Operator-equivalent method

In this way, once the results for any given LS state have been obtained via the operator equivalent technique, those for any of the J sub-levels thereof may easily be derived. This procedure could of course be extended to evaluate matrix elements between states of different J, arising from the same LS state, but for calculations, whether in the LSMlMs) or the LSJMj) basis, in which it is required to incorporate mixing of different LS states the resulting cross-product matrix elements cannot be found by the operator equivalent method but must be determined directly from the wave functions. 

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]. 

The alternative parameters P and P refer directly to an operator equivalent method 26), an irreducible tensor operator method (75) of handling the general first-order perturbation model. 

The odd 3-1 symbols can be evaluated as outlined in Sect. 4b by an operator equivalent method which ultimately brings the calculation into the form of Eq. (76). However, although this method has considerable theoretical interest, it is laborious and not feasible for high /-values. We refer to the following paper (70) for a general and useful way to calculate any 3-1 symbol in terms of a 3 s5mibol. 

It has been shown that the cubic splitting A is related to the coefficient of the cubic term in the ligand field potential by Eq. (58). In analogous manner it is possible to relate the tetragonal splitting 8 and the rhombic splitting [r to the coefficients B and respectively. This is best accomplished by the operator equivalent method (40) which makes it possible to obtain matri.x elements of Vt and Vr within the set of d-orbitals. It is found that... 

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... 

In the operator-equivalent method, one is concerned with matrix elements of eq. (36) within a given free-ion state denoted by The Vk" polynomials may be replaced as follows... 

Crystal-field calculations based on the operator-equivalent method are performed by diagonalizing small matrices containing matrix elements of Hcep within a given state. The dimensions of these matrices are at most... 

The notation of the operator-equivalent method has persisted although the method itself has largely been discarded. Thus, many of the reported results of fitting experimental data are given in the form A r ). These are related to the Bfem of eq. (33) by simple coefficients. A simple relation also exists between the Bfcm of eq. (38) and the... 

Stevens called the operator equivalent method. In this approach the CF Hamiltonian for an ion in 3 w symmetry, appropriate to the A site in pyrochlores is given as ... 

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