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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
In X-ray diffraction, the structure factor is defined as the ratio of the amplitude of X-rays scattered by a unit cell to that by an electron in the Bragg condition, or F=AcellAe
Sophisticated Analytical Techniques for Investigation of Building Materials
Published in A. Bahurudeen, P.V.P. Moorthi, Testing of Construction Materials, 2020
When the X-rays diffract a crystal based on the arrangement of atoms in the crystal, two or more rays can interfere. Interference is the result of the superposition of two or more waves. Superposition results in either a constructive interference or destructive interference based on the amplitude of the diffracted waves under consideration. When constructive interference takes place, then the intensity of the superposition will be the maximum and vice versa. Bragg’s law forms the basis for indexing X-ray diffraction pattern of any specimen under investigation. Therefore, for any crystal structure based on the arrangement of the atoms in the inter-plane, the diffraction pattern will vary, resulting in a constructive as well as destructive interference. This is given by a factor called a structure factor, F, which describes the effect of crystal structure on the intensity of the diffracted beam.
Theory and Simulation for Dynamics of Polymerization-Induced Phase Separation in Reactive Polymer Blends
Published in Boris A. Rozenberg, Grigori M. Sigalov, Marina Z. Aldoshina, Yurii B. Scheck, Heterophase Network Polymers, 2020
Thein Kyu, Hao-Wen Chiu, Jae-Hyung Lee
Figure 4 shows the temporal evolution of the corresponding scattering patterns obtained by Fourier transforming the domain structures (patterns) of Fig. 2. The structure factor initially shows a diffused scattering pattern without a clear maximum, suggestive of a heterogeneous nucleation process (e.g. see t = 100). Later, it transforms into a scattering ring, while the diameter increases with progressive polymerization (t = 1500). The increase in diameter of the scattering ring at t = 2000 may be attributed to the formation of newer fluctuations as opposed to the Ostwald ripening observed in some thermal quenched systems. Another possibility is that the difference between the coexistence point and the reaction temperature (i.e., supercooling) becomes larger due to the progressive shift of the UCST to a higher temperature (or the LCST to a lower temperature) by virtue of increasing molecular weight. The PIPS tends to afford smaller domain sizes because the larger the supercooling the smaller the domain size, i.e., ξ ∝ 1 / ΔT (Figure 1). The increase of the intensity (structure factor) may be caused by the increasing number of fluctuations (scattering centers) as well as by the increase in the magnitude of the fluctuations (scattering contrast).
Molecular Physics Early Career Researcher Prize 2021 winner’s profile
Published in Molecular Physics, 2023
In our work, we combined advanced molecular dynamics (MD) simulations and machine learning (ML) techniques to study solid-liquid phase transitions of a molecular system, carbon dioxide (CO2), and the table salt, sodium chloride (NaCl). In the solid phase (molecular or salt crystal), the molecules or ions are arranged in a specific pattern giving rise to a beautifully ordered structure whereas, in the liquid (molten) state, they are disordered. A fundamental property of a crystal, called the structure factor, can describe the arrangements of atoms and molecules in the crystalline phase and distinguish it well from the disordered liquid phase. For a crystal system, one gets several high-intensity peaks in the structure factor profile. These peaks correspond to the arrangement of atoms along specific crystal lattice directions. In our work, we have taken a large number of structure factor peaks and trained a neural network (NN) that learns the molecular arrangements in the solid and the liquid phase, and after successful training, the NN (used in dimensionality reduction) provides a one-dimensional order parameter (collective variable) which can identify a crystalline phase and differentiate it from the disordered liquid state. Subsequently, we used this NN collective variable in the on-the-fly probability-based enhanced sampling (OPES) method (the latest enhanced sampling method) which samples the solid-liquid phase transitions (see Figures 2 and 3 of our Mol. Phys. 2021 article).
Stealth and equiluminous materials for scattering cancellation and wave diffusion
Published in Waves in Random and Complex Media, 2021
S. Kuznetsova, J. P. Groby, L. M. Garcia-Raffi, V. Romero-García
This discrete system can be characterized by the following scalar function defined in the spatial (direct) domain Ω as where * is the convolution operator, is the Dirac's delta and is the transparency of the scatterer, defined without loss of generality as Under these assumptions, the amplitude of the scattered wave is proportional to the spatial Fourier transform of , , where is a vector of the reciprocal space. This follows from the well-known theory in optics that the diffraction pattern of a structure is equal to the product of the diffraction pattern of the base element and that of the array [64]. Through this work, we assume a time harmonic dependence of the type where ω the angular frequency. With this, we simply end with Therefore, the scattered intensity is given by where is known as the atomic structure factor and only depends on the geometry of the scatterer as our scatterers are considered rigid. Thus, the scattered intensity can be simply written as where is the structure factor. It should be noted here that the structure factor only depends on the position of the scatterers in Ω. Moreover, we notice that (see Appendix 2 for more details and additional demonstrations). The structure factor is extensively used in condensed matter or wave physics to describe the scattering of an incident wave by a given structure made of a distribution of scatterers. However, multiple scattering effects are neglected, although this approach has the benefit of allowing fast predictions.