# X-ray diffraction

The measurement of X-ray wavelengths proved to be very difficult because they are so short. In 1912, however, von Laue used a crystal in an attempt to diffract the X-rays after failing to do this with ordinary optical-style diffraction gratings, and was successful.

His work was followed up by Friedrich and Knipping, and good X-ray diffraction patterns were produced. To give diffraction the obstacles must be only a few wavelengths apart and so the atoms in a crystal lattice were ideal for X-ray diffraction since their separation is about 10-10 m (0.1 nm). Sir William and his son Sir Lawrence Bragg used a crystal as a reflection diffraction grating.

In the diagram a beam of X-rays of wavelength λ is incident at a glancing angle θ on a crystal where the atomic planes are separated by a distance d.

The path difference between the waves reflected at the top plane and those reflected at the second plane is ABC = 2dsinθ.

Constructive interference occurs when this path difference is equal to a whole number of wavelengths. Therefore:

X ray diffraction:      2dsinθ = mλ

This equation is useful in determining the structure of a crystal, because if the X-ray wavelength is known the atomic spacing in the crystal can be found.

If the inter-atomic spacing can be found from X-ray diffraction, then the Avogadro constant (NA) can be calculated. The electron charge can then be found, from F = NAe.

Alternatively X-ray diffraction can provide a means of calculating X-ray wavelengths if the inter-atomic spacing is known.

Example problem
A beam of X-rays of wavelength 0.3 nm is incident on a crystal, and gives a first-order maximum when the glancing angle is 9.0 degrees. Find the atomic spacing.

d = λ/2sinθ = 0.3/2x0.156 = 0.96 nm

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© Keith Gibbs 2016