## Abstract

This paper is a sequel to an earlier one (ter Haar & Lines 1962 referred to as A) in which we applied a molecular-field treatment to anisotropic antiferromagnetics. In the present paper we apply spin-wave theory to investigate the influence of anisotropy of nearest-neighbour interactions and of the occurrence of next-nearest-neighbour interactions on the stability of the types of order found in A. After a brief introduction, face-centred cubic antiferromagnetics are considered in the second section. We find that there is no type of f.c.c. order which is stable for nearest-neighbour isotropic exchange interactions only. For the case of type 1 order with all spins along the direction of the unique cubic axis the order is stabilized by a small amount of anisotropy in the nearest-neighbour interaction. This is the only f.c.c. order which we found to be stable for nearest-neighbour interactions only. The influence of the more-remote-neighbour interactions is probably small for this case. For the case of type 1 order with all spins perpendicular to the unique cubic axis, we find that this type of order is only stable, provided interactions more remote than the nearest-neighbour ones occur. As far as type 2 order is concerned, the case where the preferred direction of order is in one of the ferromagnetically ordered planes turned out to be too complicated to be treated, but the case where the preferred direction is perpendicular to the ferromagnetic planes and the isotropic case can be treated. The orders in the latter cases are stable, provided the next-nearest-neighbour interactions are not too weak. If they are too weak, type 3A order is the stable one. Type 3A order with the spins oriented along the unique cubic axis is stable, provided there is a small amount of isotropic antiferromagnetic next-nearest-neighbour interaction present. Type 3A order with spins perpendicular to the unique cubic axis is stable only if we include second and third nearest-neighbour interactions of sufficient magnitude. For most of these cases we have computed the spin-wave ground-state energy and the average value in this ground state of the total sublattice spin-component along the preferred direction; this value should be close to its maximum for the spin-wave treatment to be reliable. We observe that for all orders considered here there is a general rule: the order is not stable, if it is possible to single out a plane in the structure for which the average interactions between atoms within the plane and those outside is zero. In Section 2 we discuss the body-centred tetragonal lattice. We find that type 1 order is stable, provided the isotropic next-nearest-neighbour exchange interaction is larger than the nearest-neighbour exchange interaction. If their ratio is less than $\frac{1}{2}$ the so-called rutile type diagonal order-or type 2 order-is stable whenever its existence is predicted by the molecular-field theory. In the latter case one must introduce four sets of spin-waves rather than the two sets occurring for the other types of order considered in the present paper. In the last section we consider antiferromagnetic resonance. We find that the resonance frequency observed for MnO agrees rather better with the exchange interaction deduced from susceptibility measurements than with the value of this interaction deduced from mixed-salt paramagnetic-resonance measurements. For the case of MnF$_{2}$ we find a resonance wavelength of about 0$\cdot $95 mm as against the experimental wavelength of 1$\cdot $15 mm. We finally predict resonance frequencies of 15$\cdot $0 and 19$\cdot $1 cm$^{-1}$ for (NH$_{4}$)$_{2}$IrCl$_{6}$ and K$_{2}$IrCl$_{6}$ if they should show type 1 order and of 10$\cdot $6 and 13$\cdot $5 cm$^{-1}$, if the order should be type 3A.

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