EDP Sciences logo
Subscriber Authentication Point
Search
Menu
Home All issues Volume 667 (November 2022) A&A, 667 (2022) A3 Full HTML
Open Access
Issue
A&A
Volume 667, November 2022
Article Number A3
Number of page(s) 8
Section Atomic, molecular, and nuclear data
DOI https://doi.org/10.1051/0004-6361/202243883
Published online 28 October 2022

© R. D. Rohrmann and M. V. Rueda 2022

Licence Creative CommonsOpen Access article, published by EDP Sciences, under the terms of the Creative Commons Attribution License (https://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

This article is published in open access under the Subscribe-to-Open model. Subscribe to A&A to support open access publication.

1 Introduction

Rayleigh scattering is potentially relevant to several areas of astronomical spectroscopy including cool stars, star-forming regions, mass-losing stars, exoplanets, the circumgalactic medium, and cosmology. This process significantly affects opacity and emission of the monochromatic radiation field in stellar atmospheres at temperatures of a few thousand Kelvin (Marigo & Aringer 2009). It also provides a diagnostic tool for determining geometrical parameters of double stellar systems containing a giant star (Isliker et al. 1989; González-Riestra et al. 2003; Skopal & Shagatova 2012), and the scale height and composition of exoplanetary atmospheres (Lecavelier Des Etangs et al. 2008; Sing et al. 2015; Dragomir et al. 2015). Rayleigh scattering is especially important in spectroscopy for gaseous nebulae and emission regions of active galactic nuclei (Nussbaumer et al. 1989; Ferland et al. 2017). In addition, scattering of light from isolated hydrogen atoms has an impact on the cosmic microwave background anisotropies. The coupling of photons to neutral hydrogen through Rayleigh scattering during the recombination epoch may have had significant effects on the microwave background fluctuations’ spectrum, providing in this way information about the formation of atoms in the early Universe (Scheuer 1965; Gunn & Peterson 1965; Peebles & Yu 1970; Yu et al. 2001; Bach & Lee 2015; Alipour et al. 2015; Beringue et al. 2021).

A coherent scattering cross section including resonances is required for radiation hydrodynamics simulations of late-type stars and accretion disks irradiated by the central star (Hayek et al. 2010; Hirose et al. 2022). It is an important ingredient in self-consistent magneto-hydrodynamical models of solar-type atmospheres which include a chromosphere (Hansteen et al. 2007; Hayek et al. 2010), and in detailed hydrodynamical models of giant stars (Collet et al. 2008; Hayek et al. 2011). Scattering resonances of the Lyman series may affect the temperature structure in the upper atmospheres of cool stars. Specifically, coherent line scattering reduces the temperature by several hundred degrees in the high atmosphere of solar-type stars (Hayek et al. 2010, Hayek et al. 2011). This effect could be even the most severe in the low-density giant star atmospheres, resulting in a significantly steeper temperature mean gradient. On the other hand, resonant scattering processes affect the transfer of stellar radiation in protoplanetary disks and have implications on their photochemistry (Neufeld 1991; Bethell & Bergin 2011; Heays et al. 2017).

The present work is motivated by the incompleteness in the compiled set of theoretical Rayleigh cross sections from atomic hydrogen, which is typically limited to wavelengths redward of the Lyman-α resonance (Mittleman & Wolf 1962; Isliker et al. 1989; Lee & Kim 2004; Lee 2005; Fišák et al. 2016; Colgan et al. 2016; Hirose et al. 2022). An exact theoretical description of the Rayleigh scattering by one-electron systems was developed by Gavrila (1967). However, his calculations partially covered only a few resonances and they did not consider damping effects, yielding unphysical singularities. Scattering cross sections from atomic hydrogen have also been calculated by other authors (Nussbaumer et al. 1989; Sadeghpour & Dalgarno 1992; McNamara et al. 2018), but their results have only been presented in a graphical form or over a limited range of photon frequencies and, therefore, they are not useful for computing purposes. In addition, such evaluations neglected fine-structure energy shifts due to relativistic and spin-orbit interaction effects.

The aim of the present work is to expand on the availability of the Rayleigh cross section of neutral hydrogen atoms in the high-photon-energy regime, including an arbitrary number of resonances with excited bound states and taking into account the fine structure and finite lifetimes of such states. Current evaluations correspond to an isolated atom at rest and, therefore, they do not include perturbations arising from collisions with other atoms or Doppler shifts either due to translational motion of the radiating atom.

Rayleigh scattering represents a second-order photon-electron process in the Kramers-Heisenberg dispersion theory (Kramers & Heisenberg 1925; Waller 1929; Chen & Kotlarchyk 2007). The cross section for photons scattered by 1s bound electrons can be computed using the complex atomic polarizability (Bonin & Kresin 1997). Fundamental properties of polarizability and scattering process were first treated by Placzek (1934). Real and imaginary contributions to polarizability are expressed as sums of allowed dipole transitions to intermediate np states, whose well-known oscillator strength values allow for an exact analytic evaluation (Penney 1969). The imaginary part of polarizability has two terms related through the optical theorem to the spectral line absorptions and the photoionization process. Appropriate expressions for the scattering resonance result when level broadening effects are considered (Heddle 1964).

The nonrelativistic dipole approximation adopted here is valid for photon energy much less than 2/α ≈ 275 Rydberg (Ry; Bethe & Salpeter 1957), with α = 1/137.036 being the fine-structure constant. In this energy regime, relativistic effects on the dynamical polarizability can be neglected (Johnson & Feiock 1968; Thu et al. 1996; Zapryagaev 2011). For instance, the leading relativistic correction to the static dipole polarizability (nonrelativistic value of 9/2 bohr3), that is to say in the zero-photon-frequency limit, is 143α2≈1.81×10−6${{14} \over 3}{\alpha ^2} \approx 1.81 \times {10^{ - 6}}$ (Kaneko 1977). However, corrections to the dynamic dipole polarizability become substantial near resonance peaks due to the corresponding energy shifts and splitting of bound states. Therefore, a semirelativistic model that accounts for the fine-structure effects is sufficient for calculating the cross section of Rayleigh scattering at energies both below and above the ionization threshold.

In Sect. 2, we briefly describe the elastic photon scattering from isolated hydrogen atoms in the ground state through the complex polarizability and the use of transition oscillator strengths. We consider cross section corresponding to unpolarized incident radiation and outgoing radiation averaged over all directions. Section 3 shows the results from numerical calculations done in the infinite level lifetime approximation and neglecting the fine structure. Section 4 is devoted to the analysis of the cross section in the neighborhood of resonances when the natural broadening of excited bound states and the fine structure are taken into account. In Sect. 5, we give functional forms which fit the atomic polarizability. Section 6 presents some evaluations of the Rayleigh cross section. Conclusions are given in Sect. 7.

2 Evaluation of the atomic polarizability

The nonrelativistic cross section of Rayleigh scattering of unpolarized light by a nonoriented hydrogen atom in the ground state, expressed in atomic units (bohr2), is given by

σRa(ϵ)=π6α4ϵ4| αpol(ϵ) |2,${\sigma _{{\rm{Ra}}}}\left( \right) = {\pi \over 6}{\alpha ^4}{^4}{\left| {{\alpha _{{\rm{pol}}}}\left( \right)} \right|^2},$(1)

with ϵ being the photon energy in Rydberg units (R = 13.60569 eV), and αpol(ϵ) being the dynamic polarizability measured in bohr3. The polarizability is constituted by real and imaginary parts

αpol=αR+iαI,| αpol |2=αR2+αI2.$\matrix{{{\alpha _{{\rm{pol}}}} = {\alpha _{\rm{R}}} + i{\alpha _{\rm{I}}},} \hfill & {{{\left| {{\alpha _{{\rm{pol}}}}} \right|}^2} = \alpha _{\rm{R}}^{\rm{2}} + \alpha _{\rm{I}}^{\rm{2}}.} \hfill \cr} $(2)

They can be expressed in terms of the oscillator strengths of bound states (f1n) and the continuum (df1k/dϵ),

αR0(ϵ)=4[ ∑n=2∞f1nϵ1,n2−ϵ2+P∫1∞df1k/dϵ1,kϵ1,k2−ϵ2dϵ1,k ],$\alpha _{\rm{R}}^{\rm{0}}\left( \right) = 4\left[ {\sum\limits_{n = 2}^\infty {{{{f_{1n}}} \over {_{1,n}^2 - {^2}}} + P\int_1^\infty {{{{{{\rm{d}}{f_{1k}}} \mathord{\left/{\vphantom {{{\rm{d}}{f_{1k}}} {{\rm{d}}{_{1,k}}}}} \right.\kern-\nulldelimiterspace} {{\rm{d}}{_{1,k}}}}} \over {_{1,k}^2 - {^2}}}{\rm{d}}{_{1,k}}} } } \right],$(3)

αI0(ϵ)=∑n=2∞2πϵ1,nf1nδ(ϵ1,n−ϵ)+2πϵ(df1kdϵ),$\alpha _{\rm{I}}^0\left( \right) = \sum\limits_{n = 2}^\infty {{{2\pi } \over {{_{1,n}}}}{f_{1n}}\delta \left( {{_{1,n}} - } \right) + {{2\pi } \over }\left( {{{{\rm{d}}{f_{1k}}} \over {{\rm{d}}}}} \right),}$(4)

where Ƥ denotes the principal value of the integral, δ(χ) is the Dirac function, and the superscript zero means that bound states are assumed to have infinite lifetimes. In addition, ϵ1,n and ϵ1,k are the energies (measured in Rydberg) over the ground state n′ = 1

ϵn′,n=1n′2−1n2,ϵn,′k=1n′2+1k2,$\matrix{{{_{n\prime ,n}} = {1 \over {n{\prime ^2}}} - {1 \over {{n^2}}},} \hfill & {{_{n,\prime k}}} \hfill \cr} = {1 \over {n{\prime ^2}}} + {1 \over {{k^2}}},$(5)

where n (= 2,3,…, ∞) represents the main quantum number of bound states, and k is a positive real number (0 < k < ∞) labeling states in the continuum.

The real part of the polarizability (αR) is associated with the refractive index of the medium and can have positive or negative values. The imaginary part of the polarizability (αI) is a positive quantity related to the absorption cross section through the optical theorem. In the continuum (ϵ > 1), it corresponds to the photoionization cross section given in bohr2 by

σPh(ϵ)=2πα ϵ α1.${\sigma _{{\rm{Ph}}}}\left( \right) = 2\pi \alpha \,\,{\alpha _1}.$(6)

As a reference, the Thomson cross section in the same units is

σTh=8π3 α4.${\sigma _{{\rm{Th}}}} = {{8\pi } \over 3}\,{\alpha ^4}.$(7)

Exact expressions for the mean oscillator strength of transitions 1s → np and 1s → kp are well known (Sugiura 1927; Menzel & Pekeris 1935)1,

f1n=2563n5(n−1)2(n−2)(n+1)2(n+2),${f_{1n}} = {{256} \over 3}{{{n^5}{{\left( {n - 1} \right)}^{2\left( {n - 2} \right)}}} \over {{{\left( {n + 1} \right)}^{2\left( {n + 2} \right)}}}},$(8)

df1kdϵ1,k=1283k8(1+k2)4exp[ −4k arctn(1/k) ]1−exp[ −2πk ].${{{\rm{d}}{f_{1k}}} \over {{\rm{d}}{_{1,k}}}} = {{128} \over 3}{{{k^8}} \over {{{\left( {1 + {k^2}} \right)}^4}}}{{\exp \left[ { - 4k\,{\rm{arctn}}\left( {{1 \mathord{\left/{\vphantom {1 k}} \right.\kern-\nulldelimiterspace} k}} \right)} \right]} \over {1 - \exp \left[ { - 2\pi k} \right]}}.$(9)

The oscillator strength has analytic continuation through the ionization threshold (Fano & Cooper 1968),

limn→∞(n32f1n)limk→∞(df1kdϵ1k)=1283exp(4)≈0.78146725925.$\mathop {\lim }\limits_{n \to \infty } \left( {{{{n^3}} \over 2}{f_{1n}}} \right)\mathop {\lim }\limits_{k \to \infty } \left( {{{{\rm{d}}{f_{1k}}} \over {{\rm{d}}{_{1k}}}}} \right) = {{128} \over {3\exp \left( 4 \right)}} \approx 0.78146725925.$(10)

Moreover, the asymptotic behavior of the oscillator strength is expressed by the series

n32fl n=1283exp(4) [ 1+83n2 +21445n3+201922835n6+41168342525n8+173695841403325n10+O(n−12) ].$\matrix{{{{{n^3}} \over 2}{f_{{\mathop{\rm l}\nolimits} \,n}}} \hfill &amp; = \hfill &amp; {{{128} \over {3\exp \left( 4 \right)}}\left[ {1 + {8 \over {3{n^2}}}} \right. + {{214} \over {45{n^3}}} + {{20192} \over {2835{n^6}}} + {{411683} \over {42525{n^8}}}} \hfill \cr {} \hfill &amp; {} \hfill &amp; {\left. { + {{17369584} \over {1403325{n^{10}}}} + O\left( {{n^{ - 12}}} \right)} \right].} \hfill \cr}$(11)

The same expression is valid for df1k/dϵ1,k with the substitution n2 = −k2 (n = ik) on the right-hand side of Eq. (11). Direct use of Eqs. (8) and (9) yields numerical errors for high quantum numbers, so we employed Eq. (11) to evaluate f1n and df1k/dϵ1,k for n, k ≫ 1.

The principal value integral in Eq. (3) reduces to a regular integral for ϵ < 1 and the imaginary pole term can be ignored. For ϵ > 1 , Cauchy principal value reads as follows:

P∫1∞⋯limδ→0(∫1ϵ−δ⋯+∫ϵ+δ∞⋯).$P\int_1^\infty { \cdots \mathop {\lim }\limits_{\delta \to 0} } \left( {\int_1^{ - \delta } { \cdots + \int_{ + \delta }^\infty \cdots } } \right).$(12)

In practice, the evaluation of this term is split into three subdomains:

P∫1∞⋯=∫1ϵ−δ⋯+∫ϵ−δϵ+δ⋯+∫ϵ+δ∞⋯,(δ≪ϵ).$\matrix{{P\int_1^\infty { \cdots = } \int_1^{ - \delta } { \cdots + \int_{ - \delta }^{ + \delta } \cdots + } \int_{ + \delta }^\infty \cdots ,} \hfill &amp; {\left( {\delta \ll } \right).} \hfill \cr}$(13)

The second-term integrand on the right-side of Eq. (13) is approximated by its Laurent expansion, where the odd terms about ϵ are removed. Each even term in ϵ can be analytically integrated yielding the following series:

∫ϵ−δϵ+δdf1k/dϵ1,kϵ1,k2−ϵ2dϵ1,k=(−fϵ+2ϵfϵ(1))δ2ϵ2+(−3fϵ+6ϵfϵ(1)−6ϵ2fϵ(2)+4ϵ3fϵ(3))δ372ϵ4+( −15fϵ+30ϵfϵ(1)−30ϵ2fϵ(2)+20ϵ3fϵ(3)− 10ϵ4fϵ(4)+4ϵ5fϵ(5) )δ52400ϵ+O(δ7),$\matrix{{\int_{ - \delta }^{ + \delta } {{{{{{\rm{d}}{f_{1k}}} \mathord{\left/{\vphantom {{{\rm{d}}{f_{1k}}} {{\rm{d}}{_{1,k}}}}} \right.\kern-\nulldelimiterspace} {{\rm{d}}{_{1,k}}}}} \over {_{1,k}^2 - {^2}}}d{_{1,k}} = \left( { - {f_} + 2f_^{\left( 1 \right)}} \right){\delta \over {2{^2}}}} } \hfill \cr {\matrix{{} \hfill &amp; { + \left( { - 3{f_} + 6f_^{\left( 1 \right)} - 6{^2}f_^{\left( 2 \right)} + 4{^3}f_^{\left( 3 \right)}} \right){{{\delta ^3}} \over {72{^4}}}} \hfill \cr} } \hfill \cr {\matrix{{} \hfill &amp; { + \left( { - 15{f_} + 30f_^{\left( 1 \right)} - 30{^2}f_^{\left( 2 \right)} + 20{^3}f_^{\left( 3 \right)}} \right.} \hfill \cr} } \hfill \cr {\matrix{{} \hfill &amp; { - \left. {10{^4}f_^{\left( 4 \right)} + 4{^5}f_^{\left( 5 \right)}} \right){{{\delta ^5}} \over {2400}} + {\cal O}\left( {{\delta ^7}} \right),} \hfill \cr} } \hfill \cr}$(14)

where fϵ(l)$f_^{\left( l \right)}$ is evaluated with (9) as follows:

fϵ(l)≡δl(df1k/dϵ)∂ϵl.$f_^{\left( l \right)} \equiv {{{\delta ^l}\left( {{{{\rm{d}}{f_{1k}}} \mathord{\left/{\vphantom {{{\rm{d}}{f_{1k}}} {{\rm{d}}}}} \right.\kern-\nulldelimiterspace} {{\rm{d}}}}} \right)} \over {\partial {^l}}}.$(15)

The first and third integrals on the right-hand side of Eq. (13) were calculated in a standard way using Gaussian quadratures.

3 Results for infinite lifetimes

Expressions (3) and (4) provide high-precision values of αpol far away from resonance cores. This section shows the results of their evaluation. In practice, the sum over bound states in Eq. (3) is truncated to some upper number N:

∑N≡∑n=2Nf1nϵ1,n2−ϵ2.${\sum _N} \equiv \sum\limits_{n = 2}^N {{{{f_{1n}}} \over {_{1,n}^2 - {^2}}}.}$(16)

Convergence in the evaluation of αR0(ϵ)$\alpha _{\rm{R}}^0\left( \right)$ was reached by increasing N and the number of points in the quadratures. Figure 1 shows the sensitivity of Eq. (16) to N. Precision in the sum increases roughly two orders of magnitude for each one-order increase in the number of bound states.

Determining the accuracy by using the use of spectral distribution of oscillator strengths can be done through two simple tests: (i) the Thomas–Reiche–Kuhn f-sum rule

∑n=2∞f1n+∫1∞df1kdϵ1,kdϵ1,k=1,$\sum\limits_{n = 2}^\infty {{f_{1n}} + \int_1^\infty {{{{\rm{d}}{f_{1k}}} \over {{\rm{d}}{_{1,k}}}}{\rm{d}}{_{1,k}}} = 1,}$(17)

and (ii) the static polarizability value which is exactly know (Wentzel 1926; Waller 1926; Epstein 1926)2:

αpol(0)=4.5 bohr3.${\alpha _{{\rm{pol}}}}\left( 0 \right) = 4.5\,{\rm{boh}}{{\rm{r}}^3}.$(18)

As shown in Table 1, both tests can be verified within machine precision. Here, an upper quantum number N = 106 is adopted.

Figure 2 shows different contributions to the dynamical polarizability as functions of the photon energy. The value ϵ = 1 corresponds to the photoelectric threshold for transitions from the 1s state. When the natural broadening of the levels is neglected, real and imaginary parts of the polarizability due to bound states become singulars over an infinite sequence of resonances located at energies {ϵ1,n (n = 1,2,…,∞), which are distributed at 0.75 ≤ e ≤ 1 and accumulate on the photoionization edge (upper panel of Fig. 2). In this approach, the contribution of bound states to α10(ϵ)$\alpha _{\rm{1}}^0\left( \right)$ vanishes for all energies outside of resonances (i.e., ∀ ϵ ≠ ϵ1,n) according to the set of Dirac functions in Eq. (4).

Contributions of the continuum to polarizability exhibit simple forms (lower panel of Fig. 2). The real part increases monotonically with energy up to ϵ = 1, where it diverges. For energies higher than 1.144210 Ry, αR from the continuum becomes negative and reaches a minimum value of – 0.7712916 bohr3 for ϵ ≈ 1.4540. On the other hand, the imaginary part mimics – with a multiplicative factor proportional to ϵ – the behavior of the photoabsorption cross section, according to Eq. (6).

The absolute magnitude of αpol as a function of ϵ is illustrated in logarithmic scales in the lower panel of Fig. 3, where our results (solid line) are compared with those from Gavrila (1967) which are represented by symbols. Relative differences are plotted in the upper panel of Fig. 3. In the limit ϵ → 1+, contributions to αR from discrete states and the continuum diverge with opposite signs (see Fig. 2), but they compensate for each other in such a way that total αR remains finite. Therefore, observed divergences of αpol in our evaluations are only produced by resonances, and as a consequence of neglecting the level broadening.

thumbnail Fig. 1

Accuracy in the evaluation of bound states’ contribution to real polarizability, as a function of the photon energy and for different numbers of sum terms in Eq. (16). The reference values ΣNo correspond to N0 = 106. Vertical lines are located on resonances and show a fast convergence effect.
thumbnail Fig. 2

Variation of the real and imaginary parts of the polarizability due to contributions from bound and continuum states.
Table 1

Contributions to the f -sum rule and static polarizability αpol(0) coming from bound and continuum states.

thumbnail Fig. 3

Dynamic polarizability. Lower panel: comparison between the values calculated in the present work (line) and the results of Gavrila (1967, symbols), as a function of the light energy. Upper panel: absolute relative errors in a logarithmic scale.

4 Fine-structure and damping effects

When the effects of a fine structure and finite lifetimes of excited bound states are considered, real and imaginary parts of polarization take the forms

αR(ϵ)=4[ ∑njf1,nj(ϵ1,nj2−ϵ2)(ϵ1,nj2−ϵ2)2+ϵ2Δn2+P∫1∞df1k/dϵ1kϵ1,k1−ϵ2dϵ1k ],${\alpha _{\rm{R}}}\left( \right) = 4\left[ {\sum\limits_{nj} {{{{f_{1,nj}}\left( {_{1,nj}^2 - {^2}} \right)} \over {{{\left( {_{1,nj}^2 - {^2}} \right)}^2} + {^2}{\rm{\Delta }}_n^2}} + P\int_1^\infty {{{{{{\rm{d}}{f_{1k}}} \mathord{\left/{\vphantom {{{\rm{d}}{f_{1k}}} {{\rm{d}}{_{1k}}}}} \right.\kern-\nulldelimiterspace} {{\rm{d}}{_{1k}}}}} \over {_{1,k}^1 - {^2}}}{\rm{d}}{_{1k}}} } } \right],$(19)

α1(ϵ)=∑nj4f1,njϵΔn(ϵ1,nj2−ϵ2)2+ϵ2Δn2+2πϵ(df1kdϵ),${\alpha _1}\left( \right) = \sum\limits_{nj} {{{4{f_{1,nj}}{{\rm{\Delta }}_n}} \over {{{\left( {_{1,nj}^2 - {^2}} \right)}^2} + {^2}{\rm{\Delta }}_n^2}} + {{2\pi } \over }\left( {{{{\rm{d}}{f_{1k}}} \over {{\rm{d}}}}} \right)} ,$(20)

where Δn is the natural breadth of the level n and j refers to the two components of each state np (np1/2 and np3/2). Absorption oscillator strengths of fine-structure transitions are given by the following (Wiese & Fuhr 2009):

f1,nj=1/2=13f1n,f1,nj=3/2=23f1n.$\matrix{{{f_{1,nj = {1 \mathord{\left/{\vphantom {1 2}} \right.\kern-\nulldelimiterspace} 2}}} = {1 \over 3}{f_{1n}},} \hfill &amp; {} \hfill &amp; {{f_{1,nj = {3 \mathord{\left/{\vphantom {3 2}} \right.\kern-\nulldelimiterspace} 2}}} = {2 \over 3}{f_{1n}}.} \hfill \cr}$(21)

The natural breadth is basically the same for np1/2 and np3/2 levels, and it can be written in Rydberg units as

Δn=ℏRΓn,Γn=∑n′=1n−1Ann′,$\matrix{{{{\rm{\Delta }}_n} = {\hbar \over R}{{\rm{\Gamma }}_n},} \hfill &amp; {} \hfill &amp; {{{\rm{\Gamma }}_n} = \sum\limits_{n\prime = 1}^{n - 1} {{A_{nn\prime }},} } \hfill \cr}$(22)

with Γn [s−1] being the total probability rate of spontaneous decay from n to any lower level, and Ann [s−1 ] is the Einstein coefficient,

Ann′=α3Rℏgn′gnϵn′,n2fn′n=8.03250×109[ s−1 ]gn′gnϵn′,n2fn′n,${A_{nn\prime }} = {{{\alpha ^3}R} \over \hbar }{{{g_{n\prime }}} \over {{g_n}}}_{n\prime ,n}^2{f_{n\prime n}} = 8.03250 \times {10^9}\left[ {{{\rm{s}}^{ - 1}}} \right]{{{g_{n\prime }}} \over {{g_n}}}_{n\prime ,n}^2{f_{n\prime n}},$(23)

with gn being the statistical weight of the level n. In practice, we calculated Δn using accurate Ann′ values compiled by Wiese & Fuhr (2009), which expand n′ ≤ 19 and n ≤ 20 with an uncertainty of less than 0.3%. A precise (within data errors) Δn fitting expression for n ≥ 2 is given by

Δnfit=α3n5(−0.187+2.915lnn).${\rm{\Delta }}_n^{{\rm{fit}}} = {{{\alpha ^3}} \over {{n^5}}}\left( { - 0.187 + 2.915\ln n} \right).$(24)

Table 2 shows calculated values for natural breadths and probabilities of a spontaneous transition for a selection of levels. Errors of are lower than 0.2% for n ≥ 2. As a reference, evaluations of spontaneous decay probabilities from the well-known Kramers approximation,

Ann′K=323π3α3Rℏ1n′3n5ϵn′n,$A_{nn\prime }^{\rm{K}} = {{32} \over {3\pi \sqrt 3 }}{{{\alpha ^3}R} \over \hbar }{1 \over {n{\prime ^3}{n^5}{_{n\prime n}}}},$(25)

yield Δn values with errors between 14% and 40% for n ≤ 20.

Due to relativistic corrections and spin-orbit interaction, each np state splits into two levels with energies (Sobelman 1979)

ϵnp1/2=−1n2+α2n3(34n−1),ϵnp3/2=1n2+α2n3(34n−12),${_{n{p_{{1 \mathord{\left/{\vphantom {1 2}} \right.\kern-\nulldelimiterspace} 2}}}}} = - {1 \over {{n^2}}} + {{{\alpha ^2}} \over {{n^3}}}\left( {{3 \over {4n}} - 1} \right),{_{n{p_{{3 \mathord{\left/{\vphantom {3 2}} \right.\kern-\nulldelimiterspace} 2}}}}} = {1 \over {{n^2}}} + {{{\alpha ^2}} \over {{n^3}}}\left( {{3 \over {4n}} - {1 \over 2}} \right),$(26)

with the zero-energy point in the continuum edge. The ground state (1s) remains single (1s1/2), but its energy changes from

ϵ1s=−1  Ry toϵ1s1/2=−1.0000133128 Ry.$\matrix{{{_{1s}} = - 1\,\,{\rm{Ry}}\,{\rm{to}}} \hfill \cr {{_{1{s_{{1 \mathord{\left/{\vphantom {1 2}} \right.\kern-\nulldelimiterspace} 2}}}}} = - 1.0000133128\,{\rm{Ry}}{\rm{.}}} \hfill \cr}$(27)

As a consequence, each resonance 1 s–np splits into a doublet with energies

ϵ1,nj=ϵnpj−ϵ1s1/2,(j=12,32),$\matrix{{{_{1,nj}} = {_{n{p_j}}} - {_{1{s_{{1 \mathord{\left/{\vphantom {1 2}} \right.\kern-\nulldelimiterspace} 2}}}}},} \hfill &amp; {\left( {j = {1 \over 2},{3 \over 2}} \right),} \hfill \cr}$(28)

which are slightly higher than that from Eq. (5). They are listed for n ≤ 20 in Table 3 along with the corresponding transition wavelengths (compare them with Kramida 2010).

Polarizability contributions (19) and (20) were evaluated in the way described in Sect. 2. They provide different results than those given by Eqs. (2)–(4) in the neighborhood of each resonance. The first pair of resonances occur around ϵ = 0.75 and correspond to Lyman-α1/2 and Lyman-α3/2 transitions. Figure 4 shows the module of the polarizability with (solid line) and without (dot-dashed line) a fine structure and damping effects for these resonances. Resonance polarizability given by Eqs. (3) and (4) is redshifted in 1.0816 × 10−5 Ry (line center in ϵ1,n = 0.75 Ry) and its real and imaginary contributions become singulars. On the contrary, imaginary polarizability α1 with damping and fine-structure effects (dotted line) presents sharp Lorentzian peaks centered at ϵ1,2p1/2${_{1,2{p_{{1 \mathord{\left/{\vphantom {1 2}} \right.\kern-\nulldelimiterspace} 2}}}}}$ and ϵ1,2p3/2${_{1,2{p_{{3 \mathord{\left/{\vphantom {3 2}} \right.\kern-\nulldelimiterspace} 2}}}}}$, with a half width Δn. The maximum values of ∣αpol∣ coincide with the α1 peaks since the real part (dashed line) vanishes there. In fact, the real part of the polarizability tends to be antisymmetric about each resonance center (Fig. 4 shows its absolute value).

Table 2

Selection of transition probabilities and natural level breadths.

Table 3

Resonances energies (Ry) and wavelengths (Å) of transitions 1s1/2 − np1/2 and 1s1/2 − np3/2 for 2 ≤ n ≤ 20.

thumbnail Fig. 4

Polarizability in the Lyα1/2 (1s1/2 − 2p1/2) and Lyα3/2 (1s1/2 − 2p3/2) resonances (solid line). We note that ϵ1,2*$_{1,2}^*$ denotes the mean energy of these transitions. Real and imaginary parts of the polarizability are detailed on the plot (dashed and dotted lines, respectively). The inner graph shows details of the Lyα3/2 core. Evaluation of Lyman-α resonance without the effects of a fine structure and finite lifetime is represented by a dot-dashed line.

5 Analytic fits

In this section, we provide expressions to evaluate the absolute value of the polarizability. As has been shown, the imaginary part of the polarizability is significant in a very small region around each resonance (Fig. 4) and in the photoionization region due to the continuum contribution (Fig. 2). Therefore, for energies lower than the ionization threshold (ϵ<ϵ1s1/2)$\left( { &lt; {_{1{s_{{1 \mathord{\left/{\vphantom {1 2}} \right.\kern-\nulldelimiterspace} 2}}}}}} \right)$ and outside resonance cores, the magnitude of the polarizability (∣αpol∣) is very well approximated by its real part

| αpol(ϵ) |=| αR(ϵ) |,(ϵ<ϵ1s1/2,ϵ≠ϵ1,nj).$\matrix{{\left| {{\alpha _{{\rm{pol}}}}\left( \right)} \right| = \left| {{\alpha _{\rm{R}}}\left( \right)} \right|,} \hfill &amp; {} \hfill &amp; {\left( { &lt; {_{1{s_{{1 \mathord{\left/{\vphantom {1 2}} \right.\kern-\nulldelimiterspace} 2}}}}}, \ne {_{1,nj}}} \right)} \hfill \cr} .$(29)

Preresonance region (ϵ ≲ ϵ1,2 = 0.75 Ry)

Redward of Lyman-α, the polarizability is a wellbehaved monotonic function of ϵ and can be approximated with high precision (relative error less than 0.006% at ϵ < 0.7496) by

αR(ϵ)=11−s(1.464860.950713−ϵ2.172+1.66478ϵ1.2*2−ϵ2).${\alpha _{\rm{R}}}\left( \right) = {1 \over {1 - s}}\left( {{{1.46486} \over {0.950713 - {^{2.172}}}} + {{1.66478} \over {{{_{1.2}^*}^2} - {^2}}}} \right).$(30)

being ϵ1,2*=12(ϵ1,2j=1/2+ϵ1,2j=3/2)$_{1,2}^* = {1 \over 2}\left( {{_{1,2j = {1 \mathord{\left/{\vphantom {1 2}} \right.\kern-\nulldelimiterspace} 2}}} + {_{1,2j = {3 \mathord{\left/{\vphantom {3 2}} \right.\kern-\nulldelimiterspace} 2}}}} \right)$ and

s={ 0.0017sin(8.2ϵ1.33)−0.000093(ϵ≤ϵa)−0.00163sin(16.86| ϵ−ϵa |1.2)(ϵa<ϵ<0.73),0(0.73<ϵ<0.745),−10−4.9+0.205(0.7501−ϵ)−0.3(0.745<ϵ<0.75), $s = \left\{ {\matrix{{0.0017\sin \left( {8.2{^{1.33}}} \right) - 0.000093} \hfill &amp; {\left( { \le {_a}} \right)} \hfill \cr { - 0.00163\sin \left( {16.86{{\left| { - {_a}} \right|}^{1.2}}} \right)} \hfill &amp; {\left( {{_a} &lt; &lt; 0.73} \right),} \hfill \cr 0 \hfill &amp; {\left( {0.73 &lt; &lt; 0.745} \right),} \hfill \cr { - {{10}^{ - 4.9 + 0.205{{\left( {{{0.7501}_{ - }}} \right)}^{ - 0.3}}}}} \hfill &amp; {\left( {0.745 &lt; &lt; 0.75} \right),} \hfill \cr} } \right.$(31)

ϵa = 0.48083 (for s ≡ 0, Eq. (30) has a precision of 0.2%).

Resonance region (ϵ1,2≲ϵ</ϵ1s1/2/)$\left( {{_{1,2}} \mathbin{\lower.3ex\hbox{$\buildrel&lt;\over{\smash{\scriptstyle\sim}\vphantom{_x}}$}} &lt; /{_{1{s_{{1 \mathord{\left/{\vphantom {1 2}} \right.\kern-\nulldelimiterspace} 2}}}}}/} \right)$

The total polarizability in the resonance region can be reasonably well approximate in the following way. In the neighborhood of a resonance 1s ↔ np, the polarizability is well represented keeping only the contributions of 1s1/2 − np1/2 and 1s1/2 − np3/2 transitions,

αR(ϵ)=4f1,nj=1/2(ϵ1,nj=1/22−ϵ2)(ϵ1,nj=1/22−ϵ2)2+ϵ2Δn2+4f1,nj=3/2(ϵ1,nj=3/22−ϵ2)(ϵ1,nj=3/22−ϵ2)2+ϵ2Δn2,${\alpha _{\rm{R}}}\left( \right) = {{4{f_{1,nj = {1 \mathord{\left/{\vphantom {1 2}} \right.\kern-\nulldelimiterspace} 2}}}\left( {_{1,nj = {1 \mathord{\left/{\vphantom {1 2}} \right.\kern-\nulldelimiterspace} 2}}^2 - {^2}} \right)} \over {{{\left( {_{1,nj = {1 \mathord{\left/{\vphantom {1 2}} \right.\kern-\nulldelimiterspace} 2}}^2 - {^2}} \right)}^2} + {^2}{\rm{\Delta }}_n^2}} + {{4{f_{1,nj = {3 \mathord{\left/{\vphantom {3 2}} \right.\kern-\nulldelimiterspace} 2}}}\left( {_{1,nj = {3 \mathord{\left/{\vphantom {3 2}} \right.\kern-\nulldelimiterspace} 2}}^2 - {^2}} \right)} \over {{{\left( {_{1,nj = {3 \mathord{\left/{\vphantom {3 2}} \right.\kern-\nulldelimiterspace} 2}}^2 - {^2}} \right)}^2} + {^2}{\rm{\Delta }}_n^2}},$(32)

αI(ϵ)=4f1,nj=1/2ϵΔn(ϵ1,nj=1/22−ϵ2)2+ϵ2Δn2+4f1,nj=3/2ϵΔn(ϵ1,nj=3/22−ϵ2)2+ϵ2Δn2.${\alpha _{\rm{I}}}\left( \right) = {{4{f_{1,nj = {1 \mathord{\left/{\vphantom {1 2}} \right.\kern-\nulldelimiterspace} 2}}}{{\rm{\Delta }}_n}} \over {{{\left( {_{1,nj = {1 \mathord{\left/{\vphantom {1 2}} \right.\kern-\nulldelimiterspace} 2}}^2 - {^2}} \right)}^2} + {^2}{\rm{\Delta }}_n^2}} + {{4{f_{1,nj = {3 \mathord{\left/{\vphantom {3 2}} \right.\kern-\nulldelimiterspace} 2}}}{{\rm{\Delta }}_n}} \over {{{\left( {_{1,nj = {3 \mathord{\left/{\vphantom {3 2}} \right.\kern-\nulldelimiterspace} 2}}^2 - {^2}} \right)}^2} + {^2}{\rm{\Delta }}_n^2}}.$(33)

For ϵ1,nj = 3/2 < ϵ < ϵ1,n + 1,j = 1/2 (n = 2,3,…), we adopted a fitting formula similar to that one used in Rohrmann (2018)

αR(ϵ)=δnβntan[ βn(ϵ−ϕn) ]f(ϵ),${\alpha _{\rm{R}}}\left( \right) = {{{\delta _n}} \over {{\beta _n}}}\tan \left[ {{\beta _n}\left( { - {\phi _n}} \right)} \right]f\left( \right),$(34)

where

βn={ π2(ϕn−ϵ1,nj=3/2)(ϵ1,nj=3/2<ϵ<ϕn),π2(ϵ1,n+1j=1/2−ϕn)(ϕn<ϵ<ϵ1,n+1j=1/2), ${\beta _n} = \left\{ {\matrix{{{\pi \over {2\left( {{\phi _n} - {_{1,nj = {3 \mathord{\left/{\vphantom {3 2}} \right.\kern-\nulldelimiterspace} 2}}}} \right)}}} \hfill &amp; {\left( {{_{1,nj = {3 \mathord{\left/{\vphantom {3 2}} \right.\kern-\nulldelimiterspace} 2}}} &lt; &lt; {\phi _n}} \right),} \hfill \cr {{\pi \over {2\left( {{_{1,n + 1j = {1 \mathord{\left/{\vphantom {1 2}} \right.\kern-\nulldelimiterspace} 2}}} - {\phi _n}} \right)}}} \hfill &amp; {\left( {{\phi _n} &lt; &lt; {_{1,n + 1j = {1 \mathord{\left/{\vphantom {1 2}} \right.\kern-\nulldelimiterspace} 2}}}} \right),} \hfill \cr} } \right.$(35)

δn={ 315.49655(n−2),15.5183449×n2.9769922×(1−An)−1(n>2), ${\delta _n} = \left\{ {\matrix{{315.49655} \hfill &amp; {\left( {n - 2} \right),} \hfill \cr {15.5183449 \times {n^{2.9769922}} \times {{\left( {1 - {A_n}} \right)}^{ - 1}}} \hfill &amp; {\left( {n > 2} \right),} \hfill \cr} } \right.$(36)

ϕn=ϵ1,n+1j=1/2−(ϵ1,n+1j=1/2−ϵ1,nj=3/2)Bn,${\phi _n} = {_{1,n + 1j = {1 \mathord{\left/{\vphantom {1 2}} \right.\kern-\nulldelimiterspace} 2}}} - \left( {{_{1,n + 1j = {1 \mathord{\left/{\vphantom {1 2}} \right.\kern-\nulldelimiterspace} 2}}} - {_{1,nj = {3 \mathord{\left/{\vphantom {3 2}} \right.\kern-\nulldelimiterspace} 2}}}} \right){B_n},$(37)

f(ϵ)=(1+Cnξ){ 1−Dn[ 1−(2ξ−1)2 ] },$f\left( \right) = \left( {1 + {C_n}\xi } \right)\left\{ {1 - {D_n}\left[ {1 - {{\left( {2\xi - 1} \right)}^2}} \right]} \right\},$(38)

ξ={ ϕn−ϵϕn−ϵ1,nj=3/2(ϵ1,nj=3/2<ϵ<ϕn),ϵ−ϕnϵ1,n+1j=1/2−ϕn(ϕm<ϵ<ϵ1,n+1j=1/2). $\xi = \left\{ {\matrix{{{{{\phi _n} - } \over {{\phi _n} - {_{1,nj = {3 \mathord{\left/{\vphantom {3 2}} \right.\kern-\nulldelimiterspace} 2}}}}}} \hfill &amp; {\left( {{_{1,nj = {3 \mathord{\left/{\vphantom {3 2}} \right.\kern-\nulldelimiterspace} 2}}} &lt; &lt; {\phi _n}} \right),} \hfill \cr {{{ - {\phi _n}} \over {{_{1,n + 1j = {1 \mathord{\left/{\vphantom {1 2}} \right.\kern-\nulldelimiterspace} 2}}} - {\phi _n}}}} \hfill &amp; {\left( {{\phi _m} &lt; &lt; {_{1,n + 1j = {1 \mathord{\left/{\vphantom {1 2}} \right.\kern-\nulldelimiterspace} 2}}}} \right).} \hfill \cr} } \right.$(39)

Quantities An, Bn, Cn, and Dn are given as follows:

An={ 0.1412(2−logn)2.83(n≤100),0(n>100), ${A_n} = \left\{ {\matrix{{0.1412{{\left( {2 - {{\log }_n}} \right)}^{2.83}}} \hfill &amp; {\left( {n \le 100} \right),} \hfill \cr 0 \hfill &amp; {\left( {n > 100} \right),} \hfill \cr} } \right.$(40)

Bn={ 0.214657809(n=2),0.268−10−1.2−0.45(logn)2−1.61×10−7(logn)22(n>2), ${B_n} = \left\{ {\matrix{{0.214657809} \hfill &amp; {\left( {n = 2} \right),} \hfill \cr {0.268 - {{10}^{ - 1.2 - 0.45{{\left( {\log n} \right)}^2} - 1.61 \times {{10}^{ - 7}}{{\left( {\log n} \right)}^{22}}}}} \hfill &amp; {\left( {n > 2} \right),} \hfill \cr} } \right.$(41)

For ϵ1,nj = 3/2 < ϵ < ϕ and ϕn < ϵ < ϵ1,n + 1,j = 1/2,

Cn={ 0.7346−10−0.12−1.95logn+0.035(logn)−2,0.928−10−0.35−1.65logn+0.243(logn)−0.43 ${C_n} = \left\{ {\matrix{{0.7346 - {{10}^{ - 0.12 - 1.95\log n + 0.035{{\left( {\log n} \right)}^{ - 2}}}},} \hfill \cr {0.928 - {{10}^{ - 0.35 - 1.65\log n + 0.243{{\left( {\log n} \right)}^{ - 0.43}}}}} \hfill \cr} } \right.$(42)

respectively. For ϵ1,nj = 3/2 < ϵ < ϕn

Dn={ 0.245(n=2),0.255(3≤n≤6),0.256−10−1.15−0.22(logn−0.6)−1.12(n>6), ${D_n} = \left\{ {\matrix{{0.245} \hfill &amp; {\left( {n = 2} \right),} \hfill \cr {0.255} \hfill &amp; {\left( {3 \le n \le 6} \right),} \hfill \cr {0.256 - {{10}^{ - 1.15 - 0.22{{\left( {\log n - 0.6} \right)}^{ - 1.12}}}}} \hfill &amp; {\left( {n > 6} \right),} \hfill \cr} } \right.$(43)

and for ϕn < ϵ < ϵ1,n+1,j=1/2,

Dn=0.053−10−0.44−1.965(logn)0.6,${D_n} = 0.053 - {10^{ - 0.44 - 1.965{{\left( {\log n} \right)}^{0.6}}}},$(44)

The quantity ϕn represents the energy between two resonances where the real polarizability vanishes. Eq. (41) gives Bn (the position of ϕn relative to ϵ1,nj = 3/2 and ϵ1,n + 1,j = 1/2, see Eq. (37)) with an error <0.5% ∀ n and below 0.02% for n ≥ 30 (precision increasing with n). Just outside of resonance cores, where imaginary polarizability is not significant, Eq. (34) describes the total polarizability redward of ϵ1,nj =1/2 with a precision better than 0.9% ∀n, 0.5% for n > 9 and 0.2% for n > 16. Blueward of ϵ1,n j = 3/2, the relative error is 1.8% for n = 2, 1.6% for n = 3, <0.85% for n ≥ 4, and <0.2% for n ≥ 20.

Postresonance region (ϵ>/ϵ1s1/2/$ > /{_{1{s_{{1 \mathord{\left/{\vphantom {1 2}} \right.\kern-\nulldelimiterspace} 2}}}}}/$ = 1.0000133128 Ry)

For large values of ϵ, the nonrelativistic Rayleigh cross section converges to the Thomson scattering cross section. This means

| αpol(ϵ) |→4ϵ−2,(ϵ≫1).$\matrix{{\left| {{\alpha _{{\rm{pol}}}}\left( \right)} \right| \to 4{^{ - 2}},} \hfill &amp; {} \hfill &amp; {\left( { \gg 1} \right).} \hfill \cr}$(45)

A precision better than 0.4% (relative error) for the polarizability above the ionization threshold was obtained with

| αpol(ϵ) |=4ϵ2[ 1+0.62621+2.8179ϵ0.6776(1+0.0216672ϵ1.4745)logϵ ].$\left| {{\alpha _{{\rm{pol}}}}\left( \right)} \right| = {4 \over {{^2}}}\left[ {1 + {{0.6262} \over {1 + 2.8179{^{0.6776}}\left( {1 + 0.0216672{^{1.4745}}} \right)\log }}} \right].$(46)

thumbnail Fig. 5

Rayleigh cross section in units of the Thompson cross section as a function of the photon energy. The solid line represents evaluations with Eq. (1) combined with Eqs. (19) and (20). Symbols represent the results from Gavrila (1967).

6 Scattering cross section

The Rayleigh scattering cross section was obtained simply by multiplying ∣αpol∣2 by the factors appearing in Eq. (1). The Rayleigh cross section for hydrogen atoms obtained in this work is displayed in Fig. 5. Current calculations (solid line) include about one hundred resonances which have finite amplitudes. These results are in very good agreement with those derived from Gavrila (1967) in the limited number of energies presented there (symbols), which do not include resonance cores. For high enough energies, in the regime where the dipole approximation still holds, the cross section slowly approaches the expected Thomson formula.

It should be noted that the energy interval between successive n states, the separation between fine-structure components, and the natural breadth of the levels scale with α and n in the form

En+1−En≈2n3,ΔEfine  structure=α22n3,Δn≈3α3ln(n)n5.$\matrix{{{E_{n + 1}} - {E_n} \approx {2 \over {{n^3}}},} \hfill &amp; {{\rm{\Delta }}{E_{{\rm{fine}}\,\,{\rm{structure}}}} = {{{\alpha ^2}} \over {2{n^3}}},} \hfill &amp; {{{\rm{\Delta }}_n} \approx {{3{\alpha ^3}\ln \left( n \right)} \over {{n^5}}}.} \hfill \cr}$(47)

The first of these relations describes the distribution of resonances 1s–np and their accumulation on the photoionization edge (Fig. 5). The other two relations characterize the shape of each of these (double) resonances, as shown in Fig. 6. The natural width of the resonances becomes small enough and decreases very quickly as the main quantum number n of the excited state increases faster than the energy separation between fine structure components. Consequently, the profiles of successive resonances are progressively narrower and the magnitude of ∣αpol∣ in their peaks increases with n. Relative intensities of ls1/2 − np1/2 and ls1/2 − np3/2 resonances are proportional to the ratio 1:2 of their oscillator strengths, which are in turn proportional to the statistical weights of sublevels np1/2and np3/2, see Eq. (21).

As an illustration, Fig. 7 compares the use of polarizability fits in the evaluation of the Rayleigh cross section within the resonance region. Fits based on Eq. (34) give satisfactory results where the cross section changes many orders of magnitude over energy intervals between successive resonances. On the other hand, Eqs. (32) and (33) match – with high accuracy – the resonance cores including fine-structure details. In astrophysical conditions where the fine-structure splitting can be considered negligible, Eqs. (32) and (33) can be substituted by

| αpol(ϵ) |=4f1n[ (ϵ1,n*−ϵ2)2+(ϵΔn)2 ]−1/2,$\left| {{\alpha _{{\rm{pol}}}}\left( \right)} \right| = 4{f_{1n}}{\left[ {{{\left( {_{1,n}^* - {^2}} \right)}^2} + {{\left( {{{\rm{\Delta }}_n}} \right)}^2}} \right]^{ - {1 \mathord{\left/{\vphantom {1 2}} \right.\kern-\nulldelimiterspace} 2}}},$(48)

where ϵ1,n*$_{1,n}^*$ is the mean energy of the transition 1s − np.

Current calculations were performed for an isolated atom. It is worth noting that in a realistic plasma, where broadening mechanisms are present due to particle perturbations (collisional broadening) and thermal motions (Doppler broadening), resonance profiles are expected to be significantly broader than those of an isolated radiating atom (Omont et al. 1972, 1973; Nienhuis & Schuller 1977; Burnett 1985). Moreover, interactions with surrounding ions and electrons particularly affect highly excited np states and introduce modifications in the cross section close to the photoionization threshold (Griem 2005).

thumbnail Fig. 6

Rayleigh cross section for the first four double resonances, from Lyman-α to Lyman-δ, computed with a fine structure and damping effects. Profiles are horizontally offset from each other in steps of 4 × 10−6 Ry.
thumbnail Fig. 7

Rayleigh cross section for photon energies including the resonances Lyman-γ and Lyman-δ. Solid lines represent full solutions based on Eqs. (19) and (20). Dashed and dotted lines correspond to fitting evaluations with Eqs. (34) and (32)–(33), respectively. The inner graph shows the core of Lyman-δ.

7 Conclusions

We have performed an accurate numerical evaluation of the Rayleigh scattering cross section for hydrogen atoms in the ground state, including resonances and incident photon energies above the ionization threshold. Current evaluations were carried out using the nonrelativistic dipole approximation in the second-order standard quantum perturbative approach. Due to symmetries of the hydrogen ground state, the calculation can be focused on the atomic polarizability which is expressed in terms of the oscillator strengths’ distribution. The method is valid for incident photon energies above and below the ionization threshold. It involves a summation over all intermediate electron states which is split into a sum over bound states and a Cauchy principal value integral over the continuum with an imaginary pole term. Convergence in evaluations is achieved by increasing the number of intermediate bound states and quadrature points.

Our results for Rayleigh scattering are in good agreement with available theoretical data and they expand upon them with a detailed representation of the resonances’ region and the incorporation of a fine structure of the bound levels and damping effects due to finite lifetimes of the excited bound states. We provide fitting formulas to obtain the Rayleigh scattering cross sections in the full nonrelativistic domain, as is required for opacity calculations and their use in astrophysical computer codes.

Acknowledgements

We wish to thank Shigenobu Hirose, who put our attention on current issue. We also thank the anonymous referee for the constructive remarks. This work was supported by MINCYT (Argentina) through Grant No. PICT 2016-1128.

References

  1. Alipour, E., Sigurdson, K., & Hirata, C. M. 2015, Phys. Rev. D, 91, 083520 [CrossRef] [Google Scholar]
  2. Bach, K., & Lee, H.-W. 2015, MNRAS, 446, 264 [NASA ADS] [CrossRef] [Google Scholar]
  3. Beringue, B., Meerburg, P. D., Meyers, J., & Battaglia, N. 2021, J. Cosmology Astropart. Phys., 2021, 060 [CrossRef] [Google Scholar]
  4. Bethe, H. A., & Salpeter, E. E. 1957, Quantum Mechanics of One- and Two-Electron Atoms, 298 [Google Scholar]
  5. Bethell, T. J., & Bergin, E. A. 2011, ApJ, 739, 78 [NASA ADS] [CrossRef] [Google Scholar]
  6. Bonin, K.D. & Kresin, V.V. 1997, Electric-dipole polarizabilities of atoms, molecules, and clusters, 20 [Google Scholar]
  7. Burnett, K. 1985, Phys. Rep., 118, 339 [NASA ADS] [CrossRef] [Google Scholar]
  8. Chen, S.-H., & Kotlarchyk, M. 2007, Interactions of Photons and Neutrons with Matter (World Scientific), 425 [Google Scholar]
  9. Colgan, J., Kilcrease, D. P., Magee, N. H., et al. 2016, ApJ, 817, 116 [Google Scholar]
  10. Collet, R., Asplund, M., & Trampedach, R. 2008, Mem. Soc. Astron. Italiana, 79, 649 [NASA ADS] [Google Scholar]
  11. Dalgarno, A., & Kingston, A. E. 1960, Proc. Roy. Soc. Lond. A, 259, 424 [NASA ADS] [CrossRef] [Google Scholar]
  12. Dalgarno, A., & Lewis, J. T. 1955, Proc. Roy. Soc. Lond. A, 233, 70 [NASA ADS] [CrossRef] [Google Scholar]
  13. Dragomir, D., Benneke, B., Pearson, K. A., et al. 2015, ApJ, 814, 102 [NASA ADS] [CrossRef] [Google Scholar]
  14. Epstein, P. S. 1926, Phys. Rev., 28, 695 [NASA ADS] [CrossRef] [Google Scholar]
  15. Fano, U., & Cooper, J. W. 1968, Rev. Mod. Phys., 40, 441 [NASA ADS] [CrossRef] [Google Scholar]
  16. Ferland, G. J., Chatzikos, M., Guzmán, F., et al. 2017, Rev. Mexicana Astron. Astrofis., 53, 385 [Google Scholar]
  17. Fišák, J., Krtička, J., Munzar, D., & Kubát, J. 2016, A&A, 590, A95 [NASA ADS] [CrossRef] [EDP Sciences] [Google Scholar]
  18. Gavrila, M. 1967, Phys. Rev., 163, 147 [NASA ADS] [CrossRef] [Google Scholar]
  19. González-Riestra, R., Rossi, C., & Viotti, R. F. 2003, A&A, 399, 681 [NASA ADS] [CrossRef] [EDP Sciences] [Google Scholar]
  20. Griem, H. R. 2005, Principles of Plasma Spectroscopy, 147 [Google Scholar]
  21. Gunn, J. E., & Peterson, B. A. 1965, ApJ, 142, 1633 [Google Scholar]
  22. Hansteen, V. H., Carlsson, M., & Gudiksen, B. 2007, in Astronomical Society of the Pacific Conference Series, 368, The Physics of Chromospheric Plasmas, eds. P. Heinzel, I. Dorotovič, & R. J. Rutten, 107 [Google Scholar]
  23. Hayek, W., Asplund, M., Carlsson, M., et al. 2010, A&A, 517, A49 [NASA ADS] [CrossRef] [EDP Sciences] [Google Scholar]
  24. Hayek, W., Asplund, M., Collet, R., & Nordlund, Å. 2011, A&A, 529, A158 [NASA ADS] [CrossRef] [EDP Sciences] [Google Scholar]
  25. Heays, A. N., Bosman, A. D., & van Dishoeck, E. F. 2017, A&A, 602, A105 [NASA ADS] [CrossRef] [EDP Sciences] [Google Scholar]
  26. Heddle, D. W. 1964, Journal of the Optical Society of America, 54, 264 [NASA ADS] [CrossRef] [Google Scholar]
  27. Hirose, S., Hauschildt, P., Minoshima, T., Tomida, K., & Sano, T. 2022, A&A, 659, A87 [NASA ADS] [CrossRef] [EDP Sciences] [Google Scholar]
  28. Isliker, H., Nussbaumer, H., & Vogel, M. 1989, A&A, 219, 271 [NASA ADS] [Google Scholar]
  29. Johnson, W. R., & Feiock, F. D. 1968, Phys. Rev., 168, 22 [NASA ADS] [CrossRef] [Google Scholar]
  30. Kaneko, S. 1977, J. Phys. B At. Mol. Phys., 10, 3347 [NASA ADS] [CrossRef] [Google Scholar]
  31. Kramers, H. A., & Heisenberg, W. 1925, Z. Physik, 31, 681 [NASA ADS] [CrossRef] [Google Scholar]
  32. Kramida, A. E. 2010, At. Data Nucl. Data Tables, 96, 586 [CrossRef] [Google Scholar]
  33. Lecavelier Des Etangs, A., Pont, F., Vidal-Madjar, A., & Sing, D. 2008, A&A, 481, L83 [NASA ADS] [CrossRef] [Google Scholar]
  34. Lee, H.-W. 2005, MNRAS, 358, 1472 [NASA ADS] [CrossRef] [Google Scholar]
  35. Lee, H.-W., & Kim, H. I. 2004, MNRAS, 347, 802 [NASA ADS] [CrossRef] [Google Scholar]
  36. Marigo, P., & Aringer, B. 2009, A&A, 508, 1539 [CrossRef] [EDP Sciences] [Google Scholar]
  37. McNamara, K., Fursa, D. V., & Bray, I. 2018, Phys. Rev. A, 98, 043435 [NASA ADS] [CrossRef] [Google Scholar]
  38. Menzel, D. H., & Pekeris, C. L. 1935, MNRAS, 96, 77 [NASA ADS] [Google Scholar]
  39. Mittleman, M. H., & Wolf, F. A. 1962, Phys. Rev., 128, 2686 [NASA ADS] [CrossRef] [Google Scholar]
  40. Neufeld, D. A. 1991, ApJ, 370, L85 [Google Scholar]
  41. Nienhuis, G., & Schuller, F. 1977, Physica B+C, 92, 397 [NASA ADS] [CrossRef] [Google Scholar]
  42. Nussbaumer, H., Schmid, H. M., & Vogel, M. 1989, A&A, 211, L27 [NASA ADS] [Google Scholar]
  43. Omont, A., Smith, E. W., & Cooper, J. 1972, ApJ, 175, 185 [Google Scholar]
  44. Omont, A., Smith, E. W., & Cooper, J. 1973, ApJ, 182, 283 [NASA ADS] [CrossRef] [Google Scholar]
  45. Peebles, P. J. E., & Yu, J. T. 1970, ApJ, 162, 815 [NASA ADS] [CrossRef] [Google Scholar]
  46. Penney, C. M. 1969, J. Opt. Soc. Am., 59, 34 [NASA ADS] [CrossRef] [Google Scholar]
  47. Placzek, G. 1934, Handbuch der Radiologie, VI (Akademische Verlagsgesellschaft, Leipzig), 2, 209 [Google Scholar]
  48. Rohrmann, R. D. 2018, MNRAS, 473, 457 [Google Scholar]
  49. Sadeghpour, H. R., & Dalgarno, A. 1992, J. Phys. B At. Mol. Phys., 25, 4801 [NASA ADS] [CrossRef] [Google Scholar]
  50. Scheuer, P. A. G. 1965, Nature, 207, 963 [NASA ADS] [CrossRef] [Google Scholar]
  51. Sing, D. K., Wakeford, H. R., Showman, A. P., et al. 2015, MNRAS, 446, 2428 [NASA ADS] [CrossRef] [Google Scholar]
  52. Skopal, A., & Shagatova, N. 2012, A&A, 547, A45 [NASA ADS] [CrossRef] [EDP Sciences] [Google Scholar]
  53. Sobelman, I. I. 1979, Atomic spectra and radiative transitions (Berlin: Springer) [CrossRef] [Google Scholar]
  54. Sugiura, Y. 1927, J. Phys. Radium, 8, 113 [CrossRef] [EDP Sciences] [Google Scholar]
  55. Thu, L., Hoang, L., Komarov, L. I., & Romanova, T. S. 1996, J. Phys. B At. Mol. Phys., 29, 2897 [NASA ADS] [CrossRef] [Google Scholar]
  56. Waller, I. 1926, Z. Physik, 38, 635 [NASA ADS] [CrossRef] [Google Scholar]
  57. Waller, I. 1929, Z. Physik, 58, 75 [NASA ADS] [CrossRef] [Google Scholar]
  58. Wentzel, G. 1926, Z. Physik, 38, 518 [NASA ADS] [CrossRef] [Google Scholar]
  59. Wiese, W. L., & Fuhr, J. R. 2009, J. Phys. Chem. Ref. Data, 38, 1129 [NASA ADS] [CrossRef] [Google Scholar]
  60. Yu, Q., Spergel, D. N., & Ostriker, J. P. 2001, ApJ, 558, 23 [NASA ADS] [CrossRef] [Google Scholar]
  61. Zapryagaev, S. 2011, Physica Scripta T, 144, 014053 [NASA ADS] [CrossRef] [Google Scholar]
*

It is worth noting that, for the ground state n′ = 1, the averaged oscillator strength fn′n is equal to that from the sublevel transition, fn′s,np.

All Tables

Table 1

Contributions to the f -sum rule and static polarizability αpol(0) coming from bound and continuum states.

In the text
Table 2

Selection of transition probabilities and natural level breadths.

In the text
Table 3

Resonances energies (Ry) and wavelengths (Å) of transitions 1s1/2 − np1/2 and 1s1/2 − np3/2 for 2 ≤ n ≤ 20.

In the text

All Figures

thumbnail Fig. 1

Accuracy in the evaluation of bound states’ contribution to real polarizability, as a function of the photon energy and for different numbers of sum terms in Eq. (16). The reference values ΣNo correspond to N0 = 106. Vertical lines are located on resonances and show a fast convergence effect.
In the text
thumbnail Fig. 2

Variation of the real and imaginary parts of the polarizability due to contributions from bound and continuum states.
In the text
thumbnail Fig. 3

Dynamic polarizability. Lower panel: comparison between the values calculated in the present work (line) and the results of Gavrila (1967, symbols), as a function of the light energy. Upper panel: absolute relative errors in a logarithmic scale.
In the text
thumbnail Fig. 4

Polarizability in the Lyα1/2 (1s1/2 − 2p1/2) and Lyα3/2 (1s1/2 − 2p3/2) resonances (solid line). We note that ϵ1,2*$_{1,2}^*$ denotes the mean energy of these transitions. Real and imaginary parts of the polarizability are detailed on the plot (dashed and dotted lines, respectively). The inner graph shows details of the Lyα3/2 core. Evaluation of Lyman-α resonance without the effects of a fine structure and finite lifetime is represented by a dot-dashed line.
In the text
thumbnail Fig. 5

Rayleigh cross section in units of the Thompson cross section as a function of the photon energy. The solid line represents evaluations with Eq. (1) combined with Eqs. (19) and (20). Symbols represent the results from Gavrila (1967).
In the text
thumbnail Fig. 6

Rayleigh cross section for the first four double resonances, from Lyman-α to Lyman-δ, computed with a fine structure and damping effects. Profiles are horizontally offset from each other in steps of 4 × 10−6 Ry.
In the text
thumbnail Fig. 7

Rayleigh cross section for photon energies including the resonances Lyman-γ and Lyman-δ. Solid lines represent full solutions based on Eqs. (19) and (20). Dashed and dotted lines correspond to fitting evaluations with Eqs. (34) and (32)–(33), respectively. The inner graph shows the core of Lyman-δ.
In the text

Current usage metrics show cumulative count of Article Views (full-text article views including HTML views, PDF and ePub downloads, according to the available data) and Abstracts Views on Vision4Press platform.

Data correspond to usage on the plateform after 2015. The current usage metrics is available 48-96 hours after online publication and is updated daily on week days.

Initial download of the metrics may take a while.