Bragg’s Law - Definition, Derivation, Equation, Applications, Examples, FAQs

Introduction:
Xrd principle is used for structural determination of any crystalline substance. Where xrd full form is X-ray diffraction. It is based on the concept of Bragg's law.
In this article we will study in detail and try to understand what Braggs law, Bragg’s equation is and how to derive Bragg’s equation. We will try to understand about it through x-ray diffraction examples.

History:

Bragg diffraction was first proposed by Laurence Bragg and his father William Henry Bragg in 1913 and for this they got Nobel Prize in 1915. This was the first time when a father son duo got the Nobel Prize.

L. Bragg (1913) showed that scattered radiation from a Crystal behaves as if the diffracted beam were reflected from a plane passing through points of the crystal lattice in a method that makes these Crystal lattice planes related to mirrors. Lawrence Bragg defined this result by rotating crystal method as a set of distinct corresponding planes separated by a constant factor d. It was anticipated that the incident x-ray radiation would yield a Bragg peak of their reflections off the several planar interfered constructively. Now let us move on to what is Bragg’s law and braggs equation?

What is Bragg’s law of diffraction?

Bragg’s law statement is that “when the x-ray beam hits the crystal surface its incidence angle is reflected at the same scattering angle which is θ. And when the path difference factor d also known as d spacing is equal to the integer value n of wavelength then the constructive interference follows”.

Braggs equation formula,

2d sinθ=nλ

In which,

λ is the X-ray beam wavelength, d is the distance between crystal layers (optical path difference), θ is the incident angle (the angle between the incident beam and the scattering plane) and n is an integer. Characteristic X-rays are used to study crystal structure by X-ray diffraction. Bragg's law can be used in the Bragg spectrometer to determine the lattice size.

The principle of Bragg's law is applied to the manufacture of apparatuses such as Bragg spectrometers, which are widely used to study the arrangement of crystals and molecules.

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Why is Bragg’s spectrometer used?

Most of our knowledge of crystal structure and molecular structure as complex as DNA in crystal form comes from the use of X-rays in X-ray diffraction studies. The basic instrument for this kind of research is the Bragg spectrometer. In order to obtain nearly monochromatic X-rays, an X-ray tube is used to generate characteristic X-rays. In order to eliminate as much as possible the continuum of Brehms radiation, a filter suitable for the X-ray beam is used to optimize the energy component in the K-alpha line.

This type of filter uses elements directly above and below the metal in the X-ray target, and uses strong "absorption edges" directly above and below the target metal. The X-rays are collimated through an initial in a strong X-ray absorber (usually lead), and the resultant narrow X-ray beam is permitted to strike the crystal which is under study. The spectrometer device pair the rotation of the crystal with the rotation of the detector so that the rotation angle of the detector is two times the rotation angle of the crystal. This satisfies the Bragg's law condition of X-ray diffraction in the lattice plane.

With Bragg spectrometer, if we know the wavelength of X-Ray we can find out the distance (d) between two consecutive lattice layers of crystal structure or vice versa that is we can find the unknown wavelength of spectrometer if we know the distance d between two consecutive lattice layers of crystal structure. Hence we can say that the Bragg’s law tells us about the spacing between atomic planes d based on the wavelength of X-Ray and the angle of incidence of the X-ray θ.

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Bragg’s law derivation:

Consider homogeneous x-rays of wavelength λ incident on a Crystal at a glancing bragg angle θ. The incident rays AB and DE afterwards deflecting from the lattice plane Y and Z travel along BC and EF respectively. Let the spacing between the lattice planes be ‘d’. BP and BQ are perpendicular drawn from B on DE and EF respectively. Therefore the path difference between the two waves ABC and DEF is equal to PE plus EQ.

Path difference = PE + EQ

In the triangle PBE,

Sinθ = PE/BE

Or,

PE=BE sinθ = d sinθ

In the triangle QBE,

Sin θ= EQ/BE

Or,

EQ=BE sinθ = d sinθ

Therefore,

Path difference = PE + EQ = d sinθ + d sinθ = 2d sinθ

In this Path difference 2d sineθ is equal to integral multiple of wavelength of x-ray that is n λ, then constructive interference will occur between the reflected beams and they will reinforce with each other. Hence, the intensity of the reflected beam is extreme.

Therefore,

2d sineθ = n λ

Where n= 1, 2, 3.... So on

This is known as Bragg's law

Application of Bragg's Law.

• X-ray diffraction (XRD) uses crystal interplanar spacing (d spacing) for identification and characterization. In this situation, the wavelength of the incident X-ray is recognized, and the angle of incidence (θ) at which constructive interference occurs is calculated. On resolving the Bragg equation, gives the distance between the lattice layers of the atoms that yield constructive interference. A given unknown crystal is estimated to have many practical atomic energy levels in its structure; thus, the "reflection" set of all levels can be used to uniquely detect the unknown crystal. Commonly speaking, crystals with high symmetry (such as isometric systems) have relatively few atomic layers, while crystals with low symmetry (triclinic or monoclinic systems) often have a large number of possible atomic layers in their structure.
• In Wavelength Dispersive Spectroscopy (WDS) or X-ray Fluorescence Spectroscopy (XRF), a crystal with a recognized distance is used as the systematic crystal in a spectrometer. As the sites of the sample and the detector are secure in these applications, the angular position of the reflective crystal is transformed as per to Bragg's law so that the exact wavelength of interest can be focused to the detector for quantifiable analysis. Each element in the periodic table has a distinct energy difference between the orbital shells (such as K, L, and M), so each element produces X-rays of a steady wavelength. Therefore, by using a spectrometer crystal (with a fixed crystal pitch) and placing the crystal at a exclusive and secure angle (θ), the element of interest can be identified and quantified based on the characteristic X-ray wavelength generated in every element.

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Problem based on Bragg’s law:

The utilized reflecting plane of Lif crystal has a d value of 2.014 A°. Calculate the wavelength of 2nd order diffracted line which has a value of 50.1°.

Ans. Bragg's formula

n λ= 2d sinθ

Given data

n=2

d=2.014 A°

= 50.1°

n λ= 2d sinθ

λ= 2d sin θ / n

λ= 2×2.014×sin 50.1°/2

λ= 2.014×0.767

λ=1.544 A°

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