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An $n$-gonal antiprism is a polyhedron with two parallel congruent $n$-gon bases, one rotated relative to the other, and a band of $2n$ triangular side faces joining them. For the usual convex regular realization, both bases are regular and all side triangles are equilateral triangles.

Its combinatorial counts are $V=2n$, $E=4n$, and $F=2n+2$: the bases contribute $2n$ edges and the zigzag side band another $2n$. Hence $V-E+F=2$, agreeing with Euler characteristic. A triangular antiprism is an octahedron; a square antiprism has eight vertices, two square bases and eight triangular side faces.

A prism instead joins corresponding base vertices with a band of quadrilateral faces. The relative rotation and triangular band are the defining differences in the antiprism construction.

Place equal masses at the six vertices $\pm {\mathbf{e}}_i$ of a regular octahedron and identical unstressed axial springs on its twelve edges. For the invariant axial displacement family ${\mathbf{u}}_{\pm i}=\pm q_i{\mathbf{e}}_i$, the quadratic potential and kinetic energies areHence $m\ddot{\mathbf{q}}=-k(I+11^{\mathsf{T}})\mathbf{q}$. The breathing vector $1=(1,1,1)$ has squared frequency $4k/m$, while every traceless axial vector with $q_1+q_2+q_3=0$ has squared frequency $k/m$. Orthogonal projection onto these subspaces solves any initial condition in this family without an eighteen-dimensional characteristic polynomial.

A cuboctahedron is a convex uniform polyhedron with eight equilateral triangular faces and six square faces. Around each of its twelve vertices the faces alternate triangle, square, triangle, square. It has twenty-four edges, and the Euler characteristic check is $12-24+14=2$.

A concrete model is the convex hull of all coordinate permutations of $(\pm 1,\pm 1,0)$. The six planes $x=\pm 1$, $y=\pm 1$, $z=\pm 1$ cut out the square faces; the eight planes $\pm x\pm y\pm z=2$ cut out the triangular faces. All edges have length $\sqrt{2}$. Equivalently, join the midpoints of neighbouring edges of a cube: its vertices become triangular faces, and its faces become squares. The construction makes the name and the shared symmetries with the cube and octahedron visible.

An octahedron is a polyhedron with eight faces. The regular octahedron has six vertices, twelve equal edges and eight equilateral triangular faces. One realization has vertices $\pm {\mathbf{e}}_1,\pm {\mathbf{e}}_2,\pm {\mathbf{e}}_3$; two vertices are adjacent exactly when they are not an opposite pair. Its edge graph is the octahedral graph.

Formal ionic counting gives La(III), N(−III), and therefore Re(VI), with a $5d^1$ configuration. An isolated localized $d^1$ center would carry a saturation magnetic moment of order $1{\mu }_{\mathrm{B}}$, much larger than the value quoted. It is not a $d^0$ ion made diamagnetic simply by an octahedral ligand field.

The spatially extended Re $5d$ orbitals overlap strongly with N $2p$ orbitals. Re–N $\sigma$ interactions raise the $e_g$-derived antibonding states, while $\pi$ interactions mix the $t_{2g}$ states with the nitrogen network. Repetition through corner-sharing octahedra turns these combinations into broad energy bands. The nominal $d$ electron occupies a delocalized, partially filled band rather than an isolated magnetic orbital.

Only electrons close to the Fermi energy change their occupations appreciably in an applied magnetic field. Consequently a Pauli paramagnetism has a small, nonzero field-induced magnetization:when $g$ is the density of states including both spin directions, expressed per formula unit and per unit energy. This band model explains a very small moment without requiring the absence of electrons in the Re-derived band. The reported material has metallic transport and a Pauli-like response.^[ref]

The description “saturated” in the question should not be interpreted as complete alignment of one localized spin per Re: a nearly linear metallic response over a laboratory field range need not be a true fully spin-polarized saturation moment.

An ideal cubic perovskite structure is obtained by putting La at the cube corners, Re at the body center, and N at the face centers. In fractional crystal coordinates one convenient basis isThe counting is $8/8=1$ La, one Re and $6/2=3$ N per cell. Corner-sharing $\mathrm{Re}{\mathrm{N}}_6$ octahedra form the network, with La in the larger interstices. The ideal cell illustrates the connectivity; octahedral tilting can lower the actual crystal symmetry.

Three suitable examples are cubes bounded mainly by $\{100\}$ facets, octahedra bounded by $\{111\}$ facets, and truncated octahedra bounded by both $\{111\}$ and $\{100\}$ facets. These are crystallographically distinct shapes rather than three arbitrary rounded particle outlines.

At fixed volume the thermodynamic equilibrium shape minimizes $G_{\mathrm{surf}}={\sum }_i{\gamma }_i{A}_i$, with an edge/corner correction when particles are very small. In the Wulff construction, facet support-plane distances scale as their interfacial free energies. Silver's clean low-index energies, selective adsorption of ligands or ions, solvent chemical potential and possible support contact alter those energies. Crystal twinning and elastic strain can add further costs or stabilize multiply twinned forms. A face with slow attachment kinetics can also survive during growth even when that morphology is not the thermodynamic equilibrium Wulff shape. Homogeneous nucleation describes where nuclei form; it does not guarantee an thermodynamic equilibrium final shape.

The Archimedean solid obtained by truncating an octahedron, with eight hexagonal faces and six square faces. In an fcc nanocrystal the analogous shape can expose both $\{111\}$ and $\{100\}$ facets; the relative facet areas need not equal those of the regular Archimedean solid.