Why is there reflection in 222 plane of Silicon?

[unscientific]

We know that silicon has a FCC lattice with basis at [0,0,0] and [1/4,1/4,1/4]. The structure factor is

$$S_{(hkl)} = \left[1 + e^{i\pi(h+k)} + e^{i\pi(l+k)} + e^{i\pi(h+l)} \right] \left[ 1+e^{i\frac{\pi}{2}(h+k+l)} \right]$$

which should be zero for (222) plane. But measuring it carefully reveals a non-zero intensity. Why is this the case?

 

[nasu]

Maybe this will help
http://journals.aps.org/pr/abstract/10.1103/PhysRev.77.271

 

[unscientific]

Maybe this will help
http://journals.aps.org/pr/abstract/10.1103/PhysRev.77.271

So something like fine structure?

 

[M Quack]

There are 3 ways to get the (222) reflection in the Diamond structure:

1. Note that Carbon, Silicon or Germanium are covalently bonded. This implies that there is some finite electron density along the bond. What is the structure factor of (222) if you put, say, 1/10 of an electron half way between the atoms, say at (1/8, 1/8, 1/8) and equivalent positions?

2. Through multiple scattering, e.g. the Renninger effect, as described in the reference cited above.

3. In anomalous scattering of x-rays (ATS), by tuning the photon energy to a resonance such as the K-edge of Germanium.