.. currentmodule:: sasmodels
.. Wim Bouwman, DUT, written at codecamp-V, Oct2016
.. Reference added, Steve King, Oct 2021

.. _SESANS:

SANS to SESANS conversion
=========================

The conversion from SANS into SESANS in absolute units is a simple Hankel
transformation when all the small-angle scattered neutrons are detected [#Bakker2020]_.
First we calculate the Hankel transform including the absolute intensities by

.. math:: G(\delta) = 2 \pi \int_0^{\infty} J_0(Q \delta) \frac{d \Sigma}{d \Omega} (Q) Q dQ \!,

in which :math:`J_0` is the zeroth order Bessel function, :math:`\delta`
the spin-echo length, :math:`Q` the wave vector transfer and :math:`\frac{d \Sigma}{d \Omega} (Q)`
the scattering cross section in absolute units.

Out of necessity, a 1-dimensional numerical integral approximates the exact
Hankel transform. The upper bound of the numerical integral is :math:`Q_{max}`,
which is calculated from the wavelength and the instrument's maximum acceptance
angle, both of which are included in the file. While the true Hankel transform
has a lower bound of zero, most scattering models are undefined at :math: `Q=0`,
so the integral requires an effective lower bound. The lower bound of the
integral is :math:`Q_{min} = 0.1 \times 2 \pi / R_{max}`, in which :math:`R_{max}`
is the maximum length scale probed by the instrument multiplied by the number
of data points. This lower bound is the minimum expected Q value for the given
length scale multiplied by a constant. The constant, 0.1, was chosen empirically
by integrating multiple curves and finding where the value at which the integral
was stable. A constant value of 0.3 gave numerical stability to the integral, so
a factor of three safety margin was included to give the final value of 0.1.


From the equation above we can calculate the polarisation that we measure in a
SESANS experiment:

.. math:: P(\delta) = e^{t \left( \frac{ \lambda}{2 \pi} \right)^2 \left(G(\delta) - G(0) \right)} \!,

in which :math:`t` is the thickness of the sample and :math:`\lambda` is the
wavelength of the neutrons.

References
----------

.. [#Bakker2020] JH Bakker, AL Washington, SR Parnell, AA van Well, C Pappas,
   WG Bouwman, *Analysis of SESANS data by numerical Hankel transform
   implementation in SasView*, *Journal of Neutron Research*, 22 (2020) 57-70.
   `DOI 10.3233/JNR-200154 <https://doi.org/10.3233/JNR-200154>`_.