\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} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%