A metallic crystal has the bcc type stacking pattern. What percentage of volume of this lattice is empty space ?

Options

A. `68%`
B. `32%`
C. `26%`
D. `74%`

Explanation

To find the percentage of volume that is empty space in a body-centered cubic (BCC) metallic crystal, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the number of atoms per unit cell (z)**:
In a BCC structure, there are 2 atoms per unit cell. This is because there is one atom at the center and one atom contributed by the eight corners (each corner atom is shared by eight unit cells).

\[ z = 2 \]

2. **Calculate the volume occupied by the atoms**:
The volume occupied by one atom (considered as a sphere) is given by the formula for the volume of a sphere:

\[ V_{\text{atom}} = \frac{4}{3} \pi r^3 \]

Therefore, the total volume occupied by the atoms in the unit cell is:

\[ V_{\text{occupied}} = z \times V_{\text{atom}} = 2 \times \frac{4}{3} \pi r^3 = \frac{8}{3} \pi r^3 \]

3. **Determine the edge length (a) of the unit cell**:
In a BCC structure, the relationship between the radius (r) of the atom and the edge length (a) of the cube is given by:

\[ a = \frac{4r}{\sqrt{3}} \]

4. **Calculate the total volume of the unit cell**:
The volume of the cubic unit cell is given by:

\[ V_{\text{total}} = a^3 \]

Substituting the expression for a:

\[ V_{\text{total}} = \left(\frac{4r}{\sqrt{3}}\right)^3 = \frac{64r^3}{3\sqrt{3}} \]

5. **Calculate the packing efficiency**:
Packing efficiency (PE) is defined as the ratio of the volume occupied by the atoms to the total volume of the unit cell, expressed as a percentage:

\[ \text{Packing Efficiency} = \left(\frac{V_{\text{occupied}}}{V_{\text{total}}}\right) \times 100 \]

Substituting the values we calculated:

\[ \text{Packing Efficiency} = \left(\frac{\frac{8}{3} \pi r^3}{\frac{64r^3}{3\sqrt{3}}}\right) \times 100 \]

Simplifying this gives:

\[ \text{Packing Efficiency} = \left(\frac{8\pi \sqrt{3}}{64}\right) \times 100 = \left(\frac{\pi \sqrt{3}}{8}\right) \times 100 \approx 68\% \]

6. **Calculate the percentage of empty space**:
The empty space in the lattice can be calculated as:

\[ \text{Empty Space} = 100\% - \text{Packing Efficiency} \]

Substituting the packing efficiency we found:

\[ \text{Empty Space} = 100\% - 68\% = 32\% \]

### Final Answer:
The percentage of volume of the BCC lattice that is empty space is **32%**.

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