# X-Rays

- Page ID
- 35135

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 *n*th 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.

## Questions

- Hint: 30 degrees
- Hint: \(\ce{NaCl}\)
**Discussion -**

The larger the interplanar distance*d*, the smaller the angle.

## Contributors and Attributions

Chung (Peter) Chieh (Professor Emeritus, Chemistry @ University of Waterloo)