Tensor structures background

These are the numbers that evidently show the role of the J-O Theory in the field with undefined limits due to the broad applications of its achievements. In order to understand why this theory is so important, its physical background must be presented. It is possible to conclude briefly that the J-O Theory is a simple application of the outstanding beauty of tri-positive lanthanide ions, and in particular their unusual electronic structure. Its features are defined in the language of Racah algebra applied for the concept of effective tensor operators. The simplicity and clarity of this approach, including the well-known Judd-Ofelt parametrization scheme of the/-spectra based on (10.1), when successfully applied to very complex systems makes one wonder how is it possible that this tool works so well in fact this query is its power. 

In summary, the 3D Hessian tensor may be visualised in 3D by an ellipsoid, as shown in Figure 3, when all its eigenvalues have the same sign. The structures that may be detected are also summarised in Table 2, where the conditions on the eigenvalues concern the detection of a black structure against a white background. Similar conditions exist for the case, where the contrast is reversed. 

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