X-Ray Study of Crystals

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In a crystalline solid, the constituent particles (atoms, ions or molecules) are arranged in a regular order. An interaction of a particular crystalline solid with X-rays helps in investigating its actual structure.

Crystals are found to act as diffraction gratings for X-rays and this indicates that the constituent particles in the crystals are arranged in planes at close distances in repeating patterns.

The phenomenon of diffraction of X-rays by crystals was studied by W.L.Bragg and his father W.H.Bragg in 1913.

They used crystals of zinc sulphide (ZnS) for this purpose. The experimental setup is shown in the figure.

fig 2.21 - Study of the X-ray diffraction by crystalThe diffraction patterns were analysed to determine the interplanar distances in a particular crystal.

A simple representation of the X-ray diffraction is shown as follows:

fig 2.22 - A simple representation of X-ray diffraction

The process was based upon the principle that a crystal may be considered to be made up of a number of parallel equidistant atomic planes, as represented by lines AB, CD and EF in the below figure.

fig 2.23

Suppose two waves (Y and Z) of X-ray beams, which are in phase falls on the surface of the crystal. If the ray Y gets reflected from the first layer i.e., AB line and the ray Z is reflected from the second layer of atoms i.e., CD line, then it is evident that as compared to the ray Y, ray Z has to travel a longer distance, equal to QRS in order to emerge out of the crystal. If the waves Y and Z are in phase after reflection, the difference in distance travelled by the two rays must be equal an integral number of wavelength (nl), for constructive interference.

Thus,

Distance QRS = nl ........(i)

It is obvious from the figure that

QR = RS = PR sinq

Therefore QRS = 2 PR sinq .........(ii)

If the distance between the successive atomic planes is = d

Then, PR = d ...……(iii)

So, from equations (i), (ii) and (iii)

nl = 2d sinq

Thus, Bragg gave a mathematical equation to establish a relationship between wave length of the incident X-ray, the distance between the layers and the angle of diffraction.

• Here, l = wavelength of x-ray used

• q = Angle between incident x-rays and plane of the crystal. The diffracted beam makes an angle 2q.

• d = Distance between planes of the constituent particles in a crystal.

• n = An integer (1, 2, 3, 4, …etc) which represents the serial order of diffracted beams.

Bragg's equation can be used to calculate the distances between repeating planes of the particles in a crystal. Similarly, if interplanar distances are given, the corresponding wavelengths of the incident beam of X-ray can be calculated.