Form Factor

From GISAXS
Jump to: navigation, search
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:

  • 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.
  • |F(\mathbf{q})|^2, the form factor intensity; whereas the amplitude cannot be measured experimentally, the form factor intensity in principle can be.
  • 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, \rho(\mathbf{r}), the form factor is computed by integrating over all space:


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 \Delta \rho, and the form factor is simply:


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:


\begin{alignat}{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{alignat}

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 q=0, we expect:


\begin{alignat}{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{alignat}

And so:


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

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 (2^3)^2 = 64-fold increase in scattering intensity.

Form Factor Equations

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:

  • 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

Form factors for classes of geometric shapes

Specific form factors

See Also