Linear Density and Planar Density

LINEAR DENSITY AND PLANAR DENSITY

Linear Density (L.D.):

It is defined as number of atoms per unit length whose centres on the direction vector for specific crystallographic direction.

This is defined as the number of atoms per unit length along a specific crystal direction.

i.e., Number of atoms centred on direction vector / Length of direction vector ...(1)

The unit of linear density is m^-1, nm¹)

Planar density (PD):

The planar density of a crystal is the density of atoms in a crystal plane.

This is defined as the number of atoms per unit area on a crystal plane. This affect significantly the rate of plastic deformation.

It is defined as the number of atoms per unit area.

= Number of atoms in a plane / The area of the plane

The unit of planar density is (e.g., m^-2, nm^-2)

Problem 1.1

What is the linear density of equivalent points in the [112] direction of copper (fcc)?

Solution:

Since it is fcc structure and

The length of the lattice vector in [112] direction

Problem 1.2    

Calculate the planar density of (fcc) nickel in (100) plane. The radius of nickle atom is 1.245 Å.

Solution:

Number of atoms in (100) plane = 1 + (1/4) x 4 = 2

The radius of atom, r = 1.245 Å

But, Lattice constant, a = 4r/√2

= 2√2 r = 2√2 × 1.245 Å [fcc]

= 3.52 Å

Area of plane = a²

Planar density = No of atoms / Area of plane

Relation between lattice constant (a) and density (p)

Consider a cubic crystal of lattice constant a.

Density of the crystal = p

Volume of the unit cell = a³

.. Mass of the unit cell = p a³ ..................(1)

[Density = Mass / Volume]

Number of atoms per unit cell = n

Atomic weight of the material = M

Avogadro's number = N

Avogadro's number is the number of atoms or molecules per kg mole of the substance

Mass of each atom = M/N

Mass of each unit cell = Mass of each atom in unit cell × Number of atoms per unit cell

Mass of each unit cell = (M/N) x n ..................(2)

(for n atoms per unit cell)

From equations (1) and (2), we have

ρa³ = n M / N

ρ = n M / N a³

ρ = (Number of atoms per unit cell x Atomic weight)/(Avogadro's number × Lattice constant³)

a³ = n M / ρ N

From the above expression, the lattice constant 'a' is calculated. as

a = (nM/ρN)^1/3

ANNA UNIVERSITY SOLVED PROBLEMS

Problem 1.3

Calculate the lattice constant of Fe.

Given: density of iron 7860 kg/m³, atomic weight 55.85 and Avagadro's number 6.023 x 10²⁶ atoms/mol (A.U. May 2013)

Given data

Density of iron p = 7860 kg m^-3

Atomic weight M = 55.85

Avagadro's number N = 6.023 x 10²⁶ mol ^{̄ 1}

Number of atoms per unit cell (BCC) = 2

Solution

We know that

ρ = n M / N a³

a³ = n M / N ρ

Substituting the given values, we have