Adiabatic demagnetisation

Adiabatic demagnetisation, or magnetic refrigeration, cools a paramagnetic salt by exploiting the entropy of its spins. The key fact is that the spin entropy depends on the applied field B and the temperature T only through their ratio: S/R = ln(2 cosh x) − x tanh x with x = μB/(k_B T). At small x thermal jostling wins and the spins are disordered, carrying R ln 2 per mole — one bit each. At large x the field wins, the spins align, and the entropy falls toward zero.

The cycle has three steps. First, magnetise isothermally: with the salt still touching a pumped-helium bath near 1 K, switch on a strong field; the spins align, their entropy drops, and the difference leaves as heat into the bath, with no change in temperature. Second, decouple: break the thermal link, so that entropy now has nowhere to go. Third, demagnetise adiabatically: turn the field down at constant entropy. Since S depends only on B/T, holding S fixed holds B/T fixed, so T_f = T_i · B_f/B_i — the temperature falls in exactly the proportion the field does. The spins re-randomise, and the entropy they reclaim is paid for out of the lattice vibrations.

Setting B_f = 0 in that formula appears to deliver absolute zero in a single step, and it is the cleanest illustration of why the third law is needed. No sample sits in zero field: each spin feels the residual dipole field of its neighbours, a small b_res that no external coil cancels. The honest result, T_f = T_i √(B_f² + b_res²)/√(B_i² + b_res²), is bounded strictly above zero however far the magnet is turned down. Repeating the cycle from the new floor yields a smaller drop, and then a smaller one — a staircase with infinitely many steps converging on a temperature that is not zero.

Peter Debye proposed the method in 1926 and William Giauque independently in 1927. Giauque built it at Berkeley and reached 0.25 K in 1933, breaking a decade-long deadlock: evaporative cooling of liquid helium had stalled near 0.83 K when the vapour pressure collapsed and there was nothing left to pump away. He was awarded the 1949 Nobel Prize in Chemistry. Successive demagnetisation stages later reached the millikelvin range, and nuclear demagnetisation — using the far smaller nuclear moments — reaches microkelvin. The technique was largely superseded for continuous operation by the ³He/⁴He dilution refrigerator in the 1960s, but survives in space missions, where it needs no gravity and no pumped liquids.