Question

Perovskite is a mineral containing calcium, titanium, and oxygen. The following diagrams represent the unit cell.

a. What is the formula of perovskite?

b. An alternative way of drawing the unit cell of perovskite has calcium at the center of each cubic unit cell. What are the positions of the titanium and

oxygen atoms in this representation of the unit cell? Show that the formula for perovskite is the same for both unit cell representations.

c. How many oxygen atoms surround a given Ti atom in each representation of the unit cell?

Short answer

a)

For the unit with Ti in the center of the cube:

1 Ti at body center:

$\begin{array}{c}8\mathrm{corners}\\ \begin{array}{l}=1/8\mathrm{Ca}/\mathrm{Corner}\times 8\mathrm{corners}\\ \text{=}1\mathrm{Ca}\mathrm{atom}\end{array}\end{array}$

$\begin{array}{c}\\ 6\mathrm{face}\mathrm{centers}\times (1/2\mathrm{Oxygen})/\mathrm{face}\mathrm{center}=3\mathrm{O}\mathrm{atoms},\\ \mathrm{Formula}={\mathrm{CaTiO}}_3\end{array}$

b)

The Ti atoms are on the corner of each unit, and oxygen atoms are at the center of each edge of the unit cell.

Because each of the 12 cube edges is shared by 4 unit cells:

$12\times 1/4=3\mathrm{O}\mathrm{atoms};8\times 1/8=1\mathrm{Ti}\mathrm{atom};1\mathrm{Ca}\mathrm{at}\mathrm{center};\mathrm{formula}-{\mathrm{CaTiO}}_3$

c)

Six oxygen atoms surround each Ti atom in the extended lattice of both representations.

Step-by-step solution

Perovskite material

Perovskite is a calcium titanium oxide mineral with the chemical formula CaTiO_3.

Perovskite is mainly found in the Earth's mantle, and it has been mined by different types of countries. Each variety of perovskite has a different chemical makeup because of different physical characteristics.

Superconductors

Superconductors are materials that transport electric charge without resistance 1 and with the display of associated macroscopic quantum phenomena, such as persistent electrical currents and magnetic flux quantization.

Cubic unit cell

The cubic unit cell is known as the smallest repeating unit, and it applies when all angles are 90⁰and all the lengths are equal, with each axis being defined by a coordinate.

The simple cubic unit cell is denoted by eight atoms, which are marked with the actual cube.

Explanation

a)

For the unit with Ti in the center of the cube:

1 Ti at the body center:

$\begin{array}{c}8\mathrm{corners}\\ \begin{array}{l}=1/8\mathrm{Ca}/\mathrm{Corner}\times 8\mathrm{corners}\\ \text{=}1\mathrm{Ca}\mathrm{atom}\end{array}\end{array}$

$\begin{array}{c}\\ 6\mathrm{face}\mathrm{centers}\times (1/2\mathrm{Oxygen})/\mathrm{face}\mathrm{center}=3\mathrm{O}\mathrm{atoms},\\ \mathrm{Formula}={\mathrm{CaTiO}}_3\end{array}$

b)

The Ti atoms are on the corner of each unit, and oxygen atoms are at the center of each edge of the unit cell.

Because each of the 12 cube edges is shared by 4 unit cells:

$12\times 1/4=3\mathrm{O}\mathrm{atoms};8\times 1/8=1\mathrm{Ti}\mathrm{atom};1\mathrm{Ca}\mathrm{at}\mathrm{center};\mathrm{formula}-{\mathrm{CaTiO}}_3$

c)Six oxygen atoms surround each Ti atom in the extended lattice of both representations.