`color{green}("Crystal Lattice :")` The diagrammatical representation of three dimensional arrangement of constituent particles in a crystal is called crystal lattice. In this each particles is depicted as a point. .

`color{green}("Bravias Lattices :")` There are only 14 possible three dimensional lattices. These are called Bravias Lattice.

(a) Each point in a lattice is called lattice point or lattice site.

(b) Each point in a crystal lattice represents one constituent particle which may be an atom, a molecule (group of atoms) or an ion.

(c) Lattice points are joined by straight lines to bring out the geometry of the lattice.

`=>` It is the smallest portion of a crystal lattice. Repetition of this unit cell in different directions generates the entire lattice.

`color{green}("Characteristics of Unit Cell")` :
(i) its dimensions along the three edges `color{red}(a)`, `color{red}(b)` and `color{red}(c)`. These edges may or may not be mutually perpendicular.

(ii) angles between the edges, `color{red}(α)` (between `color{red}(b)` and `color{red}(c)`), `color{red}(β)` (between `color{red}(a)` and `color{red}(c)`) and `color{red}(γ)` (between `color{red}(a)` and `color{red}(b)`). Thus, a unit cell is characterised by six parameters, `color{red}(a)`, `color{red}(b)`, `color{red}(c)`, `color{red}(α)`, `color{red}(β)` and `color{red}(γ)`.

(i) `color{green}("Primitive Unit Cell :")` In this, constituent particles are present only on the corner positions of a unit cell.

(ii) `color{green}("Centred Unit Cell :")` In this unit cell, constituent particles are present at positions other than corners. These are of three types :

(a) `color{green}("Body-Centred Unit Cells :")` In this one constituent particle is present at its body centre besides the one present at the corners.

(b) `color{green}("Face-Centred Unit Cells :")` In this, one constituent particle is present at the centre of each face besides the one present at the corners.

(c) `color{green}("End-Centred Unit Cells :")` In this, one constituent particle is present at the centre of any two opposite faces besides the one present at the corners.

There are seven types of primitive unit cells (Fig 1.7).

The characteristics of primitive unit cells along with the centred unit cells is listed in Table 1.3.

Unit Cells of 14 types of Bravias Lattice.

(i) `color{green}("Primitive Cubic Unit Cell :")` It has atoms only at its corner. Each atom at a corner is shared between eight adjacent unit cells.

In Fig.1.9 (a), the constituent particle is shown by only the centre of the particle and not its actual size. This is called open structure.

In Fig.1.9 (b), the constituent particles are shown by their actual size. It is called space-filling structure.

In Fig 1.9 (c), the actual portions of different atoms present in a cubic unit cell is shown.

Since each cubic unit cell has `color{red}(8)` atoms on each corner

`therefore` `color{red}("No. of atoms in unit cell" = 8xx1/8 = 1 "atom")`


(ii) `color{green}("Body-centred Cubic Unit Cell :")` In this there are `color{red}(8)` atoms at each corner and one atom at the body centre of the cube.

Thus in a BCC (i) `color{red}(8 "corner" xx 1/8 "per corner atom" = 8xx1/8 = 1 "atom")`

(ii) `color{red}(1 "body centre atom" = 1xx1 = 1 "atom")`

`therefore` Total number of atoms per unit cell `color{red}(= 2 "atoms")`


(iii) `color{green}("Face-Centred Cubic Unit Cell :")` In this there are `color{red}(8)` atoms at each corner and `6` atom at each face of the cube.

Atoms present at face-centre is shared between two adjacent unit cells.

Thus, (i) `color{red}(8 "corner atoms" xx 1/8 "atom per unit cell "= 8xx1/8 = 1 "atom")`

(ii) `color{red}(6 "face-centred atoms" xx 1/2 "atom per unit cell" = 6xx1/2 = 3 "atoms")`

`therefore` Total number of atoms per unit cell `color{red}(= 4)` atoms