A moment between two integers

A moment counts electrons, not orbitals starts from a genuinely striking fact. Most chemical measurements return real numbers that have to be interpreted; a magnetic moment returns something close to an integer’s worth. Feed the count of unpaired electrons into √(n(n+2)) and nine first-row ions come out within a tenth of a Bohr magneton, several of them within a hundredth.

That essay is careful to say where the agreement fails and why — the orbital contribution, which the g-value is the orbital coming back then computes. This one is about a different failure, which is not a correction to the number but a change in what kind of thing the number is.

The integer is not being disputed. What is being disputed is the assumption that makes it an integer, which is never stated because it is usually true.

The condition the integer needs #

The count works when one state lies far below every other. Then the sample is entirely in that state, the moment is that state’s, and the count of its unpaired electrons is a property of it.

Nothing guarantees that condition. The pairing energy decides the moment is the essay about the competition that fixes which state is lower — the ligand field splitting against the cost of pairing two electrons in one orbital — and it finds the crossover at exactly Δ = P for all eight configurations that have a choice.

A crossover is a place where two states are equal in energy. Approaching it from either side means two states within a few hundred wavenumbers of each other, and a few hundred wavenumbers is a few times kT at room temperature.

What a susceptibility measurement actually averages #

The quantity measured is a magnetic susceptibility, and a susceptibility is a sum over the states of the sample weighted by their populations. What the sum contains is the square of each state’s moment, because that is what a susceptibility is built from, so the effective moment reported is

$$\mu_{\text{eff}}(T) = \sqrt{\frac{\sum_i g_i\,\mu_i^{2}\,e^{-E_i/kT}}{\sum_i g_i\,e^{-E_i/kT}}}$$

with gᵢ the degeneracy of each state. It is not an average of the moments; it is the root of an average of their squares, which is a different number and always the larger one.

For iron(II) the two candidates are as far apart as any pair can be: low spin has all six electrons paired and no moment at all, and high spin has four unpaired and a moment of 4.90. So the average has the widest possible range to move over.

The curve #

At 80 K the moment is 0.300 — the sample is almost entirely low spin. At 400 K it is 3.609. At 200 K it is 2.296, and the nearest integer’s value is 2.83, which is more than half a Bohr magneton away.

There is no count of unpaired electrons that reproduces 2.296, and more importantly there is none that reproduces the variation: a count is a fixed number and the measurement moves by more than three Bohr magnetons across a range of temperature any laboratory can reach.

The number is a temperature, read the other way #

The most useful reading of the curve is backwards, and it is the reading that makes the effect a tool rather than a nuisance.

Given the two moments — zero and 4.90, both known from the counts — and given the degeneracies, one measured moment determines the population ratio, and the population ratio at a known temperature determines the gap. So a single susceptibility measurement on a crossover compound is a thermometer for the gap, and a series of them across temperature over-determines it.

That turns a quantity a one-electron model cannot compute into one that can be measured to a few wavenumbers with a magnet and a cryostat. It is the same trade the electrons repel less inside the complex makes for the interelectron repulsion — two measured bands determine two parameters in closed form — and it is the usual way a ligand field problem gets its numbers.

The degeneracies do half the work #

The five to one is the part that is easy to leave out and it is half the physics.

A spin-2 state has five orientations and a spin-0 state has one. So the Boltzmann sum weights the high-spin form by five before any exponential is applied, and five states at 400 wavenumbers above one state are already equally populated at about 250 K — where the exponential alone would give them a sixth of the population.

In the limit of high temperature the exponentials go to one and the populations are five to one regardless of the gap. So the higher state wins at high temperature, always, whenever it is the more degenerate — which for a spin crossover it always is, because more unpaired electrons means more orientations.

That is the entropy term in a free energy, arriving as a count of states, and it is why a spin crossover is a phenomenon at all rather than a fixed ground state with an unpopulated excited one above it.

One consequence of that is a limit worth stating in advance: the maximum moment a crossover compound can show is not the high-spin value. At any finite temperature some of the sample is low spin, so the average is below 4.90 always, and the curve approaches it only as the population ratio approaches its high-temperature limit of five to one — which caps the moment at √(5/6) × 4.90 = 4.47 for this pair, before any other state is considered.

That is a testable statement and a strong one: a compound whose moment saturates well below its high-spin value at high temperature has not failed to convert, it has converted as far as the degeneracies permit.

The gap is the thing a chemist changes #

The gap is a difference between the ligand field splitting and the pairing energy, and the splitting is what a ligand controls. The spectrochemical series is not electrostatics is this collection’s essay about the order of ligands by how hard they split a d shell, and moving a complex along that series by one step moves its gap by more than the range in the figure.

So a chemist choosing ligands is choosing where on the temperature axis the transition sits, and the compounds that are useful — the ones that switch in a convenient range — are the ones whose gap has been tuned to a few hundred wavenumbers. That is a narrow target, which is why they are rare and why finding them is a research programme rather than a table lookup.

Why the square matters #

The formula averages the squares of the moments rather than the moments themselves, and the choice is not a convention — getting it wrong changes the answer in a direction that is easy to check.

A susceptibility is the response of a sample’s magnetisation to a field, and for a state of spin S that response goes as S(S+1), which is the moment squared. Averaging responses over a population is therefore averaging squares, and the effective moment quoted is the root of the average.

The consequence is that the high-spin form counts for more than its population share. At 200 K it is 22 per cent of the sample and the moment is 2.296 — which is 47 per cent of the high-spin value, not 22 per cent. A minority of strongly magnetic molecules dominates a susceptibility, and any reading that treats the moment as a linear average will overestimate how much has converted, by roughly a factor of two in the middle of the transition.

That is worth having explicitly, because the population curve and the moment curve in the figures above look alike and are not the same shape.

Where the crossover exists at all #

Only four of the ten d configurations have a spin choice in an octahedral field, and the pairing energy decides the moment counts them: d⁴ to d⁷, exactly. Everything else has one arrangement and no crossover to sit at.

The splitting the whole competition is measured against is computed from a reduction rather than taken from a table, and the pairing energy it competes with is a property of the free ion. Where those two are comparable, both spin states are populated at ordinary temperatures — which is the whole of the condition for the moment to sit between two integers.

Within those four, d⁶ is the one this essay uses because it has the widest range: zero unpaired against four, so the moment runs from 0 to 4.90. A d⁷ crossover runs from 1.73 to 3.87, which is a smaller range and a smaller signal, and the same arithmetic applies to it with everything compressed.

That is also why iron(II) is the element the whole subject of spin crossover is built on. It is not that iron is special chemically; it is that d⁶ is the configuration where the two candidates differ by the most.

What survives, and it is the formula rather than the reading #

Nothing above says the spin-only formula is wrong. It is exact for a state of given spin, and every value it feeds into the average is one of its own outputs.

What fails is the inference: measuring a moment and reading a count off it. That inference has an unstated premise — that the sample is in one state — and the premise is true for most complexes and false for the interesting ones.

The check that separates the two cases is not a better measurement at one temperature. It is a measurement at two. A moment that is the same at 100 K and at 300 K is a count; a moment that moves is an average, and the amount it moves says how far apart the states are.

The thermodynamics, in one paragraph #

The whole of the temperature dependence is a competition between an energy and a count of states, and it is worth writing it that way because it is the general form of a great many chemical equilibria.

The low-spin state is lower in energy by the gap and has one arrangement. The high-spin state is higher and has five. Which dominates is decided by the free energy — the energy minus the temperature times the entropy — and the entropy difference is the logarithm of the ratio of the counts. At the temperature where the gap equals kT ln 5, the two are equally populated, and above it the more numerous state wins.

For a gap of 400 wavenumbers that temperature is about 360 kelvin. The half-way point in the population figure is near 380, which is that estimate with the correction for the two states not being sharply defined.

So a spin crossover is an ordinary entropy-driven transition and the entropy is not a vague one: it is a count of five against one, available by inspection from the spins.

Two ways for a moment not to be a count #

Both are now on the table and they should not be confused.

The orbital contribution is a correction to the moment of a single state. It arises when the ligand field leaves the orbital angular momentum unquenched, it is largest for cobalt(II) at 4.80 against a spin-only 3.87, and it is computed in an orbital carries no angular momentum and the g-value is the orbital coming back. The moment is still a property of one state and still temperature-independent to first order; it just is not √(n(n+2)).

The thermal average is not a property of any state. The sample contains several and the number reported is a weighted mean over them, which moves when the weights move.

Distinguishing them experimentally is straightforward once the distinction is drawn — one is temperature-independent and the other is not — and confusing them means attributing a temperature dependence to an orbital effect that has none, or a large moment to a count that never existed.

The same shape of argument, elsewhere in this field #

Two other quantities in this field are averages over populated states rather than properties of one, and gathering them makes the pattern easier to spot.

The double hump and what removes it is about hydration enthalpies across the first transition series, where the deviation from a smooth line is the ligand field stabilisation — and that quantity, too, depends on which spin state is occupied, so a compound near a crossover contributes an average.

Where a d–d band falls is about the colour, and the same caution applies: a sample with two spin states populated has two sets of transitions and its spectrum is a superposition weighted by the populations, which changes with temperature.

The general form is that any property computed for a state becomes, for a real sample, a Boltzmann average of that property over the states available. Where one state dominates, the average is the property and nobody notices. Where two states are within a few times kT, the average is a new quantity with a temperature in it, and it should not be given the name of the property.

What makes a real crossover sharp #

The curve computed here is smooth because the molecules are independent, and real spin-crossover solids are often not smooth at all: the conversion can happen over a few kelvin, and it can happen at a different temperature going up than coming down. What supplies the sharpness is a structural quantity, and naming it says exactly what an isolated-molecule calculation is missing.

The two spin states are not the same size. In an iron(II) complex the low-spin state puts all six electrons in the lower set, which points away from the ligands, and the metal–ligand distance is about 1.97 ångström. The high-spin state puts two of them in the upper set, which points at the ligands, and the distance grows to about 2.17 — two tenths of an ångström, and a molecular volume several per cent larger.

In a solid that expansion has to be accommodated by the neighbours. A molecule that converts pushes on the ones around it, which makes their own conversion easier — so the transitions stop being independent and start being cooperative, and a cooperative transition is sharp rather than gradual. Push the coupling far enough and the material has two stable states over a range of temperature, which is what a hysteresis is.

The prediction that follows is checkable and holds. Dilute the compound and the sharpness goes. A spin-crossover complex in solution, or diluted into a host lattice of a non-converting analogue, gives a smooth gradual curve of exactly the shape computed here; the same compound as a pure solid, and especially one in which the molecules are linked into a network, gives an abrupt transition with hysteresis.

So the smooth curve is not the wrong answer. It is the answer for one molecule, it is what the measurement returns when the molecules are kept apart, and the deviation from it in a solid is a measure of how strongly they are coupled — through a lattice rather than through anything magnetic.

What is left #

The model here has two states and one gap. A real complex has more: the ligand field levels above the ground state, the spin–orbit components below each of them, and in a solid the neighbouring molecules, which interact and can make the transition abrupt rather than gradual. Everything in the figures is the smooth, isolated-molecule case, and a real spin-crossover compound often shows something much sharper — sometimes with a hysteresis, which no calculation on one molecule can produce.

The moments fed into the average are the spin-only values, so the orbital contribution above is left out of both states. Including it would raise the high-spin value above 4.90 and leave the low-spin one at zero, so the curve would end higher and would keep its shape.

And nothing here computes the gap. It is a stated parameter, as the pairing energy is throughout this field, because the pairing energy is a two-electron quantity and a ligand-field model is a one-electron model. What is computed is what a given gap does, which is the half of the problem that is arithmetic — and the smallest many-electron calculation is where a pairing energy can be computed rather than quoted.