In a hexagonal crystal:

Options

A. `alpha = beta = gamma ne 90^(@), a=b=c`
B. `alpha = beta = gamma = 90^(@), a=bne c`
C. `alpha = beta = gamma = 90^(@), a ne b ne c`
D. `alpha = beta =90^(@), gamma= 120^(@), a = b ne c`

Answer

To determine the cell parameters of a hexagonal crystal system, we need to understand the relationships between the edge lengths and the angles. Here’s a step-by-step breakdown of the solution:

### Step 1: Identify the Edge Lengths
In a hexagonal crystal system, the edge lengths are defined as:
- \( a \) = length of one edge
- \( b \) = length of the adjacent edge
- \( c \) = length of the vertical edge

For a hexagonal crystal:
- \( a = b \) (the two horizontal edges are equal)
- \( c \) is different from \( a \) and \( b \) (the vertical edge is not equal to the horizontal edges).

### Step 2: Identify the Angles
The angles in a hexagonal crystal system are defined as:
- \( \alpha \) = angle between edge \( b \) and edge \( c \)
- \( \beta \) = angle between edge \( a \) and edge \( c \)
- \( \gamma \) = angle between edge \( a \) and edge \( b \)

For a hexagonal crystal:
- \( \alpha = 90^\circ \) (the angle between edge \( a \) and edge \( c \))
- \( \beta = 90^\circ \) (the angle between edge \( b \) and edge \( c \))
- \( \gamma = 120^\circ \) (the angle between the two equal edges \( a \) and \( b \)).

### Step 3: Summarize the Parameters
From the above analysis, we can summarize the cell parameters for a hexagonal crystal system as follows:
- Edge lengths: \( a = b \) and \( c \neq a \)
- Angles: \( \alpha = 90^\circ \), \( \beta = 90^\circ \), \( \gamma = 120^\circ \)

### Step 4: Evaluate the Options
Now we can evaluate the given options based on the parameters we derived:
1. **Option 1**: \( \alpha = \beta = \gamma \neq 90^\circ \) - Incorrect
2. **Option 2**: All angles are equal - Incorrect
3. **Option 3**: All angles are equal to \( 90^\circ \) - Incorrect
4. **Option 4**: \( a = b \neq c \), \( \alpha = \beta = 90^\circ \), \( \gamma = 120^\circ \) - Correct

### Conclusion
The correct option that describes the cell parameters of a hexagonal crystal system is the fourth option.

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