Crystal-field interaction equivalent operators

In calculations involving higher J multiplets, matrix elements of the crystal field Hamiltonian between states belonging to different J multiplets are needed. Although these can be calculated by the method of operator equivalents extended to elements non-diagonal in J, it is convenient to use a more general approach, utilizing Racah s tensor operator technique (26). In this method the crystal field interaction may be written as... 

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... 

Substitution (101) appears to be the simplest way in which spin-dependent phenomena can be incorporated into the theory. The operator (100) commutes with S and is thus consistent with effects produced by such spin-independent interactions as the Coulomb and crystal-field terms in the Hamiltonian. Moreover, the effect of (101) turns out to be equivalent to adding to each reduced matrix element of F " a part (proportional to c,) that involves the reduced matrix element of the double tensor where = s (Judd 1977b). Such double tensors are straightforward to evaluate furthermore the proportionality = /4 holds for all terms of... 

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-... 

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... 

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