•The positive-definite conditions for 1D QCs (Laue 4–10) have been deduced.•The independent elastic constants and the positive-definite conditions for 2D QCs (Laue 6, 8–10) have been obtained.•The positive-definite conditions for cubic 3D QCs have been achieved.•All point groups for 2D QCs with aperiodic symmetry have been drawn explicitly.Mechanical stability is essential for solids and its stability criterion may date back to 75 years ago. Recently, the closed form necessary and sufficient conditions for elastic stability in all crystal classes have been investigated. Quasicrystals (QCs) are solids with long-range order and crystallographically forbidden rotational symmetries but without translational symmetry, attracting intense attentions in the last 30 years. In this work, we have explored the elastic constants and the elastic stability in detail for 1D, 2D and 3D QCs. All independent elastic constants and the closed form of necessary and sufficient conditions for elastic stability in all QCs classes are obtained, as a concise and pedagogical reference to stability criteria in aperiodic materials. Meanwhile, symmetry positions and stereographic projections of each QCs class are given as well.

Using ab initio evolutionary simulations, we have explored the potential crystal structures with up to 18 atoms in the unit cell for all possible stoichiometries of the Zr–B system. In addition to the reported ZrB, ZrB2, ZrB12, oP8-ZrB and Zr3B4, two extraordinary Zr2B3 and Zr3B2 have been found to be both mechanically and dynamically stable, as verified by the calculated elastic constants and phonon dispersions. The calculated formation enthalpies reveal that both the new phases may be synthesized and found in experiments. It also reveals that pressure is beneficial for the synthesis of all available phases, except for the ZrB phase. In addition, the values of the Vickers hardness for five Zr–B compounds are predicted utilizing two different models. Contrary to the known hard phases of ZrB2 and ZrB12, the two new compounds both have low values of hardness (less than 10 GPa), which is consistent with their excellent ductility deduced from the bulk and shear moduli. Electronic structure calculations suggest that the new phases are both metallic, while electronic density maps show strong ionic bonding characteristics between the Zr and B atoms. The crystal orbital Hamilton population (COHP) diagrams have also been calculated in order to further analyze the bonding features of the uncovered two novel phases.

On the basis of the evolutionary methodology for crystal structure prediction, the potential crystal structures of magnesium carbide with a chemical composition of Mg2C are explored. Except the known cubic phase (Fmm), two novel tetragonal structures (P42/mnm and I41/Amd) and two novel hexagonal structures (P63/mmc and PM2) of Mg2C are found. All these four new phases are mechanically and dynamically stable by the calculated elastic constants and phonon dispersions. Furthermore, the effects of pressure and temperature on the phase transitions among different Mg2C polymorphs are investigated, implying that some new phases especially the P42/mnm phase may be synthesized in future. The ratio values of B/G are also calculated in order to analyze the brittle and ductile nature of these Mg2C phases. In addition, electronic structure calculations suggest that the I41/Amd phase is semimetallic and the other three new phases are all metallic, which is different from the previously proposed magnesium carbides. Meanwhile, the calculated electronic density maps reveal that strong ionic bonding exists between the Mg and C atoms.

The potential structures of magnesium nitride with a chemical composition of Mg3N2 are examined by utilizing a widely adopted evolutionary methodology for crystal structure prediction. In addition to the previously proposed phases (α-, β- and γ-Mg3N2), we find five high-pressure phases for Mg3N2: (1) a Cmc21 symmetric structure (ε-Mg3N2) at 27 GPa, (2) a R3c̅ symmetric structure (τ-Mg3N2) at 30 GPa, (3) a Cmcm symmetric structure (ω-Mg3N2) at 53 GPa, (4) a Ima2 symmetric structure (λ-Mg3N2) at 68 GPa, and (5) a Ibam symmetric structure (μ-Mg3N2) at 115 GPa. All these phases are mechanically and dynamically stable by checking the elastic constants and phonon dispersion. All phases are direct band gap semiconductors except that the ω-Mg3N2 phase has a nontrivial indirect band gap. Mechanical properties calculations reveal that the γ-Mg3N2 phase has superior ductility than other phases. The Vickers hardness of each phase has been evaluated to be about 15 GPa based on an empirical relation.

On the basis of the evolutionary methodology for crystal structure prediction, the potential crystal structures of magnesium carbide with a chemical composition of Mg2C are explored. Except the known cubic phase (Fmm), two novel tetragonal structures (P42/mnm and I41/Amd) and two novel hexagonal structures (P63/mmc and PM2) of Mg2C are found. All these four new phases are mechanically and dynamically stable by the calculated elastic constants and phonon dispersions. Furthermore, the effects of pressure and temperature on the phase transitions among different Mg2C polymorphs are investigated, implying that some new phases especially the P42/mnm phase may be synthesized in future. The ratio values of B/G are also calculated in order to analyze the brittle and ductile nature of these Mg2C phases. In addition, electronic structure calculations suggest that the I41/Amd phase is semimetallic and the other three new phases are all metallic, which is different from the previously proposed magnesium carbides. Meanwhile, the calculated electronic density maps reveal that strong ionic bonding exists between the Mg and C atoms.

Using ab initio evolutionary simulations, we have explored the potential crystal structures with up to 18 atoms in the unit cell for all possible stoichiometries of the Zr–B system. In addition to the reported ZrB, ZrB2, ZrB12, oP8-ZrB and Zr3B4, two extraordinary Zr2B3 and Zr3B2 have been found to be both mechanically and dynamically stable, as verified by the calculated elastic constants and phonon dispersions. The calculated formation enthalpies reveal that both the new phases may be synthesized and found in experiments. It also reveals that pressure is beneficial for the synthesis of all available phases, except for the ZrB phase. In addition, the values of the Vickers hardness for five Zr–B compounds are predicted utilizing two different models. Contrary to the known hard phases of ZrB2 and ZrB12, the two new compounds both have low values of hardness (less than 10 GPa), which is consistent with their excellent ductility deduced from the bulk and shear moduli. Electronic structure calculations suggest that the new phases are both metallic, while electronic density maps show strong ionic bonding characteristics between the Zr and B atoms. The crystal orbital Hamilton population (COHP) diagrams have also been calculated in order to further analyze the bonding features of the uncovered two novel phases.