Trapped two-component Fermi gases with up to six particles: Energetics, structural properties, and molecular condensate fraction

We investigate small equal-mass two-component Fermi gases under external spherically symmetric confinement in which atoms with opposite spins interact through a short-range two-body model potential. We employ a non-perturbative microscopic framework, the stochastic variational approach, and determine the system properties as functions of the interspecies s-wave scattering length a, the orbital angular momentum L of the system, and the numbers N1 and N2 of spin-up and spin-down atoms (with N1-N2 =0 or 1 and N < 7, where N=N1+N2). At unitarity, we determine the energies of the five- and six-particle systems for various ranges r0 of the underlying two-body model potential and extrapolate to the zero-range limit. These energies serve as benchmark results that can be used to validate and assess other numerical approaches. We also present structural properties such as the pair distribution function and the radial density. Furthermore, we analyze the one-body and two-body density matrices. A measure for the molecular condensate fraction is proposed and applied. Our calculations show explicitly that the natural orbitals and the momentum distributions of atomic Fermi gases approach those characteristic for a molecular Bose gas if the s-wave scattering length a, a>0, is sufficiently small.Comment: 21 pages, 15 figures; accepted for publication in special issue of CRA

Natural capital, ecosystem services, and soil change: why soil science must embrace an ecosystems approach

Soil is part of the Earth's life support system, but how should we convey the value of this and of soil as a resource? Consideration of the ecosystem services and natural capital of soils offers a framework going beyond performance indicators of soil health and quality, and recognizes the broad value that soil contributes to human wellbeing. This approach provides links and synergies between soil science and other disciplines such as ecology, hydrology, and economics, recognizing the importance of soils alongside other natural resources in sustaining the functioning of the Earth system. We articulate why an ecosystems approach is important for soil science in the context of natural capital, ecosystem services, and soil change. Soil change is defined as change on anthropogenic time scales and is an important way of conveying dynamic changes occurring in soils that are relevant to current political decision-making time scales. We identify four important areas of research: (i) framework development; (ii) quantifying the soil resource, stocks, fluxes, transformations, and identifying indicators; (iii) valuing the soil resource for its ecosystem services; and (iv) developing decision-support tools. Furthermore, we propose contributions that soil science can make to address these research challenges