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update beta approx docs. Refs sasview/sasview#1164. #203

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update beta approx docs. Refs sasview/sasview#1164. #203

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220 changes: 168 additions & 52 deletions doc/guide/fitting_sq.rst
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Expand Up @@ -15,78 +15,194 @@ Fitting Models with Structure Factors

.. figure:: p_and_s_buttons.png

**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.
**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.

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

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

.. math::
I(\mathbf Q) = F(\mathbf Q, \mathbf\xi) F^*(\mathbf Q, \mathbf\xi)
\big/ V(\mathbf\xi)

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

.. math::
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

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

.. math::
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,

.. math:: n = V_f/V(\mathbf\xi)

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 $\mathbf\xi$ with probability
$P_\mathbf{\xi}$,

.. math::

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

.. math:: I(Q) = n \langle F F^* \rangle S(Q)

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

.. math::

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

Or equivalently,

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

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, $\langle \cdot \rangle_\mathbf\xi = \int_\Xi P_\mathbf\xi \langle\cdot\rangle\,\mathrm d\mathbf\xi \big/ \int_\Xi P_\mathbf\xi \,\mathrm d\mathbf\xi$ 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 described
in \cite{bresler_sasfit:_2015}.

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

* *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.
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.
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 $I(Q)$).
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, 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.
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*:

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.:
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.:

$I(Q)$ = $(scale$ / $volume)$ x $P(Q)$ x $S(Q)$ + $background$
.. math::
I(Q) = \text{scale} \frac{V_f}{V} P(Q) S(Q) + \text{background}

If **structure_factor_mode = 1** then the $\beta(q)$ correction will be
used, i.e.:
where $P(Q) = \langle F(Q)^2 \rangle$.

$I(Q)$ = $(scale$ x $volfraction$ / $volume)$ x $( <F(Q)^2>$ + $<F(Q)>^2$ x $(S(Q)$ - $1) )$ + $background$
If **structure_factor_mode = 1** then the $\beta(Q)$ correction will be
used, i.e.:

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

This is equivalent to:

$I(Q)$ = $(scale$ / $volume)$ x $P(Q)$ x $( 1$ + $\beta(Q)$ x $(S(Q)$ - $1) )$ + $background$
.. math::
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.
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.
More mode options may appear in future as more complicated operations are
added.

References
^^^^^^^^^^
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