\sect{Spin1 bosons in harmonic trap with Zeeman interaction}
$N$ Bosons that have spin1 are placed in a 3D harmonic trap.
The harmonic trap frequency is $\Omega$.
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.
\Dn
(1) Write an expression for the density of one-particle states~$g(\epsilon)$.
\Dn
(2) Write an expression for the ${B=\infty}$ condensation temperature~$T_c$.
\Dn
(3) Write an equation for $T_c(B)$. It should be expressed in terms
of the appropriate polylogarithmic function.
\Dn
(4) Find the leading correction in ${T_c(B)/T_c \approx 1+\cdots}$
assuming that $B$ is very large.
It should be expressed in terms of an elementary function.
\Dn
(5) Find what is $T_c(B)/T_c$ for $B=0$, and what is the first-order
correction term if $B$ is very small.
\Dn
(6) Sketch a schematic plot of $T_c(B)/T_c$ versus $B$.
Indicate by solid line the exact dependence, and by dashed and dotted lines the approximations.
It should be clear from the figure whether the
approximation under-estimates or over-estimates
the true result, and what is the $B$ dependence of the slope.
\Dn
{\bf Tips:} The prefactors are important in this question. Do not use numerical substitutions. \\
Use the notation $L_{\alpha}(z)$ for the polylogarithmic function,
and recall that ${L_{\alpha}(1) = \zeta(\alpha)}$. \\
Note also that $L_{\alpha}'(z)=(1/z)L_{\alpha{-}1}(z)$, and that $\Gamma(n)=(n-1)!$ for integer $n$.