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-thetaThe angle the diffractometer scans and reports, which is twice the Bragg angle theta and must be halved before use., 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-spacingThe distance between neighbouring parallel planes of atoms in a crystal, calculated from a peak position with Bragg's law.: 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 lawThe condition n * lambda = 2 * d * sin(theta) under which X-rays reflected from parallel atomic planes interfere constructively. 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 absencesReflections that are missing because of the lattice type, which is why SC, BCC and FCC show different sequences of h^2 + k^2 + l^2. (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 refinementA method that fits a calculated pattern to the whole measured XRD pattern to refine lattice parameters and phase fractions. 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 equationAn estimate of crystallite size from peak width, tau = K * lambda / (beta * cos(theta)), reliable roughly below 100 nm. 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.