Diffraction by Organized Layers — Question 2

Question 2

Adjacent Bragg orders determine both wavelength and order number

The same plane family of spacing $d=0.250\,\mathrm{nm}$ gives two consecutive positive Bragg orders at angles whose sines are 0.300 and 0.400. The illumination is monochromatic, and no order lies between these two peaks.

1. Find the wavelength without initially knowing either order.

2. Identify both order numbers and the highest nongrazing order available for these planes.

Question 2 - Solution

Adjacent Bragg orders determine both wavelength and order number

(a) Method and calculation

$$2d\sin\theta_1=n\lambda,\quad2d\sin\theta_2=(n+1)\lambda.$$$$\boxed{\lambda=2d(0.400-0.300)=0.0500\,\mathrm{nm}}.$$

(b) Interpretation and further calculation

$$\boxed{n=2d(0.300)/\lambda=3,\quad n+1=4}.$$$$n_{\max}\lambda<2d\Longrightarrow n_{\max}<10\Longrightarrow\boxed{n_{\max}=9}.$$

Physical check

Knowing that the orders are consecutive is essential; without it, a common integer multiplier would remain undetermined.