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

Atomic Distributions in the γ-Brass Structure of the Cu−Zn System: A Structural and Theoretical Study

The crystal structures, atomic distributions, and theoretical electronic structures of five different Cu5-xZn8+x γ-brass compounds (x = −0.59(3), −0.31(3), 0.00(3), 0.44(3), and 0.79(3)) are reported with the goal of identifying chemical influences on the observed phase width. These structures have been refined by both neutron and X-ray powder diffraction to obtain accurate crystal chemical parameters. All compounds crystallize in the space group I4̄3m (No. 217) (Z = 4), and the unit cell parameters are a = 8.8565(4), 8.8612(5), 8.8664(3) , 8.8745(4), and 8.8829(7) Å, respectively, for Cu5.59Zn7.41, Cu5.31Zn7.69, Cu5.00Zn8.00, Cu4.56Zn8.44, and Cu4.21Zn8.79. The results indicate specific site substitutions on both sides of the ideal composition “Cu5Zn8”. In all cases, the 26-atom cluster building up the γ-brass structure shows a constant inner [Cu4Zn4] tetrahedral star with compositional variation occurring at the outer octahedron and cuboctahedron. First principles and semiempirical electronic structure calculations using both a COHP and Mulliken population analysis were performed to understand the observed compositional range and to address the “coloring problem” for the site preferences of Cu and Zn atoms for this series of compounds

Tuning magnetic frustration on the diamond lattice of the A-site magnetic spinels CoAl2−x_{2-x}Gax_xO4_4: Lattice expansion and site disorder

The spinels CoB$_2$O$_4$ with magnetic Co$^{2+}$ ions on the diamond lattice A site can be frustrated because of competing near-neighbor ($J_1$) and next-near neighbor ($J_2$) interactions. Here we describe attempts to tune the relative strengths of these interactions by substitution on the non-magnetic B-site. The system we employ is CoAl$_{2-x}$Ga$_x$O$_4$, where Al is systematically replaced by the larger Ga, ostensibly on the B site. As expected, Ga substitution expands the lattice, resulting in Co atoms on the A-site being pushed further from one other and thereby weakening magnetic interactions. In addition, Ga distributes between the B and the A site in a concentration dependent manner displacing an increasing amount of Co from the A site with increasing $x$. This increased inversion, which is confirmed by neutron diffraction studies carried out at room temperature, affects magnetic ordering very significantly, and changes the nature of the ground state. Modeling of the magnetic coupling illustrates the complexity that arises from the cation site disorder.Comment: 9 pages, 10 figure

Local structures of polar wurtzites Zn_{1-x}Mg_{x}O studied by Raman and {67}Zn/{25}Mg NMR spectroscopies and by total neutron scattering

Local compositions and structures of Zn_{1-x}Mg_{x}O alloys have been investigated by Raman and solid-state {67}Zn/{25}Mg nuclear magnetic resonance (NMR) spectroscopies, and by neutron pair-distribution-function (PDF) analyses. The E2(low) and E2(high) Raman modes of Zn_{1-x}Mg_{x}O display Gaussian- and Lorentzian-type profiles, respectively. At higher Mg substitutions, both modes become broader, while their peak positions shift in opposite directions. The evolution of Raman spectra from Zn_{1-x}Mg_{x}O solid solutions are discussed in terms of lattice deformation associated with the distinct coordination preferences of Zn and Mg. Solid-state magic-angle-spinning (MAS) NMR studies suggest that the local electronic environments of {67}Zn in ZnO are only weakly modified by the 15% substitution of Mg for Zn. {25}Mg MAS spectra of Zn_{0.85}Mg_{0.15}O show an unusual upfield shift, demonstrating the prominent shielding ability of Zn in the nearby oxidic coordination sphere. Neutron PDF analyses of Zn_{0.875}Mg_{0.125}O using a 2x2x1 supercell corresponding to Zn_{7}MgO_{8} suggest that the mean local geometry of MgO_{4} fragments concurs with previous density functional theory (DFT)-based structural relaxations of hexagonal wurtzite MgO. MgO_{4} tetrahedra are markedly compressed along their c-axes and are smaller in volume than ZnO_{4} units by ~6%. Mg atoms in Zn_{1-x}Mg_{x}O have a shorter bond to the $c$-axial oxygen atom than to the three lateral oxygen atoms, which is distinct from the coordination of Zn. The precise structure, both local and average, of Zn_{0.875}Mg_{0.125}O obtained from time-of-flight total neutron scattering supports the view that Mg-substitution in ZnO results in increased total spontaneous polarization.Comment: 12 pages, 14 figures, 2 table

Local structural origins of the distinct electronic properties of Nb-substituted SrTiO3_3 and BaTiO3O_3

Near or less than 10% Nb substitution on the Ti site in perovskite SrTiO$_3$ results in metallic behavior, in contrast to what is seen in BaTiO$_3$. Given the nearly identical structure and electron counts of the two materials, the distinct ground states for low substitution have been a long-standing puzzle. Here we find from neutron studies of average and local structure, the subtle yet critical difference that we believe underpins the distinct electronic properties in these fascinating materials. While SrTi$_{0.875}$Nb_${0.125}$O$_3$ possesses a distorted non-cubic structure at 15 K, the BO$_6$ octahedra in the structure are regular. BaTi$_{0.875}$Nb$_{0.125}$O$_3$ on the other hand shows evidence for local cation off-centering whilst retaining a cubic structure.Comment: 4 pages, 5 figure

Precise implications for real-space pair distribution function modeling of effects intrinsic to modern time-of-flight neutron diffractometers

Total scattering and pair distribution function (PDF) methods allow for detailed study of local atomic order and disorder, including materials for which Rietveld refinements are not traditionally possible (amorphous materials, liquids, glasses and nanoparticles). With the advent of modern neutron time-of-flight (TOF) instrumentation, total scattering studies are capable of producing PDFs with ranges upwards of 100–200 Å, covering the correlation length scales of interest for many materials under study. Despite this, the refinement and subsequent analysis of data are often limited by confounding factors that are not rigorously accounted for in conventional analysis programs. While many of these artifacts are known and recognized by experts in the field, their effects and any associated mitigation strategies largely exist as passed-down `tribal' knowledge in the community, and have not been concisely demonstrated and compared in a unified presentation. This article aims to explicitly demonstrate, through reviews of previous literature, simulated analysis and real-world case studies, the effects of resolution, binning, bounds, peak shape, peak asymmetry, inconsistent conversion of TOF to d spacing and merging of multiple banks in neutron TOF data as they directly relate to real-space PDF analysis. Suggestions for best practice in analysis of data from modern neutron TOF total scattering instruments when using conventional analysis programs are made, as well as recommendations for improved analysis methods and future instrument design

Precise implications for real-space pair distribution function modeling of effects intrinsic to modern time-of-flight neutron diffractometers

Total scattering and pair distribution function (PDF) methods allow for detailed study of local atomic order and disorder, including materials for which Rietveld refinements are not traditionally possible (amorphous materials, liquids, glasses and nanoparticles). With the advent of modern neutron time-of-flight (TOF) instrumentation, total scattering studies are capable of producing PDFs with ranges upwards of 100–200 Å, covering the correlation length scales of interest for many materials under study. Despite this, the refinement and subsequent analysis of data are often limited by confounding factors that are not rigorously accounted for in conventional analysis programs. While many of these artifacts are known and recognized by experts in the field, their effects and any associated mitigation strategies largely exist as passed-down `tribal' knowledge in the community, and have not been concisely demonstrated and compared in a unified presentation. This article aims to explicitly demonstrate, through reviews of previous literature, simulated analysis and real-world case studies, the effects of resolution, binning, bounds, peak shape, peak asymmetry, inconsistent conversion of TOF to d spacing and merging of multiple banks in neutron TOF data as they directly relate to real-space PDF analysis. Suggestions for best practice in analysis of data from modern neutron TOF total scattering instruments when using conventional analysis programs are made, as well as recommendations for improved analysis methods and future instrument design