\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~$T_c$.

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(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. 

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(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 $n$. 
