The expansion theorem on the density matrix, introduced by one of the authors (Busseiron-Kenkyu 38 (1951) 24), is applied to the problem of ferromagnetism.The Heisenberg model with spin value of 1/2 per atom is treated. The free energy can be expanded in power series of x=4 tanh β/ (2+tanh β), where β=J/kT, and we have calculated up to the fourth power of x. The critical temperatures of various types of lattices are investigated. In comparison with the results obtained by Opechowski, who developed the free energy in powers of β, the present method yields much better results. Namely, 1) The critical temperature varies little as we proceed a step further in the approximations, which seems to indicate that the series obtained here converges rapidly. 2) With a single exception, our results fulfil the requirement of the Block spin wave theory that only three-dimensinal lattices can be ferromagnetic. 3) The Curie temperatures, obtained in every step of approximations, are found to lie between the values predicted by the two Heisenberg approximations. 4) In the third approxmation, the calculated.Curie temperatures.almost coincide with those obtained by P.R.Weiss.
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