Section 2: Building Mathematical Models¶
With your environment set up and version control in place, it is time to start on the science. This section introduces X-ray scattering from first principles and builds a complete computational model of scattering from a sphere — the simplest non-trivial geometry and the foundation for everything in Section 3.
By the end of this section you will have:
- An understanding of what SAXS measures and how the scattering vector \(q\) is defined
- A working Python function that computes the form factor \(P(q, R)\) for a solid sphere
- A reusable log-log plotting routine for scattering data
- Physical intuition for how sphere size is encoded in the scattering curve
- A polydisperse model that accounts for a distribution of sphere sizes
All of this code lives in a single Jupyter notebook — chapter_02_sphere.ipynb —
that you will build incrementally as you work through the pages.
A note on units
All scattering vectors in this section are in Å\(^{-1}\) (inverse angstroms). If you are reading papers alongside this tutorial, check their units — nm\(^{-1}\) is also common. The conversion is \(1 \text{ nm}^{-1} = 0.1 \text{ Å}^{-1}\).