Simple Bayesian method for improved analysis of quasi-two-dimensional scattering data

A new method is presented for the analysis of small angle neutron scattering data from quasi-two-dimensional systems such as flux lattices, Skyrmion lattices, and aligned liquid crystals. A significant increase in signal to noise ratio and a natural application of the Lorentz factor can be achieved by taking advantage of the knowledge that all relevant scattering is centered on a plane in reciprocal space. The Bayesian form ensures that missing information is treated in a controlled way and can be subsequently included in the analysis. A simple algorithm based on Gaussian probability assumptions is provided which provides very satisfactory results. Finally, it is argued that a generalized model-independent Bayesian data analysis method would be highly advantageous for the processing of neutron and x-ray scattering data.

Figure 1

(a) Schematic diagram of a typical experimental setup. The incoming neutrons have a wave vector ${\mathbf{k}}_i$, outgoing neutrons have wave vector ${\mathbf{k}}_f$, where $|{\mathbf{k}}_f|=|{\mathbf{k}}_i|=2\pi /\lambda$, and $\lambda$ is the neutron de Broglie wavelength. Each pixel in the multidetector corresponds to a different direction of ${\mathbf{k}}_f$. (b) Section through the Ewald sphere and reciprocal lattice in the scattering plane. The field and hence reciprocal lattice is rotated by an angle, $\omega$, with respect to ${\mathbf{k}}_i$. (c) Reciprocal lattice plane showing circles where the Ewald sphere intersects for different rotations $\omega$ around ${\mathbf{q}}_y$. If all scattered intensity is in this plane, then only pixels close to a circle will contain relevant data. The inset in panel (c) shows the region of reciprocal space for which relevant data is collected when the field is “rocked” from $-\omega$ to $+\omega$ about a single axis in the $y$ direction.

Figure 2

(a) Data taken at a single angle $\omega$. Only certain spots fulfill the Bragg condition to within the experimental resolution. (b) Sum of data taken over a range of angles encompassing all first-order Bragg reflections attainable by rotation about a vertical axis. (c) Result of Bayesian weighted method, using individual pixel priors, as described in Sec. 3c. (d) Sum of intensity inside boxes shown in panel (b) plotted as a function of rocking angle. The width of the peaks is a measurement of the longitudinal coherence of the flux lattice, and the total integrated intensity under the peaks is proportional to the square of $B\left(\mathbf{q}\right)$. The data in both panels (a) and (b) have backgrounds taken in the normal state subtracted from them. A small amount of smoothing (convolution with a $3\times 3$ pixel Gaussian) has been applied to panels (a)–(c) after all other processing. The direct beam in the center of the detector is masked.

Figure 3

Weighting factors $w_{ij}$ for a single frame using (a) uniform prior and (b) individual pixel priors, as described in Sec. 3c. Note the crescent shape where the Ewald sphere cuts the lattice place, but also the maxima at the top and bottom, which are the result of the denominator in Eq. (15). In panel (b) these are less prominent as the prior is suppressed by the low counts in these regions, though more noise is evident. A $3\times 3$ pixel Gaussian smoothing has been applied to both figures.

Figure 4

(a) Result of Bayesian weighted sum of data from Fig. 2, with prior mean $\mu$ set to zero, and uniform s.d. $\xi$ determined by the maximum spot intensity. Note that spots are recovered at the top and bottom, even though they never reach the Bragg criterion. Gaussian smoothing has been applied to the weighted sum with a $3\times 3$ pixel resolution. (b) The same as panel (a) without smoothing. (c) Areas with $\sigma >0.2\xi$ are masked, where $\sigma$ is the error on the posterior intensity. (d) Sum of intensity inside boxes in panel (c) as a function of angle. These converge on the final result at different positions, depending on the angle at which the peak appears. Data taken at angles after passing through the peak of the rocking curve at the Bragg condition make very little difference to the final result.