Purpose of Calculation

“Hf is experimentally observed to be an hcp metal with c/a = 1.58. Perform calculations to predict the lattice parameters for Hf and compare them with experimental observations.” [1]

Fig. 1 The unit cell for hafnium in the hexagonal close-packed crystal structure

Calculation Methodology

The lattice parameter ratio (c/a) and the lattice constant a are predicted for hafnium in the hcp unit cell, by calculating the minimum energy of the system.

All calculations are performed with the Perdew-Burke-Ernzerhof (PBE) [2] exchange-correlation functional, a Generalized Gradient Approximation (GGA) functional. Pseudopotentials were  calculated on the fly, with the cutoff the 4f electrons and above used as the interacting electrons (4f14 5s2 5p6 5d2 6s2), while lower energy electrons were designated as core electrons (1s2 2s2 2sp 3s2 3p6 3d10 4s2 4p6 4d10). The Koelling-Harmon relativistic treatment was used for atomic solutions. [3]0.1 eV Gaussian smearing was used. Calculations were performed with Castep [4].

Convergence Calculations

Calculations to check the convergence of the minimum energy output with respect to the energy cutoff and the k-point mesh were performed, as shown in figures 2 and 3.

Fig. 2 Convergence calculations of the energy with respect to the k-point number

Fig. 3 Convergence calculations of the energy with respect the energy cutoff

Calculations performed with respect to the energy cutoff were performed with a = 3.1946 angstroms, c = 5.0511 angstroms, and a k-point mesh of 9x9x6. Between 480 eV and 500 eV, the free energy varies less than 0.005 eV, so we select 480 eV as our cutoff energy. Similarly, while varying the k-point mesh, we hold the energy cutoff at 480 eV. At a k-point mesh of 9x9x6, the free energy is similarly converged to 0.005 eV. The remaining calculations were performed at 480 eV and a k-point mesh of 9x9x6.

Calculation Results

Figure 4: Calculation of the internal energy of hcp hafnium as a function of lattice constant a for different ratios of the lattice constants, c/a. Included straight lines are meant to guide the eye and aid in identifying data curves. They do not represent the data themselves.

The minimization of the energy with respect to variation in the lattice constant ratio occurs at c/a = 1.581, which matches experimental observations, and helps support that these calculations are properly converged. The value of a that minimizes the free energy for this ratio is 3.20 angstroms, giving a value of 5.059 angstroms for the predicted value of c. These values for the lattice constants predict the expected unit cell for hafnium in the hcp crystal structure, and agree well with reference values from experiment [5].

References

[1]  D. Sholl and J. Steckel, Density Functional Theory: A Practical Introduction. (Wiley 2009)

[2]  John P. Perdew, Kieron Burke, and Matthias Ernzerhof, “Generalized Gradient Approximation Made Simple”, Phys. Rev. Lett. 77, 3865 – Published 28 October 1996; Erratum Phys. Rev. Lett. 78, 1396 (1997)

[3]  D D Koelling and B N Harmon 1977 J. Phys. C: Solid State Phys. 10 3107

[4]  S. J. Clark, M. D. Segall, C. J. Pickard, P. J. Hasnip, M. J. Probert, K. Refson, M. C. Payne, “First principles methods using CASTEP”, Zeitschrift fuer Kristallographie 220(5-6) pp. 567-570 (2005)

[5]  https://www.webelements.com/hafnium/crystal_structure.html