Diffraction by Organized Layers — Question 10

Question 10

Scattering-vector magnitude is independent of the chosen probe wavelength at a fixed reflection

Elastic scattering has incident and outgoing wave-vector magnitudes $k=2\pi/\lambda$ and angle $2\theta$ between them. Define $\mathbf Q=\mathbf k_f-\mathbf k_i$. Consider first-order scattering from planes spaced $d=0.200\,\mathrm{nm}$.

1. Use vector geometry and Bragg’s law to derive Q at the reflection.

2. Find the scattering angles for wavelengths $0.100\,\mathrm{nm}$ and $0.150\,\mathrm{nm}$, and verify that both give the same Q.

Question 10 - Solution

Scattering-vector magnitude is independent of the chosen probe wavelength at a fixed reflection

(a) Method and calculation

$$Q^2=2k^2[1-\cos(2\theta)]=4k^2\sin^2\theta\Longrightarrow Q=4\pi\sin\theta/\lambda.$$$$2d\sin\theta=\lambda\Longrightarrow\boxed{Q=2\pi/d=10\pi\,\mathrm{nm^{-1}}}.$$

(b) Interpretation and further calculation

$$\boxed{2\theta_{0.100}=2\sin^{-1}(0.25)=28.9550^\circ}.$$$$\boxed{2\theta_{0.150}=2\sin^{-1}(0.375)=44.0486^\circ};\quad4\pi(0.25)/0.100=4\pi(0.375)/0.150=10\pi.$$

Physical check

Changing wavelength moves the measured angular position but leaves the reciprocal-space location of the same lattice reflection unchanged.