Diffraction of X-Rays by Crystals

Introduction

Diffraction of X-rays by crystal is explained by Bragg’s law. This tutorial unwraps the unique diffraction observation in crystal lattice materials when X-rays pass through them. Diffraction is a universal principle when any wave passes from one medium to another.

But X-ray diffraction (known short as XRD) has unique strength and is used in many applications. The underlying principle in this phenomenon is Bragg’s law which is addressed extensively in this tutorial.

What Is Bragg’s Law?

When X-rays are allowed to pass through a crystal lattice, the rays are scattered due to refraction. The Bragg’s law states that the scattered intensity of rays is found to be at peak as observed in the below conditions

• The incident angle and the scattering angle are equal.

• Between two rays, the path length difference exists and is equal to an integer multiplied by wavelength.

Bragg Equation

The Bragg equation is stated as

$$\mathrm{nλ=2d\:sinθ}$$

Here,

n is the integer

λ is the wavelength of X-rays

d is the distance spacing of atoms in the lattice structure

θ is the angle of incidence which is the same as the angle of scattering

Derivation of Bragg’s Law

Please refer to the figure below for the derivation of Bragg's law.

• Some rays of X-ray light will get reflected in the first plane and some deep in the crystal. Some of the rays will have the second plane interaction and get diffracted at the same angle θ, and the process continues

.

• The diffracted beams will have constructive interference only if the path length difference, as seen in the figure, is an integer multiple of the wavelength.

• In the figure, the difference in the path length of the beam hitting the first and second plane is BO+OA.

So, the two beams will be in phase only if BO+AO=nλ⇒Equation (1) .

As per trigonometric calculations, BO=CO×sinθ.

Since CO is d, the interplanar distance

$$\mathrm{ BO=d\times sinθ\Rightarrow \Rightarrow Equation (2)}$$

$$\mathrm{ Again\: BO=OA\Rightarrow Equation (3).}$$

So, combining equations 1, 2, and 3 we get

$$\mathrm{2d\: sinθ=nλ}$$

This completes the derivation of Bragg’s equation.

Applications Of Bragg’s Law

• Bragg’s law is used by XRD (X- Ray diffraction) instruments to know the internal structure of crystalline materials.

• It is a non-destructive technique.

• XRD is used to ascertain the orientation of atoms and the thickness of films.

Bragg’s Diffraction

• Diffraction as a phenomenon is the interference or bending of waves around obstacle corners, including aperture.

• The diffracting object thus forms a secondary source of the wave.

• Diffracting waves possess an angle different from the incident angle due to the refractive index of the diffracting material. This angle is called the angle of diffraction.

• Bragg’s diffraction occurs when X-rays pass through a crystal lattice forming a unique constructive interference resulting in bright light.

Solved Examples

Q. The spacing of NaCl crystals is given as d = 0.282 nm. X-rays produce a Bragg’s maximum at an angle of 7 degree. What is the wavelength of X-rays?

Ans. Here d = 0.282

θ = 7 degrees

It is a first order maximum and so n = 1

Applying Bragg’s equation, 2×d×Sinθ=n×λ

λ=2×d×Sinθ = 2 x 0.282 x Sin 7 = 0.069 nm

Bragg’s Law Conclusion

When the angle of incidence of X-rays in a crystal lattice is equal to the gradient of scattering, constructive interference exists from the rays reflecting from different atomic planes of the crystal, which follows Bragg’s equation. This constructive scattered light is bright. The beams that do not follow Bragg’s equation will follow a normal traversal in the crystal. The law takes advantage of the fact that X-ray’s wavelength strikes a balance with the lattice spacing of crystalline materials, which is 1 Angstrom, which means this law is not generally applicable to any waves, but only to X-rays and crystalline materials.

Conclusion

This tutorial describes the diffraction of X-rays by crystals, which demonstrates a unique property of diffraction, refractive index, atomic spacing, and wavelength. After a brief introduction, an explanation of Bragg’s law and an equation of the applications of the same is explained. The tutorial includes illustrative figures to give a proper understanding of needed areas. Then the explanation of Bragg’s diffraction, a unique principle, is provided with a comparison with normal diffraction. Finally, we present the conclusion of Bragg’s law in a separate section.

FAQs

1.How did the science of x-ray crystallography originate?

The arrangement of atoms in crystalline solids was of great interest to scientists, and producing it in three-dimensional space was a scientific method called X-ray crystallography. This scientific method takes advantage of the fact that X-rays have a wavelength the same as the interatomic spacing of most crystalline solids. (1 Angstrom - 10^(-8) cm). With the clear proof of X-rays causing diffraction patterns in crystalline Cu SO₄.5H₂ O, crystallography as a science started progressing.

2.What is the outcome of crystallography? What does it try to find out?

Before understanding the outcome, we should understand what a crystal lattice is and how the arrangement of atoms is in the structure. The periodic arrangement of the unit cells is a property of crystals. Since this structure forms a three-dimensional repeated and regular formation of atoms, it is often called a lattice. This repeated atom structure can be seen with “d” as the distance between atoms. Within a unit cell, the distance of atoms in three planes is referred to as a, b and c. The respective angles are referred to as $\mathrm{\alpha, \beta, and\: \gamma}$. These six parameters are called lattice constants. X-ray diffraction can identify these lattice constants accurately and thus reveal the crystalline structure of the crystal.

3.What are constructive interference and destructive interference of waves?

Constructive interference occurs when the lines (representing peaks) cross over each other.

When two waves are in sync or phase, the interference happens constructively. When two waves are out of phase (180-degree phase difference), the interference is termed destructive, and the resulting wave is a straight line.

4.Will crystals reflect X-rays? Which law defines this concept?

It is Bragg’s law which states that at a particular value of the angle of incidence of X-rays, the crystal lattice structure will reflect X-rays. And that angle theta is given by $\mathrm{θ=Sin^{-1} (\frac{nλ}{2d}).}$. So, the answer is yes, the crystals reflect X-rays when Bragg’s condition is met.

5.What is XRD? What are its advantages and disadvantages?

XRD is the short form for X-ray Diffraction. The main use of XRD is to identify unknown crystalline materials. The use of XRD is in different areas like geology, engineering, biology, materials science, etcetera.

Advantages of XRD

• The technique of XRD is a very fast, minimal preparation and powerful way to identify a material's underlying structure.

• XRD instruments are easily available.

Disadvantages of XRD

• For best results, the sample should be homogeneous.

• If the sample is non-isometric, then the indexing of the unit cell pattern of the crystal is difficult to achieve.