If you’re interested in crystallography, you might have come across Bragg’s Law, which is one of the fundamental principles in this field of study. Bragg’s Law has revolutionized the way we understand the atomic structure of crystals, paving the way for countless discoveries and technological advancements.
In this article, we’ll delve deeper into the principles and applications of Bragg’s Law and how it has impacted various fields of science and engineering.
Introduction to Bragg’s Law
👉🏼 In 1913, two scientists William Henry Bragg and William Lawrence Bragg showed that constructive interference takes place between the rays scattered by the atoms in different Bragg planes when the condition \boxed{\mathbf{n\lambda = 2D\sin\theta}} is obeyed. Where ‘n’ is an integer that represents that order of reflection. The above condition is known as Bragg’s Law.
👉🏼 Thus, Bragg’s low is a simple fundamental equation that gives the relation between the wavelength \lambda of X-rays, the interplanar spacing d and the glancing angle \theta(the angle at which the incident rays make with the plane).
👉🏼 According to this law, when an X-ray beam is incident on a crystal, the X-rays diffract off the atoms in the crystal lattice, leading to constructive interference. This means that the waves of the X-rays reinforce each other, creating a distinct pattern of diffracted beams.
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Derivation of Bragg’s Law
👉🏼 Let us consider a set of parallel atomic planes in a crystal. The atomic planes are often called Bragg planes. The Bragg planes are shown by the horizontal lines and the dots on the line indicate the deposition of atoms. Let the planes be assumed to be separated by distance ‘d’ apart, as shown in the figure below.
👉🏼 Let a narrow beam of monochromatic X-rays of wavelength \lambda be incident on the first plane with a glancing angle \theta. This incident beam undergoes multiple reflections between parallel planes of the crystal.
👉🏼 Consider a wave-2 ray which is part of the incident beam scattered by atom A in the first plane which is reflected along the direction AE.
👉🏼 Similarly consider another parallel wave-1 ray which is also part of the same incident beam scattered by atom B in the next adjacent parallel plane which is reflected along the direction BF.
👉🏼 Constructive interference will take place when the path difference between the reflected rays from two different planes is an integral multiple of \lambda.
👉🏼 To determine the path difference between the two rays wave 1 and wave 2, two normals from point A to the line wave-1 and BF are drawn. Let the normals be AC and AD. From the figure, it is observed that up to AC, the path covered is the same during incidence. From AD onwards the parallel rays AE and DF will travel keeping step with each other.
👉🏼 Hence, the excess path \delta traveled by the ray along wave-1 over that along wave-2 is given by,
\delta =CB + BD
👉🏼 But in the figure from triangle ABC, \sin \theta = \frac{CB}{AB} and from triangle ABD \sin \theta = \frac{BD}{AB}.
\therefore \delta =AB \sin \theta + AB \sin \theta
Or, \delta = 2AB \sin \theta
But AB = d (the interplanar spacing)
\therefore \delta = 2d \sin \theta
👉🏼 The condition for constructive interference to take place is \delta = n \lambda, where n = 1, 2, 3, …
\therefore \boxed{\mathbf{n\lambda = 2D\sin\theta}}
👉🏼 This equation is called Bragg’s Law and \theta is called the glancing angle. This relation is similar to the relation of the ordinary plane diffraction grating.
👉🏼 For n=1, we get the first-order spectrum, for n=2 we get the second-order spectrum, and so on.
Applications of Bragg’s Law
👉🏼 Bragg’s Law has several applications in various fields of science and engineering. Some of the most notable ones are:
1️⃣ X-ray Crystallography
👉🏼 Perhaps the most well-known application of Bragg’s Law is in X-ray crystallography, which is a technique used to determine the atomic and molecular structure of crystals. X-ray crystallography relies on the diffraction pattern created by X-rays passing through a crystal, which can be used to calculate the position of atoms in the crystal lattice.
👉🏼 X-ray crystallography has been instrumental in numerous scientific discoveries, including the determination of the structure of DNA and the development of drugs for various diseases.
2️⃣ Materials Science
👉🏼 Bragg’s Law has also played a significant role in materials science, particularly in the study of the crystal structure of materials. By analyzing the diffraction patterns of X-rays passing through a material, scientists can determine the arrangement of atoms in the crystal lattice, as well as any defects or imperfections in the material.
👉🏼 This information is crucial in the development of new materials and the optimization of existing ones for various applications, such as in the aerospace, automotive, and electronics industries.
3️⃣ Non-destructive Testing
👉🏼 Another application of Bragg’s Law is in non-destructive testing, which is a technique used to inspect materials for defects or damage without causing any harm to the material itself. X-ray diffraction is one of the methods used in non-destructive testing, as it can provide valuable information about the crystal structure and composition of the material being tested.
👉🏼 Non-destructive testing is used in various industries, including aerospace, automotive, and construction, to ensure the safety and reliability of materials and structures.
Conclusion
👉🏼 In conclusion, Bragg’s Law is a fundamental principle in crystallography that has had a profound impact on various fields of science and engineering. By understanding the relationship between the angle of incidence, the wavelength of X-rays, and the interatomic spacing in a crystal, scientists and engineers have been able to make countless discoveries and advancements in their respective fields. From X-ray crystallography to materials science and non-destructive testing, Bragg’s Law continues to be a crucial tool in research and development.
FAQs on Bragg’s Law
1. How do you determine the interatomic spacing using Bragg’s Law?
The interatomic spacing can be calculated using Bragg’s Law by measuring the angle of diffraction and the wavelength of X-rays. Once the Bragg angle is known, the interatomic spacing can be calculated using the following formula: \therefore \boxed{\mathbf{n\lambda = 2D\sin\theta}}. where n is an integer representing the order of diffraction, λ is the wavelength of the X-rays, θ is the Bragg angle.
2. Can Bragg’s Law be applied to other types of waves?
While Bragg’s Law was initially derived for X-rays, it can also be applied to other types of waves, such as electrons and neutrons. This has led to the development of electron and neutron diffraction techniques, which are used in various scientific fields.
3. What are some of the limitations of Bragg’s Law?
Bragg’s Law assumes that the crystal is perfect and that there are no defects or imperfections in the crystal lattice. This is often not the case in real-world crystals, which can have impurities, dislocations, and other defects that can affect the diffraction pattern. Additionally, Bragg’s Law only applies to crystalline materials and cannot be used to study amorphous materials.
4. How has Bragg’s Law impacted the field of medicine?
Bragg’s Law has been instrumental in the field of medicine, particularly in the development of new drugs. By using X-ray crystallography to determine the structure of proteins and other molecules, scientists have been able to design drugs that can interact with these molecules in specific ways. This has led to the development of new drugs for various diseases, including cancer, HIV, and Alzheimer’s disease.
5. What is the future of Bragg’s Law in scientific research?
Bragg’s Law continues to be a valuable tool in scientific research, with new applications and techniques being developed all the time. With advancements in technology, such as the development of synchrotron radiation sources and free-electron lasers, scientists and engineers will be able to study crystals with even greater precision and accuracy. This will undoubtedly lead to new discoveries and advancements in various fields of science and engineering.
