Fitting Models with Structure Factors

Product models, or $P@S$ models for short, multiply the structure factor $S(Q)$ by the form factor $P(Q)$, modulated by the effective radius of the form factor.

Many of the parameters in $P@S$ models take on specific meanings so that they can be handled correctly inside SasView:

• scale:

  In simple $P(Q)$ models scale often represents the volume fraction of material.

  In $P@S$ models scale should be set to 1.0, as the $P@S$ model contains a volfraction parameter.

• 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. SasView scales the user-specified volume fraction by the ratio form:shell computed from the average form volume and average shell volume returned from the $P(Q)$ calculation (the original volfraction is divided by the shell volume to compute the number density, and then $P@S$ is scaled by that 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, or may not, be the same as any "size" parameters 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 half-length of the cylinder. It may be sensible to tie or constrain radius_effective to one or other of these "size" parameters.

  If just part of the $S(Q)$ calculation, the value of radius_effective may be polydisperse. If it is calculated by $P(Q)$, then it will be the weighted average of the effective radii computed for the polydisperse shape parameters.

• structure_factor_mode:

  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.:

  $I(Q)$ = $(scale$ / $volume)$ x $P(Q)$ x $S(Q)$ + $background$

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

  $I(Q)$ = $(scale$ x $volfraction$ / $volume)$ x $( <F(Q)^2>$ + $<F(Q)>^2$ x $(S(Q)$ - $1) )$ + $background$

  where $P(Q)$ = $<|F(Q)|^2>$.

  ?

  This is equivalent to:

  $I(Q)$ = $(scale$ / $volume)$ x $P(Q)$ x $( 1$ + $beta(Q)$ x $(S(Q)$ - $1) )$ + $background$

  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.