Diffraction by Organized Layers — Question 1

Question 1

The Bragg angle is measured from the reflecting planes

Two rays scatter elastically from neighboring parallel planes separated by d. The incident and outgoing rays each make angle $\theta$ with the planes. The extra inward and outward distances each equal $d\sin\theta$. Let $d=0.200\,\mathrm{nm}$ and $\lambda=0.154\,\mathrm{nm}$.

1. Derive the constructive-scattering condition and find the first-order Bragg angle.

2. Find the detector scattering angle and the largest positive order with $\theta<90^\circ$.

Question 1 - Solution

The Bragg angle is measured from the reflecting planes

(a) Method and calculation

$$\Delta\ell=d\sin\theta+d\sin\theta=2d\sin\theta\Longrightarrow\boxed{2d\sin\theta=n\lambda}.$$$$n=1:\quad\boxed{\theta_1=\sin^{-1}(0.154/0.400)=22.6437^\circ}.$$

(b) Interpretation and further calculation

$$\boxed{2\theta_1=45.2875^\circ}.$$$$n\lambda<2d\Longrightarrow n<2.5974\Longrightarrow\boxed{n_{\max}=2}.$$

Physical check

Confusing the measured scattering angle $2\theta$ with the glancing angle $\theta$ gives an incorrect plane spacing.