Category Archives: Hafnium

Lattice parameter for Hf and comparison to experimental measurements

Author: Naveen Agrawal

Introduction: Hafnium is a chemical element with atomic number 72 and electronic configuration [Xe] 4f145d26s2. Hafnium has attracted great technological interest in nuclear science because of its exceptional corrosion resistance and high thermal neutron capture cross-section. Hafnium has been found to form an HCP like crystal as shown in the figure below structure with c/a ratio of 1.58 experimentally[1]. This study involves the Density Functional Theory calculations to predict the optimum lattice parameter for Hf using the Density Functional Theory with plane-waves basis set program ‘CASTEP’ [4] available in the Materials Studio.

 

Hf crystal structure and the lattice parameters

Methods: CASTEP is used to perform plane-wave based electronic structure calculations which in general requires the selection of certain input parameters such as KPOINTS and ENCUT to optimize the computational effort and the accuracy of electronic structure results. In addition, we choose GGA (Generalized Gradient Approximation) based PBE ( Perdew Burke Ernzerhof) exchange-correlation functional  [2,3] and ultra-soft pseudo-potential with core radius 2.096 Bohrs (1.109 Angstrom) generated  with panel of 26 valence electrons (4f14 5s2 5p6 5d2 6s2) [5]. We first optimize the KPOINTS for an energy cutoff of 500 eV and later do the convergence for energy cut-off (ENCUT) which determines the maximum energy plane-wave included in the solution. These convergence tests and further calculations were performed with the SCF (Self-consistent-field) cycle convergence criteria of 2E-06 eV per atom, using finer criteria improved the accuracy of total energy insignificantly w.r.t to the desired accuracy for the study (1 meV). Optimized ENCUT and KPOINTS are used consistently for determining the optimum lattice parameters through variation in lattice constant ‘a’ for a chosen c/a ratio. Later, we compare the respective minimums obtained at different c/a ratios to determine the most energetically favorable lattice parameters.

 

KPOINTS convergence: Hafnium crystal structure (HCP) and lattice parameters (c/a = 1.5811  and a =3.194 A) imported from Materials Studio library were used for initial convergence testing of KPOINTS and ENCUT. Selection of KPOINTS grid was made in such a way to keep the density of KPOINTS in reciprocal space is uniform. Specifically, for Hafnium the number of KPOINTS were kept in the inverse ratio of unit cell vector length or reciprocal cell vector lengths. For the considered lattice, the ratio of KPOINTS to the nearest integers in X and Y directions to Z directions were kept equal to 1.58 approximately. Based on the above criteria, following grids as shown in the table below were considered for testing.

KPOINT gridIrreducible KpointsTotal energy per atom in eVRelative energy to the energy for highest KPOINTS in meV
5x5x310-7866.6951.504
7x7x416-7866.6942.952
9x9x536-7866.700-2.623
11x11x764-7866.6961.004
14x14x9120-7866.6970.000

KPOINTS convergence for the considered Hf lattice w.r.t relative total energy determined w.r.t to energy at highest number of KPOINTS (a =3.194 Angstrom and c/a = 1.581) with ENCUT = 500 eV

The figure above shows the convergence of total energy per atom in eV relative to same determined at the highest number of KPOINTS against irreducible KPOINTS available for the considered grid of KPOINTS. With the tolerance of 1meV, KPOINTS grid of 11X11X7 seems to be a reasonable choice.

ENCUT convergence: ENCUT( Energy cutoff) is the kinetic energy of the highest kinetic energy plane wave that needs to be considered to obtain a converged solution such that any higher energy cutoff would not lead to an energy difference for the tolerance considered (1meV).

ENCUT convergence for the considered Hf lattice (a= 3.194 Angstrom and c/a = 1.581) and optimized KPOINTS (11x11x7)

The figure above shows the variation in the total energy per atom in eV for the different values of ENCUT considered. Based on the desired tolerance, an ENCUT of 500 eV was determined to be optimum.

Optimization of lattice parameters: As Hafnium is an HCP metal, it needs two lattice constants ‘c’ and ‘a’ to specify the crystal structure. I chose to systematically vary both for a chosen c/a to find the minimum energy configuration for the considered c/a. I also compared the minimums obtained for different c/a ratios to find the most favorable set of lattice parameters.

 

Lattice parameter optimization based on cohesive energy per atom with KPOINTS grid (11x11x7) and ENCUT = 500 eV

The figure above shows the variation in cohesive energy per atom relative to the global minimum obtained in eV for different sets of lattice parameters. The systematic variation in lattice constant ‘a’ for a chosen c/a led to different minimum energy configurations indicated by solid black curve. Based on the cohesive energies per atom minimums obtained at different c/ratios, c/a =1.583 with a =3.2011 Angstrom were determined to be optimum lattice parameters.

c/aa in AngstromCohesive energy per atom in eVRelative cohesive energy per atom in meV (millielectronvolt)
1.5903.1968-8.845900.0749
1.5853.1997-8.845980.0019
1.5833.2011-8.845980.0000
1.5813.2024-8.845960.0124
1.5783.2045-8.845910.0641
1.5753.2065-8.845830.1453
1.5703.2097-8.845620.3547

Conclusion: Optimum Lattice parameters obtained through the DFT calculations (a = 3.2011 Angstrom  and c/a = 1.583) using the CASTEP code with the tolerance of  1meV in the total energy came in close agreement with the experiments ( c/a =1.58, a =3.1964 Angstrom)[1].  However, it should be noted the energetic difference between several minimums obtained are of a lesser order than the tolerance specified, therefore, stricter convergence criteria should be useful to resolve the energy scale to such order confidently.

References:

[1] http://periodictable.com/Elements/072/data.html

[2] J.P. Perdew, K. Burke, M. Ernzerhof, Generalized gradient approximation made simple, Phys. Rev. Lett., 77 (1996) 3865-3868.

[3] J.P. Perdew, J.A. Chevary, S.H. Vosko, K.A. Jackson, M.R. Pederson, D.J. Singh, C. Fiolhais, Atoms, Molecules, Solids, And Surfaces – Applications of theTHE Generalized Gradient Approximation for Exchange and Correlation, Phys. Rev. B, 46 (1992) 6671-6687.

[4] J. Clark Stewart, D. Segall Matthew, J. Pickard Chris, J. Hasnip Phil, I.J. Probert Matt, K. Refson, C. Payne Mike, First principles methods using CASTEP,  Zeitschrift für Kristallographie – Crystalline Materials, 2005, pp. 567.

[5] http://www.physics.rutgers.edu/~dhv/uspp/

Optimizing the structure of hcp Hafnium

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/

 

Lattice Parameter Prediction of Hafnium

For this problem we want to predict the lattice parameters of Hafnium and to compare them with the experimental values, given that Hafnium is observed to be a hcp metal, see Figure 1, with a ratio of c to a of 1.58.

Figure 1: Structure of Hf lattice with a =3.1 and c/a=1.58

Methods

To find the optimal values for the lattice parameters, energy calculations were be preformed at differing values of lattice parameter a and c. To turn this into a pseudo one parameter optimization the ratio of c to a was be held fixed while a was varied, then this was repeated for multiple c/a ratios.

The exchange-correlation functional used was PBE, the pseudopotential used was OTFG ultrasoft and the relativistic treatment was that that of Koelling-Harmon.

Convergence tests with respect to the number of k-points and the energy cutoff energy were preformed and the results of which can be found in the Appendix. Using those results the energy cutoff was set to 435.4eV, and the k-point grid was set to 10x10x6. Also, the SCF tolerance was set to 1.0e-6eV/atom with a convergence window of three steps, all other parameters where kept to the quality fine preset for CASTEP.

Material Studio was used to create the crystal of Hafnium as a hcp metal of group P63/MMC and change the lattice parameters, while CASTEP was used to perform the calculations.

Atomic electron configuration for Hf is:
1s2 2s2 2p6 3s2 3p6 3d10 4s2 4p6 4d10 4f14 5s2 5p6 5d2 6s2

And the pseudo atomic electron configuration for Hf used was:
4f14 5s2 5p6 5d2 6s2

Optimization at c/a=1.58

As a starting point, the c/a ratio of was set to 1.58 and the lattice parameter a was set to the range from 2.5 Å to 4.0 Å in increments of 0.1 Å to determine the range of a where the optimal value for a should lie. The results of those calculations can be found in Table 1 and Figure 2.

Table 1: Results of the calculations for c/a=1.58

Figure 2: Cohesive Energy per atom versus Lattice Parameter a, for c/a =1.58

From Table 1 and Figure 2 it can be seen that the minimum cohesive energy occurs at a = 3.2 Å, a volume per atom of 22.419 Å3.

Optimization of c/a

To optimize the ratio of c to a, energy calculations were preformed for a c to a ratio of 1.56 and 1.60 for lattice parameter a ranging from 3.0 Å to 3.4 Å, with the resulting cohesive energy versus volume plots are shown in Figure 3.

 

Figure 3: Cohesive energy per atom versus Volume per atom plots for the different c/a ratios tested.

From Figure 3 we can see that the cohesive energy was at it’s minimum with a c to a ratio of 1.58.

To obtain the optimum value for a, and by extension c, a quadratic function was fit to the plot of the cohesive energy versus lattice parameter a for the c to a ratio of 1.58. The minimum point of the function was used to find the optimal value of a.

Figure 4: Plot of Cohesive Energy per atom vs lattice parameter a with a c/a=1.58.

From the quadratic fit the optimal values of a and c were found to be 3.20 Å and 5.06 Å, respectively.

From experiment the lattice parameters a and c of Hafnium were found to be 3.1964 Å and 5.0511 Å, respectively.  This corresponds to a relative error for a and c of  0.2% and 0.2%, respectively.

Appendix

Validation of the Cutoff Energy

To validate that the cutoff energy was low enough to obtain the level of precision we want from the calculations we can look at the convergence of the free energy with respect to the cutoff energy. While this calculation was done at the start of this project, it was redone for an a of 3.2Å  and a c/a ratio of 1.58 as it will serve as a stronger validation for the results presented.

Figure 5: Free Energy vs. Cutoff Energy for a=3.2, c/a=1.58 and k-points: 10x10x6

From Figure 5, one can see that the free energy for the cutoff used, 435.4eV, was within 0.05eV of the converged value.

Validation of k-points

To validate that the number of k-points was high enough for the level of precision we want from the calculations, we can look at the convergence of the free energy with respect to the number of k-points.

Figure 6: Free Energy vs. Number of k-points, a=3.2Å, c/a=1.58 and Ecut=480eV

Figure 6 shows that the k-point values used, 10x10x6, was within 0.001eV of the converged value.

Cohesive Energy

The DFT program used, CASTEL does not report the cohesive energy, it reports the energy of the free atom and then the free energy of the crystal.  To correct for that we define the cohesive energy as the free energy of the crystal divided by the number of atoms in the crystal’s cell minus the energy of the free atom.

Determining the lattice constants for Hf

   For Project 1,  the lattice constants for Hf are calculated. Since Hf is an hcp metal with c/a=1.58, we need to calculate the energy by changing the lattice constant c while keeping c/a fixed. And the optimized a, c can be obtained by minimizing the free energy.

1. The hcp structure of Hf

   Figure 1 shows the unit cell of Hf hcp structure. OA, OB, OC are three lattice vectors. The lengths and angles of the lattices vectors (OA, OB, OC) are a, b, c and α, β, γ, respectively. To keep the symmetry, the constraints for the unit cell are a=b, α=β=90 degrees, γ=120 degrees.

Figure 1 The unit cell of Hf

2. The parameters for the calculations

Some important parameters and inputs for the calculation are listed as following:

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

The core configuration of the pseudopotential: 1s2 2s2 2p6 3s2 3p6 3d10 4s2 4p6 4d10

Functional: GGA Perdew Burke Ernzerhof (PBA) functional

Pseudopotential: OTFG ultrasoft

Smearing width: 0.1

Origin shift of K points: 0, 0, 0

The k points grid: investigated in the next section

The energy cutoff: investigated in the next section

3. The convergence of the calculations

   Before starting the actual calculation for the lattice constant, it is important to explore the convergence of the calculations with respect to the number of k points and the energy cutoff. So the most accurate and efficient results can be obtained. For the following calculation, the lattice constant is fixed at c= 5.0511Å, a= 3.1946Å, while the size of k point grid and the energy cutoff are varied.

   (1) The number of k-points in the irreducible part of the BZ

     In this section 3.1, the energy cutoff is fixed at 400 eV, and the size of k point grid changes and Number of k-points in the irreducible part of the BZ (the number of k points) varies from 1 to 162, shown in Table 1. According to Table 1, the relations between Energy/atom, calculation time and the number of k points can be plotted, which are shown in Figure 2. As shown in Figure 2, the energy/atom becomes stable with the increase of k points. However, the calculation time also increases rapidly when the number of k points increases. By combining the results of these two graphs, 9*9*6 k points with 36 k points is chosen to obtain an accurate result with less calculation time.

   A little discussion here, since my calculation involves unit cells with different volumes, the k points need to be chosen so the density of k points in reciprocal space is comparable for the different supercells. But as shown in section 4, since the volumes of unit cell change just a little during the calculation, fixing the number of k points will not make much difference. So in section 4, the number of k points is chosen as 36 for all calculations.

Table 1 The calculation of different sizes of k point grid

Figure 2 The plots between Energy/atom, calculation time and the number of k points. The solid lines are just used for guide the eye.

(2) The energy cutoff

In this section, the number of k point is fixed at 36, and the energy cutoff varies from 200 to 800, shown in Table 2. And Figure 3 shows the relation between energy/atom, calculation time and energy cutoff. Although the energy/atom keeps increasing with the increase of energy cut off, the differences between each energy/atom are becoming smaller and smaller. On the other hand, the calculation time increases rapidly with the increase of energy cut off. In order to save the time and keep the accuracy, the energy cutoff is chosen as 435.4 eV.

Table 2 The calculation of different energy cutoff

Figure 3 The plots between energy/atom, calculation time and energy cutoff. The solid lines are used to guide the eye.

4 The calculation of the lattice constants

Using the results from section 3, the k points and the energy cutoff are chosen as 9*9*6 and 435.4 eV. Then lattice constants c and a are changed to obtain the minimum energy, shown in Table 3. As shown in Figure 4, the unit cell and the lattice constants can be predicted. As shown in Figure 4a, the quadratic relation between energy/atom and volume/atom only valid for a small range of lattice constants around the equilibrium value. A small range of volume/atom is chosen and the relation is re-plotted in Figure 4b. According to the fitting equation in Figure 4b, the fitting volume/atom is 22.64 Å^3 and the lattice constant are c=5.0673 Å, a=3.2072 Å. And for the experiment data, the lattice constants are a=3.1964 Å, c=5.0511 Å (1). So the calculation is accurate with an error of 0.3%.

Table 3 The calculation of energy with different lattice constants.

Figure 4 The plots between energy/atom and volume/atom. the solid lines are used to guide the eye. The dashed line is the fitting line.

5. Some discussion and future works

Although the calculation result is accurate with an error of 0.3%, there are still some ways to further increase the accuracy. First, the energy cut off can be increased so the energy/atom can be more precise. Second, we can calculate more points around the value c=5.05 Å and maybe a more accurate result can be obtained. Third, instead of using the quadratic fitting, we can use a higher order polynomial fitting.

For some future works: first, it is worthwhile changing the ratio c/a to determine the optimum value of c/a. Second, it should be interesting to analyze the behavior of energy at small or large lattice constants.

6. Reference:

(1) https://www.webelements.com/hafnium/crystal_structure.html

Determining the Lattice Constants of Hafnium

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.

[1] D. Sholl and J. Steckel, Density Functional Theory: A Practical Introduction. (Wiley 2009)
[2] 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)
[3] K. Hermann, Crystallography and Surface Structure: An Introduction for Surface Scientists and Nanoscientists. (Wiley-VCH, 2011)

Determining the Lattice Parameters of Hf

DFT and Lattice Parameters

For many metals (simple cubic, body-centered cubic, and face-centered cubic) using DFT to calculate the lattice parameter for a metal or crystal is rather straightforward as there is only one parameter to vary. One may begin by assuming the total energy of the system is a Taylor expansion of the lattice parameter, a, as below:

\begin{equation}E_{tot}(a) = E_{tot}(a_{0}) + \alpha (a-a_{0}) + \beta (a-a_{0})^2\end{equation}

Following Sholl (1), we may reduce this equation to:

\begin{equation}E_{tot}(a) = E_{tot}(a_{0}) + \beta (a-a_{0})^2\end{equation}

Thus, the lattice parameter for many metals and crystals can be determined by sampling various values of a. The energy of the system can then be calculated at each value of a and should look like a quadratic with a minimum at the true value of the lattice parameter.

Lattice Parameters for hcp Crystals

For hexagonal-close packed (hcp) structures, determining the lattice parameters is not so straightforward. For sc, fcc, and bcc metals there is only one parameter to optimize the energy. When considering an hcp metal, there are two lattice parameters on which the total energy depends: a, and c.

It is still possible, however, to manage this multi-variable minimization using our technique from above. First, we fix the ratio c/a to some value, r and sample various values of a at which to calculate the energy. With c/a fixed we will also know the value of c and we may construct our energy curves at varying values of r. The curve with the lowest total energy at its minima will be considered the “theoretical values” of the lattice parameters.

Lattice Parameters of Hf

Hafnium, element 72 on the periodic table, is a d-block transition metal with an hcp crystal structure. We wish to use DFT, as outlined above, to determine the equilibrium (ground state) lattice parameters of Hf.

Below are results obtained from CASTEP single point calculations for r = 1.40, 1.48, 1.58, 1.72, and 1.85 with an energy cut-off of 290 eV and a k-point grid of 8x8x6.

From this plot we can determine that the optimum value of r is about 1.58. Using this, a refined set of calculations may be performed at = c/a=1.58 for various a to calculate an accurate estimate for both a and c. The results of these calculations are found below.

Fig 2. Refined determination of a.

In the above plot the blue points are CASTEP results and ther orange line is the harmonic approximation of the total system energy around the true lattice parameter. From this data we can observe a couple of things. First, we can say that the lattice parameters of Hf are approximately:

$$a_{0} = 3.22$$ $$c_{0} = 5.10$$

We may also see that the behavior of the energy is not truly quadratic with respect to the lattice parameter(s). The harmonic potential is a very good approximation for a near a0, but for a > a0 the harmonic potential is an overestimate of the energy and an underestimate for a < a0. This is because at larger separations (a > a0) the energy decreases as it should approach the dissociation energy of Hf (a → ∞) while at shorter separations (a < a0) the energy increases due to strong repulsive forces.

Fig. 3 Optimized Hf Cell

 

 

WebElements, an online reference for chemical elements, reports the lattice constants for Hafnium as (2):

$$a_{0}^{ref} = 3.20$$

$$c_{0}^{ref} =  5.05$$

Thus our calculations agree quite well with available data and we are satisfied.

Convergence

The above results were found using a relatively small energy cut-off (290 eV) and k-point grid (9x9x6) to allow for quick calculations that give an idea of the behavior or the energy with respect to the lattice constant. We now wish to see if we were converged with respect to the energy cut-off and k-point grid.

For the energy cut-off:

Fig 4. Energy Cut Off Convergence

While it appears that a rather high energy cut-off (~600 eV) is needed for convergence, it is important to realize that the energy differences, even between Ecut=250-270 are within chemical accuracy (~0.04 eV) and therefore using an Ecut of 290 should produce rather reliable results (as we have confirmed with literature values). Chemical accuracy is a standard used by computational chemists as a benchmark for making reliably accurate chemical predictions. Essentially, if energies are within ~0.04 eV then we can make confident predictions; this is exactly the case for our Ecut convergence.

Similarly for the k-points (where we have fixed the ratio kx/kz = ky/kz = 4/3) we find the following. Here our convention for “# of KPoints” is just to add the # of KPoints in each direction (e.g. 8x8x6 = 22 KPoints in the plot)

Fig. 5 K-Point Convergence

Again, we see that not very many k-points are needed (~20) before our energies are within chemical accuracy. This means that our use of an 8x8x6 k-point grid is sufficient for our needs.

Single Point Calculations

For the single point calculations performed for Hf with varying lattice constants the following (ultra-fine) calculation settings were used:

Exchange Correlation Functional Type: Generalized Gradient Approximation (GGA)

Exchange Correlation Functional: PBE

Plane-Wave Energy Cut-Off: 290.0 eV

K-Point Grid: 9x9x6

Pseudopotentials: Ultrasoft

Electronic Energy Convergence Criteria: 5.0 10-7 eV/atom

Geometry Optimization

For the geometry optimization of Hf, the following (ultra-fine) calculation settings were used:

Exchange Correlation Functional Type: Generalized Gradient Approximation (GGA)

Exchange Correlation Functional: PBE

Plane-Wave Energy Cut-Off: 290.0 eV

K-Point Grid: 9x9x6

Pseudopotentials: Ultrasoft

Electronic Energy Convergence Criteria: 5.0 10-7 eV/atom

Ionic Energy Convergence Criteria: 5.0 10-6 eV/atom

Ionic Force Convergence Criteria: 0.01 eV/Å

Future Work:

In the future it would be of interest to investigate the effect of the type of pseudo-potential used (e.g. ultrasoft, soft, etc.) as well as the psuedo-potential cut-off for treating the core and valence electrons. Also of interest would be how our results change using different XC Functionals.

References:

(1) Sholl, D. & Steckel, J. A. Density Functional Theory: A Practical Introduction. (John Wiley & Sons, 2011).

(2) https://www.webelements.com/hafnium/crystal_structure.html

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

Determining the Lattice Constants of Hf in an hcp Crystal Structure

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