# Atom-Rydberg-atom chemi-ionization processes in solar and DB white-dwarf atmospheres in the presence of (n – n')-mixing channels

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

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A. Mihajlov, V. Srećković, Lj. Ignjatović, M. Dimitrijević. Atom-Rydberg-atom chemi-ionization processes in solar and DB white-dwarf atmospheres in the presence of (n – n')-mixing channels. Monthly Notices of the Royal Astronomical Society, 2016, 458 (2), pp.2215-2220. 10.1093/mnras/stw308. hal-02318252

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# Atom–Rydberg-atom chemi-ionization processes in solar and DB white-dwarf atmospheres in the presence of (n – n')-mixing channels

A. A. Mihajlov,<fnref n="1" />,<fnref n="2" /> V. A. Srećković,<fnref n="1" />,<fnref n="2" />* Lj. M. Ignjatović<fnref n="1" />,<fnref n="2" />*
and M. S. Dimitrijević<fnref n="2" />,<fnref n="3" />,<fnref n="4" />,<fnref n="5" />*

Accepted 2016 February 5. Received 2016 January 21; in original form 2015 December 2

## ABSTRACT

In this paper, the rate coefficients of the chemi-ionization processes in H(1s) + H*(n, l) and He(1s<fnref n="2" />) + He*(n, l) collisions (where the principal quantum number n ≫ 1) are determined for the first time, taking into account the influence of the corresponding (n – n')-mixing processes. It is demonstrated that the inclusion of (n – n') mixing in the calculation influences the values of chemi-ionization rate coefficients significantly, particularly in the lower part of the block of Rydberg states. The interpretation of this influence is based on two existing methods of describing inelastic processes in symmetrical atom–Rydberg-atom collisions. The calculations of the chemi-ionization rate coefficients are performed for the temperature region that is characteristic of solar and DB white-dwarf atmospheres.

**Key words:** atomic processes – molecular processes – Sun: atmosphere – Sun: photosphere – stars: atmospheres – white dwarfs.

## 1 INTRODUCTION

Our main aim in this paper is to consider of two types of atomic collision processes, which occur simultaneously in stellar atmospheres and have an influence on one another. This is about the chemi-ionization processes

$$A^*(n, l) + A \Rightarrow e + A + A^+, \qquad (1a)$$

$$A^*(n, l) + A \Rightarrow e + A_2^+, \qquad (1b)$$

and the (n – n')-mixing excitation–de-excitation processes

$$A^*(n, l) + A \Rightarrow A + A^*(n', l), \qquad (2)$$

where A and A⁺ are atoms and their positive ions in their ground states, A*(n, l) is the atom in a highly excited (Rydberg) state with the principal quantum number n ≫ 1 and orbital quantum number l, A₂⁺ is the corresponding molecular ion in the ground electronic state, and e is a free electron. In this paper, we consider the most significant astrophysical cases A = H and A = He.

Because the history of the investigations of the processes (1) and (2) and the current state of this body of literature have recently

been presented in Mihajlov et al. (2012), only the articles directly connected to the material of this work will be cited here. Until now, these processes have always been considered separately, although both processes are caused by the same mechanism. Here we consider the dipole resonant mechanism, which has been described in the literature several times and has been discussed in detail in the review paper Mihajlov et al. (2012). For the first time, in this paper, we consider the chemi-ionization processes (1) in the presence of (n – n')-mixing processes.

Fig. 1 illustrates the above-mentioned dipole resonant mechanism as a resonant energy exchange within the electronic component of the collision system A*(n, l) + A: the transition of the subsystem A + A⁺, from the excited electronic state with the energy U₂(R) to the ground electronic state with the energy U₁(R) (long arrow pointing downward), is followed by the simultaneous transition of the Rydberg electron from the initial bound state with energy εₙ < 0 to the free state with the energy ε(k), under the resonance condition, U₂(R) – U₁(R) = ε(k) – ε(n, l) (long arrow pointing upward). Similarly, in the case of (n – n')-mixing, the same mechanism causes transitions of the subsystem A + A⁺, between the excited and ground electronic states (in the domain of larger R), together with transitions of the Rydberg electron from the state with energy ε(n, l) to the states ε(n', l'), where n' ≠ n. These transitions are illustrated in Fig. 1 with the corresponding short arrows.

As we can see from Fig. 1, the atomic collision that we consider proceeds in accordance with the excited molecular term U₂(R).

<footnote n="1">¹University of Belgrade, Institute of Physics, PO Box 57, 11001 Belgrade, Serbia</footnote>

<footnote n="2">²Isaac Newton Institute of Chie, Yugoslavia Branch, Volgina 7, 11060 Belgrade, Serbia</footnote>

<footnote n="3">³Astronomical Observatory, Volgina 7, 11160 Belgrade 74, Serbia</footnote>

<footnote n="4">⁴IHIS-Technoexperts, Bežanijska 23, 11080 Zemun, Serbia</footnote>

<footnote n="5">⁵Observatoire de Paris, F-92195 Meudon Cedex, France</footnote>

<footnote>*E-mail: vlada@ipb.ac.rs (VAS); ljuba@ipb.ac.rs (LMI); mdimitrijevic@ aob.rs (MSD)</footnote>
<figure><img src="image_3.png" /><figcaption>**Figure 1.** Schematic illustration of the resonant transitions that cause the processes (1) and (2).</figcaption></figure>

Before it enters the zone where the chemi-ionization processes (1) are possible, the system $A^*(n, l) + A$ passes through the zone where the processes (2) take place. Because of this, a way to include processes (2) in the calculation of the rate coefficients of the chemi-ionization processes (1) is developed in this paper.

The processes (1) are characterized by the partial rate coefficients, which are a function of quantum numbers $n$ and $l$ as well as a function of plasma temperature $T$. In this work, these rate coefficients are calculated in the cases of both hydrogen and helium, for $3 \le n \le 15$. Bearing in mind the application of the results obtained here to the photosphere of the Sun and to some DB white dwarfs, the averaged rate coefficients are determined in the corresponding temperature ranges: for $4000 \le T \le 10000$ K in the case of hydrogen, and for $7000 \le T \le 24000$ K in the case of helium.

The values of chemi-ionization rate coefficients obtained here are compared with the corresponding values of the rate coefficients determined earlier, for the same atmospheres, in Mihajlov et al. (2003a, 2011a), where $(n - n')$-mixing processes were not taken into account. In the paper, all expressions are given in atomic units. As a final note, the presented material consists of five tables in the supporting material.

## 2 THEORY

### 2.1 Description and general formulae

Let $K_A^a(n, l; T)$ and $K_A^b(n, l; T)$ be the partial rate coefficients of processes (1a) and (1b) determined separately for given $n, l$ and $T$, and let $K_A^{ab}(n, l; T)$ be the total rate coefficient of processes (1) defined by

$$K_A^{ab}(n, l; T) = K_A^a(n, l; T) + K_A^b(n, l; T). \qquad (3)$$

The rate coefficients $K_A^a(n, l; T)$ and $K_A^b(n, l; T)$ are determined on the basis of standard expressions

$$K_A^{ab}(n, l; T) = \int_{E_{n,i}}^{\infty} \sigma_A^{ab}(n, l; E) \sqrt{\frac{2E}{m_{\text{red}}}} f_T(E) dE, \qquad (4)$$

$$K_A^b(n, l; T) = \int_{E_{n,i}}^{\infty} \sigma_A^b(n, l; E) \sqrt{\frac{2E}{m_{\text{red}}}} f_T(E) dE, \qquad (5)$$

where $E$ is impact energy, $\sigma_A^{ab}(n, l; E)$ and $\sigma_A^b(n, l; E)$ are the corresponding cross-sections, $m_{\text{red}}$ is the reduced mass of the sub-

system $A + A^+$ and $f_T(E)$ is the Maxwell distribution function, $f_T(E) = \exp(-E/kT)\sqrt{E}$. The parameter $E_{n,i}$ is given here by the relation $E_{n,i} = U_2(R_{n,i})$ where $R_{n,i}$ is the upper limit of the chemi-ionization zone, defined as the root of the equation

$$U_{12}(R) \equiv U_2(R) - U_1(R) = \frac{1}{2n^2}. \qquad (6)$$

The above-mentioned cross-sections are determined here within the semi-classical approximation, by means of the standard expressions

$$\begin{align}
\sigma_A^{ab}(n, l; E) &= 2\pi \int_0^{\rho_{\text{max};A}^{ab}} P_A^{ab}(n, l; \rho; E) \rho \, d\rho, \nonumber \\
\sigma_A^b(n, l; E) &= 2\pi \int_0^{\rho_{\text{max};A}^b} P_A^b(n, l; \rho; E) \rho \, d\rho \tag{7}
\end{align}$$

where $\rho$ is the impact parameter, $\rho_{\text{max};A}^{ab}$ and $\rho_{\text{max};A}^b$ are its corresponding maximal values of this parameter, and $P_A^{ab}(n, l; \rho; E)$ and $P_A^b(n, l; \rho; E)$ are the total probability of chemi-ionization and the probability of associative ionization, respectively, determined for the given values of $n, l, \rho$ and $E$. These probabilities are taken here in the forms

$$P_A^{ab}(n, l; \rho; E) = \frac{1}{2} p_0(n, l; \rho; E) p_{i;A}^{ab}(n, l; \rho; E) \qquad (8)$$

and

$$P_A^b(n, l; \rho; E) = \frac{1}{2} p_0(n, l; \rho; E) p_{i;A}^b(n, l; \rho; E), \qquad (9)$$

where $1/2$ is the probability that the subsystem $A + A^+$ develops according to the term $U_2(R)$. The quantity $p_0(n, l; \rho; E)$ is the probability that in the domain of values of $R$ where the processes (2) with $n' > n$ are possible, the mentioned subsystem remains in the same electronic state with the energy $U_2(R)$. Finally, $p_{i;A}^{ab}(n, l; \rho; E)$ and $p_{i;A}^b(n, l; \rho; E)$ are the corresponding ionization probabilities determined under the condition that the subsystem $H(1s) + H^+$ or $He(1s^2) + He^+(1s)$ enters in the ionization zone in the mentioned state with a probability equal to one.

Now, we determine the average total rate coefficient

$$K_A(n, T) = \frac{1}{n^2} \sum_{l=0}^{n-1} (2l + 1) K_A^{ab}(n, l; T), \qquad (10)$$

as well as the average rate coefficient of the associative ionization $K_A^b(n, T)$ given by

$$K_A^b(n, T) = \frac{1}{n^2} \sum_{l=0}^{n-1} (2l + 1) K_A^b(n, l; T), \qquad (11)$$

(e.g. Ryabtsev et al. 2005; Golubkov & Devdariani 2011; O’Keeffe et al. 2012). The efficiency of associative ionization is characterized by the corresponding branch coefficients

$$X_A^b(n; T) = \frac{K_A^b(n; T)}{K_A(n; T)}. \qquad (12)$$

### 2.2 Ionization decay: the probability

The probabilities $p_{i;1}^{ab}(n, l; \rho; E)$ and $p_{i;1}^b(n, l; \rho; E)$ are determined here within the quasi-static approximation. These probabilities are taken in the form (Mihajlov et al. 2007, 2011a,b)

$$\begin{align}
p_{i;A}^{ab}(n, l; \rho; E) &= 1 - e^{-2q_{i;A}^1}, \nonumber \\
p_{i;A}^b(n, l; \rho; E) &= e^{-q_{i;A}^b} [1 - e^{-2q_{ias;A}}]. \tag{13}
\end{align}$$
The variables $q_{i;A}^I$, $q_{i;A}^{II}$ and $q_{ias;A}$ are given by expressions

$$ 
\begin{aligned}
q_{ias;A} &= q_{i;A}^I - q_{i;A}^{II}, \\
q_{i;A}^I &= \int_{R_0}^{R_{n,i}} \frac{W_i(n, l; R)}{\nu_{\text{rad}}(E, \rho; R)} \, dR, \\
q_{i;A}^{II} &= \int_{R_{A;max}^{b}}^{R_{n,i}} \frac{W_i(n, l; R)}{\nu_{\text{rad}}(E, \rho; R)} \, dR,
\end{aligned} 
\quad (14) $$

where the rate coefficients of ionization decay $W_i(n, l; R)$ and radial ion–atom velocity $\nu_{\text{rad}}(E, \rho; R)$ are given by expressions

$$ 
\begin{aligned}
W_i(n, l; R) &= \frac{1}{2\pi} c U_{12}^3(R) D_{12}^2(R) \sigma_{\text{ph},i}(n, l, \varepsilon_{\text{ph}}), \\
\nu_{\text{rad}}(E, \rho; R) &= \sqrt{\frac{2}{m_{\text{red}}}} \left[ E - U_2(R) - \frac{E \rho^2}{R^2} \right].
\end{aligned} 
\quad (15) $$

In these expressions, c is the speed of light, $D_{12} = |<1|\hat{d}_{m,i}|2>$ is the molecular-ion dipole matrix element, $\sigma_{\text{ph},i}(n, l, \varepsilon_{\text{ph}})$ is the cross-section for photoionization of excited hydrogen atom $H^*(n, l)$ or helium atom $He^*(n, l)$ by a photon with energy $\varepsilon_{\text{ph}} = U_{12}(R)$. In the expression for the dipole matrix element, $\hat{d}$ denotes the operator of the corresponding ion dipole momentum where $|1>$, $|2>$ denote the ground and first excited state of this ion.

In equation (14), $R_0$ is the lower limit of the domain R that is reached during the collision with a given $\rho$ and E, and $R_{1,2;\text{man}}^b$ denotes the upper limit of the domain R where only the process of associative ionization (1b) is possible. Thus, the parameters $R_0$ here represent the roots of the equation $U_2(R) = E(1 - \rho^2/R^2)$, and $R_{A;\text{max}}^b$ represents the root of the equation $U_{12}(R) = E$. Let us note that in expressions (13) and (14) it is assumed that $R_{A;\text{max}}^b < R_{n,i}$. In the case of $R_{A;\text{max}}^b > R_{n,i}$ we have the quantity $q_{i;A}^{II} = 0$ and $q_{ias} = q_{i;A}^I$.

Here, we draw attention to the fact that at this point there is already a difference from the previous works of Mihajlov et al. (2007, 2011a,b) concerning the chemi-ionization processes in stellar atmospheres. There, the chemi-ionization rate coefficients were determined with the average ionization decay rate, obtained by averaging the partial rates over the whole shell with given n. This makes it possible to use the Kramers photoionization cross-section averaged over the shell adjusted by means of the approximate Gaunt factor. In contrast, here the rate coefficients $K_A^{ab}(n, l; T)$ and $K_A^{b}(n, l; T)$ were determined on the basis of equations (4) and (5) using the partial cross-sections for photoionization, determined here on the basis of the exact expressions from Sobelman (1979).

## 2.3 Pre-ionization decay: the probability

We can see from equations (8) and (9) that the basic difference from the previous papers is that we directly take into account the effect of the decay of the initial electronic state of the considered atom–Rydberg-atom system, because of the possibility of the occurrence of excitation processes (2) with $n' > n$. This is done by introducing the probability of maintenance of this state $p_0(n, l; \rho; E)$. This probability is determined through a modified version of the approximate method described in Mihajlov et al. (2004) dedicated to the $(n - n')$-mixing processes. Note that the essence of this method is that, at given n, each block of Rydberg states from $n' = n + p_1$ to $n' = n + p_2$ is spreading in a part of the quasi-continuum limited by the values $n + p_1 - \delta_n$ and $n + p_2 + 1 - \delta_n$, where the parameters $\delta_n$ are determined from the condition of maintenance of the total number of states and total oscillator strengths for transitions from

<figure><img src="image_4.png" /><figcaption>**Figure 2.** A diagram showing the partial spreading of the discrete Coulomb spectre into the continuum.</figcaption></figure>

the initial state of Rydberg electron to all states of the separated block (see Fig. 2).

This modification is conditioned by the fact that in Mihajlov et al. (2004) an approximate average rate was used for the transition of the Rydberg electron of the considered system from a whole shell with the given n to a whole shell with $n' = n + p$, where $p \ge 1$. In contrast, here we must consider transitions of the Rydberg electron from an individual state $|n, l>$ to individual states $|n + p, l - 1>$ and $|n + p, l + 1>$. Note that this difference does not affect the size of the pre-ionization zone. Therefore, similarly to Mihajlov et al. (2004), we accept that the pre-ionization zone generally forms the domain of internuclear distances such that

$$ R_{n,i} < R < R_{n,n'+1-\delta_n}, \quad (16) $$

where $\delta_n = 0.5 \times [1 - (1/3) \times O(1/n)]$. Note that, here and later in the paper, $R_{n,n'}$, where $n' > n$ means the root of the equation

$$ U_{12}(R) = 0.5 \times [1/n^2 - 1/n'^2]. \quad (17) $$

Finally, in Mihajlov et al. (2004), it was taken that the domains of R corresponding to the partial transitions with $p > 1$ make up the intervals $(R_{n,n'+p-\delta_n}, R_{n,n'+p-1-\delta_n})$. However, with the method used here, it is taken that the domain of R corresponding to a partial transition with $p > 1$ in the general case only belongs to the mentioned intervals, and does not coincide with them.

In this paper, we take into account the transitions with $1 \le p \le 5$. Therefore, the probability $p_0(n, l; \rho; E)$ could be represented as

$$ p_0(n, l; \rho; E) = \prod_{p=1}^{5} p_{p,0}(n, l; \rho; E), \quad (18) $$

where $p_{p,0}(n, l; \rho; E)$ is the probability of the maintenance of the initial state of the system remaining within the interval $(R_{n,n'+p+1-\delta_n}, R_{n,n'+p-\delta_n})$.

Because the mechanism of the pre-ionization decay is the same as the one for the ionization, we assume that probabilities $p_{p,0}(n, l; \rho; E)$ are given by the relations

$$ 
\begin{aligned}
p_{p,0}(n, l; \rho; E) &= e^{-x_p}, \\
x_p &= \int_{R_{n,n'+p+1-\delta_n}}^{R_p} \frac{w_{n,n'+p}(n, l; R)}{\nu_{\text{rad}}(E, \rho, R)}. 
\end{aligned} 
\quad (19) $$

Here, the decay rate $w_{n,n'+p}(n, l; R)$ is conditioned by the dipole mechanism within the interval $(R_{n,n'+p+1-\delta_n}, R_{n,n'+p-\delta_n})$. The
upper limit is $R_p = R_{n;n+p-\delta_n}$ if $R_{n;n+p-\delta_n} \le R_{up;mix}(E, \rho)$ and $R_p = R_{up;mix}(E, \rho)$ if $R_{n;n+p-\delta_n} > R_{up;mix}(E, \rho) \ge R_{n;n+p}$, where the resonant distance is $R_{n;n+p}$ for the process (2) at given $n$ and $n' = n + p$. This distance is determined as the root of the equation, $U_{12}(R) = 0.5 \times [1/n^2 - 1/(n+p)^2]$. The meaning of parameter $R_{up;mix}(E, \rho)$ is discussed separately later in Section 2.3.1. Let us point out that here, when $R_{up;mix}(E, \rho) < R_{n;n+p}$ is considered, $p_{p;0}(n, l; \rho; E) = 0$. Thus, the decay rate $w_{n;n+p}(n, l; R)$ is given here by the relations

$$w_{n;n+p}(n, l; R) = \frac{2\pi}{3} U_{12}^4 (R_{n;n+p}) \tilde{n}^3 D_{12}^2 r_{n,l;n+p}^2, \\ \tilde{n} = n[1 - 2n^2 U_{12}(R)]^{-1/2}, \qquad (20)$$

where $r_{n,l;n+p}^2 = |<n, l|\hat{d}_{at}|n, l-1>|^2 + |<n, l|\hat{d}_{at}|n, l+1>|^2$, $\hat{d}_{at}$ is the operator of the dipole moment of the hydrogen atom, and $|n, l>$, $|n, l-1>$ and $|n, l+1>$ denote the corresponding states of the Rydberg electron.

### 2.3.1 The parameter $R_{up;mix}$

The characteristic length $R_{up;mix}$ is defined here as the upper limit of the domain $R$ where at given $E$ and $\rho$ we can consider that the inner electron is in the subsystems $H^+$ + $H(1s)$ and $He^+(1s) + He(1s^2)$ and that it is sufficiently delocalized, so that these subsystems can be treated as quasi-molecular complexes. As a qualitative characteristic of the mentioned delocalization, we take here the probability of resonant charge exchange $P_{c;exc}(R; E; \rho)$ in the mentioned subsystems as a function of $R$ at given $\rho$ and $E$. The basis for this is found in the theory of the resonant charge exchange of the processes in the symmetric ion-atom collisions, developed in Firsov (1951) and Bates & Boyd (1962).

From this theory, it follows that

$$P_{c;exc}(R; E; \rho) = \sin^2[\varphi(R; E; \rho)], \qquad (21)$$

where the phase $\varphi(R; E; \rho)$ is given by the relation

$$\varphi(R; E; \rho) = \frac{1}{2} \int_{R}^{\infty} \frac{U_{12}(R')}{\nu_{\text{rad}}(R', \rho, E)} dR'. \qquad (22)$$

This can be used in the case considered because $P_{c;exc}(R; E; \rho)$ becomes noticeably different from zero only deeply inside the orbit of the Rydberg electron at given $n$. Taking into account data from Firsov (1951) and Bates & Boyd (1962), we can consider that when $P_{c;exc}(R; E; \rho)$ reaches the value of $1/2\pi$, the corresponding $R$ can be treated as the upper limit of the charge exchange zone at given $\rho$ and $E$, and thus as the upper limit of the domain with a sufficient degree of delocalization of the electron in the subsystems $H^+$ + $H(1s)$ and $He^+(1s) + He(1s^2)$. Consequently, the parameter $R_{up;mix}$ is determined here as the root of the equation

$$\sin^2[\varphi(R; E; \rho)] = \frac{1}{2\pi}, \qquad (23)$$

where $\varphi(R; E; \rho)$ is given by equation (22) under the condition that this root is in the domain of monotonic increase of the left-hand side of equation (23).

The behaviour of the phase $\varphi(R; E; \rho)$ in the hydrogen and helium cases is illustrated in Table 1 in the supporting material, where its values for $E = E_{n,i}$, $\rho = 0$ and $R = R_{n,n+1}$ within the range $3 \le n \le 15$ are shown. Note that the values for the helium case in Table 1 are connected with the excited states of the $He^*(n, l)$ atom with $l > 0$. Of course, these data should be treated as the qualitative data as equations (21) and (22) make strict sense when $E \gg U_{12}(R)$, while in the hydrogen case this condition is fulfilled only for $n > 7$ and for helium.

## 3 RESULTS AND DISCUSSION

From the above presented material, it follows that the total average rate coefficients of the processes (1) as well as the rate coefficients for the associative ionization (1b), $K_A(n; T)$ and $K_A^b(n; T)$, are determined by means of equations (3)–(20). Note that the chemi-ionization processes (1) can be described on the basis of the dipole resonant mechanism only in the case of a state with $n \ge 5$, for which the potential curves of the system $H^*(n, l) + H(1s)$ lie above the potential curve of the system $H^+ + H^-(1s^2)$, where $H^-(1s^2)$ is the stable negative hydrogen ion. However, it can be shown that the points of the intersection of the potential curves of the system $H^*(n, l) + H(1s)$ with $n = 3$ and 4 with the potential curve of the system $H^+ + H^-(1s^2)$ are located at internuclear distances, which are several times larger than the average atomic radius $H^*(n, l)$. Therefore, the existence of these intersections cannot significantly affect the values of the corresponding rate coefficients of the processes (1a). From this aspect, the application of the dipole resonant mechanism can be completely justified in the case of states with $n = 3$ and 4.

In Tables 2 and 4 in the supporting material, the values are presented of the total average rate coefficients of chemi-ionization processes in the hydrogen and helium cases $K_H(n; T)$ and $K_{He}(n; T)$ within the range $3 \le n \le 15$. Bearing in mind that the main applications of the results obtained here are to the photosphere and lower chromosphere of the Sun (hydrogen case) and to the corresponding parts of the atmospheres of DB white dwarfs (helium case), the calculations of these rate coefficients were performed here for temperatures $4000 \le T \le 10000$ K in Table 2 and $7000 \le T \le 24000$ K in Table 4. The values of the corresponding branch coefficients $X_A^b(n; T)$ for the same $n$ and $T$ are presented in the supporting material in Tables 3 and 5. In accordance with the above, the rate coefficients have been calculated by summing the probability of the decay of the initial state of the collisional system in the pre-ionization zone with the Rydberg electron's transitions from a state $|n\rangle$ to a state $|n+p\rangle$, where $1 \le p \le 5$.

To allow an estimation of significance of the changes introduced by this work to the method of determining the rate coefficients of the considered chemi-ionization processes and to allow a comparison with the method from Mihajlov et al. (2003a, 2011a), not only are the values of the total rate coefficient $K_A(n; T)$ determined here, but also the values of the rate coefficient $K_A^*(n; T)$. This is obtained if we take $p_0(n, l; \rho; E) = 0$, where $p_0(n, l; \rho; E)$ is the total probability of the pre-ionization decay, given by equations (18)–(20). All these quantities are presented in Fig. 3 for the hydrogen case and $T = 5000$ K and in Fig. 4 for the helium case and $T = 10000$ K. We can see from these figures that there are considerable differences between the values of the rate coefficients determined in Mihajlov et al. (2003a, 2011a) and the values $K_A^*(n; T)$, while the differences with respect to the rate coefficients $K_A(n; T)$ are very large for $n \le 6$ and decrease quickly with the increase of $n$ in the area $n > 6$. This is a consequence of the way of $K_A^*(n; T)$ is determined without the simplifications in the previous papers.

In the above-mentioned works Mihajlov et al. (2003b) and Mihajlov et al. (2007), related to the photosphere of an M red dwarf with a temperature close to 4000 K, it has been shown that only the chemi-ionization processes (1a) and (1b) with $4 \le n \le 8$ influence strongly the populations of hydrogenic Rydberg states in this photosphere, as well as its other characteristics. It is clear that, even for this reason alone, it is crucial and necessary to take into account the changes of the rate coefficients of these processes, which, according to our results, are particularly large for $n \le 6$.
<figure><img src="image_5.png" /><figcaption>**Figure 3.** Comparison of the calculated values of rate coefficients of the chemi-ionization processes (1a) and (1b) with the data from Mihajlov et al. (2011a).</figcaption></figure>

<figure><img src="image_6.png" /><figcaption>**Figure 4.** Comparison of the calculated values of rate coefficients of the chemi-ionization processes (1a) and (1b) with the data from Mihajlov et al. (2003a).</figcaption></figure>

From the presented material, we can see that it is also very important to further investigate the properties of decay of the initial state of the collision system H*(n, l) + H(1s) or He*(n, l) + He(1s²) in the pre-ionization zone. Additionally, the results obtained here suggest that the rate coefficients of the chemi-ionization processes (1) could be affected also through other channels of influence by the processes (2), by which we mean the processes of (n - n') mixing, taking place in two or more steps.

In order to show the importance of the investigated chemi-ionization processes, we compared their efficiencies with the efficiencies of the relevant concurrent processes, that is, electron-Rydberg-atom impact ionization

$$H^*(n) + e \rightarrow H^+ + e + e, \qquad (24)$$

in the hydrogen case and

$$He^*(n) + e \rightarrow He^+ + e + e, \qquad (25)$$

in the helium case, which are treated in the same range of principal quantum numbers.

<figure><img src="image_7.png" /><figcaption>**Figure 5.** Parameter $F_i[n, T; H^*(n)]$ as a function of the height $h$, for principal quantum numbers $n = 3–8$, for a model of the solar photosphere (Vernazza, Avrett & Loeser 1981); the wider bold lines denote present calculations, and the fainter lines denote calculations from Mihajlov et al. (2003a).</figcaption></figure>

Fluxes generated in atom-Rydberg-atom and electron-excited-atom impact ionization are denoted by $I_A^{aa}(T; A^*)$ and $I_A^{ea}(T; A^*)$ and given by the expressions

$$I_A^{aa}(T; A^*) = K_A(n, T; A^*) N(A^*) N(A), \qquad (26)$$

$$I_A^{ea}(T; A^*) = \alpha_A^{ea}(n, T; A^*) N(A^*) N(e), \qquad (27)$$

where $A^* = H^*(n)$ or $He^*(n)$, the rate coefficient $K_A(n, T; A^*)$ is given by equation (10) and the ionization rate coefficient $\alpha_A^{ea}(n, T; A^*)$ is determined by means of semi-empirical expressions from Vriens & Smeets (1980).

The relative importance of the chemi-ionization processes in comparison with electron-excited-atom impact ionization can be characterized by parameters $F_A(n, T)$ defined as $F_A(n, T) = I_A^{aa}(T; A^*)/I_A^{ea}(T; A^*)$. According to equations (26) and (27) we have

$$F_A(n, T; A^*) = \frac{K_A(n, T; A^*) N(A)}{\alpha_A^{ea}(n, T; A^*) N(e)}. \qquad (28)$$

Illustrative calculations were carried out for the photosphere of the Sun, for the hydrogen case, and for some DB white dwarfs with $\log g = 8$ and the temperatures $T_{eff} = 12 000$ K for the case of helium.

Figs 6 and 5 show the results of these calculations (wider bold lines) as well as the results of our previous work presented in Mihajlov et al. (2003a, 2011a) (fainter lines). As we can see, the influence of processes (2) decreases with the increase of $n$, and for $n > 10$ it practically does not exist. With the increase of temperature, the influence of chemi-ionization processes decreases, which is manifested in Figs 6 and 5. These parameters are shown as functions of height $h$ for the case of the solar photosphere, and of $\log(\tau)$ for DB white dwarfs, where $\tau$ is the Rosseland optical depth. Temperatures and atom and electron densities necessary to determine $F_A[n, T; A^*(n)]$ have been taken from DB white-dwarf models (Koester, private communication) and in the case of the solar photosphere from Vernazza et al. (1981).

Our results show that for lower temperatures the chemi-ionization processes are still dominant over electron-excited-atom ionization processes, for $n = 3, 4$ and 6 almost in the whole observed
<figure><img src="image_8.png" /><figcaption>**Figure 6.** Parameter $F_i[n, T]$; He$^*(n)$] as a function of the logarithm of Rosseland optical depth log (τ), for principal quantum numbers $n = 3–10$, with $T_{eff} = 12 000$ K and log g = 8; the wider bold lines denote present calculations, and the fainter lines denote calculation from Mihajlov et al. (2011a).</figcaption></figure>

atmosphere, as illustrated by Figs 6 and 5. For $n = 6, 7$ and 8 and in the whole observed atmosphere, chemi-ionization processes are comparable with electron-excited-atom ionization processes. This is illustrated in Fig. 6, where $T_{eff} = 12 000$ K in the helium case, and in Fig. 5 for hydrogen. The behaviour of the curves in these figures indicates the effective temperature range where it is particularly important to take into account the studied processes.

## 4 CONCLUSIONS

From the presented material, it is shown that the processes of (n - n')-mixing (2) have considerable influence on the rates of chemi-ionization processes (1). Direct calculations have been performed, which show this influence on the quantitative level. The results obtained here are presented in tabular form as supporting material, where the values of total constants for rates of the processes (1), and also just the rates for the process of associative ionization (1b) for the hydrogen and helium cases are presented. Tables 2 and 4 (provided as supporting information) cover the range of values of principal quantum numbers of the Rydberg states of hydrogen and helium atoms from n = 3 to n = 15 and the temperature range from T = 4000 K to T = 10 000 K and T = 7000 K to T = 24 000 K respectively, so that they can be directly applied in connection with the modelling of the photosphere and the lower chromosphere of the Sun and the atmosphere of DB white dwarfs. The obtained results show that the efficiency of the chemi-ionization processes, in spite of the influence of (n - n')-mixing processes (2) with n' > n in most parts of the photosphere of the Sun and most parts of white-dwarf atmospheres, remains dominant or at least comparable to the efficiency of the concurrent processes (24) and

(25). Note that the presented results point to further directions of the investigation of the influence of (n - n')-mixing processes on the chemi-ionization processes. Principally, this means taking into consideration the processes (2) with n' < n (i.e. processes that take place in several steps).

## ACKNOWLEDGEMENTS

The authors are grateful to the Ministry of Education, Science and Technological Development of the Republic of Serbia for supporting this work within the projects 176002 and III44002.

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## SUPPORTING INFORMATION

Additional Supporting Information may be found in the online version of this article:

diffS2.pdf (http://www.mnras.oxfordjournals.org/lookup/suppl/doi:10.1093/mnras/stw308/-/DC1).

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