Why are rectangular and centered rectangular lattices considered two different Bravais lattices?

Question

They are both lattices, i.e., all the vectors are in the form of $m \vec{e}_1 + n\vec{e}_2 $ with $m$, $n$ integral.

The point groups are the same, i.e., D2

In which sense are they different? A primitive cell vs a composite cell?

Answer 1

A primitive cell, by definition, contains only one lattice point. A Bravais lattice in 2D is the set of all the points obtained by all the integer combinations of two independent displacement vectors $\bf a$ and $\bf b$ in the plane $$ {\bf R} = n_1{\bf a} + n_2{\bf b} ,~~~~~~n_1, n_2 \in {\mathbb Z}. $$

A two-site lattice like the oc lattice in your picture cannot be considered immediately as a Bravais lattice because the centers of the rectangles are not integer combinations of the displacements along the two sides of the rectangle. However, it turns out that it is one of the five Bravais lattices in 2D because, with references to the vectors $\bf a$ and $\bf b$ as in the figure, the set $\bf a$ and ${\bf b'} = \frac12 {\bf a} + \frac12 {\bf b}$ allows a description of all the lattice points as integer combination of these two displacements.

The reason why it is a Bravais lattice different from the general oblique lattice obtained with two generic displacements $\bf a$ and $\bf b$ with $|{\bf a}| \neq |{\bf b}| $ and forming an angle different from $\pm 90^{\circ}$ is that the oc lattice has two mirror reflections, as symmetry elements, absent in the general oblique lattice. On the other hand, it is distinct from the rectangular lattice because $\bf a$ and $\bf b$ are not orthogonal.