What is the packing efficiency of simple cubic lattice?

\(52.4\%\)

The correct answer is option 3. \(52.4\%\).

The cubic unit cell is the simplest form of packing seen in a simple cubic structure. The packing efficiency of the simple cubic lattice can be found based on the length of the edge. The edge length can be derived as follows:

First, we have to consider the following image: we can see that the halves of two atoms lie along the edge of the cube. This means that if we consider the radius of each atom to be r

we can then concur that the length of the edge will be represented as follows,

\(a = 2r\)

Here \(a\) is the edge length of the simple cubic lattice.

Using the formula mentioned above it will now be easy to derive the packing efficiency of this lattice. This is demonstrated below:

\(\text{Packing Efficiency} = \frac{\frac{4}{3}\pi r^3}{a^3} × 100\)

Where \(r\) is the radius of the atom and \(a\) is the edge length of the simple cubic cell.

we have to plug the value for edge length.

\(\text{Packing Efficiency} = \frac{\frac{4}{3}\pi r^3}{(2r)^3} × 100\)

Since \(r\) is present both in the numerator and denominator, we can cancel that term to get the equation given below.

\(\text{Packing Efficiency} = \frac{\frac{4}{3}\pi }{(2)^3} × 100\)

or \(\text{Packing Efficiency} = \frac{4\pi }{3 × 8} × 100\)

or \(\text{Packing Efficiency} = \frac{\pi }{3 × 2} × 100\)

or \(\text{Packing Efficiency} = \frac{\pi }{6} × 100\)

or \(\text{Packing Efficiency} = 0.524 × 100\)

or \(\text{Packing Efficiency} = 52.4\%\)