Using the singular value decomposition to extract 2D correlation functions from scattering patterns

Abstract

The truncated singular value decomposition (TSVD) is applied to extract the underlying 2D correlation functions from small-angle scattering patterns. The approach is tested by transforming the simulated data of ellipsoidal particles and it is shown that also in the case of anisotropic patterns (i.e. aligned ellipsoids) the derived correlation functions correspond to the theoretically predicted profiles. Furthermore, the TSVD is used to analyze the small-angle X-ray scattering patterns of colloidal dispersions of hematite spindles and magnetotactic bacteria in the presence of magnetic fields, to verify that this approach can be applied to extract model-free the scattering profiles of anisotropic scatterers from noisy data.

Unraveling Nanostructured Spin Textures in Bulk Magnets

“…In principle, the real‐space 2D magnetic correlation function , with being the autocorrelation function in case of nuclear scattering, can be extracted from the experimental reciprocal scattering data via a direct Fourier transform. [ 27 ] For the analysis of nuclear scattering patterns, however, usually indirect approaches are applied where the inverse problem is solved, [ 28,44,45 ] and which can be readily adapted to magnetic SANS. The challenge is to extract good and robust estimations for from the noisy data, as well as in case of restricted q ranges as is usually the case in experiment.…”

“…The angle φ specifies the orientation of r in the ‐ z ‐plane, and the extracted 2D distribution function is given by . [ 44 ] The matrix A in Equation () is the data transfer matrix, which, in case of the 2D indirect Fourier transform, has the elements [ 27,28,44,45 ] …”

Using small-angle scattering to guide functional magnetic nanoparticle design

“…(d) The top panel shows the 2D SAXS pattern of a colloidal suspension of magnetotactic bacteria that were aligned by applying an external magnetic field (in the horizontal direction) from which the 2D correlation function (bottom) was extracted by an inverse Fourier transform using a singular value decomposition. 132 The real-space correlation function reflects the nearest and next-neighbors distance distribution of the linear chain of MNPs and the average size of the isometric MNPs. Reproduced with permission of the International Union of Crystallography.…”

Unraveling Nanostructured Spin Textures in Bulk Magnets

“…being the autocorrelation function in case of nuclear scattering, can be extracted from the experimental reciprocal scattering data I m (q) via a direct Fourier transform. 28 For the analysis of nuclear scattering patterns, however, usually indirect approaches are applied where the inverse problem is solved, [29][30][31] and which can be readily adapted to magnetic SANS. The challenge is to extract good and robust estimations for P (r) from the noisy data, also in case of restricted q-ranges as is usually the case in experiment.…”

“…The angle ϕ specifies the orientation of r in the yz-plane, and the extracted 2D distribution function P (r) is given by P (r, ϕ) = C(r, ϕ)r. 30 The matrix A in Eq. 2 is the data transfer matrix, which, in case of the 2D indirect Fourier transform, has the elements [28][29][30][31] A ij = cos (q i r j cos (Θ i − ϕ j )) ∆r j ∆ϕ j .…”