China
Africa
Explore chapters and articles related to this topic
Structure Prediction from Scattering Profiles: A Neutron-Scattering Use-Case
Published in Anuj Karpatne, Ramakrishnan Kannan, Vipin Kumar, Knowledge-Guided Machine Learning, 2023
Cristina Garcia-Cardona, Ramakrishnan Kannan, Travis Johnston, Thomas Proffen, Sudip K. Seal
Crystallographic structure determination and refinement has been the cornerstone of materials science and our understanding of the atomic structure for many decades. The ability to design customized material with targeted mechanical and chemical properties relies on their internal structure. Neutron scattering is a state-of-the-art experimental technique that allows scientists to probe material structures with atomic resolutions by scattering beams of neutrons from them. While calculating the scattering intensities of a given crystal structure is straight forward, obtaining the atomic structure from the scattering intensities is not due to the so-called “crystallographic phase problem”. In a nutshell, the scattering intensities we measure only give us the amplitude of the structure factor F but not the phase value.
The Intensity of Diffracted X-ray Beams
Published in Dong ZhiLi, Fundamentals of Crystallography, Powder X-ray Diffraction, and Transmission Electron Microscopy for Materials Scientists, 2022
The structure factor, Fhkl, is a complex number (Fhkl = |Fhkl|eiαhkl), and the phase problem is the central problem in single crystal analysis (Luger, 2014). There are different methods for the solution of the phase problem.
Picometer Detection by Adaptive Holographic Interferometry
Published in Klaus D. Sattler, Fundamentals of PICOSCIENCE, 2013
Since the result of the diffraction experiment is only the structure factor's amplitude, the calculation of the phases has been a major problem (the "phase problem") in crystallography during several decades, leading to developing methods such as somorphous replacement (Green et al. 1954), molecular replacement (MR; Rossmann and Blow 1962), density modification (DM; Read 2001), direct methods (Miller et al. 1993; Sheldrick and Schneider 2001), and multiple (or single) anomalous dispersion (MAD or SAD) (Karle 1980; Hendickson and Ogata 1997). Currently, the combined use of anomalous scattering methods and density modification has changed the situation to the point that the phases can be considered as an experimental result in most cases.
Complexing properties of 2-pyridylphoshonate and 2-pyridylsulfonate ligands for Zn2+ and Ag+ central atoms
Published in Journal of Coordination Chemistry, 2022
Zuzana Vargová, Michaela Rendošová, Silvia Saksová, Róbert Gyepes, Mária Vilková
Single-crystal X-ray diffraction data for {[Zn(H2pypo)2(H2O)2]}n(ClO4)2n·2nH2O (CCDC number 2157021) were obtained on a Bruker D8 VENTURE diffractometer. The primary radiation used was MoKα (λ = 0.71073 Å). The phase problem was solved by intrinsic phasing. The structure model was refined on F2 using the SHELXL program [36]. Non-hydrogen atoms were refined anisotropically and all hydrogen atoms isotropically. Hydrogen atoms on the aromatic rings were included in idealized positions. Crystal data are listed in Table S1 in Supplementary Material. The structure figures were drawn using DIAMOND [37].
Existence of least energy nodal solutions for a double-phase problem with nonlocal nonlinearity
Published in Applicable Analysis, 2023
In contrast to (5), the double-phase problem (1) is much more complicated. To be more precise, the difficulties involve two aspects. On one hand, the behavior of the double-phase operator depends on the values of the function . In fact, on the set , the operator is controlled by the p-Laplacian, and in the case , the leading term is the weighted q-Laplacian. The nonstandard growth of the two quasilinear differential operators prevents the usual Sobolev space from being applied, and we will work on the Musielak-Orlicz Sobolev space (see its definition in Section 2). On the other hand, the appearance of the nonlocal nonlinearity in (1) makes the techniques to get nodal solutions by the Nehari manifold method as in Refs. [12, 13, 15, 16] not to be directly applicable. In fact, based on the direct observation, denoting I as the corresponding functional, the following decompositions play an important role in their approach. Here with , . However, in our case, these decompositions no longer hold true. Define the energy functional corresponding to (1) by By direct computations, we get This shows and , which implies that (6) does not hold.
A simplified coupling model of carbonation and chloride ingress based on Stefan-like condition
Published in European Journal of Environmental and Civil Engineering, 2022
Xingji Zhu, Xuerui Dai, Lu Liu, Wenfeng Bian, Longjun Xu, Zhaozheng Meng
For the mathematical modelling of boundary movement in the freezing-melting problem of ice, Stefan proposed an equation for calculating the critical state of phase transition in 1891. That is named as Stefan State Problem in thermodynamics. And in the 1930s, some scholars extended the Stefan condition to the two-phase problem.