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The low-temperature magnetization of the S = i Heisenberg ferromagnet has been investigated by the method of double-time temperature-dependent Green functions. The equation for the lowest order Green function and the equation for the next higher order Green function, truncated to order {n) = \-(S z ), were solved. The low-temperature magnetization so obtained was found to agree with that obtained by Dyson. In particular, an argument is presented which suggests that the T z term, which has previously plagued this method, does indeed vanish.T HE magnetization of a Heisenberg ferromagnet in the low-temperature region has been rigorously studied by Dyson 1 using the method of spin waves. More recently various authors have applied the method of double-time temperature-dependent Green functions to this problem. [2][3][4][5][6][7][8] In the case S-% there is a discrepancy of order T s in the magnetization which arises from an error in the decoupling approximation used to solve the Green-function equation of motion. Tanaka and Morita 9 have reported the elimination of the T z term by solving the equation of motion for the higher order Green function. While their final result appears to be correct, we feel that they did not make the correct approximations in deriving their equation of motion, and hence lost some insight into the results.Expressing the spin operators of the /th site by the Pauli operators S f *+iS f "=Sf+=bf,satisfying the anticommutation relations {M/ f >=i; {*/,*/> = {*/W)=o, and the commutation relations C*/,V]=«./(l-2» / ); [b f ,n g~] = 8 gf bf; [b f \n g~] = -8 gf b/, (2) and defining an exchange sum /(k) = Le ik,^/ (f-m), where 7(0) = 0, (3) / the Heisenberg exchange Hamiltonian with an external field H becomes 3C=D*ff+i/(0)]2>/We will consider the two Green functions G g /= {{b g ) bg/)) and Ggi m f= {{bjbib m \ b/)). The departure of the magnetization from saturation is then given byThe reader is referred to Zubarev 3 for details of the Green-function method. The Green functions are determined from their equations of motion:m and \_E-ixH-y{