Generalized calculation of magnetic coupling constants for Mott-Hubbard insulators: Application to ferromagnetic Cr compounds

Using a Rayleigh-Schr\"odinger perturbation expansion of multi-band Hubbard models, we present analytic expressions for the super-exchange coupling constants between magnetic transition metal ions of arbitrary separation in Mott-Hubbard insulators. The only restrictions are i) all ligand ions are closed shell anions and ii) all contributing interaction paths are of equal length. For short paths, our results essentially confirm the Goodenough-Kanamori-Anderson rules, yet in general there does not exist any simple rule to predict the sign of the magnetic coupling constants. The most favorable situation for ferromagnetic coupling is found for ions with less than half filled d shells, the (relative) tendency to ferromagnetic coupling increases with increasing path length. As an application, the magnetic interactions of the Cr compounds Rb$_2$CrCl$_4$, CrCl$_3$, CrBr$_3$ and CrI$_3$ are investigated, all of which except CrCl$_3$ are ferromagnets.Comment: 13 pages, 6 eps figures, submitted to Phys Rev

Resilience and Challenges of Marine Social–Ecological Systems Under Complex and Interconnected Drivers

In this paper, we summarize the contributions made by an interdisciplinary group of researchers from different disciplines (biology, ecology, economics, and law) that deal with key dimensions of marine social–ecological systems. Particularly, the local and global seafood provision; the feasibility and management of marine protected areas; the use of marine ecosystem services; the institutional dimension in European fisheries, and the affordable models for providing scientific advice to small-scale fisheries. This Special Issue presents key findings from selected case studies around the world available to educators, policy makers, and the technical community. Together, these papers show that a range of diverse ecological, economic, social, and institutional components often mutually interact at spatial and temporal scales, which evidence that managing marine social–ecological systems needs a continuous adaptability to navigate into new governance systems