Lattice structures of two and 3 atoms basis

Question

I seem to be having a real issue in how people instinctively know how to draw a unit cell lattice structure for 2 and 3 atom basis from vector points given.

So two typical example that I have been trying to do is sodium chloride and fluorite.

When I looked up the lattice points structure for the two compunds each was given like so.

Sodium Chloride $$Cl=begin{pmatrix}0 0 0end{pmatrix},:Na=begin{pmatrix}0.5 0.5 0.5end{pmatrix}$$

Fluorite

$$Ca^{2+}=begin{pmatrix}0 0 0end{pmatrix},:F^-=begin{pmatrix}0.25 0.25 0.25end{pmatrix},:F^-=begin{pmatrix}0.75 :0.75 :0.75end{pmatrix}$$

From wiki and the links I got the information from

http://www.ilpi.com/inorganic/structures/nacl/index.html
  
  http://www.kayelaby.npl.co.uk/chemistry/3_7/3_7_7.html

They are both FCC structure, what I cant seem to do however is draw what these cells look like from this information, now I have seen on google what these structure look like but if I was given this as a question to draw the cells from just these postions vectors, I am not sure I could. When I have seen these structures in google I cant see how the resemble FCC structures, I assume that there like a sub FCC between the lattice points but I honestly cannot see it.

Elena Pascal · Answer

Face centered cubic refers to the Bravais lattice not to the atom positions. The lattice has FCC symmetry. Your base (motif) for the lattice for NaCl has two atoms one Cl and one Na such that the distance between the two is a/2. It is convenient to put one of the two atoms in the motif at the points in the lattice. For instance Cl at $(0,0,0)$ and Na at (0, 0.5, 0), or any other choice. Once you draw the first motif (two atoms for NaCl) you then repeat this motif according to the FCC lattice symmetry in order to get the entire crystal.

There is a second layer of complexity when it comes to unit cell that have more than one motif like FCC. People like to reduce it to an even smaller, primitive cell and then give atom positions in that smaller cell. That is how you end up with information for only one pair of atoms, these are the atoms in the primitive cell. Na is now in the middle of this red cell.

Lattice structures of two and 3 atoms basis

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