## 3 Results
### Negative Poisson Ratio  
For a stress directed along a certain direction applied to a solid material, the Poisson ratio [16; 17; 18] is defined as the negative value of the ratio of the resulting differential transverse and longitudinal strains, \(\
u=-\ \delta\epsilon_{trans}\ /\ \delta\epsilon_{long}\). In this relationship, the strain, \(\epsilon_{i}\), is the variation of the solid size, \(\Delta L_{i}\), along direction \(i\) divided by the original size, \(L_{i}\), \(\epsilon_{i}=\Delta L_{i}\ /\ L_{i}\). A positive sign is expected for the Poisson ratio because when a positive pressure is applied along a certain direction to the solid, a contraction along the longitudinal direction and an extension along all the transverse directions is usually found. However, negative values of the Poisson ratio are theoretically possible and some materials exhibit anomalous NPRs [16; 17; 18] for certain longitudinal and transverse directions. These solids laterally expand when they are stretched or laterally shrink when compressed. These materials are known as auxetic [59].  
The visualization of the variation of the mechanical properties with the strain orientation is generally quite complicated. The EIAM code was developed by [PERSON] et al. [4] in order to reduce this difficulty to a large extent. 3D representations of the most important elastic properties of rutherfordine were obtained employing this code and are displayed in Figure 1. For the case of the shear modulus and Poisson ratios, depending on two directions (the longitudinal and transverse directions [4]), Figure 1C,D give the representation of the surface of maximum \(G\) and \(\
u\), respectively; that is, the surface formed with the maximum values of these properties for the given direction of the longitudinal strain and all possible transverse directions. These maximum surfaces are usually very similar for compounds with related structures [14; 15] and are very useful to identify shearing effects in the phase transformations between related materials [14; 60]. While the surfaces of the minimum Poisson ratio are very different even for related materials, their analysis is also very useful, since it may reveal the existence of negative values of the Poisson ratio. The surface of the minimum Poison ratio of rutherfordine obtained from the computed elasticity matrix is displayed in Figure 2, and shows that rutherfordine is an NPR material, the value of the lowest Poisson ratio being \(\
u_{min}=-0.29\). The direction of the minimum Poisson ratio is \(U^{L}_{min}=\) (1.0, 0.0, 0.0), and the corresponding transverse  
\begin{table}
\begin{tabular}{c c c} \hline \hline \multirow{2}{*}{_ij_} & \multicolumn{2}{c}{\(C_{ij}\)} \\ \cline{2-3}  & **Previous Work [1]** & **Present Work** \\ \hline
11 & 259.88 & 272.60 \\
22 & 23.03 & 22.69 \\
33 & 165.27 & 178.07 \\
44 & 9.85 & 6.08 \\
55 & 65.88 & 66.90 \\
66 & 15.06 & 9.87 \\
12 & \(-8.33\) & \(-7.61\) \\
13 & 72.39 & 77.03 \\
23 & \(-5.18\) & \(-6.12\) \\ \hline \hline \end{tabular}
\end{table}
Table 1: Computed elastic constants, \(C_{ij}\), for rutherfordine mineral.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline \multicolumn{2}{c}{**Property**} & \multicolumn{2}{c}{**Previous Work [1]**} & \multicolumn{2}{c}{**Present Work**} \\ \hline \(B\) & Bulk modulus & 17.97 & 17.90 \\ \(G\) & Shear modulus & 19.47 & 14.12 \\ \(E\) & Young modulus & 42.92 & 33.53 \\ \(\
u\) & Poisson ratio & 0.10 & 0.19 \\ \(D\) & Ductility index & 0.92 & 1.27 \\ \(H\) & Hardness index & 9.47 & 4.13 \\ \(A^{U}\) & Universal anisotropy index & 8.82 & 13.28 \\ \hline \hline \end{tabular}
\end{table}
Table 2: Computed mechanical properties of rutherfordine. The values of the bulk, shear, and Young moduli (\(B\), \(G\), and \(E\)) are given in in GPa.  
direction is \(U^{T}_{min}=\) (0.0, 1.0, 0.0). The same effect was observed when the elasticity data obtained in the previous work [1] was analyzed and the corresponding value of the lowest Poisson ratio was \(\
u_{min}=-0.27\).