Later in the article, I explain step-by-step how to calculate them. You can think of this as a volume density, or as an indication of how tightly-packed the atoms are.įor quick reference, I have created a table below of atomic packing factor (APF) values for common crystal structures. Take for example the spinel structure which contrasts with inverse spinels notwithstanding various almost-spinels etc.Atomic Packing Factor (APF) tells you what percent of an object is made of atoms vs empty space. So far, this answer has barely scraped the surface. So unfortunately, these have to be learnt by the student or researcher individually. In nickelarsenide, the more electronegative partner (although probably not strictly an anion) forms the hexagonal structure while the electropositive partner occupies the octahedral voids. In fluorite, the cations form the cubic lattice while the anions occupy the tetrahedral voids. Similarly, anti-nickelarsenide, anti-platinumsulfide and anti-lead-oxide structures can be determined.Īs can be seen when comparing the fluorite and the nickelarsenide structures, it is not always the same atom type that forms the lattice. As this is opposite from the original fluorite, it is known as the anti-fluorite structure. Trivially, this is the case with the fluorite structure ( $\ce$ the positions of cations and anions are reversed: it is now the oxide anions that form the fcc substructure and the sodium cations occupying the tetrahedral voids. This unfortunately changes rather quickly once you leave octahedral voids in the face and body-centred cubic structures. Orthocresol’s answer is good at highlighting why it doesn’t make sense to distinguish between a rock-salt structure and an *anti-rock-salt structure: in the most basic cubic AB salt structures, the positions of cations and anions are symmetry-equivalent and thus either can be used to construct the unit cell and packing with the other sitting in the appropriate void.
† Similarly, the NaCl structure is similar to the face-centred cubic, but fcc has the same atom at all the lattice points. Thanks to Karsten Theis for pointing that out. * That's similar to the body-centred cubic structure, but is not the same: in the bcc structure the atom in the middle is the same as the one at the edges. moving our "point of view" backwards / forwards) and we get the same CsCl unit cell but with the atoms swapped round.Ī similar exercise may convince you of the NaCl / rock salt case.† Now, all we need to do is to subtract 0.5 from every number (this is OK because it's just shifting the zero, i.e. The second arrow is just a tiling in the other two dimensions. The first arrow shows us how to extend the unit cell in the third dimension, which has been collapsed: essentially, we need to add 1 to every number (which corresponds to adding a new unit cell behind the current one). What we need to do is to extend this unit cell a couple of times in each dimension. So, 0 and 1 indicates that there is a blue atom at the front and at the back, whereas 0.5 indicates the orange atom right in the middle of the unit cell. However, I think it's easier to use the 2D depiction on the left, where the numbers indicate the positions of the atoms (lattice points) in the third dimension, as a fraction of the unit cell length in that dimension. You can draw it out in 3D and try to convince yourself again using the argument above. That should already suggest to you the answer to the 3D case in your question. Both of them can be repeated (and translated) to form exactly the same crystal structure. This shows that there is no difference between the two representations of the unit cell. However, I could just as easily have drawn this:Īgain, this is a valid unit cell but this time, the orange dots are at the vertices and the blue dot is in the centre.
VOLUME OCCUPIED BY ATOMS HCP FULL
It's obvious that this is a valid unit cell, as you can repeat it as many times as you wish to generate the full crystal structure.
the black lines joining the dots), it appears that the blue dots should be at the vertices and the orange dot in the centre of the unit cell. The following could be considered to be a 2D version of the cesium chloride (CsCl) structure.* The blue and orange dots represent cations and anions respectively (or the other way round it doesn't matter, the point is that they're different things).įrom the way I've drawn the unit cells (i.e. To see why this is the case, it is helpful to look at a 2D analogue first. By symmetry, both representations are entirely equivalent. For many of the 1:1 solid-state structures, either the cations or the anions may be considered to be at the vertices (i.e. The actual answer is that it doesn't matter.