord , \[G^{ord}\ =\ \frac{1}{2}{\Delta}G^{0}_{exch}\ Q\ {-}\ TS^{ord}_{conf}.\] ΔG 0 exch = ΔH 0 exch − TΔS 0 exch relates to the exchange reaction Fe M2 + Mg M1 ↔ Fe M1 + Mg M2 . Since for the investigated olivine both ΔH 0 exch and ΔS 0 exch are positive (ΔH 0 exch = 1.2 kJ/mol, ΔS 0 exch = 3.7 J/mol K), an ordered Fe 2+ ,Mg configuration is favoured by the enthalpic part of ΔG 0 exch whereas the vibrational entropic part favours anti-ordering. As a result, at low temperatures, where ΔH 0 exch > TΔS 0 exch , Fe 2+ prefers M2. Since, however, the energy TΔS 0 exch steadily increases with increasing temperature it promotes Fe 2+ into M1 and full disorder is attained at a crossover temperature T co where ΔH 0 exch = T co ΔS 0 exch . Above T co , TΔS 0 exch becomes progressively larger than ΔH 0 exch and stimulates further fractionating of Fe 2+ into M1 corresponding to increasing anti-order. The unusual phenomenon of anti-order increasing at increasing temperatures is due to ΔH 0 exch being relatively small in FeMg olivine compared to the temperature proportional energy TΔS 0 exch . In other AB olivines (A, B = Mn, Fe, Co, Ni, Mg) the exchange enthalpies are much larger, between 9 and 20 kJ/mol, so that they dominate ΔG 0 exch to a degree that precludes a crossover from ordered to anti-ordered states up to the melting point. The exchange enthalpies reported for MnMg, FeMg, CoMg, NiMg and MnFe olivines can be rationalized in terms of cation radius (r) and electronegativity (χ) ratios of the A and B cations. In a novel approach, both radii and electronegativities have been derived from topological analyses of the procrystal electron density distributions of pure M 2 [SiO 4 ] olivines (M = Mn, Fe, Co, Ni, Mg) yielding a very satisfactory description by \[{\Delta}H^{0}_{exch}\ =\ 252.6({\pm}6.1)\ [r(A)/r(B)\ {-}\ 1]\ {-}\ 75.8({\pm}1.9)\ [{\chi}(A)/{\chi}(B)\ {-}\ 1].\] Accordingly, the small value of ΔH 0 exch found for FeMg olivine is a consequence of opposite radius and electronegativity contributions which almost cancel. In MnFe olivine, although both contributions are small, they cooperate resulting in a moderate value of ΔH 0 exch . In MnMg olivine, it is the radius ratio that dominates, contrary to CoMg and NiMg olivine where the electronegativity ratios control ΔH 0 exch . Consequently, Mn prefers M2, and Co and Ni segregate into M1. ΔS 0 exch can be split into vibrational, ΔS 0,vib exch , and electronic exchange entropies, ΔS 0,el exch . Describing the first in terms of a new octahedral distortion parameter, D f , and estimating the second from the Boltzmann distribution of the 3d-electrons, ΔS 0 exch can be satisfactorily modelled by \[{\Delta}S^{0}_{exch}\ =\ 35.76({\pm}\ 0.34)\ {\{}[D_{f}\ (A)^{M1}\ +\ D_{f}\ (B)^{M2}]/[D_{f}\ (B)^{M1}\ +\ D_{f}\ (A)^{M2}]\ {-}\ 1{\}}\ +\ {\Delta}S^{0,el}_{exch}\] The resulting lnK D = −(ΔH 0 exch − T ΔS 0 exch )/RT allows, for the first time and to the best of our knowledge, an exclusively electron density based description of the experimentally observed temperature variations of the site occupancies in AB olivines. This modelling of lnK D allows also for predicting the temperature variation of equilibrium cation distributions in AB olivines not investigated so far.

Analysis of multiple solutions in powder pattern indexing: the common reciprocal metric tensor approach

Powder pattern indexing routines frequently yield multiple solutions, i.e. different reciprocal lattices and unit cells. Here, a method is suggested that reveals whether or not there are numerical and geometric relationships between the solutions. It is based on the detection of a reciprocal vector triplet that is common to two or more proposed reciprocal lattices. Hence, the method can be termed a common reciprocal metric tensor approach. If no such common tensor exists, the different reciprocal lattices are unrelated, but if one exists the lattices are either in a sublattice/superlattice or in a coincidence-site lattice relationship, depending on the character of the respective orientation matrix. Furthermore, the approach can also be used to generate, from a given indexing solution, further valid indexing solutions that could also be produced by indexing routines.

Relationship between Guinier-Preston zones and the kinetics of the intracrystalline Fe2+, Mg exchange reaction in Johnstown meteoritic orthopyroxene

The Fe2+, Mg order/disorder process in orthopyroxenes from the Johnstown meteoritic diogenite is extremely slow. It is controlled by an activation energy of ≈ 100 kcal/mol, whereas in other orthopyroxenes with similar compositions the activation energy is only 60 and 65 kcal/mol. The microstructure of the orthopyroxenes is characterized by the exsolution of abundant coherently intergrown Guinier Preston (GP) zones that have been suggested to be responsible for the slow exchange kinetics. In order to test this hypothesis, orthopyroxene crystals were homogenized by dry heating at 1075 °C for 4 days at Fe/FeO buffer conditions. The variation of the distribution of Fe2+ and Mg on the M1 and M2 was followed by site refinements employing conventional X-ray intensity data. The kinetic behaviour of the homogenized crystals was studied in six controlled continuous cooling experiments in which the crystals were cooled from 850 °C to 250 °C at various rates. The quenched ordering states could be best reproduced by the Arrhenius relation \batchmode \documentclass[fleqn,10pt,legalpaper]{article} \usepackage{amssymb} \usepackage{amsfonts} \usepackage{amsmath} \pagestyle{empty} \begin{document} \[ln\ \mathit{k}_{\mathit{dis}}[d^{{-}1}]\ =\ 34.0\ {-}71.1\ [kcal/mol]\ /\mathit{RT},\] \end{document}(1) where k dis is the rate constant for the microscopic disordering step. It appears that homogenization of the GP zones has in fact lowered the activation energy from ≈ 100 kcal/mol to 71 kcal/mol. Equilibrium experiments were performed at three different temperatures. Comparing the refined Fe2+, Mg distribution coefficients, K D , with data previously obtained on non-homogenized Johnstown orthopyroxenes (Heinemann et al ., 2000) shows that the presence or absence of GP zones has no effect on the equilibrium distribution. Regression analysis of the combined ln K D data yielded \batchmode \documentclass[fleqn,10pt,legalpaper]{article} \usepackage{amssymb} \usepackage{amsfonts} \usepackage{amsmath} \pagestyle{empty} \begin{document} \[ln\ \mathit{K}_{\mathit{D}}\ =\ 0.467(77)\ {-}2535(87)/\mathit{T}.\] \end{document}(2) Using the derived k dis and K D relations, the cooling rate of the original Johnstown orthopyroxenes was calculated from the Mueller (1967, 1969) rate law yielding dT / dt ≈ −0.4 K/My near the temperature of apparent equilibrium T ae ≈ 325 °C. This rate is the maximum rate at which the orthopyroxenes have cooled, and T ae , albeit low, is the maximum temperature at which the GP zones have nucleated.

A system of metrically invariant relations between the moduli squares of reciprocal-lattice vectors in one-, two- and three-dimensional space

Given the background of trial-and-error methods employed in recent automatic powder pattern indexing, an alternative route is suggested based on a generalization of the original Runge–de Wolff approach. For this purpose, a system of five metrically invariant relations between the squared moduli (Q values) of reciprocal-lattice vectors is developed that encompasses the earlier special relations. The five invariant relations correspond to a line, a zone, a bizone, a cone and a pencil configuration of reciprocal-lattice vectors. In particular, the zone configuration relates four vectors being arbitrarily distributed in a plane and as such allows one to identify among a set of measured Q values all quadruples that define reciprocal-lattice planes intersecting in space.