Diffraction by Organized Layers — Question 6

Question 6

Different basis atoms can restore a reflection that equal atoms cancel

A cubic lattice has a two-atom basis at $(0,0,0)$ and $(1/2,1/2,1/2)$, with real amplitudes $f_A$ and $f_B$. Its structure factor is $F=f_A+f_B(-1)^{h+k+l}$. Take $f_A=2f_B=2f$. Compare structure factors alone, neglecting angle-dependent experimental factors.

1. Find the even-sum and odd-sum reflection intensities and their ratio.

2. Derive the general odd-to-even intensity ratio and its equal-atom limit.

Question 6 - Solution

Different basis atoms can restore a reflection that equal atoms cancel

(a) Method and calculation

$$h+k+l\ \text{even}:\ F=3f;\qquad h+k+l\ \text{odd}:\ F=f.$$$$\boxed{I_{\mathrm{even}}\propto9f^2,\quad I_{\mathrm{odd}}\propto f^2,\quad I_{\mathrm{odd}}/I_{\mathrm{even}}=1/9}.$$

(b) Interpretation and further calculation

$$\boxed{I_{\mathrm{odd}}/I_{\mathrm{even}}=\left|\frac{f_A-f_B}{f_A+f_B}\right|^2}.$$$$f_A=f_B\Longrightarrow\boxed{I_{\mathrm{odd}}=0}.$$

Physical check

With unlike atoms the half-cell translation no longer reproduces an identical scattering environment; the equal-atom extinction need not survive.