If the unit cell of a mineral has cubic close packed (ccp) array of oxygen atoms with m fraction of octahedral holes occupied by aluminium ions and n fraction of tetrahedral holes occupied by magnesiums ions, m and n respectively, are

Options

A. `1/2, 1/8`
B. `1, 1/4`
C. `1/2, 1/2`
D. `1/4, 1/8`

Explanation

To solve the problem, we need to analyze the cubic close-packed (ccp) structure of oxygen atoms and determine the fractions of octahedral and tetrahedral holes occupied by aluminum and magnesium ions, respectively.

### Step-by-Step Solution:

1. **Understanding the CCP Structure**:
- In a cubic close-packed (ccp) structure, which is equivalent to face-centered cubic (FCC), oxygen ions are arranged at the corners and face centers of the cube.
- Each unit cell of ccp contains 4 oxygen ions.

2. **Identifying the Types of Holes**:
- In a ccp structure, there are two types of voids: octahedral and tetrahedral.
- The total number of octahedral voids in a ccp unit cell is 4.
- The total number of tetrahedral voids in a ccp unit cell is 8.

3. **Occupancy of Octahedral Holes**:
- The problem states that a fraction \( m \) of the octahedral holes is occupied by aluminum ions (Al³⁺).
- Since there are 4 octahedral voids, if \( m \) fraction is occupied, then the number of octahedral holes occupied by aluminum is \( 4m \).
- Given that aluminum is a trivalent cation (Al³⁺), it occupies half of the octahedral voids. Therefore, \( m = \frac{1}{2} \).

4. **Occupancy of Tetrahedral Holes**:
- The problem states that a fraction \( n \) of the tetrahedral holes is occupied by magnesium ions (Mg²⁺).
- Since there are 8 tetrahedral voids, if \( n \) fraction is occupied, then the number of tetrahedral holes occupied by magnesium is \( 8n \).
- Given that magnesium is a divalent cation (Mg²⁺), it occupies one of the tetrahedral voids. Therefore, \( n = \frac{1}{8} \).

5. **Final Values**:
- From the above analysis, we find:
- \( m = \frac{1}{2} \)
- \( n = \frac{1}{8} \)

### Conclusion:
Thus, the fractions of octahedral and tetrahedral holes occupied by aluminum and magnesium ions are:
- \( m = \frac{1}{2} \)
- \( n = \frac{1}{8} \)