For the high pressure region, the tetragonal and trigonal
phases are found to be the most stable ones in structure searches at 2-5 GPa and 10-20 GPa, respectively. Interestingly, within the whole pressure range (up to 20 GPa), we did not find the
structure proposed by Filinchuk et al. for the
phase 126, but instead found the
phase with four formula units (44 atoms) per cell and
phase with 2 formula units per cell at pressures below 5 GPa (see Fig. 4.2). Given that the
structure is dynamically unstable at ambient pressure, and based on our enthalpy calculations, we hypothesized that the
and
structures might correspond to the experimentally observed
and
phases. Further investigation confirmed this suggestion, as we will show below.
Lattice constants of the most interesting candidate structures are listed in Table I. The calculated zero-pressure lattice constant ( =
=
= 15.79 ) of the
phase is in excellent agreement with the experimental value (15.76 ) 126, which gives a benchmark of the typical accuracy to expect of DFT simulations for this system. For the
structure, the Mg atom occupies the crystallographic
site at (0.5, 0, 0), the B atom occupies the
site at (0.533, 0.25, 0.375), and the H atoms are at the
sites with coordinates (0.439, 0.247, 0.295) and (0.627, 0.123, 0.369). For the
structure, the Mg atoms occupies the
site at (0, 0, 0) and
site at (0.5, 0.5, 0.5), the B atom occupies at the
site at (0.25, 0.25, 0.25), and the H atoms are at the
sites with coordinates (0.459, 0.282, 0.210), (0.139, 0.327, 0.089), (0.223, 0.038, 0.286), and (0.174, 0.354, 0.416). From Table I, one can see that relaxing the experimental
structure, one gets unexpectedly large changes in the lattice constants – so large that, in fact, the relaxed lattice constants of the
structure are the closest match to the experimental ones 126. One has to keep in mind that what is called the “experimental" cell parameters in many cases is a non-unique result of indexing powder XRD spectra, and this is the case here.
Symmetry |
|
|
|
|
|
(Theory) |
(Expt) |
||||
|
7.69 |
8.32 |
5.58 |
5.79 |
5.44 |
|
7.69 |
8.32 |
5.58 |
5.79 |
5.44 |
|
12.30 |
10.52 |
5.99 |
5.73 |
6.15 |
|
0.986 |
0.984 |
0.963 |
0.933 |
0.987 |
B |
24.0 |
31.0 |
31.7 |
28.5 |
28.5 |
|
4.3 |
3.6 |
3.6 |
3.6 |
5.8 |
|
-17.0 |
-15.5 |
-13.6 |
0 |
|
|
-21.2 |
-19.3 |
-15.4 |
0 |
Mg(BH
)
becomes more stable than the
phase at pressures above 0.7 GPa (Fig. 4.3). In the room-temperature experiment, a pressure-induced structural transformation is observed for the porous
phase, and occurs in two steps: The
phase turns into a diffraction-amorphous phase at 0.4-0.9 GPa, and then at approximately 2.1 GPa into the
phase 126. The calculated phase transition pressure from the
phase to the proposed
phase with
symmetry is 1.2 GPa (the corresponding phase transition pressure for
phase is 0.8 GPa), which are in good agreement with the experimental values (0.4-0.9 GPa). We note a tiny enthalpy difference between
and
structures at pressures around 1 GPa. As pressure increases to 9.8 GPa, the
structure becomes the most stable one, in agreement with earlier predictions 128; 136. Bil et al. 131 indicated that it is important to treat long-range dispersion interactions to get the ground state structures of magnesium borohydrides correctly. We have examined the energetic stability of the considered structures through a semi-empirical Grimme correction to DFT energies, stresses and forces 67 (see the inset of Fig. 4.3). When this correction is included, the
and
structures once again come out as more stable than the
structure, by 21.2 kJ/mol and 15.4 kJ/mol, respectively. Energetic stability seems to correlate with the degree of disparity of bond lengths and atomic Bader charges. The
structure has two inequivalent Mg-H distances, 2.26 and 2.07 , compared to 2.11 and 2.07 in the
structure, and 2.12 and 2.06 in the
structure. As we can see, the more homogeneous bond lengths, the greater stability. Bader charges show the same picture: for H atoms, we find them to be -0.63 and -0.59
in the
structure, -0.63 and 0.62
in the
structure, and -0.63 and -0.61
in the
structure 137. More homogeneous Bader charges and bond lengths in the
and
structures correlate with their greater thermodynamic stability at ambient pressure, in agreement with proposed correlations between local bonding configurations and energetic stability 135.
Our calculations suggest that the structure, proposed by experiment for the
phase, is unstable. This implies that either density functional theory calculations are inaccurate for this system, or experimental structure determination was incorrect. To assess these possibilities, we simulated the XRD patterns of the
and
structures, and compared them with the experimental XRD pattern of the
phase at ambient pressure (see Fig. 4.4a). One observes excellent agreement, both for the positions and the intensities of the peaks (including both strong and weak peaks), of the
structure with experiment 126. The situation is very peculiar: two structures,
and
, have nearly identical XRD patterns, both compatible with the experiment – but one,
, is the true thermodynamic ground state (global minimum of the enthalpy), whereas the other,
, is not even a local minimum of the enthalpy (dynamically unstable structure, incapable of sustaining its own phonons). In this situation, the true structure is clearly
. This example gives a clear real-life example of the fact that very different structures can have very similar powder XRD patterns, making structure determination from powder data dangerous, and in such cases input from theory is invaluable. The
structure also has a rather similar XRD pattern, but the peak positions are slightly shifted. Comparison with an independent experimental XRD pattern collected at 10 GPa (Fig. 4.4b) shows that the peak positions and intensities of the
structure are once again in excellent agreement with the experimental data 128, while the strong peaks of the
structure at
,
, and
obviously deviate from the observed ones. This reinforces our conclusion that the
structure is the best candidate for the high pressure
phase. At pressures below 10 GPa a mixture of
and
phases is possible, as the XRD peaks of these two structures are quite similar. We remind that in the experiment, the
and
phases are nearly indistinguishable 126.
Filinchuk et al. demonstrated the bulk modulus of the phase (28.5 GPa) is almost three times higher than that (10.2 GPa) reported by George et al. by fitting the Murnaghan equation of state 126; 128. Our third-order Birch-Murnaghan 138 fits of the equation of state yielded bulk moduli of the
and
structures equal to 24 GPa and 31.7 GPa, respectively, consistent with the measured value (28.5 GPa) 126. The observed large density difference with respect to the
phase at ambient conditions (44%) is equally consistent with 45% (43%) for
(
) structures 126. Therefore, it is difficult to discriminate between the
and
structures by their compression behavior, density or bulk modulus. Our calculations show that the
structure does not only match all experimental observations for the
phase and has the lowest enthalpy among all sampled structures at the relevant pressure range, but is also dynamically stable – phonons were computed at 0, 5 and 10 GPa. The phonon densities of states (PDOS) of
and
phases at ambient pressure are shown in Fig. 4.5, and once again we see a great degree of similarity. The similarity of all characteristics of these two phases parallels the observed similarity of characteristics of the
and
phases and invites one to propose that while the I41/acd structure corresponds to the
phase, the
phase may have the
structure.