The ratio of number of rectangular plane and diagonal plane in a cubic unit cell

Options

A. `1:2`
B. `3:1`
C. `2:3`
D. `3:4`

Answer

To determine the ratio of the number of rectangular planes to diagonal planes in a cubic unit cell, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Types of Planes in a Cubic Unit Cell:**
- A cubic unit cell has two types of planes: rectangular planes and diagonal planes.

2. **Count the Rectangular Planes:**
- In a cubic unit cell, there are three rectangular planes. These planes correspond to the three pairs of opposite faces of the cube:
- Plane 1: ABCD (front face)
- Plane 2: EFGH (back face)
- Plane 3: ABEF (side face)
- Plane 4: CDGH (opposite side face)
- Plane 5: ADHE (top face)
- Plane 6: BCFG (bottom face)
- However, when counting distinct rectangular planes, we only consider the unique orientations, which gives us a total of **3 rectangular planes**.

3. **Count the Diagonal Planes:**
- Diagonal planes are formed by connecting the diagonals of the cube. In a cubic unit cell, there are four diagonal planes:
- Plane 1: ACGH (diagonal from A to C and G to H)
- Plane 2: BDFE (diagonal from B to D and F to E)
- Plane 3: ABEG (diagonal from A to B and E to G)
- Plane 4: CDHF (diagonal from C to D and H to F)
- Thus, there are a total of **6 diagonal planes**.

4. **Calculate the Ratio:**
- The ratio of the number of rectangular planes to diagonal planes is given by:
\[ \text{Ratio} = \frac{\text{Number of Rectangular Planes}}{\text{Number of Diagonal Planes}} = \frac{3}{6} = \frac{1}{2} \]

### Final Answer:
The ratio of the number of rectangular planes to diagonal planes in a cubic unit cell is **1:2**.