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The spin evolution of a Bose-Einstein condensate starting from a mixture oftwo or three groups of $^{52}$Cr (spin-3) atoms in an optical trap has beenstudied theoretically. The initial state is so chosen that the system does notdistinguish up and down. In this choice, the deviation caused by thesingle-mode approximation is reduced. Moreover, since the particle number isgiven very small (N=20), the deviation caused by the neglect of the long-rangedipole force is also reduced. Making use of these two simplifications, atheoretical calculation beyond the mean field theory is performed. Thenumerical results are help to evaluate the unknown strength $g_0$.

The spin evolution of spin-3 $^{52}$Cr Bose-Einstein condensate