Bragg's Law is a fundamental principle in physics that explains how X-rays and other types of electromagnetic radiation can be used to determine the structure of crystals. Bragg’s Law is used to calculate the angles at which X-rays are diffracted by a crystal to determine the positions of atoms within the crystal lattice. This law was put forward by Sir William Lawrence Bragg and his father, Sir William Henry Bragg in 1913. It was developed as a simple explanation for the characteristic diffraction pattern observed when X-rays were diffracted using various crystals. The Braggs studied the diffraction patterns of X-rays in crystals and noticed that the angles at which the X-rays diffracted were related to the distance between the atoms in the crystal lattice. Hence, this law helps to determine and explain the atomic arrangement in crystals. The interaction of X-rays with atoms in crystals is shown in figure 1.

Figure 1: Interaction of x-ray with atoms in crystals

When X-rays are incident on a crystal structure, it is absorbed by constituent atoms and produces oscillations in the electronic cloud around them. These oscillations will cause the re-emission of X-rays in certain directions based on Rayleigh scattering. For simplicity, it can be considered that the X-rays are reflected from crystal planes formed by the regular arrangement of atoms in them. These reflected X-rays will interfere constructively or destructively to form a diffraction pattern on the detector which can be explained by Bragg’s Law. The X-rays will favor certain directions that can produce constructive interference between the waves reflected from various parallel crystal planes as the optical path difference between these waves is an integral multiple of the wavelength.

Figure 2: X-rays incident in a crystal lattice

The path difference between the ray reflected at the first plane (at lattice point A) and reflected from the second plane in figure 2 is IB + BJ = 2IB (due to reflection). 

Considering the triangle IAB, with AB = d (lattice constant),

So,

Hence, 

For constructive interference to take place between both waves, the path difference should be an integer multiple of the wavelength (or 2IB = mλ). When this condition is satisfied the X-ray will appear to have undergone reflection and an X-ray spot is visible in the given direction. Else, destructive interference will take place. This condition is given by Bragg’s Law and can be expressed mathematically by the equation:

where m is the order of diffraction, 

λ is the incident wavelength of the X-ray, 

d is the distance between the atoms in the crystal lattice or the lattice constant,

θ is the incident angle of the X-ray 

Applications

Bragg’s Law can be used to identify materials. For example, in an X-ray diffractometer, X-ray diffraction is used to characterize the molecular structure of crystalline material by irradiating the sample with X-rays. X-rays of wavelength comparable to that of lattice constant (of a few angstroms) are used. The interference patterns, showing intense peaks of scattered radiations, produced are recorded while varying the angle of incidence or while rotating the sample to find various characteristic X-ray spots. Multiple X-ray spots at varying angles will be obtained as the crystal will have various parallel planes of different orientations. As each crystalline material will have its own structure, these X-ray diffraction spots can be considered as a fingerprint of the material and hence can be used for identification purposes. The interatomic distance can be calculated from these observations which help to deduce atomic/molecular arrangement in the crystal and hence determine the entire crystal structure.

The law is also important in other fields, such as materials science, where it is used to analyze the crystal structures of materials. Bragg's law has also been applied to the study of biological molecules, where it has been used to determine the structures of proteins and other complex molecules.