What is the packing efficiency of simple cubic lattice?

The correct option is 52.4%.

To understand why this is correct, let us calculate the packing efficiency of a simple cubic lattice step-by-step.

Step 1: Understand the Geometry of a Simple Cubic Unit Cell
In a simple cubic unit cell, atoms are located only at the eight corners of the cube. These corner atoms touch each other along the edges of the cell.
If the radius of each spherical atom is represented by $r$ and the edge length of the cubic unit cell is represented by $a$, then the relationship between the edge length and the atomic radius is:
$a=2r$

Step 2: Find the Number of Atoms Per Unit Cell
Each of the 8 corners of the cube contains an atom, but each corner atom is shared among 8 adjacent cubic unit cells. Therefore, the net number of atoms ($Z$) belonging exclusively to one unit cell is:
$Z=8\times \frac{1}{8}=1$

Step 3: Calculate the Volume of the Atoms in the Unit Cell
Since there is only 1 atom (sphere) per unit cell, the volume occupied by the atoms ($V_{\text{atoms}}$) is equal to the volume of one sphere of radius $r$:
$V_{\text{atoms}}=\frac{4}{3}\pi r^3$

Step 4: Calculate the Total Volume of the Unit Cell
The volume of the cubic unit cell ($V_{\text{cell}}$) is the cube of its edge length $a$. Substituting $a=2r$:
$V_{\text{cell}}=a^3={\left(2r\right)}^3=8r^3$

Step 5: Calculate the Packing Efficiency
Packing efficiency is the percentage of the total space inside the unit cell that is occupied by the atoms:
$\text{Packing Efficiency}=\frac{V_{\text{atoms}}}{V_{\text{cell}}}\times 100$
Substituting the volume values:
$\text{Packing Efficiency}=\frac{\frac{4}{3}\pi r^3}{8r^3}\times 100$
Simplifying the expression by canceling $r^3$:
$\text{Packing Efficiency}=\frac{\pi }{6}\times 100$
Using $\pi \approx 3.14159$:
$\text{Packing Efficiency}\approx 0.5236\times 100\approx 52.4\%$

Thus, the packing efficiency of a simple cubic lattice is 52.4%.