Diffraction by Organized Layers — Question 3

Question 3

A cubic lattice parameter predicts another plane-family peak

A cubic crystal obeys $d_{hkl}=a/\sqrt{h^2+k^2+l^2}$. With wavelength $0.154\,\mathrm{nm}$, its first-order (111) reflection occurs at scattering angle $2\theta=60.0^\circ$. Assume both (111) and (200) are allowed reflections.

1. Infer the cubic lattice parameter a.

2. Predict the first-order (200) scattering angle.

Question 3 - Solution

A cubic lattice parameter predicts another plane-family peak

(a) Method and calculation

$$\theta_{111}=30^\circ,\quad d_{111}=\lambda/(2\sin30^\circ)=0.154\,\mathrm{nm}.$$$$\boxed{a=\sqrt3d_{111}=0.266736\,\mathrm{nm}}.$$

(b) Interpretation and further calculation

$$d_{200}=a/2,\quad\sin\theta_{200}=\lambda/(2d_{200})=1/\sqrt3.$$$$\boxed{2\theta_{200}=70.5288^\circ}.$$

Physical check

Different plane families are labeled by their Miller indices; they need not be successive diffraction orders of a single spacing.