# Bragg's Law: How X-rays Are Reflected by the Crystal Lattice

## Important Formula

## What do the formula symbols mean?

## Diffraction order

`$$ m $$`Unit

`$$ - $$`

*integer*number (\(m\) = 0,1,2, ...) indicating a multiple of the wavelength \( m \, \class{violet}{\lambda} \) at which

*constructive interference*occurs.

## Wavelength

`$$ \lambda $$`Unit

`$$ \mathrm{m} $$`

## Lattice constant

`$$ \class{blue}{d} $$`Unit

`$$ \mathrm{m} $$`

## Angle

`$$ \class{gray}{\theta} $$`Unit

`$$ \mathrm{rad} $$`

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.