.. algorithm::

.. summary::

.. relatedalgorithms::

.. properties::

Description
-----------

Algorithm Reference Discussion
------------------------------

Transmission for scattering at large angles should be modified to account for the longer
path length after the scattering event [#Brulet]_ [#Ghosh]_.

The longer path length after scattering will also slightly increase the
probability of a second scattering event, but this is not dealt with
here.

If our on-axis transmission is :math:`T_0` through a sample of thickness
:math:`d`, then the transmission at some other thickness :math:`x` is
:math:`\exp(-\mu x)` where attenuation coefficient
:math:`\mu = -\ln( \frac{T_0}{d})`.

If a neutron scatters at angle :math:`2\theta` at distance :math:`x`
into the sample, its total transmission is then:

.. math:: T^{''} = \exp(-\mu x) \exp( \frac{-\mu(d-x)}{\cos(2\theta)})

:math:`T^{''}` should be integrated and averaged between :math:`x = 0`
and :math:`x = d`.

The resulting integral can be approximated, see page 208 of [#Hammouda]_, as:

.. math::

    T^{'} = \frac{T_0(T_0^A - 1)}{A \ln(T_0)}

With :math:`A = \frac{1}{\cos(2\theta)} - 1`.

For example, if :math:`T_0 = 0.2` and :math:`2\theta = 40` then
:math:`T^{'} = 0.158`, a shift of :math:`~20`\ % of the SANS curve.
Note that the result is independent of sample thickness.

:math:`T_0` is a function of neutron wavelength, whilst :math:`A` is a
function of detector pixel location.

The output of this algorithm is:

:math:`\text{Output Workspace}=\frac{T^\prime}{T_0}`

Error Propagation
#################

The error propagation follows this formula:

:math:`\text{OutputWorkspace}_{error} = \frac{T_{0E} ^A - 1}{A\ln(T_0E)}`

Which means, that we do not consider the error in the definition of
:math:`2\theta` (the parameter A)

Enabling Wide Angle Correction for Reduction of SANS ISIS
---------------------------------------------------------

The Wide Angle Correction was set on the legacy user file with the command:

``SAMPLE/PATH/ON``

For TOML user files, the ``wide_angle_correction`` boolean flag can be set on the transmission field, if the
flag is not present on the TOML file, the default value is ``false``:

..  code-block:: yaml

  [transmission]
    wide_angle_correction=true

More information on :ref:`sans_toml_v1-ref`.

SANS ISIS Reduction
-------------------

The output of SANSWideAngleCorrection is used as WavePixelAdj parameter
at :ref:`algm-Q1D`.

Wide Angle Correction and the SANS Reduction
--------------------------------------------

The equation for the reduction is (see :ref:`algm-Q1D`)

.. math::

    P_I(Q) = \frac{\sum_{\{i, j, n\} \supset \{I\}} S(i,j,n)}{\sum_{\{i, j, n\} \supset \{I\}}M(n)\eta(n)T(n)\Omega_{i j}F_{i j}}

But :math:`T(n)` is now a function of the detector position to account for wide angles, :math:`T(n,theta)` or :math:`T(n,i,j)`.

Thus, the correction changes to:

.. math::

    P_I(Q) = \frac{\sum_{\{i, j, n\} \supset \{I\}} S(i,j,n)}{\sum_{\{i, j, n\} \supset \{I\}}M(n)\eta(n)T(n)\Omega_{i j}F_{i j}Corr(i,j,n)}

Where Corr (Correction factor) in this case will be:

:math:`\text{Corr}(i,j,n) = \frac{T_0^A - 1}{A \ln(T_0)}`

Which is the OutputWorkspace of SANSWideAngleCorrection.

This workspace enters inside :ref:`algm-Q1D` as the workspace property ``WavePixelAdj``, but this step is handled automatically inside
the reduction workflow.

Comparison with Wide Angle Correction at SNS
--------------------------------------------

The transmission correction applied at SNS is described
:ref:`here <algm-HFIRSANSReduction-v1>`,
and it is applied through the
:ref:`algm-ApplyTransmissionCorrection` algorithm.
The correction applied there is an approximation for the same equations
described here. The picture above compare their results

.. figure:: /images/SNS_ISIS_WideAngleCorrections.png
   :alt: SNS_ISIS_WideAngleCorrections.png

Note a difference among them is when they are applied. At SNS, the
correction is applied before averaging the counters per bin inside
:ref:`algm-Q1D` algorithm, while at ISIS, it is used after, inside the
:ref:`algm-Q1D` algorithm, for the division of the counters per bin
normalized by the transmission counters.

References
----------

.. [#Brulet] Annie Brulet et al. - Improvement of data treatment in small-angle neutron scattering - `J. Appl.Cryst. (2007). 40 <http://dx.doi.org/10.1107/S0021889806051442>`_

.. [#Ghosh] Ghosh, Egelhaaf & Rennie - Computing guide for Small Angle Scattering

.. [#Hammouda] Hammouda B. The SANS toolbox. NIST Center for Neutron Research, 2008


Usage
-----

**Example - Correcting Some Dummy Data**

.. testcode:: ExCorrection

   # Create some dummy data, but crop it for quick demonstration purposes.
   sample = CreateSimulationWorkspace(Instrument='SANS2D', BinParams=[5,500,100005], UnitX='TOF')
   sample = CropWorkspace(sample,StartWorkspaceIndex=0,EndWorkspaceIndex=20)

   # Create a dummy transmission workspace.
   transmission = CropWorkspace(sample,StartWorkspaceIndex=10,EndWorkspaceIndex=10)
   transmission *= 2

   corrected_data = SANSWideAngleCorrection(sample, transmission)

   print("{:.6f} was corrected to {:.6f}.".format(sample.readY(19)[0], corrected_data.readY(19)[0]))

Output:

.. testoutput:: ExCorrection

   1.000000 was corrected to 1.004997.

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.. sourcelink::