The Ising model of an antiferromagnet exhibits an anomalous entropy peak at a critical magnetic field which remains even at the absolute zero of temperature, in violation of the third law of thermodynamics. By investigating numerically the behaviour of finite chains of spins interacting through the more general anisotropic Heisenberg coupling, described by the Hamiltonian provided it is shown that the zero point anomalous entropy is peculiar to the Ising model (γ = 0). If the anisotropy is sufficiently great, however, an appreciable entropy peak does persist down to low temperatures although in the anisotropic (pure Heisenberg) limit the entropy has only a broad maximum which falls steadily in height as T approaches zero.