# Neutron Diffraction

A crystalline sample contains atoms which are organized in parallel planes with distances that can be measured by diffraction experiments. For each distance between atomic planes, Bragg’s law gives us the scattering angle (deflection angle away from the direct beam) for a ray with a certain wavelength. If we perform a diffraction experiment on a single crystal sample we would on a detector see a single spot for each wavelength of neutrons in the beam. If we have many different wavelengths in the beam we will get a pattern a so-called Laue pattern with many spots which represents all the different atomic plane distances in the crystal, see figure 1.a.

Figure 1. a; The Laue pattern. b; circle pattern from a powder sample. c; incoming neutron beam with and without rotating atomic plane. d; circle pattern from neutron scattering

In a powder sample, there are many crystal grains with random orientations. In figure 1.b you can see the resulting ring-like diffraction pattern from a powder sample as taken with a single neutron wavelength. The radius of each ring now corresponds to a specific distance between the atomic planes inside each crystal grain of the sample. You can imagine the ring pattern as a vinyl record with hills and valleys. In neutron scattering the hills are called peaks.

To get a deeper understanding of why the patterns for a single neutron wavelength changes form a spot to a ring we can get help from figure 1.c and 1.d showing the direction of the neutron incoming ray (dotted line) and scattered ray (marked kf). One atomic layer of the sample is shown as a square. As the atomic layer (i.e. the crystal grain) is rotated around the incoming beam (tilted outwards in the bottom left figure) it will eventually trace a circle of neutrons on the detector, as shown in figure 1.d.

Professional neutron scatterers often speak of the neutron scattering vector q from a sample rather than the atomic layer distances. They find the neutron scattering vector (q) by subtracting the outgoing neutron wave-vector (kf) from the incoming (ki). The three vectors are connected geometrically in the so-called scattering triangle as shown by the three connected full-line arrows in the drawings on the left.”