X-Ray Diffraction and Bragg’s Law - UPSC PDF Download

X-Ray Diffraction and Bragg’s Law X-Ray Diffraction
The wavelength of x-ray is of the order of Angstrom (A). Hence optical grating cannot be ˚ used to diffract X-rays. But the dimension of atoms is of the order of few angstrom and also atoms are arranged perfectly and regularly in the crystal. Hence crystals provide an excellent facility to diffract x-rays.

Bragg considered crystal in terms of equidistant parallel planes in which there is regularity in arrangement of atoms. These are called as Bragg planes. There are different families of such planes that exist in the crystal and are inclined to each other with certain angle.

In Braggs diffraction the crystal is mounted such that an X-ray beam is inclined on to the crystal at an angle θ. A detector scans through various angles for the diffracted X-rays. It shows peaks for (maximum current) for those angles at which constructive interference takes place. Braggs law gives the condition for constructive interference

Derivation of Bragg’s Law:
Consider Monochromatic beam of X-Rays. It is incident on the crystal with glancing angle θ. Ray AB, which is a part of the incident beam, is scattered by an atom at B along BC. Similarly the ray DE is scattered by an atom in the next plane at E along EF. The two scattered rays undergo constructive interference if path difference between the rays is equal to integral multiple of wavelength.

Construction:BP and BQ are the perpendicular drawn as shown in fig. The path difference
δ = P E + EQ = nλ

From Right angled triangle PBE
sin θ = P E/BE

Where BE=d(Interplanar spacing)=d_hkl
Therefore P E = BE sin θ = d sin θ
Similarly from Right angled triangle QBE
QE = BE sin θ = d sin θ
Substituting in (7.1) δ = d sin θ + sin θ = nλ
δ = 2d sin θ = nλ

Therefore the condition for constructive interference is integral multiple of wavelength of X-rays which is given byHence Bragg’s Law.
Since Bragg diffraction satisfies the laws of reflection it is also called Bragg reflection

Bragg’s X-ray Spectrometer(Determination of wavelength and Interplanar spacing)
It is an instrument devised by Bragg to study the diffraction of X-rays using a crystal as Grating. It is based on the principle of Bragg Reflection. Construction: Monochromatic

X-Ray Beam from an X-Ray tube is collimated by slits s₁ and s₂ and is incident on the crystal mounted on the turntable at a glancing angle θ. The crystal can be rotated using the turntable. The reflected X-Ray beam is again collimated by slits s₃ and s₄ and allowed to pass through ionization chamber fixed on the Mechanical Arm. Due to ionization in the medium current flows through the external circuit, which is recorded by the Quadrant Electro Meter (E). In order to satisfy the laws of reflection the coupling between the turntable and the mechanical arm is so made that, if the turntable is rotated through an angle q then mechanical arm rotates through an angle 2θ.

Experiment:
Rotating the turntable increases glancing angle. Ionization current is measured as a function of glancing angle. The Ionization current is plotted versus glancing angle. It is as shown below. The angles corresponding to intensity maximum are noted

down. The lowest angle θ corresponding to maximum intensity corresponds to the path difference λ.
2d sin θ₁ = n₁λ = λ
Similarly for next higher angles
2d sin θ₂ = n₂λ = 2λ
2d sin θ₃ = n₃λ = 3λ   and so on

sin θ₁ : sin θ₂ : sin θ₃ = 1 : 2 : 3

If equation (7.2) is satisfied for θ₁, θ₂, θ₃ etc, then the Bragg’s law is verified.

By determining θ using Bragg’s Spectrometer and by knowing the value of interplanar separation (d), Wavelength (λ) of X-ray beam can be calculated.
By determining θ using Bragg’s Spectrometer and by knowing the value of Wavelength (λ) of X-ray beam Interplanar separation (d) can be calculated.