Unit Cells

Introduction

A unit cell is the smallest portion of a crystal lattice that shows the three-dimensional pattern of the entire crystal. A crystal can be thought of as the same unit cell repeated over and over in three dimensions. The Figure below illustrates the relationship of a unit cell with the entire crystal lattice.

 

 

Origin of unit cells

Let us consider a sphere(s) of equal size. In arranging these spheres, we must ensure that only the single layers between the lattices come in close contact with each other. During this process, two layers are formed. They include square and hexagonal packing layers. In square packing layers, the spheres are arranged in a manner that the rows are both horizontally and vertically arranged. Solids containing square packing layers have a coordination number of four as shown in the diagram. Hexagonal packing on the other hand, do not form a similar row and column of cells. 

 

 

Packing of atoms in a unit cell

There are 2 types of packing of unit cells; square packing and hexagonal packing. In a square packing, both the horizontal and verticle units are uniformly placed.  On the other hand, hexagonal packing layers are well-organized. Solids made up of hexagonal packing layers have a coordination number of six, and the voids present are smaller than compared to those in square packing layers.

 

 

 

The unit cells are distinguished based on 4 important parameters. They are;

  1. The fraction of volume occupied by spheres in the cells

  2. The difference between edge length and the radius of one sphere.

  3. The effective number of spheres inside the cell(s).

  4. Location of spheres inside unit cells

 

 

Types of Unit Cell

Many unit cells together make a crystal lattice. Constituent particles like atoms, molecules are also present. Each lattice point is occupied by one such particle.

  1. A simple or primitive cubic unit cell

  2. The body-centred cubic unit cell

  3. The face-centred cubic unit cell

 

 

Simple or primitive cubic unit cells

Simple cubic unit cells are widely used and are normally made up of 3 locations for the atoms/spheres. They are, face centres, body centres and the corners. The table below shows some contributions of spheres placed at various locations.

 

 

 

Body-centred Cubic Unit Cell (BCC)

In short, it is called BCC. A BCC unit cell contains atoms at each corner of the cube along with an atom at the centre of its structure. According to this structure, the atom at the body centre wholly belongs to the unit cell in which it is present. In a BCC unit cell, every corner has atoms, additionally,  one atom present at the centre of the structure.

 

 

Number of Atoms in BCC Cell:

Thus, in a BCC cell, we have:

  • 8 corners × 1/8 per corner atom = 8 × 1/8 = 1 atom

  • 1 body center atom = 1 × 1 = 1 atom

Therefore, the total number of atoms present per unit cell = 2 atoms.

 

 

Face-centred Cubic Unit Cell (FCC)

An FCC unit cell contains atoms at all the corners of the crystal lattice and as well as at the center of all the faces of the cube. The atom present at the face-center is shared between 2 adjacent unit cells and only 1/2 of each atom belongs to an individual cell.

 

 

  1. In FCC unit cell atoms are present in all the corners of the crystal lattice

  2. Also, there is an atom present at the center of every face of the cube

  3. This face-centered atom is shared between two adjacent unit cells

  4. Only 12 of each atom belongs to a unit cell

 

 

Characteristics of cubic unit cells

Face-centered cubic

Body-centered cubic

Simple cubic

It consists of one constituent particle present at the center of each face in addition to those present at the corners.

It consists of one constituent particle at its body center in addition to those present at the corners.

It consists of one constituent particle present at the center of any two opposite faces in addition to those present at the corners.

 

 

The density of cubic crystals

The density of a cubic crystal is calculated by

                        M×Za3×NA

Where, Z- is the rank of the unit cell.

            M- the molar mass of the solid.

            a -the edge length of the unit cell

            NA- Avogadro number

 

 



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