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Bravais Lattice - Unit Cell Definition, Types, Primitive unit cells and Centered unit cells

Introduction

Bravais Lattice is an endless number of distinct points created by a series of individual translation operations described by:

R= n1a1+ n2a2+ n3a3
where n1, n2, and n3 are any numbers and a1, a2, and a3 are fundamental integers.
Auguste Bravais studied this concept in the mid-19th century. The Bravais lattice is a three-dimensional grid with 14 fundamental unit cells.

Definition

Lattice: In maths, a lattice is a partly ordered sequence where any two elements have a distinct supremum and an infimum. A supremum is defined as the lowest upper limit of the element, also called its join. In contrast, an infimum is the most significant lower limit of the element, also called its meet.
Space lattice: A point lattice on which atoms are strung.
Unit cell: An essential building component that should be repeated regularly.

Unit Cell

The smallest part of a crystal lattice, a unit cell, creates the complete lattice when replicated in different orientations. Its proportions along the three sides a, b, and c define it. These sides might or might not be perpendicular to one another.

Angles occur between the edges. Greek alphabets represent them. α exists between edges b and c, ß exists between edges a and c, and γ exists between a and b. As a result, a unit cell is defined by six parameters.

Types of Unit Cells

Primitive and centered unit cells are the two types of unit cells that can be found.

Primitive unit cells: When component particles are only found at the unit cell's corner positions, the primitive unit cell is the name of this type of cell.

Centered unit cells: A centered unit cell has one or more component particles in locations in addition to those at the corners.
Centered unit cells can be subdivided into three types.

  • Body-centered unit cells
  • Face centered unit cells
  • End centered unit cells

1. Body-centered unit cells are cells that are centered on the body. Apart from the ones in the corners, such a unit cell has one component particle (atom, molecule, or ion) at its body center.
2.  Face-centered unit cells contain a constituent particle at the midpoint of each face aside from the corners. Hence, each face of such a unit cell includes one component particle in the center.
3. End-centered Unit Cell: This type of unit cell has one component particle in the middle of two opposed faces, in addition to the particles at the corners. It is also referred to as a base-centered unit cell.

Lattice Systems
There are seven lattice systems: triclinic, monoclinic, orthorhombic, tetragonal, trigonal, hexagonal, and cubic. The crystal families and lattice systems are identical except for the trigonal and hexagonal systems, which are merged into one hexagonal crystal family.

The six crystal families are – triclinic, monoclinic, orthorhombic, tetragonal, hexagonal, and cubic.

 

Crystal System

Shape of unit cell

Bravais Lattices

P

B

F

E

Cubic 

Cube 

 

Tetragonal 

Square Prism 

   

Orthorhombic

Rectangular Prism 

Monoclinic

Parallelogram Prism 

   

Triclinic 

Parallelepiped (general)

     

Hexagonal 

120° Rhombic Prism 

     

Rhombohedral

Parallelepiped (equilateral)

     

In the above table, the initials P, B, F, and E stand for the primitive, body-centered, face-centered, and end-centered cube. Cubic The cubic system includes Bravais lattices whose point group is the same as a cube's symmetry. The essential cube, the body-centered cubic, and the face-centered cubic are the three types of cubes. Examples of cubic lattice systems are NaCl and Cu.
In this case, a = b = c

1. Tetragonal
The basic tetragonal shape is created by extending a rectangular pyramid with a square base by tugging on opposing faces of a primitive cube. Thus, tetragonal polygons are simple polygons with four sides. Examples of tetragonal lattice systems are SnO2 and TiO2.
a = b ≠ c

2. Orthorhombic
The simple orthorhombic transforms the tetragonal's square bases into rectangles, resulting in an object with three unequally long perpendicular sides. Examples include KNO3, BaSO4.
a ≠ b ≠ c

3. Monoclinic
Altering the rectangular surfaces opposite to one of the orthorhombic axes into generic parallelograms yields the primitive monoclinic. The base-centered monoclinic is created by extending the base-centered orthorhombic in the same way. Examples of monoclinic lattice systems include Na2SO4.
a ≠ b ≠ c

4. Triclinic
The cube is destroyed by shifting the orthorhombic parallelograms so that none of the axes are perpendicular to the other two. An example of this lattice structure includes CuSO4.5H2O.
a ≠ b ≠ c

5. Rhombohedral
Stretching a cube along one of its axes yields the basic trigonal (or rhombohedral) lattice system. An example is Calcite (CaCO3).
a = b = c

6. Hexagonal
The basis of the hexagonal crystal lattice is a prism with a regular hexagon. Graphite and ZnO are examples.
a = b ≠ c

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