Interplanar cystal spacing of cubic crystal families is defined as
$$d_{hkl} = \frac{a}{\sqrt{h^2+k^2+l^2}}.$$
This source says that the interplanar spacing of the $(111)$ plane in FCC is $\frac{a}{\sqrt3}$, which is in agreement with the formula above.
However for BCC, interplanar spacing of $(111)$ is said to be $\frac{a}{2\sqrt3}$, which doesn't agree with the formula.
My question is: Does this mean that the formula for interplanar spacing of cubic structures is limited to special cases of crystal plane, for example $(111)$?
Best Answer
The formula $d_{hlk} = \frac{a}{\sqrt{h^2+k^2+l^2}}$ is a special case of the general formula:
$$d_{hkl} = \frac{2\pi}{|\vec{G}|}$$
for a reciprocal lattice vector $\vec{G}$ to the case of a cubic lattice. For FCC and BCC lattices you can express the reciprocal lattice vectors (usually written in terms of the Miller indices $(h,k,l)$) in terms of the reciprocal lattice vectors for a SC system BUT this is not quite correct!
In fact, the Miller index $(1,0,0)$ for a BCC lattice does not describe a family of lattice planes, because if you look at a BCC lattice, there are extra atoms at the centre of each SC cube. This means that the the correct lattice vector is in fact $(2,0,0)$.
This is summarised by the selection rules for these lattices which describe the Miller indices that are not valid $\vec{G}$ vectors. (These are important in x-ray diffraction since x-rays are scattered by exactly some $\vec{G}$ so these indices not being valid reciprocal vectors gives rise to systematic absences in the x-ray diffraction pattern. These are:
BCC: $h+k+l$ must be even
FCC: $h,k,l$ must have the same parity.
If you look at your table, you see your formula in the OP disagrees precisely when these conditions are not met.