To determine the desired points of phase transition we performed the computations for each lattice at 23, 29, 35, 41, 47, 50, 53, 56,  61 and 65 K. The potential energies for FCC were computed at the equilibrium lattice spacing of 5.649 A, and also at smaller lattice spacings to simulate the increase of pressure to find its phase transition with the tetragonal phase. The potential energies of the HCP lattice were computed at the equilibrium lattice spacing of 4.039 A, c/a ratio of 1.651. We also computed the energies at a spacing of 3.627 A to simulate its phase transition with the tetragonal phase. This lattice spacing was determined through linear extrapolation of the isothermal compressibility, and was, unfortunatley, much too small, producing a pressure far higher than we had anticipated. The tetragonal phase was computed with a lattice spacing of
5.596 A, with c/a ratio 0.9044, which corresponds to the physical lattice at aprroximately 4 GPa. The energies determined within 20% of those measured in experiment and from other calculations.

Our best result came from the FCC/HCP data. The graph below shows the results of the free energy calculations of the FCC and HCP lattices. The top curve is that of the FCC, while the bottom two are of the same HCP data, fit to the two free energy differences we found. The error on the free energies of specific data points has been left off the graph because these errors are on the order of .1 K, while the error in the free energy differences is larger than the calculated differences. The shape of the free energy curves roughly follow those of other results, although we would like to see the FCC free energy fall of faster than the HCP at low energies. The temperature of the phase transition has been found experimentally to be around 34 degrees.
  One difficulty we ran into out calculations was that our volume was a fixed quantity, while in the actual physical system the pressure is the fixed quantity. For the low pressure calculation between FCC and HCP this was not a problem, but on our other calculations this was. We expected to see the pressure of the FCC phase rise faster with decreased lattice parameter and did not reach the desired phase of the phase transition with the tetragonal phase. At best we could only roughly extrapolate this, assuming a linear dependence, as shown below. Pressures in this case were calculated by a numeric derivative of the potenial energies with respect to volume.

Using our extrapolated HCP lattice constant the HCP lattice energy was far too high to hope to see the phase transition with tetragonal. We assume this means that the pressure resulting from the change in lattice parameter was too high.
 

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