A moment that is not an integer's worth of anything

One of the figures on a d shell in a field: What a set of ligands does to five degenerate orbitals — computed twice, from an integrated point-charge potential and from an angular overlap matrix, which agree on every ratio.

Two essays draw this figure, each at the values its own argument needs rather than at the setting shown above. What each one uses it to show is below, in the words of its own caption.

A moment between two integers

What fraction of the sample is in each state at a gap of eight hundred wavenumbers rather than four. The populations cross at twice the temperature and the moment follows them, so the whole curve is a Boltzmann factor read through a moment — and the crossing this essay lives near is a crossing of populations rather than of energies.

The effective moment of an iron(II) complex whose two spin states lie 400 wavenumbers apart, from 80 K to 400 K, with the values belonging to whole numbers of unpaired electrons drawn across. The curve starts near zero, ends above three and a half, and passes each of the integer values without stopping.

The same complex with the two states twice as far apart. The curve keeps its shape and moves along the temperature axis: at a gap of 800 wavenumbers the moment is still under one Bohr magneton at 200 K, and a measurement at room temperature would report a low-spin compound.

The model is what is fitted

A magnetic moment that is a thermal average and therefore a real number moving with temperature. Every quantity in this essay is of the same kind — a Boltzmann sum over a spectrum — and the fitting problem is always the same one: a smooth function of temperature, and more than one model that can follow it.