Figure 8: Calculated projected density of states using PBE
functional
Mechanical properties
For LiBH4 with the orthorhombic crystal structure, there
are nine independent-elastic-constants namely C11,
C12, C13, C22,
C23, C33, C44,
C55, and C66. They should satisfy the
following Born’s stability criterions [33] for mechanical stability
which is found to be satisfied rigorously according to our outcomes as
given in table-3.
\(C_{11}>0\), \(C_{22}>0\), \(C_{33}>0\), \(C_{44}>0\),\(C_{55}>0\), \(C_{66}>0\) (1)
\(\left[C_{11}+\ C_{22}+C_{33}+{2C}_{12}{+2C}_{13}+{2C}_{23}\right]>0\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (2)\)
\(C_{11}+C_{22}-{2C}_{12}>0\),\(C_{11}+C_{33}-{2C}_{13}>0\),\(C_{22}+C_{33}-{2C}_{23}>0\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (3)\)
Moreover, other mechanical parameters are calculated using
Voigt-Reuss-Hill’s approximation [34-35]. It is well-known that
elastic properties are closely related to the basic solid state
properties such as Bulk, Young and Shear moduli, thermal expansion,
Debye temperature etc. The values of elastic parameters are summarized
under table-3. It is worth mentioning that neither theoretical nor
former experimental data is available to compare with our outcomes.
Thus, mechanical behavior of the studied compound is being explored for
the first time with varying pressure prior to report its fitness for
hydrogen storage devices.
\begin{equation}
B_{V}=\ \frac{[C_{11}+\ C_{22}+C_{33}+{2(C}_{12}{+C}_{13}+C_{23})]}{9}\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (1)\nonumber \\
\end{equation}\begin{equation}
B_{R}=\ \begin{matrix}[C_{11}\left(C_{22}+C_{33}-{2C}_{23}\right){+C}_{22}\left(C_{33}-2C_{13}\right)-2C_{33}C_{12}\\
+C_{12}\left(2C_{33}-C_{12}\right)+C_{13}\left(2C_{12}-C_{13}\right)+C_{23}\left(2C_{13}-C_{23}\right)]^{-1}\\
\end{matrix}\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (2)\nonumber \\
\end{equation}Where B is Hill’s bulk modulus which is average of Voigt-Reuss bulk
modulus and
\begin{equation}
B=\ {[B}_{V}+B_{R}]/2\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (3)\nonumber \\
\end{equation}And\(=C_{13}\left(C_{12}C_{23}-{C_{13}C}_{22}\right)+C_{23}\left(C_{12}C_{13}-C_{23}C_{11}\right){+C}_{33}\left(C_{11}C_{22}-C_{12\ }^{2}\right)\ \ \ \ \ \ \ \ \ \ \ \ \ \ (4)\)
\begin{equation}
G_{V}=\ \frac{\left[C_{11}+\ C_{22}+C_{33}+{3(C}_{44}{+C}_{55}+C_{66}\right)-{(C}_{12}{+C}_{13}+C_{23}]}{15}\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (5)\nonumber \\
\end{equation}\begin{equation}
G_{R}=\ \begin{matrix}15\{4[C_{11}\left(C_{22}+C_{33}+C_{23}\right){+C}_{22}\left(C_{33}+C_{13}\right)+C_{33}C_{12}\\
-C_{12}\left(C_{23}+C_{12}\right)-C_{13}\left(C_{12}+C_{13}\right)-C_{23}\left(C_{13}+C_{23}\right)]/\\
+3[\left(\frac{1}{C_{44}}\right)+\left(\frac{1}{C_{55}}\right)+\left(\frac{1}{C_{66}}\right)\}^{-1}\\
\end{matrix}\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (6)\nonumber \\
\end{equation}\begin{equation}
G=\ {[G}_{V}+G_{R}]/2\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (7)\nonumber \\
\end{equation}Hill’s Shear modulus G (GPa) is an average of Voigt-Reuss shear moduli.
While the ratio of Hill’s bulk and shear moduli be recognized as Pugh’s
ratio which is an essential parameter to predict whether the studied
material associates to brittle or ductile nature [36].
Table 3: Values of the elastic constants C11,
C12, C13, C22,
C23, C33, C44,
C55, and C66 in \((GPa)\), Bulk, Young
and Shear moduli in \((GPa)\), Poisson’s coefficient \(\nu\), Anisotropy
factor A and Pugh’s ratio \(\frac{B}{G}\).