Calculate the percentage of vacant space in a Si unit cubic cell. The unit-cell content for Si is 8 and `r=(sqrt(3)a)/8`.

Answer

To calculate the percentage of vacant space in a silicon (Si) unit cell, we will follow these steps:

### Step 1: Understand the given data
- The unit-cell content for Si is 8 (this means there are 8 atoms per unit cell).
- The relationship between the radius (r) and the edge length (a) of the unit cell is given as \( r = \frac{\sqrt{3}a}{8} \).

### Step 2: Calculate the volume of one atom
The volume \( V \) of one atom can be calculated using the formula for the volume of a sphere:
\[ V_{\text{atom}} = \frac{4}{3} \pi r^3 \]
Substituting \( r = \frac{\sqrt{3}a}{8} \):
\[ V_{\text{atom}} = \frac{4}{3} \pi \left(\frac{\sqrt{3}a}{8}\right)^3 \]
Calculating \( r^3 \):
\[ r^3 = \left(\frac{\sqrt{3}a}{8}\right)^3 = \frac{3\sqrt{3}a^3}{512} \]
Thus,
\[ V_{\text{atom}} = \frac{4}{3} \pi \cdot \frac{3\sqrt{3}a^3}{512} = \frac{4\pi \cdot 3\sqrt{3}a^3}{1536} \]

### Step 3: Calculate the total volume of atoms in the unit cell
Since there are 8 atoms in the unit cell:
\[ V_{\text{total atoms}} = 8 \times V_{\text{atom}} = 8 \times \frac{4\pi \cdot 3\sqrt{3}a^3}{1536} = \frac{32\pi \cdot 3\sqrt{3}a^3}{1536} = \frac{8\pi \cdot 3\sqrt{3}a^3}{384} \]

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

### Step 5: Calculate packing efficiency
The packing efficiency (PE) can be calculated using the formula:
\[ \text{Packing Efficiency} = \left(\frac{V_{\text{total atoms}}}{V_{\text{unit cell}}}\right) \times 100 \]
Substituting the values:
\[ \text{Packing Efficiency} = \left(\frac{\frac{8\pi \cdot 3\sqrt{3}a^3}{384}}{a^3}\right) \times 100 \]
This simplifies to:
\[ \text{Packing Efficiency} = \left(\frac{8\pi \cdot 3\sqrt{3}}{384}\right) \times 100 \]
Calculating the numerical value:
\[ \text{Packing Efficiency} = \left(\frac{24\pi\sqrt{3}}{384}\right) \times 100 = \left(\frac{\pi\sqrt{3}}{16}\right) \times 100 \]

### Step 6: Calculate the percentage of vacant space
The percentage of vacant space can be calculated as:
\[ \text{Vacant Space} = 100 - \text{Packing Efficiency} \]
Substituting the packing efficiency:
\[ \text{Vacant Space} = 100 - \left(\frac{\pi\sqrt{3}}{16} \times 100\right) \]
Assuming the packing efficiency calculated is approximately 34% (as stated in the video):
\[ \text{Vacant Space} = 100 - 34 = 66\% \]

### Final Answer
The percentage of vacant space in a silicon unit cell is **66%**.

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