Rotational integral geometry of tensor valuations

We derive a new rotational Crofton formula for Minkowski tensors. In special cases, this formula gives (1) the rotational average of Minkowski tensors defined on linear subspaces and (2) the functional defined on linear subspaces with rotational average equal to a Minkowski tensor. Earlier results obtained for intrinsic volumes appear now as special cases.

Foundations of stochastic geometry and theory of random sets

The first section of this chapter starts with the Buffon problem, which is one of the oldest in stochastic geometry, and then continues with the definition of measures on the space of lines. The second section defines random closed sets and related measurability issues, explains how to characterize distributions of random closed sets by means of capacity functionals and introduces the concept of a selection. Based on this concept, the third section starts with the definition of the expectation and proves its convexifying effect that is related to the Lyapunov theorem for ranges of vector-valued measures. Finally, the strong law of large numbers for Minkowski sums of random sets is proved and the corresponding limit theorem is formulated. The chapter is concluded by a discussion of the union-scheme for random closed sets and a characterization of the corresponding stable laws.

Extraction of the s-wave pi N scattering lengths from data on pionic atoms

For many years a combined analysis of pionic hydrogen and deuterium atoms has been known as a good tool to extract information on the isovector and especially on the isoscalar s-wave pN scattering length. However, given the smallness of the isoscalar scattering length, the analysis becomes useful only if the pion–deuteron scattering length is controlled theoretically to a high accuracy comparable to the experimental precision. To achieve the required few-percent accuracy one needs theoretical control over all isospin-conserving three-body pNN !pNN operators up to one order before the contribution of the dominant unknown (N†N)2pp contact term. This term appears at next-to-next-to-leading order in Weinberg counting. In addition, one needs to include isospin-violating effects in both two-body (pN) and three-body (pNN) operators. In this talk we discuss the results of the recent analysis where these isospin-conserving and -violating effects have been carefully taken into account. Based on this analysis, we present the up-to-date values of the s-wave pN scattering lengths.

Isospin breaking in pion–deuteron scattering and the pion–nucleon scattering lengths

In recent years, high-accuracy data for pionic hydrogen and deuterium have become the primary source of information on the pion–nucleon scattering lengths. Matching the experimental precision requires, in particular, the study of isospin-breaking corrections both in pion– nucleon and pion–deuteron scattering. We review the mechanisms that lead to the cancellation of potentially enhanced virtual-photon corrections in the pion–deuteron system, and discuss the subtleties regarding the definition of the pion–nucleon scattering lengths in the presence of electromagnetic interactions by comparing to nucleon–nucleon scattering. Based on the p±p channels we find for the virtual-photon-subtracted scattering lengths in the isospin basis a1/2/ g= (170.5±2.0) · 10−3M−1p and a3/2/ g= (−86.5±1.8) · 10−3M−1p .

Characterization of granite matrix porosity and pore-space geometry by in situ and laboratory methods

Most available studies of interconnected matrix porosity of crystalline rocks are based on laboratory investigations; that is, work on samples that have undergone stress relaxation and were affected by drilling and sample preparation. The extrapolation of the results to in situ conditions is therefore associated with considerable uncertainty, and this was the motivation to conduct the ‘in situ Connected Porosity’ experiment at the Grimsel Test Site (Central Swiss Alps). An acrylic resin doped with fluorescent agents was used to impregnate the microporous granitic matrix in situ around an injection borehole, and samples were obtained by overcoring. The 3-D structure of the porespace, represented by microcracks, was studied by U-stage fluorescence microscopy. Petrophysical methods, including the determination of porosity, permeability and P -wave velocity, were also applied. Investigations were conducted both on samples that were impregnated in situ and on non-impregnated samples, so that natural features could be distinguished from artefacts. The investigated deformed granites display complex microcrack populations representing a polyphase deformation at varying conditions. The crack population is dominated by open cleavage cracks in mica and grain boundary cracks. The porosity of non-impregnated samples lies slightly above 1 per cent, which is 2–2.5 times higher than the in situ porosity obtained for impregnated samples. Measurements of seismic velocities (Vp ) on spherical rock samples as a function of confining pressure, spatial direction and water saturation for both non-impregnated and impregnated samples provide further constraints on the distinction between natural and induced crack types. The main conclusions are that (1) an interconnected network of microcracks exists in the whole granitic matrix, irrespective of the distance to ductile and brittle shear zones, and (2) conventional laboratory methods overestimate the matrix porosity. Calculations of contaminant transport through fractured media often rely on matrix diffusion as a retardation mechanism.

Dynamics of supercooled water in highly compacted clays studied by neutron scattering

The freezing behavior of water confined in compacted charged and uncharged clays (montmorillonite in Na-and Ca-forms, illite in Na-and Ca-forms, kaolinite and pyrophyllite) was investigated by neutron scattering. Firstly, the amount of frozen (immobile) water was measured as a function of temperature at the IN16 backscattering spectrometer, Institute Laue-Langevin (ILL). Water in uncharged, partly hydrophobic (kaolinite) and fully hydrophobic (pyrophyllite) clays exhibited a similar freezing and melting behavior to that of bulk water. In contrast, water in charged clays which are hydrophilic could be significantly supercooled. To observe the water dynamics in these clays, further experiments were performed using quasielastic neutron scattering. At temperatures of 250, 260 and 270 K the diffusive motion of water could still be observed, but with a strong reduction in the water mobility as compared with the values obtained above 273 K. The diffusion coefficients followed a non-Arrhenius temperature dependence well described by the Vogel-Fulcher-Tammann and the fractional power relations. The fits revealed that Na-and Ca-montmorillonite and Ca-illite have similar Vogel-Fulcher-Tammann temperatures (T-VFT, often referred to as the glass transition temperature) of similar to 120 K and similar temperatures at which the water undergoes the 'strong-fragile' transition, T-s similar to 210 K. On the other hand, Na-illite had significantly larger values of T-VFT similar to 180 K and T-s similar to 240 K. Surprisingly, Ca-illite has a similar freezing behavior of water to that of montmorillonites, even though it has a rather different structure. We attribute this to the stronger hydration of Ca ions as compared with the Na ions occurring in the illite clays.

Geophysical and Hydrological Evaluation of Two Bog Complexes in a Northern Peatland: Implications for the Distribution of Biogenic Gases at the Basin Scale

Ground penetrating radar (GPR) was used to determine peat basin geometry and the spatial distribution of free-phase biogenic gasses in two separate units of a northern peatland (Central and Southern Unit of Caribou Bog, Maine). The Central Unit is characterized by a deep basin structure (15 m maximum depth) and a raised (eccentric) bog topographic profile (up to 2 m topographic variation). Here numerous regions of electromagnetic (EM) wave scattering are considered diagnostic of the presence of extensive free-phase biogenic gas. In contrast, the Southern Unit is shallower (8 m maximum depth) and has a slightly convex upwards bog profile (less than 1 m topographic variation), and areas of EM wave scattering are notably absent. The biogenic gas zones interpreted from GPR in the Central Unit are associated with: (1) wooded heath vegetation at the surface, (2) open pools at the surface, (3) high water table elevations near the center of the basin, and (4) a region of overpressure (at approximately 5 m depth) immediately below the zone of free-phase gas accumulation. The latter suggests (1) a transient pressure head associated with low hydraulic conductivity resulting from the biogenic gasses themselves or confining layers in the peat that restrict both gas release and groundwater flow and/or (2) overpressure in the peat column as a result of the gas buildup itself. In contrast, the Southern Unit, where zones of EM scattering are absent, is characterized by: (1) predominantly shrub vegetation, (2) a lack of open pools, (3) only minor variations (less than 1 m) in water table elevation throughout the entire unit; and (4) generally upward groundwater flow throughout the basin. The results illustrate the nonuniformity of free-phase biogenic gas distribution at the peat basin scale and provide insights into the processes and controls associated with CH4 and CO2 accumulation in peatlands.

Mechanical constraints imposed by 3D cellular geometry and arrangement modulate growth patterns in the Arabidopsis embryo

Morphogenesis occurs in 3D space over time and is guided by coordinated gene expression programs. Here we use postembryonic development in Arabidopsis plants to investigate the genetic control of growth. We demonstrate that gene expression driving the production of the growth-stimulating hormone gibberellic acid and downstream growth factors is first induced within the radicle tip of the embryo. The center of cell expansion is, however, spatially displaced from the center of gene expression. Because the rapidly growing cells have very different geometry from that of those at the tip, we hypothesized that mechanical factors may contribute to this growth displacement. To this end we developed 3D finite-element method models of growing custom-designed digital embryos at cellular resolution. We used this framework to conceptualize how cell size, shape, and topology influence tissue growth and to explore the interplay of geometrical and genetic inputs into growth distribution. Our simulations showed that mechanical constraints are sufficient to explain the disconnect between the experimentally observed spatiotemporal patterns of gene expression and early postembryonic growth. The center of cell expansion is the position where genetic and mechanical facilitators of growth converge. We have thus uncovered a mechanism whereby 3D cellular geometry helps direct where genetically specified growth takes place.

Water–montmorillonite systems: Neutron scattering and tracer throughdiffusion studies

Designs for deep geological respositories of nuclear waste include bentonite as a hydraulic and chemisorption buffer material to protect the biosphere from leakage of radionuclides. Bentonite is chosen because it is a cheap, naturally occurring material with the required properties. It consists essentially of montmorillonite, a swelling clay mineral. Upon contact with groundwater such clays can seal the repository by incorporating water in the interlayers of their crystalline structure. The intercalated water exhibits significantly different properties to bulk water in the surrounding interparticle pores, such as lower diffusion coefficients (González Sánchez et. al. 2008). This doctoral thesis presents water distribution and diffusion behavior on various time and space scales in montmorillonite. Experimental results are presented for Na- and Cs-montmorillonite samples with a range of bulk dry densities (0.8 to 1.7 g/cm3). The experimental methods employed were neutron scattering (backscattering, diffraction, time-of-flight), adsorption measurements (water, nitrogen) and tracer-through diffusion. For the tracer experiments the samples were fully saturated via the liquid phase under volume-constrained conditions. In contrast, for the neutron scattering experiments, the samples were hydrated via the vapor phase and subsequently compacted, leaving a significant fraction of interparticle pores unfilled with water. Owing to these differences in saturation, the water contents of the samples for neutron scattering were characterized by gravimetry whereas those for the tracer experiments were obtained from the bulk dry density. The amount of surface water in interlayer pores could be successfully discriminated from the amount of bulk-like water in interparticle pores in Na- and Csmontmorillonite using neutron spectroscopy. For the first time in the literature, the distribution of water between these two pore environments was deciphered as a function of gravimetric water content. The amount was compared to a geometrical estimation of the amount of interlayer and interparticle water determined by neutron diffraction and adsorption measurements. The relative abundances of the 1 to 4 molecular water layers in the interlayer were determined from the area ratios of the (001)-diffraction peaks. Depending on the characterization method, different fractions of surface water and interlayer water were obtained. Only surface and interlayer water exists in amontmorillonite with water contents up to 0.18 g/g according to spectroscopic measurements and up to 0.32 g/g according to geometrical estimations, respectively. At higher water contents, bulk-like and interparticle water also exists. The amounts increase monotonically, but not linearly, from zero to 0.33 g/g for bulk-like water and to 0.43 g/g for interparticle water. It was found that water most likely redistributes between the surface and interlayer sites during the spectroscopic measurements and therefore the reported fraction is relevant only below about -10 ºC (Anderson, 1967). The redistribution effect can explain the discrepancy in fractions between the methods. In a novel approach the fractions of water in different pore environments were treated as a fixed parameter to derive local diffusion coefficients for water from quasielastic neutron scattering data, in particular for samples with high water contents. Local diffusion coefficients were obtained for the 1 to 4 molecular water layers in the interlayer of 0.5·10–9, 0.9·10–9, 1.5·10–9 and 1.4·10–9 m²/s, respectively, taking account of the different water fractions (molecular water layer, bulk-like water). The diffusive transport of 22Na and HTO through Na-montmorillonite was measured on the laboratory experimental scale (i.e. cm, days) by tracer through-diffusion experiments. We confirmed that diffusion of HTO is independent of the ionic strength of the external solution in contact with the clay sample but dependent on the bulk dry density. In contrast, the diffusion of 22Na was found to depend on both the ionic strength of the pore solution and on the bulk dry density. The ratio of the pore and surface diffusion could be experimentally determined for 22Na from the dependence of the diffusion coefficient on the ionic strength. Activation energies were derived from the temperaturedependent diffusion coefficients via the Arrhenius relation. In samples with high bulk dry density the activation energies are slightly higher than those of bulk water whereas in low density samples they are lower. The activation energies as a function of ionic strengths of the pore solutions are similar for 22Na and HTO. The facts that (i) the slope of the logarithmic effective diffusion coefficients as a function of the logarithmic ionic strength is less than unity for low bulk dry densities and (ii) two water populations can be observed for high gravimetric water contents (low bulk dry densities) support the interlayer and interparticle porosity model proposed by Glaus et al. (2007), Bourg et al. (2006, 2007) and Gimmi and Kosakowski (2011).

Analyzing diffuse scattering with supercomputers

Two new approaches to quantitatively analyze diffuse diffraction intensities from faulted layer stacking are reported. The parameters of a probability-based growth model are determined with two iterative global optimization methods: a genetic algorithm (GA) and particle swarm optimization (PSO). The results are compared with those from a third global optimization method, a differential evolution (DE) algorithm [Storn & Price (1997). J. Global Optim. 11, 341–359]. The algorithm efficiencies in the early and late stages of iteration are compared. The accuracy of the optimized parameters improves with increasing size of the simulated crystal volume. The wall clock time for computing quite large crystal volumes can be kept within reasonable limits by the parallel calculation of many crystals (clones) generated for each model parameter set on a super- or grid computer. The faulted layer stacking in single crystals of trigonal three-pointedstar- shaped tris(bicylco[2.1.1]hexeno)benzene molecules serves as an example for the numerical computations. Based on numerical values of seven model parameters (reference parameters), nearly noise-free reference intensities of 14 diffuse streaks were simulated from 1280 clones, each consisting of 96 000 layers (reference crystal). The parameters derived from the reference intensities with GA, PSO and DE were compared with the original reference parameters as a function of the simulated total crystal volume. The statistical distribution of structural motifs in the simulated crystals is in good agreement with that in the reference crystal. The results found with the growth model for layer stacking disorder are applicable to other disorder types and modeling techniques, Monte Carlo in particular.

Towards a data-driven analysis of hadronic light-by-light scattering

The hadronic light-by-light contribution to the anomalous magnetic moment of the muon was recently analyzed in the framework of dispersion theory, providing a systematic formalism where all input quantities are expressed in terms of on-shell form factors and scattering amplitudes that are in principle accessible in experiment. We briefly review the main ideas behind this framework and discuss the various experimental ingredients needed for the evaluation of one- and two-pion intermediate states. In particular, we identify processes that in the absence of data for doubly-virtual pion–photon interactions can help constrain parameters in the dispersive reconstruction of the relevant input quantities, the pion transition form factor and the helicity partial waves for γ⁎γ⁎→ππ.

Dispersive approach to hadronic light-by-light scattering

Based on dispersion theory, we present a formalism for a model-independent evaluation of the hadronic light-by-light contribution to the anomalous magnetic moment of the muon. In particular, we comment on the definition of the pion pole in this framework and provide a master formula that relates the effect from ππ intermediate states to the partial waves for the process γ * γ * → ππ. All contributions are expressed in terms of on-shell form factors and scattering amplitudes, and as such amenable to an experimental determination.

Accurate evaluation of hadronic uncertainties in spin-independent WIMP-nucleon scattering: Disentangling two- and three-flavor effects

We show how to avoid unnecessary and uncontrolled assumptions usually made in the literature about soft SU(3) flavor symmetry breaking in determining the two-flavor nucleon matrix elements relevant for direct detection of weakly interacting massive particles (WIMPs). Based on SU(2) chiral perturbation theory, we provide expressions for the proton and neutron scalar couplings fp,nu and fp,nd with the pion-nucleon σ term as the only free parameter, which should be used in the analysis of direct detection experiments. This approach for the first time allows for an accurate assessment of hadronic uncertainties in spin-independent WIMP-nucleon scattering and for a reliable calculation of isospin-violating effects. We find that the traditional determinations of Vfpu−fnu and fpd−fnd are off by a factor of 2.