X-ray Diffraction

X-ray Diffraction

When an x-ray beam is passed through a crystal, the scattering can be treated on the basis of the interference of the waves reflected from the crystal planes.

The general principle of diffraction methods depends on the phenomenon of interference, which occurs when any wave motion is scattered from a number of centers. This phenomenon is exhibited by visible radiation when a beam of light passes through a series of closely spaced slits. If the light is monochromatic, i.e. consists of radiation of only a single wavelength, the waves that we ascribe to the light emerging from the slits will add in only certain directions. In these directions constructive interference is said to occur, and at these directions a beam of diffracted light will appear. At other directions the diffracted waves will be out of phase to various extents, destructive interferences will occur, and les light will be seen. For the pattern of scattering by the slits it is easy to see, that constructive interference between occurs in directions defined by the angle θ at which constructive interference yields several values of θ at which constructive interference occurs when such is the case, the angles for constructive interference can be measured and if λ is which are related to the spacing d between the slits and the wavelength λ of the light by the relation

sin θ = n λ/d

where n is an integer. One sees from this, furthermore, that d and λ must be one of the same order of magnitude if n λ/d is to take on a number of values between 0 and 1 when n assumes various small integral values. Under these conditions yields several values of θat which constructive interferences occurs. When such is the case, the angles for constructive interference can be measured, and if λ is known, can be turn around to give:

d = n λ/ sin θ

Diffraction by crystals: studies of the internal structure of crystals depend on a penetrating radiation that will enter the crystal and will display interference effect as a result of scattering from the ordered array of scattering centers. 

In the Bragg method, the phenomenon is observed when nearly monochromatic x-rays are deflected from a crystal. A beam of x-rays is passed into a crystal which is represented by layers of particles. The reflection is in fact, not simple and is greatly distributed by the interference effect. The incoming effect of x-rays can be represented with all the waves in phase. The nature of the outgoing beams must be investigated. We must consider the effect of the perpendicular set of crystal planes, with spacing d and x-rays with some particular wavelength.

Single crystal diffraction: for a crystal arranged and subjected to a rotation about one of the axes, the a axis, the various crystal planes parallel to the a axis will satisfy Bragg’s law and will produce reflections along the “equatorial” plane by the Miller index:

H = 0

Arrangement of the Bragg expression to:

2 d/n sinθ = 2d hkl sinθ

Shows that the higher order reflections from planes with a spacing d can be treated as if they due to first order reflections from planes with spacing dhkl = d/n

Powder diffraction: the crystals in a powdered sample present all possible orientations to the x-ray beam. The diffraction obtained will be just like that which would result from mounting a single crystal and turning it through all possible angles. For each crystal plane there will be some, one angle at which the Bragg law will be satisfied, and some of the crystals will have this orientation; therefore a diffracted beam will result at the suitable angle, as is depicted in particles. Since there are quite a few crystal planes with a fairly high density of particles, there will be reflections from each of these, and the pattern will show scattering at a large number of angles.

Example: a powder pattern produced by x-rays from a chromium target, which have a wavelength of 229 pm, contains a line at θ = 65˚ that is indexed as 200. What is the value of d200/ at what angle would the 100 line occur, if it occurs and what would be the value of d100?

Solution: the use of an index 200 shows that we are treating all reflection as first order and are assigning fractions of the actual plane spacings to the higher order reflections. The Bragg relation is then written as λ = 2dhkl sinθ. With sinθ= 0.906 we obtain;

D200 = 229pm/2(0.906) = 126 pm

The reflection indexed as 100 has, since the Miller indices show the reciprocals of the plane spacings, twice this spacing, or 252 pm. Its powder pattern line would occur such that sinθ = 229 pm/[2(252 pm)] = 0.454 and θ = 27˚. Notice that we could have treated the reflection at θ = 65˚ as a second order reflection from the 100 planes. Then we would have obtained the spacing of these 100 planes from n λ = 2d sin θ with n = 2 as d100 = 29229 pm)/ (2sin 65˚) = 252 pm.