Theoretical investigations of hardness in hexaborides ReB6 (Re = La, Ce, Nd) compounds and their solid solutions, La1_xCexB6 and Nd1_xCexB6: A DFT + U plus SO study

The search for new hard and extra-hard materials is of great interest for many industrial applications. Therefore, in this study, we aim to investigate the mechanical properties, particularly the hardness, of hexaboride com-pounds ReB6 (Re = La, Ce, Nd) and their solid solutions La1_xCexB6 and Nd1_xCexB6 for x = 0.00, 0.25, 0.50, 0.75, and 1.00. We use the DFT + U + SO method to predict the macro and micro-hardness of these materials through the Vickers macro-hardness model and the S?imunek micro-hardness model. The calculated hardness values of binary compounds and their alloys indicate that they are both macroscopically and microscopically hard, and can withstand high melting temperatures. The macro-hardness values of the binary compounds LaB6 (25.147 GPa), CeB6 (25.954 GPa), and NdB6 (25.006 GPa) are comparable to those of semiconductors, silica, and ceramics. On the other hand, the micro-hardness values of S?imunek LaB6 (7.76 GPa), CeB6 (9.88 GPa), and NdB6 (11.69 GPa) are lower than their respective Vickers macro-hardness values. The solid solutions with x = 0.25 (La0.75Ce0.25B6 and Nd0.75Ce0.25B6) are found to be the macroscopically hardest alloys, with a hardness of 26.28 GPa and 25.79 GPa, respectively. Conversely, the solid solutions with x = 0.75 (La0.25Ce0.75B6 and Nd0.25Ce0.75B6) are found to be the microscopically hardest solutions, with a hardness of 9.82 GPa and 9.88 GPa, respectively.