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

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 affects 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 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

rCu = 1.278 Å, a = 2 √2 rCu = 3.614 Å

The length of the lattice vector in [112] direction

= 2.26 × 109 atoms / m.

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 × 4 = 2

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 = a × a

Relation between lattice constant (a) and density (ρ).

Consider a cubic crystal of lattice constant a.

Density of the crystal = ρ

Volume of the unit cell = a3

⸫ Mass of the unit cell = ρ 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 × n                …..(2)

(for n atoms per unit cell)

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

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