(a) Calculate the packing efficiency in c ccp crystal lattice.
(b) What is the number of particles per unit cell of a simple cube.

Explanation

(a) The number of atoms per unit cell in fee structures is four. Each atom is considered as one sphere.
So, the volume of four atoms (four spheres) = `4 xx 4/3 pir^(3) = 16/3 pi r^(3)`
From this figure it is clear that there are three spheres touching each other along the face diagonal. Let .a. be the edge length of the cube and .r. be the radius of the sphere.The face diagonal AC = b, then b = 4r.
In ABC, `AC^(2) = BC^(2) + AB^(2)`
`b^(2) = a^(2) + a^(2) b^(2) = 2a^(2)`
`b = sqrt(2a) = 4r`
`a = 4r/sqrt(2) = 2sqrt(2)r`
The volume of the cubic unit cell =`a^(3) = (2sqrt(2)r)^(3)`
Therefore,Packing efficiency = Volume of four atoms (four spheres) in unit cell/Volume of the unit cell`xx` 100%
Packing efficiency = `16/3 pir^(3)/(2sqrt(2)r)^(3) xx 100%`
Packing efficiency = `(16/3 xx 22/7 r^(3))/8(sqrt(2))^(3)r^(3) xx100%`
Packing efficiency = `(16/3 xx 22/7 )/8(sqrt(2))^(3) xx100% = 74%`
Therefore 74% of unit cell is occupied by atoms and the rest 26% is empty space in fcc structures.
The packing fraction in fcc structures (ccp) = 0.74
The fraction of empty space in fcc structures (ccp) 0.26
(b) One particle.