Bulk modulus transition metals

The trends in several ground state properties of transition metals have been shown in Figs. 2, 3 and 15 of Chap. A and Fig. 7 of Chap. C. The parabolic trend in the atomic volume for the 3-6 periods of the periodic table plus the actinides is shown in Fig. 3 of Chap. A. We note that the trend for the actinides is regular only as far as plutonium and that it is also broken by several 3 d metals, all of which are magnetic. Similar anomalies for the actinides would probably be found in Fig. 15 of Chap. A - the bulk modulus - and Fig. 7 of Chap. C - the cohesive energy if more measurements had been made for the heavy actinides. 

The cohesive energy, equilibrium atomic volume, and bulk modulus across a transition metal series may now be evaluated by choosing the following simple exponential forms for ( and h(R), namely... 

Fig. 7.12 The theoretical ( ) and experimental (x) values of the equilibrium band width, Wigner-Seitz radius, cohesive energy, and bulk modulus of the 4d transition metals. (From Pettifbr (1987).)...

To make this point quantitatively requires a knowledge of the radial interactions other than tho.se from the d states (Eq. 20-13). Assuming such terms are the same across a transition. series, and fitting them to the bulk modulus for one metal, does, in combination with Eq. (20-13), predict a minimum in Iq at the center of the... 

Figure 16 Young s modulus of bulk and layered group 4 transition metal nitrides. (Reprinted from Ref 73. 1987, with permission from Elsevier)...

Figure 13.3 shows the equilibrium nuclear separation, cohesive energy, and bulk modulus for 3d and 4d rows of transition metals. The atomic number increases in step of one from 19 to 31 in the left-hand column and from 37 to 49 in the right-hand column. Figure 13.3a shows the expected paraboUc dependence of in-ternuclear separation on the d shell filling. The computed values are correct within a few percent. The structures of metals with shells of 3d 4s (scandium), 3d 4s (titanium), 4d 3s (zirconium), 4d 5s (silver) are more loose. ... 

Fig. 2. Calculated properties of the 3d and 4d transition metals-cohesive energy, lattice constant, and bulk modulus- compared with experiment (crosses). This represents a milestone in the development of the methods to calculate with sufficient accuracy to find these quantities. (From refs. 25 and 123, figure courtesy of V. MoruzzO... 

In summary we can show, in a qualitative way, the transition of valency from divalent over intermediate valent towards trivalency by plotting the bulk modulus as function of pressure as shown in fig. 99. The different regions (semiconductor, intermediate-valent semiconductor, intermediate-valent metal, metal) are clearly indicated (Boppart and Wachter 1984c). 

The relation between a and ay depends on the crystal structure of material. In the case of cubic system, these are cormected by a very simple relation, y = 3a. Although the values of a show a parabolic dependence on the atomic nmnber of the transition metal (number of d-electrons in the system) as shown in Figure 27A (White, 1979), the a and bulk modulus Bq show a more complex behavior as functions of the atomic nmnber for the rare earth elements as shown in Figme 27B (Benedict and Holzapfel, 1993 ... 

FIGURE 27 (A) d electron number versus linear expansion coefficient transition metals (White, 1979). (B) Linear thermal expansion coefficient a (closed circles) and bulk modulus (solid line) Bo of the rare earth elements (Benedict and Holzapfel, 1993 Spedding et al., 1961). 

The bulk modulus may be obtained from the calculated equation of state by differentiation. This quantity is more difficult to calculate accurately than is the equilibrium volume, both because it involves the difference in two calculated numbers and it is sensitive to the volume at which it is calculated. The calculated volumes of the transition metals (Andersen et al. 1985) are usually a little lower than those measured (LDA tends to overbond), and bulk moduli evaluated at the calculated volumes tend to be higher. The bulk moduli of the lanthanide metals calculated by Skriver (1983b) for the fee structure at the calculated equilibrium radii, are shown in fig. 28. The observed slow increase across the series, broken only by the low values for the divalent metals, is reproduced well, although the calculated bulk moduli are typically 30% too high. 

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