Diffraction by Organized Layers — Question 7

Question 7

A small Bragg-peak shift reveals the sign and magnitude of lattice strain

At fixed wavelength, a first-order peak initially occurs at $2\theta=60.0^\circ$. After deformation its scattering angle decreases by $0.100^\circ$. Assume a small uniform change in the relevant plane spacing, with no change in reflection index.

1. Differentiate Bragg’s law to relate fractional spacing change to angular shift.

2. Calculate the strain and determine whether the planes moved farther apart or closer together.

Question 7 - Solution

A small Bragg-peak shift reveals the sign and magnitude of lattice strain

(a) Method and calculation

$$2d\sin\theta=\lambda\Longrightarrow2\sin\theta\,\Delta d+2d\cos\theta\,\Delta\theta\simeq0.$$$$\boxed{\frac{\Delta d}{d}\simeq-\cot\theta\,\Delta\theta}.$$

(b) Interpretation and further calculation

$$\Delta\theta=-0.0500^\circ=-8.72665\times10^{-4}\,\mathrm{rad}.$$$$\boxed{\Delta d/d=+0.00151150=+0.151150\%:\ \text{spacing increases}}.$$

Physical check

The angular shift in the derivative is half the measured $2\theta$ shift and must be expressed in radians.