Packing Efficiency of Two Dimensional Square Unit Cell

To find the packing efficiency of the given two-dimensional square unit cell, we calculate the ratio of the area occupied by the circles inside the unit cell to the total area of the unit cell.

Step 1: Determine the relationship between the edge length ($L$) and the radius of the circles ($r$).

Let $r$ be the radius of each identical circle and $L$ be the side length of the square unit cell.

From the given figure:

• The circles at the corners touch the central circle along the diagonal of the square.
• The length of the diagonal of a square with side length $L$ is given by $\sqrt{2}L$.

Along the diagonal, the length consists of:
$\text{Diagonal length} = r + 2r + r = 4r$

Equating the diagonal lengths:
$\sqrt{2}L = 4r \implies L = \frac{4r}{\sqrt{2}} = 2\sqrt{2}r$

Step 2: Calculate the total area of the square unit cell.

$\text{Total Area} = L^2 = (2\sqrt{2}r)^2 = 8r^2$

Step 3: Calculate the effective number of circles inside the unit cell.

• Each of the $4$ corner circles contributes a quarter ($\frac{1}{4}$) of its area to the unit cell.
• The $1$ central circle lies completely inside the unit cell.

$\text{Effective number of circles } (N) = \left(4 \times \frac{1}{4}\right) + 1 = 2$

Step 4: Calculate the total area occupied by the circles.

$\text{Area occupied by circles} = N \times \pi r^2 = 2\pi r^2$

Step 5: Calculate the packing efficiency.

$\text{Packing Efficiency} = \frac{\text{Area occupied by circles}}{\text{Total Area of the unit cell}} \times 100\%$

$\text{Packing Efficiency} = \frac{2\pi r^2}{8r^2} \times 100\% = \frac{\pi}{4} \times 100\%$

Substituting $\pi \approx 3.1416$:

$\text{Packing Efficiency} \approx \frac{3.1416}{4} \times 100\% = 78.54\%$

Thus, the correct option is D.