Bragg’s Law and its Applications

STATEMENT: The angle of incidence of an X-ray incident on a crystal surface will reflect with the same angle of scattering. Constructive interference occurs when the path difference, d, is equal to a full number of wavelengths nλ .

Assume an X-ray beam is incident on a solid, forming an angle with the atom’s  planes. Different atoms diffract these X-rays, causing the diffracted rays to interfere. The interference is constructive in some directions, resulting in powerful reflected X-rays. Only if 2dsin 𝜃=nλ  will there be a powerful reflected X-ray beam, according to the research.

where n is a positive integer

 Bragg’s law is the name given to this equation.

X-rays, for example, are electromagnetic radiation with a wavelength of around 1Å.

When accelerated electrons collide with a target inside an evacuated tube. It is commonly known that an electron gains energy eV when it is accelerated over a potential difference of V.

 hf=eV

 if all of this energy is utilized to produce one quantum of X-radiation. 

Here h = planck’s constant

f = frequency of the EM-wave (X-ray)

Before generating, the electron is likely to have lost part of the energy it had gained.

Bragg’s Law of Diffraction

William Lawrence Bragg and William Henry Bragg proposed Bragg diffraction (also known as the Bragg formulation of X-ray diffraction) in 1913 in reaction to their observation that crystalline substances created unexpected patterns of reflected X-rays (in contrast to that of, say, a liquid). They discovered that at particular wavelengths and incident angles, powerful peaks of reflected radiation (known as Bragg peaks) were produced in these crystals. This conclusion was explained by W. L. Bragg, who modeled the crystal as a series of discrete parallel planes separated by a constant parameter d. It was proposed that incident X-ray photons would form a Bragg peak if their reflections off different planes interfere constructively, as shown above.

Bragg diffraction is a notion that applies to both neutron and electron diffraction processes. X-ray diffraction studies, which are explained by Bragg’s Law, are frequently used to identify the structures of crystals and molecules. The relationship between an X-ray light firing into and its reflection off of a crystal surface is explained by this law.

The strength of dispersed waves as a function of scattering angle is used to create a diffraction pattern. When scattered waves satisfy Bragg’s Law, very intense intensities known as Bragg peaks appear in the diffraction pattern.

According to Bragg’s law, as the scattering angle increases, the size of each dot (or reflection) in the diffracted beam diminishes continuously, and the overall pattern consists of a sequence of concentric undulations on a background.

Equation of Bragg’s Law

                                            nλ = 2d sin 𝜃

                    Where,

                    n   =   an integer

                    λ   =   wavelength of incident X-ray beam

                    d   =   distance between atomic layers

                   𝜃   =   angle of incidence

The Bragg’s Law describes the relationship between the spacing of atomic planes in crystals and the angles of incidence at which these planes produce the most intense reflections of electromagnetic radiations like X rays and gamma rays, as well as particle waves like those associated with electrons and neutrons. 

To induce constructive interference, where comparable points of a wave (e.g., its crests or troughs) arrive at a spot at the same time, reflected wave trains must stay in phase. Lawrence Bragg, an English physicist, was the first to formulate the Bragg law.        

Bragg’s X-ray diffraction:

• When incident photons are reflected by atoms on separate parallel planes, the reflected rays interfere constructively and form a diffraction pattern
• We can figure out the lattice characteristics, size, shape, crystal orientation, inter planar distance, and so on by analyzing the diffraction patterns

Applications:

1. Computed Tomography(CT) uses many X-ray slices at different angles to build up three dimensional images inside the body with the help of Bragg’s law.
2. Ultrasound uses echoes from sound waves to locate different organs inside the body, and can be used to monitor the growth of unborn babies.
3. Magnetic Resonance Images(MRI ) uses powerful magnets to measure the water content of different types of body tissue. It is a slow process, but generates incredibly detailed images of the body.
4. Positron Emission Tomography(PET) detects radioactive traces that are injected into the body and Electroencephalography(EEC) detects electrical activity in the brain.
5. A Bragg’s spectrometer used to determine the wavelength of X rays. The construction of a Bragg’s spectrometer is similar to that of an optical spectrometer.

Advantages:

• It is useful for measuring wavelengths and for determining the lattice spacings of crystals
• To measure a particular wavelength, the radiation beam and detector are both set at some arbitrary angle
• The angle is then modified until a strong signal is received

Disadvantages:

• It provides no information on the scattering intensity for the spatial distribution of electrons in the unit cell
• For X-ray reflection, it solely examines the lattice planes, the interactions of X-rays with the crystal’s constituents are not considered
• It has been assumed that X-rays with wavelengths of the order of 1Å can be reflected from crystal planes generated by atoms, ions, or molecules with spacing of the same order of magnitude as 1Å, which is not true

Conclusion

• The wavelength of X rays, the angle of diffraction, and the angle of diffraction are the three parameters of diffraction
• The angle defines the crystal orientation
• The distance between crystal planes is defined by d

Diffraction can be made to happen for a specific wavelength and set of planes. For example, continuously shifting the orientation, i.e. adjusting the angle 𝜃  until Bragg’s Law is satisfied.