Magnetic moments, diamagnetism and paramagnetism

Each unpaired electron is a tiny compass needle

Intuition

A spinning charge acts like a small magnet. An electron by itself is a tiny compass needle. If electrons come in pairs that point opposite ways, the needles cancel. Magnetism in a material is about whether those needles cancel or not.

The electrons and nuclei of an atom or ion can have net spin, and so a magnetic moment. The magnetic moment is inversely proportional to the mass of the particle. Nuclear moments are very small next to those of unpaired electrons, by a factor of about 1800. The slide gives the Bohr magneton, $\mu_B = 9.27 \times 10^{-24}\ \text{A m}^2$, and the nuclear magneton, $\mu_N = 5.05 \times 10^{-27}\ \text{A m}^2$.

Electrons have moments linked to their spin quantum number, $s = \pm\tfrac{1}{2}$. The total spin $S$ of an atom or ion is the sum of the spin quantum numbers of the outer shell electrons. The lecture works Cr²⁺ (d⁴) in two ligand fields.

Try it: worked example

Find the total spin S of Cr2+ (d4) with weak-field ligands (high spin) and with strong-field ligands (low spin).

Show answer and steps

AnswerHigh spin S = 2. Low spin S = 1.

1. Weak field, $[\text{Cr}(\text{H}_2\text{O})_6]^{2+}$: the splitting $\Delta$ is small, so all four electrons stay unpaired with the same spin.
2. $S = 4\left(+\tfrac{1}{2}\right) = 2$.
3. Strong field, $[\text{Cr}(\text{CN})_6]^{4-}$: $\Delta$ is large, so the $t_{2g}$ level holds a paired electron and two unpaired ones. That is three up and one down.
4. $S = 3\left(+\tfrac{1}{2}\right) + \left(-\tfrac{1}{2}\right) = 1$.

Exam detail

Online Test 1 lists “key parameters in the measurement of dielectric and magnetic materials”. For magnetism, the lecture defines magnetisation $M$, susceptibility $\chi$, field $H$, the sign and temperature behaviour of $\chi$, and (next page) the Curie and Néel temperatures and the hysteresis loop. The slide does not name them as the tested parameters, so know all of them.

Susceptibility tells you how a material answers a magnetic field

A magnetic material is made of many magnetic atoms. The magnetic moment per unit volume is the magnetisation, $M$. In a magnetic field $H$ it follows

$\mathbf{M} = \chi \mathbf{H}$

where $\chi$ is the magnetic susceptibility. Magnetic materials are characterised by measuring magnetisation against applied field, and susceptibility against temperature.

Diamagnets all repel, paramagnets lean in and weaken when hot

Compare

Diamagnetism

Every material has it. Paired electrons are polarised by the field. $\chi < 0$, a repulsive force. No magnetisation at zero field. $\chi$ is constant, independent of $T$.

Paramagnetism

Needs unpaired electrons, which the field aligns. $\chi > 0$, an attractive force. No magnetisation at zero field. Follows Curie law, $\chi = C_{\text{Curie}}/T$.

Diamagnetic examples from the lecture table (susceptibility in units of $10^{-5}$ SI): superconductor −105, pyrolytic carbon −40.9, bismuth −16.6, copper −1.0 and water −0.91. Superconductors repel a magnetic field strongly. Most electrons in living things are paired, so living things are diamagnetic.

For a paramagnet, $\chi = C_{\text{Curie}}/T$. Plain words: heating shakes the electron spins around and works against the field aligning them, so the response weakens as $T$ rises. The lecture phrases it as thermal energy overcoming the energy gap between states. On the slide, $\chi$ against $T$ is a hyperbola, and $\chi^{-1}$ against $T$ is a straight line through the origin.

Pitfall

Do not mix up the signs. Diamagnetic $\chi$ is negative and constant. Paramagnetic $\chi$ is positive and falls as $1/T$. Both give no magnetisation at zero field, so zero-field magnetisation points to ferromagnetism instead (next page).

Recall

Why are all materials at least a little diamagnetic?

Paired electrons are slightly polarised by an applied field in a way that opposes it. This weak effect exists whether or not other magnetism is present.