### Relation between edge length and radius in simple unit, Fcc, Bcc, and Hcp

Tuesday, 15 December 2020

Chemistry
Here you find the relation between edge length (a) and radius (r) for different case. means relation between 'a' and 'r' in bcc, fcc, Ecc etc.

In solid or any substance that have mass made up of small particles. Particles called atoms and molecules. If we see a solid with magnifying it by thousands of time. It looks like the above picture. This above dotted point is called lattice point. And the whole dotted graph called crystal lattice. And each square box is called unit. But in chemistry, it is also called the unit cell. A unit cell is the smallest part of the lattice point which repeated again and again in space lattice.

Note:- Space lattice and crystal lattice are same meaning. We can also use space lattice in place of crystal lattice. Here are also somewhere used space lattice in terms of crystal lattice.

If we define lattice point, the definition will be The point in a crystal lattice where atoms, molecules or ions are present is known as lattice point.

Points to be remember before establishing the relation between edge length and radius a unit cell. We can also use these points as the properties of unit cell.

- A unit cell has eight corner.
- A unit cell has six-face.
- A unit cell has twelve edge.
- It has six parameters that is a, b, c, ⍺, β and γ.

## Relation between edge length and radius in simple unit cell

First of all we should know the meaning of simple unit cell, then we give the relation between edge length and radius for simple unit cell.

A simple unit cell is define as the unit cell in which particles are present all corner of the cube, called simple or primitive unit cell.

As we can see in above fig. There are total 12 edges and eight corners in simple unit cell. Let us consume the edge length be 'a' and radius be 'r'. Then,

The relation between edge length (a) and radius of unit cell (r) in simple unit cell be r = a / 2. i.e, radius of unit cell is equal to the half of edge length. See fig. 3

If we derive this relation by mathematically then,

AB = a ------------ (1)

AB = 2r -----------(2)

a = 2r

r = a / 2

Hence, the mathematical relation between edge length and radius is r = a / 2.

## Relation between edge length and radius in FCC

In face centered cubic unit cell particles are present in each corner as well as in each faces of the cube. That means the total number of particles present in fcc is 14.

In above fig. the unit cell have eight corner and six faces each has one particles. These particles are known as face centered cubic unit cell.

Now, we have to determine the relation b/w 'a' and 'r' in fcc. take a square from this unit cell. see below fig. The relation between edge length and radius of cation and anion in fcc is a = 2√2. lets see the derivation.

Remember the relation between edge length and radius of cation and anion in fcc are same as the relation between edge length and radius in fcc.

**Derivation of relation between edge length (a) and radius (r) in fcc.**

AB = √2a ----------------- (1)

AB = 4r ------------------- (2)

From equ. (1) and (2) we get,

4r = √2a

a = 4r / √2

a = 2√2

Hence, the relation between edge length (a) and radius (r) in fcc is a = 2√2.

## Relation between edge length and radius in Bcc

Bcc means body centered cubic unit cell. This is define as the unit cell in which atoms are present each corner as well as one atom are present at the center of any face of unit cell. So that means in bcc there are total 9 particles are present. see fig.

In above fig. there are one atom present in the center of the unit cell called body centered cubic unit cell. If we match a line passing through this atoms, that line called the diagonal of the cube. So that diagonal will be equal to the 4 times radius of single atom. that means diagonal is equal to 4r. see below fig.

The relation between edge length (a) and radius (r) for bcc is a = 4 √3r / 3 . See the derivation below.

AB = 4r

AB = diagonal of cube = √3a

√3a = 4r

a = 4r / √3

Multiplying by √3 in numerator and denominator.

a = 4 √3r / 3

Hence, the relation between edge length (a) and radius (r) in bcc is a = 4 √3r / 3.

## Relation between edge length and radius in hcp

Hcp means hexagonal closed packed structure. In hcp there are total 16 edge length. see fig. Let the edge length of hcp is 'a' and height be 'c' then the relation between edge length and radius in hcp is c = 2 √2/3a.