Bragg’s Law

Bragg’s Law is a law that helps in understanding coherent and incoherent scattering from a crystal lattice. In this article, we will see Bragg’s Law, its equation, derivation, application, etc.

In this article, we will learn about how to study Bragg’s Law.

What is Bragg’s Law?

Bragg’s Law is a physical law explaining the relationship between an X-ray light shooting into and its reflection off a crystal surface. It is a special case of Laue diffraction, which determines the angles of coherent and incoherent scattering from a crystal lattice. Bragg’s Law was introduced by Sir W.H. Bragg and his son Sir W.L. Bragg.

Bragg’s Law Statement

Bragg’s Law states that,

“When the X-ray is incident onto a crystal surface, its angle of incidence, θ, will reflect with the same angle of scattering, θ”

Bragg’s Equation

Equation for Bragg’s Law is:

nλ = 2d.sin θ

where,

• n is an Integer
• λ is the Wavelength of Incident X-ray
• d is the Distance between Crystal Planes
• θ is the Angle of Incidence

Bragg’s Equation

This equation tells why the faces of crystals reflect X-ray beams at a particular incidence angle.

Derivation of Bragg’s Law

The derivation can be illustrated with a diagram showing the phases of the beams that coincide when the incident angle equals the reflecting angle. The diagram shows waves 1 and 2, in phase with each other, glancing off atoms z and B of a crystal with a separation distance (d) between its atomic, or lattice, planes.

Derivation of Bragg’s Law

The incident beams parallel until they reach point z. At point A, they strike the surface and move upwards. At B, the second beam scatters. AB + BC is the distance covered by the second beam. The extra distance is the integral multiple of the wavelength.

nλ = AB + BC

We know that AB = BC

nλ = 2CB —— 1st equation

d is the Hypotenuse of Right triangle ABz. AB is opposite of the angle θ.

AB = d sinθ —— 2nd equation

Putting equation 2 in 1, we get,

nλ = 2dsinθ

Expression is the Bragg’s Law.

Some some concepts of light help us to easily explain the concept of Bragg’s Law that are,

Scattering of Lights

Scattering of light refers to the process in which light deviates from its original path due to interactions with particles or inhomogeneities in a medium. This phenomenon is responsible for various natural effects, such as the blue color of the sky, the white color of clouds, and the Tyndall effect.

There are different types of light scattering, including Rayleigh scattering, which is the predominantly elastic scattering of light by particles much smaller than the wavelength of the light.

X-Rays

X-rays are a form of high-energy electromagnetic radiation that can pass through most objects, including the human body. When used in medical imaging, different tissues absorb X-rays at different rates, creating images that show the body’s internal structures.

Bragg’s Diffraction

Bragg’s Diffraction, also known as Bragg’s law, is a fundamental principle in X-ray and neutron diffraction. It describes the conditions under which a set of discrete parallel planes in a crystal will produce intense peaks of reflected radiation at specific wavelengths and incident angles.

Principle of Bragg’s Diffraction

Bragg’s diffraction occurs when radiation of a wavelength comparable to atomic spacings is scattered in a specular fashion by the atoms of a crystalline system, leading to constructive interference. This phenomenon applies to various types of matter waves, including X-rays, neutrons, and electrons when the wavelengths are comparable to inter-atomic distances.

Bragg’s Spectrometer

William Bragg’s spectrometer, also known as the ionization spectrometer, was designed and built by Sir William Henry Bragg in 1912-13. This spectrometer was the prototype of all modern X-ray diffractometers and was used to measure the scattering of X-rays.

It allowed the Braggs to conduct groundbreaking research in X-ray crystallography, leading to their Nobel Prize in Physics in 1915. The spectrometer was a crucial tool in understanding the structure of crystals and advancing the fields of chemistry, physics, and biology.

Applications of Bragg’s Law

The applications of Bragg’s Law are diverse and significant, particularly in the field of crystallography. Some of the critical applications include:

• X-ray Fluorescence Spectroscopy (XRS): In XRS, crystals with known interplanar spacings are used to analyze the crystals in the spectrometer, leveraging the principles of Bragg’s Law.

• Determining Crystal Structures: Bragg’s Law is used to determine a crystal’s interplanar spacing or d-spacing, which is essential for understanding the atomic and molecular arrangements in crystals.

• Measuring Wavelengths and Lattice Spacings: The law helps measure wavelengths and determine the lattice spacings of crystals, making it a valuable tool in material science and other fields.

• Medical Science: Bragg’s Law is used in medical imaging techniques, such as X-ray diffraction, to study the structure of biological molecules and tissues, providing insights into their function and interactions.

• Astronomy: In astronomy, Bragg’s Law is used to study the mineralogical composition of celestial bodies, such as planets and moons, by analyzing the diffraction patterns of X-rays or other electromagnetic waves.

• Geology: Bragg’s Law is used to study the crystal structures of minerals and rocks, helping geologists understand their composition and formation processes.

Bragg’s Law – Conclusion

• The law explains the relationship between the incident X-ray, its reflection off the crystal, and the resulting constructive or destructive interference. It provides a fundamental principle for understanding the structures of crystals and molecules through X-ray diffraction studies.
• One of the key conclusions drawn from Bragg’s Law is that diffraction is an “arranged event,” where the wavelength of the radiation, the interplanar spacing, and the angle of incidence need to be harmonized for diffraction to occur, as described by the equation is:

nλ = 2dsinθ

Read More,

• Discovery of Electrons
• Electromagnetic Field
• List of Physics Scientists and Their Inventions

Bragg’s Law – Solved Examples

Example 1: A beam of X-rays of wavelength 0.071 nm is diffracted by 110 nm² plane of rock salt with lattice constant of 0.28 nm. Find the glancing angle for the second-order diffraction.

Solution:

Given,

• Wavelength (λ) of X-rays = 0.071 nm
• Lattice constant (a) = 0.28 nm
• Plane = 110 nm²
• Order of diffraction = 2

Glancing Angle θ = ?

Bragg’s Law is 2d sin θ = nλ

d = a / √ (h² + k² + l²) , because rock salt is FCC

d = 0.28 x 10^-9/ √(1² + 1² + 0²) = 0.28 x 10^-9/√2 m

Substitute in Bragg’s Equation

2 × 0.28 × 10^-9/ √2 sinθ = 2 × 0.071 × 10^-9

sin θ = √2 × 0.071/ 0.28

sin θ = 0.3586

θ = sin^-1 (0.3586) = 21.01° ~ 21°

Example 2: Wavelengths of first-order X-rays are 2.20 Å at 27°8′. Find the distance between the adjacent Miller planes.

Solution:

Using Bragg’s law,

2d sin Ө = nλ

where,

• n = 1
• λ = 2.20 Å
• Ө = 27°8’

Substituting the values, we get

d = 2.20 Å

Example 3: X-rays of 173 pm wavelength are reflected by the (111) plane of a cubic primitive crystal at θ = 30°. The unit cell length (in pm) is closest to,

1. 173
2. 300
3. 346
4. 600
5. None of the Above/More than One of Above

Solution:

Substituting this into Bragg’s law and solving for a, we get:

a = d x √3/ sin theta

a = (λ / 2 x sinθ) x √3

a = (173pm/ 2 x sin30° ) x √3

a ≈ 299.636 pm

Therefore, the length of the unit cell is closest to 300 pm Option (2) is correct

Frequently Asked Questions on Bragg’s Law

What does Bragg’s Law of Diffraction?

Bragg’s Law describes the conditions for constructive interference of X-rays or other waves reflected by crystal lattice planes.

What is meant by Bragg’s Equation?

The expression for Bragg’s Law is,

nλ = 2d sin(θ)

Why is Bragg’s Law Important?

Bragg’s Law is crucial in X-ray crystallography for determining crystal structures and aiding in understanding molecular arrangements in materials.

Bragg’s law Experiment is based on Which Scattering of Waves?

Bragg’s Law experiment is based on X-ray diffraction, which involves the scattering of X-rays by the crystal lattice.

What is Scattering of Light?

Scattering of light refers to the redirection of light waves in different directions due to interactions with small particles or obstacles.

How is Bragg’s Law used in XRD?

X-Ray Diffraction (XRD) uses Bragg’s Law to analyze crystal structures by measuring the angles at which crystals diffract X-rays.

What are Applications of Bragg’s Law in Geology?

Bragg’s Law is applied in geology to study crystal structures of minerals, aiding in identifying and understanding geological formations.

Who Developed Bragg’s Law?

Bragg’s law was discovered by Lawrence Bragg.