Scattering models

Unfortunately, x-ray and neutron scattering data cannot be simply inverted to yield a realspace structure. Instead, the experimental scattering intensity must be fit to a candidate model, which in turn should be selected based on known physical constraints. Concordance between experimental data and a particular scattering model gives confidence that the selected model is correct. However, this is not absolute: scattering is an ill-posed problem (phase problem), so a variety of different models may all fit the data. Again, the experimenter must use their knowledge and judgment to select plausible models. (It is very helpful to have secondary measurements, such as electron microscopy, to guide model selection.)

Once a particular model has been selected and validated, fitting scattering data is a robust way to extract precise values of a variety of structural parameters (size, size distribution, shape, orientation, aggregation, etc.). There are a wide variety of literature models available.

Scaling Analysis

The scaling of scattering intensity can sometimes be fit with a very general and abstract model, from which one can measure a property of the size or shape of the structuring.

• Porod plot ($\scriptstyle I\sim q^{-4}$)
• Fresnel plot ($\scriptstyle R\sim q^{-4}$)
• Kratky plot ($\scriptstyle I\sim q^{-2}$)
• Guinier plot ($\scriptstyle \ln(I)\sim 1-(R_{g}^{2}/3)q^{2}$)
• Zimm plot

Form factor

Any given object shape has a particular form factor (reciprocal-space). If the objects are randomly ordered (e.g. in solution), then the scattering is dominated by this form factor. Thus, one can fit solution-phase data using form factor equations.

• Refer to form factor for a list of equations for different shapes.

Other Solutions Equations

• 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

Structure Factor

The structure factor is the scattering contribution from the organization of constituents (atomic crystal lattice, aggregation state in solution, etc.). In order to fit data to a given structure factor, one must disentangle the contributions from the form factor. (Or fit the data at the level of the total scattering intensity.) A variety of structure factor equations have been documented.

• Refer to structure factor for a list of equations.

Lattices of nano-objects

• Specific models
• General models