# Why is a centered parallelogram not a 2D Bravais cell, but a centered rectangle is?

We had a disagreement regarding 2D Bravais lattices during a lecture. The lecturer told us that a centered rectangle forms a Bravais lattice in 2D, but a centered parallelogram isn't:

We couldn't come up with a satisfying definition of a Bravais lattice though. It appears to be a lattice with certain periodic/translation/etc. symmetries; but intuitively, putting a dot in the center of a rectangle symmetry-wise has the same effect putting one in a parallelogram does. Why is the distinction then? So, our questions are:

• Does a centered rectangle form a Bravais cell? (Wikipedia lists it as one.)
• Does a centered parallelogram not? (Wikipedia doesn't list any.)
• Why, and/or why not?

Edit: The question (Is the centered parallelogram a Bravais-cell?) was in a test we discussed, so an exact answer to this is encouraged.

A centred lattice is defined [1, section 1.3.2.4.] as a sublattice formed by the integral linear combinations of independent vectors ($\renewcommand{\vec}[1]{\boldsymbol{#1}}\vec{a}$ and $\vec{b}$ on your diagrams) plus the translation of this sublattice by a finite number of so-called centring vectors. In your case, we only need one, $(\vec{a}+\vec{b})/2$, which goes from the "starting point" of $\vec{a}$ and $\vec{b}$ on your diagram to the centre of either your rectangle or your parallelogram. So, yes, both of the cases you consider fit with the definition of a centred lattice.

However, in the parallelogram case, the centred and the non-centred cells belong to the same Bravais class because there is no geometrical difference between them: they both have an arbitrary angle, as opposed to the rectangular case where the primitive cell defined by $(\vec{a}+\vec{b})/2$ and $(\vec{a}-\vec{b})/2$ has a different geometry (no right angle). So the statement on your lecture notes is wrong: "not Bravais lattice" should be replaced by "not distinct Bravais lattice".

[1] International Tables for Crystallography (2016). Vol. A, [online access]

I think this is mainly a matter of convention If you take the monoclinic 2D lattice then you certainly can draw a centred unit cell on it:

but there is nothing to be gained by doing so because the primitive cell is just as useful.

As a general guide we tend to use a primitive cell as our first preference, but an exception is made where a rectangular cell can be used instead. Rectangular cells make the geometry a lot simpler so we choose them even when the resulting cell isn't a primitive one. That's we we choose a centred (non-primitive) cell for the orthorhombic lattice.

But when it comes to the monoclinic lattice choosing a centred cell offers no advantage since it doesn't give us a cell with right angles. That's why for the monoclinic lattice we stick with the primitive cell.