The unit cell with crystallographic dimensions , `a= b nec " and " alpha = beta=gamma = 90^0`

Options

A. Cubic
B. Tetragonal
C. Monoclinic
D. Hexagonal

Answer

To solve the problem, we need to identify the type of unit cell based on the given crystallographic dimensions. The conditions provided are:

- \( a = b \neq c \)
- \( \alpha = \beta = \gamma = 90^\circ \)

### Step-by-Step Solution:

1. **Understanding the Conditions**:
- The condition \( a = b \) means that the lengths of the edges along the x and y axes are equal.
- The condition \( c \neq a \) (or \( c \neq b \)) indicates that the height (length along the z-axis) is different from the lengths along the x and y axes.
- The angles \( \alpha, \beta, \gamma = 90^\circ \) indicate that all angles between the axes are right angles.

2. **Identifying the Unit Cell**:
- **Cubic Unit Cell**: In a cubic unit cell, all sides are equal (\( a = b = c \)) and all angles are \( 90^\circ \). This does not satisfy our conditions since \( c \) must be different.
- **Tetragonal Unit Cell**: In a tetragonal unit cell, \( a = b \) and \( c \neq a \). Additionally, all angles \( \alpha, \beta, \gamma \) are \( 90^\circ \). This matches our conditions perfectly.
- **Monoclinic Unit Cell**: In a monoclinic unit cell, all sides can be different, and only two angles are \( 90^\circ \) while the third angle is not. This does not satisfy our conditions.
- **Hexagonal Unit Cell**: In a hexagonal unit cell, \( a = b \) but \( c \neq a \) and one angle (\( \gamma \)) is \( 120^\circ \). This also does not satisfy our conditions.

3. **Conclusion**:
- The only unit cell that meets all the specified conditions is the **Tetragonal unit cell**.

### Final Answer:
The correct unit cell is **Tetragonal**.

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