CaTiO₃ has what is known as the ideal perovskite structure. Calcium atoms are at the corners of cube, a titanium atom is at cube body center, and oxygen atoms are at cube face centers. Take the cube edge length to be $a$.

a) What is the Bravais lattice type?

b) Write down a set of primitive lattice vectors.

c) Verify that there are 3 oxygen atoms, 1 titanium atom and 1 calcium atom the basis of this crystal structure.

d) Give the positions of the atoms in the basis in fractional coordinates of the primitive lattice vectors.

Solution

<hr /> <p>(a) The Bravais lattice is simple cubic.</p> <p>(b) The primitive lattice vectors are,</p> \begin{equation} \vec{a}_1 = a \begin{bmatrix} 1\\ 0 \\ 0 \end{bmatrix}; \quad \vec{a}_2 = a \begin{bmatrix} 0\\ 1 \\ 0 \end{bmatrix}; \quad \vec{a}_3 = a \begin{bmatrix} 0\\ 0 \\ 1 \end{bmatrix}. \end{equation} <p>(c) In the conventional unit cell that is drawn, there are $\frac{1}{2}\times 6 = 3$ oxygen atoms, $\frac{1}{8}\times 8 = 1$ calcium atoms, and 1 titanium atom. The crystal structure can be constructed by repeating a basis of one Ca, one Ti, and three O atoms.</p> <p>(d) In fractional coordinates of the primitive unit cell the basis is:<br /> Ca (0,0,0)<br /> Ti (0.5,0.5,0.5)<br /> O (0.5,0.5,0)<br /> O (0.5,0,0.5)<br /> O (0,0.5,0.5)