In this project, the lattice constant and structure of crystalline platinum were found by performing CASTEP energy calculations and finding where the energy was minimized [1]. These calculations were performed in Materials Studio using the GGA Perdew Burke Ernzerhof (PBA) functional. The calculations used an energy cutoff of 321.1eV for the rougher calculations and 420eV for the finer calculations. The pseudopotential was solved using the Koelling-Harmon atomic solver with the interior shells 1s2 2s2 2p6 3s2 3p6 3d10 4s2 4p6 4d10, giving an energy of -13041.2296 eV. The outer shells used in the calculations consisted of the 4f14 5s2 5p6 5d9 6s1 electrons.
1. Simple Cubic Lattice (P23 Point Group)
Fig. 1. A primitive cell of the simple cubic lattice
For the simple cubic lattice rough calculations, 6x6x6 k points were sampled with 0.1eV Gaussian smearing and no shift. After applying symmetry, this sampling reduced to 11 total k points.
Table 1. The roughly calculated energy of the simple cubic lattice for varying values of the lattice constant.
Fig. 2. The roughly calculated energy of the simple cubic lattice configuration for different lattice constants. The minimum occurs at 2.6 Angstroms, which will help determine a starting value for both the finer, more time consuming simple cubic calculations and the rough calculations for other lattice structures. The lines connecting the data points act as a guide to the eye and are not data.
Fig. 3. K space sampling vs. energy (eV) for the P23 simple cubic lattice. The energy converges to 0.01eV beyond 176 points.
Upon finding the approximate lattice constant with an 11 point k space sampling and a 321.1eV energy cutoff, we can perform the same calculations around that point with a higher k space sampling and cutoff energy to find a more accurate result. For these calculations, 176 k points were sampled (16x16x16) and a cutoff of 420eV was used. These results should converge to 0.01eV. Ultimately, these calculations resulted in a lattice constant of approximately 2.62 Angstroms.
Table 2. The finer lattice constant vs. energy calculations.
Fig 4. Finer lattice constant vs. energy graph. The lines connecting the data points act as a guide to the eye and are not data.
2. Hexagonal Close-Packed Lattice (D3H-3 Point Group)
Fig. 5. A primitive cell of the hexagonal close-packed lattice.
For the hexagonal close-packed lattice, 24x24x18 k points were sampled with 0.1eV Gaussian smearing and a shift of (0.021inverse Angstroms ,0.021 inverse Angstroms,0). The calculation was ultimately performed using 882 k points. For this lattice structure, only calculations with a 420eV cutoff were performed resulting in approximately a 2.60 Angstrom lattice constant. For these calculations, the ratio c/a was set at 1.633 because it gives ideal hard sphere close packing.
Table 3. Table of the HCP lattice constant (angstroms) and corresponding energy. The minimum lies at approximately 2.60 and 2.61 Angstroms.
Fig. 6. Graph of the Pt HCP lattice constant (Angstroms) vs. energy (eV). The lines connecting the data points act as a guide to the eye and are not data.
Fig. 7. K space sampling vs. energy (eV) for the HCP lattice. The energy converges to 0.01eV beyond 882 points.
3. Face Centered Cubic Lattice
Fig. 8. A conventional cell of the FCC cubic lattice.
3a. F23 Point Group
For the first F23 FCC lattice calculations, 8x8x8 k points were sampled with 0.1eV Gaussian smearing and no shift. This resulted in the rough calculation being performed on 88 total k points and the finer calculations being performed on 176 k points. The finer calculation were performed on 16x16x16 k points, resulting in a total sample of 688 points, with an energy cutoff of 420eV. Both the rough and fine calculations resulted in a lattice constant of 2.80 Angstroms.
Table 4. Shows the roughly calculated energy of the F23 FCC lattice compared to the lattice constant.
Fig. 9. Shows a rough calculation of the energy of the F23 FCC lattice vs. the lattice constant. The minimum occurs around 2.8 Angstroms, which will be used as a starting point for the finer calculations. The lines connecting the data points act as a guide to the eye and are not data.
Fig. 10. K space sampling vs. energy (eV) for the F23 FCC lattice. The energy converges to 0.01eV beyond 76 points.
Table 5. Shows the lattice constant (Angstroms) vs. energy for the F23 FCC lattice with a finer k space sampling and energy cutoff. The minimum lies at approximately 2.8 Angstroms.
Fig. 11. Graph of the lattice constant (Angstroms) vs. the finely calculated energy (eV) of the configuration. The lines connecting the data points act as a guide to the eye and are not data.
3b. FM-3M Point Group
For the second FCC lattice calculation, the FM-3M point group was used, as platinum has been shown to form an FCC lattice under this point group in nature [2]. For the rough calculation, 8x8x8 k points were sampled with a 0.1eV Gaussian smearing and no shift, for a total of 20 k points. For the finer calculations, 16x16x16 k points were sampled, for a total of 120 points.
Table 6. Shows the energy of the FM-3M FCC lattice compared to the lattice constant.
Fig. 12. Shows the lattice constant vs. the energy of the FM-3M FCC Lattice for a rough calculation. The minimum occurs at 3.95 Angstroms, which will be used as a starting point for finer calculations.The lines connecting the data points act as a guide to the eye and are not data.
Fig. 13. K space sampling vs. energy (eV) for the F23 FCC lattice. The energy converges to 0.01eV beyond 35 points.
Table 7. Results for the fine calculations of the energy of the FM-3M FCC lattice at various lattice constants.
Fig. 14. Graph of the fine calculations of the lattice constant (Angstroms) vs. energy (eV) for the FM-3M FCC Lattice. The lines connecting the data points act as a guide to the eye and are not data.
For the FM-3M FCC lattice, the fine calculations were performed with a 420eV energy cutoff and a k space sampling of 16x16x16, resulting in a total of 120 k points being used. These results gave a minimum energy of -52203.84eV at 3.96 Angstroms.
Final Results
Fig. 15. (a) Comparing the rough results from the F23 FCC lattice, the P23 simple cubic lattice, and the D3H-3 HCP lattice with c/a=1.633, the F23 lattice has the lowest energy minimum. (b) Comparing all four lattice structures, the FM-3M FCC lattice energy is approximately 4 times lower and reaches a minimum at about 1.16 Angstroms higher lattice constant. The lines connecting the data points act as a guide to the eye and are not data.
For the first three lattice configurations, the F23 FCC lattice had the lowest energy minimum at lattice constant 2.8 Angstroms. However, the FM-3M FCC lattice has a lower energy by approximately a factor of 4, with a minimum at 3.96 Angstroms. This lattice configuration closely matches with experimental data, which shows that platinum forms a lattice in the FM-3M point group with lattice constant 3.92 Angstroms [2].
Appendix: Energy Cutoff Convergence
For the finer energy vs. lattice constant calculations, a 420eV cutoff energy was chosen based on the energy’s convergence to 0.01eV beyond this point.
Fig. 16. Graph of the energy cutoff (eV) vs. energy (eV) for platinum in an F23 FCC lattice. The energy converges beyond a cutoff of approximately 390eV and 420eV was ultimately used for the finer energy calculations.
Bibliography
2. Povarennych, A. & Povarennyck, A. Crystal chemical classification of minerals. 192 (Plenum Press, 1972).
