BPS black hole entropy and attractors in very special geometry. Cubic forms, gradient maps and their inversion

We consider Bekenstein-Hawking entropy and attractors in extremal BPS black holes of $$ \mathcal{N} $$ N = 2, D = 4 ungauged supergravity obtained as reduction of minimal, matter-coupled D = 5 supergravity. They are generally expressed in terms of solutions to an inhomogeneous system of coupled quadratic equations, named BPS system, depending on the cubic prepotential as well as on the electric-magnetic fluxes in the extremal black hole background. Focussing on homogeneous non-symmetric scalar manifolds (whose classification is known in terms of L(q, P, Ṗ) models), under certain assumptions on the Clifford matrices pertaining to the related cubic prepotential, we formulate and prove an invertibility condition for the gradient map of the corresponding cubic form (to have a birational inverse map which is given by homogeneous polynomials of degree four), and therefore for the solutions to the BPS system to be explicitly determined, in turn providing novel, explicit expressions for the BPS black hole entropy and the related attractors as solution of the BPS attractor equations. After a general treatment, we present a number of explicit examples with Ṗ = 0, such as L(q, P), 1 ⩽ q ⩽ 3 and P ⩾ 1, or L(q, 1), 4 ⩽ q ⩽ 9, and one model with Ṗ = 1, namely L(4, 1, 1). We also briefly comment on Kleinian signatures and split algebras. In particular, we provide, for the first time, the explicit form of the BPS black hole entropy and of the related BPS attractors for the infinite class of L(1, P) P ⩾ 2 non-symmetric models of $$ \mathcal{N} $$ N = 2, D = 4 supergravity.

Representing squares by cubic forms

Let [Formula: see text] be an arbitrary cubic form. In this paper, we show that for non-homogeneous cubic equations like [Formula: see text], there exists a non-trivial solution when [Formula: see text].

Framboid Microcrystal Growth

Framboid microcrystals grow through surface reaction of S2(-II) or H2S with =FeS moieties at defect sites on the pyrite crystal surface. The surface energies of pyrite vary from the most stable cubic through octahedral to pyritohedral and dodecahedral surfaces. Microcrystals commonly develop as truncated octahedra as the supersaturation decreases during crystal growth in sedimentary environments, although cubic forms may be favored under hydrothermal conditions. Screw dislocation growth followed by surface nucleation growth are the normal growth modes in sediments, whereas surface nucleation growth is likely to dominate in hydrothermal systems. The rate of crystal growth of framboids is unknown but appears to be very fast and normally diffusion-limited. Linear approximations to the diffusion equations show that average 6 μ‎m diameter framboids form in five days in sediments, and formation times increase exponentially from a few hours for ca. 2 μ‎m framboids to three years for the largest 250 μ‎m framboids.

Hydride-based antiperovskites with soft anionic sublattices as fast alkali ionic conductors

Most solid-state materials are composed of p-block anions, only in recent years the introduction of hydride anions (1s2) in oxides (e.g., SrVO2H, BaTi(O,H)3) has allowed the discovery of various interesting properties. Here we exploit the large polarizability of hydride anions (H–) together with chalcogenide (Ch2–) anions to construct a family of antiperovskites with soft anionic sublattices. The M3HCh antiperovskites (M = Li, Na) adopt the ideal cubic structure except orthorhombic Na3HS, despite the large variation in sizes of M and Ch. This unconventional robustness of cubic phase mainly originates from the large size-flexibility of the H– anion. Theoretical and experimental studies reveal low migration barriers for Li+/Na+ transport and high ionic conductivity, possibly promoted by a soft phonon mode associated with the rotational motion of HM6 octahedra in their cubic forms. Aliovalent substitution to create vacancies has further enhanced ionic conductivities of this series of antiperovskites, resulting in Na2.9H(Se0.9I0.1) achieving a high conductivity of ~1 × 10–4 S/cm (100 °C).