At room temperature all crystals have

Options

A. `Deltas=-ve`
B. `Deltas=+ve`
C. `Deltas=0`
D. `Deltas=-ve or +ve`

Answer

To solve the question regarding the packing fraction of polonium in a simple cubic unit cell, we will follow these steps:

### Step 1: Understand the Concept of Packing Fraction
The packing fraction is defined as the ratio of the volume occupied by the atoms in a unit cell to the total volume of the unit cell, expressed as a percentage.

### Step 2: Identify the Structure of Polonium
Polonium has a simple cubic unit cell structure. In a simple cubic unit cell, atoms are located only at the corners of the cube.

### Step 3: Determine the Number of Atoms in the Unit Cell
In a simple cubic unit cell, there is one atom per unit cell. This is calculated as:
- Each corner atom contributes \( \frac{1}{8} \) of its volume to the unit cell.
- There are 8 corners in a cube.
- Therefore, the total number of atoms in the unit cell is:
\[ \text{Total atoms} = 8 \times \frac{1}{8} = 1 \]

### Step 4: Calculate the Volume Occupied by Atoms
The volume occupied by one atom can be calculated using the formula for the volume of a sphere:
\[ \text{Volume of one atom} = \frac{4}{3} \pi r^3 \]
Where \( r \) is the radius of the atom. Since there is one atom in the unit cell, the total volume occupied by atoms in the unit cell is:
\[ \text{Volume occupied by atoms} = \frac{4}{3} \pi r^3 \]

### Step 5: Calculate the Volume of the Unit Cell
The edge length \( a \) of the simple cubic unit cell is related to the radius \( r \) of the atoms:
\[ a = 2r \]
The volume of the unit cell is given by:
\[ \text{Volume of unit cell} = a^3 = (2r)^3 = 8r^3 \]

### Step 6: Calculate the Packing Fraction
Now we can calculate the packing fraction using the formula:
\[ \text{Packing Fraction} = \left( \frac{\text{Volume occupied by atoms}}{\text{Volume of unit cell}} \right) \times 100 \]
Substituting the values:
\[ \text{Packing Fraction} = \left( \frac{\frac{4}{3} \pi r^3}{8r^3} \right) \times 100 \]
Simplifying this expression:
\[ = \left( \frac{4 \pi}{3 \times 8} \right) \times 100 = \left( \frac{\pi}{6} \right) \times 100 \]
Calculating the numerical value:
\[ \approx 52.36\% \]

### Final Answer
Thus, the packing fraction of polonium in a simple cubic unit cell is approximately **52%**.

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