Diffusion of monochromatic classical waves

Abstract

We study the diffusion of monochromatic classical waves in a disordered acoustic medium by scattering theory. In order to avoid artifacts associated with mathematical point scatterers, we model the randomness by small but finite insertions. We derive expressions for the configuration-averaged energy flux, energy density, and intensity for one-, two-, and three-dimensional (3D) systems with an embedded monochromatic source using the ladder approximation to the Bethe-Salpeter equation. We study the transition from ballistic to diffusive wave propagation and obtain results for the frequency dependence of the medium properties such as mean free path and diffusion coefficient as a function of the scattering parameters. We discover characteristic differences of the diffusion in 2D as compared to the conventional 3D case, such as an explicit dependence of the energy flux on the mean free path and quite different expressions for the effective transport velocity.

Enhancing Photocatalysis: Understanding the Mechanistic Diversity in Photocatalysts Modified with Single‐Atom Catalytic Sites

“…The EPR spectrum after UV irradiation could be simulated using the same components as described above (see Figure 2a ); however, the contributions of the R‐1 and Ti‐1 centers were higher, corresponding to the electrons trapped at the oxygen vacancies [ 21 , 25 ] and Ti 3+ centers, respectively. [ 24a ] Similar to the case of a‐TiO 2 (400)‐V SA , no change in the EPR signal was observed when irradiation was discontinued. However, in stark contrast to the behavior of a‐TiO 2 (400)‐V SA , after exposing the sample to dioxygen, the signal intensity was only partially recovered and reached only ≈85% of the initial intensity.…”

“…[ 5 , 22 ] The complex EPR spectrum of a‐TiO 2 ‐Cu SA ( Figure 2 a ) consists of five components, as determined by computer simulations. The magnetic parameters of the main constituent (≈85%), denoted as Cu‐1, are as follows: g || ≅ 2.353, A || ≅ 113 G, and g ⊥ ≅ 2.067, with an unresolved perpendicular hyperfine structure (HFS) typical of surface‐bound single Cu(II) ions ( d 9 electronic configuration) [ 5 , 23 ] or Cu(II) ions substituting lattice Ti(IV), [ 6 , 24 ] both expected to be spectroscopically the same. The next two components, Cu‐2 (≈13%, g || ≅ 2.354, A || ≅ 114 G and g ⊥ ≅ 2.065, A ⊥ ≅ 8 G) and Cu‐3 (∼2%, g || ≅ 2.353, A || ≅ 114 G and g ⊥ ≅ 2.050, A ⊥ ≅ 23 G), can also be undoubtedly assigned to isolated copper centers in slightly different environments.…”

“…The last two components with very low signal intensities can be assigned to the electrons trapped at the oxygen vacancies (R‐1, g = 2.002) [ 21 , 25 ] and the Ti(III) centers present in the TiO 2 matrix (Ti‐1, g || ≅ 1.924, and g ⊥ ≅ 1.945). [ 24a ] Similarly, EPR analysis of the a‐TiO 2 ‐Fe SA sample (Figure S11 , Supporting Information) revealed a main signal that is typically attributed to isolated Fe(III) ions at the surface or near the surface of TiO 2 , [ 5 , 13 , 26 ] although a minor amount of small iron oxide‐type clusters wrapped up by the anatase matrix is also possible (for details see Note S1 , Supporting Information). These EPR results for a‐TiO 2 ‐Cu SA and a‐TiO 2 ‐Fe SA are fully in line with the X‐ray absorption data (XANES/EXAFS), which effectively rules out the presence of larger CuO x or FeO x clusters and indicates that the Cu(II) and Fe(III) ions are strongly bound, rather than weakly physisorbed, to the surface of anatase TiO 2 (for details see Note S2 and Figures S12 and S13 , Supporting Information, note that due the overlap of the V K α lines with the Ti K β lines, XAS analysis of the vanadium‐containing sample a‐TiO 2 (400)‐V SA could not be performed).…”

Acoustic Waves in Bubbly Soft Media

“…Otherwise the wave propagation will be predominantly ballistic as the ratio between the MFP and the sample size is large [22]. At low frequencies from 0 kr =0 to 0.1, a set of numerical experiments have been carried out to estimate the MFPs of the elastic waves in various bubbly soft media for all the structure parameters employed in the present study.…”

Simulation of radiation transfer and coherent backscattering in nematic liquid crystals

“…* aksev@mail.ru † dmitry@kokorin.org ‡ vpromanov@mail.ru An effective description of the radiative transfer in a strongly inhomogeneous media provides the method of the diffusion approximation. This approach was comprehensively studied in a number of papers [14][15][16][17][18][19][20][21]. The speed of photon diffusion and diffusion coefficients are obtained in the experiments on the passage of the short pulses through the medium [22].…”