A problem of considerable physical interest, wherein a tracer of arbitrary species diffuses against a dynamic background of double occupancy avoiding classical particles of concentration x hopping on regular lattices, is studied. The theory is exact to the leading order in vacancy concentration, u =1x, and to two leading orders in x. Moreover, in the intermediate concentration, it incorporates all the dominant fluctuations from the mean field.Results are worked out for a variety of quantities of interest, such as the tracer diffusion coefficient, the dynamic response, as well as the generalized diffusional mass operator for all the Bravais cubic lattices. New insights are obtained regarding the rapid variation of the response and the mass operator near the Brillouin-zone edge as a function of the vacancy concentration when u «1. Inversion of the K-dependent characteristics of these quantities, not noticed heretofore, is reported and analyzed.
We treat Dyson's ideal-spin-wave Hamiltonian by Green's function methods. In the second section we consider only noninteracting Dyson spin waves. In the third section we consider interacting spin waves, using the simplest possible decoupling assumption. Our results lead then to the same expressions for the magnetization and the spin-wave specific heats as Dyson in the limit as S -> °o up to terms of order JT 4 . The difference is of order T 4 /S 2 . If we introduce a higher-order decoupling, we obtain results identical with those of Oguchi, which in turn agree fairly closely with those of Dyson. We also obtain a spin-wave dispersion law which is a slight improvement of the one obtained by Brout and Englert, especially for small values of S. In three appendices we discuss, respectively, the decoupling process, the shifts in the spin-wave energies, and their "damping."
A formulation is given for calculating magnetic-single-site density of states for a completely random dilute Heisenberg ferromagnet, with isotropic nearest-neighbor exchange, in the limit of very low temperatures.In conformity with Kohn's suggestion that the dynamics of sufficiently random many-body systems may be approximated by that of typical small neighborhoods, a consistent hierarchy of truncation schemes for the spatial matrix elements of the T matrix is described. The case of a drastically truncated Kohn neighborhood, consisting only of two neighboring sites, is worked out in detail. It is shown that for lattices without nearest-neighbor triangles, the given density of states exactly preserves the first four frequency moments. Moreover, for Bravais lattices with z nearest neighbors, all frequency moments of the density of states are given exactly to the two leading orders in z~. By analyzing the renormalization of the K -0 spin-wave energy, estimates for the critical temperature are obtained. In the present approximation, the magnetic long-range order cannot occur for magnetic concentrations which are~2/z. For the simple-cubic lattice, numerical computations of the magnetic-singlesite density of states and the real and imaginary parts of the coherent exchange are given for several concentrations. I~INTRODUCTIONThe ground state of a dense Heisenberg ferromagnet is exactly known. However, as soon as finite concentrations of nonmagnetic impurities are introduced, the system becomes a random coupled many-body system which cannot be solved exactly in arbitrary dimensionality.Brout' seems to have been the first one to seriously address himself to the question of the behavior of such a dilute Heisenberg ferromagnet as a function of the dilution. His analysis was rather formal, and although no precise results were recorded, a qualitative picture of the dependence of the Curie temperature Tc(m) as a function of the magnetic concentration m was predicted. In the limit that the exchange interactions are extremely long ranged, the dependence of Tc(m) on m was conjectured to be linear. For finite-range interactions, the linear dependence was conjectured to be confined to the concentrated re-gion, while in the vicinity of a certain nonzero critical concentration m, [m, is the highest relative concentration of magnetic ions fcr which magnetic long-range order (LEO) does not occur] the behavior was expected to be more complicated. The problem was later studied by Elliott~and Smart. ' Elliott used the constant-coupling twoparticle cluster approximation of Kasteleijn and van Kranendonk.For spin S and coordination number z, he estimated the critical concentration m, as m, = (S+1)/S(z -1) .Smart generalized the Bethe-Peierls-Weiss method for application to the classical spin case (i. e. , S~) and found the same result.Charap argued that because of the neglect of concentration fluctuations in the environment of the nearest-neighbor shell, the physics of the problem had been inadequately represented in this treat-
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