# Influence of the (n-n')-mixing processes on the optical properties of the hydrogen clouds in the broad-line region of AGNs

M.S. Dimitrijević, V.A. Srećković, Lj.M. Ignjatović

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M.S. Dimitrijević, V.A. Srećković, Lj.M. Ignjatović. Influence of the (n-n')-mixing processes on the optical properties of the hydrogen clouds in the broad-line region of AGNs. Contributions of the Astronomical Observatory Skalnaté Pleso, 2020, 50 (1), 10.31577/caosp.2020.50.1.179. hal-03253544

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# Influence of the (n-n')-mixing processes on the optical properties of the hydrogen clouds in the broad-line region of AGNs

M.S. Dimitrijević¹,², V.A. Srećković³ and Lj.M. Ignjatović³

¹ Astronomical Observatory, Volgina 7, 11060 Belgrade, Serbia

² Sorbonne Université, Observatoire de Paris, Université PSL, CNRS, LERMA, F-92190 Meudon, France (E-mail: mdimitrijevic@aobrs)

³ Institute of Physics Belgrade, Pregrevica 118, 11080 Zemun, Belgrade, Serbia, (E-mail: vlada@ipb.ac.rs)

Received: July 16, 2019; Accepted: October 29, 2019

**Abstract.** The role of (n-n')-mixing processes in active galactic nuclei broad-line region (BLR) clouds has been studied. Our investigation indicates that these collisional processes must have influence on the populations of the excited hydrogen atoms in moderately ionized layers of dense parts of the BLR clouds and are of interest for its modelling and research. We provide results potentially important for the spectroscopy and modeling of the BLR clouds, as well as for the study of hydrogen Rydberg states.

**Key words:** Galaxies: active – atomic processes –Plasmas

## 1. Introduction

In the papers Mihajlov et al. (2005); Gnedin et al. (2009); Mihajlov et al. (2016) authors drew attention to the importance of inelastic processes in slow collisions of Rydberg state atoms with ground state hydrogen atoms (i.e. excitation/de-excitation processes) in solar and stellar atmospheres

$$H^*(n) + H \rightarrow H + H^*(n'), \quad (1)$$

where H are hydrogen atoms in their ground states, H*(n) and H*(n') are hydrogen atoms in a highly excited (Rydberg) state.

These authors concluded that (n-n')-mixing processes (Eq. 1) significantly influence on the populations of hydrogen Rydberg atoms (RA) in weakly ionized layers of stellar atmospheres and particularly in Solar photosphere and the lower chromosphere and need to be added in any modelling and investigation of these regions. In Barklem (2007) and Mashonkina (2009) the (n-n')-mixing processes have also been analyzed together with others, and the conclusion was that there is a great need for their further investigation, since in many cases there are uncertainties in the rate coefficients of hydrogen collisions (see also Srećković et al., 2018a,b).
In active galactic nuclei (AGN), especially in the region of the moderately ionized layers of dense parts of the broad-line region (BLR), in which physical conditions are closer to stellar atmospheres than to photoionized nebulae (Ilić et al., 2007; Negrete et al., 2012; Ilić et al., 2017; Bon et al., 2018; Osterbrock & Ferland, 2006). In the BLR clouds temperature may drop to the much lower values where the gas is weakly ionized with huge number of hydrogen atoms in excited states, due to the high density and large optical depths (Crosas & Weisheit, 1993; Negrete et al., 2012; Netzer, 2013). Consequently, it is of interest to investigate the influence of the mentioned collisional processes Srećković et al. (2018c).

That is the reason why we have extended the investigation of these pro-
cesses and calculated their rate coefficients for a wider region of plasma pa-
rameters and for quantum numbers n up to 20 in order to be potentially used
in AGN modeling. The results presented here cover the range of temperatures
$2000 \text{ K} \le T \le 30000 \text{ K}$, broader than the regions relevant for weakly ionized lay-
ers of dense parts of the BLR clouds (see e.g. Crosas & Weisheit, 1993; Negrete
et al., 2012), so that the data presented here can be also used for investigation of
the atmospheres of Sun and Sun-like stars etc. (Mihajlov et al., 2011; Mihajlov
et al., 2011; Srećković et al., 2014). Also, study of these processes are of inter-
est for some laboratory spectroscopical investigation and technical applications
(Dubernet et al., 2016; Marinković et al., 2017).

We describe in Section 2 the methods of calculation and then in Section 3
we give the results of the calculation. In Section 4 the conclusions are presented.

2. Method

2.1. The resonant mechanism

On the basis of the resonant mechanism (Mihajlov et al., 2012; Srećković et al., 2013, 2018c) the (n-n')-mixing processes (Eq. 1) are characterized as a product of the resonant energy exchange between the outer electron *e* and the electronic component of the subsystem H⁺ + H (Fig.1). The system H*(n)+H is described as: *e* + (H⁺ + H), where *e* is the outer electron of the hydrogen RA H*(n). For describing the subsystem (H⁺ + H) electronic states, the adiabatic electronic ground state, or the first excited state of the molecular ion were used.

## 2.2. The (n-n')-mixing rate coefficients

The procedure of obtaining the rate coefficients for the processes (Eq. 1) is
described in detail in Mihajlov et al. (2004); Srećković et al. (2013) and only the
corresponding final expressions are given here. The rate coefficients $K_{n;n+p}^{mix}(T)$
for the mixing processes between two excited levels with main quantum numbers
<figure><img src="image_2.png" /><figcaption>**Figure 1.** The schematic presentation of the (n-n')-mixing processes. R presents the internuclear distance, $U_1(R)$ and $U_2(R)$ are the potential energy curves of the initial and final electronic state of considered system.</figcaption></figure>

$n$ and $n + p$, can be expressed as a function of the temperature $T$ as

$$ K_{n;n+p}^{mix}(T) = \frac{2\pi}{3\sqrt{3}} \frac{(ea_0)^2}{\hbar} \cdot n^{-5} \cdot g_{n;n+p} \times \int_{R_{min}(n,n+p)}^{R_{max}(n,n+p)} X(R) \cdot \exp\left[-\frac{U_2(R)}{kT}\right] \frac{R^4 \cdot dR}{a_0^5}, \quad (2) $$

where $T_e = T_a = T$, $T_e$ and $T_a$ are the electron and atom temperatures and $T$ is their common value, $g_{n;n+p}$ is the Gaunt factor, $R_{min}(n, n+p)$ and $R_{max}(n, n+p)$ are quantities defined in Mihajlov et al. (2008), $a_0$ is the atomic unit of length, $e$ is the value of electron charge, $R$ is the internuclear distance, $\hbar$ and $k$ are the Planck and Boltzmann constants and $U_2(R)$ is the potential curve of the first excited electronic states of the considered system. $X(R)$ is defined as the function $X(R) = \frac{\Gamma(3/2; \frac{|U_1(R)|}{kT})}{\Gamma(3/2)}$, where $\Gamma(3/2; x)$ and $\Gamma(3/2)$ are the corresponding values of the incomplete and complete Gamma functions. The rate coefficients for the corresponding inverse deexcitation processes are

$$ K_{n;n-p}^{mix}(T) = K_{n-p;n}^{mix}(T) \cdot \frac{(n-p)^2}{n^2} \cdot \exp\left(\frac{\epsilon_{n-p;n}}{kT}\right), \quad (3) $$

where $\epsilon_{n-p;n} = \epsilon_n - \epsilon_{n-p}$ and $\epsilon_n = -Ry/n^2$, $Ry$ is the Rydberg constant (for details see Mihajlov et al., 2008).
## 2.3. The relative efficiency

To estimate the relative efficiency of processes defined in Eq. (1) in comparison to concurrent electron-atom mixing processes i.e. electron-excited atom collisional excitation/de-excitation processes

$$e + H^*(n) \rightarrow e' + H^*(n' = n \pm p), \qquad (4)$$

we use the parameter $F_n^{(\pm)}(T)$

$$F_n^{(\pm)}(T) = \frac{\sum_{p=1}^{5} K_{n;n \pm p}^{mix}(T) N(n) N(1)}{\sum_{p=1}^{5} K_{n;n \pm p}^{ea}(T) N(n) N_e} = \frac{\sum_{p=1}^{5} K_{n;n \pm p}^{mix}(T)}{\sum_{p=1}^{5} K_{n;n \pm p}^{ea}(T)} \eta, \qquad (5)$$

where $N(n)$ is the population of the excited atom states, $N_e$ is the density of free electrons and the parameter $\eta = N(1)/N_e$. Here quantity $K_{n;n \pm p}^{ea}(T_e = T)$ is the rate coefficient for the electron-atom process (Eq.4) taken from Vriens & Smeets (1980). The expressions $K_{n;n \pm p}^{mix}(T)N(n)N(1)$ and $K_{n;n \pm p}^{ea}(T)N(n)N_e$ are the partial atom- and electron-Rydberg atom excitation and deexcitation fluxes.

## 3. Results and Discussion

The rate coefficients $K_{n;n \pm p}^{mix}(T)$ for mixing processes (Eq.1) are calculated for the following parameters $4 \le n \le 20$, $1 \le p \le 5$ and $2000 \text{ K} \le T \le 30000 \text{ K}$. Here we give the results only for temperatures $\le 10000 \text{ K}$, which are relevant for the moderately ionized layers of dense parts of the BLR clouds and data for higher temperatures are of interest for other moderately ionized astrophysical plasmas. In Fig. 2 the excitation rate coefficients are presented for selected excited states $n = 4, \dots, 13$ and temperature $T$ in the range $3000 - 10000 \text{ K}$ relevant for the moderately ionized layers of dense parts of the BLR clouds.

We have calculated the values of the parameter $F_n^{(+)}$ for $4 \le n \le 8$ in order to compare the relative influence of $(n-n')$-mixing processes (Eq.1) and influence of concurrent electron-atom excitation processes (Eq.4) for the physical conditions which could be relevant in the BLR of AGNs (see Fig. 3). It can be noted that the $(n-n')$-mixing processes for $n \le 5$ are dominating over the concurrent ones in the lower temperature regime (Eq.4, bottom panel) whereas for $6 \le n \le 8$ both processes are comparable (see Fig. 3). Also the processes (Eq.1) are comparable with concurrent excitation processes (Eq.4) only for lower $n$ in the parts of the BLR with higher temperatures. On Fig. 3 the shaded area ($2 \le n \le 4$) present extrapolated values due to the missing reliable data for the corresponding rate coefficients. The only existing rate coefficients in the literature for $n = 2$ and 3 are data from Drawin (1968) which are not recommended (see Barklem 2007, Srećković et al. 2018c). We note that these rate coefficients for $n = 2$ and 3 are of great importance and the impact of processes (see Eq.1) should be much
<figure><img src="image_3.png" /><figcaption>**Figure 2.** Plot of excitation rate coefficients for processes H*(n) + H → H + H*(n'), for selected excited states n = 4, ...13 and temperature T in the range 3 000 - 10 000K.</figcaption></figure>

higher on the populations of hydrogen excited atoms of the AGNs in comparison

to the concurrent processes.

The presented results demonstrate that these processes must have a notice-
able influence on the optical properties of the weakly ionized regions of dense
parts of the BLR clouds and that they should be used in the models. In addi-
tion, they could be helpful for modeling of various stellar atmospheres with low
effective temperatures, as well as for the research of Rydberg states of hydrogen
and the analysis of their influence during the cosmological recombination epoch
<figure><img src="image_4.png" /><figcaption>**Figure 3.** Efficiencies $F_n^{(+)}$ of the analyzed processes for $4 \le n \le 8$ and temperatures 6000 K (upper panel), 4000 K (middle panel), and 2000 K (bottom panel). The extrapolated values below $n = 4$ (see text) are shown in the left shaded region.</figcaption></figure>

(Fontenla et al., 2009; Przybilla & Butler, 2004; Barklem, 2007; Mashonkina, 2009).

**4. Conclusion**

Our primary task in this paper was determination of rate coefficients for parameters that exists in BLR clouds in order to be used in the corresponding models. The results demonstrate the fact that the considered (n-n')-mixing processes must have a significant influence on the populations of excited hydrogen atoms in cooler and denser parts of the BLR clouds in AGNs in comparison to the concurrent processes and must have influence on the optical properties.

**Acknowledgements.** This work is made within projects of the Ministry of Education, Science and Technological Development of Serbia, 176002, III44002.

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