.. currentmodule:: sasmodels .. Wim Bouwman, DUT, written at codecamp-V, Oct2016 .. _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. 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.