4f charge density

Fig. 8. Angular distribution of the 4f charge density of lanthanide atoms for Jz = J (effective moment parallel to the z-axis). After Thole in Coehoom (1990). In Ce, Pr, Nd, Tb, Dy, Ho the charge density is oblate (aj < 0), in Pm, Sm. Er, Tm, Yb it is prolate (aj > 0). In Gd, Lu (L = 0), the charge density has spherical symmetry.

Fig. 3. Contouis of equal 4f charge densities for a Ce ion in the cubic F, and Fg states. (After Walter 1985.)...

Fig. 2. The 4f charge densities in the fourteen trivalent lanthanide ions in the absence of a crystal field. After Sievers (1982) (by permission of Springer Verlagj.

Narrow bands arise when the overlap of the atomic wave functions is small (as for 5 f s). In this case, the dispersion E(k) is strongly reduced and the bandwidth W becomes very small (zero, in the case of no overlap). The electron charge density, caused by these wave functions, is high in the core region of Fig. 12, and the quasi-particles spend most of their life there, nearly bound to the atom. In case the charge density is all confined within the core region (as for 4f in lanthanides), then the bond description loses its meaning and the atomic description holds. 

It was noted earlier that the charge density of a narrow resonance band lies within the atoms rather than in the interstitial regions of the crystal in contrast to the main conduction electron density. In this sense it is sometimes said to be localized. However, the charge density from each state in the band is divided among many atoms and it is only when all states up to the Fermi level have contributed that the correct average number of electrons per atom is produced. In a rare earth such as terbium the 8 4f electrons are essentially in atomic 4f states and the number of 4f electrons per atom is fixed without reference to the Fermi level. In this case the f-states are also said to be locaUzed but in a very different sense. Unfortunately the two senses are often confused in literature on the actinides and, in order not to do so here, we shall refer to resonant states and Mott-localized states specifically. 

The Ln(III) cations of the series Ce-Lu possess the extended Xe-core electronic configuration [Xe] 4/1 (n = 1-14), a symbol which perfectly pictures the limited radial extension of the f-orbitals The 4f shell is embedded in the interior of the ion, well shielded by the 5s2 and 5p6 orbitals [63], A plot of the radial charge densities for the 4f, 5s, 5p and 6s electrons for Gd+ visually explains why Ln(III) cations are commonly thought as a tripositively-charged closed shell inert-gas electron cloutf (Fig. 1) [63]. 

The lanthanide series of elements differ from the transition metal series in that 4f shell inner electrons are shielded by the 5s2,5p6 closed shells. Consequently f shell electrons interact much less strongly with their environment than the d electrons in the transition series. The radial charge density for Pr3+ is shown in Fig. 8.1. The electronic structure of the f lanthanide ion is dominated by many different interactions than for more familiar d transition metal ions. 

It is worthwhile to demonstrate the competition between interactions by means of a qualitative evaluation of the strengths of the various interactions. This ev iluation is based on the properties of the radieil wavefunctions Rni(r) of the 4f, 5d, 6s and 6p electrons. In fig. 1.20 the radial charge densities Rh(r) are plotted as functions of r for the 4f, 5s, 5p, 5d, 6s and 6p electrons of Ce I 4f5d6s6p. These charge distributions, which are characteristic of all lanthanides were obtained by Z.B. Goldschmidt (1972) by performing Hartree-Fock calculations. 

Myron and Liu (1970) studied the energy bands of fee La and the hypothetical fee Pr by the relativistic APW method. The potentials were constructed from the relativistic atomic charge densities of the configurations 5d 6s 4f for La and 4d 6s 4f for Pr. The 4f charge distribution for Pr was assumed to be paramagnetic. The three lowest energy bands for La are shown in fig. 3.13. The resemblance of these bands to those for Th (Gupta and Loucks, 1969) is... 

Fig. 3.63. The radial charge densities of 4f, 5d, and 6s electrons in the Wigner-Seitz sphere of Gd metal (Harmon and Freeman, 1974b).

The increase of the net hydration, h, with decreasing ionic radii, r, for both the 4f and 5f elements (Fourest et al. 1984) shown in fig. 4 can be attributed to the increase in the surface charge density as the atomic number increases. The break in the lanthanide series for the ions from Nd to Gd is also observed for other bulk physical data such as... 

Fig. 9. The calculated radial charge densities of the 4f, 5d and 6s electrons in the Wigner-Seilz heie of gadolinium metal, taken from Harmon and Freeman (I974a,b).

---