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The classical Heisenberg type spin Hamiltonian is widely used for simulationsof finite temperature properties of magnetic metals often using parametersderived from first principles calculations. In itinerant electron systems,however, the atomic magnetic moments vary their amplitude with temperature andthe spin Hamiltonian should thus be extended to incorporate the effects oflongitudinal spin fluctuations (LSF). Although the simple phenomenological spinHamiltonians describing LSF can be efficiently parameterized in the frameworkof the constrained Local Spin Density Approximation (LSDA) and its extensions,the fundamental problem concerning the integration in classical spin spaceremains. It is generally unknown how to integrate over the spin amplitude. Twointuitive choices of integration measure have been used up to date; theMurata-Doniach scalar measure and the simple three dimensional vector measure.Here we derive the integration measure by considering a classical limit of thequantum Heisenberg spin Hamiltonian under conditions leading to the properclassical limit of the commutation relations for all values of the classicalspin amplitude and calculate the corresponding ratio of the number of quantumstates. We show, that the number of quantum states corresponding to theconsidered classical spin amplitude is proportional to this amplitude and thusa non-trivial integration measure must be used. We apply our results to thefirst-principles simulation of the Curie temperatures of the two canonicalferromagnets bcc Fe and fcc Ni using a single-site LSF Hamiltonian withparameters calculated in the LSDA framework in the Disordered Local Momentapproximation and a fixed spin moment constraint. In the same framework wecompare our results with those obtained from the scalar and vector measures.

journal of magnetism and magnetic materials

Longitudinal integration measure in classical spin space and its  application to first-principle based simulations of the high temperature  magnetism of Fe and Ni