The volume occupied by an atom in a simple cubic unit cell is:

Options

A. `a^(3)`
B. `(4pia^(3))/3`
C. `(pia^(3))/6`
D. `(sqrt(3)a)/8`

Answer

To find the volume occupied by an atom in a simple cubic unit cell, we can follow these steps:

### Step 1: Understand the Structure of a Simple Cubic Unit Cell
In a simple cubic unit cell, there is one atom located at each of the eight corners of the cube. However, each corner atom is shared among eight adjacent unit cells.

### Step 2: Calculate the Contribution of Atoms in the Unit Cell
Since each corner atom is shared by eight unit cells, the contribution of one atom to the unit cell is:
\[ \text{Contribution of one atom} = \frac{1}{8} \]
Thus, the total number of atoms in a simple cubic unit cell is:
\[ \text{Total atoms} = 8 \times \frac{1}{8} = 1 \text{ atom} \]

### Step 3: Determine the Volume of the Atom
To find the volume occupied by the atom, we need to know the radius (r) of the atom. The volume \( V \) of a single atom can be calculated using the formula for the volume of a sphere:
\[ V = \frac{4}{3} \pi r^3 \]

### Step 4: Relate the Radius to the Unit Cell Edge Length
In a simple cubic unit cell, the edge length \( a \) is equal to twice the radius of the atom:
\[ a = 2r \]
From this, we can express the radius in terms of the edge length:
\[ r = \frac{a}{2} \]

### Step 5: Substitute the Radius into the Volume Formula
Now, substituting \( r \) into the volume formula:
\[ V = \frac{4}{3} \pi \left(\frac{a}{2}\right)^3 \]
Calculating this gives:
\[ V = \frac{4}{3} \pi \left(\frac{a^3}{8}\right) = \frac{4\pi a^3}{24} = \frac{\pi a^3}{6} \]

### Final Answer
Thus, the volume occupied by an atom in a simple cubic unit cell is:
\[ V = \frac{\pi a^3}{6} \]

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