How is a non-primitive unit cell/lattice helpful?

Question

I am starting with the basics of X-ray crystallography, and I have encountered something I'm not able to rationalize.

As I understand it, the unit cell is the smallest parallelepiped enclosing the (a?) motif that can be translated along the lattice vectors to (re)construct a crystal.

The primitive unit cell is a unit cell that encloses at most one lattice point within its confines (the single lattice point corresponds to the unit cell possessing a single copy of the motif). However, when the primitive unit cell does not capture certain symmetries of the crystal (e.g., mirror planes), a non-primitive unit cell is chosen. This is essentially a larger unit cell (that would enclose more lattice point) that shares the symmetries the crystal possesses. The non-primitive unit cell can then be translated to construct the crystal.

What I don't understand here is why it is necessary to capture this symmetry using a non-primitive unit cell. If every crystal system (and therefore crystal) can be described by a primitive unit cell anyway (and given that lattices described by non-primitive unit cells can be described using a different primitive unit cell), why bother using a non-primitive unit cell when the end result is the same (i.e., construction of the crystal)?

I hope my description of the source of my confusion is at least somewhat clear. I would be very grateful for an explanation!

Answer 1

One strong reason for choosing a unit cell is that shares crystal symmetries is that it gives us more intuition about the crystal structure, which might be very helpful in doing practical calculations. Good examples are diamond-like lattices and graphene (which is easier to visualize, since it is two-dimensional).

Moreover, the choice of the primitive unit cell in such materials is non-unique - there are the alternative unit cells related to the chosen one by the crystal symmetry transformations.

More generally, exploiting symmetry often simplifies thinking about a problem and doing calculations, occasionally being the only possible path towards finding a mathematical solution - just think about how spherical or rotational symmetries are exploited in quantum mechanics and electrodynamics!

Update
The particular example of graphene is very telling in this context:
(image source)

As we see, the primitive cell has a rhombus shape (I do not call it diamond here to avoid confusion with the namesake carbon structure.) Due to the rotational symmetry there might be different choices of the primitive cell, oriented along different directions. Each atom has three nearest neighbors, but one of them is in the same primitive cell and thee two others are not. While mathematically using primitive cell is a foolproof way for getting correct results, it might be difficult to use one's intuition, it might make the calculation cumbersome and tedious, and it might make hard to interpret the results. Using a hexagonal cell solves some of these problems.