# Incomplete definition of the unit cell of a lattice

All of the books/online sources that I have found cite the following definition of the unit cell of a lattice :

The unit cell is defined as the smallest repeating unit having the full symmetry of the crystal structure.

source : wikipedia

My question is that all they have mentioned is the word 'small' without giving additional details about it like 'small in terms of what?'. Is it in terms of volume/area, number of lattice points in it or is it anything else?

I think that it is in terms of volume because when we reject 'end centered cubic' to be actually ' primitive tetragonal' the number of lattice points remain same but volume decreases however I would still like to confirm.

• Lattice points are not defined until after we have the unit cell. You are right, this is about the smallest volume. The definition still remains ambiguous, but that's another story. – Ivan Neretin May 21 '18 at 10:16
• Wigner-Seitz is a pretty clear definition of a unit cell (there are an infinite number of possible unit cells for any given crystal), and by construction one can see that it defines the ‘smallest’ volume possible. – Jon Custer May 21 '18 at 12:40
• Careful, it is not the smallest, but the smallest that has highest possible symmetry (="the full symmetry of the crystal structure"). – Karl May 21 '18 at 14:58
• @Karl - There is not a unit cell consistent with the lattice symmetry that can have a smaller volume. Wigner-Seitz is not a 'conventional' unit cell that often contains more than one unit cell to make it human recognizable. Since it is a minimal unit cell that can be translated with the lattice vectors to fill space, it is by definition the smallest unit cell possible. Smaller and it would not fill space. Larger and it would not be a single unit cell. – Jon Custer May 21 '18 at 22:15
• @JonCuster The nomenclature is a bit confusing, or am I lost in translation? I understand that Wigner-Seitz is a primitive cell, but is it a unit cell? Because the same minimal cell type can have totally different powder patterns, and crystallographs have defined the unit cell so they can recognize it via powder diffractometry. Or so I remember, but it's been a long time. ;-) – Karl May 22 '18 at 5:54