Two kinds of response to a magnet
In the last guide, colour gave us a direct readout of delta-o — the size of the d-orbital gap. Magnetism gives us a second, independent readout, and a remarkably blunt one: it counts how many electrons in the complex are left unpaired. The trick is that every electron is a tiny magnet, because a spinning charge makes a magnetic field. When two electrons pair up in one orbital, the rules force their spins to point opposite ways, so their two little magnets cancel exactly. Only an unpaired electron leaves a magnet sticking out for the world to feel.
This splits all matter into two camps. A substance whose electrons are all neatly paired is diamagnetic: it has no net electron magnet of its own and is in fact very weakly pushed *out* of a magnetic field. A substance with one or more unpaired electrons is paramagnetic: each unpaired spin is a free magnet that lines up with an applied field and gets pulled *into* it. The pair of behaviours together is what we call diamagnetism and paramagnetism, and the practical point is that paramagnetism is hundreds of times stronger and easy to measure, while diamagnetism is a faint background present in everything.
The classic instrument is the Gouy balance: you hang a sample between the poles of a magnet and simply weigh it. A paramagnetic complex is tugged into the field and appears to gain weight; a diamagnetic one is nudged out and appears to lose a little. From that apparent weight change you back out the strength of the sample's magnetism, and from that strength — as the rest of this guide shows — you can count its unpaired electrons. A balance, a magnet, and some careful bookkeeping let you see straight into the d shell.
The spin-only formula
To turn the measurement into a count, we need a formula linking the magnetic moment to the number of unpaired electrons. For most first-row transition-metal complexes a beautifully simple one works: the **spin-only magnetic moment**. If n is the number of unpaired electrons, the moment is mu = sqrt(n(n+2)), measured in Bohr magnetons (the natural unit of an electron's magnetism). The deeper form writes it as mu = sqrt(4S(S+1)) where S = n/2 is the total spin, but since each unpaired electron adds 1/2 to S, the two are the same thing dressed differently.
Spin-only moment mu = sqrt( n(n+2) ) Bohr magnetons
n unpaired mu (calc) typical ion
--------- --------- ------------------
0 0.00 low-spin d6, e.g. Co3+ in [Co(NH3)6]3+
1 1.73 Ti3+ (d1), Cu2+ (d9)
2 2.83 V3+ (d2), Ni2+ (d8, octahedral)
3 3.87 Cr3+ (d3), Co2+ (low-spin d7? -> 1)
4 4.90 Cr2+ high-spin (d4), Fe2+ low-spin? -> 0
5 5.92 Mn2+ / Fe3+ high-spin (d5)Read the table the way a chemist actually uses it — backwards. You measure a moment in the lab, then ask which n it sits closest to. A salt of [Mn(H2O)6]2+ reads about 5.9 Bohr magnetons, which can only mean five unpaired electrons; a Cr3+ alum reads near 3.8, meaning three; a copper(II) complex sits near 1.7 to 1.9, meaning one. Notice how widely spaced the rungs are — 1.73, 2.83, 3.87, 4.90, 5.92 — so even a rough measurement rarely lets you confuse one n with its neighbour. That spacing is exactly why this crude-looking formula is so trusted in practice.
From a measured moment to oxidation state, geometry, and spin
This is where one cheap number becomes a structural detective. The moment fixes n, the unpaired-electron count; but you reached that count by filling the split d levels from the earlier guides, and the way they split depends on the geometry and on whether the complex went high- or low-spin. So the same measurement, read against what you already know, can pin down all three at once. Watch how it untangles the d6 ion Fe(II), which has two famously different faces.
- Start from the formula. Take [Fe(H2O)6] with chloride counter-ions; the water and chloride bookkeeping says the metal is Fe2+. From its group position, Fe2+ is a d6 ion — six electrons to place in the d shell.
- Measure the moment. The hexaaqua salt reads about 5.2 to 5.4 Bohr magnetons. Compare against the table: that sits near the n = 4 rung (4.90), so four electrons are unpaired.
- Read off the spin state. For d6, four unpaired electrons is only possible by spreading out as t2g^4 eg^2 — the high-spin arrangement. So [Fe(H2O)6]2+ is octahedral, high-spin, with weak-field water giving a small delta-o that loses to the pairing energy.
- Swap the ligand and re-measure. The complex [Fe(CN)6]4- reads essentially zero — diamagnetic. The only way a d6 ion has no unpaired electrons is t2g^6 eg^0: low-spin, the strong-field cyanide opening a delta-o large enough to force all six electrons to pair.
The same number can also distinguish geometry. Nickel(II) is d8, and that is the textbook case. Octahedral [Ni(H2O)6]2+ fills as t2g^6 eg^2 with two unpaired electrons, giving roughly 2.8 to 3.2 Bohr magnetons — paramagnetic. But square-planar Ni(II), as in [Ni(CN)4]2-, stacks all eight electrons into the four lower orbitals and leaves the high dx2-y2 empty, so n = 0 and the complex is diamagnetic. Read the moment and you instantly know which shape you have: a paramagnetic d8 nickel salt is octahedral or tetrahedral, a diamagnetic one is square planar. The magnet has reported the geometry without your ever seeing the molecule.
When spin-only is not the whole story
Be honest about the word "spin-only": it deliberately ignores a second source of magnetism. An electron is not only spinning, it can also be circulating around the nucleus, and that orbital motion is itself a current loop with its own magnetic field. When the d orbitals are arranged so that an electron can hop between them and effectively "orbit," you get an extra orbital contribution to the moment that the simple formula leaves out. This is why measured values often run a touch higher than sqrt(n(n+2)) — for instance, Co2+ complexes that the formula puts at 3.87 are routinely found near 4.3 to 5.2.
There is a satisfying reason the formula still works as well as it does for the first row. The ligands' electric field largely freezes that orbital circulation — chemists say the orbital moment is "quenched" — because once the d orbitals are split into fixed t2g and eg sets pointing in definite directions, an electron can no longer freely circle from one into an equivalent one. The quenching is usually strong but rarely total, so a small orbital leftover survives. For the heavier second- and third-row metals, and especially the f-block lanthanides, orbital effects are large and spin-only is no longer a safe shortcut at all.
When the magnets start talking to each other
Everything so far treats each metal centre as a lone island whose unpaired spins ignore their neighbours — a fine picture for a dilute solution or an isolated complex. But pack paramagnetic ions into an extended solid, close enough that their d orbitals couple through bridging atoms, and the little magnets stop acting independently. They begin to influence one another's alignment, and the whole solid develops a collective magnetic personality. This is cooperative magnetism, and it is a property of the lattice, not of any single ion.
Two main patterns emerge. In ferromagnetism, the coupling makes neighbouring spins want to point the *same* way; below a critical temperature they lock into giant aligned domains and the material becomes a permanent magnet — this is what iron, cobalt, and nickel metal do, and it is far stronger than ordinary paramagnetism because billions of spins reinforce. In antiferromagnetism, the coupling makes neighbours prefer to point *opposite* ways; the spins cancel in a checkerboard and the solid shows almost no net magnetism even though every ion is paramagnetic. Manganese(II) oxide, MnO, is the classic antiferromagnet.
There is a third twist worth a nod: in ferrimagnetism, neighbours point opposite ways like an antiferromagnet, but the two sets carry unequal moments, so they do not fully cancel and a net magnetism survives — this is the secret of magnetite, Fe3O4, the lodestone that first revealed magnetism to humankind, where Fe2+ and Fe3+ on different lattice sites pull against each other unevenly. The thread connecting all of these back to our balance: the very same unpaired d electrons you learned to count in a single complex are what, multiplied across a crystal and made to talk to one another, give us refrigerator magnets, compass needles, and the read heads of every hard drive.