Editorial Feature

# Understanding Bragg's Law in X-Ray Diffraction

Determining the chemical constitution and crystalline configuration of materials is essential in all scientific fields, including material science, chemistry, and physics. X-ray diffraction (XRD) has established itself as a non-destructive and accurate technique. It can provide critical structural information on powder, solid, and liquid samples using Bragg’s law.1

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Materials are made of tiny crystallites and non-crystallites; the former gives distinct patterns on interaction with X-rays, as the X-ray wavelength is identical to the inter-atomic spacing in crystals. Bragg’s law explains the reflection of X-rays by crystals only at a specific angle of incidence.1

This article explores the principles behind Bragg’s law and its applications in XRD.

## Understanding Bragg's Law

X-rays are invisible electromagnetic radiations with high energy and a wavelength of a few angstroms. In 1912, Van Lave demonstrated that crystals could diffract X-rays.1 The following year, Sir William H. Bragg and his son Sir W. Lawrence Bragg proposed Bragg’s Law to describe the relation of X-ray diffraction with the incidence angle. This law identified constructive interference as the reason behind the reflection of X-rays by crystals at specific angles of incidence (q).2

The usual inter-atomic distance in crystalline solids is similar to X-ray wavelengths (few angstroms), making X-rays ideal for diffraction by atoms of crystalline materials. When interacting with a crystalline solid, interference occurs among the scattered waves. Constructive interference arises among the waves traveling in the same phase relative to each other.2

The equation defining the angle at which an X-ray beam of a particular wavelength diffracts from a crystal surface is known as Bragg’s law.2 Mathematically, it is represented as 2d sinq = nl, where d is the inter-atomic spacing of the crystal, l is the incident X-ray beam wavelength, and n is an integer indicating the reflection order.1

The diffracted X-rays from a crystal surface are detected, processed, and counted. By scanning the sample across the 2q angle range, all probable diffracted beams are accounted for, even those due to the random orientation of the sample.1

Every material possesses a particular set of lattice spacings, so transforming the diffraction peaks to lattice spacings allows material recognition.2 This is generally done by matching the obtained X-ray pattern to the standard data to identify crystal phases in the material.1 A comparison with microscopy or other material characterization methods can verify the XRD results.2

Bragg’s law was initially derived to describe the interference pattern arising from X-ray scattering by crystals. Currently, XRD can analyze the structure of all states of matter. It can employ beams of ions, electrons, neutrons, and protons with wavelengths identical to the space between atomic and molecular structures under investigation.1

## Applications of Bragg's Law in XRD

XRD is a versatile and non-invasive method that provides details of the chemical constitution and crystallography of materials. Moreover, XRD data can help determine crystalline phases, lattice parameters, strain, grain size, epitaxy, phase composition, crystal orientation, and overall atomic arrangement. It is applied in several fields, including materials science, chemistry, physics, geology, and biological sciences.2

Every material has an exclusive combination of interplanar distances and related intensities, giving an individual XRD pattern that serves as its fingerprint. Thus, XRD reveals diffraction powers and peak positions of a material.

Bragg’s law is then applied to estimate the interplanar distances. Hence, the crystalline phases are labeled using the JCPDS (Joint Committee on Powder Diffraction Standard) database. Additionally, the intensity of the greatest diffraction peak can be utilized for phase quantification.3

A material’s physical and chemical characteristics are governed by the atomic organization in its crystalline arrangement or the lattice (or unit cell) parameters. While the diffraction peak intensities represent the atomic position in crystals, their positions illustrate the unit cell’s shape and size.

Thus, peak position data over the 2q range in an XRD diffractogram can help determine lattice parameters. Finally, Bragg’s law helps mathematically express these unit cell parameters.3

In industrial processes, the details about a material’s structure enable identifying the effect of different reactions on the material. It helps standardize processes and products such as pharmaceutical formulations.2

Bragg’s law also reveals structural information on materials in the micrometer to nanometer regions and from surface to phase boundary. This high-accuracy information is unobtainable by conventional methods.1

## Innovations and Future Directions

Modern scientific advances and technologies are enhancing XRD analysis. For instance, a recent study in the World Journal of Physics applied artificial neural networks (ANNs) to determine crystal size and peak shape from XRD data using the Gaussian function. This helps overcome the challenging interpretation of XRD data because of the complex diffraction patterns and background noise.4

Researchers in the above study estimated the average crystal size and assessed it using the figure of merit parameter. ANNs coupled with XRD have significant potential in different fields for precise crystal structure and size characterization, comprehending material properties, and assisting in the design of novel materials.4

Bragg's law is being applied to emerging technologies and scientific discoveries. For instance, a recent article in Advanced Functional Materials demonstrated its application in computational modeling of the response of holographic sensors.

These dual-photopolymerized holographic sensors were developed for glucose monitoring. Due to interaction with glucose, the intrinsic fringe spacing of the hydrogel (sensing material) expanded, altering the reflected wavelength according to Bragg’s law. This was used as the sensing principle for the spectroscopic sensor.5

Among several technological advances in XRD equipment, two-dimensional detectors are developed for single crystals. These detectors enable the quick collection of low-noise data and have gathered significant attention in powder diffraction. Such rapid data collection is favorable for in situ analysis of structural variations, including phase transitions, and these advances rely heavily on Bragg’s law.3

Machine learning is penetrating almost every research field, and XRD is no exception. It helps determine crystal phases and space groups using XRD data, eliminating the need for manual tuning and achieving an accuracy of over 90 %. Thus, machine learning and other artificial intelligence approaches can improve XRD proficiencies in phase detection and quantification in complex materials.3

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1. Prasad, RD. et al. (2024). A Review on Modern Characterization Techniques for Analysis of Nanomaterials and Biomaterials. ES Energy & Environment. doi.org/10.30919/esee1087

2. Ashokrao, JU., Lagad, CE., Ingole, RK. (2023) A Review on Application and Importance of Analytical Methods for Metallic Preparations (Incinerated Ash) W.S.R. To XRF And XRD. World Journal of Pharmaceutical and Medical Research. https://www.wjpmr.com/home/article_abstract/4664

3. Ali, A., Chiang, YW., Santos, RM. (2022). X-ray Diffraction Techniques for Mineral Characterization: A Review for Engineers of the Fundamentals, Applications, and Research Directions. Minerals. doi.org/10.3390/min12020205

4. Sang, ND., Thi, HHQ. (2024). Prediction of Crystal Size and Microstrain Using Artificial Neural Network From Gaussian Peak Shape Analysis of X-Ray Diffraction Data. World Journal of Physics. doi.org/10.56439/WJP/2024.1115

5. Davies, ST., Hu, Y., Blyth, J., Jiang, N., Yetisen, AK. (2023). Reusable Dual‐Photopolymerized Holographic Glucose Sensors. Advanced Functional Materials. doi.org/10.1002/adfm.20221419

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Written by

### Nidhi Dhull

Nidhi Dhull is a freelance scientific writer, editor, and reviewer with a PhD in Physics. Nidhi has an extensive research experience in material sciences. Her research has been mainly focused on biosensing applications of thin films. During her Ph.D., she developed a noninvasive immunosensor for cortisol hormone and a paper-based biosensor for E. coli bacteria. Her works have been published in reputed journals of publishers like Elsevier and Taylor & Francis. She has also made a significant contribution to some pending patents.

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