Crystal field three-parameter theory

These difficulties may be alleviated by imposing theoretical constraints. First of all, a theoretical model of the crystal-field interaction can be compared with crystal-field parameters that correspond to the various minima, and the best set selected on the basis of agreement with the model. The simple point charge model extended by means of the three-parameter theory (Leavitt et al., 1975) is a step in this direction. Additional guidance at a more phenomenological level can be provided by the superposition model (Bradbury and Newton, 1967). Theoretical models can also be used to provide a set of starting parameters in the search for the correct minimum. 

Leavitt, R.P., C.A. Morrison and D.E. Wortman, 1975, Three parameter theory of crystal fields, Harry Diamond Laboratories Report HDL-TR 1673. Leavitt, R.P, J.B. Gruber, N.C. Chang and C.A. 

In the AOM formalism, a is 4e A, b is 2ezA + 2e B, c is 3effA, and d is eoA + 2e B. Again, since only energy differences are determined spectroscopically, there are only three recoverable degrees of freedom. We denote this by saying that matrix M3 has three spectroscopically independent parameters. In crystal field theory these are defined as... 

In the scope of the Judd-Ofelt theory three parameters .22, 24, and 2 are commonly used to describe the transitions between J multiplets. For this case, the contributions from individual crystal field split levels of a given multiplet are simply summed up. To account for individual transitions, effective transition operators can be used to derive a parametrization... 

The most straightforward modifications of simple- crystal field-theory-that make allowance for orbital overlap involve using all parameters of interelectronic interactions as variables rather than taking them equal to the values found for the free ions. Of these parameters, three are of decisive importance, namely, the spin-orbit coupling constant, A, and the interelectronic repulsion parameters, which may be the Slater integrals, F , or certain usually more convenient linear combinations of these called the Racah parameters, B and C. 

Since the position of the levels is a function of two parameters, no simple relationship can be deduced for the relative position of the crystal-field levels. If Bl is accidentally equal to zero, an equal spacing will be found between the three levels. The degenerate I 2) level will be at the intermediate position in all cases. Because one can determine the Bq parameter from the splitting of a 7 = I level, the experimental position of two of the three crystal-field levels in the 7=2 level is in theory sufficient to determine Bq. The position of the crystal-field levels in the octagonal D4d symmetry ( =8), is the same as in the hexagonal crystal field, due to the absence of non-zero off-diagonal matrix elements. 

The most precise measurements of the fine-structure parameters D and E have in fact been carried out using zero-field resonance. Figure 7.6 shows the three zero-field transitions in the Ti state of naphthalene molecules in a biphenyl crystal at T = 83 K. In these experiments, the absorption of the microwaves was detected as a function of their frequency [5]. The lines are inhomogeneously broadened and nevertheless only about 1 MHz wide. Owing to the small hnewidth of the zero-field resonances, the fine-structure constants can be determined with a high precision. This small inhomogeneous broadening is due to the hyperfine interaction with the nuclear spins of the protons (see e.g. [M2] and [M5]). For triplet states in zero field, the hyperfine structure vanishes to first order in perturbation theory, since the expectation value of the electronic spins vanishes in all three zero-field components (cf Sect. 7.2). The hyperfine structure of the zero-field resonances is therefore a second-order effect [5]. 

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