\sect{Cooling by adiabatic demagnetization}

%Baruch's  A16

Consider a system of ${N}$ spins on a lattice at temperature
${T}$, each spin has a magnetic moment  . In presence of an
external magnetic field each spin has two energy levels, ${\muħ
H}$.
\begin{itemize}
\item [(a)]

Evaluate the changes in energy $\delta E$ and in entropy $\delta
S$ as the magnetic field increases from ${0}$ to ${H}$. Derive the
magnetization $M(H)$ and show that
\[\delta E=T\delta S - \int_{0}^{H}M\left(H'\right)dH '.\]\\
Interpret this result.

\item [(b)]
Show that the entropy ${S\left(E, N\right)}$ can be written as
$S(M,N)$. Deduce the temperature change when ${H}$ is
reduced to zero in an adiabatic process. Explain how can this
operate as a cooling machine to reach ${T\approx 10^{-4} K}$.
(Note: below ${10^{-4} K}$ in realistic systems spin-electron or
spin-spin interactions reduce ${S\left(T, H=0\right)\rightarrow
0}$ as ${ T\rightarrow 0}$. This method is known as cooling by
adiabatic demagnetization.\\

\end{itemize}

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