One of my readers asked recently what was this “fractional coordinates” thing which I write of when I do unit cells. The fractional coordinates is a wonderful time saving thing which also allows us to reserve our brain power for useful things.
If we start in one corner of the unit cell at the origin (0 0 0) then the first number in the three is the fraction that we move along the x axis (a axis). The second number is the fraction that we move along the y axis (b axis) and the last number is the fraction of the cell we move along the z axis (c axis).
For example (0.5 0.5 0) is a spot which is in the middle of a face and one the edge of the cell. By translation it appears also at 0.5 0.5 1.0.
While (0 0 0) is at the corner of the unit cell, by translation it also appears at all the other corners.
Another example is 0.5 0 0 which is half way along an edge of the cell, by translation it also appears in three other places.
Lastly 0.5 0.5 0.5 is at the centre of the cell.