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/10.1051/00046361/202243883  
Published online  28 October 2022 
Rayleigh scattering from hydrogen atoms including resonances and high photon energies
Instituto de Ciencias Astronómicas, de la Tierra y del Espacio (CONICETUNSJ),
Av. España 1512 (sur),
5400
San Juan, Argentina
email: rene.rohrmann@conicet.gov.ar
Received:
27
April
2022
Accepted:
29
July
2022
The nonrelativistic cross section from Rayleigh scattering by hydrogen atoms in the ground state was calculated over a wide range of photon energies (<0.8 keV). Evaluations were performed in terms of the real and imaginary components of the atomic polarizability. The sum over intermediate states that characterizes this secondorder radiative process was performed using exact analytic expressions for oscillator strengths of bound and continuum states. Damping terms associated with the finite lifetimes of excited states and their splitting into two finestructure levels (p_{1/2} and p_{3/2}) are taken into account in resonance cross sections. Fitting formulas required for crosssection evaluation are presented for incident photon energy (i) redward of the first resonance (Lymanα_{1/2}), (ii) in the spectral region corresponding to resonances (for an arbitrary number of them), and (iii) above the ionization threshold.
Key words: scattering / atomic processes / opacity
© R. D. Rohrmann and M. V. Rueda 2022
Open 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.
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1 Introduction
Rayleigh scattering is potentially relevant to several areas of astronomical spectroscopy including cool stars, starforming regions, masslosing 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; GonzálezRiestra 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 latetype stars and accretion disks irradiated by the central star (Hayek et al. 2010; Hirose et al. 2022). It is an important ingredient in selfconsistent magnetohydrodynamical models of solartype 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 solartype stars (Hayek et al. 2010, Hayek et al. 2011). This effect could be even the most severe in the lowdensity 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 Lymanα 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 oneelectron 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 finestructure energy shifts due to relativistic and spinorbit 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 highphotonenergy 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 secondorder photonelectron process in the KramersHeisenberg 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 wellknown 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 finestructure 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 bohr^{3}), that is to say in the zerophotonfrequency limit, is (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 finestructure 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 (bohr^{2}), is given by
with ϵ being the photon energy in Rydberg units (R = 13.60569 eV), and α_{pol}(ϵ) being the dynamic polarizability measured in bohr^{3}. The polarizability is constituted by real and imaginary parts
They can be expressed in terms of the oscillator strengths of bound states (f_{1n}) and the continuum (df_{1k}/dϵ),
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, ϵ_{1,n} and ϵ_{1,k} are the energies (measured in Rydberg) over the ground state n′ = 1
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 (α_{R}) is associated with the refractive index of the medium and can have positive or negative values. The imaginary part of the polarizability (α_{I}) 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 bohr^{2} by
As a reference, the Thomson cross section in the same units is
Exact expressions for the mean oscillator strength of transitions 1s → np and 1s → kp are well known (Sugiura 1927; Menzel & Pekeris 1935)^{1},
The oscillator strength has analytic continuation through the ionization threshold (Fano & Cooper 1968),
Moreover, the asymptotic behavior of the oscillator strength is expressed by the series
The same expression is valid for df_{1k}/dϵ_{1,k} with the substitution n^{2} = −k^{2} (n = ik) on the righthand 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 f_{1n} and df_{1k}/dϵ_{1,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:
In practice, the evaluation of this term is split into three subdomains:
The secondterm integrand on the rightside 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:
where is evaluated with (9) as follows:
The first and third integrals on the righthand side of Eq. (13) were calculated in a standard way using Gaussian quadratures.
3 Results for infinite lifetimes
Expressions (3) and (4) provide highprecision values of α_{pol} 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:
Convergence in the evaluation of 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 oneorder 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 fsum rule
and (ii) the static polarizability value which is exactly know (Wentzel 1926; Waller 1926; Epstein 1926)^{2}:
As shown in Table 1, both tests can be verified within machine precision. Here, an upper quantum number N = 10^{6} 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 {ϵ_{1,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 vanishes for all energies outside of resonances (i.e., ∀ ϵ ≠ ϵ_{1,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, α_{R} from the continuum becomes negative and reaches a minimum value of – 0.7712916 bohr^{3} 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 α_{pol} 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 α_{R} 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 α_{R} remains finite. Therefore, observed divergences of α_{pol} in our evaluations are only produced by resonances, and as a consequence of neglecting the level broadening.
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 Σ_{No} correspond to N_{0} = 10^{6}. Vertical lines are located on resonances and show a fast convergence effect. 
Fig. 2 Variation of the real and imaginary parts of the polarizability due to contributions from bound and continuum states. 
Contributions to the f sum rule and static polarizability α_{pol}(0) coming from bound and continuum states.
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 Finestructure 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
where Δ_{n} is the natural breadth of the level n and j refers to the two components of each state np (np_{1/2} and np_{3/2}). Absorption oscillator strengths of finestructure transitions are given by the following (Wiese & Fuhr 2009):
The natural breadth is basically the same for np_{1/2} and np_{3/2} levels, and it can be written in Rydberg units as
with Γ_{n} [s^{−1}] being the total probability rate of spontaneous decay from n to any lower level, and A_{nn} [s^{−1} ] is the Einstein coefficient,
with g_{n} being the statistical weight of the level n. In practice, we calculated Δ_{n} using accurate A_{nn}′ 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) Δ_{n} fitting expression for n ≥ 2 is given by
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 wellknown Kramers approximation,
yield Δ_{n} values with errors between 14% and 40% for n ≤ 20.
Due to relativistic corrections and spinorbit interaction, each np state splits into two levels with energies (Sobelman 1979)
with the zeroenergy point in the continuum edge. The ground state (1s) remains single (1s_{1/2}), but its energy changes from
As a consequence, each resonance 1 s–np splits into a doublet with energies
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 Lymanα_{1/2} and Lymanα_{3/2} transitions. Figure 4 shows the module of the polarizability with (solid line) and without (dotdashed 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 ϵ_{1,n} = 0.75 Ry) and its real and imaginary contributions become singulars. On the contrary, imaginary polarizability α_{1} with damping and finestructure effects (dotted line) presents sharp Lorentzian peaks centered at and , with a half width Δ_{n}. The maximum values of ∣α_{pol}∣ coincide with the α_{1} 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).
Selection of transition probabilities and natural level breadths.
Resonances energies (Ry) and wavelengths (Å) of transitions 1s_{1/2} − np_{1/2} and 1s_{1/2} − np_{3/2} for 2 ≤ n ≤ 20.
Fig. 4 Polarizability in the Lyα_{1/2} (1s_{1/2} − 2p_{1/2}) and Lyα_{3/2} (1s_{1/2} − 2p_{3/2}) resonances (solid line). We note that 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 Lyα_{3/2} core. Evaluation of Lymanα resonance without the effects of a fine structure and finite lifetime is represented by a dotdashed 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 and outside resonance cores, the magnitude of the polarizability (∣α_{pol}∣) is very well approximated by its real part
Preresonance region (ϵ ≲ ϵ_{1,2} = 0.75 Ry)
Redward of Lymanα, 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
being and
ϵ_{a} = 0.48083 (for s ≡ 0, Eq. (30) has a precision of 0.2%).
Resonance region
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 1s_{1/2} − np_{1/2} and 1s_{1/2} − np_{3/2} transitions,
For ϵ_{1,nj = 3/2} < ϵ < ϵ_{1,n + 1,j = 1/2} (n = 2,3,…), we adopted a fitting formula similar to that one used in Rohrmann (2018)
where
Quantities A_{n}, B_{n}, C_{n}, and D_{n} are given as follows:
For ϵ_{1,nj = 3/2} < ϵ < ϕ and ϕ_{n} < ϵ < ϵ_{1,n + 1,j = 1/2},
respectively. For ϵ_{1,nj = 3/2} < ϵ < ϕ_{n}
and for ϕ_{n} < ϵ < ϵ_{1,n+1,j=1/2},
The quantity ϕ_{n} represents the energy between two resonances where the real polarizability vanishes. Eq. (41) gives B_{n} (the position of ϕ_{n} relative to ϵ_{1,nj = 3/2} and ϵ_{1,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 ϵ_{1,nj} =_{1/2} with a precision better than 0.9% ∀n, 0.5% for n > 9 and 0.2% for n > 16. Blueward of ϵ_{1,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 ( = 1.0000133128 Ry)
For large values of ϵ, the nonrelativistic Rayleigh cross section converges to the Thomson scattering cross section. This means
A precision better than 0.4% (relative error) for the polarizability above the ionization threshold was obtained with
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 ∣α_{pol}∣^{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 finestructure components, and the natural breadth of the levels scale with α and n in the form
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 ∣α_{pol}∣ in their peaks increases with n. Relative intensities of ls_{1/2} − np_{1/2} and ls_{1/2} − np_{3/2} resonances are proportional to the ratio 1:2 of their oscillator strengths, which are in turn proportional to the statistical weights of sublevels np_{1/2}and np_{3/2}, 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 finestructure details. In astrophysical conditions where the finestructure splitting can be considered negligible, Eqs. (32) and (33) can be substituted by
where 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).
Fig. 6 Rayleigh cross section for the first four double resonances, from Lymanα to Lymanδ, computed with a fine structure and damping effects. Profiles are horizontally offset from each other in steps of 4 × 10^{−6} Ry. 
Fig. 7 Rayleigh cross section for photon energies including the resonances Lymanγ and Lymanδ. 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 Lymanδ. 
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 secondorder 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 20161128.
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All Tables
Contributions to the f sum rule and static polarizability α_{pol}(0) coming from bound and continuum states.
Resonances energies (Ry) and wavelengths (Å) of transitions 1s_{1/2} − np_{1/2} and 1s_{1/2} − np_{3/2} for 2 ≤ n ≤ 20.
All Figures
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 Σ_{No} correspond to N_{0} = 10^{6}. Vertical lines are located on resonances and show a fast convergence effect. 

In the text 
Fig. 2 Variation of the real and imaginary parts of the polarizability due to contributions from bound and continuum states. 

In the text 
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 
Fig. 4 Polarizability in the Lyα_{1/2} (1s_{1/2} − 2p_{1/2}) and Lyα_{3/2} (1s_{1/2} − 2p_{3/2}) resonances (solid line). We note that 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 Lyα_{3/2} core. Evaluation of Lymanα resonance without the effects of a fine structure and finite lifetime is represented by a dotdashed line. 

In the text 
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 
Fig. 6 Rayleigh cross section for the first four double resonances, from Lymanα to Lymanδ, 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 
Fig. 7 Rayleigh cross section for photon energies including the resonances Lymanγ and Lymanδ. 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 Lymanδ. 

In the text 
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