Crystal superposition modeling

Fig. 7. Variation of the crystal-field parameters and the crystal-field strength of / OCl Pr3+ (R = La, Pr, Gd) under pressure. The dashed fines show a superposition model calculation of die crystal-field strength.

A further possibility to derive local distortions can be based on models like the superposition model, which explicitly relate the crystal-field parameters to structural parameters. This has been done for CaI Cl Sm2+ (Shen and Holzapfel, 1995b) and MFCl Sm2+ (M = Ba, Sr, Ca) (Shen and Holzapfel, 1996). These results will be discussed in the next section. 

The crystal-field parameters introduced in sect. 4.1 still contain all the structural information about the local environment. Therefore, a direct comparison of crystal-field parameters derived from different hosts, even with the same site symmetry, is not reasonable. In addition, the crystal-field parameters cannot be directly related to the distance and angle variations induced by the high-pressure application. Widely used models which extract the structural information from the crystal-field parameters are the angular-overlap (Jprgensen et al., 1963) and superposition model (Bradbury and Newman, 1967). In the case of f elements, the superposition model has been employed widely for the analysis of crystal-field parameters. 

The underlying assumption of the superposition model is that the one-electron crystal field is additive and can be regarded as a superposition of the contributions from individual ions... 

In contrast to the LaCb case, for ROC two different ligands must be taken into account. In this host the Pr3+ ion is surrounded by four O2- and five Cl" ions. The four O2- ions are located in a plane below the central ion at a distance Ro, four Cl- ions in a plane above the central ion at a distance Rcu and one Cl- ion above the Cl- plane at a distance RcY. Within the superposition model, the sum of the contributions from these ligands for the crystal-field parameters Bk can be written as ... 

According to the superposition model the intrinsic parameters found for SrFCl Sm2+ were simply transferred to either CaFCl Sm2+ or BaFCl Sm2+. Then, combining the intrinsic parameters with crystal-field parameters gained from optical high pressure studies on CaFCl Sm2+... 

In many cases the crystal structure of a rare-earth compound studied under high pressure is a priori known. In such studies the quality of the theoretical link between structure and spectra can be tested. However, a different possibility would be to use the experimentally determined spectral variations in connection with a theoretical approach to derive information about the local structure of the rare-earth ions. Such an attempt has been made in sect. 4.4.2, where the local distortions have been derived either directly from the spectra or by applying the superposition model. Similarly, high pressure studies have been used to get information about the structure in more complicated cases of multiple sites or glasses. In addition, the spectra of rare-earth ions have been used to detect phase transitions that often occur under pressure. Results of such studies will be discussed in the next two sections. 

Newman [31] proposed a model known as the superposition model which simplified the calculations of different contributions to the crystal field parameters and facilitated their analysis. According to the superposition model, the crystal fields are built up from separate... 

The superposition model has also been applied to experimental crystal field parameters obtained for lanthanides [31] substituted into host lattices of oxides, zircons, anhydrous trihalides, oxysulphides, alkaline earth fluorides and some other cubic crystals. The intrinsic parameters obtained from the analysis are given in Table 8.23. The solution spectrum of Er3+ aquo ion is given in Fig. 8.29. 

The superposition model (Newman, 1970) provided a framework for understanding crystal-field effects without the necessity of considering specific mechanisms such as electrostatic interactions. In the late sixties and the seventies, the possibility of technological applications in the laser field spurred experimental and theoretical research to the point that today we have been able to identify at least two hundred host crystals that have been studied, with greater than four hundred ion-host combinations whose spectra have been reported in the literature. This review is an attempt to systematize some of this material. 

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. 

In an attempt to parametrize the crystal-field interaction, Bradbury and Newman (1967) introduced the superposition model. In this model, it is assumed that the A4 and are dominated by ligand (nearest-neighbor) contributions, which may be written as follows... 

Two well-defined crystal-field excitations at 12.2 and 18.1 meV and a quasi-elastic excitation centered at zero energy were derived from inelastic neutron scattering experiments of ferromagnetic YbNiSn. A crystal-field potential based on a superposition model was presented in that work (Adroja et al. 1998). 

In analysis of the physical properties of lanthanide crystals the superposition model (Bradbury and Newman 1968, Newman 1971) is widespread. In this model a) the crystal field is formed only by the nearest neighbours (ligands) of an R ion b) the interaction of the 4f electron with a ligand is axially symmetric, so the Hamiltonian of an ion in the field of the v-th ligand is written as follows (the quantization axis is directed along the radius-vector Ry of the ligand, the origin placed on the R ion) ... 

Thus in the superposition model six parameters are used for describing a certain type of the ligand field in a particular lattice. If the sum (5) is restricted to the nearest neighbours of the R ion, artificial overestimation of the short-range interaction occurs. Disregard of the electrostatic component of the crystal field brings about inadequate results when estimating parameters B29 of tbe crystal field quadrupole components and the parameters of electron-phonon interaction (Newman 1978). 

The idea of building the crystal field of transition-metal and lanthanide compounds as a superposition of single ligand contributions was first expressed by Griffith (1964), but it was effectively introduced by Bradbury and Newman (1967) and developed in subsequent work. It is utilized to standardize the analysis of crystal field data. A large amount of information on and beyond the subject appears in a review paper by Newman and Ng (1989) which stresses the relationship of the superposition model with the angular overlap model often preferred for d electrons. 

The main hypothesis of the superposition model is that the cfp may be written as a sum of individual contributions from the ions in the crystal, each one being the product of a purely angular part [C (J ), where A stands for the angular coordinates 0 and of the vector J ] and a radial part which depends on the rare earth/ion distance. The resulting... 

A number of crystal-field data sets have been analysed by the superposition model, for rare-earth substituted garnets by Newman and Stedman (1969), for CaF2, SrF2 and LaCh by Newman (1971, 1978), for LaAlOs, La203 and La202S by Linares and Louat (1975), for vanadates, arsenates and phosphates by Linares et al. (1977), etc. Some examples (intrinsic parameters and power law when determined) are listed in table 3. 

For low site symmetries systematic angular discrepancies sometimes occur between the experimental field parameters and those deduced from the superposition model. It is namely the case for rare earth substituted Y2O3 in which 5 is far too strong for all the members of the series. The distortion cannot be accounted for by any power law since the crystal structure is precisely known at low temperature and shows that the six oxygen first neighbours are nearly equidistant from the rare earth. 

The cross relations due to Kibler (1971, 1974, 1975) between the coefficients of the Angular Overlap model e (up to (p effects), the electrostatic model (EM) crystal field parameters, and the superposition model (SM), translate in familiar cjp terminology the contribution of one overlap type A with ligand L to the 5. It is written as... 

---