Explanation:

The unit cell of the mineral has a cubic close packed (CCP) array of oxygen atoms. The CCP structure consists of a cube of oxygen atoms, with another layer of oxygen atoms stacked on top of it, such that the oxygen atoms in the second layer occupy the tetrahedral holes formed between the oxygen atoms in the first layer. The third layer consists of oxygen atoms that occupy the octahedral holes formed between the oxygen atoms in the first and second layers.

M fraction of octahedral holes are occupied by aluminum ions. Therefore, the number of aluminum ions in the unit cell is equal to the number of octahedral holes occupied by them, which is m times the total number of octahedral holes in the unit cell.

Similarly, n fraction of tetrahedral holes are occupied by magnesium ions. Therefore, the number of magnesium ions in the unit cell is equal to the number of tetrahedral holes occupied by them, which is n times the total number of tetrahedral holes in the unit cell.

The total number of octahedral holes in the unit cell is equal to the number of atoms in the unit cell. Since the CCP structure has one oxygen atom at each corner of the cube and one oxygen atom at the center of each face, the total number of atoms in the unit cell is:

8 × 1/8 (for the corner atoms) + 6 × 1/2 (for the face atoms) + 1 (for the atom at the center) = 4

Similarly, the total number of tetrahedral holes in the unit cell is twice the number of atoms, since each atom has two tetrahedral holes associated with it. Therefore, the total number of tetrahedral holes in the unit cell is:

2 × (8 + 6) = 28

The number of aluminum ions in the unit cell is m times the total number of octahedral holes in the unit cell, which is 4m. The number of magnesium ions in the unit cell is n times the total number of tetrahedral holes in the unit cell, which is 28n.

Since the mineral is electrically neutral, the total number of positive charges (due to the aluminum and magnesium ions) must be equal to the total number of negative charges (due to the oxygen ions) in the unit cell. Therefore, we have:

2+ charge of aluminum ion × number of aluminum ions + 2+ charge of magnesium ion × number of magnesium ions = 2- charge of oxygen ion × number of oxygen ions

Substituting the values we have obtained, we get:

3m + 2n = 4

Dividing both sides by 2, we get:

3/2 m + n = 2

Since m and n are fractions, we can write them as m = a2 and n = b8, where a and b are integers.

Substituting these values in the equation above, we get:

3a + b/4 = 2

Multiplying both sides by 4, we get:

12a + b = 8

Since a and b are integers, the only possible solution to this equation is a = 2 and b = 4.

Therefore, m = a2 = 1/2 and n = b8 = 1/8.

Hence, the correct answer is option 'A': 1/2