Experimental results have reported that hafnium is a hcp metal with a lattice parameter ratio of c/a=1.58. [1] We verify this result with ab initio calculations; our calculations suggest that the lattice parameters of hafnium are a=3.12Å, c=4.94Å, assuming that hcp is the stable crystal structure of hafnium.

Figure 1 – The unit cell of hafnium

Methods and Results

We used the CASTEP code with the CA-PZ functional (Ceperley and Alder 1980, Perdew and Zunger 1981) for density functional theory calculations in the local density approximation (LDA). Ultrasoft pseudopotentials with core radii of 2.096Å were generated on-the-fly, and calculations in reciprocal space used a cutoff energy of 500eV and a 13×13×10 k-point mesh.

Atomic calculations were performed for the following core orbitals: 1s2 2s2 2p6 3s2 3p6 3d10 4s2 4p6 4d10 4f14 5s2 5p6 5d2 6s2

Pseudopotentials were used for the following valence orbitals: 4f14 5s2 5p6 5d2 6s2

Using twice the empirical atomic radius (1.59Å) as a starting point, we began searching for a minimum in the free energy with an energy cutoff of 435.4eV and 9×9×6 k-point mesh until we found an approximate minimum at a=3.12Å, to three significant figures. c was initially fixed with respect to a by the ratio c/a=1.58 and was updated as a varied to maintain this ratio. Once we had found this minimum, we increased energy cutoff and k-point sampling fineness until the final free energy had converged with a spead of less than ~0.001eV. We checked convergence by increasing cutoff energy first, then by increasing k-point sampling, and then checked that the cutoff energy had not been consequentially altered by the change in k-point sampling. We then returned to the minimization process to check that the same value of a minimized free energy. Having confirmed this, we minimized free energy by varying c with fixed a. After finding a minimum at c=4.94Å, as shown in Figure 2, we checked whether a=3.12Å still minimized free energy, and we confirmed that it did up to the level of precision established earlier. This is shown in Figure 3.

Figure 2 – Final minimization of the free energy over c

Figure 3 – Final minimization of the free energy over a

Discussion

Previously published results [3] indicate that a=3.20Å and c/a=1.582; our results corroborate the latter ratio (our c/a=1.583) but disagree significantly with the actual magnitudes of the lattice parameters. Given that our colleagues matched published values much more closely by using different functionals and parameters, such as the generalized-gradient approximation, we suspect that the LDA functional is inadequate for precision calculations, but a closer examination of LDA parameters reveal a way for this functional to be comparatively useful.