The total number of elements of symmetry in a cubic crystal is

Options

A. 9
B. 23
C. 10
D. 14

Answer

To determine the total number of elements of symmetry in a cubic crystal, we will break down the components of symmetry into three categories: axes of symmetry, planes of symmetry, and center of symmetry.

### Step-by-Step Solution:

1. **Identify Axes of Symmetry**:
- In a cubic crystal, there are three types of axes of symmetry: C2, C3, and C4.
- **C3 Axes**: There are 4 C3 axes that pass through the diagonally opposite corners and the body center. Each C3 axis allows for a rotation of 120°.
- **C4 Axes**: There are 3 C4 axes that pass through the centers of opposite faces. Each C4 axis allows for a rotation of 90°.
- **C2 Axes**: There are 6 C2 axes that pass through the midpoints of edges and the body center. Each C2 axis allows for a rotation of 180°.

**Total Axes of Symmetry**:
\[ \text{Total} = 4 (C3) + 3 (C4) + 6 (C2) = 13 \]

2. **Identify Planes of Symmetry**:
- In a cubic crystal, there are two types of planes of symmetry: diagonal planes and rectangular planes.
- **Diagonal Planes**: There are 6 diagonal planes.
- **Rectangular Planes**: There are 3 rectangular planes.

**Total Planes of Symmetry**:
\[ \text{Total} = 6 (Diagonal) + 3 (Rectangular) = 9 \]

3. **Identify Center of Symmetry**:
- A cubic crystal has one center of symmetry located at the center of the body.

**Total Center of Symmetry**:
\[ \text{Total} = 1 \]

4. **Calculate Total Elements of Symmetry**:
- Now, we sum all the elements of symmetry:
\[ \text{Total Elements of Symmetry} = \text{Total Axes} + \text{Total Planes} + \text{Total Center} \]
\[ = 13 + 9 + 1 = 23 \]

### Final Answer:
The total number of elements of symmetry in a cubic crystal is **23**.

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