What is SAXS?

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What is SAXS?

Small-angle X-ray scattering (SAXS) is a technique for measuring the structure of materials at length scales between roughly 1 and 100 nanometers — the scale of proteins, lipid vesicles, nanoparticles, and polymer chains. Rather than imaging a structure directly, SAXS measures how a beam of X-rays is scattered when it passes through a sample. The pattern of scattered photos carries a fingerprint of the underlying structure, and our job is to decode it.

The experiment

A SAXS measurement involves three components:

flowchart LR
    A["X-ray\nsource"] -->|"incident beam"| B["Sample"]
    B -->|"scattered X-rays"| C["Detector"]
    B -.->|"transmitted\nbeam"| C

X-rays from the source pass, at the Advanced Light Source this is the synchrotron, through the sample. Most of them continue in a straight line (the transmitted beam, which is blocked by a beamstop). A small fraction are scattered and the scattering angle, with respect to the transmitted beam, depends on the size and arrangement of structures in the sample. The detector records the intensity of these scattered X-rays as a function of angle. This measurement provides information on the entire ensemble of materials present in the path of the beam. Compared to forms of microscopy which typically image tens-to-hundreds of objects, scattering provides insight into the structure of millions of particles.

The scattering vector q

Rather than working directly with the scattering angle \(2\theta\), we use the scattering vector \(q\). It combines the angle and the X-ray wavelength \(\lambda\) into a single quantity with units of inverse length:

\[q = \frac{4\pi}{\lambda}\sin\theta\]

where \(2\theta\) is the angle between the transmitted beam and the scattered beam, and \(\lambda\) is the wavelength of the X-rays.

Key Concept: What q measures

\(q\) has units of inverse length (in this tutorial, Å\(^{-1}\)). It is a spatial frequency — large \(q\) corresponds to small length scales, and small \(q\) corresponds to large length scales. A rough rule of thumb is that a feature of size \(d\) produces a signature at \(q \approx 2\pi / d\).

Typical SAXS measurements cover \(q\) from about \(0.001\) to \(1\) Å\(^{-1}\), which corresponds to length scales from roughly \(6\) to \(6000\) Å.

What we measure: I(q)

Most scattering measurements use a 2-dimensional detector where each pixel is sensitive to the number of X-ray photons that hit it. Using a series of standards, the relative position of each pixel with respect to the sample can be determined which allows us to calculate the scattering angle subtended by each pixel which can then be converted into the \(q\)-vector. This provides a measure of the scattering intensity \(I(q)\) — how much X-ray signal arrives at each pixel value of \(q\). Because samples are usually isotropic (the same in every direction), the two-dimensional detector image is averaged into a one-dimensional \(I(q)\) curve. This curve is what we compute, fit, and interpret.

Form factor and structure factor

The intensity \(I(q)\) has two contributions:

  • The form factor \(P(q)\) encodes the shape and size of individual particles.
  • The structure factor \(S(q)\) encodes how those particles are arranged relative to each other — whether they are randomly distributed, packed closely, or ordered.

In this tutorial we will always assume the sample is dilute: particles are far apart and do not interact. Under that assumption \(S(q) = 1\), and the intensity simplifies to:

\[I(q) \propto N \, (\Delta\rho)^2 \, V^2 \, P(q)\]

where \(N\) is the number of particles, \(\Delta\rho\) is the contrast between the particle and the surrounding medium (introduced on the next page), and \(V\) is the particle volume. Structure factors for concentrated systems will be covered in a later section.

Units in this tutorial

All scattering vectors in this tutorial are in Å\(^{-1}\) (inverse angstroms). You will also encounter nm\(^{-1}\) (inverse nanometers) in the literature — the conversion is \(1 \text{ nm}^{-1} = 0.1 \text{ Å}^{-1}\). Mixing units is one of the most common sources of errors in scattering calculations, so always check which convention a paper is using before comparing numbers.

Most BioSAXS experiments use dilute samples to remove the added complication of analyzing a structure-factor when the morphology of a single particle is what is of interest. These solutions are typically fabricated with less than 10% weight by volume to be considered a dilute suspension.

What's next: Scattering Length Density — the material property that determines how strongly a particle scatters X-rays.

Overview Scattering Length Density