which of the following dimension of unit cell represent a cubic unit

Options

A. `a = b = c, alpha = beta = gamma= 90^@`
B. `a = b = c , alpha = beta = gamma ne90^@ `
C. `a = b ne c , alpha = beta = gamma = 90^@`
D. `a ne b ne c, alpha ne beta ne gamma ne 90^@ `

Answer

To determine which dimensions of a unit cell represent a cubic unit cell, we need to analyze the properties of a cubic unit cell.

### Step-by-Step Solution:

1. **Understanding the Unit Cell Dimensions**:
- A unit cell is defined by its dimensions (a, b, c) and the angles between them (α, β, γ).
- For a cubic unit cell, all three dimensions must be equal, and all angles must be equal to 90 degrees.

2. **Setting Up the Conditions**:
- For a cubic unit cell, the conditions are:
- \( a = b = c \) (the lengths of the edges are equal)
- \( \alpha = \beta = \gamma = 90^\circ \) (the angles between the edges are all right angles)

3. **Evaluating the Options**:
- If the options provided are:
- **Option A**: \( a = b = c \) and \( \alpha = \beta = \gamma = 90^\circ \) (This represents a cubic unit cell)
- **Option B**: \( a = b = c \) but \( \alpha, \beta, \gamma \neq 90^\circ \) (This represents a trigonal unit cell)
- **Option C**: \( a = b \neq c \) and \( \alpha, \beta, \gamma = 90^\circ \) (This represents a tetragonal unit cell)
- **Option D**: Any other combination that does not satisfy the cubic conditions.

4. **Conclusion**:
- The only option that satisfies both conditions of equal edge lengths and right angles is **Option A**: \( a = b = c \) and \( \alpha = \beta = \gamma = 90^\circ \).

### Final Answer:
The dimensions of a unit cell that represent a cubic unit cell are \( a = b = c \) and \( \alpha = \beta = \gamma = 90^\circ \).