When a molecule forms a crystalline solid we can use X-ray diffraction to determine the positions of the atoms within the molecule. X-rays are very short wavelength light with wavelengths of roughly the diameter, or less, of an atom. The Bragg equation below gives the scattering angles θ where bright “reflections” of monochromatic X-rays are detected from layers of equivalent atoms within the crystal.

n λ = 2 d sin (θ)

Where λ is the wavelength of the X-rays, d is the spacing between the lattice planes of equivalent atoms in the crystal, θ is the scattering angle and n = 1, 2, 3, is the order of the reflection (quantum numbers such as n are dimensionless, they are pure numbers). Calculate the lattice spacing d when using radiation of wavelength λ = 0.154 nm for n = 1 at a scattering angle θ = 11°. Specify the units for your calculated d value. The symbol nm stands for a nanometre or 10-9 m.

Solution

Rearrange the Bragg equation so that d is the subject and then substitute in the values of the variables.

n λ = 2 d sin (θ);     d = n λ / 2 sin (θ);      d = (1 *  0.154 nm) / 2 sin 110

d = 0.404 nm

Note that we do not need to change the units to metres, they may be left as nanometres (nm) but whichever unit of length is used it will need to have been inserted in the equation in order to know the units of the lattice spacing.