\sect{Bosons with Spin in magnetic field} 

$N$ Bosons that have mass $m$ and spin1 are placed
in a box that has volume $V$. A magnetic field B is
applied, such that the interaction is $-\gamma B S_z$,
where $S_z=1,0,-1$ and $\gamma$ is the gyromagnetic ratio.
In items (c-f) assume the Boltzmann approximation
for the occupation of the $S_z\ne1$ states.

\Dn

(a) Find an equation for the condensation temperature $T_c$.

\Dn

(b) Find the condensation temperature $T_c(B)$ for $B=0$
and for $B\rightarrow \infty$.

\Dn

(c) Find the critical $B$ for condensation if $T$ is set
in the range of temperatures that has been defined in item(b).

\Dn

(d) Describe how $T_c(B)$ depends of $B$ in a qualitatively manner.
Find approximate expressions for moderate and large fields.

\Dn

(e) Find the condensate fraction as a function of $T$ and $B$.

\Dn

(f) Find the heat capacity of the gas assuming large but finite field.






\ \\ \ \\

OLD: 


\sect{Baruch's  B06.}

Consider an ideal Bose gas  with spin 1 in a magnetic field $B$.
The Hamiltonian for each particle is ${\cal
H}=\frac{{\bf p}^2}{2m}-\gamma BS_z$ where $S_z=1,0,-1$ and $\gamma$ is
the gyromagnetic ratio. The average density is $n$ and particle
mass is $m$.
\begin{itemize}
\item[(a)]  Particles of which $S_z$ can
condense? Find an equation for the condensation temperature $T_c$.
Solve this equation explicitly for (i) $B=0$, and (ii) large $B$,
$\gamma B\gg k_BT_c$; keep the lowest order correction.
\item[(b)] Plot, qualitatively, $T_c(B)/T_c(\infty)$  as function of $B$.
If $T$ is below but close to the value of
$T_c(B\rightarrow \infty)$  describe what happens as $B$ is
increased from $B=0$. Find the critical $B$ for condensation in
the limit of (a-ii).
\item[(c)] Evaluate the specific heat in the limit of (a-ii).
\item[(d)] Evaluate the condensate  fraction $\langle n_0\rangle/n$ as function of $T$ in the case of (a-ii).
\\
\end{itemize}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%