Question

Lithium oxide crystallizes with the anti-fluorite structure, with lithium ions filling all the tetrahedral holes in a close-packed array of oxide ions. (i) Describe the meaning of the underlined words. (ii) Explain why the cations and anions lie in the opposite positions to those of calcium and fluoride in the fluorite structure. (iii) What are the coordination numbers of lithium and oxygen? Using the 'atom counting' principle, determine the number of formula units per unit cell.

Short answer

In the anti-fluorite structure of lithium oxide, lithium ions fill the tetrahedral holes in a close-packed array of oxide ions. This arrangement is the inverse of the fluorite structure due to the difference in ion sizes and charge balance. The coordination numbers for lithium and oxygen are 4 and 8, respectively. Using the 'atom counting' principle, there are 6 formula units of lithium oxide (\(Li_2O\)) per unit cell.

Step-by-step solution

(i) Define Terms

Anti-fluorite structure is a type of crystal structure that is the inverse of the fluorite structure. In anti-fluorite structure, cations occupy all the tetrahedral holes in a close-packed arrangement of anions. Tetrahedral holes are the spaces inside a crystal lattice that have a tetrahedral shape, created by the arrangement of anions. In the anti-fluorite structure, these holes are filled by cations. A close-packed array of oxide ions refers to a highly efficient arrangement of oxide ions in a crystal lattice, in which the oxide ions are as closely packed as possible.

(ii) Opposite positions relative to fluorite structure

The anti-fluorite structure is the inverse of the fluorite structure. In the fluorite structure, the anions form a close-packed array with cations filling the tetrahedral holes, whereas in the anti-fluorite structure, cations form a close-packed array with anions filling the tetrahedral holes. The reason for this switching of positions is due to the difference in sizes and charge balance between the cations and anions in lithium oxide compared to calcium fluoride. In lithium oxide, the lithium ions are smaller and more numerous than the oxide ions, so it is energetically favorable for the lithium ions to occupy the tetrahedral holes, instead of the oxide ions as in the fluorite structure.

(iii) Coordination numbers and formula units per unit cell

In the anti-fluorite structure, the smaller ions (lithium) occupy the tetrahedral holes, making 4 contacts with the close-packed oxide ions. Thus, the coordination number for lithium is 4. Conversely, each oxide ion is in contact with 8 lithium ions, so the coordination number for oxygen is 8. To determine the number of formula units per unit cell, first count the number of cations and anions in the unit cell. The anti-fluorite structure has 8 tetrahedral holes filled by lithium ions - these correspond to the corners and the centers of the faces of the unit cell. There are also 4 oxide ions in the unit cell, one at each corner of a tetragonal cell that sits within the cubic lattice. Each corner ion (4 for Li and 1 for O) is shared by 8 adjacent cells, while each face-centered ion (4 for Li) is shared by 2 adjacent cells. So, the number of lithium ions per unit cell is: \(\frac{4}{8}\times 4 + \frac{1}{2}\times 4 = 3\) The number of oxide ions per unit cell is: \(\frac{1}{8}\times 4 = \frac{1}{2}\) Since the formula for lithium oxide is \(Li_2O\), the number of formula units per unit cell is the ratio of the number of lithium ions to the number of oxide ions in the cell, which is: \(3 \div \frac{1}{2} = 6\). Thus, there are 6 formula units of lithium oxide per unit cell.