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Interference Patterns and Multiple Scatterers

Where does a scattering pattern come from? That is the question we will answer in this theory page. Firstly, we have to remember that a neutron behaves both as a wave and a particle. The wave part is the important aspect to understand the scattering pattern, and therefore we shall treat all neutrons as waves for most of the following text.

A scattering pattern is actually an interference pattern, much like the interference pattern in the surface of a lake if you throw two stones in close to each other. The pattern is caused by constructive and destructive interference. When two waves are on top of each other they will be added together. If they are in phase they will create constructive interference where a lower wave valley and a taller wave top will emerge. If the two waves are totally out of phase they will create destructive interference at that spot with no wave top or valley at all. Of course, if the two waves cancel out completely then their amplitudes (height of their wave top and depth of their valley) need to be the same.

To the left is an illustration of constructive and destructive interference. At the top of the illustration are two waves that are in phase with each other, and when they meet, they add together and create one wave with a higher amplitude. This is referred to as constructive interference. At the bottom are two waves that are 180 degrees out of phase with each other, which causes the wave to cancel out completely. This is referred to as deconstruction interference. To the right is an illustration of the structure of atoms in sodium chloride. The structure is presented as a three-dimensional cube, where the chloride anion atoms, presented in green, are linked to sodium cation atoms, which are presented in gray.

Figure 1. a; Schematic view of two types of interference. b; The repeated structure of atoms in Sodium Chloride.

This is also how neutrons would be scattered from two atoms that are placed close to each other. The neutrons are scattered coherently in the directions where constructive interference emerges. However, in a real sample there are a lot more atoms than two! For example, in one gram of sodium chloride there are ~3.5*1025 atoms, so if they were just randomly distributed through a sample of salt, the pattern would be very complicated. In fact, the pattern from totally randomly distributed atoms will result in an equal amount of scattering in all directions in so-called incoherent scattering. However, if the sample atoms are arranged in some pattern with characteristic distances, say between a sodium and a chloride atom or two chloride atoms, then the neutrons scattering from atoms with these characteristic distances will add up to an overall pattern in so-called coherent scattering. To quatify the elastic coherent scattering of neutrons we need to be familiar with Bragg’s Law.