Magnetic dipole interactions in crystals

The influence of magnetic dipole interactions (MDIs) on the magnetic properties of local-moment Heisenberg spin systems is investigated. A general formulation is presented for calculating the eigenvalues λ and eigenvectors μ ˆ of the MDI tensor of the magnetic dipoles in a line (one dimension, 1D), within a circle (2D) or a sphere (3D) of radius r surrounding a given moment μ → i for given magnetic propagation vectors k for collinear and coplanar noncollinear magnetic structures on both Bravais and non-Bravais spin lattices. Results are calculated for collinear ordering on 1D chains, 2D square and simple-hexagonal (triangular) Bravais lattices, 2D honeycomb and kagome non-Bravais lattices, and 3D cubic Bravais lattices. The λ and μ ˆ values are compared with previously reported results. Calculations for collinear ordering on 3D simple tetragonal, body-centered tetragonal, and stacked triangular and honeycomb lattices are presented for c/a ratios from 0.5 to 3 in both graphical and tabular form to facilitate comparison of experimentally determined easy axes of ordering on these Bravais lattices with the predictions for MDIs. Comparisons with the easy axes measured for several illustrative collinear antiferromagnets (AFMs) are given. The calculations are extended to the cycloidal noncollinear 120 ° AFM ordering onmore » the triangular lattice where λ is found to be the same as for collinear AFM ordering with the same k. The angular orientation of the ordered moments in the noncollinear coplanar AFM structure of GdB 4 with a distorted stacked 3D Shastry-Sutherland spin-lattice geometry is calculated and found to be in disagreement with experimental observations, indicating the presence of another source of anisotropy. Similar calculations for the undistorted 2D and stacked 3D Shastry-Sutherland lattices are reported. The thermodynamics of dipolar magnets are calculated using the Weiss molecular field theory for quantum spins, including the magnetic transition temperature T m and the ordered moment, magnetic heat capacity, and anisotropic magnetic susceptibility χ versus temperature T . The anisotropic Weiss temperature θ p in the Curie-Weiss law for T>T m is calculated. A quantitative study of the competition between FM and AFM ordering on cubic Bravais lattices versus the demagnetization factor in the absence of FM domain effects is presented. The contributions of Heisenberg exchange interactions and of the MDIs to T m and to θ p are found to be additive, which simplifies analysis of experimental data. Some properties in the magnetically-ordered state versus T are presented, including the ordered moment and magnetic heat capacity and, for AFMs, the dipolar anisotropy of the free energy and the perpendicular critical field. The anisotropic χ for dipolar AFMs is calculated both above and below the Neel temperature T N and the results are illustrated for a simple tetragonal lattice with c/a>1, c/a=1 (cubic), and c/a<1 , where a change in sign of the χ anisotropy is found at c/a=1 . Finally, following the early work of Keffer [Phys. Rev. 87, 608 (1952)], the dipolar anisotropy of χ above T N =69 K of the prototype collinear Heisenberg-exchange-coupled tetragonal compound MnF 2 is calculated and found to be in excellent agreement with experimental single-crystal literature data above 130 K, where the smoothly increasing deviation of the experimental data from the theory on cooling from 130 K to T N is deduced to arise from dynamic short-range collinear c -axis AFM ordering in this temperature range driven by the exchange interactions.« less