11.20:
X-ray Crystallography
In 1913, the father and son scientists William Henry Bragg and William Lawrence Bragg noticed that when X-rays strike a crystalline solid at a certain angle, the X-rays diffract and produce a pattern of regularly spaced spots.
This led to the development of X-ray crystallography, which uses this phenomenon to determine the structures of crystalline solids ranging from simple ionic compounds to complex macromolecules like nucleic acids and proteins.
Recall that diffracted electromagnetic waves undergo constructive and destructive interference. This produces interference patterns, or diffraction patterns, showing the varying intensity of the diffracted waves at different points in space.
X-rays are diffracted by the electrons of atoms if the atoms are regularly spaced and the X-ray wavelength is similar to the interatomic distance.
When X-rays diffract from atoms in different planes, the diffracted waves may or may not be in phase. This depends on the interplanar spacing, d, and the angle at which the X-rays struck the atoms, or the incidence angle, theta.
This is because the paths that the X-rays take from the source to the detector have different lengths. If the path difference is an integer multiple of the wavelength of the X-rays, then the X-rays constructively interfere.
This results in the pattern of regularly spaced spots of diffracted waves observed by the Braggs, where each spot represents a diffraction angle resulting in constructive interference.
The relationship between the angle of diffraction, the interplanar spacing, and the X-ray wavelength is expressed with Bragg’s equation. This relationship provides information about the underlying highly ordered arrangement of the atoms in the crystal.
Ultimately, the lattice parameters can be derived from this information via a series of calculations. Modern instruments collect diffraction patterns from many different orientations and use the patterns and spot intensities to identify the crystal structure that is most likely to produce the observed combination of results.
11.20:
X-ray Crystallography
The size of the unit cell and the arrangement of atoms in a crystal may be determined from measurements of the diffraction of X-rays by the crystal, termed X-ray crystallography.
Diffraction
Diffraction is the change in the direction of travel experienced by an electromagnetic wave when it encounters a physical barrier whose dimensions are comparable to those of the wavelength of the light. X-rays are electromagnetic radiation with wavelengths about as long as the distance between neighboring atoms in crystals (on the order of a few angstroms). When a beam of monochromatic X-rays strikes a crystal, its rays are scattered in all directions by the atoms within the crystal. When scattered waves traveling in the same direction encounter one another, they undergo interference, a process by which the waves combine to yield either an increase or a decrease in amplitude (intensity) depending upon the extent to which the combining waves’ maxima are separated.
Bragg’s Law and Bragg’s Equation
When X-rays of a certain wavelength, λ, are scattered by atoms in adjacent crystal planes separated by a distance, d, they may undergo constructive interference when the difference between the distances traveled by the two waves prior to their combination is an integer factor, n, of the wavelength. This is Bragg's law. This condition is satisfied when the angle of the diffracted beam, θ, is related to the wavelength and interatomic distance by the equation: nλ = 2d sin θ. This relation is known as the Bragg equation in honor of W. H. Bragg and W. L. Bragg, the English physicists who explained this phenomenon. They were awarded the Nobel Prize in Physics in 1915 for their contributions.
This text has been adapted from Openstax, Chemistry 2e, Section 10.6: Lattice Structures in Crystalline Solids.