Zbigneiew Gajek

Chapter 17 – Analysis of the experimental data. Interpretation of crystal field parameters with additive models

Interpretation of Crystal Field Parameters with Additive Models This chapter discusses an effective method of the interpretation of experimental data in f- electron ionic systems, paying a special attention to simplified crystal field (CF) models based on the partitioning of the CF potential. These simplified models are of invaluable help in the burdensome ambiguity problem connected intimately with the standard parameteric analysis. Several further areas of application are indicated that are equally promising. The theoretical grounds of the models, their shortcomings and advantages are discussed, and the general frames of the interpretation routine are sketched in the chapter. The list of possible applications of the simplified models is certainly not complete, but it conveys an idea of their usefulness. Efficiency of the twin superposition model may be similar in some cases, but in light of the ab initio calculations, the angular overlap model provides the parameters that seem to be more natural and more appropriate for both comparison and classification, as well as for anticipation of their values.

Chapter 19 – Extension of the crystal field potential beyond the one-electron model

Within the one-electron model extended by every possible interconfiguration interactions, the entire crystal field (CF) effect can be reconstructed. However, one ought to employ the electron configurations of the whole ionic complex and consider among the excited configurations those of charge-transfer type and those referring to the ligands themselves. The initial one-configurational model corresponding to absolutely fixed configurations leads to the crystal field effect representation. All the remaining contributions are expressible by means of appropriate interconfigurational interactions. In practice, the calculations are carried out under some simplifying assumptions. From these calculations, the spectrum of energy levels in the crystal field is available. This result can be compared with the experimental data to verify adequacy of the method, but the spectrum can be put to test the effective parameterization within the one-electron model.

Virtual bound state contribution to the crystal field potential

This chapter provides an overview of virtual bound state contribution to the crystal field potential. The contribution of the so-called virtual bound states or semilocalized states within atomic core area is one of the three main mechanisms specific for the metallic state being responsible for the crystal field potential in metallic phases. The two remaining are the electrostatic contribution of atomic cores screened by the conduction electrons and hybridization effect of the localized and conduction band states. Two features distinguish this mechanism from the others. First is its two-stage, indirect nature leading in consequence to the effect being opposite to that produced by the immediate interaction and the second is a collective character of this effect hindering its decomposition into the intrinsic contributions. The contribution of the 5 d virtual bound state into the crystal field potential of the 4 f open-shell electrons in the lanthanide compounds is a model example of this mechanism, and this chapter analyzes the example in detail.

Ionic complex or quasi-molecular cluster. Generalized product function

This chapter presents an account of the construction of effective potentials for various models, their development and improvement, the room for simplification, and the physical meaning of the defined contributions. Setting up a theoretical model is preceded by suggesting, and an advisable partition of the whole surroundings of the chosen ion into two separate areas―the close vicinity of the central ion labeled as ionic complex or cluster, and the outer surroundings. In the close vicinity, the quantum–chemistry methods are rather necessary; whereas, the contribution from the outer space may be well enough described with classical electrostatics. The focus is on crystal field effect. Despite the limitation, a system of electrons and nuclei forming the cluster remains in its nature a complex many-body problem. Unattainability of an exact solution of the problem on the one hand and the inadequacy of the simplified models on the other suggest an intermediate approach as the only practical resolution. In such an approach, only a part of the electrons of the considered cluster is treated explicitly; whereas, the remaining are taken into account as point charges in the electrostatic outcome.

Parameterization of crystal field Hamiltonian

This chapter discusses crystal field potential parameterization in the context of zero-field-splitting (ZFS) in the spin Hamiltonian commonly used in EPR. The essence of the crystal field effect lies in the aspherical part of environment interaction on localized states of unpaired electrons from an unfilled shell of a paramagnetic ion or atomic core in metals. The nature of this interaction is multi-sourced and highly sophisticated, far exceeding that of the simple electrostatic field. In the spin Hamiltonian, ZFS is a term of effective nature having often indirect and secondary connection with the crystal field sensu stricto . Usually, it is a combined effect of various higher order interactions. Therefore, direct setting the crystal field interpretation to Ĥ ZFS is incorrect and leads to misunderstandings.

Screening the crystal field in metallic materials

There are two approaches to the screening problem in metals. The first one, called the Thomas-Fermi method, is a long-wave approximation suitable for slow variable perturbing potentials. The second one, known as the Lindhard method, consists of a more exact calculation of the charge density by summing all Fourier components of the potential including large q too. Taking into account only the linear response of the system to the perturbation remains an essential simplification of the approach. However, the two approaches possess serious constitutional shortcomings. Saying nothing of other simplifying assumptions, they make allowances only for linear response of the system to the perturbation. Besides, these methods include the screening effect, and ignore the self-consistent procedure that turns out to be crucial in the screening mechanism. On the other hand, a direct dealing with the net crystal-field potential is a strong point of these approaches.

Specific mechanisms of metallic state contributing to the crystal field potential

The superposition model of crystal field (CF) potential in metals seems to be false. Although several non-metallic compounds come under the model, there is one mechanism arising from the presence of conduction electrons, making their condensation in the form of the virtual bound states possible, which breaks loose from the model. Moreover, this virtual bound state mechanism affecting the second and fourth order terms is the predominant component, which determines the sign of the crystal field parameters. The prevailing theoretical approach based on the density functional theory and local density approximation is formulated in terms of the one-electron model leaving the superposition problem aside. Nevertheless, this global calculated potential can be decomposed into on-site, nearest neighbor shell and remaining lattice contributions. Construction of the complete theoretical model of crystal-field in metals is nicely contained within the frame of the perturbational scheme by employing projection operators and the second quantization techniques convenient for degenerate states.

Chapter 6 – One-configurational model with neglecting the non-orthogonality. The charge penetration and exchange effects

The mutual penetration of the charge density clouds should not be identified with the overlap of the corresponding wave functions in the strict sense of the notion. There are a lot of convincing experimental data evidencing a mutual compensation of the penetration and exchange effects by the overlap effect, and more strictly by the so-called contact covalency effect. This compensation effect was probably the reason of some earlier striking successes of the point charge model. An analogous effect makes the pseudopotential in metals so weak that the almost free-electron model works very well. The effective repulsion due to the Pauli exclusion counteracts the attractive potential acting on the metal ion electrons due to their penetration into the outer ligand shells. In general, the crystal field parameters of higher orders corrected for these effects are still far from their experimental values. It thus appears that important contributions to these parameters must also arise from mechanisms, which perturb the free-ion states.

Lattice Dynamics Contribution

The lattice dynamics has no first order influence on the electronic energy levels system in the static crystal field. A certain constant shift of all levels is inessential from the crystal field splitting point of view. On the other hand, the lattice dynamics modifies the force constant of the nuclear oscillator part of the eigen states, individually diminishing the energy of the lower states and increasing the energy of excited ones―enlarging somewhat the total splitting. The basic assumption of the analysis is that it is possible to write down the total potential energy of the system of electrons and nuclei as an explicit differentiable function of the coordinates, V(r, Q), where r and Q denote the whole set of coordinates of the electrons and nuclei.

Point Charge Model (PCM)

The concept of the crystal field and the point charge model (PCM) was introduced to the solid-state physics, pointing out the correspondence between the energy levels of an unfilled electron shell of a metal ion in crystal, and the irreducible representations of its point symmetry group. Based on this simplified electrostatic model, the interpretation of the magnetic properties of transition metal ions in their salts is given. Within the framework of PCM, the crystal field is identified with the electrostatic field generated by point charges attributed to the lattice sites. The first of the two main imminent features of the PCM is the additivity of the crystal field potential in the model with respect to its sources, which means that the contribution of each lattice site to the global potential can be calculated separately. All unpaired electrons of the central ion are assumed to be equivalent and their potential energies in the crystal field independent.