Fitting Models with Structure Factors

Note

This help document is under development

Interaction models (previously called product models), or \(P@S\) models for short, multiply the form factor \(P(Q)\) by the structure factor \(S(Q)\), modulated by the effective radius of the form factor. For the theory behind this, see Theory later.

Parameters

Except for volfraction, when writing your own \(P\) and \(S\) models, DO NOT give your model parameters these names!

Many parameters are common amongst \(P@S\) models, but take on specific meanings:

• scale:

  Overall model scale factor.

  To compute number density \(n\) the volume fraction \(V_f\) (parameterised as volfraction) is needed. In most \(P(Q)\) models \(V_f\) is not defined and scale is used instead. Some \(P(Q)\) models, such as the vesicle, do define volfraction and so can leave scale at 1.0.

  Structure factor models \(S(Q)\) contain volfraction. In \(P@S\) models this is also used as the volume fraction for the form factor model \(P(Q)\), so these models can also leave scale at 1.0. If \(P(Q)\) already has a volfraction parameter, it is tied to the volfraction for \(S(Q)\).

  If the volume fraction required for \(S(Q)\) is not the volume fraction needed to compute the \(n\) for \(P(Q)\), then leave volfraction as the \(V_f\) for \(S(Q)\) and use scale to define the \(V_f\) for \(P(Q)\) as \(V_f\) = scale \(\cdot\) volfraction. This situation may occur in a mixed phase system where the effective volume fraction needed to compute the structure is much higher than the true volume fraction.

• volfraction:

  The volume fraction of material, \(V_f\).

  For hollow shapes, volfraction still represents the volume fraction of material but the \(S(Q)\) calculation needs the volume fraction enclosed by the shape. To remedy this the user-specified volfraction is scaled by the ratio form:shell computed from the average form volume and average shell volume returned from the \(P(Q)\) calculation when calculating \(S(Q)\). The original volfraction is divided by the shell volume to compute the number density \(n\) used in the \(P@S\) model to get the absolute scaling on the final \(I(Q)\).

• radius_effective:

  The radial distance determining the range of the \(S(Q)\) interaction.

  This may be estimated from the “size” parameters \(\mathbf \xi\) describing the form of the shape. For example, in a system containing freely-rotating cylinders, the volume of space each cylinder requires to tumble will be much larger than the volume of the cylinder itself. Thus the effective radius of a cylinder will be larger than either its actual radius or half- length.

  If radius_effective_mode = 0 (see below) it may be sensible to tie or constrain radius_effective to one or other of the “size” parameters describing the form of the shape (although the parameter cannot then be polydisperse). But radius_effective may also be specified directly, independent of the estimate from \(P(Q)\).

  If radius_effective is calculated by \(P(Q)\), it will be the weighted average of the effective radii computed for the polydisperse shape parameters, and that average is used to compute \(S(Q)\). When specified directly, the value of radius_effective may be polydisperse, and \(S(Q)\) will be averaged over a range of effective radii. Whether this makes any physical sense will depend on the system.

Note

The following additional parameters are only available in SasView 4.3 and later.

radius_effective_mode:

Defines how the effective radius (parameter radius_effective) should be computed from the parameters of the shape.

When radius_effective_mode = 0 then the unconstrained radius_effective parameter in the \(S(Q)\) model is used. This is the default in SasView versions 4.2 and earlier. Otherwise, in SasView 4.3 and later, radius_effective_mode = k represents an index in a list of alternative radius_effective calculations.

In SasView 4.3 and later k must be entered as an integer (and it will be necessary to read the source code file to discover what calculations the modes represent), but in SasView 5.0 and later the options appear in a drop-down box.

For example, the ellipsoid model defines the following radius_effective_modes:

1 => average curvature
2 => equivalent volume sphere
3 => min radius
4 => max radius

Note: radius_effective_mode will only appear in the parameter table if the model defines the list of modes, otherwise it will be set permanently to 0 for the user-defined effective radius.

WARNING! If \(P(Q)\) is multiplied by \(S(Q)\) in the FitPage, instead of being generated in the Sum|Multi dialog, the radius_effective used is constrained (equivalent to radius_effective_mode = 1).

• structure_factor_mode:

  The type of structure factor calculation to use.

  If the \(P@S\) model supports the \(\beta(Q)\) decoupling correction [1] then structure_factor_mode will appear in the parameter table after the \(S(Q)\) parameters.

  If structure_factor_mode = 0 then the local monodisperse approximation will be used, i.e.:

  \[I(Q) = \text{scale} \frac{V_f}{V} P(Q) S(Q) + \text{background}\]

  where \(P(Q) = \langle F(Q)^2 \rangle\). This is the default in SasView versions 4.x and earlier.

  If structure_factor_mode = 1 then the \(\beta(Q)\) correction will be used, i.e.:

  \[I(Q) = \text{scale} \frac{V_f}{V} P(Q) [ 1 + \beta(Q) (S(Q) - 1) ] + \text{background}\]

  The \(\beta(Q)\) decoupling approximation has the effect of damping the oscillations in the normal (local monodisperse) \(S(Q)\). When \(\beta(Q) = 1\) the local monodisperse approximation is recovered. This mode is only available in SasView 4.3 and later.

  More mode options may appear in future as more complicated operations are added.

Theory

Scattering at vector \(\mathbf Q\) for an individual particle with shape parameters \(\mathbf\xi\) and contrast \(\rho_c(\mathbf r, \mathbf\xi)\) is computed from the square of the amplitude, \(F(\mathbf Q, \mathbf\xi)\), as

\[I(\mathbf Q) = F(\mathbf Q, \mathbf\xi) F^*(\mathbf Q, \mathbf\xi) \big/ V(\mathbf\xi)\]

with the particle volume \(V(\mathbf \xi)\) and

\[F(\mathbf Q, \mathbf\xi) = \int_{\mathbb R^3} \rho_c(\mathbf r, \mathbf\xi) e^{i \mathbf Q \cdot \mathbf r} \,\mathrm d \mathbf r = F\]

The 1-D scattering pattern for monodisperse particles uses the orientation average in spherical coordinates,

\[I(Q) = n \langle F F^*\rangle = \frac{n}{4\pi} \int_{\theta=0}^{\pi} \int_{\phi=0}^{2\pi} F F^* \sin(\theta) \,\mathrm d\phi \mathrm d\theta\]

where \(F(\mathbf Q,\mathbf\xi)\) uses \(\mathbf Q = [Q \sin\theta\cos\phi, Q \sin\theta\sin\phi, Q \cos\theta]^T\). A \(u\)-substitution may be used, with \(\alpha = \cos \theta\), \(\surd(1 - \alpha^2) = \sin \theta\), and \(\mathrm d\alpha = -\sin\theta\,\mathrm d\theta\). Here,

\[n = V_f/V(\mathbf\xi)\]

is the number density of scatterers estimated from the volume fraction \(V_f\) of particles in solution. In this formalism, each incoming wave interacts with exactly one particle before being scattered into the detector. All interference effects are within the particle itself. The detector accumulates counts in proportion to the relative probability at each pixel. The extension to heterogeneous systems is simply a matter of adding the scattering patterns in proportion to the number density of each particle. That is, given shape parameters \(\mathbf\xi\) with probability \(P_\mathbf{\xi}\),

\[I(Q) = \int_\Xi n(\mathbf\xi) \langle F F^* \rangle \,\mathrm d\xi = V_f\frac{\int_\Xi P_\mathbf{\xi} \langle F F^* \rangle \,\mathrm d\mathbf\xi}{\int_\Xi P_\mathbf\xi V(\mathbf\xi)\,\mathrm d\mathbf\xi}\]

This approximation is valid in the dilute limit, where particles are sufficiently far apart that the interaction between them can be ignored.

As concentration increases, a structure factor term \(S(Q)\) can be included, giving the monodisperse approximation for the interaction between particles, with

\[I(Q) = n \langle F F^* \rangle S(Q)\]

For particles without spherical symmetry, the decoupling approximation is more accurate, with

\[I(Q) = n [\langle F F^* \rangle + \langle F \rangle \langle F \rangle^* (S(Q) - 1)]\]

Or equivalently,

\[I(Q) = P(Q)[1 + \beta\,(S(Q) - 1)]\]

with the form factor \(P(Q) = n \langle F F^* \rangle\) and \(\beta = \langle F \rangle \langle F \rangle^* \big/ \langle F F^* \rangle\). These approximations can be extended to heterogeneous systems using averages over size, \(\langle \cdot \rangle_\mathbf\xi = \int_\Xi P_\mathbf\xi \langle\cdot\rangle\,\mathrm d\mathbf\xi \big/ \int_\Xi P_\mathbf\xi \,\mathrm d\mathbf\xi\) and setting \(n = V_f\big/\langle V \rangle_\mathbf\xi\).

Further improvements can be made using the local monodisperse approximation (LMA) or using partial structure factors as done in [2], but these are not implemented in this code.

For hollow shapes, volfraction is computed from the material in the shell rather than the shell plus solvent inside the shell. Using \(V_e(\mathbf\xi)\) as the enclosed volume of the shell plus solvnt and \(V_c(\mathbf\xi)\) as the core volume of solvent inside the shell, we can compute the average enclosed and shell volumes as

\begin{align*} \langle V_e \rangle &= \frac{ \int_\Xi P_\mathbf\xi V_e(\mathbf\xi)\,\mathrm d\mathbf\xi }{ \int_\Xi P_\mathbf\xi\,\mathrm d\mathbf xi } \\ \langle V_s \rangle &= \frac{ \int_\Xi P_\mathbf\xi (V_e(\mathbf\xi) - V_c(\mathbf\xi))\,\mathrm d\mathbf\xi }{ \int_\Xi P_\mathbf\xi\,\mathrm d\mathbf xi } \end{align*}

Given \(n\) particles and a total solvent volume \(V_\text{out}\) outside the shells, the volume fraction of the shell, \(\phi_s\) and the shell plus enclosed solvent \(\phi_e\) are

\begin{align*} \phi_s &= \frac{n \langle V_s \rangle}{n \langle V_s \rangle + n \langle V_c \rangle + V_\text{out}} = \frac{n \langle V_s \rangle}{V_\text{total}} \\ \phi_e &= \frac{n \langle V_e \rangle}{n \langle V_e \rangle + V_\text{out}} = \frac{n \langle V_e \rangle}{V_\text{total}} \end{align*}

Dividing gives

\[\frac{\phi_S}{\phi_P} = \frac{\langle V_e \rangle}{\langle V_s \rangle}\]

so the enclosed volume fraction can be computed from the shell volume fraction and the form:shell volume ratio as

\[\phi_S = \phi_P \langle V_e \rangle \big/ \langle V_s \rangle\]

References

[1]	Kotlarchyk, M.; Chen, S.-H. J. Chem. Phys., 1983, 79, 2461

[2]	Bressler I., Kohlbrecher J., Thunemann A.F. J. Appl. Crystallogr. 48 (2015) 1587-1598

Document History

2019-03-31 Paul Kienzle, Steve King & Richard Heenan