Spin Hamiltonian of Low-Symmetry Crystalline Systems

The spin Hamiltonian has been developed for paramagnetic species found in crystalline sites of low (less than axial) symmetry. A tensorial form of the Hamiltonian has been stressed in order to emphasize its primarily mathematical origin and to clarify symmetry constraints. The treatment of these constraints, due to both Kramers (time-inversion) and spatial symmetry, is general without restrictive assumptions concerning the strength of the applied magnetic field. It is only assumed that the field-dependent interactions of the states described by the spin Hamiltonian with all other states can be treated as small perturbations. It has been shown that the spin Hamiltonian possesses the symmetry of the actual Hamiltonian if and only if the transformation matrices of the actual and fictitious spin states differ by at most a phase for every symmetry operation. Specific applications are to those terms of the spin Hamiltonian usually sufficient to describe the (approximately) orbital singlet states of a transition-metal ion with a fictitious electronic spin $s\le \frac{5}{2}$. However, the treatment may easily be extended to cover other terms, including those involving external electric fields and nuclear moments. Guidelines for the analysis and solution of the Hamiltonian matrix have been set out. Certain special reference frames of this Hamiltonian have been discussed, including and in particular the one designated by the symmetry axes of the orthorhombic point groups. The form of the Hamiltonian appropriate for this symmetry has been developed in detail. An operational definition of magnetic axes has been given and it has been shown that rotational studies can be effective in differentiating between orthorhombic, monoclinic, and triclinic site symmetries, when the paramagnetic ions substitute into several crystallographically equivalent sites.