X-Rays

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Learning Objectives
• Explain X-rays.
• Interpret the symbols used in the Bragg equation.

Like light, X-rays are electromagnetic radiation with very short wavelengths. Thus, X-ray photons have high energy, and they penetrate opaque material, but are absorbed by materials containing heavy elements.

X-ray Diffraction

When light passes through a series of equal-spaced pinholes, it gives rise to a pattern due to wave interference, and such a phenomenon is known as diffraction. X-rays have wavelengths comparable to the interatomic distances of crystals, and the interference patterns are developed by crystals when a beam of X-rays passes a crystal or a sample of crystal powder. The phenomena are known as diffraction of X-rays by crystals. More theory is given in Introduction to X-ray Diffraction.

X-ray diffraction, discovered by von Laue in 1912, is a well established technique for material analysis. This link is the home page of Lambda Research, which provide various services using X-ray diffractions. For example:

• Residual Stress Measurement
• Qualitative Phase Analysis
• Quantitative Phase Analysis
• Precise Lattice Parameter Determination

In 1913, the father and son team of W.H. Bragg and W.L. Bragg gave the equation for the interpretation of X-ray diffraction, and this is known as the Bragg equation.

$$2\, d\, \sin \theta = n\, \lambda$$

where d is the distance between crystallographic planes, $$\theta$$ is half the angle of diffraction, n is an integer, and $$\lambda$$ is the wavelength of the X-ray. A set of planes gives several diffraction beams; each is known as the nth order.

Example 1

The X-ray wavelength from a copper X-ray is 154.2 pm. If the inter-planar distance from $$\ce{NaCl}$$ is 286 pm, what is the angle $$\theta$$?

Solution

\begin{align*} \sin \theta &= \dfrac{\lambda}{2 d}\\ &= \dfrac{154}{2\times282}\\ &= 0.273 \end{align*}

$$\theta = 15.8^\circ$$

Example 2

An X-ray of unknown wavelength is used. If the inter-planar distance from $$\ce{NaCl}$$ is 286 pm, and the angle $$\theta$$ is found to be 7.23°, what is $$\lambda$$?

Solution

\begin{align*} \lambda &= 2\, d\, \sin\theta\\ &= 2\times282\times\sin(7.23^\circ)\\ &= \mathrm{71\: pm} \end{align*}

Example 3

The X-ray of wavelength 71 pm is used. If the inter-planar distance from $$\ce{KI}$$ is 353 pm, what is the angle $$\theta$$ for the second order diffracted beam?

Solution
The calculation is shown below:

\begin{align*} \sin \theta &= \dfrac{\lambda}{2 d}\\ &= \dfrac{71}{2\times353}\\ &= 0.100\\ \theta &= 5.8^\circ \end{align*}

These examples illustrate some example of the applications of X-ray diffraction for the study of solids.

Exercise $$\PageIndex{1}$$

If the wavelength is 150 pm and the interplanar distance d is 300 pm, what is the angle $$\theta$$ in the Bragg equation, for n = 2?