Stacking Characteristics of Close Packed Materials

It is shown that the enthalpy of any close packed structure for a given element can be characterized as a linear expansion in a set of continuous variables ${\alpha }_n$, which describe the stacking configuration. This enables us to represent the infinite, discrete set of stacking sequences within a finite, continuous space of the expansion parameters $H_n$. These $H_n$ determine the stable structure and vary continuously in the thermodynamic space of pressure, temperature, or composition. The continuity of both spaces means that only transformations between stable structures adjacent in the $H_n$ space are possible, giving the model predictive as well as descriptive ability. We calculate the $H_n$ using density functional theory (DFT)and interatomic potentials for a range of materials. Some striking results are found: e.g., the Lennard-Jones potential model has 11 possible stable structures and over 50 phase transitions as a function of cutoff range. The very different phase diagrams of Sc, Tl, Y, and the lanthanides are understood within a single theory. We find that the widely reported 9R-fcc transition is not allowed in equilibrium thermodynamics, and in cases where it has been reported in experiments (Li, Na), we show that DFT theory is also unable to predict it.

Figure 1

(left) Physically realizable stackings projected onto the ${\alpha }_2\text{−}{\alpha }_3$ plane. Configurations for up to 25 atomic-layer repeats are shown in red. Blue points indicate the 43 structures used in our calculations. (right) Tukay box plots of normalized enthalpy $H_n^{\prime }$ vs $n$ showing the rapid convergence of Eq. (2). Data are taken from DFT calculations across all elements and pressures. The structure-independent $H_0$ are omitted. The normalization is defined by $H_n^{\prime }=(|H_n|/\sum _{i=2}^9|H_i|)$.

Figure 2

(a) Figure showing close packed materials plotted against their $(H_2,H_3)$. Lines show the movement under pressure according to DFT calculations. Blue dots show the position of interatomic potentials at equilibrium volume. The outlying interatomic potential is Fortini’s Ru EAM potential [8]. The regions of fcc, hcp, and dhcp stability shown assume that $H_4$ and higher terms are zero. (b) Expanded view of the position of interatomic potentials in the region of $H_2\text{−}{H}_3$ space bound by the rectangle in (a). The lines again show the effects of compression.

Figure 3

(top) DFT calculated enthalpies for phases of yttrium with pressure. (bottom) Fitted $H_n$ values with pressure. The insets are colored to show the stable phase for given ($H_2$, $H_3$) using the same color scheme; when $H_4$ is positive (left), all six phases appear, for negative $H_4$ (right) only fcc, hcp, and dhcp are possible. The line shows changing values of ($H_2$, $H_3$) with pressure. Because $H_4$ for Y is also pressure dependent, this is a projection onto the plane of constant $H_4$ which it intersects: the line is colored green when the $H_4>0$ and yellow when $H_4<0$ to show that it passes through the wedge of $hhf$ stability, but not $hff$. Small dots indicate 10 GPa intervals.

Figure 4

Correlation between the stability of hcp over fcc ($H_2$) and the divergence from the ideal close packed ratio of $(c/a{)}_0=\sqrt{\frac{2}{3}}$. The effect of pressure up to 20 GPa is again shown as paths colored to correspond to the relative volume.

Figure 5

Zero-pressure $H_2$, $H_3$, and $H_4$ for the Lennard-Jones potential as a function of the interaction range. The diagonal dotted line demonstrates the regular introduction of new $H_i$ series at intervals of the interplanar spacing. The upper of the two ribbons at the top of the graph shows the minimum enthalpy structure at each value of the cutoff, the lower shows the minimum enthalpy structure predicted by Eq. (2) using the $H_n$ values up to $n=4$. The different colors represent different structures described using the hf notation as follows: red, $\mathtt{f}$; blue, $\mathtt{h}$; green, $\mathtt{h}\mathtt{f}$; purple, $\mathtt{h}\mathtt{h}\mathtt{f}$; yellow, $\mathtt{h}\mathtt{h}\mathtt{h}\mathtt{f}$; pink, $\mathtt{h}\mathtt{h}\mathtt{f}\mathtt{f}$; white, $\mathtt{hhfff}$; olive, $\mathtt{hhhhhf}$; lime, $\mathtt{hhhhff}$; cyan, $\mathtt{hhhhhhf}$; brown, $\mathtt{hhhhfff}$; black, $\mathtt{hhffhhf}$.