Bragg's Law

Introduction

Bragg's Law refers to a fundamental principle describing the diffraction of X-rays by crystal lattices. It is named after William Lawrence Bragg and his father, William Henry Bragg. The work of these physicists is very important in the area of X-ray crystallography. This law defines how X-rays will set up constructive interference at particular angles, which form part of the atomic planes in crystals. Mathematically, this interaction is represented by the equation nλ=2dsinθ, whereθ is the angle of incidence and reflection. Associated terms with Bragg's Law are the wavelength of the X-rays and spacing between crystal planes. This theory has a huge application in the determination of crystal structures, and identifying materials; the areas of applicability of this principle involve Physics, Chemistry, and Biology. The knowledge of Bragg's Law is always necessary for the analysis and interpretation of the patterns recorded from the X-ray diffraction experiment.

Superconductivity
Kammerlingh Onnes observed this phenomenon at 4 K in mercury. A substance is said to be superconducting when it offers no resistance to the flow of electricity. There is no substance that is superconducting at room temperature. Superconductors are widely used in electronic power transmission, building supermagnets, etc. For example, YBa₂Cu₃O₇, Nb₃Ge alloy, (TMTSF)₂PF₆, etc.

Cohesive Energy
It is the energy needed to achieve infinitely separated gaseous ions from one mole of an ionic crystal lattice that is negative of the Lattice energy.
$\underset{(\mathrm{s})}{\mathrm{MX}}\to \underset{(\mathrm{g})}{{\mathrm{M}}^{+}}+\underset{(\mathrm{g})}{\mathrm{X}}$

Bragg’s Equation
This equation gives a simple relationship between the wavelength Of X-rays the distance between the planes in the crystal and the angle Of reflection. This equation can be written as:
$\begin{array}{rl}& \mathrm{n}\lambda =2\mathrm{d}\sin \theta \\ & \text{Here}\\ & \mathrm{n}=\text{Order of reflection; in general it is taken as}1.\\ & \lambda =\text{Wavelength of}\mathrm{X}\text{- rays}\\ & \mathrm{d}=\text{Distance between two layers of the crystals}\\ & \theta =\text{Angle of incident light}\end{array}$

As for a given set of lattice planes, the value of 'd' is fixed so the possibility of getting maximum reflection depends only on θ. If we increase θ gradually a number of positions will be observed at which there will be maximum reflection.

Bragg’s Application

• Bragg's observation has proved highly beneficial in determining the structures and dimensions of ionic crystalline solids.
• It also helped in describing many properties of X-rays.
• It helped in the construction of an X-ray spectrometer by which the crystalline structure of crystals can be easily described. For example, the face-centered cubic structure of NaCl.

For a better understanding of the topic and to learn more about Bragg's Law with video lesson we provide the link to the

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Some Solved Examples

Example 1: In superconductivity, the electrical resistance of material becomes?

1) Zero

2)Infinite

3)Finite

4)All of the above

Solution

Superconductors are those materials that have zero electrical resistance or infinite conductance.
Their conductivity is known as superconductivity.
Hence, the answer is the option (1).

Example 2: Which of the following relation is correct for first order Bragg’s diffraction?

1)$\sin \mathrm{\Theta }=\frac{2a}{\lambda }(h^2+k^2+l^2)$

2)$\sin \mathrm{\Theta }=\frac{2a}{\lambda }{(h^2+k^2+l^2)}^{\frac{1}{2}}$

3) $\sin \mathrm{\Theta }=\frac{\lambda }{2a}{(h^2+k^2+l^2)}^{\frac{1}{2}}$

4)$\sin \mathrm{\Theta }=\frac{\lambda }{2a}{(h^2+k^2+l^2)}^2$

Solution

Bragg’s Equation

This equation gives a simple relationship between the wavelength Of X-rays the distance between the planes in the crystal and the angle Of reflection. This equation can be written as:
$\mathrm{n}\lambda =2\mathrm{d}\sin \theta$

Here $\mathrm{n}=$ Order of reflection; in general it is taken as 1.
$\lambda =$ Wavelength of $\mathrm{X}-$ rays
$\mathrm{d}=$ Distance between two layers of the crystals
$\theta =$ Angle of incident light

As for a given set of lattice planes the value of 'd' is fixed so the possibility of getting maximum reflection depends only on θ. If we increase θ gradually a number of positions will be observed at which there will be maximum reflection.

Now,

$n\lambda =2d\sin \mathrm{\Theta }$

Thus, $d=\frac{\lambda }{2\sin \mathrm{\Theta }}\dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots (i)$

Again, we have

$d=\frac{a}{\sqrt{h^2+k^2+l^2}}\dots \dots \dots \dots \dots \dots \dots \dots \dots \dots (ii)$

On combining both these equations, we have

$\frac{\lambda }{2\sin \mathrm{\Theta }}=\frac{a}{\sqrt{h^2+k^2+l^2}}$

Thus, $\sin \mathrm{\Theta }=\frac{\lambda }{2a}{(h^2+k^2+l^2)}^{\frac{1}{2}}$

Therefore, Option(3) is correct.

Example 3: Calculate the wavelength of X-rays, which produces a diffraction angle $2\theta$ equal to ${16.8}^0$ for a crystal. Assume first order diffraction with inter particle distance in crystal of 0.2nm.

1)$58.4\times {10}^{-11}\mathrm{m}$

2)$4.3\times {10}^{-11}\mathrm{m}$

3)$3.7\times {10}^{-11}\mathrm{m}$

4) $5.8\times {10}^{-11}\mathrm{m}$

Solution

We have given,
n = 1, $d=0.2\times {10}^{-9}\mathrm{m}$

Thus, $\theta =16.8/2={8.4}^{\circ }$

Therefore $\lambda =\frac{2\times 0.2\times {10}^{-9}\times \sin 8.4}{1}$
Thus, $\lambda =5.84\times {10}^{-11}\mathrm{m}$
Therefore, Option(4) is correct

Example 4: Pure silicon is an insulator but on heating it becomes a semi-conductor because:

1)On heating, electrons occupy higher energy states

2)No effect on electrons on increasing temperature

3)On heating, electrons move freely in crystals

4) Both (1) and (3)

Solution

Pure silicon has its electrons in a lower energy state. An increase in temperature promotes the electrons to occupy higher energy states. These electrons move freely in crystals and are responsible for electrical conduction.
Hence, the answer is the option (4).

Example 5: In Bragg's equation for diffraction of X-rays, n represents:

1)Quantum number

2) An integer

3)Avogadro's number

4)moles

Solution

As we learned,

Bragg's Equation

$\mathrm{n}\lambda =2\mathrm{d}\sin \theta$

Here, n represents an integer that is multiplied by the wavelength and represents phase waves for the determination of the distance between the planes.

Hence, the answer is the option (2).

Summary

This takes the basic form in which Bragg's Law interconnects the angle of incidence with that of the reflection of X-rays, their wavelength, and the distance between planes within a crystal:
nλ=2dsinθ. Here, one can see immediately that the equation is a description of when X-rays reflected from different planes in a crystal come into reinforcement with one another, resulting in constructive interference. It is thus central to all the X-ray diffraction techniques that have aided greatly in the correct analysis of crystal structures. Bragg's Law made immense contributions to scientific discovery—like the discovery of the DNA double helix structure—thus remaining one of the cornerstones of modern material science. We can think of X-rays as waves that not only interact with several layers of atoms inside a crystal but in such a way that all the reflected waves add up to produce very strong signals at specific angles, thus giving evidence for the periodicity of crystal structures and going deep into the atomic arrangements.