Diffraction by Organized Layers — Question 4

Question 4

Squared-sine peak ratios identify the first allowed face-centered families

For a monatomic face-centered cubic crystal, allowed reflections have h,k,l all odd or all even. First-order Bragg peaks obey $\sin^2\theta=\lambda^2(h^2+k^2+l^2)/(4a^2)$. The first three peaks have $\sin^2\theta=0.120,0.160,0.320$ at $\lambda=0.154\,\mathrm{nm}$.

1. Identify the first three plane families, using the stated selection rule.

2. Calculate a independently from the first and second peaks and check agreement.

Question 4 - Solution

Squared-sine peak ratios identify the first allowed face-centered families

(a) Method and calculation

$$0.120:0.160:0.320=3:4:8.$$$$\boxed{(111):3,\quad(200):4,\quad(220):8}.$$

(b) Interpretation and further calculation

$$\boxed{a_{111}=\frac{\lambda\sqrt3}{2\sqrt{0.120}}=0.385\,\mathrm{nm}}.$$$$\boxed{a_{200}=\frac{\lambda\sqrt4}{2\sqrt{0.160}}=0.385\,\mathrm{nm}}.$$

Physical check

Peak ratios should be combined with reflection conditions; unrestricted sums of squares would predict additional peaks that are absent in this lattice.