#### 3.2.2. Anisotropic Elastic Properties

In this section, the anisotropic elastic moduli bulk, shear and Young’s modulus, along with the Vickers hardness of Mo

_{2}ZrB

_{2} and Mo

_{2}HfB

_{2}, will be discussed. The discussion starts with the anisotropic bulk modulus shown in

Figure 4.

The minimum of the anisotropic bulk modulus of Mo_{2}ZrB_{2} and Mo_{2}HfB_{2} is 227.14 GPa and 239.41 GPa, respectively, and can be found along the {010} direction. Along the {100}, the bulk modulus of Mo_{2}ZrB_{2} and Mo_{2}HfB_{2} is 257.32 GPa and 263.00 GPa, respectively, which is the maximum.

Taking into account the maximum and minimum of the elastic moduli of Mo

_{2}ZrB

_{2} and Mo

_{2}HfB

_{2}, one can make an assumption of the anisotropy (see

Table 3): For the bulk modulus, the anisotropy is 0.13 and 0.10 for Mo

_{2}ZrB

_{2} and Mo

_{2}HfB

_{2}, respectively.

The anisotropic shear modulus

G is shown in

Figure 5.

The shear modulus near the {011} direction in the b-c plane of Mo_{2}ZrB_{2} and Mo_{2}HfB_{2} is 139.93 GPa and 152.27 GPa, respectively, and it is the minimum. The maximum of the shear modulus is observed near the {101} direction in the a-c plane, as it is 153.26 GPa and 164.12 GPa, for Mo_{2}ZrB_{2} and Mo_{2}HfB_{2}, respectively.

The anisotropic shear modulus of Mo_{2}ZrB_{2} and Mo_{2}HfB_{2} within the ab-c plane, as well as the a-c plane, behaves differently. While within the ab-c plane, G of Mo_{2}HfB_{2} is only slightly changing, while G of Mo_{2}ZrB_{2} shows clear a clear minimum near the {111} direction and maxima near {110} and {001}. Within the a-c plane, G of Mo_{2}ZrB_{2} is almost constant between the {101} and {001} directions, while G of Mo_{2}HfB_{2} shows clear minima and maxima between these directions. This different behavior of the shear modulus of Mo_{2}ZrB_{2} and Mo_{2}HfB_{2} also effects the Young’s modulus and Vickers hardness in the same planes.

Using the maximum and minimum shear modulus of Mo

_{2}ZrB

_{2} and Mo

_{2}HfB

_{2}, the anisotropy of

G is 0.10 and 0.08 for Mo

_{2}ZrB

_{2} and Mo

_{2}HfB

_{2}, respectively (see also

Table 3).

The anisotropic Young’s modulus is shown in

Figure 6.

The Young’s modulus of Mo_{2}ZrB_{2} and Mo_{2}HfB_{2} near the {110} direction in the a-b-plane is the minimum, as Y is 340.83 GPa and 371.43 GPa, respectively. The maximum of the Young’s modulus of Mo_{2}ZrB_{2} and Mo_{2}HfB_{2} is 411.43 GPa and 438.21 GPa, respectively, and it is found along the {100} direction.

Taking into account the maximum and minimum Young’s modulus of Mo

_{2}ZrB

_{2} and Mo

_{2}HfB

_{2}, the anisotropy of

Y is 0.21 and 0.18 for Mo

_{2}ZrB

_{2} and Mo

_{2}HfB

_{2}, respectively (see also

Table 3).

In

Figure 7 the anisotropic Vickers hardness

H^{V} of Mo

_{2}ZrB

_{2} and Mo

_{2}HfB

_{2} is shown.

The anisotropic Vickers hardness was calculated using the bulk and shear modulus according to the formula of Tian et al. [

52]. The minimum of the Vickers hardness of Mo

_{2}ZrB

_{2} and Mo

_{2}HfB

_{2} is along the {100} direction, as

H^{V} is 16.13 GPa and 17.48 GPa, respectively. However, the maximum of the Vickers hardness of Mo

_{2}ZrB

_{2} and Mo

_{2}HfB

_{2} is along different crystallographic directions. Near the {110} direction in the

a-

b plane, the maximum Vickers hardness of Mo

_{2}ZrB

_{2} is 19.47 GPa, while for the same direction, the

H^{V} of Mo

_{2}HfB

_{2} is 20.62 GPa, which is not the maximum

H^{V} of Mo

_{2}HfB

_{2}. The maximum Vickers hardness of Mo

_{2}HfB

_{2} is near the {101} direction in the

a-

c plane as

H^{V} is 20.70 GPa. In the same direction,

H^{V} of Mo

_{2}ZrB

_{2} is 18.85 GPa. The experimental Vickers hardness of Mo

_{2}ZrB

_{2} is 19.50 GPa [

10,

14]; thus, the DFT calculated numbers of the anisotropic Vickers hardness are in very good agreement with the experiment. Unfortunately for Mo

_{2}HfB

_{2}, no experimental Vickers hardness is currently known.

Taking into account the maximum and minimum Vickers hardness of Mo

_{2}ZrB

_{2} and Mo

_{2}HfB

_{2}, the anisotropy of

H^{V} is 0.21 and 0.18 for Mo

_{2}ZrB

_{2} and Mo

_{2}HfB

_{2}, respectively, which is similar to the one of

Y (see also

Table 3).

In summary, the anisotropic bulk, shear and Young’s modulus as well as the Vickers hardness of Mo

_{2}HfB

_{2} are for the same directions in the same plane higher than the elastic moduli of Mo

_{2}ZrB

_{2}, which might be due to their respective interatomic bonding conditions. This hypothesis will be investigated in

Section 3.3. The maxima of the bulk and Young’s modulus of Mo

_{2}ZrB

_{2} and Mo

_{2}HfB

_{2} were observed for the {100} direction, while the maximum of the shear modulus of Mo

_{2}ZrB

_{2} and Mo

_{2}HfB

_{2} was found near the {101} direction. The maximum of the Vickers hardness of Mo

_{2}ZrB

_{2} was found near the {110} direction, while the maximum of the Vickers hardness of Mo

_{2}HfB

_{2} is near the {101} direction. The minimum of the bulk modulus of Mo

_{2}ZrB

_{2} and Mo

_{2}HfB

_{2} is along the {010} direction, while the minimum of the shear modulus of these borides is along the {011} direction. The minimum of the Young’s modulus of Mo

_{2}ZrB

_{2} and Mo

_{2}HfB

_{2} is along the {110} direction, while for the Vickers hardness the minimum is along the {100} direction.

Also, the elastic moduli of Mo_{2}HfB_{2} are less anisotropic than the elastic moduli of Mo_{2}ZrB_{2}. To better compare this with the Universal Anisotropy Index A^{U}, one can also define a B and G averaged anisotropy index and for Mo_{2}HfB_{2}; this is 0.09 (A^{U} = 0.02), while for Mo_{2}ZrB_{2} it is 0.12 (A^{U} = 0.03). The similarity of the calculated B and G averaged anisotropy index and the Universal Anisotropy Index indicates that both indexes can be used to determine the elastic anisotropy, and that Mo_{2}ZrB_{2} and Mo_{2}HfB_{2} show only slightly elastic anisotropic behavior.