Spin-thermodynamics of cold spin-1 atoms decoupled from spatial modes

We study the thermodynamic properties of cold spin-1 atoms with a fixed magnetization M and decoupled from spatial modes. Three temperature domains are found: (0, T1) is a domain of second condensation, namely, both the spatial and spin degrees of freedom are frozen; (T1, T2) is a T - sensitive domain, where the internal energy U \propto T, entropy SE \propto log T, and U/kBT is always less than 3/2; (T2, T3) is the third domain with a maximum entropy. When T is higher than T3, the spatial modes can not be neglected. The appearance of these domains originates from the two gaps: (i) The gap between the ground state and the first excited state, and (ii) the gap between the highest spin-state without spatial excitation and the lowest state with a spatial mode excited. These two gaps are crucial to the low temperature physics and they can be tuned.