Amorphous materials and pair distribution functions

Glasses, amorphous thin films, liquids, and highly disordered solids have no unit cell, so the crystallographic toolkit of the morning sessions (Bragg disks, lattice vectors, orientation libraries) does not apply. But these materials are far from structureless: they have well-defined bond lengths, coordination shells, and often medium-range order (MRO) extending over 1–3 nm. This module covers how to quantify that structure from nanobeam diffraction Cockayne (2007).

Describing disorder: n-body distribution functions¶

Without a lattice, structure is described statistically. The workhorse is the two-body pair distribution function g(r): the probability, relative to a random ideal gas, of finding an atom at distance r from another atom. Sharp peaks at the nearest-neighbor distance and progressively broader peaks at higher shells encode the local bonding; how quickly the oscillations decay measures the extent of order:

• A nanocrystalline material shows sharp, persistent peaks out to large r.

• A material with medium-range order shows a sharp first shell and damped oscillations that die out over a few nanometers.

• A liquid or fully amorphous structure shows a sharp minimum bond distance, a strong first shell, and essentially no structure beyond the second or third shell.

For multi-component systems the bookkeeping multiplies: two elements A and B produce three partial PDFs (A–A, B–B, A–B).

Measuring the PDF with electron diffraction (ePDF)¶

Diffraction measures the Fourier transform of g(r). From a 4D-STEM dataset of an amorphous sample the workflow is:

1. Polar transform. Convert each diffraction pattern from Cartesian (kx, ky) to polar (φ, k) coordinates. This step is exquisitely sensitive to the pattern origin: an incorrect center produces wavy azimuthal artifacts that corrupt everything downstream. Automatic origin refinement (minimizing the standard deviation of intensity along the azimuthal direction at each probe position) makes this robust at scale.

2. Azimuthal average → I(k). Average over φ to get the radial scattering intensity:

3. Background subtraction. Fit and remove the smooth single-atom scattering background, e.g. with a model of the form

   $$B(k) = c + i_0 \exp\!\left(-\frac{k^2}{2 s_0^2}\right) + i_1 \exp\!\left(-\frac{k^4}{2 s_1^4}\right)$$

   (1)

4. Reduced structure factor. Convert the background-corrected intensity to the reduced structure factor F(k) = k·(S(k) − 1).

5. Sine transform → G(r). A windowed sine transform of F(k) gives the reduced PDF G(r). Truncation of the k-range produces spurious low-r oscillations (atoms cannot be 0.5 Å apart!); damping schemes that iteratively fit S(k) and estimate the atomic number density ρ₀ suppress these artifacts Yoshimoto & Omote (2022).

6. Normalize → g(r). With the density in hand, $g(r) = 1 + \dfrac{G(r)}{4 \pi r \rho_0}$:

Because each step has failure modes, validation matters: in the tutorial we use a simulated 4D-STEM dataset of amorphous tantalum (built on the liquid/glass models of Ding et al. (2017)), so the measured g(r) can be compared against the ground-truth PDF computed directly from the atomic coordinates.

Beyond the mean: mapping disorder in 4D¶

Unlike a selected-area or powder ePDF, a 4D-STEM measurement retains spatial resolution: each probe position carries its own diffuse-scattering signal, so you can map local structure:

• Spatially resolved PDFs distinguish amorphous phases, map crystalline/amorphous phase fractions, and track devitrification.

• Fluctuation electron microscopy (FEM): the variance of the diffracted intensity between probe positions (as a function of k and probe size) is sensitive to medium-range order that is invisible in the mean pattern Treacy et al. (2005), Voyles & Muller (2002):

• Elliptic distortions of the amorphous halo are a signal, not just a nuisance: under mechanical load, the halo of a metallic glass distorts measurably, so fitting ellipses to the rings maps strain in amorphous materials, extending the morning’s strain-mapping ideas to samples with no lattice at all:

Practical notes¶

• Reach high enough k (several Å⁻¹): camera length down, and mind the detector corners.

• Energy filtering removes the inelastic background and markedly improves S(k) at low k.

• Amorphous halos also calibrate elliptical distortion for all the crystalline analyses; see the data handling module.