Probing Atomic Structure with X-Rays

When X-rays encounter a crystal, the rays scatter from planes of atoms inside the lattice. Under certain conditions, the scattered waves interfere constructively, creating intense diffraction peaks. William Lawrence Bragg and his father, William Henry Bragg, showed that the condition for constructive interference is governed by a simple relation between wavelength, interplanar spacing, and the angle of incidence. This insight laid the foundation for modern crystallography and earned them the Nobel Prize in Physics in 1915.

Bragg's Law

The Bragg condition is expressed mathematically as

$n\lambda =2d\mathrm{\backslash sin}\theta$,

where $n$ is the diffraction order (an integer), $\lambda$ is the wavelength of the incident radiation, $d$ is the spacing between crystal planes, and $\theta$ is the angle between the incident beam and the crystal plane. When this condition is met, waves reflecting from successive planes are in phase and reinforce each other, producing a strong reflected beam.

Interpreting the Equation

For a given wavelength, constructive interference occurs only at specific angles that satisfy the equation. As the wavelength shortens or the spacing increases, the required angle changes. Higher-order diffractions ($n>\; 1$) appear at larger angles. By measuring these angles in an experiment, scientists can deduce the spacing between atomic planes, revealing the crystal's structure.

Applications in Crystallography

X-ray diffraction is a primary method for determining the atomic arrangement in solids ranging from minerals to complex biological molecules. By shining X-rays of known wavelength on a crystal and recording the angles of the diffracted beams, researchers can reconstruct the three-dimensional arrangement of atoms. This technique has unveiled the double-helix structure of DNA, the arrangement of proteins, and the composition of countless minerals and alloys.

Using the Calculator

To apply Bragg's law, enter the diffraction order, wavelength of the X-rays, and the spacing between lattice planes. The calculator then solves for the angle $\theta$ using

$\theta =\mathrm{\backslash arcsin}\frac{n\lambda }{2d}$.

The output is given in degrees. If the expression inside the arcsine exceeds one, no solution exists, meaning the chosen wavelength and spacing do not produce that diffraction order. In that case, you may need a different wavelength or order.

Example Calculation

Suppose you shine 0.154 nm X-rays (typical of copper Kα radiation) on a crystal with lattice spacing 0.203 nm. For the first-order diffraction ($n=1$), the equation yields $\theta$ ≈ 23°. Higher orders occur at larger angles if the sine function allows them. Experimentally measuring such angles and comparing them to theoretical predictions enables you to deduce the crystal lattice constants with high precision.

Beyond X-Rays

Bragg's law applies not only to X-rays but also to neutron and electron diffraction. These techniques use the wave-like nature of particles to probe structures that X-rays might not reveal, such as magnetic ordering or light-element positions. By adjusting the wavelength and analyzing the diffraction pattern, scientists can explore materials on the atomic scale.

Limitations and Considerations

The equation assumes ideal, parallel planes and monochromatic radiation. Real crystals may contain defects, and X-ray sources often produce multiple wavelengths. Additionally, thermal motion can broaden diffraction peaks. Nonetheless, Bragg's law provides the starting point for more sophisticated models that include these effects. The calculator here illustrates the core relationship and helps you plan experiments or interpret diffraction data.

Final Thoughts

By exploring different wavelengths, spacings, and diffraction orders, this calculator helps you visualize how X-rays interact with crystalline materials. Bragg's simple formula reveals the periodic structure hidden within solids and remains an essential tool in physics, chemistry, and materials science.