Polarization-analyzed small-angle neutron scattering. II. Mathematical angular analysis

Abstract

Polarization‐analyzed small‐angle neutron scattering (SANS) is a powerful tool for the study of magnetic morphology with directional sensitivity. Building upon polarized scattering theory, this article presents simplified procedures for the reduction of longitudinally polarized SANS into terms of the three mutually orthogonal magnetic scattering contributions plus a structural contribution. Special emphasis is given to the treatment of anisotropic systems. The meaning and significance of scattering interferences between nuclear and magnetic scattering and between the scattering from magnetic moments projected onto distinct orthogonal axes are discussed in detail. Concise tables summarize the algorithms derived for the most commonly encountered conditions. These tables are designed to be used as a reference in the challenging task of extracting the full wealth of information available from polarization‐analyzed SANS.

Spin canting across core/shell Fe3O4/MnxFe3−xO4 nanoparticles

“…Polarization-analyzed small-angle neutron scattering (PASANS) measurements 41,42 were performed to further explore the unusual magnetic behavior of these particles, in this case organized into ordered dense assemblies. Figure 7a shows a schematic of the experimental setup in which the neutron beam was polarized either spin up (↑) or down (↓), scattered off the sample in a given field and temperature condition, and then spin-analyzed to measure all four possible scattering cross-sections (↑↑, ↑↓, ↓↑, or ↓↓).…”

Kryckaet al.Reply:

“…The M ⊥,Y ( Q) and M ⊥,Z ( Q) scattering from symmetrically-equivalent magnetization components along Y and Z are not identical in this case [2] because the net moments along Y and Z sum to zero over the nanocomposite, resulting in zero spin-flip scattering along the coordinate axes. In contrast, our nanoparticle system exhibits non-zero spin-flip scattering at all θ [3], which implies that the canted shells must have a net uniform moment throughout each particle [4] which is equivalent along Y and Z (on average) and that…”

“…4 of [3]. [3]) appear only as a difference between spin-flip cross-sections. Since our spin-flip cross-sections are equivalent within experimental resolution, we summed them together [1] as in [4], rendering these extra terms irrelevant.…”

“…The specific distribution, however, does not impact the spin-flip scattering asymmetry ratio as evidenced by the fact that the spin-flip scattering shape was preserved when viewed along any θ-cut, Fig. 4 of [3]. [3]) appear only as a difference between spin-flip cross-sections.…”

“…2b of [2] ("flower") or Fig. 2(iv) of [3] over the submicron lengthscale of a close-packed grain of nanoparticles, involving a net moment alongX H and a gradual, average spatial variation among M ⊥,Y ( Q) shell moments with mirror symmetry about the X-axis. Although we did not discuss the characteristics of the long-range, magnetic, nanoparticle structure in [1] for brevity, particle clusters with similarly canted shells were required to properly model the spin-flip scattering [4].…”