X-ray diffraction is a non-destructive analytical technique. It is mainly used in identification
of crystalline materials. In an X-ray diffraction experiment, a sample is place at the center
of the instrument and a beam of X-rays is passed through the sample. The sample absorbs
these x-rays and re-emits them in the form of new x-rays. The re-emitted radiation is
recorded by the detector in the form of a graph. The peak observed in the xrd spectrum
tells about the atomic structure of the sample. Each crystalline material has a different
atomic structure therefore gives different XRD spectrum. In crystalline material, crystals
are composed of regular arrangement of atoms and each atom is composed of a nucleus surrounded
by its electrons. When x-rays hit an atom its energy is absorbed
by the electrons. Since this is not enough energy to be released in form of heat or other
form. Electrons re-emit the energy in the form of a new x-ray but the same energy as
the original. This process is called elastic scattering. The phenomena by which X-rays
are reflected from the atoms in a crystalline solid is called diffraction. The intensity
of scattered rays depends on the scattering power of the individual atoms, which intern
depends on the number of electrons in the atom.
X-rays are high energy light with a repeating period called wavelength. The wavelength of
an X-ray is similar to the distance between atoms in a crystal, therefore when x-rays
interact with each other special interference effect occurs. If the waves are in same phase,
the waves combine together to increase the light intensity. This is called constructive
interference. If the waves are out of sync the signal is destroyed. This is called destructive
interference. In order for constructive interference to
be occur the scattered waves must be in same phase.
Now constructive interference can only occur if the incident and scattered wave travel
equal distance. Since, we know that in a crystal the repeating arrangement of atoms form distinct
planes and these planes are separated from each other by a well-defined distance, known
as d spacing. When x-ray fall on atomic planes, x-rays are scattered by the regularly spaced
atoms. When x-ray fall on top layer then both incident and scattered ray travel same distance.
But when x-ray fall on internal layer, both incident and scattered ray travel some extra
distance. Similarly if x-ray fall on third layer, both the rays have to travel some more
extra distance. This extra distance traveled by x-rays is known as path difference. Because
of this path difference it is not necessary that scattered rays will interfere constructively.
So the only possibility when scattered rays can interfere constructively is that when
incident and scattered rays must travel a whole number. Suppose if the path difference
is equal to 1 wavelength so in that case both the rays will be in same phase because ½
of the distance is traveled by the incident radiation and half of the distance is traveled
by the scattered ray. Similarly if the path difference is equal to 2 wavelengths so in
that case also both the rays will be in same phase as 1 wavelength distance will be traveled
by the incident ray and 1 wavelength distance will be traveled by the scattered ray. So
for constructive interference path difference is equal to nλ where n=0,1,2,3 etc.
The incident rays interact with the atom at an angle θ and reflect at a certain angle.
In case of reflection both incident and scattered or reflected angles are same. Therefore angle
between ACB will be θ and so for the triangle ADB.
So in triangle ACB, Sin θ = CB/AB and we know that AB = d, so CB= d sin θ
Similarly in triangle ADB Sin θ = DB/AB and AB = d, so DB= d sin θ
Now replace CB and DB by d Sin θ in path difference equation
Finally we get a new equation that is 2d sin θ =nλ. This equation is known as Bragg’s
Equation. So Bragg’s law states that diffraction will
occur only if angle of incidence is equal to angle of scattering. And the waves constructively
interfere with path difference of a whole no of wavelength. So with the help of Bragg’s
equation we can directly measure wavelength of x-ray, d spacing of crystals and diffraction
angle.