# Crystal Lattice

Crystal lattice is the periodic and systematic arrangement of atoms that are found in crystals with the exception of amorphous solids and gases. In the simplest of terms, the crystal lattice can be considered as the points of intersection between straight lines in a three-dimensional network. The physical properties of crystals like cleavage, electronic band structure and optical transparency are predominantly governed by the crystal lattice. A unit cell is the smallest component of the crystal lattice and describes the arrangement of atoms in a crystal. The unit cell is characterized by its lattice parameters which consist of the length of the cell edges and the angles between them.

There are seven crystal lattice systems namely monoclinic, triclinic, orthorhombic, rhombohedral, tetragonal, hexagonal and cubic. All these seven crystal lattice have their own variants giving rise to a total of 14 Bravais Lattices. Among these seven lattice systems, the cubic system exhibits the symmetry of a cube is considered to be the simplest and also the most symmetric of all crystal lattices. There are three different types of cubical crystal lattice systems namely Simple Cubic (SC), Body-Centered Cubic (CBC) and the Face Centered Cubic (FCC).  As the name indicates, Simple Cubic is the simplest form of the structure. The simple cubic unit cell has one lattice point at each of the eight corners of the cube. The number of lattice points is usually denoted by Z and for a SC, the value of Z is 1. The packing efficiency of a lattice is defined as the ratio of total volume of the cell occupied by the host atom to the volume of the cell. Usually, crystals with higher packing efficiency are preferred because of their favorable properties. This packing efficiency for SC works has been reported to be around 52%.

The unit cell of a BBC crystal lattice contains one host atom in the center of the cube and one each at the corner of the cube. In a BCC, the corner atoms do not overlap with one another and the value of Z is 2. The packing efficiency of BCC is higher than the SC and works out to be 68%. The unit cells in FCC have one host atom at each corner and one host atom on each face. The value of Z for FCC is 4 and has a very high packing efficiency of 74%. The FCC crystal lattice is also known as the Cubic Closest Packing (CCP) because it has the maximum efficiency for spheres of equal radius. The structure of cesium chloride (CsCl) is that of a Simple Cubic whereas Zinc Sulfide is said to have a Face Centered Cubic arrangement.

Many of the electrical and mechanical properties of these crystals are controlled by the defects and irregularities arising in the ideal geometry of the crystal lattice.  It has been reported that the presence of magnetic impurities can cause severe alteration of properties such as specific heat of the substance. Certain dislocations in the crystal could lead to a substance which allows shear at lower stress than that deserved for an ideal crystal.

It should be noted that the stability of a crystal structure cannot be predicted solely based on the chemical composition of the substance. Even the prediction of medium sized structures, an enormous computational power is required to run sophisticated algorithms such as evolutionary algorithms, random sampling and principles of metadynamics. Based on possible combinations of symmetry, there are a total of 32 crystal classes of which 20 are piezoelectric as they are devoid of a center of symmetry. Ten of these 32 groups are polar and hence pyroelectric too. Certain solid substances exhibit the phenomenon of Polymorphism wherein the same substance can exist in more than one crystalline form and have different sets of properties.