The maximal axial parameters in equivalent parametrizations of crystal-field Hamiltonians of tetragonal and cubic symmetries

The variation ranges of the axial B_{k0} crystal-field parameters, for k=2,4,6, of tetragonal including cubic crystal-field Hamiltonians H_{CF} for all possible orientations of the relevant reference frame are studied. The distinguished z-axis directions fixed by the maximal absolute values of B_{k0} are analyzed. The diagrams for any tetragonal H_{CF} parametrization depicting the maximal values of |B_{k0}|/M_{k}, where M_{k} is the 2^{k}-pole modulus, as a function of the x=B_{k4}/B_{k0} or B_{64}/B_{60} ratios, together with the distinguished directions are presented. The max|B_{k0}|/M_{k} magnitudes and the relevant distinguished directions are the discriminants of all the equivalent parametrizations. They vary within the intervals (0.7395,1], (0.6074,1] for tetragonal k=4 and tetragonal k=6 H_{CF} components, respectively. Such specified directions determine the mutual spatial orientation of the component 2^{k}-poles of the H_{CF}, and due to their rigid coupling in the H_{CF}, they also refer to the global H_{CF} parametrization. This approach demonstrates the difference in fitting capability between the real and complex isomodular H_{CF} parametrizations.