Dissemination of IT for the Promotion of Materials Science (DoITPoMS)

# Bragg′s law

The concept used to derive Bragg's law is very similar to that used for Young’s double slit experiment.

An X-ray incident upon a sample will either be transmitted, in which case it will continue along its original direction, or it will be scattered by the electrons of the atoms in the material. All the atoms in the path of the X-ray beam scatter X-rays. We are primarily interested in the peaks formed when scattered X-rays constructively interfere. (In addition, after scattering some X-rays suffer a change in wavelength. This incoherent scattering is not considered here).

Constructive interference occurs when two X-ray waves with phases separated by an integer number of wavelengths add to make a new wave with a larger amplitude.

When two parallel X-rays from a coherent source scatter from two adjacent planes their path difference must be an integer number of wavelengths for constructive interference to occur.

Path difference = n λ

Therefore;

n λ = 2 d sin θ

In order to consider the general case of hkl planes, the equation can be rewritten as:

λ = 2 dhkl sin θhkl

since the dhkl incorporates higher orders of diffraction i.e. n greater than 1.

The angle between the transmitted and Bragg diffracted beams is always equal to 2θ as a consequence of the geometry of the Bragg condition. This angle is readily obtainable in experimental situations and hence the results of X-ray diffraction are frequently given in terms of 2θ. However, it is very important to remember that the angle used in the Bragg equation must always be that corresponding to the angle between the incident radiation and the diffracting plane, i.e. θ.

The diffracting plane might not be parallel to the surface of the sample in which case the sample must be tilted to fulfil this condition. (The concept of orientation will be dealt with later in this TLP).

This is the procedure to obtain asymmetric reflections and work in transmission: