Frequency Redistribution Research Articles

Koedam β factors revisited

A Koedam β factor makes it possible to compute the total output power in line radiation from a positive column discharge using a single radiance measurement normal to an aperture in the wall. The results of analytic derivations of β factors are presented for columns withuniform (β = 1.0) and parabolic (β = 0.75) excitation rates per unit volume and with negligible opacity. A Monte Carlo code for simulating radiation trapping with a spatially uniform density of absorbing atoms is then used to determine β factors as a function of opacity. The code includes partial frequency redistribution and a Voigt line shape with radiative broadening, resonance collisional broadening, and Doppler broadening. The resulting β factors are found to be nearly independent of opacity over a wide range of column radii for spectral line shapes dominated by Doppler broadening or by resonance collisional broadening. Additional Monte Carlo simulations are used to study β factors as a function of a non-uniform density of absorbing atoms from radial cataphoresis with line shapes dominated by Doppler broadening, foreign gas broadening, and resonance collisional broadening. Radial cataphoresis is found to increase β factors in all cases. Geometrical effects, refraction, and imperfect transmission at the glass wall are studied and found to decrease β factors.

Non-LTE line transfer with spatially correlated turbulence: Fluctuation against average emergent radiation field

The increasing interest in using turbulence to explain anomalies observed in stellar spectral lines is discussed. This turbulence occurs at a finite spatial scale, a key parameter. In our previous work on non-LTE radiation transfer in the presence of a turbulent velocity field (Loucif & Magnan 1982), we calculated individual profiles representing intensity fluctuations around the averaged line shape for photon diffusion with complete redistribution in frequency (CFR), a function well-adapted to media where collisional excitations dominate. We now generalize the problem by accounting for partial redistribution in frequency (PFR). To show the effect of the spatial correlation in the velocity field on the emergent line profiles as well as the importance of fluctuations in the medium's optical properties against the average, exhaustive calculations of emergent non-LTE line profiles are performed in the frame of a systematic comparison between the two processes of redistribution for the three following cases: static medium, averaged turbulent medium, and deviation from the average. In all cases, we confirm the differences and similarities in CFR and PFR descriptions previously derived for static media. These results seem reasonable since light scattering occurs at the atomic level–a scale well below the macroscopic size of the turbulence scale length. In addition, pronounced differences in the emergent radiation field are obtained between the average case and the so-called deviation from average for both frequency redistribution functions. These systematic differences rule out classical treatment of line formation in turbulent fields that assume averaged media to model observations obtained over an exposure time shorter than the turbulent field time scale (). Beyond the different assumptions made to describe the non-LTE line formation in a turbulent medium, our model provides a simple and coherent description of realistic media applicable to different astrophysical observational backgrounds. For this purpose, a unified picture is proposed for the radiation field formation, invoking mainly the medium opacity, the turbulence correlation length, and the velocity. Observations of variable objects, or along extended lines of sights, allowing a high probability for such deviations to occur (e.g., quasars), are suitable for application of our model.

Improving the Numerical Modeling of the OiResonance Triplet in the Solar Spectrum

We present a slight modification to an existing multilevel radiative transfer code that allows us to include the effects of frequency cross redistribution (XRD) between lines sharing the same upper level. With this improved code, we calculate theoretical profiles of the O I resonance triplet lines at 130 nm through a hydrostatic model of the average quiet Sun. The width of the calculated XRD profiles show good agreement with an observed, spatially averaged disk-center spectrum obtained with the high-resolution telescope spectrometer (HRTS) spectrograph. This is in stark contrast to profiles calculated with complete frequency redistribution (CRD) and ordinary partial frequency redistribution (PRD), which have Lorentzian wings that are too broad. We find deep central reversals, contrary to most observed profiles, but we note that this discrepancy is to a large degree the result of limited instrumental spectral resolution.

The Effect of Coherent Scattering on Radiative Losses in the Solar CaiiK Line

We investigate the influence of partial frequency redistribution (PRD) on radiative cooling due to the Ca II K line in a hydrostatic model of the quiet Sun, and in a series of 20 snapshots from a chromospheric radiation-hydrodynamics simulation. The net radiative rates in the K line were calculated through these models with three different frequency-redistribution formalisms: complete redistribution (CRD), angle-averaged PRD, and angle-dependent PRD. It is found that the approximation of CRD generally leads to an overestimate of the net radiative rates because of its neglect of coherent scattering. Compared to the more realistic full angle-dependent solution, angle-averaged PRD provides accurate emergent profiles and cooling rates in the hydrostatic model and also provides accurate instantaneous profiles in the dynamical snapshots, even when considerable macroscopic velocities are present. Instantaneous angle-averaged cooling rates in the dynamical snapshots may be different at times, but on average, cooling is very similar to the angle-dependent case. In the temperature minimum, the Ca II K line provides, on average, more cooling in the dynamic chromosphere at the same column mass than in the hydrostatic model. It provides less cooling in the chromosphere. Finally, we formulate an approximation for angle-averaged redistribution in a moving atmosphere, employing the unshifted redistribution function. This approximation is easy to implement in existing numerical codes and should provide an efficient method for PRD transfer solutions in dynamic models without loss of accuracy in emergent profiles and radiative cooling rates.

Partially Coherent Scattering in Stellar Chromospheres. V. Analytic Approximations Including Background Continuum Absorption

The comoving-frame partially coherent scattering (CPCS) approximation is a heuristic approach to modeling partially coherent resonance-line scattering. It is analytic, conceptually motivated, and easy to apply, while generally maintaining roughly 20% accuracy. Here, it is extended to include the effects of continuum or dust absorption. As usual for resonance lines, a key process is Doppler diffusion, whereby photons created in the line core reach wing frequencies via the gradual accumulation of thermal Doppler shifts over many scatterings. This complex process can be idealized to yield semiquantitative results by using the single longest excursion escape model included in the CPCS approach, as described in earlier papers in this series. This paper extends the CPCS scheme by allowing the partial frequency redistribution to be truncated by continuous absorption during propagation. The result is a semianalytic approximation that reveals the mechanisms by which background opacity alters resonance-line formation and profile diagnostics. This is relevant for understanding shifts in the slope of the Wilson-Bappu effect, resonance-line profiles in quasar models, and resonance scattering in any highly opaque low-density environment with appreciable continuous absorption.

Line Formation in a Purely Scattering, Optically Thick Atmosphere

A model problem in the theory of line formation in an optically thick, purely scattering, stellar atmosphere is considered. The integral equation of radiation transfer at line frequencies is solved numerically for a two-level atom in the approximation of complete frequency redistribution in scattering. The numerical results are compared with those calculated from equations of the asymptotic theory. On the basis of the asymptotic theory, the positions of intensity maxima in a line are found for different absorption profiles.

The formation of polarized lines: Allowance for the Hanle effect

The paper formulates the standard theory for the transport of polarized radiation in the presence of resonance scattering in an atmosphere with a weak magnetic field, so that the Zeeman splitting is small compared to the Doppler line width. For an atmosphere with conservative scattering, this reduces to the Milne problem, which consists of computing a polarized radiation field in a medium with sources lying in infinitely deep layers. In the approximation of complete frequency redistribution, the problem reduces to solving a Wiener-Hopf integral equation for a (6×6)-matrix source function. Asymptotic and numerical solutions for the standard problem are obtained, including solutions for the Milne problem, for the case of a Doppler absorption profile. The line polarization profiles for the emergent flux at various angular distances from the disk center are derived, and the dependence of the limiting degree of polarization (at the line center at the disk edge) on the direction of the magnetic field is computed. For nearly conservative scattering with photon destruction probability e≪1, the limiting degree of polarization varies with e in accordance with a simple empirical law similar to that found earlier for a medium with zero magnetic field.

FT–IR and Raman spectra of Gd phosphate crystals and glasses

IR and Raman spectra of xGd 2O 3(1− x)P 2O 5 glasses ( x=0.22–0.31) are reported with emphasis on glassy Gd(PO 3) 3 and the crystalline forms emerging upon its devitrification. Increasing the concentration of Gd 2O 3 from 0.22 to 0.26 mole fractions leads to an abrupt lowering of the highest observed P–O stretching frequency of 25 cm −1. The spectral shift is related to changes in both the coordination of the lanthanide ion and the increase in metal to phosphorus ratio. The comparison of vibrational spectra of vitreous gadolinium metaphosphate ( x=0.25) with those of its crystalline analogs reveals a decrease in POP and PO 2 bond angles in glass. The changes observed are consistent with an elongation of P–O − terminal and P–O–P bridging bonds in the PO 4 tetrahedra. The redistribution of band frequencies is explained with the more covalent interaction of Gd 3+ with polyphosphate anions in glasses. The spectra of (Gd 2O 3) 0.27(P 2O 5) 0.73 and (Gd 2O 3) 0.31(P 2O 5) 0.69 glasses contain the characteristic bands of the metaphosphate structure along with features due to vibrations of PO 3 end groups. Their presence is interpreted in terms of the progressive fragmentation of the polyphosphate chains at these glass compositions.

Multilevel Radiative Transfer with Partial Frequency Redistribution

A multilevel accelerated lambda iteration (MALI) method for radiative transfer calculations with partial frequency redistribution (PRD) is presented. The method, which is based on Rybicki & Hummer's complete frequency redistribution (CRD) formalism with full preconditioning, consistently accounts for overlapping radiative transitions. Its extension to PRD is implemented in a very natural way through the use of the Ψ operator operating on the emissivity rather than the commonly used Λ operator, which operates on the source function. Apart from requiring an additional inner computational loop to evaluate the PRD emission-line profiles with fixed population numbers, implementation of the presented method requires only a trivial addition of computer code. Since the presented method employs a diagonal operator, it is easily extended to different geometries. Currently, it has been implemented for one-, two-, and three-dimensional Cartesian grids and spherical symmetry. In all cases, the speed of convergence with PRD is very similar to that in CRD, with the former sometimes even surpassing the latter. Sample calculations exhibiting the favorable convergence behavior of the PRD code are presented in the case of the Ca II H and K lines, the Mg II h and k lines, and the hydrogen Lyα and Lyβ lines in a one-dimensional solar model and the Ca II resonance lines in a two-dimensional flux-sheet model.

Gas heating inside radio sources to mildly relativistic temperatures via induced compton scattering

The measured brightness temperatures of the low-frequency synchrotron radiation from intense extragalactic sources reach 1011–1012 K. If there is some amount of nonrelativistic ionized gas within such sources, it must be heated through induced Compton scattering of the radiation. If cooling via inverse Compton scattering of the same radio radiation counteracts this heating, then the plasma can be heated up to mildly relativistic temperatures kT∼10–100 keV. In this case, the stationary electron velocity distribution can be either relativistic Maxwellian or quasi-Maxwellian (with the high-velocity tail suppressed), depending on the efficiency of Coulomb collisions and other relaxation processes. We derive several simple approximate expressions for the induced Compton heating rate of mildly relativistic electrons in an isotropic radiation field, as well as for the stationary electron distribution function and temperature. We give analytic expressions for the kernel of the integral kinetic equation (one as a function of the scattering angle, and the other for an isotropic radiation field), which describes the photon redistribution in frequency through induced Compton scattering in thermal plasma. These expressions can be used in the parameter range Open image in new window [in contrast to the formulas written out previously in Sazonov and Sunyaev (2000), which are less accurate].

Direct numerical method to solve radiation trapping problems with a Doppler-broadening mechanism for partial frequency redistribution

We present a numerical method for solving the Holstein-Biberman-Payne equation with Doppler frequency redistribution. The method is based on direct time propagation of the initial distribution of excited atoms, employing the split-propagation technique. It allows a precise study of various aspects of radiation transfer phenomena occurring in an arbitrary convex three-dimensional spatial region, driven by external conditions with arbitrary time dependencies. We present some results obtained for slab and spherically shaped geometries and compare them to results from other evaluation methods. The efficiency of the method is demonstrated by determining the intensity of radiation escaping the gas cell in afterglow experiments.

Frequency redistribution effect on emissivity and intensity of Ge22+ and C5+ X-ray lasers

This work concerns the effect of frequency redistribution on X-ray laser beams. The redistribution function, calculated by extending the frequency fluctuation model to 2-photon processes, shows coherent scattering and a redistributed contribution. The spectral emissivity, corrected in order to account for this effect, is calculated for X-ray lasers, the amplifier plasma being composed either of neon-like germanium ions or hydrogen-like carbon ions. The coupled radiative transfer equation and population equations are solved consistently, and the intensity of the X-ray laser output is given for two 3p-3s lines (19.6 and 23.6 nm) of neon-like germanium. We also study the variation with plasma length (i.e. line intensity) of the emissivity of the C 5+ Balmer-α lasing line. The physical conditions of the calculations are obtained from a modelling of experiments conducted at the Rutherford Appleton Laboratory.

Doppler shift and frequency redistribution function in the laboratoryframe. Effect on the output of neon-like germanium x-ray lasers

This work is devoted to the effect of the Doppler shiftand frequency redistribution on the output of soft-x-raylasers operating in the collisional excitation scheme.Owing to the Doppler frequency detuning, the lineshapefunction as well as the redistribution function in thelaboratory frame may differ from the same functionscalculated in the atom frame. In fact, the lasing lines0-1 (19.6 nm) and 2-1 (23.6 nm) in neon-like germaniumshow an important Doppler broadening and theredistribution functions calculated in the two framesdiffer by a large amount. The emissivity, corrected inorder to account for frequency redistribution, iscalculated, and the radiative transfer equation issolved. We show that the intensities of the above linesdepend on the frame chosen.

The Analysis of the H Lyman α Emission Line Profile from Jupiter's Aurora

We present the analysis and modeling of the emission spectra of the jovian northern auroral region taken from May 28 to June 3, 1993, with the Goddard High Resolution Spectrograph on board the Hubble Space Telescope. They extend from 1204 to 1240 Â covering H Lyman α and part of the Werner and the Lyman bands of H 2. We used the 2×2 arcsec large science aperture combined with the G160M grating (spectral resolution of 570 mÂ) centred on Jupiter's central meridian. The auroral region studied extends from 50 to 60° north latitudes and from 130 to 220° System III longitudes. Within the 1 arcsec pointing uncertainty, most of the region delineated by the theoretical ovals at 5.9 and 30 jovian radii (R J) in the VIP4 model of J. E. C. Connerney, M. H. Acuna, N. F. Ness, and T. Satoh (1998, New models of Jupiter's magnetic field constrained by the Io flux tube footprint. J. Geophys. Res. 103, 11,929–11,939), including the auroral oval derived from the Wide Field Planetary Camera 2 images by J. T. Clarke, G. E. Ballester, J. T. Trauger, R. Evans, J. E. P. Connerney, K. Stapelfeld, D. Crisp, P. D. Feldman, C. J. Burrows, S. Casertano, J. S. Gallagher, R. E. Griffiths, J. J. Hester, J. G. Hoessel, J. A. Holtzman, J. E. Krist, V. Meadows, J. R. Mould, P. A. Scowen, A. M. Watson, and J. A. Westphal (1996, Far-ultraviolet imaging of Jupiter's aurora and the Io “footprint”. Science 274, 404–409), was sampled. We derive auroral brightnesses (averaged over the slit) ranging from 20 kR in the 190–220° longitude region to 30 kR in the 130–160° longitude range. We use the theoretical model developed by D. Rego, R. Prangé, and L. Ben Jaffel (1999, Auroral Lyman α and H 2 bands from the giant planets. 3. Lyman α intensity and spectral profile including radiative effects and H 2 color ratios. J. Geophys. Res. Planets. 104, 5939–5954), which calculates self-consistently the auroral Lyman α line profile for electron and proton precipitations and a given frequency redistribution function. This model shows that Lyman α profiles are not dependent upon the identity of the particles for a given penetration depth. These profiles only constrain the atmospheric H column density above the emitting layer, H col. Best agreement with the data is found with the complete frequency redistribution (CR). We derive an auroral H col of 1.3×10 16 cm −2 for all spectra with an upper limit of 5×10 16 cm −2. Precipitating protons also produce fast H atoms by charge exchange which can also be excited, but the resulting Doppler-shifted profile was not detected. Present data thus rule out protons as being the only precipitating particles. In the auroral zones, theoretical models predict that chemical reactions induced by particle precipitations often result in the production of atomic hydrogen, and thus an enhancement of the H density is expected. However, the value of H col we derive, between ∼8 and 23 times less than the equatorial values of L. Ben Jaffel, J. T. Clarke, R. Prangé, R. Gladstone, and A. Vidal-Madjar (1993, The Lyman alpha bulge of Jupiter: Effects of a non-thermal velocity field. J. Geophys. Res. Lett. 20(8), 747–750), is in good agreement with the auroral values of R. Prangé, D. Rego, L. Pallier, L. Ben Jaffel, C. Emerich, J. T. Clarke, G. E. Ballester, and J. Ajello (1997, Detection of self-reversed Lyman α lines from the jovian aurorae with the Hubble space Telescope, Astrophys. J. 484, L169–L173). This suggests that the H abundance predicted with standard atmospheric model is unrealistically large in the auroral region.

Solution of the Holstein equation of radiation trapping by the geometrical quantization technique. III. Partial frequency redistribution with Doppler broadening

We introduce an analytical method to investigate radiation trapping problems with Doppler frequency redistribution. The problem is formulated within the framework of the Holstein-Biberman-Payne equation. We interpret the basic integro-differential trapping equation as a generalized wave equation for a four-dimensional (4D) classical system (an associated quasiparticle). We then construct its analytical solution by a semiclassical approach, called the geometrical quantization technique (GQT). Within the GQT, it is shown that the spatial and frequency variables can be separated and that the frequency part of the excited atom distribution function obeys a stationary Schr\"odinger equation for a perturbed oscillator. We demonstrate that there is a noticeable deviation of the actual spectral emission profile from the Doppler line in the region of small opacities. The problem of calculating the spatial mode structure and the effective radiation trapping factors is reduced to the evaluation of wave functions and quantized energy values of the quasiparticle confined in the vapor cell. We formulate the quantization rules and derive the phase factors, which allow us to obtain analytically the complete spectrum of the trapping factors in 1D geometries (layer, cylinder, sphere) and other (2D and 3D) geometries when the separation of space variables is possible. Finally, we outline a possible extension of our method to treat radiation trapping effects for more general experimental situations including, for instance, a system of cold atoms.

Influence of magnetic fields on the coherence effects in the Nai D1and D2lines

To clarify the physical nature of the enigmatic scattering polarization in the Na i D 1 and D 2 line cores we have explored their behavior with full Stokes vector polarimetry in regions with varying degree of magnetic activity near the solar limb. These observations represent the first time that ZIMPOL II, the second generation of our CCD based imaging polarimeter systems, has been used for a scientific program. With ZIMPOL II the four Stokes images can be demodulated and recorded with a single CCD sensor such that the resulting images of the fractional polarization $Q/I$, $U/I$, and $V/I$ are entirely free from spurious features due to seeing or flat-field effects. The polarization in the cores of the lines, in particular in D 2 , exhibits dramatic and unexpected spatial variations in both $Q/I$ and $U/I$, including polarization self-reversals of the D 2 $Q/I$ core peak. As the fluctuations in the Q , U , and V parameters appear to be relatively uncorrelated, we have parametrized the profiles and made scatter plots of the extracted parameters. Comparison with synthetic scatter plots based on different theoretical models suggests that the polarization signals in the cores of the D 2 and D 1 lines have different physical origins: While the D 1 core is likely to be governed by ground-state atomic polarization, the D 2 core is dominated by the alignment of the excited state and by effects of partial frequency redistribution.

ALBEDO-SHIFTING METHOD: SCATTERING IN A LINE WITH COMPLETE FREQUENCY REDISTRIBUTION

The albedo-shifting method is used to solve the problem of radiative transfer at line frequencies. Radiative transfer with complete frequency redistribution in a plane-parallel, semi-infinite atmosphere is considered. It is shown that the method works well in this case and enables one to considerably improve the convergence in an iterative solution of the equation for the source function.

Transfer of Polarized Radiation: Nonlinear Integral Equations for I-Matrices in the General Case and for Resonance Scattering in a Weak Magnetic Field

General equations of the Wiener-Hopf type for a matrix source function with nonsymmetrical kernel matrices are considered in the form of continuous superpositions of exponentials. Certain problems in the transfer of polarized radiation reduce to equations of this kind. In general there are two different H-matrices in the theory (which are a generalization of the Ambartsumian-Chandrasekhar scalar H-function), generated by an initial equation of the Wiener-Hopf type and its analog, but with the kernel matrix and the unknown matrix of the source function being transposed. In addition there are two corresponding I-matrices, actually consisting of Laplace transforms of the matrix source functions, through which the Stokes vector of the escaping radiation is directly determined. In the problem of diffuse reflection from a half-space, the I-matrices are expressed in terms of a product of these two H-matrices, and for the latter there is a system of nonlinear equations which is a generalization of the corresponding Ambartsumian-Chandrasekhar scalar equation. In the problem of the emission of partially polarized radiation from a half-space containing uniformly distributed internal sources we have obtained a system of two nonlinear equations for the I-matrices directly. In the special case of a symmetrical kernel matrix, this system of two equations reduces to one equation. It is shown that in the case of resonance scattering in a weak magnetic field (the Hanle effect) in the approximation of complete frequency redistribution, the system of two nonlinear equations for the I-matrices (of dimension 6×6) also reduces to one nonlinear equation, although the kernel matrix for the main integral equation for the matrix source function Ŝ(τ) is not symmetrical. For this case we have found a matrix generalization of the so-called “ $$\sqrt \varepsilon$$ law,” consisting of an equation of the type Ŝ(0)ÂŜ T (0) = $$\widehat B$$ (where T denotes transposition) at the boundary of a half-space containing uniformly distributed primary sources of partially polarized radiation.

Scattering of cyclotron radiation in a moving plasma

We consider the scattering of cyclotron radiation in a plasma moving along a homogeneous magnetic field. The equation of radiation transfer in a co-moving frame is derived and two limiting cases are pointed out. In the first case of a “small” velocity gradient, the total Doppler frequency shift due to variations in the plasma velocity over the flow is much smaller than the width of the line. The second, opposite, case of a “large” velocity gradient is analogous to the Sobolev approximation in the theory of moving stellar envelopes. The solution of the transfer equation for a wind-type flow illuminated by radiation of a given intensity is obtained in the latter case, when the influence of the plasma motion on cyclotron scattering is most important. It is shown that cyclotron scattering in a moving plasma differs from the known (and qualitatively similar) problems of resonance scattering in moving stellar envelopes and cyclotron scattering in a motionless plasma permeated by an inhomogeneous magnetic field. In particular, a symmetric absorption band with residual intensity proportional to the velocity gradient appears in the spectrum of the outgoing radiation, while in these two other problems, the depth of the corresponding spectral features cannot exceed half the continuum level. Detailed qualitative analysis reveals that this difference is due to the particular form of the frequency redistribution for cyclotron scattering.

Radiation trapping of the Hg 185 nm resonance line

The decay rate of the Hg 61P1 level was measured as a function of cold spot temperature (Hg density) and buffer gas pressure in cylindrical, sealed fused silica cells. The decay rates were obtained using a time-resolved laser-induced 185 nm fluorescence experiment with multi-step excitation. Cold spot temperatures from 25 to 100 °C were studied. The Hg densities for this temperature range and with no buffer gas yield the lowest possible decay rates due to radiation trapping with partial frequency redistribution. Decay rates with argon buffer gas pressures of 3 and 30 Torr were also studied. The results are in agreement with published data from a discharge afterglow experiment. Monte Carlo simulations of radiation transport in the cells, including the effects of hyperfine and isotope structure, the effects of foreign gas broadening, and partial frequency redistribution are compared to the experimental data. Reasonably good agreement is obtained, however there is evidence of quenching of Hg 61P1 atoms in collisions with ground state Hg and Ar atoms. An analytic formula for the fundamental mode trapped decay rate of the 61P1 level, which is applicable over a substantial region of parameter space, was devised from the Monte Carlo results.