Structure Determination By X – Rays

Structure Determination By X – Rays 

The German physicist M Von Laue (1879 – 1960) in 1913 suggested the possibility of diffraction of X – Rays by crystals. The reason for this suggestion was that the wavelength of X – Rays was of about the same order as the inter atomic distances in a crystal. Then if these X – Rays was allowed to strike the crystal the rays will penetrate into the crystal and will be scattered by the electrons of the atoms or the ions of the crystal. The rays reflected from different layers of the atoms, due to wave nature will then undergo interference ( constructive and destructive) to produce a diffraction pattern just as it happens in case of light passing through a grating containing a large number of closely spaced lines. In other words, crystals should act as a three dimensional grating for X – Rays. Bragg applied this fact in determining structure and dimensions of crystal.

W.L. Bragg and his father W.H. Bragg determined the cubic structure of NaCl using X – Rays. According to Bragg, a crystal (Composed of series of equally spaced atomic planes) could be employed not only as a transmission grating (as suggested by Laue) but also as a reflection grating. In Bragg’s treatment, the X – rays strike the crystal at angle θ, these penetrates into the crystal and are reflected by different parallel layers of particles in the crystal.

A strong reflected (constructive) beam will result only if all the reflected rays are in phase. The reflected waves by different layer planes will be in phase with one another only if the difference in the path length of the waves reflected from the successive planes is equal to an integral number of wavelengths.

It may noted from the fig. that the beams of X – rays which are reflected from deeper layers travel more to reach the detector. Two X – Ray waves in phase are shown to be approaching the crystal. One wave is reflected from the first layer of atoms while the second wave is reflected from the second layer of atoms. The wave reflected from the second layer travels more distance before emerging from the crystal than the first wave. The extra distance travelled is equal to
LN + NM. For constructive interference to take place the extra distance travelled by the more penetrating beam must be on integral multiple of the wavelength

Path difference = LN + NM

= LN = nλ (n = 1, 2, 3 ….)

Since the triangles OLN and OMN are congruent hence LN = NM

So, path difference = 2LN

LN = d Sinθ where d is the distance between two planes

So, path difference = 2 d Sinθ

For constructive interference 2 d Sinθ must be equal to nλ

Or, nλ = 2 d Sinθ

This relation is called Bragg’s equation. Distance between two successive planes d can be calculated form this equation. With X – Ray of definite wavelength, reflection at various angles will be observed for a given set of planes separated by a distance‘d’. These reflections correspond to n = 1, 2 , 3 and so on and are spoken of as first order, second order, third order and so on the
angle θ increases as the intensity of the reflected beams weakens

X – Ray reflection from crystals