Question

Calculate the percentage efficiency of packing in the case of body centered cubic crystal.

Answer 1

In body centered cubic arrangement the spheres are touching along the leading diagonal of the cube as shown in the figure.

In ∆ABC

Ac² = AB² + BC²

AC = `sqrt("AB"^2 + "BC"^2)`

AC = `sqrt("a"^2 + "a"^2)`

= `sqrt(2"a"^2)`

= `sqrt2"a"`

In ∆ACG

AG² = AC² + CG²

AG = `sqrt("AC"^2 + "CG"^2)`

AG = `sqrt((sqrt2"a")^2 + "a"^2)`

AG = `sqrt(2"a"^2 + "a"^2)`

= `sqrt(3"a"^2)`

= `sqrt3"a"`

i.e., `sqrt3"a"` = 4r

r = `sqrt3/4 "a"`

∴ Volume of the sphere with radius ‘r’

= `4/3π"r"^3`

= `4/3π (sqrt3/4 "a")^3`

= `sqrt3/16π "a"^3`

Number of spheres belong to a unit cell in bcc arrangement is equal to two and hence the total volume of all spheres

= `2 xx ((sqrt3π "a"^3)/16)`

= `(sqrt3π "a"^3)/8`

Packing fraction = `"Total volume occupied by spheres in a unit cell"/"Volume of the unit cell" xx 100`

Packing fraction = `(((sqrt3π "a"^3)/8))/(("a"^3)) xx 100`

= `(sqrt3π)/8 xx 100`

= `sqrt3π xx 12.5`

= 1.732 × 3.14 × 12.5

= 68%

i.e., 68 % of the available volume is occupied. The available space is used more efficiently than in simple cubic packing.