I'm told that in the normal cubic system there are $\frac18$ of an atom at each corner and these atoms' radius are such that $a = 2r$ ($a$ is the lattice constant, $r$ radius of an atom).

In bbc (body centered cubic system) they say the atoms in the diagonals are touching, and as a consequence the atoms at each corner are not big enough to satisfy the above relation $a = 2r$

Why is this? Couldn't it be like the normal cubic system but stick an atom in the center to fill up the remaining space? How do we know that they touch along the cube's diagonal and not along the edges?


The easiest answer in my opinion; try it! If each atom has the same radius, it is simply not possible to fit one of them into the center of the cubic system (known as a cubic hole). It is, however, possible if the atom is sufficiently small. CsCl is an example of such a compound.

One can show this geometrically (as is done here), but you can also convince yourself that it must be the case using equally-sized tennis balls or something similar to try it out in practise.

  • $\begingroup$ Consider the body centered cubic unit cell: It's either that the ions are all the same size (in that case they touch on the cube's diagonal) or that they touch along the edges (in that case the atom in the center needs to be smaller). Is my reasoning correct? Also, if we take a tetragonal unit cell, the atom in the center needs to be even smaller if we want all of them to be touching. Why is it a requirement for the atoms in a unit cell to touch? Please elaborate if you can. $\endgroup$ – Tamás May 28 '14 at 13:59
  • $\begingroup$ It seems to me that your reasoning is correct, but I would make a distinction between the BCC and the situation where there is a smaller ion in the center of a simple cubic system. In the latter case, I would call this a cubic hole - in the first, it's just a part of the BCC. I guess this in some respects can be regarded as a convention. $\endgroup$ – Kjetil Sonerud May 28 '14 at 14:58
  • $\begingroup$ When it comes to the case of atoms touching, I think one must make a distinction between a) neutral atoms in a structure and b) ions. In the latter case, touching can not be allowed, because this would lead to same-charge interactions (unfavourable energetics). In other words, the ion in the hole must be bigger than the hole itself - in must not be allowed to "rattle" around. In the former case, if the atom in the hole is bigger, the structure no longer has a closest (optimal) packing, which would not happen in practise. $\endgroup$ – Kjetil Sonerud May 28 '14 at 15:02
  • $\begingroup$ Well, i still don't get what's the relationship between Bravais lattices, coordination numbers and polarization (when the size of the atoms are not identical in the unit cell - i couldn't find any reference to this one in english, all the search results are pointing to optical crystallography and polarized light, but i'm referring to the phenomenon known from organic chemistry as polarization). It's off topic, but could someone explain briefly or at least point me to a resource about the connection between Bravais lattices, unit cells and coordination numbers. It seems to me they contradict. $\endgroup$ – Tamás May 28 '14 at 19:14
  • $\begingroup$ Can you elaborate on how you think they contradict; perhaps phrasing this as a new question? It's probably easier to follow than a comment field discussion :) That said, I guess Bravais lattices are just a way of categorizing crystal systems. The coordination number (in this context) basically refers to the number of nearest neighbors an atom/ion has in a given position (ie. an ion in a cubic hole has a CN = 8, an atom in CCP has CN=12 etc.) $\endgroup$ – Kjetil Sonerud May 29 '14 at 7:52

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