Reduction of Chandrasekhar's planetary problem in the case of a specularly reflecting boundary to the standard problem

The reduction of Chandrasekhar's planetary problem to the standard problem has been made by several authors (cf. Chandrasekhar⁽¹⁾; Sobolev;⁽²⁾Van de Hulst;⁽³⁾Coulsonet al.; ⁽⁴⁾Ueno; ⁽⁵⁾Kagiwada and Kalaba;⁽⁶⁾. However, the diffuse reflection and transmission problem in a finite anisotropically scattering atmosphere bounded by a specular reflector has not been reduced to the standard problem. In the present paper, starting with an integral equation governing the source function for Chandrasekhar's planetary problem in the case of a specular reflector, we show how to express the required scattering and transmission functions in terms of the Chandrasekhar's scattering and transmission functions. So far as we know, the reduction formulae are new. They will be useful for the numerical study of Chandrasekhar's planetary problem in the case of a specular reflector with the aid of high-speed digital computer.     

Determination de la longueur de covariance d'une surface rugueuse par l'etude de la diffusion residuelle a l'infini

We propose a simple method for directly measuring the correlation length of a slightly rough surface by using the residual scattered light. As in a previous work about strong diffusors, we use the experimental results to discuss the validity of Beckmann's theory on surface scattering, now in the case of weak diffusors.     

The modified Beckmann–Kirchhoff scattering theory for surface characteristics in-process measurement

This paper focuses on how the scattering theory can be applied to the analysis of the surface characteristics in an in-process optical measurement. The mean scattered intensity distributions from a surface without and with the additional layers are presented based on the modified Beckmann–Kirchhoff scattering theory. The results show that the introduction of the additional layers only affects rough surfaces. The light scattered from a rough surface under the additional layers seems to be scattered from a bare rough surface with a different surface parameter in the small angle approximation. The experiment is conducted with pixel gray value measurement along the main direction of light scattering stripe to verify the theoretical analysis. The experimental curves can well fit the proposed model, which testifies the correction of the modified Beckmann–Kirchhoff scattering theory.     

A comparison of electron-phonon interaction in metallic hydrogen and free electron like metals

The electron-phonon interaction 〈I)²〉 for metallic hydrogen is calculated directly from McMillan's expression and compared with the result recently obtained by Papaconstantopoulos and Klein using the Gaspari-Gyorffy theory. Comparison is also made with 〈I)²〉 for Na.     

Development of Matlab filtering techniques in digital speckle pattern interferometry

In interferometric fringe pattern analysis, specular and speckle fringe patterns are the two main divisions. While specular fringes are characterized by quality fringes, speckle (that obtains due to the diffuse scattering of the coherent radiation from an optically rough surface) fringe patterns are characterized by noisy fringes. This paper concentrates on this aspect and the Matlab based filtering methods to improve the quality of speckle fringe patterns by developing the appropriate software. Further, the newly developed software “Macurv” will be presented which can give the second order derivative (curvature) fringe information. A software with several functions is written using Matlab. The objective of the software is to provide a more effective way for the post-processing of speckle interferometric fringes. The algorithm and functions of the developed software “Macurv” will be explained.     

Rigorous estimates of the e-(μ-p) scattering amplitude below the μ- excitation threshold

Starting from an operator formulation of Kohn's variational principle, we derive an estimate of the non-relativistic e^-(μ^-p) scattering amplitude for scattering energies below the μ^- excitation threshold. Our investigations proceed via an appropriate several-step Born expansion of this scattering amplitude. In order to construct majorants of these Born expansions, we perform estimates of certain operator norms by means of rearrangement inequalities and an extension of Hilbert's inequality.     

Structure analysis of an oxidized nickel (110) surface using low energy ion scattering

Low energy (6 keV) argon and neon ion scattering in the low angle mode (θ = 30°) has been used to investigate changes in the surface structure of a Ni(110) surface caused by the adsorption of oxygen at low exposures (10^?6 Torr s). The experimental energy spectra indicate that due to adsorption of oxygen, the interatomic distance in the 〈1̄10〉 direction increases while in the 〈001̄〉 direction this distance seems to decrease. This represents strong evidence that a reconstruction process is taking place during the early stages of oxidation of the Ni(110) face, in which the interatomic distances in the 〈1̄10〉 direction doubles. The oxygen atoms were found to lie in or close to the nickel 〈001̄〉 rows. These results are not in agreement with recently published dynamical LEED calculations.     

激光光斑形状和尺寸对扫描显示中散斑对比度的影响

计算了圆形、矩形激光光斑对于激光扫描显示系统中散斑对比度的影响.对于圆形光斑,当光斑尺寸大于散射表面相关长度时,得到的散斑对比度随着光斑尺寸变小而下降;当光斑尺寸接近表面相关长度大小时,由于光斑中包含的散射颗粒变少,得到的是非高斯散斑,散斑的对比度反而会变大;随着激光斑进一步变小,由于镜面反射效果散斑的对比度会很快下降.对圆形光斑部分结果给出了实验验证.为了保证光斑中有足够的散射颗粒,只在一个方向压窄光斑,分别计算了平行于扫描方向和垂直于扫描方向压窄光斑得到的散斑对比度.平行于扫描方向压窄光斑,散斑的时间相关性会下降,平均效果变明显,散斑的对比度变小;垂直于扫描方向压窄,散斑对比度变化不大.     

Scattering of atomic beams off a single surface step: An exact solution

The scattering of atomic beams off a single rectangular step on a planar hard-wall surface is calculated exactly using the Wiener-Hopf technique. The diffuse scattering consists of a sharp peak around the specular direction which is due to interference effects between the two surface levels and which oscillates in magnitude with incident angle. A second part then describes a smooth background scattering which strongly depends on the actual shape of the step whereas the former does not. Since the step is an infinitely extended defect a finite part of the incident atoms is scattered into the peak around specular, in contrast to the case of a compact scattering object, where only an infinitesimal part is removed from the specular beam.     

Experimental study of the angular correlation function in the double passage of waves through a one-dimensional random phase screen

Abstract

Experimental results are presented for the angular correlation function of far-field speckle patterns scattered in the double passage of waves through a one-dimensional random phase screen. The experiment for the correlation measurement was set up to use a CCD camera to obtain the image of the speckle patterns in the scattering directions for each given angle of incidence; the cross-correlation function is then calculated from the digitized images. The theoretical analysis of the motion of the speckle as the source is moved, as given by Escamilla, is verified experimentally. It is found that in contrast with the memory effect line of speckle motion, the speckle pattern produced in the region of observation tracks the backscattering direction.     

Numerical results for finite order X and Y functions of radiative transfer

It is well known that, in the theory of radiative transfer, Chandrasekhar's X and Y functions play an important role in the diffuse reflection and transmission problem (cf. Chandrashekhar⁽¹⁾). In a preceding paper (cf. Bellmanet al.⁽¹⁰⁾), graphs and selected tables of these functions covering wide ranges of slab thickness and albedos for single scattering have been provided. In this paper, making use of a system of coupled integral recurrence relations for finite order X and Y functions (cf. Bellmanet al.⁽¹⁴⁾), numerical results for these basic functions are tabulated up to optical thickness τ = 2.0 from τ = 0.1, assuming the conservative case of isotropic scattering. The maximum order of these functions is taken to be fifteenth. It is shown that the accuracy obtained is satisfactory in the domain under consideration. Furthermore, numerical results for Chandrasekhar's approximation for X and Y functions are also tabulated for stabs of small optical thickness.     

Medium corrections in intermediate-energy nucleon-nucleus scattering

Medium corrections in elastic and inelastic nucleon-nucleus scattering are discussed in the framework of Watson's multiple scattering theory. We derive formulae for the effective interactions to be used in the calculation of elastic and inelastic scattering based on a consistent local-density approximation. We show the numerical results of analyses of the p + ¹⁶O reaction at E_p = 135 MeV.     

Polarization of multiple scattered light in planetary atmospheres from solution of the coupled linear integral equations derived for mixed rayleigh-isotropic scattering

A set of coupled integral equations describing nonconservative multiple scattering for a mixed isotropic and Rayleigh single scattering phase function in inhomogeneous, plane-parallel planetary atmospheres is derived. The equations are applicable for the frequency redistributions MRE (monochromatic radiative equilibrium), CFR (complete frequency redistribution), or PFR (partial frequency redistribution). Solution of the equations permits one to calculate the intensity and degree of polarization in an arbitrary direction outside or inside the plane-parallel scattering medium. The equations are readily adaptable to more complicated geometries. Solutions for several cases are presented to demonstrate the versatility and validity of the method. These include a calculation of MRE pure Rayleigh scattering of sunlight in an optically-thick planetary atmosphere to demonstrate agreement with the results of Coulson⁽¹⁾et al. (1960), calculations of the altitude profile of the degree of polarization of the earth's Ly-α 1216A and helium 584A dayglow, and the center to limb variation of the degree of polarization of the sunlight diffusely reflected from a distant planet such that the scattering is pure Rayleigh and conservative at the top of the planet's atmosphere, varying smoothly to conditions of nonconservative and pure isotropic scattering deep in the planet's atmosphere. Tables of functions that one might utilize (without resort to a digital computer) to obtain solutions in the escape function approximation are also given.     

Inverse multiple scattering problems—II. Limited information content of partially fitted planetary curves with application to the Venusian visual phase curve

The random grid search method and the minimization search method recently developed by Fymat (1972) for solving inverse multiple-scattering problems of planetary atmospheres are employed for a study of the extent and nature of true information yielded by theoretical partial fits of observed planetary curves. The study is conducted with the Venusian visual phase curve as a background example. It is shown that such fits cannot be used toinfer the model scattering parameters of the planet's atmosphere. They cannot provide any indication of the shape of the scattering diagram, even in the region of the fit. It is also shown that the measured Bond albedo can provide neither a criterion for sorting out different possible scattering model candidates nor a means for accurately determining the single scattering albedo. A high value of the latter albedo for Venus' atmosphere at visible wavelengths(ω̃_o ≈ 0·99) is also found, and Euler's scattering model is definitely discarded for this planet.     

Search for M1 strength in 51V and the difference in 51V(e,e′) and 51V(p,p′) spectra

Following the studies of the distribution of M1 strength in the even-even N = 28 isotones ⁴⁸Ca, ⁵⁰Ti, ⁵²Cr and ⁵⁴Fe by inelastic electron scattering, the result of a search for M1 transitions in the odd-even N = 28 nucleus ⁵¹V is reported. No strong M1 excitation has been detected, in contrast to a recent (p, p') experiment. There is no immediate explanation for this discrepancy. Shell-model calculations indicate that a part of it might be accounted for by an interference between the spin and orbital term of the electromagnetic transition operator in the (e, e') experiment.     

Disordered clean surfaces and adsorbates investigated with atom-surface scattering

In this paper two different physical situations are considered which can be treated with the same method: a fluid adsorbate (disordered in the x, y plane) and a clean surface with random steps (disordered in the z direction). The hard corrugated wall model is used in the eikonal approximation; the differences between the two cases arise only from the different statistical properties of the two physical situations. The differential scattering probability is evaluated. For the fluid adsorbate the latter splits into a coherent (purely specular) contribution and an incoherent one (which is, in fact, weakly inelastic and related to classical diffusion on the surface). For stepped “rough” surfaces only incoherent scattering is present and the differential scattering probability for hexagonal lattices is given.     

Theory of cyclotron resonance halfwidth for electrons scattered by impurities and phonons

A general theory of the cyclotron resonance halfwidth for electrons scattered by impurities and phonons is developed on the basis of the proper connected diagram expansion of the current-correlation-function formula for the dynamic conductivity. The theory is applied to the cases of Ge samples at extremely high magnetic fields and different temperatures. The usual form of Matthiessen's rule Γ = Γ₁ + Γ₂ + …, where Γ and Γ_j are the total and component energy-dependent resonance widths, is valid only if the component widths Γ_j computed separately for each cause of scattering depend linearly on the densities of scatterers. The resonance width Γ_I due to the charged impurities at very low electron densities ($\text{\$}\text{?}$10¹² cm^?3) and at very low temperatures is known to vary in proportion to the square-root of the impurity density. Large deviations from the Matthiessen's rule occur in such a case. The theory is in good quantitative agreement with currently available experimental data. In order to test the generalized form of Matthiessen's rule, however, the high-field resonance experiments around 15 K is desirable where both phonon and impurity scatterings contribute in a comparable manner.     

Impulse approximation in QCD,Parton's pT-distribution and application to deep inelastic process

The recently formulated theory of the impulse approximation in QCD is developed further. The integro-differential equation describing the evolution of the parton's distribution function is solved. The formalism is applied to compute R = σ_L/σ_T in deep inelastic electron scattering and the average transverse momentum 〈k_T〉 of the Drell-Yan pairs. The cross section formula for the latter process is significantly different from the one recently conjectured. Agreement with recent data is good in both cases.     

Orientation of chemisorbed species from electron impact and dipole selection rules: The formate ion on Pt{110}

Selection rules in high resolution electron energy loss spectroscopy are shown to permit a determination of the structure of formate ions adsorbed on Pt{110} from data obtained on and off specular along three different azimuthal directions. Long-range dipole scattering indicates that the ion is adsorbed upright bonding, through the two oxygen atoms. Shorter-range impact scattering provides the further information that the ion is coplanar with the 〈110〉 direction and tilted (in the same plane) through at most 10^° from the surface normal.     

Effect of satellite Q-branches on temperatures computed from oxygen vibrational raman intensities

Splitting of the ³Σ^-_g ground state of molecular oxygen has a small but noticeable effect on the intensity distribution in the fundamental and hotband Raman Q-branches. An analysis of oxygen Q-branch spectra [computed for Hund's coupling case (b) and pure anisotropic scattering], with expressions corresponding to ¹Σ theory, results in temperature errors of -2.0% to -0.5% in the range 500–2000 K.