## 4 Results and Discussion
### Mechanical Properties  
Mechanical properties are essential for any materials with technological applications. However, there is very little information on the mechanical properties for pyrophosphate crystals in contrast to their vibrational properties. To fill this gap, we have calculated the elastic coefficients from the VASP relaxed structure for the five crystals. From the elastic coefficients, the mechanical parameters for these crystals are obtained using the VRH approximation for poly-crystals [17; 18]. They are bulk modulus (K), shear modulus (G), [PERSON]'s modulus (E), and [PERSON]'s ratio (\(\mathrm{\SIUnitSymbolMicro}\)), and are listed in Table 4. Also included are the Pugh ratio G/K and the universal elastic anisotropy parameter A\({}^{\mathrm{U}}\)[54]. A\({}^{\mathrm{U}}\) equal to 0 implies perfect isotropy and is usually much less than 2 and seldom goes beyond 4 for most crystals with different crystalline symmetries [54]. There is a large variations of A\({}^{\mathrm{U}}\) among the five pyrophosphate crystals with a low value of 0.7182 for C4 and high value of 3.610 for C1, thus C1 is far more elastically anisotropic than C4. Generally speaking, low symmetry triclinic crystal should have higher A\({}^{\mathrm{U}}\) compared to orthorhombic crystals, however, it is still disconcerting that C5, which is also orthorhombic, has much larger A\({}^{\mathrm{U}}\) than C4. Each of these mechanical parameters are correlated and all are derived from the elastic coefficients. The higher the bulk modulus, the less compressible is the system. Shear modulus is related to the rigidity of the material and Young's modulus represent stiffness of the material. G/K is a parameter based on [PERSON]'s criteria [55] to estimate brittleness or ductility in pure metals from comparative analysis. i.e., if G/K ratio is high, then it is brittle and ductile if the G/K ratio is low. However, [PERSON]'s ratio may also work in other materials, as it also has relation with [PERSON]'s ratio (\(\mathrm{\SIUnitSymbolMicro}\)) i.e., higher G/K ratio lower is \(\mathrm{\SIUnitSymbolMicro}\)and vice versa. All five crystals have low K and G values, with K ranges from 22.88 GPa to 30.28 GPa and G ranges from 13.90 GPa to 15.63 GPa respectively. As a result, G/K ratio ranges from a low value of 0.4591 for C4 to a much higher value of 0.6831 for C1. This is a variation of almost 50% in the Pugh ratio signal the brittle nature for C1 and being ductile for C4. Given the fact that they have almost identical TBOD (C1: 0.02125 and C4: 0.2139), there must be a credible reason for rationalization with the use of Pugh ratio.  
In Figure 6, we plot the G vs K in the range for K from 0 to 40 GPa and G from 0 to 20 GPa for the five pyrophosphates C1-C5 together with a selected set of materials within the same range. The slopes of the dashed lines show the G/K ratio which are close to the inverse of the Poisson's ratio for the pyrophosphates. They can be divided into two groups with G/K = 0.611 (C1, C5) and 0.457 (C2, C3, C4). One is ductile and the other is brittle, according to Pugh ratio criterion of greater or larger than G/K around 0.5 [55]. We are not aware of experimental measurement for these pyrophosphate crystals, so the results presented can be considered as theoretical predictions.  
\begin{table}
\begin{tabular}{c c c c c c c} \hline  & **K(GPa)** & **G(GPa)** & **E(GPa)** & \(\mathrm{\SIUnitSymbolMicro}\) & **G/K** & **A\({}^{\mathrm{U}}\)** \\ \hline
**C1** & 22.88 & 15.63 & 38.19 & 0.2218 & 0.6831 & 3.6100 \\ \hline
**C2** & 30.28 & 14.84 & 38.26 & 0.2894 & 0.4899 & 3.1220 \\ \hline
**C3** & 28.19 & 14.66 & 37.48 & 0.2784 & 0.5199 & 2.7776 \\ \hline
**C4** & 30.26 & 13.90 & 36.15 & 0.3009 & 0.4591 & 0.7182 \\ \hline
**C5** & 24.53 & 15.58 & 38.57 & 0.2379 & 0.6351 & 1.9064 \\ \hline \end{tabular}
\end{table}
Table 4: Calculated Cij and mechanical parameters for the five crystals.