Discrete curvature and torsion from cross-ratios

Motivated by a Möbius invariant subdivision scheme for polygons, we study a curvature notion for discrete curves where the cross-ratio plays an important role in all our key definitions. Using a particular Möbius invariant point-insertion-rule, comparable to the classical four-point-scheme, we construct circles along discrete curves. Asymptotic analysis shows that these circles defined on a sampled curve converge to the smooth curvature circles as the sampling density increases. We express our discrete torsion for space curves, which is not a Möbius invariant notion, using the cross-ratio and show its asymptotic behavior in analogy to the curvature.

Solid-liquid phase equilibria of (H2O-Mn(H2PO2)2-MnCl2-NaCl), (H2O-Mn(H2PO2)2-MnCl2) and (H2O-NaCl- MnCl2) systems at 323.15 K

The solid-liquid phase equilibria (SLE) and densities of H2O-NaCl- -MnCl2-Mn(H2PO2)2 quaternary system, H2O-NaCl-MnCl2 and H2O-MnCl2- -Mn(H2PO2)2 ternary systems were investigated at 323.15 K by the isothermal solution saturation method. The analyses of the liquid and solid phases were used to determine the composition of the solid phase using the Schreinemakers graphic method. The ternary systems contain one invariant point, two invariant curves and two crystallization regions. In the quaternary system, there is one invariant point, three invariant curves, and three crystallization areas corresponding to NaCl, MnCl2 4H2O, and Mn(H2PO2)2H2O. The crystallization area of Mn(H2PO2)2 H2O, being the largest in comparison with those of other salts, occupied 80.75 % of the total crystallization area.

Static compression of Fe4N to 77 GPa and its implications for nitrogen storage in the deep Earth

Compression and decompression experiments on face-centered cubic (fcc) γ′-Fe4N to 77 GPa at room temperature were conducted in a diamond-anvil cell with in situ X-ray diffraction (XRD) to examine its stability under high pressure. In the investigated pressure range, γ′-Fe4N did not show any structural transitions. However, a peak broadening was observed in the XRD patterns above 60 GPa. The obtained pressure-volume data to 60 GPa were fitted to the third-order Birch-Murnaghan equation of state (EoS), which yielded the following elastic parameters: K0 = 169 (6) GPa, K′ = 4.1 (4), with a fixed V0 = 54.95 Å at 1 bar. A quantitative Schreinemakers' web was obtained at 15–60 GPa and 300–1600 K by combining the EoS for γ′-Fe4N with reported phase stability data at low pressures. The web indicates the existence of an invariant point at 41 GPa and 1000 K where γ′-Fe4N, hexagonal closed-packed (hcp) ε-Fe7N3, double hexagonal closed-packed β-Fe7N3, and hcp Fe phases are stable. From the invariant point, a reaction γ′-Fe4N = β-Fe7N3 + hcp Fe originates toward the high-pressure side, which determines the high-pressure stability of γ′-Fe4N at 56 GPa and 300 K. Therefore, the γ′-Fe4N phase observed in the experiments beyond this pressure must be metastable. The obtained results support the existing idea that β-Fe7N3 would be the most nitrogen-rich iron compound under core conditions. An iron carbonitride Fe7(C,N)3 found as a mantle-derived diamond inclusion implies that β-Fe7N3 and Fe7C3 may form a continuous solid solution in the mantle deeper than 1000 km depth. Diamond formation may be related to the presence of fluids in the mantle, and dehydration reactions of high-pressure hydrous phase D might have supplied free fluids in the mantle at depths greater than 1000 km. As such, the existence of Fe7(C,N)3 in diamond can be an indicator of water transportation to the deep mantle.

Phase Equilibria and Phase Diagrams for the Ternary Aqueous System Containing Lithium, Sodium, and Pentaborate Ions at 298.15 and 323.15 K and 101.325 kPa

Phase equilibria and phase diagrams for the ternary aqueous system containing lithium, sodium, and pentaborate ions at 298.15 and 323.15 K and 101.325 kPa were investigated by the methods of isothermal dissolution equilibrium. From the experimental data, the phase diagrams and the diagrams of physicochemical properties versus composition of lithium pentaborate in the equilibrium systems were plotted, respectively. The phase diagrams of the ternary system LiB5O8 + NaB5O8 + H2O at two temperatures contain one invariant point, two univariant curves, and two crystallization regions corresponding to sodium pentaborate pentahydrate (NaB5O8·5H2O) and lithium pentaborate pentahydrate (LiB5O8·5H2O). Due to the different dissolution behaviors of pentaborate salts in the aqueous systems, the component of LiB5O8 has a relatively strong effect on the solubility of NaB5O8. It was found that this system belongs to a simple eutectic type at two temperatures, and neither double salts nor solid solutions were formed. The densities and refractive indices in the ternary system at 298.15 and 323.15 K are as similar as changing regularly with the increase of LiB5O8 concentration. On the basis of empirical equations of the density and refractive index in electrolytes, the calculated values of density and refractive index agreed well with the experimental values at two temperatures.