X-ray Diffraction

We have seen that a one-dimensional periodic arrangement of coherent radiation sources (chain of sources) produces a diffraction pattern. The diffraction grating is an example. Atoms and molecules have a three dimensional periodic arrangement inside crystalline solids. A diffraction pattern is produced if the atoms or molecules act like a three dimensional grating. Inside crystalline solids the inter-atomic spacing is of the order of $1 \, {\rm A}$. Crystals can produce a diffraction pattern with X-ray whose wavelength is comparable to inter atomic spacings. The wavelength of visible light is a few thousand times larger and this does not serve the purpose.

Figure 13.1:

X-ray is incident on a crystal as shown in Figure 13.1. The oscillating electric field of this electromagnetic wave induces a oscillating dipole moment in every atom or molecule inside the crystal. These dipoles oscillate at the same frequency as the incident X-ray. The oscillating dipoles emit radiation in all directions at the same frequency as the incident radiation, this is known as Thomson scattering. Every atom scatters the incident X-ray in all directions. The radiation scattered from different atoms is coherent. The total radiation scattered in any particular direction is calculated by superposing the contribution from each atom.

For a crystal where the atoms have a periodic arrangement, it is convenient to think of the three-dimensional grating as a set of planes arranged in a one-dimensional grating as shown in Figure 13.2.

Figure 13.2:

Consider X-ray incident at a grazing angle of $\theta$ as shown in Figure 13.2. The intensity of the reflected X-ray will be maximum when the phase difference between the waves reflected from two successive planes is $\lambda$ or its integer multiple. This occurs when

$$2 d \, \sin \theta = m \lambda \,.$$ (13.1)

This formula is referred to as Bragg's Law. The diffraction can also occur from other planes as shown in Figure 13.3.

Figure 13.3:

The spacing between the planes $d$ is different in the two cases and the maxima will occur at a different angle. The first set of planes are denoted by the indices $(1,0,0)$ and the second set by $(1,1,0)$. It is, in principle, possible to have a large number of such planes denoted by the indices referred to as the Miller indices. The distance between the planes is

$$d(h,k,l)=\frac{a}{\sqrt{h^2+k^2+l^2}}$$ (13.2)

where is the lattice constant or lattice spacing. Note that the crystal has been assumed to be cubic.

Figure 13.4 shows a schematic diagram of an X-ray diffractometer. This essentially allows us to measure the diffracted X-ray intensity as a function of $2 \, \theta$ as shown in Figure 13.4.

Figure 13.4:

Figure 13.5 shows the unit cell of $La_{0.66} Sr_{0.33} MnO_3$. X-ray of wavelength $\lambda=1.542 {\rm A}$ is used in an X-ray diffractometer, the resulting diffraction pattern with intensity as a function of $2 \theta$ is shown in Figure 13.6.

Figure 13.5:

Figure 13.6:

The $2 \theta$ values of the first three peaks have been tabulated below. The question is how to interpret the different peaks. All the peaks shown correspond to $m=1$ ie. first order diffraction maximas, the higher orders $m=2,3,...$ are much fainter. The different peaks correspond to different Miller indices which give different values of $d$ (eq. 13.2). The maxima at the smallest $\theta$ arises from the largest $d$ value which correspons to the indices $(h,k,l)=(1,0,0)$. The other maxima may be interpreted using the fact that $\theta$ and $d$ are inversely related.

$h,k,l$	$2 \theta$
1,0,0	$23.10^{\circ}$
1,1,0	$32.72^{\circ}$
1,1,1	$40.33^{\circ}$

Problems

1. Determine the lattice spacing for $La_{0.66} Sr_{0.33} MnO_3$ using the data given above. Check that all the peaks give the same lattice spacing.
2. For a particular crystal which has a cubic lattice the smallest angle $\theta$ at which the X-ray diffraction pattern has a maxima is $\theta=15^{\circ}$. At what angle is the next maxima expected? [Ans: $21.5^{\circ}$]
3. For a cubic crystal with lattice spacing $a=2 {\rm A}$ and X-ray with $\lambda=1.5 \, {\rm A}$, what are the two smallest angles where a diffraction maxima will be observed?