Relevant bibliographies by topics / Bragg equation (Bragg Law)

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Academic literature on the topic 'Bragg equation (Bragg Law)'

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Published: 24 April 2022

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Journal articles on the topic "Bragg equation (Bragg Law)"

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Camacho de la Rosa, Angela, David Becerril, María Guadalupe Gómez-Farfán, and Raúl Esquivel-Sirvent. "Bragg Mirrors for Thermal Waves." Energies 14, no. 22 (November 9, 2021): 7452. http://dx.doi.org/10.3390/en14227452.

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We present a numerical calculation of the heat transport in a Bragg mirror configuration made of materials that do not obey Fourier’s law of heat conduction. The Bragg mirror is made of materials that are described by the Cattaneo-Vernotte equation. By analyzing the Cattaneo-Vernotte equation’s solutions, we define the thermal wave surface impedance to design highly reflective thermal Bragg mirrors. Even for mirrors with a few layers, very high reflectance is achieved (>90%). The Bragg mirror configuration is also a system that makes evident the wave-like nature of the solution of the Cattaneo-Vernotte equation by showing frequency pass-bands that are absent if the materials obey the usual Fourier’s law.
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Zayed, Elsayed M. E., Mohamed E. M. Alngar, Anjan Biswas, Mehmet Ekici, Padmaja Guggilla, Salam Khan, Hashim Mohammad Alshehri, and Milivoj R. Belic. "Optical Solutions in Fiber Bragg Gratings with Polynomial Law Nonlinearity and Cubic-Quartic Dispersive Reflectivity-=SUP=-*-=/SUP=-." Оптика и спектроскопия 129, no. 11 (2021): 1409. http://dx.doi.org/10.21883/os.2021.11.51648.1016-21.

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Optical solitons with ber Bragg gratings and polynomial law of nonlinear refractive index are addressed in the paper. The auxiliary equation approach together with an addendum to Kudryashov's method identify soliton solutions to the model. Singular periodic solutions emerge from these integration schemes as a byproduct. Keywords: solitons; cubic-quartic; Bragg gratings.
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Zayed, Elsayed M. E., Mohamed E. M. Alngar, Anjan Biswas, Mehmet Ekici, Abdullah Khamis Alzahrani, and Milivoj R. Belic. "Solitons in fiber Bragg gratings with cubic–quartic dispersive reflectivity having Kerr law of nonlinear refractive index." Journal of Nonlinear Optical Physics & Materials 29, no. 03n04 (September 2020): 2050011. http://dx.doi.org/10.1142/s0218863520500113.

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This paper retrieves soliton solutions to fiber Bragg ratings with dispersive reflectivity where cubic–quartic dispersive effects are considered as opposed to the usual chromatic dispersion. The auxiliary equation approach and an addendum to Kudryashov’s scheme display a complete spectrum of soliton forms to the model that is studied with Kerr effect.
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Tseng, I.-Fan, Chi-Shian You, and Chia-Cheng Tsai. "Bragg Reflections of Oblique Water Waves by Periodic Surface-Piercing and Submerged Breakwaters." Journal of Marine Science and Engineering 8, no. 7 (July 16, 2020): 522. http://dx.doi.org/10.3390/jmse8070522.

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The Bragg reflections of oblique water waves by periodic surface-piercing structures over periodic bottoms are investigated using the eigenfunction matching method (EMM). Based on the assumption of small wave amplitude, the linear wave theory is employed in the solution procedure. In the step approximation, the surface-piercing structures and the bottom profiles are sliced into shelves separated by abrupt steps. For each shelf, the solution is composed of eigenfunctions with unknown coefficients representing the wave amplitudes. Upon applying the conservations of mass and momentum, a system of linear equations is obtained and is then solved by a sparse-matrix solver. The proposed EMM is validated by several examples in the literature. Then, the method is applied to solve Bragg reflections of oblique water waves by various surface-piercing structures over periodic bottoms. From the numerical experiments, Bragg’s law of oblique waves was used to predict the occurrences of Bragg resonance.
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Smirnov, Yury G., Eugenii Yu Smol’kin, and Dmitry V. Valovik. "Nonlinear Double-Layer Bragg Waveguide: Analytical and Numerical Approaches to Investigate Waveguiding Problem." Advances in Numerical Analysis 2014 (January 22, 2014): 1–11. http://dx.doi.org/10.1155/2014/231498.

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The paper is concerned with propagation of surface TE waves in a circular nonhomogeneous two-layered dielectric waveguide filled with nonlinear medium. The problem is reduced to the analysis of a nonlinear integral equation with a kernel in the form of the Green function. The existence of propagating TE waves for chosen nonlinearity (the Kerr law) is proved using the contraction mapping method. Conditions under which k waves can propagate are obtained, and intervals of localization of the corresponding propagation constants are found. For numerical solution of the problem, a method based on solving an auxiliary Cauchy problem (the shooting method) is proposed. In numerical experiment, two types of nonlinearities are considered and compared: the Kerr nonlinearity and nonlinearity with saturation. New propagation regime is found.
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Ogilvie, Robert E. "Is there a “universal” MAC equation?" Proceedings, annual meeting, Electron Microscopy Society of America 48, no. 2 (August 12, 1990): 228–29. http://dx.doi.org/10.1017/s0424820100134740.

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The search for an empirical absorption equation begins with the work of Siegbahn (1) in 1914. At that time Siegbahn showed that the value of (μ/ρ) for a given element could be expressed as a function of the wavelength (λ) of the x-ray photon by the following equationwhere C is a constant for a given material, which will have sudden jumps in value at critial absorption limits. Siegbahn found that n varied from 2.66 to 2.71 for various solids, and from 2.66 to 2.94 for various gases.Bragg and Pierce (2) , at this same time period, showed that their results on materials ranging from Al(13) to Au(79) could be represented by the followingwhere μa is the atomic absorption coefficient, Z the atomic number. Today equation (2) is known as the “Bragg-Pierce” Law. The exponent of 5/2(n) was questioned by many investigators, and that n should be closer to 3. The work of Wingardh (3) showed that the exponent of Z should be much lower, p = 2.95, however, this is much lower than that found by most investigators.
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Leonardi, Alberto. "Whole pair distribution function modeling: the bridging of Bragg and Debye scattering theories." IUCrJ 8, no. 2 (February 10, 2021): 257–69. http://dx.doi.org/10.1107/s2052252521000324.

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Microstructure-based design of materials requires an atomic level understanding of the mechanisms underlying structure-dependent properties. Methods for analyzing either the traditional diffraction profile or the pair distribution function (PDF) differ in how the information is accessed and in the approximations usually applied. Any variation of structural and microstructural features over the whole sample affects the Bragg peaks as well as any diffuse scattering. Accuracy of characterization relies, therefore, on the reliability of the analysis methods. Methods based on Bragg's law investigate the diffraction peaks in the intensity plot as distinct pieces of information. This approach reaches a limitation when dealing with disorder scenarios that do not conform to such a peak-by-peak basis. Methods based on the Debye scattering equation (DSE) are, otherwise, well suited to evaluate the scattering from a disordered phase but the structure information is averaged over short-range distances usually accessed by experiments. Moreover, statistical reliability is usually sacrificed to recover some of the computing-efficiency loss compared with traditional line-profile-analysis methods. Here, models based on Bragg's law are used to facilitate the computation of a whole PDF and then model powder-scattering data via the DSE. Models based on Bragg's law allow the efficient solution of the dispersion of a crystal's properties in a powder sample with statistical reliability, and the PDF provides the flexibility of the DSE. The whole PDF is decomposed into the independent directional components, and the number of atom pairs separated by a given distance is statistically estimated using the common-volume functions. This approach overcomes the need for an atomistic model of the material sample and the computation of billions of pair distances. The results of this combined method are in agreement with the explicit solution of the DSE although the computing efficiency is comparable with that of methods based on Bragg's law. Most importantly, the method exploits the strengths and different sensitivities of the Bragg and Debye theories.
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LINTON, C. M. "Water waves over arrays of horizontal cylinders: band gaps and Bragg resonance." Journal of Fluid Mechanics 670 (January 25, 2011): 504–26. http://dx.doi.org/10.1017/s0022112010005471.

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The existence of a band-gap structure associated with water waves propagating over infinite periodic arrays of submerged horizontal circular cylinders in deep water is established. Waves propagating at right angles to the cylinder axes and at an oblique angle are both considered. In each case an exact linear analysis is presented with numerical results obtained by solving truncated systems of equations. Calculations for large finite arrays are also presented, which show the effect of an incident wave having a frequency within a band gap – with the amount of energy transmitted across the array tending to zero as the size of the array is increased. The location of the band gaps is not as predicted by Bragg's law, but we show that an approximate determination of their position can be made very simply if the phase of the transmission coefficient for a single cylinder is known.
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Hackley, Vincent A., Peter K. Stoimenov, Derek L. Ho, Li Piin Sung, and Kenneth J. Klabunde. "Structure development in aerogel-processed nanocrystalline alkaline earth oxides as revealed by SANS." Journal of Applied Crystallography 38, no. 4 (July 13, 2005): 619–31. http://dx.doi.org/10.1107/s0021889805015244.

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Nanocrystalline MgO, CaO and SrO were prepared according to a modified aerogel process (AP). Small-angle neutron scattering (SANS) was used to probe the nanoscale structural features of these materials after each stage of the synthetic process, including hydrolysis, supercritical drying and calcining. SANS data were interpreted using a classical analysis involving power-law and Guinier regimes, and by application of the maximum entropy method. Results are compared with previously published structural data based on X-ray diffraction, electron microscopy and gas adsorption. It is found that the gel hydrolysis product suspended in methanol and toluene exhibits rod-like scattering at small length scales. This is consistent with a filiform morphology previously reported for air-dried Mg(OH)2alcogel, yet SANS data for air-dried alcogels tested in this study indicate no evidence for low-dimensional structure on any length scale. A previous assertion of mass fractal structure in the AP aerogels and oxides was not confirmed by the present data. Instead, surface fractal scattering was found to be the most dominant characteristic feature associated with the SANS data for all AP powders examined. Additionally, MgO and CaO exhibited a correlation peak that corresponds to liquid-like ordering at Bragg length scales of 5.9 nm and 20.3 nm, respectively. These values are roughly consistent with previous independent estimates of primary particle size, suggesting that local packing of primary crystallites is facilitated by the calcination/dehydration process. An alternative interpretation treats these features as Guinier scattering regions. Fitting of results using the unified Guinier/power-law equation yields sphere-equivalent radii for the primary particles that are nearly identical to the Bragg lengths calculated from the positions of the maxima. Air-dried alcogels produced very weak maxima that could be interpreted either as correlation peaks or as Guinier regions. No maxima were observed for aerogel samples. Maximum entropy analysis using a spherical shape factor produced interesting but complex results for the calculated volume size distributions of these materials. Overall, the observed trend shows an increase in structural feature size with increasing metal cation size.
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Zayed, Elsayed M. E., Mohamed E. M. Alngar, Mahmoud El-Horbaty, Anjan Biswas, Ali Saleh Alshomrani, Salam Khan, Mehmet Ekici, and Houria Triki. "Optical solitons in fiber Bragg gratings having Kerr law of refractive index with extended Kudryashov’s method and new extended auxiliary equation approach." Chinese Journal of Physics 66 (August 2020): 187–205. http://dx.doi.org/10.1016/j.cjph.2020.04.003.

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Dissertations / Theses on the topic "Bragg equation (Bragg Law)"

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Marinho, Leonardo Ribeiro. "Análise Completa das Fibras de Bragg de Núcleo Oco." Universidade do Estado do Rio de Janeiro, 2013. http://www.bdtd.uerj.br/tde_busca/arquivo.php?codArquivo=8141.

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A evolução nos sistemas digitais de comunicação está intrinsicamente relacionada ao desenvolvimento da tecnologia de fibras ópticas. Desde a sua criação, na década de 60, inúmeras pesquisas vem sendo realizadas com o intuito de aumentar a capacidade de informação transmitida, por meio da redução da atenuação, controle da dispersão cromática e eliminação das não-linearidades. Neste contexto, as Fibras de Bragg surgem como uma estrutura de grande potencialidade para se minimizar tais inconvenientes. As fibras de Bragg possuem um mecanismo de operação diferente em relação às fibras tradicionais de suportar os modos confinados. Nelas, o núcleo possui um baixo índice de refração, e a casca é constituída por anéis dielétricos de diferentes índices de refração, alocados alternadamente. Para uma fibra de Bragg com núcleo oco, como a considerada neste trabalho, há perdas decorrentes dos modos de fuga. Portanto, a análise da dispersão destas estruturas se situa no plano complexo, tornando-a muito difícil. Esta dissertação será fundamentada em uma estratégia imprescindível à análise dos modos transversais TE0m, TM0m e dos híbridos. Os resultados encontrados são validados confrontando-os com os obtidos na literatura. O trabalho discutirá as perdas e dispersões dos modos citados, e os resultados obtidos poderão nortear as pesquisas das fibras de Bragg.
The evolution of digital communication systems is intrinsically related to the development of optical fiber technology. Since its creation in the 1960s, many studies have been conducted in order to increase the system capacity, such as the attenuation reduction, chromatic dispersion control and elimination of nonlinearities. In this context, Bragg fibers appear as a structure with great potential to mitigate these drawbacks. Bragg fibers have a different operational mechanism with respect to traditional fibers to support the confined modes. Their core has a low refractive index, and the cladding consists of dielectric rings of different refractive indices, allocated alternately. For a Bragg fiber with hollow core, as considered in this paper, there are losses due to the occurrence of leaky modes. Therefore, the dispersion analysis of these structures falls in the complex plane, making it even harder. This dissertation will be based on a strategy essential to the analysis of transverse modes: TE0m, TM0m and hybrids. The found results have been validated by comparing them with those obtained in the literature. The paper discusses the losses and dispersions of the mentioned modes, and the results obtained will serve to guide the research on Bragg fibers.
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Chuzeville, Vincent Pierre. "Amorçage en détonation des explosifs hétérogènes de type coulé fondu : Etablissement de corrélations entre microstructure et réactivité." Thesis, Université Paris-Saclay (ComUE), 2016. http://www.theses.fr/2016SACLY014/document.

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Ce travail de thèse porte sur les mécanismes d’amorçage en détonation par choc des explosifs solides de type coulé-fondu. Les explosifs solides sont des matériaux hétérogènes constitués de grains de matière énergétique dans un liant pouvant être lui-même énergétique. Si l’existence des points chauds, sites préférentiels d’initiation des réactions chimiques à l’échelle locale, est largement reconnue, la topologie de la croissance des réactions, et l’influence de la microstructure sur cette dernière n’est que peu étudiée dans les explosifs coulés-fondus. Deux familles d’explosifs ont été retenues pour cette étude : les hexolites, mélanges de grains d’hexogène (RDX) et d’un liant trinitrotoluène (TNT) et les ontalites, composées d’oxynitrotriazole (ONTA) et de TNT. Les recherches se sont orientées autour du triptyque : caractérisation – expérimentations – modélisation.Un important travail de compilation et de ré-exploitation de données issues de la littérature, associé à une modélisation des équations d’état des explosifs purs, ont permis de définir des lois permettant de calculer le comportement de ces derniers sous choc. Ces lois ont ensuite été validées par une méthode de mélange sur différentes compositions coulées-fondues et composites. Parallèlement, la microstructure des compositions d’étude a également été caractérisée via des mesures de granulométrie et de microtomographie, inédites sur ce type d’explosif.Des expérimentations d’impact plan soutenu ont été réalisées afin d’établir les diagrammes de marche des ondes de choc réactives, permettant de relier la profondeur de transition à la détonation à la pression de sollicitation. Elles ont permis de mettre en lumière l’influence de la microstructure sur la sensibilité au choc de deux hexolites et d’acquérir des données sur deux ontalites. L’utilisation de deux métrologies innovantes, la radio-interférométrie à 94 GHz et les fibres optiques à réseau de Bragg, a permis de mesurer la transition choc – détonation (TCD) de façon continue avec une résolution inédite. Enfin des essais d’impact plan non soutenu ont été réalisés à des fins de validation.Un modèle de TCD est proposé. Ce dernier, basé sur une approche de germination-croissance des fronts de déflagration à l’échelle locale, permet de prendre en compte la microstructure des explosifs. Ces travaux semblent mettre en évidence l’influence de la fracturation des grains d’explosif sous choc, qu’il conviendra d’étudier dans le futur. Enfin, une étape de terminaison des réactions lors de la TCD, associée à des calculs thermocinétiques détaillés, a été étudiée
This study deals with the detonation initiation by shock of condensed melt-cast high explosives. Solid explosives are heterogeneous materials, made of energetic material grains in a binder, which can be energetic itself. If the existence of hot-spots, preferred initiation sites for chemical reaction at the local scale, is widely recognized, the reaction growth topology, and the microstructure influence, are poorly known for melt-cast explosives. We study here two melt-cast explosive families: hexolites, a mix of hexogen (RDX) grains and trinitrotoluene (TNT) binder, and ontalites made of nitrotriazolone (NTO) and TNT. This study has been focused on the triptyque: characterization - experimentations - modeling.An important work of compilation and re-exploitation of literature data, combined with pure explosives’ equation of state modeling, allowed us to define laws to calculate the explosives’ comportment under a shock solicitation. These ones have been validated, thanks to a mixing method, on different melt-cast and cast-curd plastic bonded explosives. At the same time, the compositions’ microstructure has been also characterized via granulometry measurements and microtomographies, never published for this type of explosive.Plate impact tests have been performed in order to establish the reactive shock trajectory of these compositions, allowing us to determine the relation between the run-distance of detonation and the input pressure. It brought the microstructure influence on hexolite shock sensitivity to light, and gave us some first results for ontalites. The use of continuous and innovative measurements, as microwave interferometry and chirped fiber Bragg gratings, allowed us to study the shock to detonation transition (SDT) with a resolution never seen before. Finally, non-sustained plate impact test have been performed for a validation purpose.A SDT model is proposed. Based on a germination-growth approach of deflagration fronts at the local scale, it takes into account the explosive’s microstructure. This work seems to show the grain fragmentation under shock influence, point we will have to study in the future. Finally, a completion step of reactions, associated with chemical kinetics calculations, has been studied
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Kéfélian, Fabien. "Corrélation du bruit de phase de lasers à réseau de Bragg par injection optique : Application à la génération et au transport sur fibre de signaux radiofréquence." Phd thesis, Télécom ParisTech, 2005. http://tel.archives-ouvertes.fr/tel-00011613.

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Le mélange de deux faisceaux laser sur un photo-détecteur permet de générer un signal radiofréquence jusqu'au THz. Par corrélation des deux sources optiques, le signal obtenu peut acquérir la pureté spectrale requise pour les réseaux de communications radio sur fibre. Notre travail porte sur la méthode de corrélation par accrochage optique sur un peigne de fréquences. L'injection optique permet de transférer le bruit de phase d'un laser maître, pris comme référence, à un laser esclave. En utilisant deux harmoniques d'un laser modulé en fréquence comme sources distinctes d'injection, les bruits de phase des deux lasers esclaves sont corrélés et la différence de fréquences est multiple de la fréquence primaire. Nous avons réalisé une étude théorique générale de l'injection dans les lasers semi-conducteur à cavité complexe, en particulier les lasers DFB, en mettant notamment en évidence l'asymétrie géométrique du bruit. Nous avons relié théoriquement le degré de corrélation entre les deux lasers aux paramètres d'injection et au bruit de phase. L'expression a été confirmée par des mesures sur le contraste de franges d'interférences et le spectre du photo-courant hétérodyne. Ces battements temporels ont été mis en regard avec l'optique de Fourier et le speckle. Nous avons étudié la pureté spectrale du battement et établi les limites fondamentales de cette technique en fonction de la qualité de l'oscillateur primaire, des propriétés spectrales des lasers, des paramètres d'injection et de transport sur fibre. Les mesures de bruit de phase sur le signal généré expérimentalement, pour différentes conditions d'injection, sont en très bon accord avec les expressions analytiques.
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Goodman, Steven John. "Resonances of scattering in non-uniform and anisotropic periodic gratings at extreme angles." Queensland University of Technology, 2006. http://eprints.qut.edu.au/16429/.

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Bragg scattering of optical waves in thick gratings at extreme angles, where the scattered wave propagates parallel (extremely asymmetric scattering - EAS) or nearly parallel (grazing angle scattering - GAS) to the grating boundaries, is associated with many unique and practically important resonant phenomena. It has been demonstrated that one of the main physical mechanisms for these resonant phenomena is the diffractional divergence of the scattered wave inside and outside the grating region. This thesis fills the gaps in the theoretical and experimental understanding of Bragg scattering in gratings at extreme angles by investigating EAS and GAS in structures where diffractional divergence of waves is significantly affected by anisotropy and/or non-uniformities of the dielectric permittivity. Unusually high sensitivity of wave scattering in thick periodic gratings to small step-like variations of mean structural parameters at the grating boundaries is predicted and described for the case when the scattered wave (the +1 diffracted order) propagates almost parallel to the front grating boundary (the geometry of GAS). A unusual pattern of strong multiple resonances for bulk electromagnetic waves is predicted and analysed numerically in thick periodic holographic gratings in a guiding slab with mean permittivity that is greater than that of the surrounding media. It is demonstrated that these resonances are related to resonant generation of a new type of eigenmodes in a thick slab with a periodic grating. These eigenmodes are generically related to the grating -- they do exist not if the grating amplitude is zero. A new type of resonant coupling of bulk radiation into the conventional guided modes of a slab with a thick holographic grating is predicted and explained theoretically. It occurs in the presence of strong frequency detunings of the Bragg condition by means of interaction of the strongly non-eigen +1 diffracted order with the slab-grating boundaries. Therefore, it is only in the presence of step-like variations of the mean permittivity at the grating boundaries that this type of resonant coupling can occur. A new method for the analysis of EAS and GAS in anisotropic gratings is developed. This method is based on the consideration of the diffractional divergence of the scattered wave and the two-wave approximation in anisotropic gratings. Special efforts are focused on the analysis of EAS and GAS of extraordinary waves in uniaxial gratings. In particular, it is demonstrated that increasing curvature of the normal surface in the direction of propagation of the scattered wave results in increase of its diffraction divergence and the resonant amplitude. A theoretical model is developed for comparison of the theoretical predictions with data obtained from experimental observations of EAS in a holographic grating written in a photorefractive medium. The developed model is applied for the interpretation of experimental observations of EAS in BaTiO3 photorefractive crystals. Good agreement with the theoretical predictions is demonstrated.
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Books on the topic "Bragg equation (Bragg Law)"

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1959-, Lang S. P., and Bedore Salim H. 1961-, eds. Handbook of solitons: Research, technology, and applications. Hauppauge, NY: Nova Science Publishers, 2009.

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Solymar, L., D. Walsh, and R. R. A. Syms. The band theory of solids. Oxford University Press, 2018. http://dx.doi.org/10.1093/oso/9780198829942.003.0007.

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The solution of Schrodinger’s equation is discussed for a model in which atoms are represented by potential wells, from which the band structure follows. Three further models are discussed, the Ziman model (which is based on the effect of Bragg reflection upon the wave functions), and the Feynman model (based on coupled equations), and the tight binding model (based on a more realistic solution of the Schrödinger equation). The concept of effective mass is introduced, followed by the effective number of electrons. The difference between metals and insulators based on their band structure is discussed. The concept of holes is introduced. The band structure of divalent metals is explained. For finite temperatures the Fermi–Dirac function is combined with band theory whence the distinction between insulators and semiconductors is derived.
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Book chapters on the topic "Bragg equation (Bragg Law)"

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"Bragg Diffraction Equation or Bragg’s Law." In Encyclopedia of Microfluidics and Nanofluidics, 213. New York, NY: Springer New York, 2015. http://dx.doi.org/10.1007/978-1-4614-5491-5_200339.

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Cantor, Brian. "Bragg’s Law." In The Equations of Materials, 24–44. Oxford University Press, 2020. http://dx.doi.org/10.1093/oso/9780198851875.003.0002.

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The diffraction of X-rays is used as the main method for determining the atomic and molecular structures of inorganic and biological materials. The basic law of diffraction was discovered by Lawrence Bragg when he was a student at Cambridge University and he was just 22 years old. Bragg’s law explains how the angle of a diffracted X-ray beam varies with the wavelength of the X-rays and the spacing of the atoms and molecules in the material. This chapter examines the way X-rays are generated and scattered by electrons, atoms and crystals; the use of structure factors and Fourier transforms to calculate the intensity of the scattered X-rays; and the effect of using electrons or neutrons instead of X-rays. Bragg was born and brought up in Adelaide in Australia. He discovered Bragg’s law with the help of his father, William, after they had moved to England. Lawrence was a Professor at Manchester University, Cambridge University, and the Royal Institution; contributed to the development of range-finding, asdic, and sonar during the First and Second World Wars; and supervised Crick and Watson when they discovered the structure of DNA.
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Boothroyd, Andrew T. "Diffraction in the Static Approximation." In Principles of Neutron Scattering from Condensed Matter, 31–72. Oxford University Press, 2020. http://dx.doi.org/10.1093/oso/9780198862314.003.0002.

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The basic principles of crystallography are reviewed, including the lattice, basis and reciprocal lattice. The Bragg diffraction law and Laue equation, which describe coherent scattering from a crystalline material, are derived, and the structure factor and differential cross-section are obtained in the static approximation. It is explained how the presence of defects, short-range order, and reduced dimensionality causes diffuse scattering. For non-crystalline materials, such as liquids and glasses, the pair distribution function and density-density correlation function are introduced, and their relation to the static structure factor established. For molecular fluids, the form factor is defined and calculated for a diatomic molecule, and the separation of intra- and inter-molecular scattering is discussed. The principles of small-angle neutron scattering are described.
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Krishnan, Kannan M. "X-Ray Diffraction." In Principles of Materials Characterization and Metrology, 408–80. Oxford University Press, 2021. http://dx.doi.org/10.1093/oso/9780198830252.003.0007.

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X-rays diffraction is fundamental to understanding the structure and crystallography of biological, geological, or technological materials. X-rays scatter predominantly by the electrons in solids, and have an elastic (coherent, Thompson) and an inelastic (incoherent, Compton) component. The atomic scattering factor is largest (= Z) for forward scattering, and decreases with increasing scattering angle and decreasing wavelength. The amplitude of the diffracted wave is the structure factor, F hkl, and its square gives the intensity. In practice, intensities are modified by temperature (Debye-Waller), absorption, Lorentz-polarization, and the multiplicity of the lattice planes involved in diffraction. Diffraction patterns reflect the symmetry (point group) of the crystal; however, they are centrosymmetric (Friedel law) even if the crystal is not. Systematic absences of reflections in diffraction result from glide planes and screw axes. In polycrystalline materials, the diffracted beam is affected by the lattice strain or grain size (Scherrer equation). Diffraction conditions (Bragg Law) for a given lattice spacing can be satisfied by varying θ or λ — for study of single crystals θ is fixed and λ is varied (Laue), or λ is fixed and θ varied to study powders (Debye-Scherrer), polycrystalline materials (diffractometry), and thin films (reflectivity). X-ray diffraction is widely applied.
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Krishnan, Kannan M. "Crystallography and Diffraction." In Principles of Materials Characterization and Metrology, 220–76. Oxford University Press, 2021. http://dx.doi.org/10.1093/oso/9780198830252.003.0004.

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Crystalline materials have a periodic arrangement of atoms, exhibit long range order, and are described in terms of 14 Bravais lattices, 7 crystal systems, 32 point groups, and 230 space groups, as tabulated in the International Tables for Crystallography. We introduce the nomenclature to describe various features of crystalline materials, and the practically useful concepts of interplanar spacing and zonal equations for interpreting electron diffraction patterns. A crystal is also described as the sum of a lattice and a basis. Practical materials harbor point, line, and planar defects, and their identification and enumeration are important in characterization, for defects significantly affect materials properties. The reciprocal lattice, with a fixed and well-defined relationship to the real lattice from which it is derived, is the key to understanding diffraction. Diffraction is described by Bragg law in real space, and the equivalent Ewald sphere construction and the Laue condition in reciprocal space. Crystallography and diffraction are closely related, as diffraction provides the best methodology to reveal the structure of crystals. The observations of quasi-crystalline materials with five-fold rotational symmetry, inconsistent with lattice translations, has resulted in redefining a crystalline material as “any solid having an essentially discrete diffraction pattern”
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Blow, David. "Diffraction by crystals." In Outline of Crystallography for Biologists. Oxford University Press, 2002. http://dx.doi.org/10.1093/oso/9780198510512.003.0009.

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In Chapter 4 many two-dimensional examples were shown, in which a diffraction pattern represents the Fourier transform of the scattering object. When a diffracting object is three-dimensional, a new effect arises. In diffraction by a repetitive object, rays are scattered in many directions. Each unit of the lattice scatters, but a diffracted beam arises only if the scattered rays from each unit are all in phase. Otherwise the scattering from one unit is cancelled out by another. In two dimensions, there is always a direction where the scattered rays are in phase for any order of diffraction (just as shown for a one-dimensional scatterer in Fig. 4.1). In three dimensions, it is only possible for all the points of a lattice to scatter in phase if the crystal is correctly oriented in the incident beam. The amplitudes and phases of all the scattered beams from a three-dimensional crystal still provide the Fourier transform of the three-dimensional structure. But when a crystal is at a particular angular orientation to the X-ray beam, the scattering of a monochromatic beam provides only a tiny sample of the total Fourier transform of its structure. In the next section, we are going to find what is needed to allow a diffracted beam to be generated. We shall follow a treatment invented by Lawrence Bragg in 1913. Max von Laue, who discovered X-ray diffraction in 1912, used a different scheme of analysis; and Paul Ewald introduced a new way of looking at it in 1921. These three methods are referred to as the Laue equations, Bragg’s law and the Ewald construction, and they give identical results. All three are described in many crystallographic text books. Bragg’s method is straightforward, understandable, and suffices for present needs. I had heard J.J. Thomson lecture about…X-rays as very short pulses of radiation. I worked out that such pulses…should be reflected at any angle of incidence by the sheets of atoms in the crystal as if these sheets were mirrors.…It remained to explain why certain of the atomic mirrors in the zinc blende [ZnS] crystal reflected more powerfully than others.
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Glusker, Jenny Pickworth, and Kenneth N. Trueblood. "Diffraction." In Crystal Structure Analysis. Oxford University Press, 2010. http://dx.doi.org/10.1093/oso/9780199576340.003.0011.

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A common approach to crystal structure analysis by X-ray diffraction presented in texts that have been written for nonspecialists involves the Bragg equation, and a discussion in terms of “reflection” of X rays from crystal lattice planes (Bragg, 1913). While the Bragg equation, which implies this “reflection,” has proved extremely useful, it does not really help in understanding the process of X-ray diffraction. Therefore we will proceed instead by way of an elementary consideration of diffraction phenomena generally, and then diffraction from periodic structures (such as crystals), making use of optical analogies (Jenkins and White, 1957; Taylor and Lipson, 1964; Harburn et al., 1975). The eyes of most animals, including humans, comprise efficient optical systems for forming images of objects by the recombination of visible radiation scattered by these objects. Many things are, of course, too small to be detected by the unaided human eye, but an enlarged image of some of them can be formed with a microscope—using visible light for objects with dimensions comparable to or larger than the wavelength of this light (about 6 × 10−7 m), or using electrons of high energy (and thus short wavelength) in an electron microscope. In order to “see” the fine details of molecular structure (with dimensions 10−8 to 10−10 m), it is necessary to use radiation of a wavelength comparable to, or smaller than, the dimensions of the distances between atoms. Such radiation is readily available (1) in the X rays produced by bombarding a target composed of an element of intermediate atomic number (for example, between Cr and Mo in the Periodic Table) with fast electrons, or from a synchrotron source, (2) in neutrons from a nuclear reactor or spallation source, or (3) in electrons with energies of 10–50 keV. Each of these kinds of radiation is scattered by the atoms of the sample, just as is ordinary light, and if we could recombine this scattered radiation, as a microscope can, we could form an image of the scattering matter.
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Meurig Thomas, John. "The Birth and Initial Exploitation of X-ray Diffraction." In Architects of Structural Biology, 17–40. Oxford University Press, 2020. http://dx.doi.org/10.1093/oso/9780198854500.003.0002.

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A non-mathematical account of the discovery of X-ray diffraction by von Laue and its use as a new kind of high-resolution microscopy by W. L. Bragg is given. There follows a simple explanation of how the electron densities in various regions of any molecule that can be crystallized can be retrieved from its X-ray diffraction pattern. Also, it is explained how the molecular weight of the molecule can be determined from straightforward measurements of the diffraction and the density of the crystal. The identity of the elements in a crystal, as well as the nature of the chemical bonding between them, may also be derived from measurement of the electron density distribution within it. The importance of Bragg’s Law, relating X-ray pattern to interatomic distance, is demonstrated, and initial applications of it by Bragg and Pauling are given.
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Chukhovskii, F. N. "Exact Solution of the Takagi-Taupin Equation for Dynamical X-Ray Bragg Diffraction by a Crystal with a Transition Layer." In May 16, 69–76. De Gruyter, 1985. http://dx.doi.org/10.1515/9783112495124-008.

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Glusker, Jenny Pickworth, and Kenneth N. Trueblood. "The phase problem and electron-density maps." In Crystal Structure Analysis. Oxford University Press, 2010. http://dx.doi.org/10.1093/oso/9780199576340.003.0015.

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In order to obtain an image of the material that has scattered X rays and given a diffraction pattern, which is the aim of these studies, one must perform a three-dimensional Fourier summation. The theorem of Jean Baptiste Joseph Fourier, a French mathematician and physicist, states that a continuous, periodic function can be represented by the summation of cosine and sine terms (Fourier, 1822). Such a set of terms, described as a Fourier series, can be used in diffraction analysis because the electron density in a crystal is a periodic distribution of scattering matter formed by the regular packing of approximately identical unit cells. The Fourier series that is used provides an equation that describes the electron density in the crystal under study. Each atom contains electrons; the higher its atomic number the greater the number of electrons in its nucleus, and therefore the higher its peak in an electrondensity map.We showed in Chapter 5 how a structure factor amplitude, |F (hkl)|, the measurable quantity in the X-ray diffraction pattern, can be determined if the arrangement of atoms in the crystal structure is known (Sommerfeld, 1921). Now we will show how we can calculate the electron density in a crystal structure if data on the structure factors, including their relative phase angles, are available. The Fourier series is described as a “synthesis” when it involves structure amplitudes and relative phases and builds up a picture of the electron density in the crystal. By contrast, a “Fourier analysis” leads to the components that make up this series. The term “relative” is used here because the phase of a Bragg reflection is described relative to that of an imaginary wave diffracted in the same direction at a chosen origin of the unit cell.
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Conference papers on the topic "Bragg equation (Bragg Law)"

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Freire, J. L. F., V. E. L. Paiva, G. L. G. Gonzáles, R. D. Vieira, J. L. C. Diniz, J. E. Maneschy, and A. L. F. S. d’Almeida. "Fatigue Assessment of Dented Pipeline Specimens." In ASME 2020 Pressure Vessels & Piping Conference. American Society of Mechanical Engineers, 2020. http://dx.doi.org/10.1115/pvp2020-21854.

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Abstract This paper reports results from an investigation program launched with the objective of assessing fatigue lives of actual pipeline specimens with dents. Nine pipeline 3m-length specimens were constructed with low carbon steel pipes API 5L Gr. B. The specimens had 323mm diameter and 6.35mm wall thickness. The specimens were loaded with hydrostatic internal pressure pulsating at a 1Hz rate. Six specimens had 15% deep longitudinal smooth dents (ratio between dent depth and outside specimen diameter) and three specimens had complex longitudinal 6% deep dent shapes. Nominal and hot spot stresses and strains were determined by experimental techniques (Fiber Optic Bragg Strain Gages - FBSG, and Digital Image Correlation - DIC) and by a numerical technique (Finite Elements - FE). The stresses and strain fields determined from nominal loading conditions or from experimental measurements and from the finite element analyses were combined with different fatigue assessment methods. The estimated lives were compared with the actual test results. The fatigue assessment methods encompassed those proposed by the Pipeline Defect Assessment Manual (PDAM) and by the API 579-1/ASME FFS-1 Level 2 methods described in parts 12 (Dents) and 14 (Fatigue). Most of the predicted lives exhibited high level of conservatism. A Level 3 method that employed experimentally and numerically determined hot-spot strains in conjunction with a fatigue strain-life equation proposed by Coffin-Manson predicted fatigue lives very close to the test results.
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Mayonado, Gina, Shabbir M. Mian, Valentina Robbiano, and Franco Cacialli. "Investigation Of The Bragg-Snell Law In Photonic Crystals." In 2015 Conference on Laboratory Instruction Beyond the First Year. American Association of Physics Teachers, 2015. http://dx.doi.org/10.1119/bfy.2015.pr.015.

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Mejía-Cortés, C., Rodrigo A. Vicencio, and Boris A. Malomed. "Mobility of 1D solitons in the discrete CQ Schrödinger equation." In Bragg Gratings, Photosensitivity, and Poling in Glass Waveguides. Washington, D.C.: OSA, 2014. http://dx.doi.org/10.1364/bgpp.2014.jtu3a.15.

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Campbell, R., G. L. Oppo, and M. Borkowski. "Interaction of Breathers in the Two-Component Discrete Nonlinear Schrödinger Equation." In Bragg Gratings, Photosensitivity, and Poling in Glass Waveguides. Washington, D.C.: OSA, 2014. http://dx.doi.org/10.1364/bgpp.2014.jtu3a.28.

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Soto-Crespo, J. M., N. Devine, N. P. Hoffmann, and N. Akhmediev. "Double peak rogue waves of the Sasa-Satsuma equation in a chaotic wave field." In Bragg Gratings, Photosensitivity, and Poling in Glass Waveguides. Washington, D.C.: OSA, 2014. http://dx.doi.org/10.1364/bgpp.2014.jm5a.47.

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Shcherbakov, Alexandre S., Je Maximov, E. Tepichin Rodriguez, and Sandra E. Balderas Mata. "Collinear three-wave acousto-optical coupled states in a medium with a square-law nonlinearity and losses." In Bragg Gratings, Photosensitivity, and Poling in Glass Waveguides. Washington, D.C.: OSA, 2007. http://dx.doi.org/10.1364/bgpp.2007.jmd24.

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Renninger, W. H., A. Chong, and F. W. Wise. "Dissipative Solitons in Normal-Dispersion Fiber Lasers: Exact Pulse Solutions of the Complex Ginzburg-Landau Equation." In Bragg Gratings, Photosensitivity, and Poling in Glass Waveguides. Washington, D.C.: OSA, 2007. http://dx.doi.org/10.1364/bgpp.2007.jwbpdp3.

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Kan, K. V., and N. A. Kudryashov. "Solitary waves for the sixth order nonlinear differential equation in optical fiber Bragg grating." In INTERNATIONAL CONFERENCE OF NUMERICAL ANALYSIS AND APPLIED MATHEMATICS ICNAAM 2020. AIP Publishing, 2022. http://dx.doi.org/10.1063/5.0085931.

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Shcherbakov, Alexandre S., and Arturo Aguirre Lopez. "Revealing multi-pulse four-wave Bragg spatial solitons in periodic square-law nonlinear crystal with direct transitions." In Nonlinear Guided Waves and Their Applications. Washington, D.C.: OSA, 2005. http://dx.doi.org/10.1364/nlgw.2005.wd28.

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Chatta, Rihab, Mehdi Ammar, Mourad Zghal, and Rabah Attia. "Numerical solution to modal field equation with a finite difference beam propagation method: application to Bragg fiber." In International Symposium on Optical Science and Technology, edited by Richard C. Juergens. SPIE, 2002. http://dx.doi.org/10.1117/12.481183.

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