In this paper, we develop a spin-wave theory of the Holstein-Primakoff type for exchange-coupled crystals with an arbitrary number $n$ of magnetic ions per primitive magnetic unit cell, when the resultant electronic spin vectors of these ions are mutually parallel or antiparallel in a given domain, except for spin-wave fluctuations. A simple and systematic method is presented for finding a complete set of normal spin-wave modes. This method is used to derive a convenient formula for the component of the total electronic spin vector of the magnetic ions in a domain along the axis of spin alignment. We show that there exists at least one "acoustic" branch among the $\le n$ distinct branches of the spin-wave spectrum when the magnetic anisotropy and external magnetic field contributions vanish. For the case when all the $m\ge 1$ acoustic branches existing in the absence of these contributions are identical, we prove that the acoustic spin-wave energies corresponding to a given wave-number vector $\kappa$ are of $O\left({\left|\kappa \right|}^{\frac{2}{m}}\right)$ for $\left|\kappa \right|\to 0$. The situation in which a single acoustic branch exists when no external magnetic field or anisotropy effects are present is studied in detail and, under suitable restrictions, an explicit formula is derived for the energies of the magnons of this branch for $\left|\kappa \right|\to 0$. We apply this spin-wave theory to obtain general cross-section formulas for the one-magnon zero-phonon scattering of neutrons by the class of exchange-coupled crystals referred to in the first sentence of this abstract, when the magnetic ions in these crystals are completely quenched orbitally. The formulas in question are used to predict a spin-wave phenomenon of wide generality for polarized incident neutrons. This phenomenon is of particular experimental interest in connection with the acoustic spin-wave scattering of such neutrons by crystals of this class having a single acoustic branch and has been qualitatively confirmed by experiments on magnetite. For the last-mentioned crystals, we use an exact limit result of this paper to suggest a simple approximate form of the general cross-section equations pertaining to acoustic spin-wave scattering, when only magnons of sufficiently small $\left|\kappa \right|$ are of interest.

• Received 18 September 1961

DOI:https://doi.org/10.1103/PhysRev.125.1940