Exact Integral Equation Research Articles

The Milne problem for isotropic scattering with a backward leak

Abstract The Milne problem is studied for the case of, one-speed, time independent, plane symmetric Boltzmann equation, for which the scattering law is isotropic scattering with a backward leak. First of all, the new solution for the angular flux is expressed in terms of the solution of the Boltzmann equation with isotropic scattering. The expansion coefficients are determined by the resulting new boundary condition and orthogonality relations. The results for the expansion coefficients are given in the form of exact integral equations, but the numerical values for the extrapolated endpoint are obtained for the first order approximate coefficients. The variation of the extrapolated endpoint with β, the degree of backward anisotropy agrees in general with the previous findings.

Density of states of two interacting holes in a solid

If two holes are suddenly created in the same band and at the same atomic site e.g. by an Auger process in a solid, their density of states N(ω) will depend on their Coulomb interaction. In a tight binding model, we present the exact N(ω), in the limit of zero bandwidth. In the case of a general band, we give an exact integral equation that allows calculating N(ω) once the single electron density of states is known. The interaction is shown to produce a characteristic distortion of N(ω) and hence of the Auger spectrum.

Remarks on the connection between conventional reaction theories and the integral-equation approach to few-particle scattering

The method of exact integral equations is formulated in terms of wavefunctions in such a way that a direct comparision becomes possible to the more conventional methods for treating nuclear reactions. It is shown how to develop the wavefunction rigorously in an expansion which outside the reaction region is identical to the resonating-group ansatz (below the breakup threshold). Common and different aspects of the approximations employed in both approaches are clarified. The three-particle problem is used for the purpose of demonstration and for the execution of test calculations. Possibilities are discussed of how experience with one approach could be fruitful for the further development of the other.

Condensation on Rotating Axisymmetric Curved Bodies

Laminar film condensation on an axisymmetric generally curved rotating body is analysed. With the body rotating about a vertical axis, the condensate film drains under the combined action of gravitational and rotational accelerations. An exact integral equation is found for film thickness in terms of body shape, fluid properties, and rotational parameters. This shows the effect of curvature to be important. The theory was tested by experiments with steam condensing on a concave body generated by a 90° circular arc. Comparison of experimental results with predictions from the general theory developed are generally good, especially at high rotational speeds. It is shown that the inclusion of the effect of curvature produces a markedly better prediction than that from an analysis treating the body as a succession of truncated conical elements.

Numerical stress concentrations for stepped shafts in torsion with circular and shaped fillets

A computer programme has been developed to solve for torsion of solids of revolution by numerical solution of an exact boundary integral equation which is valid for any shape and any degree of connectivity. The programme gives solutions directly at the boundary and requires only simple boundary data defining the contour to be input. The basic integral equation has been differentiated to give internal stresses directly, when required. The computed results have been checked experimentally by photoelastic techniques and are compared with analytical solutions for the circular cylinder and the cone. Comprehensive charts are given for stress concentrations in stepped shafts with circular fillets and fillets constructed of two blending radii. Fillets giving stress concentrations close to unity have also been investigated.

Conductivity and spin-wave stiffness in disordered systems-an exactly soluble model

An exact discussion is given of the conductivity of disordered resistor networks and of the related problem of spin-wave stiffness in disordered Heisenberg ferromagnets, by considering the special case of the Bethe lattice. For arbitrary substitutional disorder an exact nonlinear integral equation is obtained for a generating function determining the conductivity and spin-wave stiffness in such branching models. An effective medium (coherent potential) approximation is shown to provide an asymptotically exact solution in the limit of high coordination number z. Its extension, to order (1/z)4, is obtained.

Connection Between Regge Behavior and Fixed-Angle Scattering

The physical origins and dynamical details of Regge behavior are studied by examining a general class of theories which are capable of describing the observed large-momentum-transfer data and extending them into the intermediate-t range. The mechanism responsible for the smooth connection between the deep-scattering region and the Regge region is discussed in detail. We derive a convenient new form of the exact integral equation which describes generalized ladder graphs containing irreducible scattering amplitudes as the rungs. The limiting behavior of the Regge trajectories and residues are then calculated in both the single-channel and the more realistic coupled-channel cases. The trajectories are found to approach negative constants for large negative momentum transfer and the residues fall as powers in the same limit. Furthermore, since the forward and backward Regge regimes must join smoothly onto the same fixed-angle behavior, there are relations between a priori unrelated trajectory functions and residues. The standard properties of Regge poles, in particular factorization and signature, are shown to be present even though the basic, fixed-angle interaction possesses neither of these properties. These general considerations are then applied to the more specific constituent-interchange model. Here we find that the asymptotic behavior of the trajectories and residues are controlled by the form factors of the particles involved in the scattering. Finally, we elucidate the relationship between the constituent-interchange diagrams and the Harari-Rosner duality diagrams.

Equivalent-circuit representation and characteristics of a radiating cylinder driven through a circumferential slot

The problem of an infinitely long coaxial cylinder with a single circumferential slot in the sheath is investigated analytically. When the slot width is small compared with the radius of the outer cylinder, an exact integral equation for the aperture field is formulated and subsequently solved by a quasi-static technique. Equivalent circuit representation of the coupling between the outside antenna and the inside coaxial line is obtained in closed form. Due to the highly localized nature of the coupling, the result thus obtained is applicable to finite cylindrical antenna with multiple feed slots.

Distribution of diffusing particles near an absorbing wall

On the basis of Boltzmann's kinetic equation we obtain and solve numerically the exact diffusion integral equation for the distribution of particles which is valid throughout the whole region of transition from macroscopic to kinetic description. It is shown that in spite of strong distortion of the angular part of the distribution function, the behavior of the density at the wall varies little by comparison with the diffusion distribution.

Explicit solution of the integral equations for fireball decay

The exact integral equations for the single particle distribution and multiplicity distribution of a decaying fireball are explicitly solved. The asymptotic behaviour of the solutions for large fireball mass is shown to be the same as in the simplified model with linear decay chain. The possibility of obtaining the many-particle distributions is also discussed.

Weak Inhomogeneity Formalism as Applied to the Bernstein Modes

Bernstein modes propagating in the direction of the number density gradient are studied. The resulting field equation is compared with that found from an exact integral equation obtained by Fourier-transform techniques. The weak inhomogeneity formalism is shown to be accurate only to the order λ/L.

A ninth-order solution for the solitary wave

Several solutions for the solitary wave have been attempted since the work of Boussinesq in 1871. Of the approximate solutions, most have obtained series expansions in terms of wave amplitude, these being taken as far as the third order by Grimshaw (1971). Exact integral equations for the surface profile have been obtained by Milne-Thomson (1964,1968) and Byatt-Smith (1970), and these have been solved numerically. In the present work an exact operator equation is developed for the surface profile of steady water waves. For the case of a solitary wave, a form of solution is assumed and coefficients are obtained numerically by computer to give a ninth-order solution. This gives results which agree closely with exact numerical results for the surface profile, where these are available. The ninth-order solution, together with convergence improvement techniques, is used to obtain an amplitude of 0.85for the solitary wave of greatest height and to obtain refined approximations to physical quantities associated with the solitary wave, including the surface profile, speed of the wave and the drift of fluid particles.

Approximate solution to the radiative equilibrium between concentric spheres

An approximate method was developed for the study of the radiative energy transfer through a gray gas layer enclosed between black concentric spheres which are maintained at uniform but different temperatures. The basic concept consists of assuming a temperature distribution through the gas layer, and assigning appropriate jump boundary conditions to evaluate the flux distribution using the exact integral equation governing the flux for this geometry. The approximate solutions have been found to agree very well (6 per cent maximum deviation) with the existing exact numerical solution throughout the entire range of optical thickness.

Exact integral equation for pion-pion scattering involving only physical region partial waves

We derive an exact relation for ππ scattering which yields the real parts of the ππ partial wave amplitudes a l ( I) (s) in the region 4 m 2 π ⩽ s ⩽ 60 m 2 π , is given i) the I = 0 and the I = 2 S wave scattering lengths and ii) the 1m a l ( I) (s) for 4 m 2 π ⩽ s ⩽ ∞, where s denotes the centre-of-mass energy, l the angular momentum and I the isotopic spin. It also provides i) a system of integral equations to determine the a l ( I) (s) for 4 m 2 π ⩽ s ⩽ 16 m 2 π , 16 2 π ⩽ s ⩽ ∞, and ii) necessary and sufficient conditions for crossing symmetry expressed in terms of the physical region partial wave amplitudes only.

Radiating hydrogen flow in axisymmetric nozzles

A considerable number of studies published in recent years have been devoted to the study of gas in channels and pipes. In view of the complexity of the question and the lack of analytic techniques, individual aspects of the problem are generally considered. The determination of the radiant field characteristics in regions of simple geometric form filled with a stationary radiating-absorbing medium has been carried out in several studies. The articles [1–3] are devoted to the calculation of the radiant field and the temperature field for a given flow of a perfect inviscid nonheat-conducting radiating gas with constant absorption coefficient. The flow is assumed to be irrotational [1, 2] or nearly potential [3]. The authors investigated the accuracy of the solution obtained with the aid of various approximate methods and found that the diffusion approximation yields a small error in calculating the radiation density field and the values of the radiant thermal fluxes for a quite broad class of wall reflecting properties. We may note also [4, 5], in which a calculation is made of one-dimensional steady flow of a viscous heat-conducting radiating perfect gas with constant transport coefficients. In [1–5] the absorption coefficient is considered constant. This assumption simplifies the solution process considerably, since as the independent variables we can take the corresponding optical thicknesses. The study [3] contains a remark that the calculation method proposed there may be used with a variable absorption coefficient. However, this possibility was not used in the calculations presented. For a constant absorption coefficient these studies yield a rather complete analysis of the methods for solving two-dimensional problems in geometrically simple regions in the absence of mechanical motion and one-dimensional problems with motion. They contain results obtained for the exact integral or integrodlfferential equations and present an analysis of the approximate methods. The study [3] considers broader possibilities of solving two-dimensional problems (using the Monte-Carlo method), but the flow is assumed known ahead of time. In the following we present a method for calculating the two-dimensional equilibrium flow of an inviscid non-heat-conducting radiating gas with variable absorption coefficient. As an example, we consider the flow of radiating-absorbing hydrogen in axisymmetric nozzles. It is assumed that the radiation is gray and is in local thermodynamic equilibrium. The transport equation is considered in the diffusion approximation. The nozzles examined have a semi-infinite cylindrical inlet section. The initial gas flow in the cylindrical section is supersonic. In the solution process we determine the radiation density field and all the flow parameters within the nozzle.

Multichannel treatment of resonant reactions using the quasiparticle method

A multichannel resonant reaction formalism is introduced using the quasiparticle method of Grassberger and Sandhas. Contrary to Feshbach's projection operator approach our method can, without difficulty, take the Pauli principle correctly into account and treat rearrangement processes. The direct transition amplitude and the closed channel bound states obey exact onedimensional integral equations. The generalized potentials occurring in them are calculable via an iteration scheme which should rapidly converge in most cases of physical interest. Thus the level shift and level width are finally expressed by accessible quantities.

Transport of Resonance Radiation in Optically Thick Media

view Abstract Citations (4) References (16) Co-Reads Similar Papers Volume Content Graphics Metrics Export Citation NASA/ADS Transport of Resonance Radiation in Optically Thick Media Doschek, G. A. ; Donahue, T. M. Abstract The mathematical assumption of complete frequency redistribution (CFR) is investigated for the problem of the transport of resonance radiation through gaseous media of large optical thicknesses. The investigation is initiated by solving the CFR integral equation of radiative transfer in a plane-parallel layer of gas. A steady-state excited-atom density is maintained by plane-wave white-light radiation produced by a source external to the medium. Computer solutions of the CFR transfer equation are obtained for various values of the natural damping coefficient and a range of total optical thicknesses. The CFR source functions for large total optical thicknesses are then used as trial solutions in the exact integral equation for the problem. Only a first iteration is performed. The resultant intensity profiles for various values of the damping coefficient are compared with CFR profiles calculated from the CFR source functions. The orders of magnitude of the total optical thicknesses and damping coefficients used in the numerical computations approximate the values that are realized in planetary nebulae. The non-CFR intensity profiles calculated by the method described above deviate from the CFR profiles in the manner predicted by qualitative argument. Publication: The Astrophysical Journal Pub Date: August 1970 DOI: 10.1086/150574 Bibcode: 1970ApJ...161..737D full text sources ADS |

An exact integral equation for steady surface waves

An exact integral equation is found for inviscid steady surface waves. Existing known approximations are all derived from the one equation. A numerical solution is attempted in the case of the solitary wave. From the numerical solution the amplitude of the solitary wave of maximum height is estimated. The numerical solution is also compared with the existing approximations and the implications drawn from the comparison are discussed.

Renormalization « groups » extended to masses; application to the structure of the renormalization constantsZ 3 andm o

By a precise elucidation of the axioms from which renormalization « groups » result we construct the most general «groups » which include mass renormalization. We discuss in detail the particular case involving only photon field and electron mass renormalizations. An exact integral equation which can be put in a form generalizing that given by Gell-Mann and Low. allows us to show that a dependence of the bare charge α0 on the observed one α is, contrary to an often quoted paradox, consistent with the renormalization group. We demonstrate that in the cases when, the bare or the observed mass of the electron is null or infinite, which should correspond to determined values of α the formal perturbation expansion in α0 of the photon vacuum polarization tensor is no more divergent than a single power of a logarithm. We finally determine wherein lies the failure of the argument leading to the paradox stated above and discuss its implication.

The pair distribution function of a quantum fluid

The theory based on an exact integral equation for the pair distribution function given by Van Leeuwen, Groeneveld and de Boer is applied to a quantum fluid with a short-range intense repulsion and a long-range weak attraction. The pair distribution function is calculated for several density values at T = 0 K. The results are compared with the experimental data for liquid helium -4 obtained by Henshaw.