Petrology and geochronology of "muscovite age standard" B4M

Muscovite B4M, distributed in 1961 as an age standard, was ground under ethanol. Five grain size fractions were obtained and characterized by X-ray diffraction. They display a mixing trend between a phengitic (enriched in the fraction <0.2 µm) and a muscovitic component (predominant in the fraction >20 µm). High-pressure phengite is preserved as a relict in retrograde muscovite. Electron microprobe analyses of the distributed mineral separate reveal at least four white mica populations based on Si, Al, Mg, Na, Fe and F. Rb/K ratios vary by one order of magnitude. Rb–Sr analyses link the mineralogical heterogeneity to variable Rb/Sr and 87Sr/86Sr ratios. The grain size fractions define no internal isochron. Relict fine-grained phengite gives older ages than coarse-grained retrograde greenschist facies muscovite. The inverse grain size–age relationship also characterizes 39Ar/40Ar analyses. Cl/K anticorrelates with step ages: Cl-rich coarse muscovite is younger than Cl-poor fine relict phengite. Sr and Ar preserve a similar isotopic inheritance despite peak metamorphism reaching 635±20 °C. A suitable mineral standard requires that its petrological equilibrium first be demonstrated. Relicts and retrograde reaction textures are a guarantee of isotopic disequilibrium and heterogeneous ages within single crystal at the micrometre scale.

Ar diffusion in K-feldspar: present and absent

The mechanisms of Ar release from K-feldspar samples in laboratory experiments and during their geological history are assessed here. Modern petrology clearly established that the chemical and isotopic record of minerals is normally dominated by aqueous recrystallization. The laboratory critique is trickier, which explains why so many conflicting approaches have been able to survive long past their expiration date. Current models are evaluated for self-consistency; especially Arrhenian non-linearity leads to paradoxes. The models’ testable geological predictions suggest that temperature-based downslope extrapolations often overestimate observed geological Ar mobility substantially. An updated interpretation is based on the unrelatedness of geological behaviour to laboratory experiments. The isotopic record of K-feldspar in geological samples is not a unique function of temperature, as recrystallisation promoted by aqueous fluids is the predominant mechanism controlling isotope transport. K-feldspar should therefore be viewed as a hygrochronometer. Laboratory degassing proceeds from structural rearrangements and phase transitions such as are observed in situ at high temperature in Na and Pb feldspars. These effects violate the mathematics of an inert Fick’s Law matrix and preclude downslope extrapolation. The similar upward-concave, non-linear shapes of Arrhenius trajectories of many silicates, hydrous and anhydrous, are likely common manifestations of structural rearrangements in silicate structures.

On restricted families of projections in ℝ3

We study projections onto non-degenerate one-dimensional families of lines and planes in R 3 . Using the classical potential theoretic approach of R. Kaufman, one can show that the Hausdorff dimension of at most 12 -dimensional sets [Math Processing Error] is typically preserved under one-dimensional families of projections onto lines. We improve the result by an ε , proving that if [Math Processing Error], then the packing dimension of the projections is almost surely at least [Math Processing Error]. For projections onto planes, we obtain a similar bound, with the threshold 12 replaced by 1 . In the special case of self-similar sets [Math Processing Error] without rotations, we obtain a full Marstrand-type projection theorem for 1-parameter families of projections onto lines. The [Math Processing Error] case of the result follows from recent work of M. Hochman, but the [Math Processing Error] part is new: with this assumption, we prove that the projections have positive length almost surely.