Numerical results for the thermal scattering functions

A recent formulation in radiative transfer defined the thermal scattering functions that characterize radiative transfer from a general, plane-parallel, finite medium driven solely by an internal distribution of thermal sources. Exiting diffuse intensities are expressed as space convolutions of the thermal scattering functions with any thermal source distribution. A parametric study is presented to obtain the basic structure of these scattering functions. The independent variables of these azimuthally independent functions are the direction consine μ and source location t, while the parameters are the single scattering albedo ω, total optical depth t₀, and the asymmetry factor g in the Henyey-Greenstein phase function. The basic functional trends are discussed using various parametric plots, and selected tabular results are given to allow numerical checks. The computational method is invariant imbedding. As a particular application, these functions are used in the following companion paper to obtain exiting intensities from inhomogeneous and nonisothermal media.     

Messung der Wirkungsquerschnitte für elastische Elektronenstreuung in polykristallinen Al-Schichten bei verschiedenen Temperaturen

Absolute intensities of electrons scattered elastically in a polycrystalline aluminium foil were measured by means of a retarding field apparatus at 51 kev and in the temperature range 150–800 °K. From the integrated intensities of the Debye-Scherrer rings (I _R), of the temperature diffuse scattering (I _TDS) and of the temperature independent diffuse scattering (I _DS) the total cross sections σ of the foil for the corresponding scattering processes have been determined. The measured values σ _R (T) for Laue-Bragg scattering show very good agreement with the calculation, based on the dynamical two beam approximation, and σ _TDS (T) for the temperature diffuse scattering agrees fairly well with the kinematical value for the Einstein model of the vibrating crystal. The temperature independent part of the elastic background intensity is mainly due to elastic scattering of electrons in the amorphous oxide layer of the aluminium foil. It turns out, that the sum of all cross sections for the elastic scattering processes mentioned increases slightly with temperature.     

Self-consistent Overhauser model for the pair distribution function of an electron gas at finite temperature

We present calculations of the spin-averaged pair distribution function g(r) in a homogeneous gas of electrons moving in dimensionality D=3 or D=2 at finite temperature. The model involves the solution of a two-electron scattering problem via an effective potential, which embodies many-body effects through a self-consistent Hartree approximation, leading to two-body wave functions to be averaged over a temperature-dependent distribution of relative momentum for electron pairs. We report illustrative numerical results for g(r) in an intermediate-coupling regime and interpret them in terms of changes of short-range order with increasing temperature.     

General aspects of beam threshold effects in LEED

We present in this paper some general aspects of beam threshold effects in LEED concerning their observation, their extension and their detection. The observation is possible on LEED intensities but also on crystal to ground current and total reflected current. We discuss the various types of profiles due to threshold effects for LEED intensities. From universal relations coming from emergence conditions of a new beam we draw universal charts for the “speed of emergence” versus E, θ or φ. These charts are used for a discussion of the range in I(E) or I(φ) profiles, where threshold effects may affect the LEED intensities. The same relations are used for a discussion of the detection of threshold effects. Assuming a Gaussian distribution (in energy and angles) for the incident beam, we derive a “total equivalent experimental resolution” and we discuss the consequence on intensity profiles and on the optimal conditions of detection.     

Probability distribution of singly and multiply scattered light near the critical point of a binary mixture

The probability density function p(I) is theoretically and experimentally investigated in the case of single, double, and multiple light scattering near a second-order transition (demixing of the binary mixture nitrobenzene and n-hexane). Experimentally we find no differences between the statistics of singly, doubly or multiply scattered light, and as expected from theory, p(I) shows an exponential variation. These results justify the use of either heterodyne or homodyne techniques for studying single or multiple scattering close to a critical point. We also describe a particularly simple way of determining whether homodyne or heterodyne detection occurs.     

Semi-classical theory of two-photon resonant third-harmonic generation

A semi-classical density matrix theory based on the Method of Averages is used to derive an expression for the third-harmonic generation intensity I^3ω as a function of the fundamental intensity I^ω in the neighbourhood of a two-photon resonance. The result takes account of optical Stark shifts, two-photon saturation and both homogeneous and inhomogeneous broadening. At high intensities, I^3ω∝I^ω, in contrast to the case where optical Stark shifts are ignored for which I^3ω∝(I^ω)^-1.     

A new efficient method of solution to radiation transfer in absorbing,emitting, isotropically scattering,homogeneous, finite or semi-infinite,plane-parallel media

A new efficient method of analysis, which utilizes the natural eigenfunctions of the problem, is developed for solving radiation transfer in an absorbing, emitting, gray, isotropically scattering, homogeneous, finite or semi-infinite, plane-parallel medium. Expressions for the forward and backward radiation intensities, the incident radiation and forward and backward radiation heat fluxes are included. Since the physical aspects of the problem are well documented, attention is focused on the presentation of the method of analysis and discussion of its convergence and accuracy. For the test case involving uniform incident radiation on the boundary surface at x = 0, it is shown that the solution converges extremely rapidly to the exact results, and that lower-order approximations are highly accurate. For example, the first-order solution predicts values of the incident radiation, reflectivity and transmissivity that are less than 1% in error in almost all cases, for optical thicknesses in the range 0.1 ⩽ L ⩽ 10 and single-scattering albedos in the range 0.1 ⩽ ω ⩽ 0.99. The present method of solution has an excellent potential for generalization to anisotropic scattering and to radiation transfer in spherical and cylindrical geometries.     

Temperatures from rotational-vibrational raman Q-branches

An iterative, least-squares analysis scheme, based on the differential of the intensity, dI = dτ?I/?τ + d? ?I/?? has been developed for obtaining the temperature θ = 1/τ and density ? from a set of rotational-vibrational Raman, Q-branch intensities. Spectral locations that cover the maximum variation in ?I/?τ were determined from a set of computed intensities and derivatives. The results for the Stokes (anti-Stokes) bands indicate for temperatures less than 2100K (2700K) that the maximum extent of ?I/?τ can be spanned by recording intensities only in the first two vibrational bands, v(1-0) and v(2-1) and, due to the multivalued nature of ?I/?τ, equally spaced values for ?I/?τ can be obtained only by programming the spectrometer to select an appropriate set of non-uniformly spaced passbands.     

Nanoparticle shape reconstruction by solving the direct and inverse small-angle scattering problems for a unit potential localized inside a torus

The simplest doubly connected surface of revolution (torus) is used as an example to demonstrate the possibility of stable reconstruction a three-dimensional homogeneous body defined by a unit potential U(r) using a spherically averaged small-angle scattering (SAS) curve I(s). Annealing Monte Carlo simulations are performed without using prior information about nanoparticle shape and size. Exact and approximate expressions are obtained for the form factor of a torus. It is shown graphically that the exact and approximate SAS form-factor curves agree for an experimentally accessible scattering region. Examples are given.     

Theory for the nonlinear optical response at noble-metal surfaces with nonequilibrium electrons

We present a theory for the electron-temperature dependence T _el of optical second harmonic generation (SHG). Such an analysis is required to study the dynamics of metallic systems with many hot electrons not at equilibrium with the lattice. Using a tight-binding theory for the nonlinear susceptibility χ ⁽²⁾(ω,T _e1) and the Fresnel coefficients we present results for the SHG intensity I ⁽²⁾(ω,T _e1) and its dependence on T _el for Cu. Note, χ ⁽²⁾(ω,T _e1) rather than the Fresnel coefficients determines essentially this temperature dependence. Most interestingly we find frequency ranges where I ⁽²⁾(ω,T _e1) increases for small light intensities, while it decreases for large light intensities. Our theory yields also that SHG probes effects due to hot electrons more sensitively than linear optics. The results of our calculations are compared with recent experiments on Cu and Au.     

Surface structure of unreconstructed W(001) from alternative leed techniques

The surface geometry of the high temperature phase W(001) is obtained by comparing relative intensities of diffraction beams at individual energies. The influence of non-structural parameters — Debye temperatures, ion core scattering potential, inner potential and incident beam angle, are examined. The results are contrasted with a comparison using conventional I(E) curve methods and a quantitative r-factor analysis. Uncertainty in the choice of non-structural parameters is shown capable of partly explaining the wide variation of results found in previous studies of this surface. The two methods are shown to give very similar results, and a surface layer contraction 6.7 ± 2% concluded.     

Field transformation in a multimode optical waveguide with random irregularities of the film surface

A self-consistent mathematical model for the transformation of the average intensity of the mode spectrum I(z) of a waveguide field in a multimode planar optical waveguide with a step profile and rough surface is developed. This model is based on the matrix model for multiple scattering of modes in an optical waveguide. The elements of the intermode scattering matrix are found, which describe the process of mutual transfer of the energy of modes along a waveguide and their transformation into radiation modes. The transformation of the I(z) modes in waveguides with large-and small-scale inhomogeneities is investigated. It is shown that the largest qualitative differences in the noted dependences manifest themselves only in the initial portions of the optical waveguide. The length z of these portions is much smaller than the characteristic scale length L _k at which the fundamental energy of the kth mode excited in the optical waveguide is renewed. The effect of self-filtration of the mode spectrum I(z) is described, as a result of which a stable (normalized), independent of distance z, distribution I* is formed. It is established that irregularities of the optical waveguide boundaries exert a depolarizing effect on a guided light beam. The specific features of the normalization of the radiative dissipation of a group of modes I_i(z) in an optical waveguide are investigated. It is ascertained that, in the case of small-scale irregularities, the attenuation coefficient is described by a nonlinear monotonic dependence α(z), which asymptotically converges to the value α*, characteristic of the normalized field I*. When the optical-waveguide film has large irregularities, the dependence α(z) is characterized by a pronounced maximum due to the formation of alternative channels of radiative dissipation of the energy of waveguide modes.     

Kinetic theory of classical liquids. I. Basic theory

A somewhat new approach to a kinetic theory of classical liquids is presented, and it is used to calculate the dynamical structure factor S(qω). It gives correctly the zeroth, second, and fourth frequency moments of S(qω), and it goes correctly over to the free particle value for large q. For small q and ω it goes over to proper hydrodynamics, including the coupling to heat diffusion. Results of numerical calculations on liquid argon are presented and they show very good agreement with available neutron scattering and molecular dynamics data.     

Mean-square displacements of metal and boron atoms in crystal lattices of rare-earth hexaborides

The intensities of the I₄₁₀ and I₄₁₁ reflections of nine rare-earth hexaborides MB₆ (M=La, Ce, Pr, Nd, Sm, Eu, Gd, Tb, Dy) are experimentally studied in the temperature range 4.2–300 K. The mean-square displacements of metal and boron atoms are calculated from the temperature dependences of the intensities I₄₁₀(T) and I₄₁₁(T). The characteristic temperatures of the metal (θ_M) and boron (θ_B) sublattices of rare-earth hexaborides are determined in the Debye approximation. It is found that the characteristic temperatures decrease with an increase in the atomic number of the metal.     

Two and three pion-cut contributions to nucleon-nucleon scattering

Examining information from NN forward scattering in terms of discrepancy functions, we show that the 2π cut contributions as calculated via dispersion methods from πN scattering are in perfect agreement with NN scattering. Furthermore, we demonstrate the need for 3π cut contributions which are quantitatively well described by a nucleon exchange model. Finally, in addition to the 2π and 3π cut contributions, we determine the coupling constants of the ω and A₁ to be: g_Vω²/4π = 8.1 ± 1.5, g_Tω/g_Vω = 0.14 ± 0.20 and g_A1²/4π = 7.3 ± 3.0. The coupling of the η turns out to be zero.     

Atomic vibrational dynamics of amorphous Fe-Mg alloy thin films

The atomic vibrational dynamics of ⁵⁷Fe in 800-Å thick amorphous Fe_xMg_1−x alloy thin films (0.3≤x≤0.7) has been investigated at room temperature by nuclear resonant inelastic X-ray scattering (NRIXS) of 14.4125 keV synchrotron radiation. The amorphous phase has been successfully stabilized by codeposition of Fe and Mg in ultrahigh vacuum onto a substrate held at −140 °C during deposition. The amorphous structure of the samples was confirmed by X-ray diffraction and conversion electron Mössbauer spectroscopy. The ⁵⁷Fe-projected partial vibrational density of states, g(E), has been obtained from the measured NRIXS vibrational excitation probability, together with thermodynamic quantities such as the probability of recoilless absorption (f-factor), the average kinetic energy per Fe atom, the average force constant, and the vibrational entropy per Fe atom. A plot of g(E)/E² versus E proves the existence of non-Debye-like vibrational excitations with a peak at E_bp∼3-5 meV (boson peak). Both the boson peak height and E_bp were found to depend linearly on the composition x. Above the boson peak, g(E)/E² exhibits an exponential decrease.     

Influence of temperature on raman spectra of the FeS2 single crystal with pyrite structure

Raman scattering in a natural FeS₂ single crystal with the pyrite structure was investigated in the temperature range of 80–300 K. All five Raman-active modes E _g , T _g (1), T _g (2), A _g , and T _g (3) were observed under normal conditions (T = 23°C). The influence of temperature on the Raman spectra was studied in the HH configurations (polarizations of the incident and scattered radiations are parallel), which made it possible to detect the strongest spectral lines A _g and E _g . Widths of modes E _g and A _g were substantially smaller than those given in previous publications. It was found that the temperature dependences of frequencies and widths (FWHM) of modes A _g and E _g are approximated well by the Klemens model, which describes the three-phonon scattering mechanism.     

Study of thermal conductivity and specific heat of amorphous and partially crystalline poly(ethylene terephthalate) in relation to its structure

The thermal conductivity k(T) and the specific heat of amorphous and partially crystalline polyethylene terephthalate) were measured in the intervals 1.2–40K and 1.2–10K, respectively. For a quantitative study of the relation between the thermal conductivity and the structure and degree of crystallinity of the samples their small angle X-ray scattering was measured. For T > 20 K, k(T) increases with increasing degree of crystallinity φ, whereas for T < 10 K, k(T) decreases when φ increases. Amorphous PET shows a temperature dependence of k(T) which is typical for all amorphous materials. These results are compared with curves which were computed from experimental small angle structure functions using a model for phonon scattering in vitreous systems obtained by Klemens. It is shown that for T < 10 K the change in conductivity in the partially crystalline samples relative to that of the purely amorphous sample can quantitatively be explained by additional scattering of phonons from static long-range order fluctuations of the sound velocity which are due to the microscopic structure of the polymer. From a measurement of the optical extinction of the samples relative values of their thermal conductivity at 50 mK are estimated. The specific heat obeys a T³-law between 1.2 K and about 7 K and decreases linearly with φ. The Debye specific heat of the amorphous sample was computed from the sound velocities. It is only 85% of the measured value.     

Second-and third-harmonic generation by carbon nanotubes irradiated with femtosecond laser pulses

Femtosecond pulses of a Cr:forsterite laser are used to study second-and third-harmonic generation in a layer of single-wall carbon nanotubes produced by low-velocity spraying. The harmonic amplitude in our experiments scales as (I _p)ⁿ as a function of the pump intensity I _p, with n=2 and 3 for the second and third harmonics, respectively. This scaling law holds up to pump intensities on the order of 10¹²W/cm². The ratio of the maximum signal to the averaged background in the spectra of the second and third harmonics is estimated as 50 and 30, respectively. The second and third harmonics produced by a linearly polarized pump field are also linearly polarized, with their polarization vectors oriented along the polarization direction of the pump field. The capabilities of nonlinear-optical methods for structural and morphological analysis of carbon nanotubes are discussed, as well as ways to create solid-state carbon-nanotube generators of optical harmonic.