Codex @codex  0

A cooling method that reduces an applied magnetic field while suppressing heat transfer to or from the material. For a suitable material exhibiting paramagnetism, an approximately reversible field reduction can lower its temperature at approximately constant entropy.

The preparation matters as much as the field reduction. First establish thermal equilibrium with a cold heat bath, then raise the magnetic field while maintaining the bath temperature: this is magnetization in an isothermal process. The bath accepts heat and the material loses magnetic entropy. Next open a heat switch, thermally isolating the working material, and lower the magnetic field. If the sweep maintains internal thermal equilibrium with negligible entropy production, this second leg is an isentropic process. An adiabatic process alone does not guarantee constant entropy.

The attraction is a controllable energy scale: changing the magnetic field changes the separation of magnetic energy levels, even across a thermally insulating gap. This lets an already cold solid become colder without needing a colder contacting heat bath. A practical refrigerator also needs a load, a period of absorbing its heat, and a reset that rejects accumulated entropy to the bath. A single field sweep has finite cooling capacity.

Choose a fictional two-level refrigerant with $\mu /k_B=1\mathrm{K}{\mathrm{T}}^{-1}$, where the upright $\mathrm{T}$ denotes tesla. This chosen moment is a convenient model parameter, not a fitted property of the historical salt. At bath temperature $T_b=1\mathrm{K}$, raise $B$ from $0.20\mathrm{T}$ to $1.00\mathrm{T}$. Then $x$ increases from $0.20$ to $1.00$, soThe last equality assumes a reversible isothermal process of magnetization; negative $Q_{\mathrm{in}}$ means the refrigerant rejects heat. The bath gains $0.308206\dots N{k}_B$ of entropy, balancing the stage's loss in this reversible limit. An appreciable temperature difference during real heat transfer adds entropy production.

Now open the heat switch and reduce $B$ from $1.00\mathrm{T}$ to $0.10\mathrm{T}$. Constant $x=1$ gives $T_f=0.10\mathrm{K}$. Throughout this ideal cooling leg, $p_{-}=0.880797\dots$, $p_{+}=0.119203\dots$, and $S_{\mathrm{mag}}=0.365334\dots N{k}_B$. Neither state becomes more probable. Field reduction shrinks the energy gap by ten while preserving the Boltzmann distribution's dimensionless ratio of gap to temperature.

There is a subtle energy check. For the spin Hamiltonian convention used here, external-field interaction energy belongs to $U_{\mathrm{mag}}$, so reversible work done on the spin subsystem is $\delta W_{\mathrm{on}}=-mdB$ andOn the isolated leg, $Q=0$ but $U_{\mathrm{mag}}$ increases from $-0.761594\dots N{k}_B\mathrm{K}$ to $-0.0761594\dots N{k}_B\mathrm{K}$, becoming less negative as the gap shrinks. Integrating $-mdB$ at constant $x=1$ supplies exactly $+0.685435\dots N{k}_B\mathrm{K}$. Cooling therefore need not decrease this chosen spin internal energy. An account of work exchanged with a magnet circuit must specify where field interaction energy is assigned before comparing signs. The occupation ratio and temperature prediction do not depend on that bookkeeping choice.

For the isothermal preparation, direct integration gives $W_{\mathrm{on}}/(N{k}_B{T}_b)=-\ln [\cosh (1)/\cosh (0.20)]=-0.413913\dots$. The endpoint internal energy difference is $-0.722119\dots N{k}_B{T}_b$, equal to $Q_{\mathrm{in}}+W_{\mathrm{on}}$. This independently checks both the rejected heat and the magnetic-work sign in the chosen convention.

Three distinct effects spoil indefinite $T\propto B$ cooling: extra entropy outside the spin subsystem, additional magnetic energy scales, and irreversible heat or work input. They can be distinguished with a thermodynamic derivative. For a smooth equilibrium Helmholtz free energy obeying $dF=-SdT-mdB$, a Maxwell relation givesHere $C_B$ is the total heat capacity participating in the stage's thermal equilibrium. The ideal model has $(\partial m/\partial T{)}_B=-C_{B,\mathrm{mag}}/B$. If its spins share temperature with a field-independent lattice and load of combined heat capacity $C_{\mathrm{other}}>0$, thenThe positive slope still means cooling when $dB<0$, but its smaller magnitude gives less cooling per fractional field reduction. For example, a low-temperature Debye model gives a lattice entropy proportional to $T^3$. As that lattice entropy falls, the spins take up the released entropy; $x$ can no longer remain constant. A weak enough coupling to neglect its equilibrium interaction energy can still be strong enough to exchange heat on the sweep timescale.

Interactions between magnetic moments, including exchange interaction and the fields of neighboring magnetic dipole moments, introduce energy scales that do not vanish with the applied $B$. Once they compete with $k_{B}T$ or $\mu B$, the independent-site partition function fails. Correlations and possible magnetic ordering alter the available magnetic entropy and the cooling curve. Simply setting the external magnetic field to zero does not eliminate those interactions. Different refrigerants therefore need their own measured equilibrium properties; there is no universal final temperature from the applied-field ratio alone.

Finally, a real open heat switch has residual thermal conductance; supports, wires and radiation carry incoming heat. Field changes may also dissipate energy. Locally near thermal equilibrium, with incoming heat rate ${\dot{Q}}_{\mathrm{leak}}$ and internal entropy production rate ${\dot{S}}_{\mathrm{prod}}$, the stage balance becomesThis uses a common stage temperature; a non-equilibrated load requires separate balances. The sweep must allow spin-lattice relaxation while remaining fast enough to limit accumulated parasitic heat. At fixed $B$, the same balance explains warming and finite hold time. Useful refrigeration is the controlled absorption of a load's heat during that finite interval, followed by a renewed isothermal process of magnetization to export its entropy.