= Limits of adiabatic demagnetization
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 $dF=-S\,dT-m\,dB$, a <Maxwell relation> gives
$$
dS=\frac{C_B}{T}\,dT+\left(\frac{\partial m}{\partial T}\right)_B dB,\qquad
\left(\frac{dT}{dB}\right)_S=-\frac{T}{C_B}\left(\frac{\partial m}{\partial T}\right)_B.
$$
Here $C_B$ is the total <heat capacity> participating in the stage's <thermal equilibrium>. The ideal model has $(\partial m/\partial T)_B=-C_{B,{\rm mag}}/B$. If its spins share <temperature> with a field-independent lattice and load of combined <heat capacity> $C_{\rm other}>0$, then
$$
\left(\frac{dT}{dB}\right)_{S_{\rm total}}
=\frac{T}{B}\frac{C_{B,{\rm mag}}}{C_{B,{\rm mag}}+C_{\rm other}}.
$$
The 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 $k_BT$ 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 $\dot Q_{\rm leak}$ and internal <entropy production> rate $\dot S_{\rm prod}$, the stage balance becomes
$$
\frac{dS}{dt}=\frac{\dot Q_{\rm leak}}T+\dot S_{\rm prod},\qquad
C_B\dot T=-T\left(\frac{\partial m}{\partial T}\right)_B\dot B
+\dot Q_{\rm leak}+T\dot S_{\rm prod}.
$$
This 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>.
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