Past exam of the chemistry course of the year 4 of the natural sciences course of the University of Cambridge / 2022 / Paper 2 / Section B / 21 / e

Define $\mathcal{H}=-2J{\sum }_{⟨ij⟩}{\mathbf{S}}_i\cdot {\mathbf{S}}_j$. Each site has $z=6$ exchange-connected neighbours. In mean-field theory, replacing neighbours by their mean spin produces the effective field that yieldsThusfor Mn, Fe, Co and Ni. Negative $J$ means antiferromagnetic exchange in this convention; a Hamiltonian written with $-J$ instead of $-2J$ would have twice these numerical values. The increasingly negative values indicate a growing effective exchange energy, although different orbital occupancy and anisotropy mean that they are not a simple universal function of $d$-electron number.

A three-dimensional six-connected framework is more favourable to mean-field reasoning than isolated chains, but the theory neglects short-range correlations, anisotropic exchange and fluctuations. The ratios $|\Theta |/T_C\simeq 1.65,1.69,3.75,2.10$ show substantial departures from the simplest unfrustrated isotropic mean-field relation. The values are therefore effective high-temperature exchange estimates, not a complete microscopic fit to the ordered phases.