A simple cubic lattice consists of eight identical spheres of radius R in contact, placed at the corners of a cube. What is the volume of the cubical box that will just enclose these eight spheres and what fraction of this volume is actually occupied by the spheres?

Answer

To solve the problem, we need to determine two things: the volume of the cubical box that encloses the eight spheres and the fraction of this volume that is occupied by the spheres.

### Step 1: Determine the side length of the cube

In a simple cubic lattice, the spheres are placed at the corners of the cube. Each corner sphere touches the adjacent corner spheres. The distance between two opposite corners of the cube is equal to four times the radius of the spheres (since there are two radii from each sphere at the corners).

Given that the radius of each sphere is \( R \), the relationship between the side length \( a \) of the cube and the radius \( R \) is:
\[ a = 2R \]

### Step 2: Calculate the volume of the cube

The volume \( V \) of a cube is given by the formula:
\[ V = a^3 \]
Substituting \( a = 2R \):
\[ V = (2R)^3 = 8R^3 \]

### Step 3: Calculate the volume occupied by the spheres

Each sphere has a volume given by the formula:
\[ \text{Volume of one sphere} = \frac{4}{3} \pi R^3 \]
Since there are 8 spheres, the total volume occupied by the spheres is:
\[ \text{Total volume occupied} = 8 \times \frac{4}{3} \pi R^3 = \frac{32}{3} \pi R^3 \]

### Step 4: Calculate the fraction of the volume occupied by the spheres

The fraction of the volume of the cube that is occupied by the spheres can be calculated as:
\[ \text{Fraction occupied} = \frac{\text{Volume occupied by spheres}}{\text{Volume of the cube}} = \frac{\frac{32}{3} \pi R^3}{8R^3} \]
Simplifying this gives:
\[ \text{Fraction occupied} = \frac{32 \pi}{3 \times 8} = \frac{4 \pi}{3} \]
Calculating the numerical value:
\[ \frac{4 \pi}{3} \approx 0.4189 \]

### Final Results

1. **Volume of the cubical box**: \( 8R^3 \)
2. **Fraction of the volume occupied by the spheres**: \( \frac{4 \pi}{3} \approx 0.4189 \)