Paramagnetism

Paramagnetism is what happens in materials whose atoms carry permanent magnetic moments — unpaired electron spins, incompletely filled d-shells or f-shells — but where those moments don't interact strongly enough with each other to spontaneously align. At zero field, thermal agitation keeps the moments pointing in random directions and the macroscopic magnetization averages to zero. Apply an external field, and each moment feels a torque that tries to align it with the field. The tendency to align is opposed by thermal noise.

The balance between field-alignment energy (−μ·B, lowest when m is parallel to B) and thermal energy (k_B T per mode) determines the equilibrium magnetization. In the weak-field, high-temperature limit where μB ≪ k_B T, statistical mechanics gives Curie's law: χ = C/T, where the Curie constant C = n μ²/(3 k_B), n is the number density of atoms, and μ is the individual atomic moment. The susceptibility falls as temperature rises — heat jostles the moments harder, and the same applied field can align a smaller fraction of them. In the opposite, low-temperature or strong-field limit, the magnetization saturates when all the moments point along the field.

Typical paramagnetic susceptibilities are small but consistently positive: aluminium χ ≈ +2 × 10⁻⁵, platinum χ ≈ +2.6 × 10⁻⁴, oxygen (at STP) χ ≈ +1.9 × 10⁻⁶. The fact that oxygen is paramagnetic while nitrogen is diamagnetic is visible as a classroom demo: liquid-air mixtures spill out of a funnel differently in the gap of a strong magnet, with the oxygen pulling toward the poles. Rare-earth salts — gadolinium, dysprosium, erbium sulfates — are strongly paramagnetic at room temperature and become the working substances of adiabatic-demagnetisation refrigerators used to reach milli-kelvin temperatures.