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Role of Bragg’s law in X-Ray Diffraction studies

X-ray Crystal diffraction
X-ray Crystal diffraction

Bragg’s law is the foundation stone on which the edifice of x-ray diffraction stands. It holds same significance in crystal structure determinations as the Beer- Lambert law(link) holds in light absorbance measurements. The law was established by Sir W.H Bragg and his son Sir W.L Bragg in 1913 to explain the diffraction of x-rays from atomic planes of crystals of sodium chloride, zinc sulphide and diamond.The Nobel prize for Physics was awarded to the Bragg duo in 1915 for their contribution in the field of crystallography.

A beam of x-rays incident to a crystal face gets partially scattered by the atoms of the crystal. The fraction that is not scattered reaches the next atomic layer where another part is scattered and remainder passes across to the next layer and so on. As a result the diffraction pattern is generated from the constructive and destructive interference of X-rays diffracted from each plane. The diffracted beams interact constructively if the beams are in phase or destructively if they are out of phase. A basic requirement for x-rays to diffract is that the sample exhibit crystallinity and the spacing between the atomic layers must lie in the wavelength range of x-ray radiation.

Diffraction pattern on film
Diffraction pattern on film

The Bragg’s law can be expressed mathematically as

nλ = 2d Sinθ

where,

 λ is the wavelength of x-ray beam

θ the angle of incidence

d  the spacing between the atomic planes

n is an integer

Diffracted beam from several randomly oriented crystals in the sample progresses in a conical shape from the crystal face and the diffraction pattern can be displayed on a photographic film as a series of concentric rings.

Based on XRD measurements you can record the distance between the atomic layers of the crystal lattice, estimate bond lengths and angles, and confirm the identity of the unknown materials by correlating their crystal lattice structure with standard reference materials.

The law though developed to  observe scattering of X-rays by  crystals is applicable to determine the structure using different beams such as electrons, ions, neutrons or protons with wavelengths in the range of distances between the atoms or molecules in the crystal.

Understanding nature of matter through knowledge of Crystallography

Scientist with crystal structure lattice model
Scientist with crystal structure lattice model

All matter is composed of atoms and their clusters called molecules. The arrangement of atoms and molecules characterize a substance and assign its unique identity. Superficially it is difficult to appreciate such arrangements but if you were to have a peek down to the atomic and molecular levels you will be amazed to see that many of the commonly occurring materials display a highly symmetric and orderly arrangement of atoms and molecules. X-ray diffraction or XRD is tool that helps you gain such an insight into structure of matter at the atomic level. An understanding of the science of crystallography will be necessary to fully exploit the potential of the XRD technique.

What is Crystallography?

Crystallography is a science dealing with study of crystals. It is an established fact that pure compounds have a definite arrangement of atoms which are responsible for their characteristic properties irrespective of the source of the material. Examples are common salt which has a cubic arrangement of sodium and chlorine atoms, diamond a pure form of carbon has tetrahedral atomic arrangement of carbon atoms whereas graphite has hexagonal ring structures.

It is obvious that crystals can exist in solid state only. Gases or liquids do not exhibit crystallinity. However, solids can exist both in  crystalline or non-crystalline states. The present article discusses some of the crystalline states and characteristic features of materials that are common in nature.

Crystal lattice

Crystals comprise of well organized array of atoms, molecules or ions. A unit cell or a crystal lattice is the smallest structure of the compound or element which repeats itself by translation throughout the crystal. It can be said that a lattice is theoretically an infinite array of atoms, molecules or ions which repeats itself to constitute a crystal. A lattice made up of same atoms as in case of elements is called monoatomic and if it has more than one type of  atoms it is called a polyatomic lattice. There are different types of lattices which are classified according to the geometrical arrangements of atoms which will be taken in a subsequent article.

Types of crystals and their properties

Single crystals

Single crystals are large enough in size and their outer boundaries display distinct geometrical shapes. Some of these are found naturally but for others the shapes can be cut artificially to make them more attractive. Typical examples are gemstones which can be cut to different shapes to improve their shapes and light reflection properties.

Ionic crystals

The atoms constituting ionic crystals have different electro- negatives and are held together in their positions by electrostatic forces. Such crystals are generally hard and have high melting points

Covalent crystals

Covalent crystals are characterized by sharing of electrons between atoms of the crystal. Strong bonding makes the crystals extremely hard and give them high melting points. A typical example of such crystals is diamond whose hardness is difficult to match.

Metallic crystals

Metallic crystals are constituted of metal atoms positioned on the lattice sites. The outer shell electrons of metal atoms are free to move around throughout the lattice. This property makes such crystals good conductors of heat and electricity and also show high melting points.

Molecular crystals

Molecular crystals comprise of individual molecules at the lattice points. Such molecules are held together by weak non-covalent forces such as van der waals forces or hydrogen bonding. Such interactions render molecular crystals with softness and low melting points. Examples of molecular crystals are proteins which exhibit crystalline features.

Polycrystalline materials

Polycrystalline materials are clusters of several crystallites or grains having different sizes and orientations. The orientations can be random in nature and influenced by their growth and processing during formation.

Allotropes or Polymorphs

Allotropes or polymorphs are different forms of an element which are made up of the atoms bound together in different geometrical arrangements. A common example is diamond which can be present in a tetragonal arrangement in diamond but in hexagonal ring arrangement in graphite. Such polymorphs show different physical properties.

Amorphous solids

Amorphous or non-crystalline solids do not exhibit long-range order characteristic of a particular crystalline material. Such materials often exhibit only a short-range order. Examples are glass, polymer films and gels

Anisotropy

Anisotropy refers to the difference in properties when measured along different axial directions in a crystal. Such physical properties include absorbance of light, reflective index, conductivity, tensile strength, etc. Anisotropy generally results from distortion and elongation of grains in some direction during the formation of crystals.

Wealth of information provided by XRD analysis technique

Internal atomic arrangements in complex molecules
Internal atomic arrangements in complex molecules

X-rays present a unique ability to the analytical chemist to understand the structure of materials. X-rays simply do not get reflected by reflecting surfaces superficially but they have the ability to penetrate and get diffracted from internal atomic layers within crystals. This possibility arises as their wavelengths are in the range of atomic dimensions of crystals that constitute most of the materials under examination.

XRD analysis has not only contributed to our understanding of arrangement of atoms within crystalline materials but has contributed immensely to establish purity of materials and identification of different phases. Such studies yield valuable details on materials ranging from polymers, metallurgical samples, pharmaceuticals, electronic components, minerals and nano materials. The scope of applications is vast and in this article some of the application areas of XRD analysis are briefly discussed.

Single crystal and Powder XRD analysis are two common techniques used in XRD laboratories. Both the techniques have a distinct advantage that they are non-destructive and it is possible to handle small amount of samples that are generally available. Single crystal studies provide the geometrical arrangement of atoms in the crystal lattice including critical information on bond lengths and angles. However, such studies take 2 to 3 days for data collection and subsequent analysis. On the other hand powder XRD analysis can be concluded in about half an hour. In addition sample preparation is simple and the information made available can be used for both qualitative and quantitative purposes.

Pharmaceuticals

XRD analysis is a valuable tool for development of new drugs, characterization of active materials and excipients, testing at different stages of manufacture for effective quality control. Such information helps to improve quality of formulations, bio -availability and improvement of stability characteristics.

 Geological applications

Powder XRD is a powerful technique for characterization and grading of minerals. Each mineral has a well defined crystal geometry and yields a unique x-ray diffraction pattern which helps in taking commercial decisions before undertaking mining operations.

Stability studies

Changes in environmental parameters such as temperature and humidity can lead to certain solid phase transitions which can be conveniently monitored using powder XRD. Such information provides useful clues on the stability of final finished products and can make contributions to evaluate their stability behaviour.

Studies on nano materials

The properties of materials depend largely on crystal size. This feature can provide useful information on nano particles which are the building blocks of pharmaceuticals, polymers and composites.

Electronics

Micro Electronics is contributing significantly to advances in computer and electronic consumer products. Silicon and gallium arsenide are two popular materials used in integrated circuits production. XRD has been used in such applications to provide useful information on crystal structures and defects in manufacturing of microelectronic components

Forensics

Powder XRD plays a significant role in criminal investigations. Samples collected from scene of crime such as broken glass, paint chips, hair and powders provide unique x-ray diffraction patterns which can help trace their origin and ultimately to lead to the prime suspects.

XRD alone or in combination with other techniques such as chromatography, FT-IR or DSC provides valuable confirmatory information on nature and characteristics of materials.

 

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