Pair distribution function and structure factor of spherical particles

The availability of neutron spallation-source instruments that provide total scattering powder diffraction has led to an increased application of real-space structure analysis using the pair distribution function. Currently, the analytical treatment of finite size effects within pair distribution refinement procedures is limited. To that end, an envelope function is derived which transforms the pair distribution function of an infinite solid into that of a spherical particle with the same crystal structure. Distributions of particle sizes are then considered, and the associated envelope function is used to predict the particle size distribution of an experimental sample of gold nanoparticles from its pair distribution function alone. Finally, complementing the wealth of existing diffraction analysis, the peak broadening for the structure factor of spherical particles, expressed as a convolution derived from the envelope functions, is calculated exactly for all particle size distributions considered, and peak maxima, offsets, and asymmetries are discussed.Comment: 7 pages, 6 figure

Geometrical Ambiguity of Pair Statistics. I. Point Configurations

Point configurations have been widely used as model systems in condensed matter physics, materials science and biology. Statistical descriptors such as the $n$-body distribution function $g_n$ is usually employed to characterize the point configurations, among which the most extensively used is the pair distribution function $g_2$. An intriguing inverse problem of practical importance that has been receiving considerable attention is the degree to which a point configuration can be reconstructed from the pair distribution function of a target configuration. Although it is known that the pair-distance information contained in $g_2$ is in general insufficient to uniquely determine a point configuration, this concept does not seem to be widely appreciated and general claims of uniqueness of the reconstructions using pair information have been made based on numerical studies. In this paper, we introduce the idea of the distance space, called the $\mathbb{D}$ space. The pair distances of a specific point configuration are then represented by a single point in the $\mathbb{D}$ space. We derive the conditions on the pair distances that can be associated with a point configuration, which are equivalent to the realizability conditions of the pair distribution function $g_2$. Moreover, we derive the conditions on the pair distances that can be assembled into distinct configurations. These conditions define a bounded region in the $\mathbb{D}$ space. By explicitly constructing a variety of degenerate point configurations using the $\mathbb{D}$ space, we show that pair information is indeed insufficient to uniquely determine the configuration in general. We also discuss several important problems in statistical physics based on the $\mathbb{D}$ space.Comment: 28 pages, 8 figure

Pair distribution function studies in cementitious systems

The analysis of amorphous/nanocrystalline phase(s) within cement matrices that contain high amounts of crystalline phase(s) is very challenging. Synchrotron techniques can be very useful to characterize such complex samples.1 This work is focused on total scattering Pair Distribution Function (PDF) quantitative phase analyses in selected real-space ranges for a better understanding of the binding gel(s). Powder diffraction data collected in BL04-MSPD beamline have been analyzed by PDF and Rietveld methodologies to determine nanocrystalline and microcrystalline phase contents. The comparison between both methodologies allows us to have a better insight about the nanocrystalline/microcrystalline components which coexist in cement pastes. Three sets of hydrated model samples have been studied: i) monocalcium aluminate, CaAl2O4, the main component of calcium aluminate cements, ii) ye’elimite, Ca4Al6SO16, the main component of calcium sulfoaluminate cements, and iii) tricalcium silicate, Ca3SiO5, the main component of Portland cements. For the CaAl2O4 paste, the PDF fit shows that the aluminum hydroxide gel has a gibbsite local structure with an average particle size close to 5 nm.2 Figure 1 shows the final fit for CaAl2O4 paste in two different real-space regions. On the contrary, for Ca4Al6SO16 paste, it has been found that the particle size of the aluminum hydroxide gel is below 3 nm. Moreover, the Ca3SiO5 paste contains a different nanocrystalline gel, C-S-H, which has also been thoroughly studied. Different crystal structures (including Tobermorite, Clinotobermorite and Jennite) have been tested to find the structural model that fits better the experimental data. The results from this ongoing investigation will be reported and discussed.Universidad de Málaga. Campus de Excelencia Internacional Andalucía Tech. This work has been supported by Spanish MINECO through BIA2014-57658-C2-1-R and BIA2014-57658-C2-2-R, which is co-funded by FEDER, research grants. We also thank CELLS-ALBA for providing synchrotron beam time at BL04-MSPD

Synchrotron Radiation Pair Distribution Function Analysis of Gels in Cements

The analysis of atomic ordering in a nanocrystalline phase with small particle sizes, below 5 nm, is intrinsically complicated because of the lack of long-range order. Furthermore, the presence of additional crystalline phase(s) may exacerbate the problem, as is the case in cement pastes. Here, we use the synchrotron pair distribution function (PDF) chiefly to characterize the local atomic order of the nanocrystalline phases, gels, in cement pastes. We have used a multi r-range analysis approach, where the ~4–7 nm r-range allows determining the crystalline phase contents; the ~1–2.5 nm r-range is used to characterize the atomic ordering in the nanocrystalline component; and the ~0.2–1.0 nm r-range gives insights about additional amorphous components. Specifically, we have prepared four alite pastes with variable water contents, and the analyses showed that a defective tobermorite, Ca11Si9O28(OH)2 8.5H2O, gave the best fit. Furthermore, the PDF analyses suggest that the calcium silicate hydrate gel is composed of this tobermorite and amorphous calcium hydroxide. Finally, this approach has been used to study alternative cements. The hydration of monocalcium aluminate and ye’elimite pastes yield aluminum hydroxide gels. PDF analyses show that these gels are constituted of nanocrystalline gibbsite, and the particle size can be as small as 2.5 nmThis work has been supported by Spanish MINECO through BIA2014-57658-C2-2-R, which is co-funded by FEDER, BIA2014-57658-C2-1-R and I3 (IEDI-2016-0079) grants. We also thank CELLS-ALBA (Barcelona, Spain) for providing synchrotron beam time at BL04-MSPD beamline. Finally, we thank Prof. Simon Billinge, Long Yang and Monica Dapiaggi for their help with the PDF script and simulations for Ca(OH)2 scattering dat