Crystal Stevens operator

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. 

For the ethylsulphates, with only four independent terms, the equivalent crystal-field operators were derived by Elliott and Stevens (1952) and used by them to interpret the magnetic resonance results (Elliott and Stevens 1953). Only minor modifications of the numerical coefficients of the terms were needed to give a reasonable fit for each member of the series. Some further interactions are required, however, for two cerium compounds. In the hexagonal crystal field of the ethylsulphate, the energy levels of the ground manifold J = f are split into three... 

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 ... 

When the matrix elements are calculated for states built from /-electron configurations it is always found that the constants A% (these quantities are related to the strength of crystal field) always occur with (the sharp brackets denote integration with respect to 4/ radial function). A parameters play an important role in crystal field calculations and can be used as parameters in describing the crystal field. For the lowest L S J state they can easily be determined by using the operator equivalent technique of Elliott and Stevens [545—547] and with the help of existing tables of matrix elements. Wybotjbne [548], however, feels that a better approach is to expand Vc in terms of the tensor operators,, as... 

Here Bk s stand for the crystal field parameters (CFP), and Ck(m) are one-electron spherical tensor operators acting on the angular coordinates of the mth electron. Here and in what follows the Wyboume notation (Newman and Ng, 2000) is used. Other possible definitions of CFP and operators (e.g. Stevens conventions) and relations between them are dealt with in a series of papers by Rudowicz (1985, 2000,2004 and references therein). Usually, the Bq s are treated as empirical parameters to be determined from fitting of the calculated energy levels to the experimental ones. The number of non-zero CFP depends on the symmetry of the RE3+ environment and increases with lowering the symmetry (up to 27 for the monoclinic symmetry), the determination of which is non-trivial (Cowan, 1981). As a result, in the literature there quite different sets of CFP for the same ion in the same host can be found (Rudowicz and Qin, 2004). 

Stevens [4] developed an operator-equivalent method for evaluating crystal field matrix elements based on the Wigner-Eckart theorem. It was shown that within a particular J (or L) manifold all operators of the same rank have matrix elements which are proportional to one another. The matrix elements of these operators along with proportionality constants for the ground terms of f" ions have been tabulated [5]. 

McCabe and Stevens applied this equation successfully to data on the growth of copper sulfate crystals suspended in a forced-convection U-tube apparatus. Their calculation method is useful in designing and operating crystallizers, within the limitations of its rather simplified empirical basis. 

Using the operator equivalent technique of Stevens (32), and following Elliott and Stevens (33), one may also express the splittings in terms of the better known A parameters (c.f. also Abragam and Bleaney (34)), which are in turn simply related to the Bk parameters which occur in the tensor operator treatment of crystal field splittings. Using the relationship given by Wyboume (35), the crystal field and Bk parameters are found to be connected by the expressions... 

Historically, the first extensive developments in crystal-field theory made use of the fact that the crystal-field coupling in lanthanide ions is small. In the operator-equivalent method (Stevens, 1952 Elliot and Stevens, 1953), the coupling of different free-ion levels by the crystal-field interaction is ignored and the crystal-field splitting of each Lj level is treated separately. Traditionally, in this method, the crystal-field Hamiltonian is written as... 

The form of the interaction of 4f moments in the lanthanides is further modified by both the crystal-field anisotropy and magnetostriction. The former is predominantly a single-ion interaction in the lanthanides and arises from the Coulomb coupling of the local spin moment (via the spin-orbit interaction) to the hexagonally symmetric charge cloud of the neighboring ions. Stevens (1952) was instrumental in providing the definitive description of this interaction via a series of operator equivalents of the spherical harmonics which describe in effect the quad-... 

The theory of CF originates from the early work of Bethe (1929). A number of complete review articles have appeared which describe theoretical models and experimental effects of CF in rare earths (Fulde 1979). For crystal fields of cubic symmetry the spin Hamiltonian with reference to the fourfold axis can be expressed conveniently in the operator-equivalent form (Hutchings 1964, Stevens 1967)... 

The crystal field potential W cf is developed in Legendre polynomials, and with the help of the Wigner-Eckhart theorem (Edmonds, 1957), can be expressed in terms of operator equivalents O which are functions of angular momentum operators J J+, J- This method of Stevens (1952) is described in detail by Hutchings (1966), whose nomenclature is adopted here. The hamiltonian as a function of angular momentum operators is... 

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