The vacant space in bcc lattice unit cell is

Options

A. `26%`
B. `48%`
C. `23%`
D. `32%`

Explanation

To find the vacant space in a body-centered cubic (BCC) lattice unit cell, we need to calculate the packing efficiency and then determine the percentage of vacant space. Here’s a step-by-step solution:

### Step 1: Determine the number of atoms in the BCC unit cell
In a BCC unit cell:
- There are 8 atoms located at the corners of the cube, and each corner atom is shared by 8 unit cells. Therefore, the contribution from corner atoms is:
\[ \text{Contribution from corner atoms} = 8 \times \frac{1}{8} = 1 \text{ atom} \]
- There is 1 atom located at the center of the cube, which is not shared. Thus, the total number of atoms in a BCC unit cell is:
\[ \text{Total number of atoms} = 1 + 1 = 2 \text{ atoms} \]

### Step 2: Calculate the volume occupied by the atoms
The volume of a single atom (considering it as a sphere) is given by the formula:
\[ V = \frac{4}{3} \pi r^3 \]
For 2 atoms, the total volume occupied by the atoms in the unit cell is:
\[ \text{Volume occupied} = 2 \times \frac{4}{3} \pi r^3 = \frac{8}{3} \pi r^3 \]

### Step 3: Relate the radius of the atoms to the unit cell length
In a BCC lattice, the relationship between the radius \( r \) of the atom and the unit cell length \( a \) can be derived from the body diagonal of the cube. The body diagonal \( d \) is given by:
\[ d = \sqrt{3}a \]
This diagonal is equal to 4 times the radius of the atom (since there are two radii at each end of the diagonal):
\[ \sqrt{3}a = 4r \implies r = \frac{\sqrt{3}}{4}a \]

### Step 4: Calculate the volume of the unit cell
The volume of the unit cell is:
\[ \text{Volume of unit cell} = a^3 \]

### Step 5: Substitute the value of \( r \) into the volume occupied
Substituting \( r = \frac{\sqrt{3}}{4}a \) into the volume occupied by the atoms:
\[ \text{Volume occupied} = \frac{8}{3} \pi \left(\frac{\sqrt{3}}{4}a\right)^3 = \frac{8}{3} \pi \frac{3\sqrt{3}}{64} a^3 = \frac{2\sqrt{3}}{8} \pi a^3 = \frac{\sqrt{3}}{12} \pi a^3 \]

### Step 6: Calculate packing efficiency
The packing efficiency (PE) is given by:
\[ \text{Packing Efficiency} = \frac{\text{Volume occupied}}{\text{Volume of unit cell}} \times 100 = \frac{\frac{2\sqrt{3}}{12} \pi a^3}{a^3} \times 100 = \frac{2\sqrt{3}}{12} \pi \times 100 \]
Calculating this gives:
\[ \text{Packing Efficiency} \approx 68\% \]

### Step 7: Calculate the percentage of vacant space
The percentage of vacant space is:
\[ \text{Vacant Space} = 100\% - \text{Packing Efficiency} = 100\% - 68\% = 32\% \]

### Final Answer
The vacant space in the BCC lattice unit cell is **32%**.

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