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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!

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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!

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  • $\begingroup$ The Wigner-Seitz primitive cell captures the symmetries... $\endgroup$
    – Jon Custer
    Sep 14, 2020 at 14:49
  • $\begingroup$ Thank you for your answer. So the predominant reason is basically to make downstream tasks more manageable? Also, you mention that the choice of the primitive unit cell is non-unique. Isn't this also true for non-primitive unit cells? $\endgroup$
    – Dunois
    Sep 14, 2020 at 15:09
  • $\begingroup$ What I mean is that one can have different primitive cells related by symmetry transformations. Thank you @JonCuster for reminding about that Wigner-Seitz is actually primitive. $\endgroup$
    – Roger V.
    Sep 14, 2020 at 16:46
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    $\begingroup$ Oh, absolutely - there are an infinite number of primitive cells, most of which give no insight into symmetries. Even the Wigner-Seitz cell can be hard to look at and see the symmetry, which is of course why non-primitive cells are used to help us humans more readily identify what is going on $\endgroup$
    – Jon Custer
    Sep 14, 2020 at 16:49

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