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

Options

A. 0.68
B. 0.32
C. 0.26
D. 0.74

Explanation

To solve the problem of finding the percentage of volume of a body-centered cubic (BCC) lattice that is empty space, we can follow these steps:

### Step 1: Understand the BCC Structure
In a body-centered cubic (BCC) lattice, there are:
- 1 atom at the center of the cube.
- 8 corner atoms, each contributing \( \frac{1}{8} \) of an atom to the unit cell.

### Step 2: Calculate the Total Number of Atoms per Unit Cell
The total number of atoms \( z \) in a BCC unit cell can be calculated as follows:
\[ z = \text{(contribution from corner atoms)} + \text{(contribution from body-centered atom)} = 8 \times \frac{1}{8} + 1 = 2 \]

### Step 3: Calculate the Volume of Atoms in the Unit Cell
The volume of a single atom (considered as a sphere) is given by the formula:
\[ \text{Volume of one atom} = \frac{4}{3} \pi r^3 \]
Thus, the total volume of atoms in the BCC unit cell is:
\[ \text{Total volume of atoms} = z \times \text{Volume of one atom} = 2 \times \frac{4}{3} \pi r^3 = \frac{8}{3} \pi r^3 \]

### Step 4: Relate the Radius of the Atom to the Edge Length of the Cube
In a BCC structure, the relationship between the radius \( r \) of the atom and the edge length \( a \) of the cube is given by:
\[ a = \frac{4r}{\sqrt{3}} \]

### Step 5: Calculate the Volume of the Unit Cell
The volume of the cubic unit cell is:
\[ \text{Volume of the cube} = a^3 \]
Substituting the expression for \( a \):
\[ \text{Volume of the cube} = \left(\frac{4r}{\sqrt{3}}\right)^3 = \frac{64r^3}{3\sqrt{3}} \]

### Step 6: Calculate the Packing Fraction
The packing fraction (the fraction of the volume occupied by the atoms) can be calculated as:
\[ \text{Packing fraction} = \frac{\text{Total volume of atoms}}{\text{Volume of the cube}} = \frac{\frac{8}{3} \pi r^3}{\frac{64r^3}{3\sqrt{3}}} \]
Simplifying this gives:
\[ \text{Packing fraction} = \frac{8 \pi \sqrt{3}}{64} = \frac{\pi \sqrt{3}}{8} \approx 0.680 \]

### Step 7: Calculate the Percentage of Empty Space
The percentage of empty space in the lattice is given by:
\[ \text{Empty space percentage} = 100\% - \text{Packing fraction} \times 100\% \]
Substituting the packing fraction:
\[ \text{Empty space percentage} = 100\% - 68\% = 32\% \]

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

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