Bragg’s law was first proposed by Sir William Bragg and his son Sir Lawrence Bragg. They studied the diffraction of X-rays on various surfaces and defined the nature of X-ray diffraction on the crystal surface. Using this law, we can categorize the crystals into different classes.
What is Bragg’s law?
When a crystal is bombarded by X-rays of a wavelength (equal to the atomic spacing between two crystal lattice planes) and at a certain incident angle (θ), intense reflected X-rays are produced after the wavelengths of the scattered X-rays interfere constructively.
Constructive interference can occur only if the differences in the travel path are equal to an integer multiple of the wavelength. When this constructive interference occurs, a diffracted beam of X-rays will leave the crystal at an angle equal to that of the incident beam.
Bragg’s law Equation
nλ = 2d sinθ
where,
- n = an integer
- λ = is the wavelength of the X-ray that incident on the crystal surface
- d = is the distance between the atomic layers
- θ = is the angle with which the X-ray incident on the crystal surface
Applications
1) In X-ray Diffraction
Usually, the values of the wavelength (λ) of the incident X-ray and the incident angle (θ) at which constructive interference occurs, are both known. So, solving Bragg’s Equation gives the d-spacing between the crystal lattice planes of atoms, which is used for identification and characterization purposes.
Read more about X-ray Diffraction spectroscopy (XRD) here.
Example:
1) Suppose x-rays of wavelength 1.6 x 10-11 m undergo first-order reflection at an angle of 5° from a crystal. Find the spacing between atomic planes.
Given:
- Wavelength (λ) = 1.6 x 10-11 m
- Glancing angle (θ) = 5°
- Order of diffraction (n) = 1
Solution:
According to Bragg’s equation:
nλ = 2dsinθ
1× 1.6 × 10 -11 = 2d sin (5)
So;
2d = 1.6 × 10 -11 / sin (5)
d = 1.8 × 10 -10 m / 2
d = 0.09 nm
Similarly,
2) the spacing between one set of crystal planes in NaCl (table salt) is 0.282 nm, where the glancing angle (θ) is 7°. Assuming that it’s the first-order maximum (n = 1), the wavelength of X-rays will be;
Given:
- Plane spacing (d) = 0.282 x 10-9 m
- Glancing angle (θ) = 7°
- Order of diffraction (n) = 1
Solution:
According to Bragg’s equation:
nλ = 2dsinθ
λ = 2 (0.282 x 10-9) sin (7°)
λ = 0.069 nm
2) In X-ray Flourescence
In the case of XRF (X-ray fluorescence spectroscopy) or WDS (Wavelength Dispersive Spectrometry), crystals of known d-spacings are used for analyzing crystals in the spectrometer.
Why is Bragg’s law important?
Bragg confirmed how X-rays passing through a crystal gives information permitting the crystal’s atomic shape to be deduced. X-ray records help scientists to construct 3-D fashions of the way atoms are organized in solids.
Concepts Berg
What is lambda in Bragg’s equation?
The variable lambda is the wavelength of the incident X-ray beam in [nλ = 2d sinθ].
Who proposed Bragg’s law?
William Lawrence Bragg proposed the equation that shows a relation between the wavelength of the X-rays, the distance between the planes, and the angles at which the X-rays are reflected. Later on, the equation was known as Bragg’s law.
Why can we not use visible light for studying the crystal structure using Bragg’s diffraction method?
The atoms in crystals are spaced at about 1 Angstrom apart. Visible light has a size (wavelength) of the order of 5000 Angstroms, so it cannot resolve the atoms in a crystal.
X-rays have wavelengths of the same order as that of interatomic separation in a crystal, therefore they are used for this purpose.
What are the limitations of Bragg’s law?
It only considers the lattice planes for the reflection of X-rays. The interactions of X-rays with the constituents of the crystal are not taken into account.
It doesn’t give any information about the intensity of scattering for the spatial distribution of electrons in the unit cell.
Reference links
- Chapter 4 (ps.uci.edu)
- X-ray reflection in accordance (serc.carleton.edu)
- Bragg’s law (www.vedantu.com)
