Can a single atom be a unit cell?

Question

I was reading a pdf online and it's author said

Consider the bcc lattice with single atoms at each lattice point, its unit cell can be reduced to two identical atoms. Atom 1 is at 000 and atom 2 is at ½ ½ ½ ...

So if two atoms can be called a unit cell for BCC structure, can't we say that a single atom can be considered a unit cell for a simple cubic lattice by the same reasoning as the author has? Is the reasoning correct? And if it's correct why don't people talk about a solitary point as a unit cell? .

The link to the pdf is http://www-dft.ts.infn.it/~peressi/diffraction.pdf

Answer 1

The answer to your questions depends on what precisely one wishes to know.

A crystal structure can be regarded as a combination of a lattice and basis. You put the basis at each lattice point; that makes the structure. However a given lattice can be looked at in more than one way. For example, the BCC lattice can be regarded as a lattice with a unit cell in a shape that just contains one point of the lattice (called primitive unit cell), or you can take as unit cell some larger region, such as a cube, which contains more than one point of the lattice. In the second case you increase the basis (the set of atoms within the chosen cell) so that the overall result is the same.

If all you wish to take an interest in are the points in the structure, say at the centre of each atom, then that's fine and in this case the simple cubic structure with a single atom basis can have a single atom as unit cell. However the actual crystal is not just a set of points, it fills the region between the points, so we have the term 'cell' to mean a three-dimensional region and such a cell has non-zero width in all three directions. It is defined in such a way that when a cell is placed at each lattice point, the result will exactly fill all of space. So in this definition the unit cell can contain just a single atom, but the cell itself is not a dimensionless point. In the case of simple cubic structure with a single-atom basis, the unit cell can be taken as a cube with a single atom in it. The atom can be placed anywhere in the cube.

Answer 2

The BCC (body-centered cubic) model theoretically contains one whole atom in the centre and an eighth of an atom in each of its eighth corners. These sum up to two whole atoms, and representing this second one as just one atom in one corner is sometimes done for simplicity.

This might feel like "skewing" the unit cell "shape" a bit from the theoretical model, but if the "shape" and dimensions isn't important, then that's fine.

In the same way, the SC (simple cubic) model theoretically contains an eighth of an atom in each of its eight corners, but could - to keep the pattern - be simplified to "contain just one atom" in one corner. I have not seen this representation used much, as it might be too simple, but again it will work in situations where the actual (theoretical and modelled) "shape" and configuration of the "slice" of the lattice that the unit cell represents is not important.

Answer 3

It is written in the article: BCC structure considered as SC with basis.

That is: instead of consider one atom, he sets a basis of 2 atoms for each lattice position. This motiv spreads as a SC, using the edges of the cube as base vectors.

But it is also possible to associate only one atom for lattice position. The primitive vectors are not orthogonal in this case. One example, departing from (0,0,0): (1,0,0), (0,1,0) and (1/2, 1/2, 1/2). Any position can be reached by a linear combination with integer coeficients. When this solution is possible, as it is for BCC, we have a Bravais lattice.