Tree-width for first order formulae

We introduce tree-width for first order formulae \phi, fotw(\phi). We show that computing fotw is fixed-parameter tractable with parameter fotw. Moreover, we show that on classes of formulae of bounded fotw, model checking is fixed parameter tractable, with parameter the length of the formula. This is done by translating a formula \phi\ with fotw(\phi)<k into a formula of the k-variable fragment L^k of first order logic. For fixed k, the question whether a given first order formula is equivalent to an L^k formula is undecidable. In contrast, the classes of first order formulae with bounded fotw are fragments of first order logic for which the equivalence is decidable. Our notion of tree-width generalises tree-width of conjunctive queries to arbitrary formulae of first order logic by taking into account the quantifier interaction in a formula. Moreover, it is more powerful than the notion of elimination-width of quantified constraint formulae, defined by Chen and Dalmau (CSL 2005): for quantified constraint formulae, both bounded elimination-width and bounded fotw allow for model checking in polynomial time. We prove that fotw of a quantified constraint formula \phi\ is bounded by the elimination-width of \phi, and we exhibit a class of quantified constraint formulae with bounded fotw, that has unbounded elimination-width. A similar comparison holds for strict tree-width of non-recursive stratified datalog as defined by Flum, Frick, and Grohe (JACM 49, 2002). Finally, we show that fotw has a characterization in terms of a cops and robbers game without monotonicity cost

Depth profile analyses by femtosecond laser ablation (multicollector) inductively coupled plasma mass spectrometry for resolving chemical and isotopic gradients in minerals

Femtosecond laser ablation (fs-LA) coupled to a multicollector inductively coupled plasma mass spectrometry (MC-ICP-MS) instrument has been proven to be a powerful means to analyze isotope ratios of “non-traditional” stable isotope systems with high spatial resolution, precision, and accuracy. The technique has been successfully applied, e.g., to investigate diffusion-generated isotopic zoning of the elements Li, Mg, and Fe in magmatic crystals. Here, we present a novel sampling technique employing a fs-LA system that is equipped with a computer numerical control (CNC) laser stage, using the open-source software LinuxCNC. Combining this laser set up with ICP-MS or MC-ICP-MS allows us to perform depth profile analyses of major and trace elements, respectively, as well as metal stable isotope variations of Fe and Mg in olivine crystals and in experimental diffusion couples. Samples are ablated in circular patterns with profile diameters of 100–200 µm using a laser spot size of 25–30 µm. Depending on the scan speed and the repetition rate of the laser, each ablated sample layer is between 300 nm and 3.0 µm thick. The integrated signal of one ablated layer represents one data point of the depth profile. We have tested this technique by analyzing 5–50 µm deep depth profiles (consisting of 15–25 individual layers) of homogeneous and chemically zoned olivine crystal cuboids. The minor and trace element analyses of the zoned cuboids, conducted by fs-LA-ICP-MS, were compared with “horizontal” profiles analyzed in polished sections of the cuboids with electron probe microanalysis (EPMA). Furthermore, we analyzed Fe–Mg isotopic depth profiles of the same cuboids with fs-LA-MC-ICP-MS, of which the chemically zoned ones also showed isotopic zoning at identical scales. Isotopic depth profiles were also conducted on an unzoned olivine cuboid that was coated with a 26Mg- and 56Fe-enriched olivine thin film (of ∼ 800 nm) in order to investigate top-to-bottom contamination during depth profiling. Our results indicate that (i) concentration data acquired by fs-LA depth profiling match well with EPMA data, (ii) precise and accurate Fe and Mg isotopic data can be obtained (i.e., precision and accuracy are ≤ 0.12 ‰ and ≤ 0.15 ‰ for both δ26Mg and δ56Fe, respectively), and (iii) potential top-to-bottom contamination during depth profiling of isotope ratios can be avoided. The technique presented herein is particularly suitable for the investigation of minerals or glasses with chemical and/or isotopic gradients (e.g., diffusion zoning) vertical to planar surfaces. It can also be applied in materials sciences in order to analyze thin films, coatings, or surface contaminations on solids

Multi-Stage Magma Evolution in Intra-Plate Volcanoes: Insights From Combined in situ Li and Mg–Fe Chemical and Isotopic Diffusion Profiles in Olivine

Understanding the timescales of magma evolution and ascent is essential for interpreting geophysical monitoring signals from active volcanoes. In this study, we explore the potential of diffusion-driven Li concentration and isotope zoning profiles recorded by magmatic olivine crystals to unravel time scales of magma evolution processes. Lithium is a fast-diffusing element and may provide the opportunity to investigate changes in magma composition during magma ascent, shortly before eruption. Lithium chemical and isotopic profiles were determined in olivines from two localities in the Massif Central volcanic region (France) that have previously been investigated for their Fe–Mg isotope systematics. The combined investigation of isotopic and chemical profiles makes it possible to distinguish between crystal growth and diffusion events. Extremely low δ7Li-values down to −30.7‰ (relative to the commonly used Li isotope standard IRMM-16) in the crystal core regions and elevated values at crystal rims (δ7Li ∼8 to 10‰), along with increasing concentrations from cores (∼3 to 1 μg/g) toward rims (12 to 6 μg/g) were found. The shape and orientation of both the chemical and isotopic profiles indicate that they were dominantly generated by Li diffusion into and within the olivine grains during magmatic differentiation. While Mg–Fe isotope and major element profiles have been modeled by a single diffusion event (Oeser et al., 2015), concentration and isotope profiles of Li indicate that a second diffusion event took place, that was not recorded by the Mg–Fe exchange diffusion couple. The first diffusion event was interpreted as reflecting the residence of the olivine crystals in a magma chamber. As diffusion coefficients for Fe–Mg exchange diffusion are very well determined, the time scales of this event are likely best quantified by Mg–Fe isotopic exchange diffusion modeling (Oeser et al., 2015). This event probably also generated the low δ7Li observed in olivine cores. Comparing the length of the Mg–Fe and Li profiles could thus be used to determine the less well-known diffusion coefficients of Li in the studied olivine crystals. The findings of this study indicate that Li diffusion at low Li concentration levels, as typically observed in natural olivine, may be not as fast as previously thought. The second diffusion event might represent a short-lived event, such as degassing, related to the ascent of the magma and/or magma cooling after emplacement of the lava. Such a process would only affect Li, which, in contrast to the refractory elements Fe and Mg, is volatile during degassing. The findings of this study show that, according to their different diffusion rates and physiochemical properties, the combined use of spatially resolved Li and Mg–Fe chemical and isotopic diffusion profiles, is a powerful tool to model even multi-stage evolution processes in magmatic systems. © Copyright © 2020 Steinmann, Oeser, Horn and Weyer

Changes in antimony isotopic composition as a tracer of hydrothermal fluid evolution at the Sb deposits in Pezinok (Slovakia)

In this work, we investigated in situ isotopic compositions of antimony (Sb) minerals from two substages of the ore deposits near Pezinok (Slovakia). The δ123Sb values of the primary Sb minerals range from −0.4 and +0.8‰ and increase progressively along the precipitation sequence. In the substage II, the early-formed gudmundite (FeSbS) shows in all sections the lowest δ123Sb values, followed by berthierite (FeSb2S4), stibnite (Sb2S3), and valentinite (Sb2O3) with the heaviest δ123Sb values. A similar trend was observed for the substage III, from the initially-formed stibnite, followed by kermesite (Sb2S2O), valentinite, senarmontite (both Sb2O3), and schafarzikite (FeSb2O4). The evolution can be rationalized by a Rayleigh fractionation model with a starting δ123Sb value in the fluid of +0.3‰, applying the same mineral-fluid fractionation factor to all minerals. Thus, the texturally observed order of mineralization is confirmed by diminishing trace element contents and heavier δ123Sb values in successively crystallized Sb minerals. Antimony in substage III was likely supplied from the oxidative dissolution of stibnite that formed earlier during substage II. The data interpretation, although limited by the lack of reliable mineral-fluid fractionation factors, implies that Sb precipitation within each substage occurred from an episodic metal precipitation, likely associated with a similar Sb isotope fractionation between fluid and all investigated Sb minerals. Large isotopic variations, induced by precipitation from a fluid as a response to temperature decrease, may be an obstacle in deciphering the metal source in hydrothermal ore deposits. However, Sb isotopes appear to be an excellent instrument to enhance our understanding on how hydrothermal systems operate

Central Tethyan platform-top hypoxia during Oceanic Anoxic Event 1a

Short-term hypoxia in epeiric water masses is a common phenomenon of modern marine environments and causes mass mortality in coastal marine ecosystems. Here, we test the hypothesis that during the early Aptian, platform-top hypoxia temporarily established in some of the vast epeiric seas of the central Tethys and caused, combined with other stressors, significant changes in reefal ecosystems. Potentially interesting target examples include time intervals characterized by the demise of lower Aptian rudist-coral communities and the establishment of microencruster facies, as previously described from the central and southern Tethys and from the proto-North Atlantic domain. These considerations are relevant as previous work has predominantly focused on early Aptian basinal anoxia in the context of Oceanic Anoxic Event (OAE) 1a, whereas the potential expansion of the oxygen minimum zone (OMZ) in coeval shallow-water environments is underexplored. Wellknown patterns in the δ13C record during OAE 1a allow for a sufficiently time-resolved correlation with previously studied locations and assignment to chemostratigraphic segments. This paper presents and critically discusses the outcome of a multi-proxy study (e.g., rare earth elements (REEs), U isotopes, and redox-sensitive trace elements) applied to lower Aptian shallow-water carbonates today exposed in the Kanfanar quarry in Istria, Croatia. These rocks were deposited on an extensive, isolated high in the central Tethys surrounded by hemipelagic basins. Remarkably, during chemostratigraphic segment C2, the depletion of redox-sensitive trace elements As, V, Mo, and U in platform carbonates, deposited in normal marine oxic waters, record the first occurrence of basinal, organic-rich sediment deposition in which these elements are enriched. During the C3 segment, seawater oxygen depletion established on the platform top as indicated by the patterns in Ce=Ce∗ and U isotopes. Shifts in redox-sensitive proxies coincide with the expansion of microencruster facies. Segment C4 witnesses the return to normal marine reefal faunas on the platform top and is characterized by patterns in redox-sensitive proxies typical of normal marine dissolved oxygen levels. It remains unclear, however, if platform-top hypoxia resulted from the expansion and upwelling of basinal, oxygen-depleted water masses or if spatially isolated, shallow hypoxic water bodies formed on the platform. Data shown here are relevant as they shed light on the driving mechanisms that control poorly understood faunal patterns during OAE 1a in the neritic realm and provide evidence on the intricate relation between basinal and platform-top water masses. © Author(s) 2019

Mantle-derived trace element variability in olivines and their melt inclusions

Trace element variability in oceanic basalts is commonly used to constrain the physics of mantle melting and the chemistry of Earth's deep interior. However, the geochemical properties of mantle melts are often overprinted by mixing and crystallisation processes during ascent and storage. Studying primitive melt inclusions offers one solution to this problem, but the fidelity of the melt-inclusion archive to bulk magma chemistry has been repeatedly questioned. To provide a novel check of the melt inclusion record, we present new major and trace element analyses from olivine macrocrysts in the products of two geographically proximal, yet compositionally distinct, primitive eruptions from the Reykjanes Peninsula of Iceland. By combining these macrocryst analyses with new and published melt inclusion analyses we demonstrate that olivines have similar patterns of incompatible trace element (ITE) variability to the inclusions they host, capturing chemical systematics on intra- and inter-eruption scales. ITE variability (element concentrations, ratios, variances and variance ratios) in olivines from the ITE-enriched Stapafell eruption is best accounted for by olivine-dominated fractional crystallisation. In contrast, ITE variability in olivines and inclusions from the ITE-depleted Háleyjabunga eruption cannot be explained by crystallisation alone, and must have originated in the mantle. Compatible trace element (CTE) variability is best described by crystallisation processes in both eruptions. Modest correlations between host and inclusion ITE contents in samples from Háleyjabunga suggest that melt inclusions can be faithful archives of melting and magmatic processes. It also indicates that degrees of ITE enrichment can be estimated from olivines directly when melt inclusion and matrix glass records of geochemical variability are poor or absent. Inter-eruption differences in olivine ITE systematics between Stapafell and Háleyjabunga mirror differences in melt inclusion suites, and confirm that the Stapafell eruption was fed by lower degree melts from greater depths within the melting region than the Háleyjabunga eruption. Although olivine macrocrysts from Stapafell are slightly richer in Ni than those from Háleyjabunga, their overall CTE systematics (e.g., Ni/(Mg/Fe), Fe/Mn and Zn/Fe) are inconsistent with being derived from olivine-free pyroxenites. However, the major element systematics of Icelandic basalts require lithological heterogeneity in their mantle source in the form of Fe-rich and hence fusible domains. We thus conclude that enriched heterogeneities in the Icelandic mantle are composed of modally enriched, yet nonetheless olivine-bearing, lithologies and that olivine CTE contents provide an incomplete record of lithological heterogeneity in the mantle. Modally enriched peridotites may therefore play a more important role in oceanic magma genesis than previously inferred.</p

Quantification of trace element contents in frozen fluid inclusions by UV-fs-LA-ICP-MS analysis

We have developed a new analytical setup for the determination of trace element concentrations in fluid inclusions by UV-fs-LA-ICP-MS. Laser ablation was performed at a low temperature of -40 degrees C by using a modified heating-freezing stage as the ablation cell. With this method it was possible to successfully analyse 53 of 55 frozen synthetic NaCl-H2O fluid inclusions in quartz, covering a size range between 8 mu m and 25 mu m down to a depth of 50 mu m. The high success rate could be achieved as the 194 nm UV-fs-laser allows excellent control over the opening procedure of frozen fluid inclusions. Trace element analyses were performed with a fast scanning magnetic sector field ICP-MS. The lower limits of detection for fluid inclusion analysis vary from 0.1 mu g g(-1) (for Bi-209) to 10 mu g g(-1) (for K-39). The typical analytical uncertainty, depending on the element and respective concentration level, ranges between 10% and 30% (1RSD), based on the reproducibility of experimentally synthesized fluid inclusions. All elements from a stock solution, which behaved inert during the HP/HT experiments (B, K, Cd, Te, Tl, Pb and Bi), could be recovered in the synthetic inclusions at concentrations that correspond within their specific analytical uncertainties to their original concentration of 53 mu g g(-1). The method represents a highly efficient tool for the determination of accurate trace element data on low concentration levels in small fluid inclusions with a high success rate of >90%. The latter is particularly advantageous considering the commonly time consuming characterization of fluid inclusions.NTH Graduate School GeoFluxe

Evaluating the role of coastal hypoxia on the transient expansion of microencruster intervals during the early Aptian

Worldwide, a growing number of modern coastal marine ecosystems are increasingly exposed to suboxic- or even anoxic conditions. Low seawater oxygen levels trigger significant ecosystem changes and may result in mass mortality of oxygen-sensitive biota. The applicability of observations from recent (anthropogenically influenced) suboxic coastal settings to fossil anoxic shallow-marine environments is, however, as yet poorly explored. The test case documented here are upper Barremian to lower Aptian strata in the Lusitanian Basin (Ericeira section, Portugal). These are characterized by the transient demise of rudist–coral communities and the rapid establishment of microencruster facies in the vacant ecological niches. The hypothesis is tested that the temporal expansion of the microencrusting organism Lithocodium aggregatum took place in response to platform-top seawater oxygen depletion. We critically discuss the outcome of a multi-proxy palaeoseawater redox approach (e.g. Rare Earth Elements (REEs), U isotopes and palaeoecology) and put the robustness of the proxies applied here to the test. This is done by considering issues with these methods in general but also emphasizing the significance of terrigenous contamination and fractionation effects. Data shown here document that evidence for coastal seawater oxygen depletion in the prelude of Oceanic Anoxic Event (OAE) 1a is lacking, and hence, anoxia was not the driving mechanism for the demise of rudist–coral ecosystems in the proto-North Atlantic platform setting studied here. In contrast, well-oxygenated early Aptian platform-top water masses are proposed for this site. Geologically short (decades to millennia) fluctuations in seawater oxygen levels cannot be excluded, however. But even if these took place, they offer no explanation for the Kyr to Myr-scale patterns discussed here. The present paper is relevant as it sheds light on the complexity of mechanisms that drive punctuated Early Cretaceous coral–rudist ecosystem turnover, and assess strengths and weaknesses of redox proxies applied to ancient shallow-marine platform carbonates. © 2020 The Authors. Lethaia published by John Wiley & Sons Ltd on behalf of Lethaia Foundatio

Stabile Cu-Isotope im Boden

Kupfer ist ein essentielles Spurenelement, kann aber in hohen Gehalten auch als Schadstoff wirken. Die Untersuchung von stabilen Cu-Isotopenverhältnissen im Boden erlaubt Rückschlüsse auf biogeochemische Prozesse im Boden. Beson-ders für bodenbildene Prozesse, die über lange Zeiträume ablaufen, könnten stabile Isotope zusätzliche Informationen über das Verhalten von Metallen im Boden geben. Es wurden acht verschiedene Bodenprofile untersucht, die vier verschiede-nen Bodentypen angehören. Die gesamte Variation der ?65Cu-Werte betrug ca. 1‰. In einem Podsol wurden Variationen zwischen Ae- und Bhs-Horizont gefunden, die sich durch die Komplexierung und Verlagerung mit organischen Säuren erklären lassen. In wasserbeeinflussten Böden (Pseudogleyen und Gleyen) führen Redoxwechsel zu einer Akkumulation von 65Cu im Unterboden, während in Braunerden der 65Cu-Gehalt mit zunehmender Tiefe abnimmt

Li–Na interdiffusion and diffusion-driven lithium isotope fractionation in pegmatitic melts

In this study, we investigate the diffusion of Li and its stable isotopes (6Li and 7Li) in flux-rich (1.8 % Li2O, 2.6 % B2O3, 2.3 % P2O5 and 3 % F) pegmatitic melts in order to contribute to the understanding of Li enrichment in such systems. Two glasses were synthesized with a model pegmatitic composition, one of which is highly enriched in Li (> 1 wt %, PEG2-blue) and the other one essentially Li-free (PEG2-Li-free). Diffusion couple experiments were performed to determine the chemical diffusivity of Li in dry pegmatitic melts. Experiments were conducted using rapid-heat and rapid-quench cold-seal pressure vessels in a temperature range of 650–940 ∘C at 100 MPa with Ar as the pressure medium. We observed rapidly formed diffusion profiles, driven by an interdiffusive exchange of the monovalent alkalis Li and Na, while the other elements are immobile on the timescale of experiments (1–30 min). From these experiments, activation energies for Li–Na interdiffusion were determined as 99 ± 7 kJ mol−1 with a pre-exponential factor of log D0 = −5.05 ± 0.33 (D0 in m2 s−1). Li and Na partitioning between the stronger depolymerized PEG2-blue and the less depolymerized PEG2-Li-free leads to a concentration jump at the interface; i.e. Na is enriched in the more depolymerized PEG2-blue. Li–Na interdiffusion coefficients in the studied melt composition are in a similar range as Li and Na tracer diffusivities in other dry aluminosilicate melts, confirming little to no effect of aluminosilicate melt composition on Li diffusivity. Thus, added fluxes do not enhance the Li diffusivity in the same way as observed for H2O (Holycross et al., 2018; Spallanzani et al., 2022). Using melt viscosity as a proxy for the polymerization of the melt shows that water has a stronger potential to depolymerize a melt compared to other fluxing elements. Faster diffusion of 6Li compared to 7Li leads to a strong Li isotope fractionation along the diffusion profile, resulting in δ7Li as low as −80 ‰ relative to the diffusion-unaffected regions. This diffusive isotope fractionation can be quantified with an empirical isotope fractionation factor (β) of 0.20 ± 0.04, similar to previously observed β values for Li diffusion in melts. This suggests in accordance with previously published data that a β value of ca. 0.2 seems to be universally applicable to diffusive Li isotope fractionation in aluminosilicate melts