D-occupation number

The high-temperature polymorphic form for most of the rare earth metals just before melting is the bcc structure. Four of the trivalent lanthanides (holmium, erbium, thulium and lutetium) are monomorphic and do not form a bcc structure before melting at atmospheric pressure (see fig. 4). However, the bcc phase can be formed in holmium and erbium by the application of pressure (< 1 GPa), see section 3.7.1. The existence of the bcc phase in the lanthanides has been correlated with the d occupation number, which decreases along the lanthanide series, but increases... 

This correlation between the stable structures and d-occupation number can be related to the crystal structure sequence as observed in the trivalent lanthanides metal series, as we will discuss in the next section. 

The d occupation number for the trivalent rare earth metals (after Skriver, 1983). 

The rapid rise of the hep-Sm-type phase boundary beyond Tm (fig. 123), and the anomalous high pressures of transformation in Y (hep - Sm-type, Sm-type dhep and dhep - fee) relative to lanthanide elements was cited by Gschneidner (1985b) as evidence for 4f valence electron hybridization having a significant role in determining the lanthanide crystal structure. He noted, however, that d occupation number probably is more important in determining which crystal structure would form, as had been proposed by others (e.g., see Duthie and Pettifor, 1977 Skriver, 1983). 

Of course in addition to the dependence on the d occupation number, second order elasticity effects also apply. This probably accounts for the fact that for the heavy lanthanide solvents the departures from Vegard s law exhibited in the c lattice spacing are only slightly negative or do not occur because the electronic and second order elasticity effects tend to cancel one another. 

Thus, the variation of lattice parameters in the intra rare earth alloys is reasonably well understood in terms of elasticity theory and electronic configurations (d occupation numbers). Exceptions to the above correlations are probably due to poor experimental data. 

Experiments (Haire and Baybarz 1979) suggest that einsteinium has an fee structure, but with an atomic volume corresponding to a divalent ion configuration. This crystal structure should therefore be compared with those of europium (bcc) and ytterbium (fee). One can also discern a crystal structure sequence for divalent metals (Johansson and Rosengren 1974, 1975a), which may also be correlated with the d occupation number (Skriver 1982). When viewed in this light, einsteinium is more similar to ytterbium than to europium. Radium, on the other hand, has a bcc phase, which should be compared with barium (bcc) and europium (bcc). Slightly compressed radium is predicted to be a superconductor (Skriver 1982). 

Fig. 15. The calculated s, p and d occupation numbers across the lanthanide series. The dashed lines correspond to divalent metals (Ba, Eu and Yb).

Fig. 24. (a) The calculated s, p and d occupation numbers across the actinide series when the 5f states are part of the core. The full line is for trivalent actinides and the dashed line is for divalent actinides, (b) The calculated s, p, d and f occupation numbers across the actinide series when the 5f states are in the energy bands. 

Brewer 1967) were qualitative and based upon the Engel (1949) correlation, involving the number of sp electrons. The first physical theories were due to Ducastelle and Cyrot-Lackman (1971) and Dalton and Deegan (1969), here relationships between crystal structure and band structure were sought. A more complete treatment for the transition metals, due to Pettifor (1970, 1972), related the hcp bcc hcp fcc sequence to the change in d occupation number across the transition metal series. 

A subsequent, more complete treatment of the band-structure problem for the lanthanides by Skriver (1983b, 1985) confirmed the essence of the Duthie-Pettifor (Duthie and Pettifor 1977) theory, at the same time adding accuracy to the computed structural energy differences. By using full LMTO calculations Skriver was also able to obtain the correct partial occupation numbers - the same as those subsequently calculated by Eriksson et al. (1990c), and shown in fig. 15. By calculating the partial occupation numbers at the pressures required to induce the structural transitions in La, Sm and Gd, Skriver (1983b) was able to associate = 2.0, 1.85 and 1.75 with the dhcp->fcc, Sm-type -> dhep and hep -> Sm-type transition, respectively. Then, calculation of the partial d occupation numbers as function of the pressure, led to fig. 30, which predicts the pressure-induced transitions for the lanthanides. 

Fig. 30. The calculated electronic pressure as a function of d occupation number for the lanthanides. The vertical lines indicate the d occupation numbers for which structural phase transitions are expected to occur (after Skriver 1983b).

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