Calculate packing efficiency in BCC lattice.

Explanation

Diagram
Relationship between .a. and .r.
`a = 4/sqrt3 r`
OR
`r = (sqrt3a)/4`
Packing efficiency formula
Substitution and answer
Detailed Answer:
In `DeltaABC`,
`b^2 = a^2 + a^2 :.b^2 = 2a^2`,
In, `Delta ABC`

`C^2 = a^2 + b^2 = a^2 + 2a^2 :. C = sqrt 3a`
Radius of the atom `=r`.
Length of the body diagonal ` C =4r`
`sqrt 3a = 4r`
`a = ( 4r)/(sqrt3)`
Edge length of the cube `= a = (4r)/sqrt(3)`
Volume of the cubic unit cell `= a^3 = ((4r)/sqrt(3))^3`
Volume of one particle (sphere) `= 4/3 pi r^3`
The number of particles per unit cell of a bcc = 2
Total volume occupied by two spheres `= 2 xx 4/3 pi r^3`
Packing efficiency `= ("Total volume occupied by the two spheres")/("VOlume of a cubic unit cell") xx 100`
`(4/3pir^3xx2)/((4/sqrt3r)^3) xx100 =(8/3pir^3)/(64/(3sqrt3)r^3)xx100 =68%`
(1) A face centred cube contains 8 lattice points at the eight corners and 6 lattice points at the centres of six faces.
(2) A particle present at the corner shares `1/8` of that particle to each unit cell.
(3) A particle present at the centre of a face provdes a share of `1/2` of that particle to each unit cell.