Elastic properties and the mechanical stability of icosahedral boron crystals

The elastic constants of the icosahedral boron crystals have been studied by the formulation of Born and Huang. First of all, a technique of symmetry decomposition has been developed for general crystals possessing molecular units in order to see the relaxation mechanism by the internal shift. It is proven that if a librational mode is Raman active, which is often the case, the mode is able to relax the external strain considerably. For alpha boron, when only central forces are assumed, the c(44) component completely vanishes. A shear strain epsilon 4 induces rotations of icosahedra, which cancel the shear strain completely. This gives a qualitative account for why this crystal is metastable. The rotations of icosahedra frequently happen in order to relax other types of strain too. This rotation-induced relaxation mechanism is looked upon as a special example of the above general property. The cancellation for epsilon 4, would remain in boron carbide, if only central forces are assumed, even though additional elements are introduced in the unit cell. In this case, the stability of the crystal has been ascribed to large noncentral forces, which emerge from the covalent bonds of the Linear chain in the unit cell. Another way of stabilizing the crystal structure of alpha boron is suggested: the surface contact of icosahedra, which is realized in the crystal of beta boron. In this family of crystals, the only direction in which a rotational motion is not induced is the z direction. The deformity of the icosahedron, instead, leads to an unexpected effect on the elasticity of boron carbide. The crystal is shown to be less stiff in the c axis than in the ab plane, despite the strongest interatomic forces being oriented parallel to the c axis. The rhombohedral site slightly deviates from the lattice vector, and this geometry gives rise to a great relaxation in the compression along the c axis.