# How many tetrahedral and octahedral holes are there in a bcc unit cell?

When I tried to count, I found 24 tetrahedral holes (each formed by 2 body centers of 2 unit cells sharing a face and 2 vertices of the same edge of that face). As each hole is being shared by 2 unit cells, there are 12 holes per unit cell.

Also,I found 6 octahedral holes (6 at face centers being shard by 2 unit cells each and 12 at every edge center being shared by 4 unit cells each).

Generally I was taught that for $n$ packed spheres, there should be $2n$ tetrahedral holes. Does the analysis above mean that this rule doesn't apply here?

• Bcc isn't a close packed structure, so I don't think one should expect that the usual 1:1:2 ratio of sphere:octahedral hole:tetrahedral hole is fulfilled. As far as I can tell, your analysis leading to 6 octahedral and 12 tetrahedral per unit cell is correct (textbooks state this when describing the structure of α-AgI). However, these holes are slightly distorted from perfect octahedral/tetrahedral symmetry. – orthocresol Aug 27 '17 at 15:21

BCC doesn't have any tetrahedral voids, however we have octahedral holes at face centres. Four adjacent corners and two closest body centres. In a homogeneous lattice the octahedral is compressed along the body-centre-atom axis. However I believe we can form a hetrogenous lattice and form a close approximation to octahedral.

We have thus $6\times1/2 = 3$ distorted octahedral voids.

A BCC has 6 octahedral holes and 12 tetrahedral holes.

At each face of the bcc, there is one octahedral hole. There is also an octahedral hole on each edge. We divide the holes found on the faces by 2 as they lay between two unit cells, and we divide the holes found on the edges by 4 because they likewise lay between 4 unit cells. There are 6 faces and 12 edges in a cubic cell.

6/2 + 12/4 = 6 octahedral voids

Likewise, there is a total of 12 tetrahedral voids found within a bbc.

If you have trouble visualizing tetrahedral holes, use a dice (or any cube) and point one corner towards you so that you are staring down the C3 axis of rotation. You should be able to see that the edges connected from the corner closest to you to the corners further from you clearly form a tetrahedron.