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It is shown that anisotropic scattering with a strong forward peak can give reasonable agreement with angular reflectance data for snow. As a result of the forward peak, solar radiation penetrates deeper into the medium, when measured in terms of photon mean free paths, than it does for isotropic scattering. The radiation transmitted directly through finite slabs can be seen to an optical depth of seven, and decreases much more rapidly with optical depth than does the diffusely transmitted (scattered) radiation.

The lineshape of the isotropic induced scattering has been measured from polarized and depolarized Rayleigh bands of CO 2 fluid. A comparison with other “collision-induced” spectra has been performed. It is argued that the scattering is dominated by two- and three-body interactions.

The rough and uneven basaltic basement in deep marine environments functions as a potential source of scattering. The source‐generated noise not only interferes with the weak and deep primary reflections, but often resembles primary reflections, thereby confusing the seismic interpretation. An analysis of the arrival time of scattered noise indicates that NMO and apparent dip of scattered noise exhibit azimuth dependence. For isotropic scattering, in‐line scattering at ranges near 0 degree azimuth and sideswipe from 90 degree azimuth should dominate the seismic section. Actual marine data displayed in various data gathers illustrate these characteristics of scattered noise and also correct a common misconception about the appearance of scattered noise. The analysis suggests the use of CDP stacking and velocity filtering to attenuate the scattered noise. The combined effect shows a total reduction of scattered noise by 21 dB for the frequency bandwidth from 20 to 32 Hz.

The neutron flux distribution has been calculated around a cylindrical, totally-absorbing rod immersed in moderators of various size and degree of absorption. Isotropic scattering was assumed as well as a uniform source distribution. From the results obtained with a standard ANISN code the linear extrapolation distance was derived. It was found necessary to introduce a correction for the fact that ANISN does not give an exact balance between production and loss of neutrons. After such a correction the linear extrapolation distance could be obtained to an accuracy of 1–2%. Through using two different definitions of the extrapolation distance it was possible to reproduce the values of Pellaud (1968) and Isakova (1968) and to explain the difference between their results.

Two-constant Kubelka—Munk (KM) theory with experimentally fitted values accounts quite well for the total reflectance of opaque mixtures of paints, but systematic errors indicate the need for better theory. No simple satisfactory alternative exists, however. Errors from the Saunderson correction for surface reflections are comparable to errors from KM theory, and opinions differ on the best values for the diffuse internal reflection coefficient. Since the surface corrections can be applied separately from diffuser theory, an extra variable is available. Even for isotropic scattering, theory shows that the internal surface reflectance ranges from 0.57 when the diffuser reflectance is high to 0.65 when it is low, while the corresponding Saunderson coefficient ranges from 0.52 to 0.65. For anisotropic scattering, the coefficient is virtually independent of the phase function for single scatter when the reflectance is high, but critically dependent when it is low. Values of around 0.4 are predicted for the absorption band, unless the particles are very small.

The critical sphere problem is studied in one speed, time independent transport theory using a scattering kernel which consists of a linear combination of forward and isotropic scattering. Numerical results indicate that the critical radius varies monotonically with anisotropy.

An approximate method to calculate the parallel neutron leakage in fast reactor slab lattices is described. It is derived from the integral transport equation and assumes isotropic scattering. By u...

The variation of the critical slab thickness from backward to forward scattering is studied for specified isotropic scattering. Numerical results are obtained from the zeroth and first-order approximate analytical expressions, derived using the method of elementary solutions. It is shown that the critical slab thickness increases monotonically with increasing forward anisotropy.

The analytical determination of the spatial and angular moments of the monoenergetic neutron-transport equation in an infinite medium initiated by Case, de Hoffmann and Placzek 30 years ago is completed. Analytical expressions for the moments that are convenient for numerical evaluation are obtained, and several high-order moments are evaluated as a demonstration.

THE problem of the radiation field in a plane-parallel multilayer system, whose outer boundary is irradiated by parallel rays, is considered. The case of non-isotropic radiation is studied. A formal solution of the problem is given, and using this, the spatial-angular distribution function of the intensity of the diffused radiation is constructed by applying the method of discrete ordinates. Consider a multilayer plane-parallel system, consisting of N layers in contact with one another which are irradiated by a parallel pencil of rays incident at some angle to the outward normal of the system. We assume that the boundaries of the layers are known and that the source functions are defined differently in each layer. It is required to determine the radiative field in any region of the system and find the law of the angular distribution of the outgoing radiation.

1. In (1) M. G. Smith solved the equationwherefor R > 0, ø( − R) = − ø(R) and 0 < ω ≤ 1. In order to determine a constant, A(R1), appearing in the solution, two expressions for q(R1) were obtained, namelyand

Unter der Voraussetzung einer isotropen Streufunktion fur interstellare Staubteilchen ergibt sich aus einer Diskussion der Hubble-Beziehung der Reflexionsnebel fur Objekte mit dem beleuchtenden Stern auserhalb des Nebels eine untere Grenze der Staubdichte, die fur die Abbildung des Nebels auf dem Palomar Observatory Sky Survey notwendig ist. Dieser untere Dichtewert ist von der absoluten Helligkeit des beleuchtenden Sterns abhangig. Die Albedo der Staubteilchen ist groser als γ = 0.6. Assuming an isotropic scattering function of interstellar grains from a discussion of the Hubble relation of the reflection nebulae it follows that for such objects with the illuminating star outside of the nebula a lower limit of the dust density depending on the absolute magnitude of the illuminating star is necessary in order to image the nebula on the Palomar Observatory Sky Survey. The albedo of the grains is larger than γ = 0.6.

An exact solution to the energy-dependent Milne problem for isotropic scattering has been obtained using a simple separable scattering kernel. The extrapolation distance and angular distribution at...

The nonabsorbing thermal-neutron Milne problem is solved for isotropic scattering in the laboratory system. The scattering kernel has been approximated by a two-term degenerate sum and the resultin...

The integral equation for the isotropic scattering of radiation in a medium bounded laterally by planes ±y0 is solved as a series in powers of exp (−y0). This series is certainly convergent for y0 large enough, and rapidly convergent for most practical purposes.

A mathematical description is given for a neutron wave propagating parallel to the surface of a semi-infinite medium or of a slab. Cercignani's method of singular eigenfunction expansions is applied under the assumption of one-speed isotropic scattering. Half-range orthogonality and closure relations for the eigenfunctions are derived, with the necessary precautions taken to account for the zeros of the dispersion function inside the area spectrum. The analog of the Milne problem is solved and the complex extrapolated endpoint evaluated. Several coefficients in an expansion of the latter quantity in terms of two characteristic parameters are computed. The results are used to derive an asymptotic solution of the slab problem, where a complex transverse buckling appears. For typical experiments on large assemblies Im(B2) is sufficiently small to be neglected.

The modified Eddington approximation proposed earlier for isotropic scattering is formulated for a general scattering law. Numerical results are given for the special case of Thomson scattering.

The heat transfer in a conducting, absorbing, emitting, scattering and heat generating medium has been considered. The specific problem of a plane parallel system with isotropic scattering was chosen. For this system, it is shown that the effects of scattering are small.

Three new approximate forms for theH-function for isotropic scattering have been developed. Each of the forms involve three unknown functions of albedo ω.

Using the Fourier transform technique, the neutron-wave-propagation problem has been studied in the presence of complete backscattering plus isotropic scattering of neutrons. The discrete and continuum solutions have been obtained in one-speed transport theory as well as in multi-velocity transport theory. A one-term degenerate kernel has been used for carrying out the analysis.

We have calculated the intensities of the transition to the two optically accessible triplet exciton states of a pyrene crystal using the Hochstrasser procedure. Using the charge-transfer contribution and taking into consideration non-orthogonality between molecular π orbitals belonging to different neighbour molecules together with multicentre integrals calculation we calculate a Davydov splitting value of approximately + 1.8 cm −1 which compares reasonably with the + 2.4 cm −1 reported by Arnold et al. The charge-transfer contribution is shown to be largely predominant. Diffusion tensor anisotropy in a simple isotropic scattering time approximation is given which shows the two dimensional structure of the pyrene crystal for the diffusion of triplets.

Two approximate methods are presented for making radiant heat-transfer computations from gray, isothermal dispersions which absorb, emit, and scatter isotropically. The integrodifferential equation of radiant transfer is solved using moment techniques to obtain a first-order solution. A second-order solution is found by iteration. The approximate solutions are compared to exact solutions found in the literature of astrophysics for the case of a plane-parallel geometry. The exact and approximate solutions are both expressed in terms of directional and hemispherical emissivities at a boundary. The comparison for a slab, which is neither optically thin nor thick (τ = 1), indicates that the second-order solution is accurate to within 10 percent for both directional and hemispherical properties. These results suggest that relatively simple techniques may be used to make design computations for more complex geometries and boundary conditions.

A modification of the Feautrier method of solving transfer problems is presented that allows advantage to be taken of such simplifications as isotropic scattering and complete frequency redistribution. The method has timing and storage requirements comparable to integral equation methods, but retains the ease of formulation of differential methods. The case of a two-level atom is treated in detail, and an appropriate generalization is sketched for the treatment of multi-level cases by means of a complete linearization technique.

A proof is given, on the basis of equilibrium fluctuation theory, of the Kubo expression for electrical conductivity. The expression is evaluated explicitly for the case of isotropic scattering (Sommerfeld model) and for non-isotropic scattering at high temperatures. For both cases the results agree with those derived by using the Boltzmann equation; for the case of non-isotropic scattering the results are valid provided the conditions ħ;/τ ⪡ E F , T ⪢ θ D , are satisfied.

Radiation from arbitrary directions incident on a thick, homogeneous, conservative atmosphere with isotropic scattering is mostly diffusely reflected. A smaller part diffuses through the atmosphere and emerges at the other side. Sobolev's equations, which give the asymptotic forms of the reflection and transmission functions for large thickness,b, in terms of the functionH(μ) known from the theory of semi-infinite atmospheres, are newly derived. It is shown by two examples that knowledge of these asymptotic forms makes it possible accurately to interpolate any quantity dependent onbbetweenb = 1 andb = ∞.

A linear problem of radiative transfer in the spectral line in a plane- parallel layer of finite thickness, under general assumptions on the function of redistribution in frequency, independent of the angle of scattering, is considered. The solution of the system of integral equations obtained is reduced to the solution of certain functional equations of Volterra type. Some calculations are carried out in the case of Doppler broadening of the line. (auth)

Cathodic vacuum emission from forward biased, rectifying contacts between n -type GaP, ZnSe and ZnS and thin Ag films covered with a monolayer of Cs agrees with a simple theory which assumes isotropic scattering of hot electrons on transmission through the interface between the semiconductor and the Ag. The maximum measured emission efficiency is 6 × 10 −3 per cent but at least 2 per cent is expected with further development.

A temperature gradient is introduced into the density matrix equation by analogy with the Boltzmann transport equation. The electric and heat currents are defined in the case of isotropic scattering, in which particular case, it is shown that the Wiedemann-Franz law holds even in the details of the oscillations of both the electric and thermal conductivities.

THE method of integral transforms is applied to the solution of problems of the one-speed theory of neutron transfer with isotropic scattering in spherically symmetrical inhomogeneous media. Functions are obtained which have the meaning of elementary solutions of Boltzmann's equation in spherical geometry.

The kernel of the integral equation for the source function in a three-dimensional homogeneous atmosphere possesses the properties of a Green's function. These properties are used to transform the integral equation into a singular integral equation for the kernel. The particular case of a homogeneous plane parallel atmosphere is discussed and a solution to the kernel equation is obtained at all points of the atmosphere.

We have studied the vibrational Raman band of the N2 molecule dissolved in various solvents. The band shape of isotropic and anisotropic scattering is discussed and compared with the one obtained from N2 gas. A collisional narrowing effect has been observed for the isotropic scattering. For N2 dissolved in SF6 the rotational correlation function is analyzed.

The reflectance of a semi-infinite atmosphere with nonconservative isotropic scattering is evaluated for single-scattering albedos = 0.1, 0.7, and 1.0 and compared with Minneart's law for μ and μ0 in the range 0.4–1.0. The reflectance for = 1.0 is well represented by Minneart's law with k = 1., whereas for small the reflectance is better represented by μR = R0[μμ0/(μ + μ0)]k, where k increases from 1.0 for = 0.0 to 1.08 for = 0.5.

Integral forms of transport equations first appeared in radiative transfer theory at the beginning of this century. If one considers radiative transfer with isotropic scattering in a layer of finite optical thickness τ, the boundary value problem may be written (cf. Section IX.1)

A random walk theory of photon diffusion in tissue is discussed, and shown to reproduce features of some recent experiments by Chance et al. The theory allows one to calculate, from the absorption and scattering coefficients of turbid medium, the time dependence of surface emission.

BS>The HERD codes furnish a numerical approximation to the one- dimensional, single energy transport equation in slab geometry. The primary purpose of HERD 2 and 3 is to compute blackness coefficients. The limitations of the codes are given and the preparation of input described. (auth)

The isotropic light scattering in liquid mixtures of small nonelectrolyte molecules was studied from both the theoretical and the experimental point of view. The necessity of the cross-term in the isotropic Rayleigh ratio resulting from the correlation of fluctuations in density and concentration is discussed. A new expression for this term is derived and compared with the previously derived expression. Isotropic scattering of various pure liquids and liquid mixtures is determined at the wavelengths 546 and 436 nm and the reliability of experimental data is discussed in great detail. Mixtures of cyclohexane, n-heptane and carbon tetrachloride (component 1) with benzene and carbon disulfide (component 2) were examined. The possibility of the determination of molecular weights from light scattering experiments on such systems is discussed. The reliability of the excess free energy of mixing data obtained by light scattering is carefully considered. It is concluded that light scattering phenomena can be explained by thermodynamics, but the light scattering method is generally inferior to vapour pressure methods both in sensitivity and accuracy, and because of the complexity of the evaluation of data.

For the kinetic theory equation (constant velocity, isotropic scattering) $$\frac{{\partial \varphi }}{{\partial t}} + \,\sum\limits_{k - 1}^{m - 1} {{u_k}} \frac{{\partial \varphi }}{{\partial {x_k}}} + \sigma \varphi = \frac{{{\sigma _s}}}{{4\pi }}\int {\varphi \left( {x,u,t} \right)\,du + S\left( {x,u,t} \right)} $$ (7.1.1) the following scheme was mentioned in the work of G. I. Marchuk and the author [69] (incomplete splitting) $$\frac{{{\varphi ^{n + 1/2}} - {\varphi ^n}}}{{r!\left( {n - r} \right)!}} = {\Lambda _1}\left( {\alpha {\varphi ^{n + 1/2}} + \beta {\varphi ^n}} \right) + \bar S,$$ (7.1.2) $$\frac{{{\varphi ^{n + 1/2}} - {\varphi ^n}}}{{r!\left( {n - r} \right)!}} = {\Lambda _2}\left( {\alpha {\varphi ^{n + 1}} + \beta {\varphi ^{n + 1/2}}} \right).$$ (7.1.3)

An exact condition for parasitic oscillation in laser elements with isotropic scattering surfaces is given in the form of an integral equation for the surface brightness. Analytic solutions valid for large gain are derived, with spherical, cylindrical and rectangular symmetry.