The Sphere Form Factor¶
The form factor \(P(q, R)\) is the function that describes the shape of the scattering curve for a given particle geometry. For a solid, homogeneous sphere of radius \(R\), the form factor is:
This function is normalized so that \(P(q \to 0) = 1\). The full scattering intensity is then:
where \(V = \frac{4}{3}\pi R^3\) is the sphere volume and \(\Delta\rho\) is the contrast introduced on the previous page.
Physical intuition¶
The expression inside the brackets is the scattering amplitude \(F(q, R)\) — a measure of how much a sphere of radius \(R\) contributes to the scattered wave at each \(q\):
Key Concept: Amplitude and Intensity
Scattering is a wave phenomenon. The amplitude \(F(q)\) describes the wave scattered by a single particle — it can be positive or negative and carries phase information. The intensity \(P(q) = F(q)^2\) is what a detector actually measures. Intensity is always non-negative.
This distinction becomes critical in Section 3, where we combine amplitudes from multiple shells. Shells are added as amplitudes (with their correct signs) before squaring, not as intensities after squaring. Getting this wrong leads to physically incorrect results.
The oscillations in \(P(q, R)\) arise from interference between X-rays scattered from different parts of the sphere. At \(q = 0\) all of these waves add up in phase, giving the maximum intensity. As \(q\) increases, waves from opposite sides of the sphere begin to cancel, and \(P(q)\) falls. The first complete cancellation — the first minimum in \(P(q)\) — occurs when:
This is a useful rule of thumb: once you can identify the first minimum in an experimental scattering curve, you can immediately estimate the particle radius.
Behavior at different length scales¶
The form factor has characteristic behavior in three regions:
| Region | Condition | Behavior |
|---|---|---|
| Guinier region | \(qR \ll 1\) | \(P(q) \approx 1\) — the plateau at low \(q\) |
| Intermediate | \(qR \sim 1\)–\(10\) | Oscillations that encode the particle size |
| Porod region | \(qR \gg 1\) | \(P(q) \propto q^{-4}\) — a power-law decay |
The Guinier approximation
At very small \(q\) (specifically when \(qR_g < 1\), where \(R_g\) is the radius of gyration), the form factor can be approximated as:
For a solid sphere, \(R_g = R\sqrt{3/5}\). Plotting \(\ln I\) versus \(q^2\) (a Guinier plot) should give a straight line whose slope yields \(R_g\) directly. This is one of the most widely used analysis tools in experimental SAXS — it requires no model assumptions beyond the dilute limit.
What the form factor does not capture¶
\(P(q, R)\) describes the scattering from a single, isolated particle. It does not account for:
- Interactions between particles — described by the structure factor \(S(q)\), which will be addressed in a later section.
- Polydispersity — real samples contain a distribution of particle sizes, which is addressed on a later page in this section.
- Internal structure — the sphere is assumed to be homogeneous. Adding layers of different SLD is the subject of Section 3.
What's next: q-Vectors in NumPy — implementing the sphere form factor in Python and generating the \(q\) arrays needed to evaluate it.