A full description of the previously proposed method of calculating low temperature specific heat and susceptibility of quantum spin chains are given. By the use of internal energy as the expansion variable the domain of convergence of the high temperature series is remarkably extended. The low temperature properties are obtained by continuing it in a natural way to polynomials in internal energy, whose form is suggested by the spin wave theory. The method is checked in the 5x= 1/2 XY model and is then applied to the Heisenberg model of S=1/2, 1, 3/2 and 2. The results are satisfactory in ferromagnets and reasonable in the S= 1/2 antiferromagnet. In antiferromagnets of S³l the results suggest inadequacy of our polynomial, which is, presumably, due to the existence of an energy gap.
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