Stevens equivalents

As the m) state is a linear combination of J,Mj) states characterizing the total kinetic momentum, so called Stevens Equivalent Operators have been introduced to calculate these component. For instance, the first component can be rewritten as... 

Stevens, K.W.H. 1952. Matrix elements and operator equivalents connected with the magnetic properties of rare earth ions. Proceedings of the Physical Society 65 209-215. 

It is worth noting here that negative q values correspond to complex operators, while Stevens parameters are always real [19]. The forms of the operator equivalents are reported in Table 1.3 [20], and the corresponding matrix elements are found tabulated in books by Abragam and Bleaney and by Altshuler... 

When treating CF parameters in any of the two formalisms, non-specialists often overlook that the coefficients of the expansion of the CF potential (i.e. the values of CF parameters) depend on the choice of the coordinate system, so that conventions for assigning the correct reference framework are required. The conventional choice in which parameters are expressed requires the z-direction to be the principal symmetry axis, while the y-axis is chosen to coincide with a twofold symmetry axis (if present). Finally, the x-axis is perpendicular to both y- and z-axes, in such a way that the three axes form a right-handed coordinate system [31]. For symmetry in which no binary axis perpendicular to principal symmetry axis exists (e.g. C3h, Ctt), y is usually chosen so as to set one of the B kq (in Wybourne s approach) or Aq with q < 0 (in Stevens approach) to zero, thereby reducing the number of terms providing a non-zero imaginary contribution to the matrix elements of the ligand field Hamiltonian. Finally, for even lower symmetry (orthorhombic or monoclinic), the correct choice is such that the ratio of the Stevens parameter is restrained to X = /A (0, 1) and equivalently k =... 

Formally the unperturbed Hamiltonian is equivalent to the Hamiltonian of the hydrogen atom in constant homogenious electric field. Chaotic dynamics of hydrogen atom in constant electric field under the influence of time-periodic field was treated earlier (Berman et. al, 1985 Stevens and Sundaraml987). To treat nonlinear dynamics of this system under the influence of periodic perturbations we need to rewrite (1) in action-angle variables. Action can be found using its standard definition ... 

The use of Eqs. (27) or (28) to evaluate VMm can become laborious when there is more than one delectron in the atom or ion. Stevens (6) and Bleaney and Stevens (7) have shown that, when all the wave functions belong to the same LS configuration, the matrix elements can all be evaluated by use of the equivalent operator... 

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... 

The synthesis of 3-benzylcyclobutanone (3) is an illustration of an overall intramolecular alkylation of an acyl anion equivalent (Section 5.9). The a,a>-dihalide is 2-benzyl-l,3-dibromopropane, and the acyl anion equivalent is methyl methylthiomethyl sulphoxide2 the product is 1-methylsulphinyl-l-methylthio-3-benzylcyclobutane which is obtained as a mixture of cis/trans isomers [(9) and (10)] (Expt 7.3). Aqueous acid hydrolysis in ethereal solution unmasks the carbonyl group. The possible mechanism of the reaction is via a Stevens-type rearrangement of the intermediate sulphur ylide, which may proceed in a pericylic, radical or ion pair fashion. 

The Shapiro Reaction, a variation on the Bamford-Stevens Reaction, is the base-induced reaction of tosylhydrazones to afford alkenes. This reaction is carried out with two equivalents of an organolithium compound. 

Bames D, Alford-Stevens A, Bimbaum L, et al. 1991. Toxicity equivalency factors for PCBs Quality assurance Good Practice, Regulation, and Law. 1 70-81. 

The hyperfine-structure from nuclear magnetic moments on the electron spin resonance curve was first interpreted by Owen and Stevens in the case of IrClg-. There is no doubt that this gives a perfect qualitative proof for the delocalization of the partly filled shell. However, it is less clear whether there is a simple equivalence between the ligand nuclear influence and b in eq. (19). The point is that the partly filled shell has to be orthogonal, in a very complicated way, on all the previously filled shells such as Is and 2s of the X atoms. 

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 foregoing argument is equivalent to the approach of Stevens and Funderburk (1972), who used a more abstract line of reasoning. Suppose that a summation of Eq. (5) is performed over all states i (i.e., all the rows are summed). Then Eq. (5) becomes... 

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