Pair Interaction Hamiltonian in Fictitious Angular Momentum Space

Abstract

We start with the Hamiltonian for the anisotropic electrostatic interaction (direct and exchange) between two magnetic ions and we derive, by using the generalized Wigner-Eckart theorem, a Hamiltonian which represents the interaction between ions in crystal field states. H̃int= ∑ ρ1=0na−1 ∑ ρ2=0nb−1 ∑ P=|ρ1−ρ2|ρ1+ρ2[δ[ρ1](a)×A[P](ρ1ρ2)×δ[ρ2](b)](0). The Hamiltonian has been written in spherical tensor notation; this form permits us to readily calculate the matrix elements of the operators and also to determine whether the interaction contains antisymmetric terms. The upper limit (nα − 1) (α = a, b) to the rank ρi (i = 1, 2) of the fictitious angular momentum operator σ[ρi] is limited only by the multiplicity nα of the crystal field level. The interaction tensor A(P) represents a combination of matrix elements of the interaction Hamiltonian. As the tensors for different crystal field levels are derived from a common Hamiltonian, we readily establish the relations among them. Although the above form of the interaction does not have a particular advantage over more conventional forms for doublet levels (nα = 2), it is very useful for studying the pair interaction between levels of higher multiplicity.