# Crystal Structure

## Unit Cell And Lattice

The ordered arrangement of atoms, molecules, or ions in a three-dimensional space showing a definite pattern can be defined as a Crystal lattice. A crystalline solid has a geometrical configuration where the atoms or molecules are placed as points in a three-dimensional space. Crystal is a repetition of a very small regular patterns called  unit cells. A unit cell is the smallest unit in a crystalline solid structure. The complete crystal lattice is designed by the periodic repetition of a single unit cell in all directions.

A basic unit cell has edges a, b, and c along the three axes and angles between these edges are given by ‘α’, i.e., the angle between b and c, ‘β’ ,i.e., angle between a and c, ‘λ’ ,i.e., the angle between a and b.

### Primitive and Non-Primitive Unit Cell

A crystal unit cell is broadly classified into two main categories, those are primitive and non-primitive unit cells.

1. Primitive Unit Cell: A primitive cell structure has particles, i.e., atoms, molecules or ions present only at the corners of the unit cell. These unit cells have a single lattice point since they symmetrically arrange themselves to share the lattice points at all corners with neighbouring unit cells. According to the primitive unit cell definition, there are 7 types of primitive unit cells. They are cubic, tetragonal, orthorhombic, hexagonal, rhombohedral, monoclinic, and triclinic.

2. Non-Primitive Unit Cell: These unit cells are different from primitive unit cells since the constituent particles are present on corners as well as other positions like faces, edges, etc., of the unit cell. The non-primitive unit cells are further divided into body-centered, face-centered, and end-centered crystal systems.

• Body Centered: These types of unit cells have an atom present at the center along with the atoms present at corners.

• Face Centered: Here, the atoms are present at the center of each face and each corner of the unit cell. The cubic and orthorhombic crystal system usually contain this kind of unit cells.

• End Centered: In addition to the constitutional particles present at corners, atoms are also present at the center of the diagonal joining two opposite faces.

### Types of Primitive Unit Cell

The seven basic unit cell shapes show different symmetrical arrangements in the three-dimensional space. These unique cell shapes consist of the seven primitive crystal systems.

1. Cubic: Here, each corner occupies an atom or ion or molecule, they do not touch along diagonally. The edges are all of equal lengths, i.e., a = b = c. The angles are α = β = γ = 900.

2. Tetragonal: This system has three mutually perpendicular axes, i.e., α = β = γ = 900. Two of the edges are equal, i.e., a = b ≠ c. White Sn and TiO2 are some examples.

3. Orthorhombic: The axes are mutually perpendicular, i.e., α = β = γ = 900. All the edges are unequal, i.e., a ≠ b ≠ c.

4. Hexagonal: Two of the edges are equal like tetragonal crystal systems, i.e., a = b ≠ c. The angles are unequal, i.e., α = β = 900; γ = 1200.

5. Rhombohedral: All edges are equal, i.e., a = b = c. The angles are all equal. However, none of the angles are perpendicular, i.e., α = β = γ ≠ 900.

6. Monoclinic: Here, none of the edge lengths are equal, i.e., a ≠ b ≠ c. Two of the angles are 90 degrees, i.e., α = γ = 900; β ≠ 900.

7. Triclinic: In this crystal system, none of the edges are equal, i.e., a ≠ b ≠ c. The angles are all unequal, i.e., α ≠ β ≠ γ ≠ 900.

### Unit Cell Examples

All crystalline solids are composed of any of the above-mentioned crystal systems.

• One example of a unit cell would be the structure of NaCl which is a face-centered cubic unit cell, i.e., it constitutes 4 atoms in 1 unit cell.

• The monoclinic shape of S8 rings is another example. It has two axes perpendicular to each other, while the third forms an oblique angle with the other two.

1. What are the Bravais lattices?

Auguste Bravais demonstrated that there are 14 different ways in which crystal lattice points can be arranged in a unit cell, in the three-dimensional space based on their geometry and without any change in their translational symmetry. For his contributions to the discovery, crystal lattice structures are described as Bravais lattices.

There are three types of Cubic Bravais lattices, that are primitive, face and body-centered, two types of tetragonal lattices, that are primitive and body-centered, four orthorhombic types, that are primitive, face, body, and end-centered, two monoclinic types, that are, primitive and end-centered.

Rhombohedral and Triclinic have only primitive types of Bravais lattice, while a simple hexagonal cell is the Bravais lattice for Hexagonal systems.

2. What is a cubic crystal system? Describe different types of cubic crystals.

The cubic crystal system is one of the seven types of crystal systems. The atoms are present at each corner of the cube. The edge lengths are all equal to each other, i.e., a=b=c. The angles along the axes are perpendicular to each other. The different types of cubic crystals are primitive or plain cubic, face-centered, and body-centered. The primitive cubic unit cell has particles at each corner which are shared by the neighbouring cells. The body-centered cube has an additional atom at the center of the cell, i.e., center of all edge diagonals. The face-centered cube has an atom at the center of each face.