Solid with cubic symmetry

Appendix E Elements of elasticity theory E.5.3 Solid with cubic symmetry 

As a final example, we discuss the case of a solid which has cubic symmetry. The minimal symmetry in such a case consists of axes of three-fold rotation, which are the major axes of the cube, as illustrated in Fig. 3.12(a). The rotations by 2n/3 around these symmetry axes lead to the following permutations of the x, y, z axes  

The simplest way to derive these relations is to consider the strain energy density expression of Eq. (E.26), which must be invariant under any of the symmetry operations. This expression for the energy is quadratic in the strains, so that the terms that appear in it can have one of the following forms  

Values are given at room temperature, in units of Mbar (10 Pa). The names of crystal structures are DIA = diamond, ZBL = zincblende, CUB = cubic, FCC = face-centered cubic, BCC = body-centered cubic (for details see chapter 1). 

The Mathematical Theory of Elasticity, A.E.H. Love (Cambridge University Press, Cambridge, 1927). 

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A series of new leucite-type compounds has been synthesized by solid-state reactions in the temperature range 800—1100 °C. With the exception of KgBeSisOia, which is tetragonal at ambient temperatures, a = 12.884, b = 13.441 A, they all crystallize with cubic symmetry (Table 34). At elevated... 

Calculations for Ceo in the LDA approximation [62, 60] yield a narrow band (- 0.4 0.6 eV bandwidth) solid, with a HOMO-LUMO-derived direct band gap of - 1.5 eV at the X point of the fee Brillouin zone. The narrow energy bands and the molecular nature of the electronic structure of fullerenes are indicative of a highly correlated electron system. Since the HOMO and LUMO levels both have the same odd parity, electric dipole transitions between these levels are symmetry forbidden in the free Ceo moleeule. In the crystalline solid, transitions between the direct bandgap states at the T and X points in the cubic Brillouin zone arc also forbidden, but are allowed at the lower symmetry points in the Brillouin zone. The allowed electric dipole... 

Thus, it is herein that we now describe the results of this analysis which we regard as the development of a general strategy for the construction of spherical molecular hosts. [11] We will begin by presenting the idea of self-assembly in the context of spherical hosts and then, after summarizing the Platonic and Archimedean solids, we will provide examples of cubic symmetry-based hosts, from both the laboratory and nature, with structures that conform to these polyhedra. 

Fourteen different unit-cell geometries occur in crystalline solids. All are parallelepipeds—six-sided geometric solids whose faces are parallelograms. We ll be concerned here only with those unit cells that have cubic symmetry that is, cells whose edges are equal in length and whose angles are 90°. 

Molecular frameworks with the shapes of each Platonic solid are known. In symmetry terms, the Platonic solids split into two families the tetrahedron, the cube and the octahedron, which have cubic symmetry, and the icosahedron and the dodecahedron, which have icosahedral symmetry. 

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