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 can break down the problem into several steps:

### Step-by-Step Solution:

1. **Understanding Elements of Symmetry**:
- Elements of symmetry in a crystal include axes of symmetry, planes of symmetry, and centers of symmetry.

2. **Counting Axes of Symmetry**:
- **C4 Axes**: These axes are present along the face centers of the cube. There are 3 C4 axes (one for each pair of opposite faces).
- **C3 Axes**: These axes are present along the diagonally opposite corners of the cube. There are 4 C3 axes (one for each set of diagonally opposite corners).
- **C2 Axes**: These axes are located at the center of the edges. There are 6 C2 axes (one for each pair of opposite edges).

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

3. **Counting Planes of Symmetry**:
- **Rectangular Planes**: These planes are present along the face centers. There are 3 rectangular planes (one for each pair of opposite faces).
- **Diagonal Planes**: These planes are formed by connecting diagonally opposite corners. There are 6 diagonal planes (one for each set of diagonally opposite corners).

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

4. **Counting Center of Symmetry**:
- There is 1 center of symmetry located at the body center of the cube.

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

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

---