source:
sasmodels/doc/guide/sesans/sans_to_sesans.rst
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SANS to SESANS conversion
.. currentmodule:: sasmodels
The conversion from SANS into SESANS in absolute units is a simple Hankel transformation when all the small-angle scattered neutrons are detected. First we calculate the Hankel transform including the absolute intensities by
in which J0 is the zeroth order Bessel function, δ the spin-echo length, Q the wave vector transfer and (dΣ)/(dΩ)(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 Qmax , 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 Qmin = 0.1 × 2π ⁄ Rmax , in which Rmax 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:
in which t is the thickness of the sample and λ is the wavelength of the neutrons.
Log Spaced SESANS
For computational efficiency, the integral in the Hankel transform is converted into a Reimann sum
However, this model approximates more than is strictly necessary. Specifically, it is approximating the entire integral, when it is only the scattering function that cannot be handled analytically. A better approximation might be
Assume that vectors qn and In represent the q points and corresponding intensity data, respectively. Further assume that δm and Gm are the spin echo lengths and corresponding Hankel transform value.
where
Also not that, for the limit as δm approaches zero,