Axiality and rhombicity

In physics and mathematics, axiality and rhombicity are two characteristics of a symmetric second-rank tensor in three-dimensional Euclidean space, describing its directional asymmetry. 
Let "A" denote a second-rank tensor in R3, which can be represented by a 3-by-3 matrix. We assume that "A" is symmetric. This implies that "A" has three real eigenvalues, which we denote by formula_1, formula_1 and formula_1. We assume that they are ordered such that 
The axiality of "A" is defined by
The rhombicity is the difference between the smallest and the second-smallest eigenvalue:
Other definitions of axiality and rhombicity differ from the ones given above by constant factors which depend on the context. For example, when using them as parameters in the irreducible spherical tensor expansion, it is most convenient to divide the above definition of axiality by formula_7 and that of rhombicity by formula_8.
Applications.
The description of physical interactions in terms of axiality and rhombicity is frequently encountered in spin dynamics and, in particular, in spin relaxation theory, where many traceless bilinear interaction Hamiltonians, having the (eigenframe) form
(hats denote spin projection operators) may be conveniently rotated using rank 2 irreducible spherical tensor operators:
where formula_12 are Wigner functions, formula_13 are Euler angles, and the expressions for the rank 2 irreducible spherical tensor operators are:
Defining Hamiltonian rotations in this way (axiality, rhombicity, three angles) significantly simplifies calculations, since the properties of Wigner functions are well understood.