Bragg's law

Introduction

Bragg's law, named after the British physicists Sir William Henry Bragg and his son Sir William Lawrence Bragg, is a fundamental principle in the field of X-ray crystallography. The law provides a condition for constructive interference of X-rays scattered by the atoms in a crystal lattice. This principle is crucial for determining the atomic structure of crystals, which has profound implications in various scientific disciplines including chemistry, physics, and materials science.

Historical Background

The discovery of Bragg's law in 1912 marked a significant milestone in the study of crystallography. The Braggs' work was instrumental in the development of X-ray diffraction techniques, which allowed scientists to probe the atomic structure of materials with unprecedented precision. Their pioneering efforts were recognized with the Nobel Prize in Physics in 1915.

Theoretical Foundation

Bragg's law can be mathematically expressed as:

\[ n\lambda = 2d \sin \theta \]

where:

\( n \) is the order of reflection,
\( \lambda \) is the wavelength of the incident X-ray,
\( d \) is the distance between crystal planes,
\( \theta \) is the angle of incidence (also known as the Bragg angle).

This equation describes the condition under which X-rays reflected from different planes of a crystal interfere constructively, leading to a detectable diffraction pattern.

Derivation of Bragg's Law

The derivation of Bragg's law involves considering the path difference between X-rays scattered by adjacent planes of atoms in a crystal. When the path difference is an integer multiple of the wavelength, constructive interference occurs. This can be visualized by considering the geometry of the crystal lattice and the angles at which X-rays are incident and reflected.

Applications

Bragg's law is fundamental to the technique of X-ray diffraction, which is used to determine the crystal structure of materials. This technique has been pivotal in numerous scientific discoveries, including the elucidation of the double helix structure of DNA, the development of new pharmaceuticals, and the design of advanced materials.

Experimental Techniques

1. 1. X-ray Diffraction

X-ray diffraction (XRD) is the most common method for applying Bragg's law. In XRD experiments, a crystal is irradiated with X-rays, and the resulting diffraction pattern is analyzed to determine the crystal structure. The angles and intensities of the diffracted beams provide information about the atomic arrangement within the crystal.

1. 1. Neutron Diffraction

Neutron diffraction is another technique that utilizes Bragg's law. Unlike X-rays, neutrons are scattered by the nuclei of atoms rather than the electron cloud, making this technique particularly useful for studying materials with light atoms, such as hydrogen, or for investigating magnetic structures.

Advanced Topics

1. 1. Anomalous Scattering

Anomalous scattering occurs when the scattering factor of an atom varies with the wavelength of the incident X-ray. This phenomenon can be exploited to obtain phase information in X-ray crystallography, which is otherwise difficult to determine.

1. 1. Multiple Scattering

In some cases, X-rays may undergo multiple scattering events within a crystal. This can complicate the analysis of diffraction patterns but also provides additional information about the crystal structure.

1. 1. Dynamic Theory of X-ray Diffraction

The dynamic theory of X-ray diffraction takes into account the multiple scattering of X-rays within a crystal. This theory is essential for understanding the diffraction patterns of perfect crystals and for interpreting high-resolution X-ray diffraction data.

References