(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=2rNo. of spheres per unit cell
=×8=1Volume of the sphere
=Ï€r3Volume of the cube
=a3=(2r)3=8r3 therefore Fraction occupied.
,i.e., packing fraction
=( πr3 ) /
8r3=0.524or % 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= r ∴‌‌ Volume of the unit cell
=a3= ( r)3= r3No. of spheres in the unit cell
=8×+6×=4Volume of four spheres
=4×πr3= πr3∴ Fraction occupied i.e., packing fraction
= =0.74or % 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=√2aand body diagonal,
AD=√AC2+CD2=√2a2+a2=√3a ∴‌‌‌√3a=4r or
a= ∴‌‌‌ Volume of the unit cell
=a3=( )3= No. of spheres per unit cell
=8×+1=2Volume of two spheres
=2×πr3= πr3therefore Fraction occupied i.e., packing fraction
==0.68or % occupied
i.e ., packing efficiency = 68%