Diffraction by Organized Layers — Question 5

Question 5

A body-centered translation eliminates odd-sum reflections

A conventional cubic cell has identical scatterers at fractional positions $(0,0,0)$ and $(1/2,1/2,1/2)$. Each has real scattering amplitude f. The structure factor is $F_{hkl}=f[1+e^{i\pi(h+k+l)}]$. Ignore all other intensity factors.

1. Derive the parity selection rule and the allowed reflection intensity in units of $f^2$.

2. Classify (100), (110), (111), and (200) by this rule.

Question 5 - Solution

A body-centered translation eliminates odd-sum reflections

(a) Method and calculation

$$e^{i\pi(h+k+l)}=(-1)^{h+k+l}.$$$$\boxed{F=2f\ \text{for even }h+k+l;\quad F=0\ \text{for odd }h+k+l}.$$

(b) Interpretation and further calculation

$$\boxed{|F|^2=4f^2\ \text{or }0}.$$$$\boxed{(100),(111):\ \text{absent};\quad(110),(200):\ \text{allowed}}.$$

Physical check

A geometrically possible Bragg angle can still be dark because the basis amplitudes cancel within every unit cell.