Fitting Models with Structure Factors

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.

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

with particle volume $V(mathbf xi)$ and

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

where $F(mathbf Q,mathbfxi)$ uses $mathbf Q = [Q sinthetacosphi, Q sinthetasinphi, Q costheta]^T$. A $u$-substitution may be used, with $alpha = cos theta$, $surd(1 - alpha^2) = sin theta$, and $mathrm dalpha = -sintheta,mathrm dtheta$. Here,

is the number density of scatterers estimated from the volume fraction 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 $mathbfxi$ with probability $P_mathbf{xi}$,

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

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

Or equivalently,

with 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, $langle cdot rangle_mathbfxi = int_Xi P_mathbfxi langlecdotrangle,mathrm dmathbfxi big/ int_Xi P_mathbfxi ,mathrm dmathbfxi$ and setting $n = V_fbig/langle V rangle_mathbfxi$. Further improvements can be made using the local monodisperse approximation (LMA) or using partial structure factors, as described in cite{bresler_sasfit:_2015}.

Many parameters are common amongst $P@S$ models, and take on specific meanings:

• scale:

  Overall model scale factor.

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

  The structure factor model $S(Q)$ has volfraction. This is also used as the volume fraction for the form factor model $P(Q)$, replacing the volfraction parameter if it exists in $P$. This means that $P@S$ models can leave scale at 1.0.

  If the volume fraction required for $S(Q)$ is not the volume fraction needed to compute the number density for $P(Q)$, then leave volfraction as the volume fraction for $S(Q)$ and use scale to define the volume fraction 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.

  For hollow shapes, volfraction still represents the volume fraction of material but the $S(Q)$ calculation needs the volume fraction enclosed by the shape. Thus 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 $P@S$ 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 will be larger than either the radius or the half-length of the cylinder. It may be sensible to tie or constrain radius_effective to one or other of these "size" parameters. radius_effective may also be specified directly, independent of the estimate from $P(Q)$.

  If it is calculated by $P(Q)$, radius_effective will be the weighted average of the effective radii computed for the polydisperse shape parameters, and that average 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.

• radius_effective_mode:

  Selects the radius_effective value to use.

  When radius_effective_mode = 0 then the radius_effective parameter in the $P@S$ model is used. Otherwise radius_effective_mode = k is the index into the list of radius_effective_modes defined by the model indicating how the effective radius should be computed from the parameters of the shape. For example, the ellipsoid model defines the following:

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

  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 user defined effective radius.

• 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) = scale(V_f)/(V)P(Q)S(Q) + background

  where $P(Q) = langle F(Q)^2 rangle$.

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

  I(Q) = scale(V_f)/(V)P(Q)[1 + β(Q)(S(Q) − 1)] +  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.

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