Point-group discrimination using the c-axis angular variation of the EPR fine-structure spectra for S-state ions

Abstract

The phenomenological and generalized spin-Hamiltonian description for the energy levels of a paramagnetic ion in an arbitrary symmetry crystalline electric field environment are discussed. The transformation properties of both the axial and polar classes of the irreducible tensor operators which may be used in description are investigated. New cosine and sine tesseral tensor operators, which are the real and imaginary linear combinations of these tensor operators for both classes, are defined and their conjugation and transformation properties are determined. The secular equation that results when the spin Hamiltonian is expressed in terms of these new tessera) tensor operators is examined under a rotation about a crystallographic c axis when it coincides with the magnetic z axis. Two distinct cases are found: (1) No angular variation of the EPR spectrum in the c plane is observed when both the cosine and sine tesseral tensors operators are allowed for every term with q ≠ 0 if these operators are of either the axial and polar class. (2) Some transitions exhibit an angular variation of the EPR spectrum in the c plane when only one of the two tesseral tensor operators is allowed if the tensor operators are of the polar class. This occurs when the point group includes one of the symmetry operations σ v , σ v ′, σ d or C 2 ′. This angular variation can be used to discriminate between similar point groups which differ by only one of these symmetry operations. It has been oberved by many EPR spectroscopists but its theoretical significance has not been appreciated since it has been believed that the spin Hamiltonian must be consistent with the local site symmetry constrained by the spherical symmetry requirement. This latter requirement arises if the spin Hamiltonian is invariant under parity conjugation which is equivalent to imposing space inversion symmetry. Releasing this constraint means that those ZFS terms for which k is odd are allowed and their matrix elements are not zero for S-state ions if the effect of excited state configurations is included. It is also shown that this c-axis angular variation can be used to determine the values of the off diagonal ZFS parameters.