X-ray Diffraction Essentials: Bragg Condition, Miller Indices, Moseley’s Law and Phase Diagrams (MIT OCW)

Overview

MIT OpenCourseWare explains how X-ray diffraction probes crystal structures by diffracting X-rays off atomic planes and reading diffraction peaks to infer lattice constants and crystal symmetry. The lecture ties Bragg diffraction to Miller indices and shows how a simple Cu target yields a readable spectrum, from which the aluminum lattice constant can be extracted.

Key insights

• Bragg condition: constructive interference occurs when the path difference equals an integer multiple of the wavelength, enabling diffraction peaks that map to crystal planes.
• From peaks to structure: peaks labeled by hkl (Miller indices) are used with HKL distances to compute the lattice constant and identify crystal symmetry via selection rules.
• Moseley’s Law grounds periodicity in atomic number: x-ray frequencies scale with (Z−1)^2, linking spectral lines to the nucleus and explaining the ordering of elements.
• Beyond Bragg: early Laue diffraction used a continuous spectrum to reveal crystal symmetry, while characteristic lines provide more precise phase identification.

Introduction to X-ray Diffraction and Bragg Condition

The video covers how X-ray diffraction arises when X-rays scatter off crystal planes separated by a distance D, and how the Bragg condition, nλ = 2D sin θ, determines when constructive interference yields observable peaks. Planes in a crystal act like mirrors, and the choice of plane is encoded in the HKL indices, which label the set of planes with a lattice spacing d_hkl.

In cubic lattices the distance between planes dhkl is tied to the lattice constant a by dhkl = a/√(h^2 + k^2 + l^2). This is the bridge between experimental angles and real-space structure.

Quote: "you shine X rays on the crystal and they can diffract and have constructive and destructive interference, but only if they line up" - Henry Moseley, physicist.

From Bragg to Miller Planes

The lecture shows how Bragg diffraction plus Miller indices gives a practical recipe: compute λ/(2a^2) sin^2 θ to relate the observed angles to the lattice geometry, and use the measured two-theta values to back out a lattice constant that should be constant across all peaks.

Quote: "If I can get H squared plus K squared plus L squared to equal these things, then I’ve kept a constant" - MIT Instructor.

Experiment and Spectra: Aluminum Pattern

Using a copper target with Cu Kα radiation (λ ≈ 1.54 Å), the spectrum for aluminum shows a series of peaks which can be indexed as 111, 200, 220, 311. The first peak yields D111 ≈ 2.37 Å, and using D111 = a/√3 gives a ≈ 4.11 Å, consistent across peaks and indicating an FCC structure for aluminum.

The instructor emphasizes the importance of reporting the X-ray wavelength when giving a diffraction pattern, as the lattice constant extraction relies on it.

Selection Rules and Crystal Symmetry

To identify the crystal structure, selection rules are used: for simple cubic, all reflections are allowed; for body-centered cubic, reflections must have h+k+l even; for face-centered cubic, odd and even reflections do not mix. The observed peaks match the FCC selection rules, confirming the crystal structure in this example.

Quote: "odd, even, even, odd. This looks like an FCC crystal to me" - MIT Instructor.

Moseley’s Law and the Periodic Table

The video recounts Moseley’s 1912–1913 work showing that the frequency of Kα x-rays scales with Z−1, yielding a straight line when plotting sqrt(nu) against Z. This provided a physical basis for atomic number as the driver of periodicity, grounding the modern understanding of the periodic table and identifying protons as the source of atomic structure. Moseley’s work is noted as foundational, and his early death at 27 during World War I is highlighted as a tragic loss to science.

Quote: "Moseley’s law was essentially him thinking about the Bohr model for these X rays" - Henry Moseley, physicist.

The Brass Anomaly and X-ray Lines

The lecturer explains that brass produces zinc lines in XRD patterns in addition to copper lines, a reminder that alloy composition can introduce extra spectral features. The discussion connects practical material choices to spectral fingerprints and how to interpret them in diffraction data.

Quote: "you'll get the zinc lines" - MIT Instructor.

Laue Diffraction vs Characteristic X-rays

The talk briefly contrasts Laue diffraction, which uses a broad spectrum with a fixed crystal orientation to produce diffraction spots, with the modern use of characteristic lines (Kα, Lβ) for precise phase and structure determination. Laue diffraction historically helped determine zinc sulfide structure and earned the Nobel Prize in that era.

Quote: "In Laue XRD you shoot the X rays which have all the wavelengths in them... you get spots" - MIT Instructor.

Phase Diagrams and Crystal Symmetry

Phase diagrams map a material's crystal structure as a function of temperature and pressure. The lecture uses water as an illustration, noting that ice has many crystalline phases and that under Earth-like conditions Ice I is important for phenomena like ice floating on water, enabling aquatic life in winter. The crystallography perspective connects to phase stability and material properties across conditions.

Quote: "There are 17 current phases of ice" - MIT Instructor.

Closing: The Power and Responsibility of XRD

The final messages stress XRD as a starting point for materials characterization, the importance of reporting wavelength, and the role of crystal structure in determining material properties. The talk ends with a nod to quasicrystals and groundbreaking symmetry concepts that challenged traditional crystallography.

Quote: "Dan Schectman was the one who discovered quasicrystals" - Dan Schectman.