The packing efficiency of a simple cubic crystal with an interstitial atom exactly fitting at the body center is :

Options

A. (a) 0.48
B. (b) 0.52
C. (c) 0.73
D. (d) 0.91

Answer

To solve the problem of finding the packing efficiency of a simple cubic crystal with an interstitial atom fitting exactly at the body center, we can follow these steps:

### Step 1: Understand the Simple Cubic Structure
In a simple cubic crystal, the atoms are arranged at the corners of a cube. Each corner atom contributes \( \frac{1}{8} \) of its volume to the unit cell, and there are 8 corners in a cube. Therefore, the total number of atoms per unit cell is:
\[ \text{Total atoms} = 8 \times \frac{1}{8} = 1 \text{ atom} \]

### Step 2: Relate the Radius of Atoms to the Edge Length
Let \( r \) be the radius of the corner atoms and \( a \) be the edge length of the cube. In a simple cubic structure, the atoms touch each other along the edge, so:
\[ a = 2r \]

### Step 3: Consider the Interstitial Atom
An interstitial atom is placed at the body center of the cube. Let the radius of this interstitial atom be \( x \). The body diagonal of the cube can be expressed as:
\[ \text{Body diagonal} = \sqrt{3} a \]
This diagonal will be equal to the sum of the radius of the interstitial atom and the diameters of the two corner atoms:
\[ \sqrt{3} a = 2r + 2x \]

### Step 4: Substitute the Edge Length
Substituting \( a = 2r \) into the body diagonal equation gives:
\[ \sqrt{3} (2r) = 2r + 2x \]
This simplifies to:
\[ 2\sqrt{3} r = 2r + 2x \]
Dividing the entire equation by 2:
\[ \sqrt{3} r = r + x \]
Rearranging gives:
\[ x = (\sqrt{3} - 1) r \]

### Step 5: Calculate the Volume of Atoms
The volume of one corner atom is:
\[ V_{\text{corner}} = \frac{4}{3} \pi r^3 \]
The volume of the interstitial atom is:
\[ V_{\text{interstitial}} = \frac{4}{3} \pi x^3 = \frac{4}{3} \pi ((\sqrt{3} - 1) r)^3 \]

### Step 6: Total Volume of Atoms
The total volume of atoms in the unit cell is:
\[ V_{\text{total}} = V_{\text{corner}} + V_{\text{interstitial}} = \frac{4}{3} \pi r^3 + \frac{4}{3} \pi ((\sqrt{3} - 1) r)^3 \]

### Step 7: Calculate the Volume of the Cube
The volume of the cube is:
\[ V_{\text{cube}} = a^3 = (2r)^3 = 8r^3 \]

### Step 8: Calculate Packing Efficiency
The packing efficiency (or packing fraction) is given by:
\[ \text{Packing Efficiency} = \frac{V_{\text{total}}}{V_{\text{cube}}} \]
Substituting the volumes we calculated:
\[ \text{Packing Efficiency} = \frac{\frac{4}{3} \pi r^3 + \frac{4}{3} \pi ((\sqrt{3} - 1) r)^3}{8r^3} \]
This simplifies to:
\[ \text{Packing Efficiency} = \frac{4\pi}{3 \times 8} \left(1 + \left(\frac{(\sqrt{3} - 1) r}{r}\right)^3\right) \]
Calculating this gives:
\[ \text{Packing Efficiency} \approx 0.732 \]

### Conclusion
The packing efficiency of the simple cubic crystal with an interstitial atom fitting at the body center is approximately 0.732, or 73.2%.