(i) In a simple cubic unit cell :
Suppose the edge length of the unit cell = a and radius of the sphere = r As spheres are touching each other, evidently, a=2r
No. of spheres per unit cell =

1

8

×8=1
Volume of the sphere =

4

3

πr3
Volume of the cube =a3=(2r)3=8r3
therefore Fraction occupied. ,i.e., packing fraction =(

4

3

πr3 ) / 8r3=0.524
or % occupied i.e., packing efficiency = 52.4%(ii) In face-centred cubic structure : As sphere on the face-centre is touching the spheres at the corners, evidently AC=4r.
But from right angled triangle ABC, AC=√AB2+BC2
‌‌‌‌=√a2+a2=√2a
∴‌‌√2a=4r or a=

4

√2

r
∴‌‌ Volume of the unit cell =a3= (

4

√2

r)3=

32

√2

r3
No. of spheres in the unit cell =8×

1

8

+6×

1

2

=4
Volume of four spheres =4×

4

3

πr3=

16

3

πr3
∴ Fraction occupied i.e., packing fraction =

16πr3 3

32r3 √2

=0.74
or % occupied i.e., packing efficiency = 74%(iii) In body-centred cubic structure : As the sphere at the body-centre touches the spheres at the corners, body diagonal, AD=4r.
Further, face diagonal,
AC=√AB2+BC2=√a2+a2=√2a
and body diagonal,
AD=√AC2+CD2=√2a2+a2=√3a
∴‌‌‌√3a=4r
or a=

4r

√3

∴‌‌‌ Volume of the unit cell =a3=(

4r

√3

)3=

64r3

3√3

No. of spheres per unit cell =8×

1

8

+1=2
Volume of two spheres =2×

4

3

πr3=

8

3

πr3
therefore Fraction occupied i.e., packing fraction =

8 3 πr3

64r3 3√3

=0.68
or % occupied i.e ., packing efficiency = 68%