Examine the unit cell of a simple cubic lacttice and show through calculation that the simple cubic packing is 51%

To determine the packing efficiency of a simple cubic lattice, we need to calculate the ratio of the volume occupied by the spheres to the total volume of the unit cell.

A simple cubic lattice consists of spheres (representing particles or atoms) arranged in a regular pattern at the corners of a cube. The unit cell of a simple cubic lattice has one particle located at each corner of the cube.

Let's assume the radius of each sphere is "r" and the edge length of the unit cell is "a."

Step 1: Determine the volume of the unit cell.
The volume of a cube is given by: V = a^3.

Step 2: Determine the volume of the spheres.
Since each sphere is located at each corner of the cube, there are a total of 8 spheres in the unit cell. The volume of each sphere is given by: V_s = (4/3) * π * r^3.

Step 3: Calculate the total volume occupied by the spheres.
Since there are 8 spheres, the total volume occupied by the spheres is given by: V_total = 8 * V_s.

Step 4: Calculate the packing efficiency.
The packing efficiency is determined by the ratio of the volume occupied by the spheres to the total volume of the unit cell. It can be calculated using the formula: Packing Efficiency = (V_total / V) * 100%.

Substituting the values, we get:
Packing Efficiency = ((8 * V_s) / V) * 100%

Now, let's calculate the packing efficiency for the given data.

Note: The question states that the simple cubic lattice has a packing efficiency of 51%. To confirm this, we need to calculate the packing efficiency using the steps mentioned above.

You need to know the values of "r" and "a" to calculate the packing efficiency for your specific case. Once you have these values, substitute them into the formula to find the exact packing efficiency.