Section 2.2.15.1. Introduction

The study of hyperfine interactions is a powerful way to characterize different atomic sites in a given sample. There are many experimental techniques, such as Mössbauer spectroscopy, nuclear magnetic and nuclear quadrupole resonance (NMR and NQR), perturbed angular correlations (PAC) measurements etc., which access hyperfine parameters in fundamentally different ways. Hyperfine parameters describe the interaction of a nucleus with the electric and magnetic fields created by the chemical environment of the corresponding atom. Hence the resulting level splitting of the nucleus is determined by the product of a nuclear and an extra-nuclear quantity. In the case of quadrupole interactions, the nuclear quantity is the nuclear quadrupole moment (Q) that interacts with the electric field gradient (EFG) produced by the charges outside the nucleus. For a review see, for example, Kaufmann & Vianden (1979).

The EFG tensor is defined by the second derivative of the electrostatic potential V with respect to the Cartesian coordinates , i = 1, 2, 3, taken at the nuclear site n, where the second term is included to make it a traceless tensor. This is more appropriate, since there is no interaction of a nuclear quadrupole and a potential caused by s electrons. From a theoretical point of view it is more convenient to use the spherical tensor notation because electrostatic potentials (the negative of the potential energy of the electron) and the charge densities are usually given as expansions in terms of spherical harmonics. In this way one automatically deals with traceless tensors (for further details see Herzig, 1985).

The analysis of experimental results faces two obstacles: (i) The nuclear quadrupole moments (Pyykkö, 1992) are often known only with a large uncertainty, as this is still an active research field of nuclear physics. (ii) EFGs depend very sensitively on the anisotropy of the charge density close to the nucleus, and thus pose a severe challenge to electronic structure methods, since an accuracy of the density in the per cent range is required.

In the absence of a better tool, a simple point-charge model was used in combination with so-called Sternheimer (anti-) shielding factors in order to interpret the experimental results. However, these early model calculations depended on empirical parameters, were not very reliable and often showed large deviations from experimental values.

In their pioneering work, Blaha et al. (1985) showed that the LAPW method was able to calculate EFGs in solids accurately without empirical parameters. Since then, this method has been applied to a large variety of systems (Schwarz & Blaha, 1992) from insulators (Blaha et al., 1985), metals (Blaha et al., 1988) and superconductors (Schwarz et al., 1990) to minerals (Winkler et al., 1996).

Several other electronic structure methods have been applied to the calculation of EFGs in solids, for example the LMTO method for periodic (Methfessl & Frota-Pessoa, 1990) or non-periodic (Petrilli & Frota-Pessoa, 1990) systems, the KKR method (Akai et al., 1990), the DVM (discrete variational method; Ellis et al., 1983), the PAW method (Petrilli et al., 1998) and others (Meyer et al., 1995). These methods achieve different degrees of accuracy and are more or less suitable for different classes of systems.

As pointed out above, measured EFGs have an intrinsic uncertainty related to the accuracy with which the nuclear quadrupole moment is known. On the other hand, the quadrupole moment can be obtained by comparing experimental hyperfine splittings with very accurate electronic structure calculations. This has recently been done by Dufek et al. (1995a) to determine the quadrupole moment of ⁵⁷Fe. Hence the calculation of accurate EFGs is to date an active and challenging research field.