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We study the thermodynamic properties of cold spin-1 atoms with a fixedmagnetization M and decoupled from spatial modes. Three temperature domains arefound: (0, T1) is a domain of second condensation, namely, both the spatial andspin degrees of freedom are frozen; (T1, T2) is a T - sensitive domain, wherethe internal energy U \propto T, entropy SE \propto log T, and U/kBT is alwaysless than 3/2; (T2, T3) is the third domain with a maximum entropy. When T ishigher than T3, the spatial modes can not be neglected. The appearance of thesedomains originates from the two gaps: (i) The gap between the ground state andthe first excited state, and (ii) the gap between the highest spin-statewithout spatial excitation and the lowest state with a spatial mode excited.These two gaps are crucial to the low temperature physics and they can betuned.

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