Magnetic Properties of Transition Metals

A substance is called paramagnetic if it is weakly attracted into an external magnetic field, and diamagnetic if it is weakly repelled by one. For transition metal ions, this behaviour is governed directly by the number of unpaired electrons in the $d$ subshell: an ion with one or more unpaired $d$ electrons is paramagnetic, while an ion whose $d$ electrons are all paired up (either because the subshell is completely filled, $d^{10}$, or completely empty, $d^0$) is diamagnetic.

The spin-only formula. For most first-row transition metal ions, the orbital contribution to the overall magnetic moment is small enough to be neglected to a good approximation, and the observed magnetic moment can be estimated using the spin-only formula,

where $n$ is the number of unpaired electrons and the result is expressed in Bohr magnetons (BM), the standard unit for atomic-scale magnetic moments. Because $\mu$ increases monotonically with $n$, a measured magnetic moment can be used, in reverse, to determine how many unpaired electrons a given ion actually has -- which is one of the most direct experimental ways of confirming a proposed electron configuration.

Worked values. For a $d^1$ ion (one unpaired electron), $\mu = \sqrt{1 \times 3} = \sqrt{3} \approx 1.73$ BM. For $d^2$, $\mu = \sqrt{2 \times 4} = \sqrt{8} \approx 2.83$ BM. For $d^3$, $\mu = \sqrt{3 \times 5} = \sqrt{15} \approx 3.87$ BM. For $d^4$ (high-spin, four unpaired electrons), $\mu = \sqrt{4 \times 6} = \sqrt{24} \approx 4.90$ BM. For $d^5$ (high-spin, all five electrons unpaired -- the maximum possible for a single $d$ subshell), $\mu = \sqrt{5 \times 7} = \sqrt{35} \approx 5.92$ BM, the largest magnetic moment shown by any first-row transition metal ion.

Free-ion (high-spin) unpaired-electron counts follow directly from Hund's rule: electrons occupy the five $d$ orbitals singly, with parallel spins, before any pairing occurs, so a free $d^n$ ion (for $n \le 5$) has $n$ unpaired electrons, while a free $d^n$ ion for $n > 5$ has $(10-n)$ unpaired electrons, since the sixth through tenth electrons must each pair up with one of the first five. …