# 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 [1]. First we calculate the Hankel transform including the absolute intensities by

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

in which $$J_0$$ is the zeroth order Bessel function, $$\delta$$ the spin-echo length, $$Q$$ the wave vector transfer and $$\frac{d \Sigma}{d \Omega} (Q)$$ 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 $$Q_{max}$$, which is calculated from the wavelength and the instrumentâ€™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 $$Q_{min} = 0.1 \times 2 \pi / R_{max}$$, in which $$R_{max}$$ 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:

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

in which $$t$$ is the thickness of the sample and $$\lambda$$ is the wavelength of the neutrons.