Calculate the packing efficiency in a simple cubic metal crystal.

(N/A) Simple cubic lattice:
In a simple cubic lattice,the particles are located only at the corners of the cube and touch each other along the edge.
Let the edge length of the cube be $a$ and the radius of each particle be $r$.
So,we can write: $a = 2r$.
Now,volume of the cubic unit cell $= a^3 = (2r)^3 = 8r^3$.
We know that the number of particles per unit cell is $1$.
Therefore,volume of the occupied unit cell $= \frac{4}{3} \pi r^3$.
Hence,packing efficiency $= \frac{\text{Volume of one particle}}{\text{Volume of cubic unit cell}} \times 100 \%$.
Packing efficiency $= \frac{\frac{4}{3} \pi r^3}{8r^3} \times 100 \% = \frac{\pi}{6} \times 100 \%$.
Using $\pi \approx 3.14159$,packing efficiency $\approx \frac{3.14159}{6} \times 100 \% \approx 52.36 \% \approx 52.4 \%$.