Rayleigh and Raman scattering cross-sections and phase matrices of the ground-state hydrogen atom, and their astrophysical implications

TL;DR

This work provides explicit Kramers-Heisenberg-Waller based expressions for Rayleigh and Raman scattering cross-sections and phase matrices of the hydrogen ground state, clarifying how Raman channels transfer UV photons near Ly resonances into optical/IR emission. It shows that Rayleigh and Raman$s$-branch scattering share Thomson-like phase behavior, while Raman$d$-branch is more isotropic, leading to depolarization in higher-order Raman features. The study demonstrates that Raman scattering of a dense UV continuum in neutral hydrogen can produce broad Balmer, Paschen, and higher hydrogen emission features with widths that can mimic Doppler-broadened lines, and it outlines observational diagnostics using line widths, flux ratios, and polarization to distinguish Raman-induced features from true Doppler broadening. These results have broad implications for interpreting spectra in symbiotic stars, planetary nebulae, star-forming regions, AGN, and damped Lyα systems, and they provide a framework for future radiative-transfer simulations with polarization.

Abstract

We present explicit expressions for Rayleigh and Raman scattering cross-sections and phase matrices of the ground $1s$ state hydrogen atom based on the Kramers-Heisenberg dispersion formula. The Rayleigh scattering leaves the hydrogen atom in the ground-state while the Raman scattering leaves the hydrogen atom in either $ns$ ($n\geq2$; $s$-branch) or $nd$ ($n\geq3$; $d$-branch) excited state, and the Raman scattering converts incident ultraviolet (UV) photons around the Lyman resonance lines into optical-infrared (IR) photons. We show that this Raman wavelength conversion of incident flat UV continuum in dense hydrogen gas with a column density of $N_{\text{H}} > 10^{21}~\text{cm}^{-2}$ can produce broad emission features centred at Balmer, Paschen, and higher-level lines, which would mimic Doppler-broadened hydrogen lines with the velocity width of $\gtrsim 1,000~\text{km}~\text{s}^{-1}$ that could be misinterpreted as signatures of Active Galactic Nuclei, supernovae, or fast stellar winds. We show that the phase matrix of the Rayleigh and Raman $s$-branch scatterings is identical to that of the Thomson scattering while the Raman $d$-branch scattering is more isotropic, thus the Paschen and higher-level Raman features are depolarized compared to the Balmer features due to the flux contribution from the Raman $d$-branch. We argue that observations of the line widths, line flux ratios, and linear polarization of multiple optical/IR hydrogen lines are crucial to discriminate between the Raman-scattered broad emission features and Doppler-broadened emission lines.

Figures (8)

• Figure 1: Illustration of the three contributions to the scattering amplitude (Equation \ref{['eqn:kramersheisenberg_nonoriented']}). Time flows from left to right. (a) The $\vec{A} \cdot \vec{A}$ 'seagull' term. The incident photon is annihilated and the scattered photon is simultaneously created. (b) Resonant process of photon absorption followed by emission. (c) Non-resonant process of photon emission followed by absorption.
• Figure 2: Illustration of the energy levels of the bound-states and continuum-states of the hydrogen atom. The horizontal axis indicates $s$, $p$, and $d$ orbitals, and the vertical axis indicates the principal quantum number or the energy level; $E_{n}=-1/2n^2$ for the bound-state and $E_{n'}=1/2n'^2$ for the continuum-state ($n$: positive integer, $n'$: positive real number). The magnetic substates are assumed to be degenerated. The dashed horizontal lines denote the excited intermediate states $|I\rangle$. The arrows denote various scattering paths from the ground $1s$ state $|A\rangle$, including Rayleigh ($1s \rightarrow 1s$) and Raman ($1s \nrightarrow 1s$) scattering. The final state $|B\rangle$ is the $s$ orbital when the incident photon energy $\hbar\omega$ is less than $E_{\text{Ly}\beta} = E_{3}-E_{1} = 12.09~\text{eV}$ (scattering into wavelengths around Ly$\alpha$, Ly$\beta$, and H$\alpha$), while the final state can fall into either $s$ or $d$ orbitals when $\hbar\omega > E_{\text{Ly}\beta}$.
• Figure 3: Top two panels: Rayleigh and Raman scattering cross-sections of the ground-state hydrogen atom up to $n_{B}=4$ as a function of the wavelength of the incident photon. The vertical lines indicate the Lyman resonance wavelengths. $\sigma_{n_{B}s}$ and $\sigma_{n_{B}d}$ respectively denote the cross-sections where the final state $|B\rangle$ is $s$ and $d$ orbitals with the principal quantum number $n_{B}$. $\sigma_{\text{tot}}$, $\sigma_{s}$, and $\sigma_{d}$ are the sums of the cross-sections up to $n_{B}=4$: $\sigma_{\text{tot}} \equiv \sigma_{s} + \sigma_{d} = \sum_{n_{B}=1}^{4}\sigma_{n_{B}s} + \sum_{n_{B}=3}^{4}\sigma_{n_{B}d}$. Bottom two panels: the scattering branching ratios. The branching ratios to $n_{B}=1$ (Lyman=Rayleigh), $2$ (Balmer), $3$ (Paschen), and $4$ (Brackett) are defined as $\sigma_{1s}/\sigma_{\text{tot}}$, $\sigma_{2s}/\sigma_{\text{tot}}$, $(\sigma_{3s}+\sigma_{3d})/\sigma_{\text{tot}}$, and $(\sigma_{4s}+\sigma_{4d})/\sigma_{\text{tot}}$, respectively. The total Raman scattering cross-section is $\sigma_{\text{tot}}-\sigma_{1s}$, and the Raman $s$-branch and $d$-branchs are defined as $(\sigma_{s}-\sigma_{1s})/(\sigma_{\text{tot}}-\sigma_{1s})$ and $\sigma_{d}/(\sigma_{\text{tot}}-\sigma_{1s})$, respectively.
• Figure 4: Geometry of the plane of scattering defined by the unit propagation vectors for the incident and scattered lights ($\vec{n}$ and $\vec{n}'$, respectively), and the meridian planes containing the unit propagation vectors. Two polarization basis sets perpendicular to $\vec{n}$ and $\vec{n}'$ are denoted by $\vec{\epsilon}$ and $\vec{\epsilon}'$, respectively, where the $\parallel-\perp$ basis is defined with respect to the scattering plane and the $l$-$r$ basis is defined with respect to the meridian plane. The three unit vectors $\vec{e}_{z}=(0,0,1)$, $\vec{n}$, and $\vec{n}'$ define a spherical triangle, whose angles at $\vec{n}$ and $\vec{n}'$ are denoted as $i_{1}$ and $i_{2}$, respectively.
• Figure 5: Top: Same as the top panel of Figure \ref{['fig:crosssections']}. The horizontal lines denote the wavelength intervals $\Delta\lambda$ in which the scattering optical depth $\tau_{\lambda} = \sigma_{\text{tot}}N_{\text{H}}$ gets greater than 1 for pure hydrogen scattering gas with $\log N_{\text{H}}(\text{cm}^{-2}) = 21, 22, 23$, and $24$. Bottom: the velocity widths corresponding to $\Delta\lambda$ of each of the Lyman series: $c(\Delta\lambda/\lambda)_{\text{Ly}}$ for $\text{Ly} = \text{Ly}\alpha$, $\text{Ly}\beta$, $\text{Ly}\gamma$, and $\text{Ly}\delta$. The overlayed thin solid lines denote the power-law model $c(\Delta\lambda/\lambda)_{\text{Ly}} \propto N_{\text{H}}^{0.5}$ fitted over $N_{\text{H}} = 10^{21-23}~\text{cm}^{-2}$ (Equation \ref{['eqn:optically_thick']}).