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Crystal field theory models the effect of surrounding ligand charges or dipoles on metal atomic orbitals. Directional repulsion splits the $d$ levels. Unlike molecular orbital theory, the simplest model does not explicitly construct covalent metal–ligand bonds.

For positive ${\Delta }_o\gg B$, expand the common radical asSubstitution into the stated crystal field theory energies givesThe leading straight-line asymptotes retain the terms linear in ${\Delta }_o$ and constant in $B$. The $B^2/{\Delta }_o$ terms show from which side the interacting $T_{1g}$ levels approach them.

A $3d^9$ shell has one hole in the otherwise filled $d$ shell. Hund's rules give $S=1/2$, $L=2$, and, for a more-than-half-filled shell, $J=L+S=5/2$: the free-ion term is ${}^2{D}_{5/2}$. Its Landé factor isDistinguish the Curie effective moment from a saturated projection:The question then explicitly asks us to use orbital quenching in the crystal. Ligand fields split and mix orbital states, suppressing their freely orientable angular momentum. The spin-only approximation has $g\simeq 2$, $S=1/2$, so ${\mu }_{\mathrm{eff}}=\sqrt{3}{\mu }_B$ but the two field-axis projections are $\pm {\mu }_B$. The latter, not $\sqrt{3}{\mu }_B$, belong in the tanh argument and saturation magnetization.

The sulfate and water environment supplies mainly closed-shell, weakly magnetic surroundings, stabilizes localized Cu2+ sites and separates their moments so that exchange can be weak on the temperature scale considered. Paramagnetism here is an approximation of weakly interacting local spins, not a claim that dilution guarantees no magnetic ordering at any temperature. Elemental copper has a filled $3d$ band and itinerant conduction electrons rather than one localized Cu2+ spin per atom. Pauli spin response and orbital/core contributions then replace a local-spin Curie law; ordinary pure copper has a weak net diamagnetic response.^[ref]

There is one Cu2+ per formula unit, hence $n_s=N_A/V_m=5.5148\times 10^{27}$ m${}^{-3}$. The spin-only weak-field SI volume susceptibility isThese neglect exchange, anisotropic $g$ factors and background diamagnetism. At high field and low temperature, saturation requires the nonlinear response in a; a field-independent Curie value then describes only the zero-field slope.

A theoretical physicist who, with Yukito Tanabe, calculated electronic energy levels of transition-metal complex ions using crystal field theory. Their treatment considered different populations of the d shell in a cubic field. It supplied a framework for interpreting absorption bands and lines, connecting the local environment of an ion with the spectrum used to identify its electronic states.^[ref]

A theoretical physicist who coauthored the electronic-level calculations underlying Tanabe-Sugano diagrams with Satoru Sugano. The work applied crystal field theory to complex ions, balancing the effect of the surrounding ligands with interactions within the d shell. This made optical absorption a way to investigate electronic states in a particular chemical environment.^[ref]