Stevens operators

Table 1.3 Stevens operators expressed in terms of J+l J, Jz polynomials.

In much the same way as Stevens operators, the summation in Equation 1.15 is limited to well-defined values for f-electrons, the restriction k <7 holds, while q is limited to those values consistent with the point symmetry of the site. Finally, the even part k = 0, 2,4,6) is responsible for the CF splitting, while the odd part k = 1,3, 5,7) is responsible for the intensity of induced electric dipole transitions in optical spectroscopy [5b, 26]. 

Hcf can be expressed in terms of extended Stevens operators, Oq [15, 16], and for a given /-multiplet it takes the general form ... 

A complete decomposition of the ab initio computed CF matrix in irreducible tensor operators (ITOs) and in extended Stevens operators. The parameters of the multiplet-specific CF acting on the ground atomic multiplet of lanthanides, and the decomposition of the CASSCF/RASSI wave functions into functions with definite projections of the total angular momentum on the quantization axis are provided. 

The quantities 6 are the Stevens factors (a, fir, y, for n = 2, 4 and 6, respectively), (rn) are Hartree-Fock radial integrals and A are crystal-field potentials. The quantities O are Stevens operators and Hm in the last term of eq. (11) represents the molecular-field acting on the rare earth moment. 

In order to proceed with the diagonalization of it is convenient to rotate the z-axis into the magnetization direction. Prescriptions for the rotation of Stevens operators have been discussed by Danielsen and Lindgard (1972). In the representation with the magnetization axis as quantization axis we have therefore... 

The remainder of may be cast into a similar form by an expansion of the Stevens operators 0 (i) belonging to the ith site into a well-ordered series of the operators at and a,. Lindg rd and Danielsen (1974) derived this expansion by matching corresponding matrix elements of Racah operators (linear combinations of Stevens operators) with the Bose operators. For example, O becomes... 

Before employing eq. 6.40 to evaluate the contributions to the macroscopic coefficients k/ (T) from the single-ion anisotropy terms of the Stevens operators of eq. 6.3 must first be transformed to a co-ordinate system with the z-axis along the magnetization direction. Equation 6.40 then yields, for example. 

The expressions assumed for the angular dependence of the free energy and the simple relations (eq. 6.42) between the coefficients k/" and Bf" are strictly valid only in the limit where the anisotropy is much smaller than the isotropic two-ion exchange interaction. Under these conditions the expectation values of the Stevens operators renormalize with temperature in the straight-forward manner calculated by Callen and Callen (1%5, 1966) ... 

Eneigy of the interaction of R ions via die phonon field may be presented by means of Stevens operators of an even rank (/ , p =2, 4, 6) in the form analogous to eq. (32) (see also Aminov and Kochelaev 1962). An explicit form of the corresponding parameto is given in sect. 2.5. 

Here the 5 are the CEF parameters determined experimentally, and the are the Stevens operators (Stevens 1952). In order to compare directly the crystal-field parameters for different rare-earth ions in an isostructural series, it is more convenient to express these parameters in terms of s, which are related to the through ... 

With Stevens operator equivalents technique (where one replaces the terms in the expansion of the electrostatic potential by suitable angular-momentum operators) the potential energy of the 4f electrons with the polar axis along... 

CEF-level occupational numbers in the presence of a molecular field (or applied field) thermal occupation number of CEF -state I impurity concentration number of RE-ions electron density of states Stevens operators... 

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