The packing fraction for a body-centred cube is

Options

A. `0.42`
B. `0.53`
C. `0.68`
D. `0.82`

Answer

To find the packing fraction for a body-centered cubic (BCC) unit cell, we can follow these steps:

### Step 1: Understand the relationship between the edge length and the radius of the atom in a BCC structure.
For a body-centered cubic unit cell, the relationship between the edge length (a) and the atomic radius (r) is given by:
\[ 4r = \sqrt{3}a \]
From this, we can express the radius in terms of the edge length:
\[ r = \frac{\sqrt{3}}{4}a \]

### Step 2: Calculate the effective 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.
- 1 atom at the center contributing 1 full atom.

Thus, the effective number of atoms (Z) in the BCC unit cell is:
\[ Z = 8 \times \frac{1}{8} + 1 = 2 \]

### Step 3: Calculate the volume occupied by the atoms in the unit cell.
The volume of one atom (considered as a sphere) is given by the formula:
\[ V_{\text{atom}} = \frac{4}{3} \pi r^3 \]
Substituting the expression for r:
\[ V_{\text{atom}} = \frac{4}{3} \pi \left(\frac{\sqrt{3}}{4}a\right)^3 \]
Calculating this gives:
\[ V_{\text{atom}} = \frac{4}{3} \pi \left(\frac{3\sqrt{3}}{64} a^3\right) = \frac{\pi \sqrt{3}}{48} a^3 \]

### Step 4: Calculate the total volume occupied by the atoms in the unit cell.
Since there are 2 effective atoms in the unit cell:
\[ V_{\text{total}} = 2 \times V_{\text{atom}} = 2 \times \frac{\pi \sqrt{3}}{48} a^3 = \frac{\pi \sqrt{3}}{24} a^3 \]

### Step 5: Calculate the volume of the unit cell.
The volume of the unit cell is given by:
\[ V_{\text{cell}} = a^3 \]

### Step 6: Calculate the packing fraction.
The packing fraction (PF) is defined as the ratio of the volume occupied by the atoms to the total volume of the unit cell:
\[ PF = \frac{V_{\text{total}}}{V_{\text{cell}}} = \frac{\frac{\pi \sqrt{3}}{24} a^3}{a^3} = \frac{\pi \sqrt{3}}{24} \]

### Step 7: Calculate the numerical value of the packing fraction.
Using the value of \(\pi \approx 3.14\):
\[ PF \approx \frac{3.14 \times 1.732}{24} \approx \frac{5.441}{24} \approx 0.226 \]

### Step 8: Convert the packing fraction to a percentage.
To express the packing fraction as a percentage, we multiply by 100:
\[ PF \approx 0.226 \times 100 \approx 22.6\% \]

### Final Result
The packing fraction for a body-centered cubic unit cell is approximately **0.68** or **68%**.

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