The fraction of volume occupied by atoms in a body centered cubic unit cell is:

Options

A. (a) 0.32
B. (b) 0.48
C. (c) 0.68
D. (d) 0.74

Answer

To find the fraction of volume occupied by atoms in a body-centered cubic (BCC) unit cell, we can follow these steps:

### Step 1: Identify the number of atoms in the BCC unit cell
In a BCC unit cell, there are:
- 8 corner atoms, each contributing \( \frac{1}{8} \) of an atom to the unit cell (since each corner atom is shared by 8 adjacent unit cells).
- 1 atom at the body center, which contributes 1 whole atom.

Calculating the total number of atoms (z):
\[ z = 8 \times \frac{1}{8} + 1 = 1 + 1 = 2 \]

### Step 2: Calculate the volume occupied by the atoms
The volume occupied by the atoms can be calculated using the formula for the volume of a sphere:
\[ \text{Volume of one atom} = \frac{4}{3} \pi r^3 \]
Thus, the total volume occupied by the atoms in the BCC unit cell is:
\[ \text{Volume occupied by atoms} = z \times \frac{4}{3} \pi r^3 = 2 \times \frac{4}{3} \pi r^3 = \frac{8}{3} \pi r^3 \]

### Step 3: Calculate the volume of the cube
The volume of the cube (unit cell) is given by:
\[ \text{Volume of cube} = a^3 \]
where \( a \) is the edge length of the cube.

### Step 4: Relate the edge length \( a \) to the radius \( r \)
In a BCC unit cell, the body diagonal can be expressed in terms of the edge length \( a \) and the radius \( r \):
\[ \text{Body diagonal} = \sqrt{3}a = 4r \]
From this, we can solve for \( a \):
\[ a = \frac{4r}{\sqrt{3}} \]

### Step 5: Substitute \( a \) in the volume of the cube
Now substituting \( a \) into the volume of the cube:
\[ \text{Volume of cube} = \left(\frac{4r}{\sqrt{3}}\right)^3 = \frac{64r^3}{3\sqrt{3}} \]

### Step 6: Calculate the fraction of volume occupied by the atoms
Now we can find the fraction of the volume occupied by the atoms:
\[ \text{Fraction of volume} = \frac{\text{Volume occupied by atoms}}{\text{Volume of cube}} = \frac{\frac{8}{3} \pi r^3}{\frac{64r^3}{3\sqrt{3}}} \]
Simplifying this expression:
\[ = \frac{8 \pi r^3}{64 r^3 / \sqrt{3}} = \frac{8 \pi \sqrt{3}}{64} = \frac{\pi \sqrt{3}}{8} \]

### Step 7: Final result
Thus, the fraction of volume occupied by atoms in a BCC unit cell is:
\[ \frac{\pi \sqrt{3}}{8} \approx 0.68 \]