Past exam of the chemistry course of the year 4 of the natural sciences course of the University of Cambridge / 2022 / Paper 1 / 2 / b

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.

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.