X-ray diffraction

Why it matters

X-ray diffraction (XRD) is the first instrument most materials labs reach for. In about half an hour, a powder pattern tells you which crystalline phases are present, their lattice parameters, whether a film is textured, and roughly how small the crystallites are. When a new alloy, catalyst or battery cathode comes out of a furnace, "run XRD" is usually step one.

The core idea: Bragg's law

A crystal is a stack of atomic planes with spacing d. X-rays reflected from neighbouring planes add up (constructive interference) only when the extra path length is a whole number of wavelengths:

n * lambda = 2 * d * sin(theta)

• lambda is the X-ray wavelength. Most lab diffractometers use copper K-alpha1, lambda = 1.5406 Å.
• theta is the angle between the beam and the planes. The instrument scans and reports 2-theta, so always halve it first.
• n is the order. In practice we use n = 1 and treat higher orders as reflections from planes with a smaller d (for example (222) instead of second-order (111)).

So every peak position becomes a d-spacing: d = lambda / (2 sin(theta)).

From d-spacings to a crystal structure

For a cubic crystal with lattice parameter a, the spacing of the (hkl) planes is

d = a / sqrt(h^2 + k^2 + l^2)

Combine this with Bragg's law and square it: sin^2(theta) = (lambda^2 / 4a^2) * N, where N = h^2 + k^2 + l^2. The ratio of sin^2(theta) between peaks is therefore a ratio of integers. Which integers appear depends on the lattice, because of systematic absences (structure-factor rules):

Worked example: indexing copper

A copper powder scanned with Cu K-alpha gives peaks at 2-theta = 43.30, 50.43, 74.13, 89.93 and 95.14 degrees.

1. First peak: theta = 21.65°, sin^2(theta) = 0.1361.
2. Divide every sin^2(theta) by the first: 1, 1.333, 2.669, 3.669, 4.003.
3. Those ratios are not integers, so multiply by 3: 3, 4, 8, 11, 12. That is the FCC sequence, so the peaks are (111), (200), (220), (311) and (222).
4. Lattice parameter from (111): d = 1.5406 / (2 * 0.3689) = 2.0879 Å, so a = 2.0879 * sqrt(3) = 3.616 Å. Averaging all five peaks gives about 3.615 Å, which matches the reference value for copper.

If multiplying by 2 had given 2, 4, 6, 8 the sample would be BCC; if the raw ratios were already 1, 2, 3, 4 it would be simple cubic. Non-cubic phases need more parameters and are indexed with software.

Phase identification in practice

Most of the time you do not index by hand. You compare the pattern against a reference database: the ICDD Powder Diffraction File (commercial), or free sources such as the Crystallography Open Database (https://www.crystallography.net). Software such as GSAS-II (free) or vendor packages match peak positions and relative intensities, and Rietveld refinement fits the whole pattern to refine lattice parameters and phase fractions.

Two more numbers come out of peak shapes:

• Crystallite size from peak broadening, using the Scherrer equation tau = K * lambda / (beta * cos(theta)), with K about 0.9 and beta the width in radians after subtracting the instrument's own broadening. It is an estimate, reliable roughly below 100 nm.
• Strain and texture: peak shifts point to residual strain or a composition change; unexpected intensity ratios usually mean preferred orientation (texture), common in films and pressed powders.

Common mistakes

• Using 2-theta in Bragg's law instead of theta. Your d-spacings will be about half as large and nothing will index.
• Forgetting degrees vs radians in code (math.sin expects radians).
• Ignoring the K-alpha2 shoulder at high angles, or the sample displacement error, which shifts every peak and biases a.
• Trusting one peak. Use several peaks, and the high-angle ones give the most precise lattice parameters.
• Over-reading intensities from a textured sample as if it were a random powder.

Safety and training

Diffractometers contain an X-ray source. Modern enclosed units are interlocked, but you still need your institution's radiation safety training and instrument sign-off before you run one, and you never defeat an interlock. In this course you analyse data only.

Recap

• Bragg's law turns each 2-theta peak into a d-spacing: halve 2-theta first.
• For cubic crystals, sin^2(theta) ratios are integer ratios that reveal SC, BCC or FCC.
• Real phase ID matches against databases; peak width and shape give size, strain and texture.

Try it

Open the Do tab to write a Bragg's-law helper that indexes a measured copper pattern and computes its lattice parameter.