Calcite crystals contain scattering planes separated by 0.300 nanometers. What is the angular separation between first- and second-order diffraction maxima when X-rays of 0.130 nanometers wavelength are used?
Let’s start by highlighting some of the vital information we’ve been given. We were told that the scattering planes in the crystal we’re considering are separated by 0.300 nanometers; we’ll call that distance 𝑑. We’re also told that the wavelength of the electromagnetic radiation used is 0.130 nanometers; we’ll call that value 𝜆. We want to know the angular separation between first- and second order-diffraction maxima; we’ll call that Δ𝜃.
Since this problem involves crystals and the regular structure that crystals are composed of, this scattering is an example of Bragg scattering. For scattering with crystals, maxima are found using the equation 𝑚𝜆 equals two 𝑑 sin 𝜃, where 𝑚 is the order number, 𝜆 is the wavelength of the light used, 𝑑 is the separation distance between layers of the crystal lattice, and 𝜃 is the angle between the scattered light and the scattering plane.
We can draw a sketch of Bragg scattering where each individual blue dot represents an atom in the crystal lattice. The two layers drawn in are separated by a distance 𝑑. When light is shined on the lattice some of it reflects off the top layer and some off the second layer. Similarly to the way it does with a thin film. Depending on the phase relationships of the reflected rays of light, there can be constructive or destructive interference. Our scattering equation is for constructive interference; that is, diffraction maxima.
When we apply this relationship to our scenario for the first-order maxima, in that case 𝑚 is one so 𝜆 equals two times the separation distance 𝑑 times the sine of an angle we’ll call 𝜃. If we then write a separate equation for the second-order maxima, in that case 𝑚 is two so two times 𝜆 equals two times the separation distance 𝑑 times the sine of a different angle, one we will call 𝜙.
Δ𝜃, the angular separation we want to solve for, is equal to the magnitude of 𝜃 minus 𝜙. So let’s solve for 𝜃 and 𝜙 now. Starting with our first-order diffraction maxima equation, when we rearrange to solve for 𝜃, we find that it’s equal to the inverse sine of 𝜆 over two times 𝑑. When we plug in for 𝜆 and 𝑑 using the given values, and being careful to use units of meters for 𝑑 and 𝜆, we find that 𝜃 is equal to 12.51 degrees. That’s the angle for the first-order maxima.
Now let’s solve for 𝜙, the angle involved in the second-order diffraction maxima. In this case, 𝜙, the angle, is equal to the arcsin of 𝜆 over 𝑑. Plugging in these values as before and entering this term on our calculator, we find that 𝜙 is equal to 25.68 degrees. Now that we know both 𝜙 and 𝜃, we can solve for Δ𝜃 which we’ve defined as the magnitude of 𝜃 minus 𝜙.
When we subtract these angles, we find that Δ𝜃 is 13.2 degrees. That’s the angular separation between the first- and second-order diffraction maxima. That’s the angular separation between the first and second order diffraction maxima.