Diffraction by Organized Layers — Question 8

Question 8

A finite stack of planes gives a finite Bragg-peak width

N equally spaced coherent planes have adjacent-plane phase $\delta=4\pi d\sin\theta/\lambda$. Near a Bragg maximum, the first array zero occurs at $|\Delta\delta|=2\pi/N$. For a large stack define thickness $T\simeq Nd$. Let $T=50.0\,\mathrm{nm}$, $\lambda=0.154\,\mathrm{nm}$, and $\theta_B=30^\circ$.

1. Derive the offset in Bragg angle from the peak to its nearest zero.

2. Find that offset in detector scattering angle $2\theta$, in radians and degrees.

Question 8 - Solution

A finite stack of planes gives a finite Bragg-peak width

(a) Method and calculation

$$\Delta\delta\simeq\frac{4\pi d\cos\theta_B}{\lambda}\Delta\theta=\frac{2\pi}{N}.$$$$\boxed{|\Delta\theta|=\frac{\lambda}{2Nd\cos\theta_B}\simeq\frac{\lambda}{2T\cos\theta_B}}.$$

(b) Interpretation and further calculation

$$\boxed{|\Delta(2\theta)|=\frac{\lambda}{T\cos\theta_B}=0.00355648\,\mathrm{rad}}.$$$$\boxed{|\Delta(2\theta)|=0.203771^\circ\quad\text{from peak to first zero}}.$$

Physical check

This is a first-zero offset, not a full width at half maximum; line-width conventions must be kept separate.