Crystal Lattices and Unit Cell | Definitions, Differences, Types | Note For Science

Crystal lattice:

A regular three dimensional arrangement of points (Atoms, Ions etc.) in space is called crystal lattice.

Unit Cell:

It is the small part of the crystal lattice which have all the properties of the entire crystal.

There are seven types of crystal systems where are:-

Crystal System & Edge Length

1. Cubic $a=b=c$

2. Tetragonal $a=b\neq c$

3. Orthogonal $a\neq b\neq c$

4. Monoclinic $a\neq b\neq c$

5. Rhombohedral $a=b=c$

6. Hexagonal $a=b\neq c$

7. Triclinic $a\neq b\neq c$

Bravais Lattices:- Bravais arrange the points in the crystal system into $14$ different ways whice are called Bravais Lattices.

Bravais Lattices For Cubic System:-

There are three Bravais Lattices for cubic system:-

1. Primitive or Simple Cubic Lattice (S.C. Lattice).

2. Body Centred Cubic Lattice (B.C.C. Lattice).

3. Face Centred Cubic Lattice (F.C.C. Lattice).

1. Primitive or Simple Cubic Lattice (S.C. Lattice):-

The unit cell of Primitive or Simple Cubic Lattice (S.C. Lattice) contain 1-atom at each 8-corners.

Actual numbers of atoms in Simple Cubic Unit Cell:-

$8 Corners ×\frac{1}{8}$ per corner atom

$8×\frac{1}{8}=1\,atom$

2. Body Centred Cubic Lattice (B.C.C. Lattice):-

The unit cell of BCC lattice contain one atom at each of 8 corners as well as one atom at the centre of the cube.

Actual numbers of atoms in Body Centred Cubic Unit Cell:-

$8 Corners ×\frac{1}{8}$ per corner atom $=8×\frac{1}{8}=1\,atom$

1 body centre atom $=1×1=1\,atom$

Total $=1+1\\=2\,atoms$

3. Face Centred Cubic Lattice (F.C.C. Lattice):-

The unit cell of F.C.C. lattice contains one atom at each of 8 corners as well as one atom at the centre of each of 6 faces.

Actual numbers of atoms in Face Centred Cubic Unit Cell (F.C.C. Unit Cell):-

$8 Corners ×\frac{1}{8}$ per corner atom.

$=8×\frac{1}{8}=1\,atom$

$6 Faces ×\frac{1}{2}$ per face centre atom.

$6×\frac{1}{2}=3\,atoms$

Total $=1+3=4\,atoms$

Definition of Simple Unit Cell or Primitive Unit Cell:-

When the constituents particles are present only at the corner position of the unit cell then it is called Simple Unit Cell or Primitive Unit Cell.

Definition of Centred Unit Cell:-

When the constituents particles are present in other position in addition to corner position. Then it is called Centred Unit Cell.

Centred Unit Cells are divided into three categories:

1. Body Centred Unit Cell

When the constituent particle present at the centre of the body then it is known as Body Centred Unit cell.

2. Face Centred Unit Cell

When the constituent particle present at the centre of each faces then it is known as Face Centred Unit cell.

3. End-Centred Unit Cell

In this unit cell one constituent particle is present at the centre of any two opposite face in addition to the corner position.

Packing Fraction:-

It is the ratio of the volume occupied by the atoms in the unit cell to the total volume of the unit cell.

i.e., $Packing\,Fraction=\frac{Vol^m\, occupied\,by\,the\,atoms\,in\,the\,Unit\,Cell}{Total\,Vol^m\,of\,the\,Unit\,Cell}$

Packing Efficiency:-

It is the percentage of the ratio of the volume occupied by the atoms in the unit cell to the total volume of the unit cell.

$Packing\,Fraction$

$=\frac{Vol^m\, occupied\,by\,the\,atoms\,in\,the\,Unit\,Cell}{Total\,Vol^m\,of\,the\,Unit\,Cell}×100\text{%}$