Vacant space in BCC unit cell

BCC (Body-centered cubic) is one of the three most common types of unit cells in crystal structures. In a BCC unit cell, each corner of the cube is occupied by an atom, and there is one additional atom located at the center of the cube. The vacant space refers to the empty spaces between the atoms within the unit cell.

Calculating the vacant space

To determine the vacant space in a BCC unit cell, we need to calculate the ratio of the empty space to the total volume of the unit cell.

Step 1: Determining the volume of the atoms

In a BCC unit cell, the atoms touch along the face diagonals. The distance between the centers of two adjacent atoms is equal to four times the atomic radius (2√2r). Therefore, the volume occupied by the two atoms in the unit cell can be calculated as:

Volume of atoms = 2 * (4/3π(2√2r)^3)

Step 2: Determining the volume of the unit cell

The length of the edge of the BCC unit cell (a) can be calculated using the atomic radius (r) as:

a = 4r

The volume of the unit cell (V) can be calculated as the cube of the edge length:

V = a^3

Step 3: Calculating the vacant space

The vacant space within the unit cell is the difference between the volume of the unit cell and the volume occupied by the atoms:

Vacant space = V - Volume of atoms

Step 4: Expressing the vacant space as a percentage

To express the vacant space as a percentage, we divide the vacant space by the volume of the unit cell and multiply by 100:

Vacant space percentage = (Vacant space / V) * 100

Applying the calculations

In a BCC unit cell, the coordination number is 8, which means that each atom is surrounded by 8 neighboring atoms. The atomic radius (r) can be calculated as half the distance between the centers of two adjacent atoms:

r = (2√2r) / 2

Plugging in the values, we can calculate the vacant space percentage:

Vacant space percentage = (Vacant space / V) * 100

Therefore, the correct answer is option 'A', 32%.