# Form Factor

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Example form factor scattering from a sphere.

The Form Factor (FF) is the scattering which results from the shape of a particle. When particles are distributed without any particle-particle correlations (e.g. dilute solution of non-interacting particles, freely floating), then the scattering one observes is entirely the form factor. By comparison, when particles are in a well-defined structure, the scattering is dominated by the structure factor; though even in these cases the form factor continues to contribute, since it modulates both the structure factor and the diffuse scattering.

When reading discussions of scattering modeling, one must be careful about the usage of the term 'form factor'. This same term is often used to describe three different (though related) quantities:

• ${\displaystyle F(\mathbf {q} )}$, the form factor amplitude arising from a single particle; this can be thought of as the 3D reciprocal-space of the particle, and is thus in general anisotropic.
• ${\displaystyle |F(\mathbf {q} )|^{2}}$, the form factor intensity; whereas the amplitude cannot be measured experimentally, the form factor intensity in principle can be.
• ${\displaystyle P(q)=\left\langle |F(\mathbf {q} )|^{2}\right\rangle }$, the isotropic form factor intensity; i.e. the orientational averaged of the form factor square. This is the 1D scattering that is measured for, e.g., particles freely distributed distributed in solution (since they tumble randomly and thus average over all possible orientations).

## Equations

In the most general case of an arbitrary distribution of scattering density, ${\displaystyle \rho (\mathbf {r} )}$, the form factor is computed by integrating over all space:

${\displaystyle F_{j}(\mathbf {q} )=\int \rho _{j}(\mathbf {r} )e^{i\mathbf {q} \cdot \mathbf {r} }\mathrm {d} V}$

The subscript denotes that the form factor is for particle j; in multi-component systems, each particle has its own form factor. For a particle of uniform density and volume V, we denote the scattering contrast with respect to the ambient as ${\displaystyle \Delta \rho }$, and the form factor is simply:

${\displaystyle F_{j}(\mathbf {q} )=\Delta \rho \int \limits _{V}e^{i\mathbf {q} \cdot \mathbf {r} }\mathrm {d} V}$

For monodisperse particles, the average (isotropic) form factor intensity is an average over all possible particle orientations:

{\displaystyle {\begin{alignedat}{2}P_{j}(q)&=\left\langle |F_{j}(\mathbf {q} )|^{2}\right\rangle \\&=\int \limits _{\phi =0}^{2\pi }\int \limits _{\theta =0}^{\pi }|F_{j}(-q\sin \theta \cos \phi ,q\sin \theta \sin \phi ,q\cos \theta )|^{2}\sin \theta \mathrm {d} \theta \mathrm {d} \phi \end{alignedat}}}

Note that in cases where particles are not monodisperse, then the above average would also include averages over the distritubions in particle size and/or shape. Note that for ${\displaystyle q=0}$, we expect:

{\displaystyle {\begin{alignedat}{2}F(0)&=\int \limits _{\mathrm {all\,\,space} }\rho (\mathbf {r} )e^{0}\mathrm {d} \mathbf {r} =\rho _{\mathrm {total} }\\&=\Delta \rho \int \limits _{V}e^{0}\mathrm {d} \mathbf {r} =\Delta \rho V\end{alignedat}}}

And so:

{\displaystyle {\begin{alignedat}{2}P(0)&=\left\langle \left|F(0)\right|^{2}\right\rangle \\&=4\pi \Delta \rho ^{2}V^{2}\end{alignedat}}}

As expected, scattering intensity scales with the square of the scattering contrast and the particle volume. For multi-component systems, this has the effect of greatly emphasizing larger particles. For instance, a 2-fold increase in particle diameter results in a ${\displaystyle (2^{3})^{2}=64}$-fold increase in scattering intensity.

## Form Factor Equations in the Literature

### Reviews/summaries of form factors

The following is a partial list of form factors that have been published in the literature:

• BornAgain form factor catalog (updated with each software release): Contains all form factors from IsGisaxs, plus the following:
• hard particles:
• Dodecahedron
• Icosahedron
• ripples:
• sinusoidal (as in FitGisaxs)
• saw-tooth (as in FitGisaxs)
• soft particles (documentation forthcoming):
• Gaussian ellipsoid
• Ornstein-Zernike
• Pedersen Review: Analysis of small-angle scattering data from colloids and polymer solutions: modeling and least-squares fitting Jan Skov Pedersen, Advances in Colloid and Interface Science 1997, 70, 171. doi: 10.1016/S0001-8686(97)00312-6
1. Homogeneous sphere
2. Spherical shell
3. Spherical concentric shells
4. Particles consisting of spherical subunits
5. Ellipsoid of revolution
6. Tri-axial ellipsoid
7. Cube and rectangular parallelepipedons
8. Truncated octahedra
9. Faceted sphere
10. Cube with terraces
11. Cylinder
12. Cylinder with elliptical cross section
13. Cylinder with spherical end-caps
14. Infinitely thin rod
15. Infinitely thin circular disk
16. Fractal aggregates
17. Flexible polymers with Gaussian statistics
18. Flexible self-avoiding polymers
19. Semi-flexible polymers without self-avoidance
20. Semi-flexible polymers with self-avoidance
21. Star polymer with Gaussian statistics
22. Star-burst polymer with Gaussian statistics
23. Regular comb polymer with Gaussian statistics
24. Arbitrarily branched polymers with Gaussian statistics
25. Sphere with Gaussian chains attached
26. Ellipsoid with Gaussian chains attached
27. Cylinder with Gaussian chains attached