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

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

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. This is a 1-dimensional integral, which can be rather fast. In the numerical calculation we integrate from \(Q_{min} = 0.1 \times 2 \pi / R_{max}\) in which \(R_{max}\) will be model dependent. We determined the factor 0.1 by varying its value until the value of the integral was stable. This happened at a value of 0.3. The have a safety margin of a factor of three we have choosen the value 0.1. For the solid sphere we took 3 times the radius for \(R_{max}\). The real integration is performed to \(Q_{max}\) which is an instrumental parameter that is read in from the measurement file. 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.