| By considering the exchange interaction and crystal field effect, a computer program is modified to model the easy direction of magnetization for rare-earth compounds under a given symmetry of crystal-field, and a calculation is made of the easy direction of magnetization of Tb1−xDyxZn. It is assumed that the spontaneous magnetization is a direct consequence of the exchange interaction and that the magnetic anisotropy arises from the crystal-field effects. The exchange interaction is treated with the mean-field theory, and the crystal-field is described in the equivalent operator method. All of the matrix elements of the Stevens operators are calculated. By transforming the crystalline coordinates to the magnetization coordinates, the definitive magnetization reorientation diagram of Tb1−xDyxZn has been established. The results of the calculation are compared to the existing experimental data; that is, for the rare-earth compound of Tb1− xDyxZn the actual low-temperature easy direction of magnetization could be neither [100] or [110], but a set of non-symmetry axes [uv0] where u and v vary with the concentration x and temperature T. In addition, the change of free energy is numerically calculated in different directions for both of TbZn and DyZn. The results show that there are large magnetic anisotropies at low temperature for TbZn and DyZn. The expected compensation in the magnetic anisotropy and in the alteration of the transition temperature is verified theoretically by the substitution of the Dy in the TbZn. By examining the effect of the crystal-field parameters, the free energy does not have a sensitive dependence on A4<r4>, but A6<r 6> is critical in deciding the value of the free energy. From the diagrams of the minimum free energy vs. temperature, we see that, at low temperature, the internal energy increases faster as the temperature increases than does the entropy for the magnetic system. At high temperature, this behavior is reversed. It is found that the excited states are not very important for the magnetic system at low-temperatures because the free energy has the same value as the lowest energy eigenvalue in that case. For the magnetization reorientation diagram of Tb1−xDyxZn, it is found that, to get agreement with experiment, the values of the crystal-field parameters must be chosen for Dy3+ that are of opposite sign to those predicted by Morin. Possible reasons for the discrepancy are discussed. |
