Bragg's Law: How X-rays Are Reflected by the Crystal Lattice
The incident X-ray wave (depicted as a beam in Illustration 1) with the wavelength \( \class{violet}{\lambda} \) is reflected at two adjacent lattice planes (points represent lattice atoms). The glancing angle \( \theta \) is the angle at which an interference maximum occurs in the interference pattern.
The X-ray wave, which is reflected at the deeper lattice plane, has to travel a greater distance. This path difference corresponds twice \( \class{blue}{d}\,\sin(\theta)\). However, the path difference must also correspond to a multiple of the wavelength so that constructive interference of the two waves can take place: \( m \, \class{violet}{\lambda} \)
The Bragg's Law states the condition under which constructive interference \( m \, \class{violet}{\lambda} \) can occur on the lattice.
The term \( \sin(\class{gray}{\theta}) \) can have a maximum value of 1.
Why is visible light unsuitable for diffraction experiments on crystals?
For example, consider the first diffraction order: \( m = 1 \). Then the Bragg condition becomes:
It follows from this that the wavelength can be at most twice the lattice constant \( \class{blue}{d} \) so that something can be observed at all in the diffraction experiment. However, since the lattice constant is very small (for silicon, for example, in the order of \(10^{-12} \, \mathrm{m} \)), the visible light (wavelength of the order of \( 10^{-9} \, \mathrm{m} \)) has far too long wavelength to fulfill the condition 2
:
To fulfill Eq. 2
, you need short-wave light, otherwise you cannot examine crystal structures such as those of silicon.
X-rays - good for examining "thick" crystals. Electron radiation - e.g. for examining small samples. Neutron radiation - e.g. for localizing hydrogen atoms in a crystal lattice.