Structural features of the oxidonitridophosphates K3 M III(PO3)3N (M III = Al, Ga)

Cubic crystals of tripotassium aluminium (or gallium) nitridotriphosphate, K3 M III(PO3)3N (M III = Al, Ga), were grown by application of the self-flux method. In their isostructural crystal structures, all metal cations and the N atom occupy special positions with site symmetry 3, while the P and O atoms are situated in general positions. The three-dimensional framework of these oxidonitridophosphates is built up from [M IIIO6] octahedra linked together via (PO3)3N groups. The latter are formed from three PO3N tetrahedra sharing a common N atom. The coordination environments of the three potassium cations are represented by two types of polyhedra, viz. KO9 for one and KO9N for the other two cations. An unusual tetradentate type of coordination for the latter potassium cations by the (PO3)3N6– anion is observed. These K3 M III(PO3)3N (M III = Al, Ga) compounds are isostructural with the Na3 M III(PO3)3N (M III = Al, V, Ti) compounds.

Mathematical modeling of the elastic properties of cubic crystals at small scales based on the Toupin–Mindlin anisotropic first strain gradient elasticity

In this work, a mathematical modeling of the elastic properties of cubic crystals with centrosymmetry at small scales by means of the Toupin–Mindlin anisotropic first strain gradient elasticity theory is presented. In this framework, two constitutive tensors are involved, a constitutive tensor of fourth-rank of the elastic constants and a constitutive tensor of sixth-rank of the gradient-elastic constants. First, $$3+11$$ 3 + 11 material parameters (3 elastic and 11 gradient-elastic constants), 3 characteristic lengths and $$1+6$$ 1 + 6 isotropy conditions are derived. The 11 gradient-elastic constants are given in terms of the 11 gradient-elastic constants in Voigt notation. Second, the numerical values of the obtained quantities are computed for four representative cubic materials, namely aluminum (Al), copper (Cu), iron (Fe) and tungsten (W) using an interatomic potential (MEAM). The positive definiteness of the strain energy density is examined leading to 3 necessary and sufficient conditions for the elastic constants and 7 ones for the gradient-elastic constants in Voigt notation. Moreover, 5 lattice relations as well as 8 generalized Cauchy relations for the gradient-elastic constants are derived. Furthermore, using the normalized Voigt notation, a tensor equivalent matrix representation of the two constitutive tensors is given. A generalization of the Voigt average toward the sixth-rank constitutive tensor of the gradient-elastic constants is given in order to determine isotropic gradient-elastic constants. In addition, Mindlin’s isotropic first strain gradient elasticity theory is also considered offering through comparisons a deeper understanding of the influence of the anisotropy in a crystal as well as the increased complexity of the mathematical modeling.

The Refraction Indices and Brewster Law in Stressed Isotropic Materials and Cubic Crystals

We study the elasto-optic behavior of stressed cubic crystals (all classes) and isotropic materials (like e.g., glasses). We obtain the explicit dependence of the refraction indices on the stress (either applied or residual), as well as a mild generalization of the Brewster law for cubic crystals. We show also that the optic indicatrix and the stress ellipsoid are coaxial only in the isotropic case. This theory allows the improvement of the measurement techniques, as photoelasticity, on cubic crystals and optically isotropic materials.

Magnetic skyrmion braids

Skyrmions are vortex-like spin textures that form strings in magnetic crystals. Due to the analogy to elastic strings, skyrmion strings are naturally expected to braid and form complex three-dimensional patterns, but this phenomenon has not been explored yet. We found that skyrmion strings can form braids in cubic crystals of chiral magnets. This finding is confirmed by direct observations of skyrmion braids in B20-type FeGe using transmission electron microscopy. The theoretical analysis predicts that the discovered phenomenon is general for a wide family of chiral magnets. These findings have important implications for skyrmionics and propose a solid-state framework for applications of the mathematical theory of braids.