How crystals turn X-rays into a ruler
In 1913, months after Max von Laue showed that crystals diffract X-rays, William Lawrence Bragg found a simple way to read those patterns. He pictured the atoms of a crystal as stacks of parallel planes, each reflecting a little of the beam. The reflections add up only when the extra distance traveled by the wave hitting the deeper plane is a whole number of wavelengths:
nλ = 2d sin θ
- λ is the X-ray wavelength, in ångströms (1 Å = 0.1 nm).
- d is the spacing between the planes, the quantity you usually want.
- θ is the Bragg angle between the beam and the planes.
- n is the diffraction order, 1 for the strongest reflection.
How to use the calculator
- Choose whether you want the spacing from a measured angle, or the angle at which a known spacing will diffract.
- Pick the X-ray source. Copper Kα (1.5406 Å) is standard in powder diffraction labs; molybdenum (0.7093 Å) is common for single crystals.
- Enter the 2θ value read from your diffractogram. The calculator halves it to get θ.
Worked example
Silicon powder measured with copper radiation shows a strong peak at 2θ = 28.44°. Then θ = 14.22°, sin θ = 0.2457, and
d = 1.5406 ÷ (2 × 0.2457) = 3.1358 Å
This is the spacing of silicon’s (111) planes, which is why silicon is a favorite calibration standard. The calculator also reports that orders up to n = 4 are geometrically possible for this spacing, since sin θ cannot exceed 1.
Reading the results
- No diffraction: when nλ is larger than 2d, sin θ would have to exceed 1. Short spacings only diffract with short wavelengths, which is one reason molybdenum sources exist.
- Peak shifts: a peak moving to lower 2θ means larger spacings, for example when a lattice expands with heat or when larger atoms substitute into it.
- Identification: a list of d-spacings and intensities acts like a fingerprint. Labs compare it with reference databases to identify minerals, drugs and corrosion products.
Beyond X-rays
The same law applies to any wave with a wavelength similar to the atomic spacing: electrons in an electron microscope, neutrons at research reactors, and even visible light in opals and photonic crystals, where the spacings are hundreds of nanometers.