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\heading{E1825: Adiabatic cooling}
\auname{Dror Moshe Aharoni}


{\bf The problem:}

\Dn

Consider an ideal gas whose $N$ atoms have mass $m$, spin $1/2$ and a magnetic moment $\mu$; the energy levels of each particle are $\frac{p^2}{2m}\pm\mu B$ in a magnetic field $B$ where $p$ is the momentum.
\begin{itemize}
\item[(a)] Calculate the entropy as $S(T,B)=S_{kin}+S_{spin}$ due to kinetic and spin terms,
respectively.
 \item[(b)] Consider an adiabatic process in which the magnetic field varied from $B$ to zero (adiabatically). Show that the initial $T_i$ and final $T_f$ temperatures are related by the equation:
\[ \ln \frac{T_f}{T_i}=\frac{2}{3N}[S_{spin}(T_i, B)-S_{spin}(T_f, 0)]\]
\item[(c)] Find the solution for $\frac{T_f}{T_i}$ in the large $B$ limit.
\item[(d)] Extend (c) to the case of space dimensionality $d$ and general spin $S$.
\end{itemize}


\Dn\Dn


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{\bf The solution:}

\Dn

(a)
The Hamiltonian is:

\[
\mathcal{H}
= \frac{p^2}{2m}\pm\mu B
\]

\Dn

\[ \lambda _T = \frac{h}{\sqrt{2\pi mT}}\]

%\Dn

\[ Z = Z_{id}\cdot Z_{spin}=\underbrace{\frac{1}{N!}\left(\frac{V}{\lambda_T ^3}\right)^N}_{ideal\  gas}\underbrace{\cdot\left(2\cosh(\beta\mu B)\right)^N}_{spin\ sistem}\]

\[ \hookrightarrow F=F_{kin}+F_{spin} \]
\[F\equiv -\frac{1}{\beta}\ln Z(\beta ; X)\]
%In our case $X$ is $ \mu $
\[ S = -\left(\frac{\partial F}{\partial T}\right)_{V,N}= S_{kin} + S_{spin} = N\ln \left(\frac{V}{N\lambda^3}\right)+\frac{5}{2} + N\frac{\partial}{\partial T}\left(T \ln (2\cosh(\beta\mu B)\right) \]
With:
\[ S_{kin} = N\ln \left(\frac{V}{N\lambda^3}\right)+\frac{5}{2}\]
\[ S_{spin} = N\ln \left(2\cosh(\beta\mu B)\right)- \beta\mu B\tanh \left(\beta\mu B\right)\]
Consider that:
\[\lim_{x\rightarrow\infty}\ln (2\cosh x) = \lim_{x\rightarrow\infty}\ln (e^x+e^{-x})= x\]
\[\lim_{x\rightarrow\infty}\tanh (x) = 1 \longrightarrow  \lim_{x\rightarrow\infty} x\tanh (x) = x\]

\Dn

That's why for $B\rightarrow\infty \ \ \  S_{spin}\rightarrow 0 $ as expected.

\Dn

And for $B\rightarrow 0 \ \ \  S_{spin}=N\ln (2)$.

\Dn
\Dn
(b)
Adiabatic process:
\[S=S_{kin}+S_{spin}=const\]
\[\Rightarrow S_{kin}\left(T_i\right)+S_{spin}\left(B,T_i\right)=S_{kin}\left(T_f\right)+S_{spin}\left(0,T_f\right)\]\\
 
From the above we get the desired equation:
\[\Rightarrow N\ln (\frac{\lambda_{T_f}}{\lambda_{T_i}})^3 = \frac{3}{2}N\ln (\frac{T_i}{T_f}) = S_{spin}\left(0,T_f\right) - S_{spin}\left(B,T_i\right)\]
\Dn

(c)
As $B\rightarrow\infty $ $S_{spin}\left(T_f,B\right)=0$
\[\rightarrow \ln \left(\frac{T_f}{T_i}\right)=-\frac{2}{3}\ln (2)\]
\[ \Rightarrow T_f=\frac{T_i}{2^{2/3}} \]


(d)
In d dimantions: $ \lambda ^3 \rightarrow \lambda ^{d} $, spin has $2s+1$ states
\[\frac{d}{2}\ln \left(\frac{T_f}{T_i}\right)=-\ln (2s+1)\rightarrow T_f=\frac{T_i}{2s+1}\frac{d}{2}\]

\Dn

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