Mechanical properties

The range of mechanical, and related, properties computed by GULP has been significantly extended for the present version. Since no article on simulations of ionic materials would be complete without a mention of the ubiquitous and evergreen perennial MgO, we choose to take this well-studied system as an example.

Magnesium oxide adopted the cubic rock salt structure and possesses the well-known characteristic of exhibiting a Cauchy violation in the elastic constants (i.e. $C_{12}\neq C_{44}$). No simple two-body forcefield is capable of reproducing this many body effect. Consequently, it is necessary to use a breathing shell model to describe this material accurately. While there have been previous breathing shell models for MgO [117], we choose to fit a new set of potentials here that reproduce the structure, elastic constants, and high and low frequency dielectric constants under ambient conditions. The resulting potential model is described in Table 2.1, while the calculated properties are given in Table 2.2.

Table 2.1: Breathing shell model for magnesium oxide. Here ``shel'' denotes a potential that acts on the central position of the shell, while ``bshel'' denotes an interaction that acts on the radius of the shell which was fixed at 1.2Å. The charge on $Mg$ is $+2$, while the core and shell charges for $O$ are $+0.8$ and $-2.8$, respectively.
Species 1 Species 2 $A$(eV) $\rho$(Å) $C_{6}$(eVÅ$^{6}$) $k_{cs}$(eVÅ$^{-2}$) $k_{BSM}$(eVÅ$^{-2}$)
Mg core O bshel 28.7374 0.3092 0.0 - -
O shel O shel 0.0 0.3 54.038 - -
O core O shel - - - 46.1524 -
O shel O bshel - - - - 351.439

Table 2.2: Calculated and experimental properties for magnesium oxide under ambient conditions. All experimental elastic properties are taken from reference [118]. Note, for the calculated bulk $\left (K\right )$ and shear $\left (G\right )$ moduli the Hill value is taken, though the variation between definitions is small.
Properties Experiment Calculated
$a$(Å) 4.212 4.2123
$C_{11}\left(GPa\right)$ 297.0 297.1
$C_{12}\left(GPa\right)$ 95.2 95.1
$C_{44}\left(GPa\right)$ 155.7 155.7
$\epsilon^{0}$ 9.86 9.89
$\epsilon^{\infty}$ 2.96 2.94
$K\left(GPa\right)$ 162.5 162.4
$G\left(GPa\right)$ 130.4 130.9
$V_{s}\left(km/s\right)$ 6.06 6.05
$V_{p}\left(km/s\right)$ 9.68 9.70
$\sigma$ 0.18 0.24

The calculated properties for magnesium oxide can be seen to be in excellent agreement with experiment under ambient conditions, with the exception of the Poisson ratio. Of course, this agreement is a consequence of fitting a model with the correct essential physics to a subset of the experimental data. The disagreement in the Poisson ratios is because the values are calculated using different expressions. If the Poisson ratio is evaluated based on the sound velocities according to:


then our calculated value becomes 0.182 in good agreement with the determination of Zha et al [118].

To provide a test of the model, it is possible to calculate the variation of the elastic properties of magnesium oxide as a function of applied pressure. The variation of the elastic constants up to 60 GPa is shown in Figure 2.1.

Figure 2.1: The variation of the elastic constants of magnesium oxide with applied pressure as determined by breathing shell model calculation. $C_{11}$, $C_{12}$, and $C_{44}$ are presented by a solid, dashed and dot-dashed line, respectively.


When compared to the experimental results of Zha et al, the calculated trend in the value of $C_{11}$ is in good agreement in that it consistently increases under pressure and approximately doubles in magnitude by the time that 60 GPa is reached. Unfortunately, the trends for the other elastic constants are not so good, since the curve for $C_{12}$ flattens with increasing pressure, rather than becoming steeper, and the curve for $C_{44}$ passes through a maximum which is not observed in the experimental data from the aforementioned group. However, the calculated trends do match the extrapolated behaviour based upon the results of ultrasonic measurements [119].