Search results for "equation"
showing 10 items of 4219 documents
Analytical solution for multisingular vortex Gaussian beams: The mathematical theory of scattering modes
2016
We present a novel procedure to solve the Schr\"odinger equation, which in optics is the paraxial wave equation, with an initial multisingular vortex Gaussian beam. This initial condition has a number of singularities in a plane transversal to propagation embedded in a Gaussian beam. We use the scattering modes, which are solutions of the paraxial wave equation that can be combined straightforwardly to express the initial condition and therefore permit to solve the problem. To construct the scattering modes one needs to obtain a particular set of polynomials, which play an analogous role than Laguerre polynomials for Laguerre-Gaussian modes. We demonstrate here the recurrence relations need…
Accurate expansion of cylindrical paraxial waves for its straightforward implementation in electromagnetic scattering
2017
Abstract The evaluation of vector wave fields can be accurately performed by means of diffraction integrals, differential equations and also series expansions. In this paper, a Bessel series expansion which basis relies on the exact solution of the Helmholtz equation in cylindrical coordinates is theoretically developed for the straightforward yet accurate description of low-numerical-aperture focal waves. The validity of this approach is confirmed by explicit application to Gaussian beams and apertured focused fields in the paraxial regime. Finally we discuss how our procedure can be favorably implemented in scattering problems.
Post-spinel transformations and equation of state inZnGa2O4: Determination at high pressure byin situx-ray diffraction
2009
Room-temperature angle-dispersive x-ray diffraction measurements on spinel ZnGa{sub 2}O{sub 4} up to 56 GPa show evidence of two structural phase transformations. At 31.2 GPa, ZnGa{sub 2}O{sub 4} undergoes a transition from the cubic spinel structure to a tetragonal spinel structure similar to that of ZnMn{sub 2}O{sub 4}. At 55 GPa, a second transition to the orthorhombic marokite structure (CaMn{sub 2}O{sub 4}-type) takes place. The equation of state of cubic spinel ZnGa{sub 2}O{sub 4} is determined: V{sub 0} = 580.1(9) {angstrom}{sup 3}, B{sub 0} = 233(8) GPa, B'{sub 0} = 8.3(4), and B''{sub 0} = -0.1145 GPa{sup -1} (implied value); showing that ZnGa{sub 2}O{sub 4} is one of the less comp…
Structural and vibrational study ofZn(IO3)2combining high-pressure experiments and density-functional theory
2021
We report a characterization of the high-pressure behavior of zinc iodate, $\mathrm{Zn}{(\mathrm{I}{\mathrm{O}}_{3})}_{2}$. By the combination of x-ray diffraction, Raman spectroscopy, and first-principles calculations we have found evidence of two subtle isosymmetric structural phase transitions. We present arguments relating these transitions to a nonlinear behavior of phonons and changes induced by pressure on the coordination sphere of the iodine atoms. This fact is explained as a consequence of the formation of metavalent bonding at high pressure which is favored by the lone-electron pairs of iodine. In addition, the pressure dependence of unit-cell parameters, volume, and bond distanc…
Crossing the boundary between face-centred cubic and hexagonal close packed: the structure of nanosized cobalt is unraveled by a model accounting for…
2014
The properties of nanostructured cobalt in the fields of magnetic, catalytic and biomaterials depend critically on Co close packing. This paper reports a structural analysis of nanosized cobalt based on the whole X-ray diffraction (XRD) pattern simulation allowed by the Debye equation. The underlying structural model involves statistical sequences of cobalt layers and produces simulated XRD powder patterns bearing the concurrent signatures of hexagonal and cubic close packing (h.c.p. and f.c.c.). Shape, size distribution and distance distribution between pairs of atoms are also modelled. The simulation algorithm allows straightforward fitting to experimental data and hence the quantitative …
Facile fabrication of flower like self-assembled mesoporous hierarchical microarchitectures of In(OH)3 and In2O3: In(OH)3 micro flowers with electron…
2016
Abstract A template and capping-reagent free facile fabrication method for mesoporous hierarchical microarchitectures of flower-like In(OH) 3 particles under benign hydrothermal conditions is reported. Calcination of In(OH) 3 to In 2 O 3 with the retention of morphology is also described. Both In(OH) 3 and In 2 O 3 microstructures were analyzed with SEM, EDX, TEM and powder X-ray diffraction. The crystal sizes for In(OH) 3 and In 2 O 3 were calculated using the Scherrer equation. In In(OH) 3 the thin flakes at the periphery of micro flowers were electron beam sensitive. The mechanism of self-assembly process was analyzed as well.
LiCrO2 Under Pressure: In-Situ Structural and Vibrational Studies
2018
The high-pressure behaviour of LiCrO2, a compound isostructural to the battery compound LiCoO2, has been investigated by synchrotron-based angle-dispersive X-ray powder diffraction, Raman spectroscopy, and resistance measurements up to 41, 30, and 10 Gpa, respectively. The stability of the layered structured compound on a triangular lattice with R-3m space group is confirmed in all three measurements up to the highest pressure reached. The dependence of lattice parameters and unit-cell volume with pressure has been determined from the structural refinements of X-ray diffraction patterns that are used to extract the axial compressibilities and bulk modulus by means of Birch&ndash
Equation of state and high-pressure/high-temperature phase diagram of magnesium
2014
The phase diagram of magnesium has been investigated to 211 GPa at 300 K, and to 105 GPa at 4500 K, by using a combination of x-ray diffraction and resistive and laser heating. The ambient pressure hcp structure is found to start transforming to the bcc structure at ∼45 GPa, with a large region of phase-coexistence that becomes smaller at higher temperatures. The bcc phase is stable to the highest pressures reached. The hcp-bcc phase boundary has been studied on both compression and decompression, and its slope is found to be negative and steeper than calculations have previously predicted. The laser-heating studies extend the melting curve of magnesium to 105 GPa and suggest that, at the h…
Melting curve and phase diagram of vanadium under high-pressure and high-temperature conditions
2019
Melting curve and phase diagram of vanadium under high-pressure and high-temperature conditions We report a combined experimental and theoretical study of the melting curve and the structural behavior of vanadium under extreme pressure and temperature. We performed powder x-ray-diffraction experiments up to 120 GPa and 4000 K, determining the phase boundary of the body-centered cubic-to-rhombohedral transition and melting temperatures at different pressures. Melting temperatures have also been established from the observation of temperature plateaus during laser heating, and the results from the density-functional theory calculations. Results obtained from our experiments and calculations a…
Experimental evidence for pressure-driven isostructural and symmetry-breaking phase transitions on Bi14CrO24
2014
Abstract We performed in situ X-ray diffraction experiments on Bi 14 CrO 24 under pressure up to 17.4 GPa. We discovered two reversible phase transitions that take place at 7.3 and 12.1 GPa. The first transition is isostructural and the second-transition involves a tetragonal-monoclinic symmetry reduction. Both transitions involve a small volume collapse indicating that they have a first-order character. For the three observed phases we determined a P – V equation of state. All of them are highly compressible having bulk moduli that range from 64 to 70 GPa. We also determined the axial compressibilities for difference phases, being the response of the structures to pressure anisotropic. Thi…