M. S.S. BROOKS
2. Chemical bonding in metals t. Assignment of valence
Since the 4f electrons are localized, or non-bonding, most lanthanides have three valence electrons and there is a smooth (approximately linear) change in volume between lanthanum and lutetium, both of which are trivalent. The exceptions are the anomalously large volumes of europium and ytterbium, which are divalent. The small volume decrease between lanthanum and lutetium is, of course, the consequence of the incomplete screening of the additional nuclear charge by the 4f electrons.
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Fig. 1. The experimental equilibrium atomic volumes of the 3d, 4d and 5d transition metals, the lanthanides and the actinides.
The parabolic decrease in volume of the earlier elements in each d transition metal series with increasing atomic number (fig. 1) is closely connected to the number of valence electrons. A parabolic trend in volume means that the relative contraction, the ratio of the volume of the Z + 1 atom to that of the Z atom, decreases approxi- mately linearly with the number of valence electrons. Furthermore, the relative volume decrease is similar for the different series, see fig. 2.
The number of bonding electrons at the beginning of the 5d series may be inferred from comparing the measured atomic volumes of rare earths and transition metals.
On the right in fig. 1, Yb is clearly a divalent lanthanide and, therefore, has two valence electrons. Similarly, lutetium is a trivalent lanthanide and has three valence electrons. On the left in the same figure, these two elements are seen to be also the first two members of the 5d transition series and the third, hafnium, will have four valence electrons.
In contrast, the assignment of valencies to the actinides, in order to derive the number of (6d7s) valence electrons (Zachariasen 1952, 1961, 1964, 1973, Cunningham and Wallmann 1964, Sarkisov 1966, Weigel and Trinkl 1968, Smith et al. 1969, Fahey et al. 1972), should be treated with caution. It was assumed, implicitly or explicitly, that the 5f electrons are, like the 4f's in the lanthanides, localized. Although such methods are useful for the heavy actinide elements, and probably also for light actinides, in connection with bonding in certain molecules and ionic compounds (i.e.
in those cases for which the f electrons are non-bonding), they are less appropriate for the light actinide metals.
2.0 - - 3d s e d e s - - ~ - - L d series . . . 5d serfes
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C O H E S I O N IN RARE E A R T H S A N D A C T I N I D E S 153
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Fig. 2. Ratio of the equilibrium atomic volumes of the Z and Z + 1 elements for the lighter 3d, 4d and 5d transition series.
Another type of analysis directs attention not to the atomic volume, but to the energetics of the valence electrons in the metallic state. Originating from the work of Engel (1949) and Brewer (1963), it is assumed that there are two components in bonding. The first is a bonding energy, the second a promotion energy, required to bring the free atom to the configuration of the atom in the solid. Irregularities in cohesive energies are usually due to the promotion-energy component, and if this can be isolated and removed, the r e m a i n d e r - t h e bonding e n e r g y - s h o u l d be a smooth function, which may be interpolated across a series. This type of analysis may be used to investigate some basic properties of the lanthanides and heavy- actinide metals in which the f electrons do not participate in bonding. It is especially useful for the heavier actinides (e.g. Brewer 1971, Nugent et al. 1973, Johansson and Rosengren 1975b), where the relative stability of the divalent and trivalent metallic phases may be expressed in terms of the free-atom promotion energy, f" + 1 .__, fù, which is measured independently (Johansson and Rosengren 1975a, b, Johansson 1977a).
The cohesive energies (Brewer 1975) of the (non-f) elements in the left part of the periodic table show representative trends, being consistently close to 40 kcal/mol for divalent metals (such as barium and strontium), about 100kcal/mol for trivalent metals (such as lanthanum and yttrium) and about 145 kcal/mol for tetravalent metals (such as hafnium and zirconium).
The cohesive energies of the trivalent lanthanides should therefore be about 100 kcal/mol relative to the trivalent atomic state (f"dsZ). However, the cohesive energy of gadolinium is slightly l e s s - 95 k c a l / m o l - although the configuration of the free gadolinium atom is f75d 6s z. The difference is due to the multiplet coupling between the open 4f and 5d shells in the gadolinium free atom - which is, of course, absent in 3d, 4d and 5d transition metals. This coupling may easily be estimated in the local
spin density approximation (LSDA) (sect. 3.7). In this approximation the multiplet 1 - s s in terms of the 4f5d exchange interaction and the coupling energy is - g a « s « / ~ 4 f #»a
spin components of the 4f and 5d moments. We calculate J4f»d to be 105 meV (or 2.4 kcal) for a free Gd atom (sect. 3.7). Therefore, with a 4f spin moment of seven and a 5d spin moment of one, the magnitude of the multiplet coupling energy is estimated to be about 8.4 kcal/mol in LSDA. Thus the corrected (i.e. with the multiplet coupling energy subtracted) cohesive energy for Gd becomes about 103kcal/mol, to be compared with observed values of 103 kcal/mol for La and 102 kcal/mol for Lu. The corrected cohesive energy is a smooth function across the series. Actually not all of the multiplet coupling energy is lost in the solid, since a small 5d moment remains.
Furthermore, the exchange integrals are slightly different in the free atom and the solid. A more exact estimate of the change in coupling energy is given in sect. 3.7.
The same atomic coupling is also present, with about the same magnitude, in other trivalent free lanthanide atoms (Johansson and Munck 1984). The cohesive energy, relative to the trivalent free atoms, should therefore be about 95 kcal/mol, rather than 100 kcal/mol, for metals with no f shell - a n e t reduction o f a b o u t 5 kcal/mol. Similarly, the measured cohesive energy of the trivalent actinide element curium is about 90 kcal/mol (Ward et al. 1975), which may be taken to be representative of the cohesive energy of the trivalent actinide metals - again relative to the corresponding trivalent (5fn6d7s 2) free-atom configuration.
Therefore, relative to the trivalent free atoms, the cohesive energy is about 50 and 55 kcal/mol greater for the trivalent metals than for the divalent metals, for lanthanides and actinides, respectively. The corresponding difference between the cohesive energies of tetravalent and trivalent metals is also about 45 kcal/mol. However, a promotion energy is required to take the free atom from the divalent to the trivalent state in preparation for a trivalent solid, but not for a divalent solid. Therefore, only if this promotion energy is less than the difference between the divalent and trivalent cohesive energies will the solid be trivalent. The free-atom promotion energies are known experimentally for most of the lanthanides (Martin et al. 1978), but only for some of the actinides. However, the missing promotion energies have been estimated (Brewer 1971, Vander Sluis and Nugent 1972). The measured promotion energies (or estimates where necessary) are plotted in fig. 3, with the critical values (55 and 50 kcal/mol for actinides and rare earths, respectively) at which the metallic state changes from trivalent to divalent.
It is clear from fig. 3 that europium and ytterbium are correctly stated to be the only divalent rare earth metals. Americium, the actinide analogue of europium, should be trivalent. Physically, this is because the free-atom promotion energy in americium is less than the extra energy gained by forming a trivalent solid, and again this agrees with experiment. The promotion energies of the actinides do show that the half-filled shell is especially stable, but for the earlier part of the series the absolute values are lower than those of the corresponding lanthanides, making the divalent condensed phase of americium in contrast to e u r o p i u m - unstable. Curium and berkelium should be trivalent metals. The elements A m - B k crystallize in the typical lanthanide dhcp structure (Lee and Waldron 1972), which is completely consistent with a trivalent metallic ground state (see sect. 4.5).
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Fig. 3. Energy comparison between the atomic excitation energy, f" + 1 s 2 ~ fùds 2, and the critical value separating the divalent and trivalent metallic states.
Figure 3 also shows that, of the heavy actinides, not only the ytterbium analogue, nobelium, but also mendelevium and probably fermium and einsteinium should be divalent metals. Unfortunately, the limited availability of these elements has so far prevented any experimental studies of No, Md and Fm in their elemental metallic states. Californium is an interesting case, where the metallic ground state, according to fig. 3, is on the borderline between the divalent and trivalent condensed phases.
Heat of vaporization measurements on large samples of californium (Ward et al.
1979) show that californium is a trivalent metal. Now, samarium metal, which is trivalent in the bulk, has been found to have a divalent surface (Wertheim and Creselius 1978, Allen et al. 1978, Johansson 1979, Rosengren and Johansson 1982).
Results of experiments on microgram quantities of californium by Haire and Asprey (1976) and Noé and Peterson (1976) were extremely dependent upon the preparation conditions, and various forms of californium metal showing divalent, intermediate and trivalent behaviour, were obtained - encouraging the hypothesis that the surface of californium is, like that of samarium, divalent. Microfilm techniques have been used to prepare thin films ofeinsteinium (Haire and Baybarz 1979). Electron diffraction studies showed that the crystal structure was fcc (as in ytterbium) and that the lattice constant was typical of a divalent metal. Finally, it is quite possible that bulk einsteinium is also divalent.
X-ray diffraction measurements on californium under high pressure have been performed (Burns and Peterson 1978, Peterson et al. 1983), and the data clearly shows that Cf is trivalent at zero pressure. If either bulk einsteinium or fermium turns out to be divalent under ambient conditions, it follows from fig. 3 that modest compression would be sufficient to induce a valence transformation to the trivalent state of either of these two elements. But considerably higher pressures would be required to induce a valence change in mendelevium of, more so, in nobelium.
One of the main differences, therefore, between actinides and rare earths is the steeper change of the f - d energy difference along the actinide series (fig. 3). This first of all makes americium trivalent, but also implies that a series of elements at the end of the actinide series will be divalent. Indeed, the divalent state of nobelium is so relatively stable that, of the possible halide compounds, the only trivalent compound it will be able to form is the trifluoride (Johansson 1977a).
The energy difference between the trivalent and tetravalent metallic states can be investigated in exactly the same way (fig. 4) as between the di- and trivalent states.
It is obvious that no lanthanide can be a tetravalent m e t a l - n o t even cerium (Johansson 1974). The trivalent state is also more stable for the actinides americium, curium and berkelium. But, in the valence picture, both plutonium and neptunium have lower energies as trivalent than tetravalent metals. Hence, tetravalent ground states for Np and Pu may be eliminated. In fact, by comparing figs. 3 and 4 it is easy
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