Contribution to the Spin-Wave Theory of a Dilute Heisenberg Ferromagnet
Progress of theoretical physics
477 A spin-wave theory of a dilute Heisenberg ferromagnet is studied using Green's function techniques. The Green's function for one-magnon excitations is calculated in two ways, a decoupling method and a diagram method. An expression for energy is applied to determine the critical concentration for which the ferromagnetic ground state is unstable with respect to the formation of long-wavelength spin waves. For a simple cubic crystal, the result lies close to that obtained using a cluster
... ion method. §I. Introduct:i.on The spm-wave theory of a ferromagnetic crystal, contammg some concentration of magnetic atoms of another sort, has been studied by several authors. 1 ) The problem is similar to the lattice vibration problem in which both the mass and the effective atomic force constants are changed. The structure of the energy spectrum which has been obtained within the framework of one-vertex approximation reflects the existence of localized or quasi-localized levels in the singleimpurity problem. On the other hand, the spin-wave theory of a dilute Heisenberg ferromagnet has not been studied using the Green's function method. The difficulty of this problem is that the random nature of the system does not allow a description of an imperfect crystal in terms of a perfect one by labelling its energy levels by the eigenvalues of quasi-momentum. Thus this problem, corresponding to the transfer of magnons through a specimen, is like the impurity band problem. 2 ) This analogy, however, is not complete. It is different from the impurity-band problem in that the spin system has an internal degree of freedom and a rota-. tional symmetry of the ferromagnetic ground state. In this paper, the Green's functions for spin waves in a dilute Heisenberg ferromagnet are derived. On account of a random arrangement of magnetic atoms, it is necessary to perform an average of the Green's function. The average procedure is studied in two ways; a decoupling method (in § 3) and a diagram method (in § 4). In § 3, an expression for the energy is applied to determine the critical concentration for which the ferromagnetic ground state is unstable with respect *> The preliminary report has been published in Prog.