4f charge distribution

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... 

Both Rhyne and Feron estimated kKO) as 2.7 0.3 x 10 Jm whilst Cock (1976) deduced a value of 3.4 0.3 x 10 Jm The basal plane anisotropy in Ho is thus larger than in any other lanthanide, a result of the 4f charge distribution associated with the large orbital moment (L = 6). As in the case of k , the temperature dependence of kI cannot be readily parameterized and requires further investigation. 

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.

The position of Ba in the periodic system just before the rare earth elements indicates that the electronic charge distribution in the outermost region will be very sensitive to changes in the effective nuclear charge. Previously, in Sect. 6.1, we studied the effect of collapse of the 4f-orbital on the behaviour of 4 s and 4p holes. In this section we shall investigate the effect of collapse of the 5d-shell, as already briefly discussed in Section 3.4 (see also1073). 

We have so far considered the single f electron case. Now we turn to the case of the unfilled 4f shell. We have already shown that the electrostatic potential V (r) which a 4f electron experiences due to a charge distribution p(r) of its surrounding is of the form... 

The calculation of the single particle 4f-electron wave functions in atomic problems shows that even if these states are occupied only after the 5s, 5p and 6s orbitals, the charge distribution of the 4f is such that most of it is inside the sphere... 

One of the most striking properties of lanthanide metals and compounds is the relative insensitivity of electrons in the unfilled 4f shell to the local environment, compared to non-f electron shells with similar atomic binding energies. Whereas the 5d and 6s electrons form itinerant electron bands in the metallic solids, the 4f electrons remain localised with negligible overlap with neighbouring ions. For the maximum in the 4f radial charge distribution lies within those of the closed 5s and 5p shells, so the 4f shell is well shielded from external perturbations on the atomic potential, such as the crystal field. 

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. 

There is almost no overlapping of the charge distribution of the 4f electron and those of the 6s and 6p electrons. 

On moving to the right hand side of the lanthanide group, the charge distributions under discussion, especially that of the 4f electron, contract ( the lanthanide contraction - see section 8). 

In fig. 3.63 we show the radial charge distribution of 4f, 5d and 6s electrons in the Wigner-Seitz sphere of Gd (Harmon and Freeman, 1974b). The 4f charge drops off rapidly and becomes vanishingly small at the WS sphere radius. 

When our unrealistic assumption of spherical symmetry for the charge distribution of 4f electrons without spin is removed, each population of this open shell gives rise to a large number of nonequivalent states with different total energies (multiplets). This is a classical problem of atomic physics which will only be sketched here in order to define the concepts used in the analysis of the spectra. The different quantum number sets characterizing the states of F, and f will be simply labelled and Q, respectively. Any state of a population N and in... 

In deep core level absorption spectroscopies the overlap of the core level wavefunction with the wavefunction of the excited state is of little or no importance. Deep core levels may be regarded as classical charge distributions, completely screened by the inner shells and the photoelectron. Most of the deep core level spectra can therefore be described neglecting many-electron processes. The large spatial overlap of the shallow 4d wayefunction with the 4f wavefunctions, however, favors relaxation processes of the excited atom involving spectacular many-electron effects as the giant resonances . The correlation of these exotic final-state structures... 

The charge distribution of the electrons of the 4f shell can be expanded in successive harmonics. The lowest-order terms of this expansion, i.e. the second-order ones, are the 5 components of the quadrupolar moment. For a charge distribution associated with a state... 

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