The scattering of line radiation—III. The source function

After discussing the problems encountered in any attempt to interpret frequency-dependent source functions in terms of the population numbers, a method is developed for doing this to a certain class of frequency-dependent source functions. It is pointed out that, unlike the frequency-independent source functions, an analysis of this nature on the frequency-dependent source functions is not generally practicable and can be done in this case only by virtue of the particular form assumed for the absorption coefficient. Solutions in the form of source functions are presented and discussed for a semi-infinite uniform atmosphere for five different forms of frequency redistribution during the scattering event. The function ∫^∞₀S(ν)φ(ν) dν is then calculated and subsequently is used to obtain the departure parameters of the populations of the energy states. The different population distributions through the atmosphere, that can be deduced from these quantities, are compared for the different cases of frequency redistribution used. Because of the nature of the results obtained, the feasibility of setting up iterative schemes to obtain frequency-dependent source functions for more complex problems is also considered.     

The redistribution function for scattering by relativistic electrons

The frequency redistribution function r(ν_i, ν_f) for scattering by relativistic electrons is derived, when this scattering is isotropic in the electron rest frame and recoil effects are neglected. Unlike previous derivations, we obtain for r(ν_i, ν_f) an expression which reduces exactly to its classical value for a non-relativistic gas.     

Spectrum line profiles: A generalized Voigt function including collisional narrowing

Reduction of collision-narrowed line profile functions to dimensionless form analogous to the standard form of the Voigt profile simplifies mathematical treatment. New quadratures are given which allow the efficient calculation of collision-narrowed profiles over the full physically realizable range of both collisional broadening and narrowing.     

Non-coherent scattering in subordinate lines: II. Collisional redistribution

A suitable form of the laboratory-frame redistribution function for the resonance scattering in subordinate lines, allowing for the radiative as well as collisional broadening of both atomic levels involved, is derived. Starting from the quantum-mechanical atomic-frame redistribution of Omont et al., we rewrite it in the form of a linear combination of two redistribution functions r_V and r_III, discussed in the preceding paper. This form, which is a direct generalization of that for resonance lines, permits both a simple physical interpretation and an effective numerical evaluation of the corresponding laboratory-frame redistribution. The case of spatial degeneracy is also discussed. Considerable attention is drawn to a discussion of the basic physical assumptions used, namely, the impact approximation and the Maxwellian velocity-distribution for the lower state atoms. It is shown that both of these assumptions are reasonably well satisfied for a variety of cases of astrophysical interest. Finally, in the Appendix we try to clarify the question of the normalization of the redistribution functions (with respect to inelastic collisions), as well as the question of practical equivalence between the astrophysical applications of the results of Omont et al. and those obtained by Yelnik and Voslamber.     

A different approach to the generalized coherent-potential aproximation; Multiple scattering formalism

The single site coherent-potential approximation (SCPA) is generalized in order to treat effects due to clusterization of scatterers, off-diagonal disorder, crystal field and/or short range order. Present formalism is a straightforward generalization of Soven's original method i.e. the multiple scattering formalism by which he derived the SCPA.     

Stimulated Raman scattering pulses. I. Weak scattering

Rate equations are used to compute a weak stimulated Raman scattering pulse in the case where the scattered radiation energy can be neglected in the over-all energy balance. Expressions for the shape, position and height of the maximum, duration, and spectral width of the scattering pulse as functions of the parameters of the driving laser, scattering material, and observation conditions are derived.     

Non-coherent scattering in subordinate lines: IV. Angle-averaged redistribution functions

It is demonstrated that a simple Gaussian quadrature over the scattering angles provides a sufficiently accurate and stable method for evaluating all the angle-averaged redistribution functions R_i(x′, x) (i = I–V). We display graphically the functions R_II,III,V and discuss in detail the behaviour of the newly calculated redistribution R_V(x′, x). It is shown that this function exhibits important coherence peaks in the line-wing region which can affect the formation of subordinate lines. An approximate form of R_V, analogous to that of Jefferies and White, is briefly discussed.     

Non-coherent scattering in subordinate lines: III. Generalized redistribution functions

The concept of the redistribution function, which was originally introduced in order to describe a correlation between frequencies and directions of the absorbed and emitted photon in resonance scattering, has been extended to other resonance two-photon processes including resonance Raman scattering, resonance two-photon absorption and emission, and inverse Raman scattering. We have derived, within the frame of the impact approximation, the appropriate form of the generalized redistribution function. Using a suitable formalism, the generalized redistribution function takes the same form for all types of two-photon processes and contains all the redistribution functions, considered previously, as various limiting cases. In analogy to Hummer's original scheme of redistribution functions, we have derived a similar set of generalized redistribution functions, denoted as p_i(i = I, II, III, IV, V), and we have shown that the most general case is described by a linear combination of p_III and p_V, analogously to the previous results. Explicit formulae for the velocity-averaged (laboratory-frame) generalized redistribution functions p_i(i = I, II, III, IV, V) are given and possible numerical methods for their evaluation are briefly indicated.     

Non-coherent scattering in subordinate lines: A unified approach to redistribution functions

Within the scope of the quantum-mechanical result of Omont, Smith and Cooper, we have derived a laboratory-frame redistribution function for the scattering of radiation in subordinate lines where both atomic levels are radiatively broadened. This new function, denoted as R_V(x′, n′; x, n), is the correct counterpart of Hummer's redistribution function R_IV. Moreover, it follows from the formalism developed here that R_V contains all known redistributions R_i(i=I, II, III, IV) as special cases and we may therefore speak in some sense of a unified redistribution function. A simple method has been tested for the evaluation of R_V and our numerical examinations of the properties of R_V served, among others, as an independent verification of the results obtained by Reichel and Vardavas for R_III and R_IV. Finally, it is shown in the conclusion to this paper that the function R_V, discussed here, can be applied not only to the radiative redistribution, but also to a more general case of a collisional redistribution in subordinate lines.     

A generalized G‐function for the Quantum Rabi Model

The analytical solution of the quantum Rabi model is based on a transcendental function , the zeros of which determine the eigenenergies. is generalized here to a function , which allows a much better numerical control of the high‐energy part of the spectrum by an appropriate choice of the complex parameter z. Additionally, it is shown that all zeros of correspond to eigenvalues of the Hamiltonian as well as the zeros of for imaginary z.     

On generalized Clifford groups. I

It is shown that using the basis elements of the generalized Clifford algebra C^m_n we can construct a group G?^m_n called the generalized Clifford group (G.C.G.) which is a generalization of the Dirac group of the 16 Dirac matrices and their negative counterparts. We have studied the irreducible representations of G.C.G. and we have found the connection with the group of linear transformations leaving invariant the expression $\text{∑}\text{i=1}\text{n}\text{(x}{}^{\mathrm{i}}\text{)}{}^{\mathrm{m}}$ for m>2.     

A generalized reggeon field theory incorporating hard scattering effects

Guided by the parton interpretation of RFT and by QCD, we propose an RFT where the pomeron field depends both onb and onQ ², the bare propagator describing an inhomogenous random walk inb and in lnQ ². Here,Q ² measures the virtuality of the inelastic processes which generate the pomeron exchange amplitude as their shadow. The asymptotic behaviour of such a theory should be different from that of standard RFT, at and above the critical point. Arguments are given in favour of a supercritical pomeron.     

Radiation scattering for a general law of frequency redistribution

Some relatively simple radiative-transfer problems are considered in the general case of non-coherent scattering. The approach proposed in the paper is based on representation of the redistribution function in the form of a bilinear expansion with respect to an appropriately orthonormalized set of functions. The new simple, but at the same time rather accurate, method is suggested for solving problems of spectral-line formation to obtain the frequency redistribution. Numerical results are given for profiles of spectral lines formed by diffuse reflection from a semi-infinite medium, as well as for those formed in an isothermal atmosphere. These results illustrate the influence of the various approximations for the redistribution law on the solution of the problem. In particular, the effect of the complete redistribution approximation on the observed characteristics of the radiation field is investigated.     

Double scattering of fast particles. I

The scattering of fast particles by two potential centers is considered in this paper. An approximation which can be used in the analysis of double scattering by weakly bound atoms is constructed.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 8, pp. 123–128, August, 1971.     

Eikonal expansion in electron scattering: I. Elastic scattering

A systematic eikonal expansion for the scattering of high-energy electrons from nuclei is derived which starts from the iterated Dirac equation. The resulting scattering amplitude is written in an impact parameter representation depending on eikonal phases which are proportional to inverse powers of the energy. The first two correction terms to the leading Glauber-Baker amplitude are calculated. For a Coulomb potential they agree with a sinθ-expansion of the relativistic Coulomb scattering amplitude. In the case of scattering from an extended charge distribution at sufficiently high energies numerical partial wave calculations are accurately reproduced.     

Redistribution effects on line formation in moving stellar atmospheres—I. The isotropic redistribution function RI−A(x′, x; v)

The effects of photon frequency redistribution in semi-infinite and slab, differentially-moving atmospheres are examined for the case of pure Doppler broadening as described by the angle-averaged redistribution function. Results are obtained for the frequency dependent non-LTE line source functions and their corresponding emergent line intensities. These emergent line intensities are compared with those obtained assuming complete redistribution and are found to be exceedingly different in the line core.     

Dielectronic recombination of positive ions—I. A generalized cascade theory

A generalized theory of dielectronic recombination is formulated, explicitly incorporating the higher-order effect of cascade transitions to all the captured final states which are stable against Auger emission. The previously-used rate formulas contain only the two-step process of target excitation followed by a single radiative transition. The formal reaction theory is employed to precisely define the contribution of the successive open channels to energy shifts and resonance widths. Correct expressions are derived for the radiative and Auger branching ratios associated with each stage of the cascade. Various angular momentum averaging procedures and the effect of the spectator electrons in the participating subshells are discussed.     

Retrieval of Green's function and generalized optical theorem for the scattering of complete dyadic fields

Green's function retrieval has been widely used in different research fields due to the fact that the Green's function can be extracted by cross-correlating the records at two receivers. In this paper, the retrieval of the dyadic Green's function is studied by investigating the representation theorem. The generalized optical theorem for the dyadic fields is derived based on the elastic dynamic interferometric equation. By addressing the cross-correlation recorded at two receivers, the important role of the generalized optical theorem and energy equipartition in retrieving the exact Green's function is shown. The presented derivation also shows the Newton-Marchenko equation holdsif the condition of equipartition is not satisfied.     

Three-dimensional radiative transfer in an anisotropically scattering, semi-infinite medium: generalized reflection function

Three-dimensional radiative transfer in an anisotropic scattering medium exposed to spatially varying, collimated radiation is studied. The generalized reflection function for a semi-infinite medium with a very general scattering phase function is the focus of this investigation. An integral transform is used to reduce the three-dimensional transport equation to a one-dimensional form, and a modified Ambarzumian's method is applied to formulate a nonlinear integral equation for the generalized reflection function. The integration is over both the polar and azimuthal angles; hence, the integral equation is said to be in the double-integral form. The double-integral, reflection function formulation can handle a variety of anisotropic phase functions and does not require an expansion of the phase function in a Legendre polynomial series. Complicated kernel transformations of previous single-integral studies are eliminated. Single and double scattering approximations are developed. Numerical results are presented for a Rayleigh phase function to illustrate the computational characteristics of the method and are compared to results obtained with the single-integral method. Agreement between the two approaches is excellent; however, as the transform variable increases beyond five the number of quadrature points required for the double-integral method to produce accurate solutions significantly increases. A new interpolation scheme produces accurate results when the transform variable is large.     

Three-dimensional radiative transfer in an isotropically scattering, semi-infinite medium: generalized reflection function

The focus of this study is the generalized reflection function of multidimensional radiative transfer. The physical situation considered is spatially varying, collimated radiation incident on the upper boundary of an isotropically scattering, semi-infinite medium. An integral transform is used to reduce the three-dimensional transport equation to a one-dimensional form, and a modified Ambarzumian's method is applied to formulate a nonlinear integral equation for the generalized reflection function. The resulting equation is said to be in double-integral form because the integration is over both angular variables. Computational issues associated with this generalized reflection function formulation are investigated. The source function and reflection function formulations are compared, and the relative merits of the two approaches are discussed.