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 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 $Q_{min} = 0.1 \times 2 \pi / R_{max}$, 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.

References

[1]

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.