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We calculate the four-wave mixing (FWM) in a Bose-Einstein condensate systemhaving multiple spin wave packets that are initially overlapping in physicalspace, but have nonvanishing relative momentum that cause them to recede fromone another. Three receding condensate atom wave packets can result inproduction of a fourth wave packet by the process of FWM due to atom-atominteractions. We consider cases where the four final wave packets are composedof 1, 2, 3 and 4 different internal spin components. FWM with 1- or 2-spinstate wave packets is much stronger than 3- or 4-spin state FWM, wherein two ofthe coherent moving BEC wave packets form a polarization-grating that rotatesthe spin projection of the third wave into that of fourth diffracted wave (asopposed to the 1- or 2-spin state case where a regular density-grating isresponsible for the diffraction). Calculations of FWM for $^{87}$Rb and$^{23}$Na condensate systems are presented.Comment: 9 pages, 8 figure

Four-wave mixing in Bose-Einstein condensate systems with multiple spin states