Bragg's Law

The structures of crystals and molecules are usually being identified with the use of X-ray diffraction studies, which are explained by Bragg’s Law. The law explains the relationship between an X-ray light shooting into and its reflection off from the crystal surface.

It reveals the structure of the crystal we used. It is a simple but justifiably famous law, it brought a new scope to Crystallography. Bragg’s law made the study of the properties of various crystals easy. Using this, we can categorise the crystals into different classes.

What is Bragg’s Law?

Lawrence Bragg found how to view the positions of atoms in solids. His discovery has had a giant effect on Chemistry, Biology, and mineralogy. Bragg confirmed how X-rays passing through a crystal acquire information permitting the crystal’s atomic shape to be deduced. X-ray records help scientists to construct 3-D fashions of the way atoms are organised in solids. Bragg’s discovery became arguably the greatest experimental leap forward in 20th-century science.

The law states that when the x-ray is incident on the surface of a crystal, with an angle of incidence $\theta$, it will reflect with the same angle of scattering. Also, constructive interference occurs under the condition when the path difference is equal to the whole number n of the wavelength. It is a special case of Laue diffraction, which determines the incoherent and coherent scattering. The waves from the crystal surface can interfere constructively or destructively. The analysis which is made as the result of the interference of waves is known as Bragg diffraction. We can write Bragg’s equation as follows:

$n\lambda=2dsin{\theta}$

Here

• n is an integer.

• $\lambda$ is the wavelength of the X-ray that incident on the crystal surface.

• d is the distance between the atomic layers.

• $\theta$ is the angle with which the X-ray incident on the crystal surface.

The law was first formulated by Lawrence Bragg, who is an English physicist. It came out from the surprising patterns produced on the crystalline solids when the x-ray is incident. Also, they found that at specific wavelengths and certain angles incidents produce sharp peaks of reflected radiation. Bragg interpreted the constructive interference geometrically such that the path difference is the multiple of the incident wavelength. The peaks that formed are known as Bragg’s peaks.

Bragg’s Law Derivation

Consider a crystal that has parallel planes of ions that are spaced at a distance d apart. The conditions for a sharp peak in the intensity of the scattered radiations are as follows:

• The x-rays should be reflected by the ions in any one of the planes.

• The reflected rays that come out from the crystal’s successive planes should interfere constructively.

Bragg’s Equation

The path difference is the phase shift. The constructive interference occurs when:

$\Delta=n\lambda$

From the diagram, we can write

$sin{\theta}=\frac{x}{d}$ and

$x= dsin{\theta}$

These criteria give the condition for constructive interference. That is, $2dsin{\theta}=n\lambda$ is the expression for Bragg’s law. Bragg’s law tells the angle at which the maximum diffraction intensity can occur.

Bragg’s Spectrometer

It is used to study the crystals using x-rays. The apparatus consists of an X-ray tube from which x-ray beams fall on the crystal that is mounted on a rotating table. The table itself consists of a scale and vernier, from which we can measure the angle of incidence. There is an arm that rotates about the same axis as the crystal table. The rays reflected from the crystal enter the ionisation chamber, as a result of ionisation current flow occurs between the electrode, and the current is measured. It gives the intensity of the x-ray reflected. At certain angles ionisation current is high. From the graph, the glancing angles for different orders of reflection are measured. By knowing the angle and spacing of a crystal, we can easily find the wavelength of the X-rays.

X-ray Spectrometer

The Ionisation Current Peaks at Specific Angles

Applications of Bragg’s Law

As we know, Bragg’s law made a breakthrough in the field of crystallography. It has some useful applications.

1. In X-ray diffraction, the interplanar spacing of a crystal is used for the identification purposes.

2. It is useful for conducting the measurements of the wavelength of different families of crystals.

3. In X-ray fluorescence spectroscopy (XRS), the crystals with known interplanar spacings are used to analyse the crystals in the spectrometer.

Interesting Fact

The principle of Bragg's law is applied within the construction of instruments like the Bragg spectrometer, which is commonly used to study the structure of crystals and molecules.

Summary

Bragg’s law is widely used in crystallographic techniques. We have discussed about the brief history and analysis of Bragg’s law. The discovery of this law is a breakthrough in the Twentieth-century science field. It has enormous applications in the field of medical science, chemistry, Physics, etc. Bragg’s law basically shows the relation between the interplanar spacing and the angle of diffraction.