Introduction

Zirconium single crystal is experimentally observed to have two different crystal structures. The high-temperature β phase zirconium is a bcc structure while the room temperature α phase zirconium is in hcp structure [1]. The experimentally observed lattice constant for hcp structure α-zirconium by Goldak et al. is a= 3.22945 Å and c= 5.14139 Å at 4.2 K [2]. The room temperature lattice parameter reported by Easton and Betterton is a= 3.2327 Å and c= 5.1471 Å [3].

Density functional theory (DFT) calculation with generalized gradient approximation (GGA) is regarded to be a powerful tool for determining properties of bulk single crystals [4,5]. In this work, we used DFT calculation to predict the lattice constant for α-zirconium. We first fix a/c ratio and calculate energy corresponding to different lattice constants. The lattice parameter corresponding to the lowest energy is the predicted lattice parameter. Our calculated result is then compared to the experimentally observed one to verify the accuracy of density functional theory in calculating lattice parameters for single crystals.

Computational details

Crystal Structure

The unit cell of Zr single crystal was built according to data from Materials Project. The space group of α-zirconium is P6₃/mmc (194). The lattice constant ratio (c/a) is set to 1.592 and the γ angle was set to be 120 degrees. There are two atoms in the unit cell of α-zirconium and the coordinates are (0.3333,0.6667,0.25) and (0.6667,0.3333,0.75). The built α-zirconium is shown in Figure 1.

Figure 1 α-zirconium in hcp structure. The blue atoms are zirconium atoms and the red line indicates the unit cell of α-zirconium.

Convergence test

Our calculation used plane-wave bases with on the fly generated ultrasoft (OTFG-ultrasoft) pseudopotentials in CASTEP [5]. PBE-GGA was used as the functional. Convergence tests were performed on both cutoff energy and k-points. The lattice constant of a= 2.6 Å and c=4.1392 Å (smallest value) is used for the convergence test since the smaller lattice constant usually requires a larger k-point number and this can ensure all the calculations performed in this work converge. K-points are tested from 1 to 198 and values of cutoff energy vary from 100 eV to 600 eV. The result of energy to cut-off energy and k-points relation was shown in Figure 2 and Figure 3. The step energy difference (energy calculated in the current step subtract energy calculated in the last step) of two convergence tests was also calculated and has been shown in Figure 2 and Figure 3. The step energy difference was under 0.01eV when k points reached 144 (14*14*11) and under 0.001eV when cut-off energy reached 475 eV.

Figure 2 Energy and step energy difference versus the k-point number. K-point number of 1 and 2 is not shown in the main figure. Due to the large difference in both energy and step energy difference for k-point number small than 3, the whole picture is shown in the small figure.

Figure 3 Energy and energy difference versus cutoff energy.

Lattice constant calculation

K-point set of 14*14*11 and cutoff energy of 475eV are used to calculate energy corresponding to different lattice parameters ranging from a=2.6Å to a=3.7Å. The core radius for ultrasoft pseudopotential for Zr is 2.1 Bohr (~1.11 Å). The ultrasoft pseudopotential was generated with 12 valence electrons (4s2 4p6 4d2 5s2). The lattice parameter c is determined by the lattice constant ratio (c/a) 1.592 and the value of the lattice parameter a. The result of our calculation is shown in Figure 4. Lattice parameter corresponding to lowest energy (-2578.849923 eV) is a= 3.235 Å and c= 5.15012 Å. Compare to the measured lattice parameter at 4.2 K, our result is a little larger with an error of 0.17%. The calculated result is relatively closer to the experimentally observed data at room temperature, with an error of less than 0.1%. The small error in lattice constant value indicates the accuracy of our DFT calculation. The difference between our calculated result and experimental results may be due to the fact that we don’t know the “true exchange-correlation functional”. Choosing other functionals and performing more calculations may give a more accurate result.

Figure 4 Energy versus lattice constant for α-zirconium. The enlarged figure of the lowest energy point is shown in the small figure. Lattice constant corresponding to the lowest energy is ~3.235 eV.

References:

[1] Versaci, R. A., and M. Ipohorski. Temperature dependence of lattice parameters of alpha-zirconium. No. CNEA–500. Comision Nacional de Energia Atomica, 1991.

[2] Goldak, J., L. T. Lloyd, and C. S. Barrett. “Lattice parameters, thermal expansions, and Grüneisen coefficients of Zirconium, 4.2 to 1130 K.” Physical Review 144.2 (1966): 478.

[3] Easton, D. S., and J. O. Betterton. “The eutectoid region of the Zr− Ga system.” Metallurgical Transactions 1.12 (1970): 3295-3299.

[4] Schnell, I., and R. C. Albers. “Zirconium under pressure: phase transitions and thermodynamics.” Journal of Physics: Condensed Matter 18.5 (2006): 1483.

[5] Sholl, David, and Janice A. Steckel. Density functional theory: a practical introduction. John Wiley & Sons, 2011.