Question

2.6 (a) Derive linear density expressions for FCC [100) and[111] directions in terms of the atomic radius R (b) Derive planar density expressions for FCC (100) and (111) planes in terms of the atomic radius R

Answer 1

Part a

i)

Linear density expression for FCC [100]

Diagram of FCC [100]

Linear density

= ( number of atoms centered to direction 100) / (length of vector)

= ( number of atoms centered to direction 100) / (unit cell edge length)

For the given diagram

At two unit cell corners, there is only 1 atom at each.

number of atoms centered to direction 100 = 1 atom

unit cell edge length = 2R(2)^0.5

Linear density = 1 / [2R(2)^0.5]

ii)

Linear density expression for FCC [111]

Diagram of FCC [111]

Linear density

= ( number of atoms centered to direction 111) / (length of vector)

Vectors pass through the centers of single atom at each end.

number of atoms centered to direction 111 = 1 atom

length of vector z = (x² + y²)^0.5

Length of bottom face diagonal x = 4R

Unit cell edge length y = 2R(2)^0.5

z = [(4R)² + {2R(2)^0.5}² ]^0.5

z = 2R(6)^0.5

Linear density = 1/[2R(6)^0.5]

Part b

i)

Planar density expression for FCC [100]

Diagram of FCC [100]

Planar density

= ( number of atoms centered to direction 100) / (area of plane)

One atom at each corner and each corner atom is shared with unit cells and the middle atom is in the unit cell.

number of atoms centered to direction 100 = 2 atoms

Length of unit cell edge = 2R(2)^0.5

Plane is a square

Area = [2R(2)^0.5]² = 8R²

Planar density = 2 / 8R² = 1/(4R²)

ii)

Planar density expression for FCC [111]

Diagram of FCC [111]

Planar density

= ( number of atoms centered to direction 111) / (area of plane)

6 atoms on the plane are A, B, C, D, E, F

1/6 atom of A, D, F equivalent to 1/2 atom

1/2 atom of B, C, E equivalent to 3/2 atom

number of atoms centered to direction 111 = 1/2 + 3/2

= 2 atoms

Area of triangle = 1/2 x base length x h

(2R)² + h² = (4R)²

h = 2R(3)^0.5

Area = 1/2 x (4R) x h = 1/2 x (4R) x 2R(3)^0.5

= 4R²(3)^0.5

Planar density = 2 / [4R²(3)^0.5]

= 1/ [2R²(3)^0.5]