The aim of this project is to determine the unit cell volume using density functional theory (DFT). This work takes advantage of the geometric optimization feature of CASTEP[1], a commercially available plane-wave DFT code. Some details of the results follow, as well as comparisons with previous findings.

Initial conditions and setup

Geometry optimization in CASTEP minimizes the total energy of the unit cell that is provided while allowing variations in the geometry of the provided cell. In the current calculation, we provided the hcp structure for Hafnium pictured below.

Initial cell provided for CASTEP calculation.

The initial values of the lattice parameters are a=b=3.1956 Å and c=5.0511 Å. These values are provided by examples within Materials Studio and are accepted experimental values [2]. We maintain Throughout the calculation, symmetry demands that a=b, but a and c are allowed to vary.

Calculation

The following settings were used for Geometry Optimization in CASTEP:

Atomic calculation performed for Hf:
1s2 2s2 2p6 3s2 3p6 3d10 4s2 4p6 4d10 4f14 5s2 5p6 5d2 6s2

Max. force: .05 eV/Å

Cell optimization: Full

Energy cutoff: 435.4 eV

k-points: 9x9x6 (36 points in the irreducible part of the Brillouin zone (IBZ))

Pseudopotentials: OTFG ultrasoft

Functionals: GGA and PBE

Both the number of k-points and the energy cutoff were varied to ensure reasonable convergence. The values above gave a well-converged energy with a reasonable runtime.

Results

The minimized energy was found to be -15.73336 keV and the corresponding lattice constants were found to be a=b=3.2117 Å and c=5.0557 Å. The unit cell volume was found to be 45.163544 Å^3. This is found to be well in agreement with a result obtained in a previous post [3]. Furthermore, it is known that the ideal packing ratio for spheres arranged in an hcp lattice is c/a≈1.633. In this work, the packing ratio was found to be 1.574, approximately a 4% difference.

References

[1] Clark, S. et al. First principles methods using CASTEP. Z. Kristallogr. 220, 567–570 (2005).

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

[3] https://sites.psu.edu/dftap/2018/02/14/determining-the-lattice-constants-for-hf/