A general formulation is presented for the calculation of static field distributions caused by randomly placed impurities in a lattice. Specific evaluations are carried out for the case of combined Ruderman-Kittel-Kasuya-Yosida (RKKY) and dipolar interactions between magnetic impurities and host nuclei. In this case the line shape is shown to approach a Lorentzian in the limit of great dilution, in agreement with earlier theories for the dipolar case. Analytic expressions for the half-widths are given. The Lorentzian shape is shown to be a consequence of the $R^{-3\mathrm{}}$ range dependence (i.e., to hold only in this case) and to hold also for an arbitrary distribution of impurity-moment values. These results are corroborated by explicit machine computations for the concentration range $0.01\le c\le 1.0$ at.%, in which no approximations of a serious nature are introduced. The linewidth law for combined RKKY and dipolar coupling is derived and verified. This technique is applied to the problem of line narrowing due to resistivity damping of the RKKY oscillations, obtaining a damping length greater by a factor of ∼2 than that originally deduced by Heeger, Klein, and Tu. The self-damping of RKKY oscillations by the Mn impurities in $\mathrm{Cu}\mathrm{Mn}$ is estimated to be significant for $c\gtrsim 1$ at.%. Further applications are discussed.

• Received 3 January 1974

DOI:https://doi.org/10.1103/PhysRevB.9.4857