\sect{Spin1 bosons in 3D box with Zeeman interaction} 

$N$ 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.


