Form Factor

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:

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

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 $q=0$, we expect:

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

And so:

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

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

Form Factor Equations

• Atomic Form Factor
• Sphere
• Ellipsoid of revolution
• Cube
• Octahedron
• Pyramid
• Superball (can be used to describe rounded cubes/octahedra/etc.)

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:

• NCNR SANS solution form factors:
  • Sphere
  • PolyHardSphere
  • PolyRectSphere
  • CoreShell
  • PolyCoreShell
  • PolyCoreShellRatio
  • Cylinder
  • HollowCylinder
  • CoreShellCylinder
  • Ellipsoid
  • OblateCoreShell
  • ProlateCoreShell
  • TwoHomopolymerRPA
  • DAB
  • TeubnerStrey
  • Lorentz
  • PeakLorentz
  • PeakGauss
  • PowerLaw
  • BE_RPA

• IsGISAXS, Born form factors (see also Gilles Renaud, Rémi Lazzari,Frédéric Leroy "Probing surface and interface morphology with Grazing Incidence Small Angle X-Ray Scattering" Surface Science Reports, 64 (8), 31 August 2009, 255-380 doi:10.1016/j.surfrep.2009.07.002):
  • Parallelepiped
  • Pyramid
  • Cylinder
  • Cone
  • Prism 3
  • Tetrahedron
  • Prism 6
  • cone 6
  • Sphere
  • Cubooctahedron
  • Facetted sphere
  • Full sphere
  • Full spheroid
  • Box
  • Anisotropic pyramid
  • Hemi-ellipsoid

• 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

• Supplementary Information of: Matthew R. Jones, Robert J. Macfarlane, Byeongdu Lee, Jian Zhang, Kaylie L. Young, Andrew J. Senesi, and Chad A. Mirkin "DNA-nanoparticle superlattices formed from anisotropic building blocks" Nature Materials 9, 913-917, 2010. doi: 10.1038/nmat2870
  1. Pyramid
  2. Cube
  3. Cylinder
  4. Octahedron
  5. Rhombic dodecahedron (RD)
  6. Triangular prism

Form factors for classes of geometric shapes

• Platonic solids: Scattering functions of Platonic solids Xin Li, Roger Pynn, Wei-Ren Chen, et al. Journal of Applied Crystallography 2011, 44, p.1 doi:10.1107/S0021889811011691
  1. Tetrahedron
  2. Hexahedron (cube, parallelepiped, etc.)
  3. Octahedron
  4. Dodecahedron
  5. Icosahedron

• Polyhedra:
  • Scattering functions of polyhedra A. Senesi and B. Lee J. Appl. Cryst. 2015, 48, 565-577. doi: 10.1107/S1600576715002964
  • Form factor of any polyhedron: a general compact formula and its singularities B. Croset Journal of Applied Crystallography 2017. doi: 10.1107/S1600576717010147
  • Form factor (Fourier shape transform) of polygon and polyhedron J. Wuttke ArXiv:1703.00255

Specific form factors

• This tutorial lists sphere, rod, disk, and Gaussian polymer coil.
• Block-Copolymer Micelles: Scattering Form Factor of Block Copolymer Micelles Jan Skov Pedersen* and Michael C. Gerstenberg, Macromolecules, 1996, 29 (4), pp 1363–1365 DOI: 10.1021/ma9512115
• Capped cylinder: Scattering from cylinders with globular end-caps. H. Kaya. J. Appl. Cryst. (2004). 37, 223-230 doi: 10.1107/S0021889804000020
• Lens-shaped disc" Scattering from capped cylinders. Addendum. H. Kaya and N.-R. de Souza. J. Appl. Cryst. (2004). 37, 508-509 doi: 10.1107/S0021889804005709
• Star polymer:
  • Benoit, H. On the effect of branching and polydispersity on the angular distribution of the light scattered by gaussian coils J. Polym. Sci. 1953, 11, 507–510. doi: 10.1002/pol.1953.120110512
  • Richter, D.; Farago, B.; Huang, J. S.; Fetters, L. J.; Ewen, B. A study of single-arm relaxation in a polystyrene star polymer by neutron spin echo spectroscopy Macromoleules 1989, 22, 468–472. doi: 10.1021/ma00191a085
  • L. Willner, O. Jucknischke, D. Richter, J. Roovers, L.-L. Zhou, P. M. Toporowski, L. J. Fetters, J. S. Huang, M. Y. Lin, N. Hadjichristidis Structural Investigation of Star Polymers in Solution by Small-Angle Neutron Scattering Macromolecules 1994, 27 (14), 3821–3829. doi: 10.1021/ma00092a022
  • Takuro Matsunaga, Takamasa Sakai, Yuki Akagi, Ung-il Chung and Mitsuhiro Shibayama SANS and SLS Studies on Tetra-Arm PEG Gels in As-Prepared and Swollen States Macromolecules 2009, 42 (16), 6245–6252. doi: 10.1021/ma901013q
  • X. Li, C. Do, Y. Liu, L. Sánchez-Diáz, G. Smith and W.-R. Chen A scattering function of star polymers including excluded volume effects J. Appl. Cryst. 2014, 47 doi: 10.1107/S1600576714022249
  • Durgesh K. Rai, Gregory Beaucage, Kedar Ratkanthwar, Peter Beaucage, Ramnath Ramachandran, and Nikos Hadjichristidis Determination of the interaction parameter and topological scaling features of symmetric star polymers in dilute solution Phys. Rev. E 2015, 012602. doi: 10.1103/PhysRevE.92.012602
• Cylinders vs. ribbons in solution: Y. Su, C. Burger, B. S. Hsiao and B. Chu Characterization of TEMPO-oxidized cellulose nanofibers in aqueous suspension by small-angle X-ray scattering J. Appl. Cryst. 2014, 47, 788. doi: 10.1107/S1600576714005020
• Hollow cylinders: Kun Lu , Jaby Jacob , Pappannan Thiyagarajan , Vincent P. Conticello , and David G. Lynn Exploiting Amyloid Fibril Lamination for Nanotube Self-Assembly J. Am. Chem. Soc. 2003, 125 (21), 6391–6393. doi: 10.1021/ja0341642
• Amino acids: D. Tong, S. Yang and L. Lu Accurate optimization of amino acid form factors for computing small-angle X-ray scattering intensity of atomistic protein structures J. Appl. Cryst. 2016, 49. doi: 10.1107/S1600576716007962
• Helical nanostructures: Max Burian and Heinz Amenitsch Dummy-atom modelling of stacked and helical nanostructures from solution scattering data IUCrJ 2018 DOI: 10.1107/S2052252518005493

See Also

• Computing the scattering for lattices of nano-objects also involves computing the form factor.
• M. Burian, G. Fritz-Popovski, M. He, M. V. Kovalenko, O. Paris and R. T. Lechner Considerations on the model-free shape retrieval of inorganic nanocrystals from small-angle scattering data J. Appl. Cryst. 2015, 48, 857-868. 10.1107/S1600576715006846
• R. A. X. Persson and J. Bergenholtz Efficient computation of the scattering intensity from systems of nonspherical particles J. Appl. Cryst. 2016, 49 doi:10.1107/S1600576716011481
• Bernard Croseta Form factor of rounded objects: the sections method J. Appl. Cryst. 2018 doi: 10.1107/S1600576718007239