Diamagnetism and Paramagnetism

Two weakly-magnetic classes of materials sit on either side of "nothing": diamagnets ($\chi < 0$, tiny) and paramagnets ($\chi > 0$, small). They differ at the microscopic level by whether atoms have induced or permanent magnetic moments — and paramagnets famously obey Curie's law $\chi \propto 1/T$.

Concept

Microscopic origin.

• Diamagnetism is universal: an applied $\vec B$ induces (by Faraday's law at the atomic level) circulating electron currents whose moment opposes $\vec B$ — a microscopic Lenz's law. In materials without permanent moments, this small effect dominates.
• Paramagnetism arises in atoms with permanent dipole moments (unpaired electron spins or orbital moments). In zero field these are randomly oriented and average to zero; an applied $\vec B$ partially aligns them, giving a small positive $\chi$.

Derivation

Curie's law. Consider $N$ permanent dipoles per unit volume, each with moment $\mu$ in a field $B$ at temperature $T$. The Boltzmann probability of alignment is $\propto e^{-U/k_B T} = e^{\mu B\cos\theta /k_B T}$. In the limit $\mu B \ll k_B T$ (very common at room temperature), the average projected moment is

$\langle \mu_\parallel\rangle \approx \frac{\mu^2 B}{3 k_B T}.$

Magnetisation: $M = N\langle\mu_\parallel\rangle = \frac{N\mu^2}{3 k_B T}\,B \approx \frac{N\mu^2 \mu_0}{3 k_B T}\,H,$

since $B \approx \mu_0 H$ for small $\chi$. Therefore

$\chi = \frac{M}{H} = \frac{\mu_0 N\mu^2}{3 k_B T} \equiv \frac{C}{T},$

with Curie constant $C = \mu_0 N\mu^2/(3 k_B)$.

Why diamagnetism is $T$-independent. Induced currents respond to $\vec B$ on the time-scale of orbital motion; thermal fluctuations of orientation do not enter. Hence $\chi_{\text{dia}}$ has no leading $T$ dependence.

Worked Example

A paramagnetic salt has $\chi = 6.0\times 10^{-4}$ at $T = 300\,K$. What is $\chi$ at $T = 100\,K$?

By Curie's law, $\chi T$ is constant:

$\chi(100) = \chi(300)\cdot\frac{300}{100} = 6.0\times 10^{-4}\times 3 = 1.8\times 10^{-3}.$

The salt becomes three times more susceptible when cooled to a third of the temperature.

Common Confusions

• "Diamagnetic" does not mean "non-magnetic"; $\chi$ is small but negative.
• Every material has some diamagnetism (it is universal), but it is masked by paramagnetism or ferromagnetism when those are present.
• Curie's law only works for paramagnets and only at high enough $T$ (small $\mu B/k_BT$).
• A paramagnet placed near a strong magnet is attracted; a diamagnet is repelled. The repulsion is famously visible in superconductors (perfect diamagnets, $\chi = -1$).

Key Takeaways

• Diamagnets: $\chi < 0$, tiny, $T$-independent, repelled.
• Paramagnets: $\chi > 0$, small, Curie's law $\chi = C/T$, attracted.
• Curie's law follows from Boltzmann statistics in the high-$T$ limit.
• Diamagnetism is universal but masked when stronger effects exist.