Metallic structures II: complex geometrically determined structures

Abstract

Let no one destitute of geometry enter my doors. Plato (427–347 BC) This chapter considers complicated metallic structures determined primarily from geometric considerations. These geometric constraints help us to understand structures with large and small metallic species. First, we will introduce the concept of topological close packing , which lies at the basis of the Frank–Kasper phases . Second, we will discuss the concept of dumbbell substitutions , illustrating how pairs of small atoms can be substituted for single large atoms, to allow for deviations from stoichiometric compositions. All of these geometrical ideas are rooted in quantum mechanical principles, which we will discuss briefly. Electronic states in metals The free electron theory of metals assumes an isotropic, uniformly dense, electron gas. This is an idealization because the charge density of crystalline solids is restricted by lattice periodicity. The free electron theory offers some guiding principles to understand metallic structures. A large portion of the cohesive energy in metals derives from the energy of the electron gas . This energy depends sensitively on the electron density and its spatial variation. The density of states describes the distribution of electron energies in a solid. In a free electron gas, the electrons occupy discrete quantum states (in pairs with opposite spins) consistent with the Pauli exclusion principle. All electrons are assigned a state in sequentially higher energy levels. “Free” refers to the approximation that conduction electrons see, on average, a zero potential in the metal, allowing us to calculate analytically the density of states using quantum mechanics.