Factors governing a Crystal Structure

Crystalline order, i.e., the three-dimensional (or in the case of quasicrystals or incommensurate phases, higher-dimensional) translationally periodic repetition of a particular atomic configuration, is the outstanding characteristic of condensed matter in thermodynamic equilibrium. Which crystal structure for a given chemical composition corresponds to the lowest Gibbs free energy, G=H-TS, depends on chemical bonding, electronic band structure and geometrical factors. Since it is not possible to solve the Schrodinger equation for a crystal and thus deduce the correct crystal structure, many approximations have been developed. Indeed, today there exist quite successful attempts to predict simpler crystal structures using one-electmn approximations: the  any-electron problem is reduced to a one-electron problem by the assumption that the electrons, surrounded by a mutual exclusion zone, are moving independently of each other in the average field of all the others (local density functional theory).

Beside this rather complicated and lengthy approach to understand and predict crystal structures, there exist a number of rules based on two factors: the chemical bond factor, which also takes into account the directionality of chemical bonds, and the geometrical factor, which considers optimum space filling, symmetry and connectivity. Especially in the case of the typical metallic elements, these structural principles work very well for predicting structures.

1. Chemical bond factor :

   1. The covalent bond
   2. The metallic bond

2. Geometrical factors :

   1. Coordination
   2. Space filling
   3. Layer stackings
   4. Polymorphism