Radiative transfer in a moving medium

We illustrate first, for a one-dimensional medium, how to generalize the concept of the probability of quantum exit to the case of a moving atmosphere. The transfer of radiation takes place in a two-level atom with total redistribution in frequency at each scattering. The values of the reflection coefficient and of the source function at the surface of a semi-infinite medium with constant properties (especially concerning the velocity gradient) are given. The results are then extended to the three-dimensional case in the Appendix.     

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.     

Absorption and emission line profile coefficients of multilevel atoms—I. Atomic profile coefficients

The line profile coefficients for absorption and emission appearing in the radiative transfer equation are formulated in terms of atomic line profile coefficients and velocity distribution functions. In order to derive the atomic profile coefficients of a multilevel atom, one defines generalized atomic redistribution functions that describe the correlations between photons involved in consecutive radiative transitions of the atom. Besides their dependence on the radiation field, the atomic line profile coefficients of a multilevel atom depend on the velocity distributions of the atoms in the various excitation states, in contrast to the case of a two-level atom where only the radiation intensity but not the velocity distributions affect the atomic emission profile. Closed expressions of the atomic profile coefficients in terms of generalized redistribution functions are obtained if stimulated emissions are neglected, and one is led to an iterative approximation scheme if stimulated emissions are taken into account. The possibility of a nonlocal character of the atomic profile coefficients is pointed out, and the effect of elastic, velocity-changing collisions with excited atoms is discussed. A major aim of this paper is to draw attention to the fact that ordinary redistribution functions that describe only the correlations between the absorbed and reemitted photons in the same spectral line are not sufficient to formulate the line profile coefficients of a multilevel atom.     

Effective deexcitation time of lithium vapor

The method of numerical simulation was used to study the effective deexcitation time of the resonance transition in lithium vapor at the wavelength λ = 670.776 nm. Taking into account the partial frequency redistribution and the cylindrical geometry of the medium, the rate equations of the population balance of a two-level atom were solved along with the equation of radiation transfer. The light scattering by an atom was simulated by a linear combination of the angle-averaged frequency distribution functions R _II and R _III. The Biberman-Holstein escape factor was calculated as a function of the optical thickness of lithium vapor for different geometries of the luminescing gas and different models of the frequency redistribution functions.     

Diffusion approximations for the scattering of resonance-line photons: Interior and boundary layer solutions

Resonance-line scattering in static low density media with large optical thickness has a diffusive behavior in both space and frequency because photons belonging to the Lorentzian wings of the line may be scattered almost monochromatically a very large number of times. This diffusive behavior holds on frequency scales and spatial scales, χ_c and τ_c, much larger than the scales associated with one elementary scattering of a wing-photon.A method developed for diffusion approximations in neutron transport theory, suitably generalized to handle diffusion in frequency space, is applied to the case of conservative scattering in a bounded medium with interior sources and zero incoming radiation. The method is to separate the line radiation field into an interior part and a boundary layer part which goes to zero in the interior. Each part is expanded in terms of a small parameter ?, which is the ratio of the mean free-path at frequency χ_c to the characteristic spatial scale τ_c.It is shown that the leading term in the interior asymptotic expansion is isotropic, zero on the boundary, and obeys a space and frequency diffusion equation. In the boundary-layer expansion, the leading term is of order ? and is a solution to a monochromatic transfer equation in a semi-infinite, plane-parallel medium. The emergent radiation field is shown to be of order ? and proportional to the gradient of the interior solution at the boundary. Its angular dependence, in the case of isotropic scattering in the atom frame, is given by the Ambartsoumian H-function. A comparison is presented between numerical solutions of the full transfer equation and asymptotic solutions. Non-conservative scattering and time-dependent problems are briefly discussed.     

Formation of the contour of a spectral line upon partial frequency redistribution

The problem of the radiation transfer in an atomic gas in a laser radiation field is numerically solved under the conditions of partial frequency redistribution and cylindrical geometry of the medium. The light scattering by atoms is modeled by the angle-averaged distribution function R _III. The influence of the partial frequency redistribution on the formation of the sodium emission line λ = 589.6 nm and on the characteristic deexcitation time of the sodium vapor is studied in relation to the vapor optical thickness.     

Radiative quenching and excitation of metastable states upon differential scattering of atoms: I. Uniform quasi-classical theory

A theory of radiative-collisional transitions between the ground and metastable ¹S states of a colliding atom is proposed. The theory uses the uniform quasi-classical approximation generalized to the case of spherically asymmetric interactions. The theory takes into account the angular momentum of an emitted (absorbed) photon and allows one to calculate the total and differential scattering cross sections in a wide range of radiation frequencies including both wings and the center of the line of a forbidden atomic transition. The range of admissible collision energies and intensities of an external radiation field is restricted by the use of the adiabatic approximation, as well as the approximation of a weak coupling between the ground and excited states of a quasi-molecule, the potentials of which are assumed to be monotonically repulsive.     

Dynamics of spontaneous radiation of atoms scattered by a resonance standing light wave

The scattering of atoms by a resonance standing light wave is considered under conditions when the lower of two resonance levels is metastable, while the upper level rapidly decays due to mainly spontaneous radiative transitions to the nonresonance levels of an atom. The diffraction scattering regime is studied, when the Rabi frequency is sufficiently high and many diffraction maxima are formed due to scattering. The dynamics of spontaneous radiation of an atom is investigated. It is shown that scattering slows down substantially the radiative decay of the atom. The regions and characteristics of the power and exponential decay are determined. The adiabatic and nonadiabatic scattering regimes are studied. It is shown that the wave packets of atoms in the metastable and resonance excited states narrow down during scattering. A limiting (minimal) size of the wave packets is found, which is achieved upon nonadiabatic scattering in the case of a sufficiently long interaction time.     

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.     

Generalized Drude scattering rate from the memory function formalism: an independent verification of the Sharapov-Carbotte result

An explicit perturbative computation of the Mori’s memory function was performed by Götzeand Wölfle (GW) to calculate generalized Drude scattering (GDS) rate for the case ofelectron-impurity and electron-phonon scattering in metals by assuming constant electronicdensity of states at the Fermi energy. In the present investigation, we go beyond thisassumption and extend the GW formalism to the case in which there is a gap around theFermi surface in electron density of states. The resulting GDS is compared with a recentone by Sharapov and Carbotte (SC) obtained through a different route. We find goodagreement between the two at finite frequencies. However, we find discrepancies in the dcscattering rate. These are due to a crucial assumption made in SC namely ω ? | Σ(?+ ω) ? Σ^?(?)|. No such high frequency assumption is made in the memory functionbased technique.     

Relationship between cross section and matrix normalization of polarized radiation scattering

A simple relationship between the total scattering cross section and the normalizing constant of the scattering matrix for the general case of an arbitrary scattering particle and elliptically polarized incident radiation is obtained. The polarized radiation is described by the Stokes parameters I, Q, U, V. The obtained relationship is a consequence of two forms of energy conservation. The first one is in terms of the total scattering cross section. The other one involves the normalizing constant of the scattering matrix. The obtained relationship contains dimensionless integrals of the radiation scattered over all directions of scattering. The integrals depend on the elements of the first row of the scattering matrix and on the relative values of the Stokes parameters of the incident radiation. In the case of cross section, the incident radiation is assumed to be a plane wave. In the case of normalization constant, the incident radiation is assumed to be a convergent beam. The possible dependence of the scattering integrals on specificities of the particle illumination is taken into account in the obtained relationship. The relationship may be helpful in the various cases. So, the relationship allows one to determine any of the two characteristics of the scattering process under investigation, cross section or normalizing constant, via the other one. The relationship can be used for obtaining the scattering integrals and for analyzing the influence of the incident radiation polarization on cross section and normalizing constant.     

Neutron optics of strongly absorbing media and interaction of long-wave neutrons with gadolinium films

The modern state of neutron optics of absorbing media is briefly surveyed. In all probability, there are no physics arguments that would constrain, in the case of strong absorption, the applicability of the commonly accepted Fermi-Foldy dispersion law for neutron waves. In accord with previously known results, it is found that the coefficient of reflection of neutrons from the boundary of a strongly absorbing medium tends to unity with decreasing velocity of neutrons incident on this medium. At low neutron energies peculiar to the case of ultracold neutrons, the complex scattering length for neutron-nucleus interaction proves to be constant, whence it follows that the cross section for neutron capture by a free nucleus obeys the 1/v law. The cross section for the analogous process on nuclei within a medium is described by the 1/v′ law, where v′=?k′/m, with k′ being the real part of the neutron wave number in the medium. As the incident-neutron velocity v decreases, the velocity v′ in a medium tends to some limiting value. From the coefficient of reflection of cold neutrons that is measured as a function of the wavelength and the angle of incidence, a refined value is found for the real part of the scattering length for neutron interaction with gadolinium nuclei. An experiment was performed where ultracold neutrons were transmitted through thin samples containing natural gadolinium. In analyzing the results of this experiment, use was made of the value found here for the real part of the neutron-nucleus scattering length. The experiment indicates that the imaginary part of the scattering length is a constant or, what is the same, that, for neutron velocities ranging from 4 to about 120 m/s, the 1/v law for the cross section for neutron capture by a free nucleus is valid to within 6%.     

Bremsstrahlung von Elektronen im Feld neutraler Atome

The process of deceleration of electrons by the field of neutral atoms is investigated by the aid of a model approximating the neutral atom by a constant potentialU in the interior of a sphere of radiusγ. The free-free emission coefficient is determined in the general case for an isothermal plasma; in the special case of an impenetrable sphere it is compared with the emission coefficient of the ion bremsstrahlung as obtained by the “soft” approximation. The results show that electron-atom radiation may not be neglegible as compared with electron-ion radiation, and this fact may be used to explain the experimentally observed increase of the emission coefficient with pressure which is more rapid than linear. This is verified numerically in the limiting case of an impenetrable sphere. In addition the bound-free absorption coefficient is determined both in general and in the special case of the H^? ion; a comparison is made with the absorption coefficient as obtained byChandrasekhar. By the aid of the method of “best approximation” given in the concluding part of the present paper the quantitiesU andγ may be determined for any neutral atom.     

Two-photon bremsstrahlung processes in atoms: Polarization effects and analytic results for the Coulomb potential

The partial wave analysis of two-photon free-free (bremsstrahlung) electron transition cross sections during scattering by a static potential U(r), as well as by an atom with a nonzero angular momentum, is carried out. The dipole interaction with radiation is taken into account in the second order of perturbation theory for the general case of elliptic polarization of photons. The polarization and angular dependences of the two-photon potential scattering amplitude is presented as a combination of the scalar product of electron momenta and photon polarization vectors and five atomic parameters containing Legendre polynomials of the scattering angle as well as radial matrix elements depending on the initial (E) and final (E′) electron energies. The results are applicable both for spontaneous double bremsstrahlung at nonrelativistic energies and for induced absorption and emission in the field of a light wave. Specific polarization effects (circular and elliptic dichroism) are analyzed for two-photon bremsstrahlung processes associated with the interference of the Hermite and anti-Hermite parts of the amplitude and depending on the sign of photon helicity. The limiting cases of high and low photon frequencies are investigated analytically, and the asymptotic forms of radial matrix elements and amplitudes for the general form of the U(r) potential are determined. Closed analytic expressions are derived for the radial matrix elements of the Coulomb potential in the form of integrals of hypergeometric function, and singularities are singled out in explicit form for E′ → E. The methods of approximate calculation of the radial matrix elements are discussed, and the results of their exact numerical calculation, as well as angular distributions and the cross sections of induced one-and two-photon emission and absorption, are given for the case of the Coulomb potential. The numerical results show that dichroism effects are quite accessible for experimental observations.     

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.     

Solid-phase reactions limited by an energy barrier

The reactions of formation and decay of defect-impurity complexes in solids are investigated theoretically on the basis of the generalized Liouville equation in phase space. The use of this equation (instead of the diffusion equation ordinarily used) yields an expression for the total rate constants of the reaction of complex formation which correctly takes into account both factors limiting the reaction rate in the general case: the diffusional mobility of the constituents and the energy barrier of the reaction. A comparison is made with the Waite theory, which is based on the diffusion equation and takes into account the presence of an energy barrier for the reaction by means of a radiation boundary condition on a reaction sphere. It is shown that the Waite method of taking the energy barrier into account leads to a systematic and substantial underestimate of the barrier-limited part of the total rate constant of the reaction. The possibility of generalizing the theory developed in this paper to the case of reactions which can involve not only atoms and defects but also the charge carriers in semiconductors (electrons and holes) is discussed. Zh. Tekh. Fiz. 67, 33–38 (November 1997)     

Velocity redistribution by standing waves

This paper considers the modifications of the atomic velocity distribution imposed by a standing wave. The recoil due to induced and spontaneous processes provides an effective force on the particles. We formulate the general problem of a two-level atom in the field of two counter-propagating waves. We derive the rate equation limit for these equations and show how it can be generalized to treat a broad-band source of radiation. The connection with generalized relaxation theory is discussed. The ensuing integral equations are solved numerically for various cases of cooling and heating.     

驻波场中相位迟后的偶极力

计算了二能级原子在一个驻波场中的辐射压力，给出了在不对空间平均时的辐射压力的普遍表达式，指出辐射压力的零级项应应于受激压力，而一级项对应于偶极力，高队项则可忽略。文中详细地计算了各种不同情况下的偶极力和相位，指出对于不同速度，相位有不同的迟后，这样的结果将严重地影响原子在驻波场中的动力学，用光子再分配模型简单地对有力的相位随速度的变化作了解释。     

A theory of a receiving antenna in a moving isotropic plasma

We analyze the response of a dipole antenna to the noise-like and/or regular (quasimonochromatic) plasma oscillations and waves. The antenna is immersed in an isotropic plasma moving with velocity greater than the electron thermal velocity. In the case of a noise field, we calculate the squared spectral power density of the noise voltage at the input of a receiving antenna for frequencies close to the electron plasma frequency. It is shown that the main contribution to the noise is made by the radiation due to the excitation of waves at anomalous Doppler frequencies. In the case of an incident monochromatic wave, the mean square voltage at the antenna input is calculated as a function of the wave frequency and angle of arrival. It is shown that the effective antenna length can differ strongly from the geometrical length of the dipole. This fact results from the dispersion of longitudinal waves ensuring that many plane waves (a continuum, in the limiting case) contribute to the re-radiated field for a given direction of propagation of the radiation energy.     

Calculation of spectral and Rosseland paths in a plasma with multiply charged ions based on a statistical approach

The semiclassical approach to determine the Fourier components of the electron dipole moment disregarding polarization (noninteracting electron model) is used for analyzing ion oscillator strengths and determining the radiation properties of plasmas consisting of multiply charged ions of heavy elements. The oscillator strength distribution df/d?? (proportional to the photoabsorption cross section) is calculated as a function of the degree of ionization and self-similar frequency ?? = ??/Z. It is found that for low degrees of ionization, function df/d?? for an ion is close to function df/d?? for a neutral atom; upon an increase in the degree of ionization, regions are formed in which df/d?? = 0 (transparency windows) and the photoabsorption cross section for high degrees of ionization differs from zero only in small frequency ranges. The resultant distribution of the ion oscillator strengths is used for calculating the polarizability of ions as a function of frequency and the cross section of radiation scattering on ions. For a gold plasma, the absorbance and opacity (both spectral and averaged according to Rosseland and Planck) are calculated. The results of computing the paths and absorption coefficients coincide in order of magnitude with the available data. The effect of scattering on the Rosseland path is estimated.