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?

Answer: 30 degrees