Binding-energy relations and equations of state for the $4d$ and $5d$ transition metals

A recently proposed six-parameter mixed power-exponential expression is tested against density-functional theory binding-energy curves and equations of state across the $4d$ and $5d$ transition-metal series. It is shown to remove known failures of the popular Birch-Murnaghan, extended Rydberg, and generalized Morse expressions in that it is able to not only reproduce the observed hard-core repulsion of the early transition metals under compression but also remove predicted spurious oscillations in the binding-energy relations under expansion. However, it is unable to fit the well-known anomalous behavior of the equation of state for lanthanum.

Figure 1

(Color online) Comparison of DFT and experimental values (Refs. 32, 33, 34, 35) of $E_0$, $V_0$, $K_0$ and $K_0^{\prime }$ across the $4d$ (left-hand panel) and $5d$ (right-hand panel) transition-metal series. Values of $E_0$ calculated with respect to NSP and SP free-atom energies are given by solid lines connected by circles and triangles, respectively.

Figure 2

(Color online) Normalized binding-energy curves for hcp Y and Lu, bcc Mo and W, and fcc Ag and Au. 4-bm: four-parameter Birch-Murnaghan expansion; 3-bm: three-parameter Birch-Murnaghan expansion; 4-er: four-parameter extended Rydberg expansion; 4-gm: four-parameter generalized Morse expression. In this figure, the 4-er and 4-gm cannot be visually distinguished for Mo, W, Ag, and Au, since their values are very close.

Figure 3

(Color online) ${\widehat{a}}_{\text{BER}}^{\left(4\right)}$ in Birch-Murnaghan binding-energy relation across the $4d$ and $5d$ transition-metal series.

Figure 4

(Color online) Trend of Rose and Vinet scaling lengths $l_v$ and $l_r$, respectively, across the $4d$ and $5d$ transition-metal series. Ground-state structure is taken for each element.

Figure 5

(Color online) Deviation between the generalized Morse and DFT values of nodal point and point of inflection across the $4d$ and $5d$ transition-metal series. Ground-state structure is taken for each element.

Figure 6

(Color online) Trend of ${\mu }_{\text{BER}}$ and logarithmic volume derivative of number of valence $d$ electrons at equilibrium across the $5d$ transition-metal series.

Figure 7

(Color online) ${\widehat{a}}_{\text{EOS}}^{\left(4\right)}$ in Birch-Murnaghan equation of state across the $4d$ and $5d$ transition-metal series.

Figure 8

(Color online) $\text{ln}{H}^{\ast }$ plots for hcp Y and Lu, bcc Mo and W, and fcc Ag and Au. 4-bm: four-parameter Birch-Murnaghan expansion; 3-bm: three-parameter Birch-Murnaghan expansion; 4-er: four-parameter extended Rydberg expansion; 4-gm: four-parameter generalized Morse expression. In this figure, the 4-er and 4-gm cannot be visually distinguished since their values are very close.

Figure 9

(Color online) Shifted binding energy, pressure, and $\text{ln}{H}^{\ast }$ plots for (a) hcp Lu and (b) bcc W. 6-pe: six-parameter power-exponential expression; 6-er: six-parameter extended Rydberg expansion; 6-bm: six-parameter Birch-Murnaghan expansion. Predicted cohesive energies $E_0^{\text{pred}}$ obtained by integrating analytically equations of state are given (in eV) in insets of middle panels.

Figure 10

(Color online) Variation in $K^{\prime }$ against pressure obtained from 6-pe expression. Asymptotic values for Lu and W of 3.39 and 2.75, respectively, are shown by horizontal dotted lines.

Figure 11

(Color online) $\text{ln}{H}^{\ast }$ plots for (a) hcp La and (b) hcp Lu with 6-pe parameters fitted to DFT values over a range of 1000 GPa to $7V_0$ (dashed curves) and also 30 GPa to $7V_0$ (full curve) for La and 584 GPa to $7V_0$ (full curve) for Lu. Insets show the logarithmic volume derivative of the number of valence $s$ electrons $N_s$ versus Vinet scaling length.