CHEMI-IONIZATION IN SOLAR PHOTOSPHERE: INFLUENCE ON THE HYDROGEN ATOM EXCITED STATES POPULATION

Here we will consider processes (1)–(4) within the regions n ⩾ 5 and 2 ⩽ n ⩽ 4 separately. Within the first region we will determine the rate coefficients for the chemi-ionization processes (1) and (2) directly, using the principle of thermodynamical balance for determination of the rate coefficients for the inverse chemi-recombination processes (3) and (4).

The parameters that are needed in further considerations are the following: r_n ∼ n²—the characteristic radius of Rydberg atom H*(n); R—the inter-nuclear distance in the collision system H*(n) + H(1s); and U₁(R) and U₂(R)—the adiabatic potential energies of the ground and the first exited electronic states of molecular ion H⁺₂.

In accordance with the previous papers (Mihajlov et al. 1997, 2003a, 2003b, 2007b) we will treat processes (1) and (2) with n ⩾ 5 on the basis of dipole resonant mechanism, which was introduced in the considerations of Smirnov & Mihajlov (1971) for inelastic processes in thermal [H*(n) + H(1s)] collisions. This means that such processes are considered as a result of resonant energy conversion within the electronic component of the collisional system H*(n) + H(1s), which is realized inside the region

$$\begin{equation} R \ll r_{n}, \end{equation} \tag{ 8 }$$

where the system H*(n) + H(1s) can be presented as [H⁺ + H(1s)] + e_n, and which is caused by the dipole part of the interaction of the outer electron e_n with the subsystem [H⁺ + H(1s)]. Already in Devdariani et al. (1978) just chemi-ionization processes in atom–Rydberg-atom collisions (the case of alkali metal atoms) were described on the basis of the same mechanism. After that, the methods based on the dipole resonant mechanism have been used in practice for investigation of chemi-ionization processes until now (see, for example, Beterov et al. 2005 and Ignjatović & Mihajlov 2005). Application of this mechanism is particularly successful within the so-called decay approximation, which was examined in Janev & Mihajlov (1980) and immediately demonstrated to be suitable for any inelastic processes in slow [H*(n) + H(1s)] collisions. Let us note that in the previous papers (Mihajlov et al. 2003a, 2003b, 2007b) a method from Mihajlov & Dimitrijević (1992) and Mihajlov et al. (1996) was used, which is based on this approximation.

Only the decay approximation will be used here for processes (1) and (2) with n ⩾ 5. First, it is assumed that in the region, Equation (8), the electronic state of the subsystem [H⁺ + H(1s)] can be approximated well by one of the two adiabatic electronic states of the molecular ion H⁺₂: the ground one $|1\Sigma _{g}^{+}\vec{r}_{\rm mi},R>$ and the first excited $|1\Sigma _{u}^{+}\vec{r}_{\rm mi},R>$, and the state of the outer electron e_n—by one of the hydrogen Rydberg states $|n,l,m,\vec{r}>$. Then, it is assumed that, in the case of chemi-ionization processes (1) and (2), we can consider that the subsystem [H⁺ + H(1s)] is in the first excited electronic state $|1\Sigma _{u}^{+}\vec{r}_{\rm mi},R>$ with a probability of 1/2, which means that we can describe the relative inter-nuclear motion as going on in the reflective potential U₂(R). Finally, it means that we can expect (as a result of the mentioned electron–dipole interaction) a decay of the initial electronic state $|n,l,m,\vec{r};1\Sigma _{u}^{+}\vec{r}_{\rm mi},R> = |n,l,m,\vec{r}> |1\Sigma _{u}^{+} \vec{r}_{\rm mi},R>$ of the system [H⁺ + H(1s)] + e_n, with a transition to the final state $|\epsilon,l^{\prime },m^{\prime },\vec{r}; 1\Sigma _{g}^{+}\vec{r}_{\rm mi},R> = |\epsilon,l^{\prime },m^{\prime },\vec{r}>|1\Sigma _{g}^{+} \vec{r}_{\rm mi},R>$ in a narrow neighborhood of the resonant point R = R_n;, which is the root of the equation

$$\begin{equation} U_{12}(R)\equiv U_{2}(R)- U_{1}(R) = \epsilon -\epsilon _{n}. \end{equation} \tag{ 9 }$$

Here, _n ≅ −I/n² and are the energies of the initial (bound) and final (free) states of the outer electron and I is the ionization potential of the ground-state hydrogen atom. Here it is important that we have almost resonant simultaneous transitions: of the subsystem [H⁺ + H(1s)] to the ground electronic state $|1\Sigma _{g}^{+} \vec{r}_{\rm mi}, R>$, and of the outer electron to one of the free states $|\epsilon,l^{\prime }, m^{\prime }, \vec{r}>$.

In accordance with Janev & Mihajlov (1980), Mihajlov & Dimitrijević (1992), and Mihajlov et al. (1996), we will characterize processes (1) and (2) by quantities W_n(R), P^(a)_ci(n; ρ, E), and P^(b)_ci(n; ρ, E). The first quantity has the meaning of mean decay velocity of the considered system's initial state and is given by relations

$$\begin{eqnarray} && W_{n}(R) = \frac{1}{n^{2}}\cdot \sum \limits _{l,m} \frac{2\pi }{\hbar } \cdot |<1u; n,l,m,\bigg|e^{2}\cdot \frac{\vec{r}\cdot \vec{r}_{\rm mi}}{r^{3}}\bigg| \epsilon _{k}, l^{\prime },m^{\prime }; \nonumber \\ && \quad 1g^{+}>|^{2} \cdot g(\epsilon), \end{eqnarray} \tag{ 10 }$$

where e is the electron charge, $|1g,\epsilon,l^{\prime },m^{\prime }> = |\epsilon,l^{\prime },m^{\prime }, \vec{r}>|1\Sigma _{g}^{+} \vec{r}_{\rm mi},R>$, $|1u,n,l,m> = |n,l,m,\vec{r}> |1\Sigma _{u}^{+} \vec{r}_{\rm mi},R>$, and g() is the density of the free single-electron states in the energy space. Following Janev & Mihajlov (1980) and Ignjatović & Mihajlov (2005), we will transform Equation (10) to the simple form

$$\begin{eqnarray} W_{n}(R) &=& \frac{4}{3\sqrt{3}n^{5}} D_{12}^{2}(R) G_{nk}, \nonumber \\ D_{12}(R) &=& |<1\Sigma _{u}^{+}\vec{r}_{\rm mi},R|r_{\rm {mi};R}|1\Sigma _{g}^{+} \vec{r}_{\rm mi},R>|, \end{eqnarray} \tag{ 11 }$$

where $r_{\rm {mi};R}$ is the projection of $\vec{r}_{\rm mi}$ on the inter-nuclear axis, $G_{nk}\equiv \sigma _{\rm ph}(n,k)/\sigma _{\rm ph}^{\rm {Kr}}(n,k)$ is the generalized Gaunt factor, defined in Johnson (1972), σ_ph(n, k) is the mean photo-ionization cross section of the atom H*(n) for the given , and σ^Kr_ph(n, k) is the same photo-ionization cross section, but in Kramers's approximation (Kramers 1923; Sobel'man 1979).

The quantities P^(a)_ci(n; ρ, E) and P^(b)_ci(n; ρ, E) are the probabilities of the realization of the chemi-ionization processes (1) and (2), respectively, for given ρ and E. In the previous papers (Mihajlov & Dimitrijević 1992; Mihajlov et al. 1996, 2003a, 2003b, 2007b), the influence of the initial state's decay (during the collision) on its amplitude was neglected in order to simplify the procedure that was used. However, it generates errors that are larger than 10% for n ⩽ 8. Consequently, in this work the said influence is taken into account and a procedure similar to the ones from Janev & Mihajlov (1980) and Ignjatović & Mihajlov (2005) is used. Therefore, we will present here only the final expressions for the ionization probabilities P^(a,b)_ci(n; ρ, E), the partial cross sections σ^(a,b)_ci(n; E), and the corresponding partial rate coefficients K^(a,b)_ci(n; T), where T is the temperature of the considered plasma, using additional parameters R_n, R₀, and R_E, which are the roots of equations

$$\begin{equation} U_{12}(R)= |\epsilon _{n}|, \quad U_{2}(R)=E, \quad U_{12}(R)=E, \end{equation} \tag{ 12 }$$

respectively.

In the case when only one of the processes (1) and 2) is realized, the ionization probabilities are obtained in the form

$$\begin{equation} P_{\rm ci}^{(a,b)}(n,\rho,E)= \frac{1}{2} \left(1-e^{-2 \int \nolimits _{R_{0}}^{R_{n}} \frac{W_{n}(R)dR}{v_{{\rm rad}}}}\right), \end{equation} \tag{ 13 }$$

and in the case of realization of both processes we have it that

$$\begin{equation} P_{\rm ci}^{(a)}(n,\rho,E)= \frac{1}{2} \left(1-e^{-2\int \nolimits _{R_{0}}^{R_{E}} \frac{W_{n}(R)dR}{v_{{\rm rad}}}}\right) e^{-\int \nolimits _{R_{E}}^{R_{n}} \frac{W_{n}(R)dR}{v_{{\rm rad}}}}, \end{equation} \tag{ 14 }$$

$$\begin{equation} P_{\rm ci}^{(b)}(n,\rho,E)= \frac{1}{2} \left(1-e^{-\int \nolimits _{R_{E}}^{R_{n}} \frac{W_{n}(R)dR}{v_{{\rm rad}}}}\right) \left(1+e^{-2\int \nolimits _{R_{0}}^{R_{n}} \frac{W_{n}(R)dR}{v_{{\rm rad}}} }\right), \end{equation} \tag{ 15 }$$

where ρ and E = m_redv²/2 are the impact parameter and the atom–Rydberg-atom impact energy, respectively (m_red being the reduced mass of the collision system).

v_rad = v_rad(ρ, E; R) is the radial inter-nuclear velocity, which is given by

$$\begin{equation} v_{{\rm rad}}(\rho,E;R)=\sqrt{\displaystyle \frac{2}{m_{{\rm red}}}\left[E-U_{2}(R)-\frac{E\rho ^{2}}{R^{2}}\right]}. \end{equation} \tag{ 16 }$$

Then, from Equations (13)–(16) the partial cross sections σ^(a,b)_ci(n; E) are determined, namely,

$$\begin{equation} \sigma _{\rm ci}^{(a,b)}(n,E) = 2 \pi \int \nolimits _{0}^{\rho _{{\rm max}}^{(a,b)}(E)} {P_{\rm ci}^{(a,b)}(n,\rho,E) \rho } d\rho, \end{equation} \tag{ 17 }$$

where ρ^max_(1a,b)(E) is the upper limit of values ρ, at which the corresponding region R is reached for a given E.

After that, the partial rate coefficients for the chemi-ionization processes (1) and (2) with n ⩾ 5 are determined by expressions

$$\begin{equation} K_{\rm ci}^{(a,b)}(n,T) = \int \nolimits _{E_{{\rm min}}^{(a,b)}(n)}^{E_{{\rm max}}} {v \sigma _{\rm ci}^{(a,b)}(n,E) f(v;T)} dv, \end{equation} \tag{ 18 }$$

where σ^(a,b)_ci(n, E) is defined by Equation (17), v is the atom–Rydberg-atom impact velocity, f(v; T) is the velocity distribution function for the given temperature T, and E^(a,b)_min(n) = 0 if U₂(R_n) ⩽ 0; E^(a,b)_min = U₂(R_n) if U₂(R_n)>0; E^(a)_max = U₂(R_0;1) = U₂(R_0;1), where R_0;1 is such a point that U₁(R_0;1) = 0; E^(b)_max = ∞.

Finally, using partial rate coefficients K^(a,b)_ci(n, T), we will determine the total one, namely,

$$\begin{equation} K_{\rm ci}(n,T)=K_{\rm ci}^{(a)}(n,T)+K_{\rm ci}^{(b)}(n,T), \end{equation} \tag{ 19 }$$

which characterizes the efficiency of the chemi-ionization processes (1) and (2) together.

Under the conditions which exist in the solar atmosphere, we can determine the chemi-recombination rate coefficients (as functions of T) from the principle of thermodynamical balance for processes (1, 2) and (3, 4), namely,

$$\begin{equation} K_{\rm ci}^{(a)}(n,T)\cdot N_{n} N_{1} = K_{\rm {dr}}(n,T)\cdot N_{\rm mi}^{(\rm eq)} N_{e} \equiv K_{\rm cr}^{(a)}(n,T)\cdot N_{1}N_{ai}N_{e} \end{equation} \tag{ 20 }$$

$$\begin{equation} K_{\rm ci}^{(b)}\cdot N_{n} N_{1} = K_{\rm cr}^{(b)}(n,T)\cdot N_{1}N_{ai}N_{e}, \end{equation} \tag{ 21 }$$

where the chemi-ionization rate coefficient K^(a)_cr(n, T) is expressed through the dissociative recombination rate coefficient $K_{\rm {dr}}(n,T)$ by relation

$$\begin{equation} K_{\rm cr}^{(a)}(n,T) \equiv K_{\rm {dr}}(n,T)\cdot \chi ^{-1}(T), \qquad \chi (T) = \left(\frac{N_{ai}N_{1}}{N_{\rm mi}^{(\rm eq)}}\right), \end{equation} \tag{ 22 }$$

where N₁ and N_n denote the densities of ground- and excited-state hydrogen atoms, respectively, while N_ai and $N_{\rm mi}^{(\rm eq)}$ are the densities of atomic ions H⁺ and molecular ions H⁺₂, respectively. The index "eq" denotes that molecular ion density corresponds to thermodynamical equilibrium condition for given T. Factor χ(T) can be determined as in Mihajlov et al. (2007a) in connection with the contribution of H⁺ + H(1s) radiative collision processes to the solar atmosphere's opacity in UV and VUV region.

Taking quantity K^(a)_cr(n, T) as the rate coefficient for process (3), we can characterize both chemi-recombination processes (3) and (4) in a similar way. Namely, in accordance with Equations (20) and (21), rate coefficients K^(a)_cr(n, T) and K^(b)_cr(n, T) are given by relations

$$\begin{eqnarray} && K_{\rm cr}^{(a,b)}(n,T) = K_{\rm ci}^{(a,b)}(n,T)\cdot S_{n}^{-1}(T), \nonumber \\ && S_{n}(T) \equiv \frac{N_{i}N_{e}}{N_{n}} = \frac{1}{n^{2}}\cdot \frac{mk_{B}T}{2\pi \hbar ^{2}} \cdot \exp \bigg({-}\frac{I_{n}}{k_{B}T}\bigg), \end{eqnarray} \tag{ 23 }$$

where m is the electron mass and k_B is the Boltzmann constant. Consequently, using such partial rate coefficients, we can introduce here the total one, i.e.,

$$\begin{equation} K_{\rm cr}(n,T)=K_{\rm cr}^{(a)}(n,T)+K_{\rm cr}^{(b)}(n,T), \end{equation} \tag{ 24 }$$

which characterizes the efficiency of processes (3) and (4) together for n ⩾ 5.

The reason why the regions n ⩾ 5 and 2 ⩽ n ⩽ 4 are being considered separately is the behavior of the adiabatic potential curves of atom–atom systems H*(n) + H(1s). Namely, in the first region the atom–atom curves lie above the adiabatic curve of the ion–ion system H⁺ + H⁻(1s²) for any R, and the dipole resonant mechanism can be applied for n ⩾ 5 without any exceptions, while in the other region there are points where the atom–atom curves cross the ion–ion one and application of this mechanism generates some errors (see Janev & Mihajlov 1979). Since the corresponding cross points for n ⩽ 4 are so far from the point R = 0 that their existence could be neglected for n = 4 and 3, and with some caution even for n = 2, the dipole resonant mechanism was applicable, for example, in Mihajlov & Dimitrijević (1992) and Mihajlov et al. (1996) for n = 4 and in Zhdanov & Chibisov (1976) for n = 3. However, now we can determine the values of rate coefficients K^(a)_cr(n, T) of dissociative recombination process (3) for n = 2, 3, and 4 using the results deduced from the experimental data of Jones (1977), presented in Janev et al. (1987).

Due to this fact and the mentioned errors, we use here semi-empirical rate coefficients K^(a)_cr(n = 3, T) and K^(a)_cr(n = 4, T), which are obtained from Janev et al. (1987), for the dominant processes of the dissociative recombination, i.e., for process (3) with n = 3 and 4. The corresponding chemi-ionization rate coefficients K^(a)_ci(n = 3, T) and K^(a)_ci(n = 4, T) are then obtained from the principle of thermodynamical balance, as it has been described above. For relatively minor chemi-ionization/recombination processes, i.e., for processes (1) and (3) with n = 2, we use here rate coefficients K^(a)_ci(n = 2, T) and K^(a)_cr(n = 2, T), which are 10%–30% greater than the corresponding coefficients obtained using the data from Janev et al. (1987), in accordance with the calculated results from Urbain et al. (1991) and Rawlings et al. (1993). It gives a possibility to compensate the decrease of rate coefficients K^(a)_ci(n ⩾ 5, T) and K^(a)_cr(n ⩾ 5, T) in comparison with the corresponding ones obtained using Janev et al. (1987), due to the fact that here, unlike Janev et al. (1987), the decay of the considered system's initial electronic state has been taken into account. For other chemi-ionization and recombination processes (2) and (4) with 2 ⩽ n ⩽ 4, whose contribution could really be neglected, the corresponding rate coefficients will be determined (in accordance with what was said above) by extrapolation of those from the region n ⩾ 5. Finally, let us note that in further considerations for chemi-ionization and recombination processes (1)–(4) with n < 5 we will use also total rate coefficients, which are given by the same expressions (19) and (24), but for 2 ⩽ n ⩽ 4.

In accordance with the aim of this work, we consider here model C of solar atmosphere from Vernazza et al. (1981). Namely, this is a non-LTE model which is still actual (see Stix 2002), and it is only for this model that all the quantities necessary for our calculations are available in tabular form as functions of height (h) in solar photosphere. In Figure 1, basic plasma parameters for this model are shown. In Figure 2, deviations of non-LTE populations of excited hydrogen atom states with 2 ⩽ n ⩽ 8 in solar photosphere within the C model of Vernazza et al. (1981) are illustrated. One can see that these deviations are particularly pronounced for n = 2. Around h = 500 km N(H*(n = 2)) is one-half of the corresponding equilibrium density, and for h larger than 1000 km it is around 10 times greater. These deviations rapidly decrease with an increase of n. However, even for n = 8 this deviation is around 40% around h = 500 km, illustrating the importance of taking into account the considered processes ab initio in the modeling of solar atmosphere.

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The values of the total chemi-ionization and recombination rate coefficients K_ci(n, T) and K_cr(n, T), obtained in the described way, are presented in Tables 1 and 2, respectively. These tables cover the regions 2 ⩽ n ⩽ 8 and $4000 \,\rm K\le T\le 10 000\, \rm K$ which are relevant for solar photosphere considered on the basis of C model from Vernazza et al. (1981).

Relative contribution of partial chemi-ionization and recombination processes for given n and T characterizes corresponding branch coefficients X^(a,b)_ci(n, T), namely,

$$\begin{equation} X_{\rm ci}^{(a,b)}(n,T)=\frac{K_{\rm ci}^{(a,b)}(n,T)}{K_{\rm ci}(n,T)}, \quad X_{\rm cr}^{(a,b)}(n,T) = \frac{K_{\rm cr}^{(a,b)}(n,T)}{K_{\rm cr}(n,T)}. \end{equation} \tag{ 25 }$$

Since $X_{\rm ci, \rm cr}^{(b)}(n,T) = 1 -X_{\rm ci, \rm cr}^{(a)}(n,T)$ and X^(a,b)_ci(n, T) = X^(a,b)_cr(n, T) ≡ X^(a,b)(n, T), it is enough to present only the values of one of the coefficients X^(a,b)(n, T). Here, the values of the coefficient X^(a)(n, T), which directly describe relative contributions of the associative ionization and dissociative recombination processes (1) and (3), are presented in Table 3.

Let I_ci(n, T) and I_cr(n, T) be the total chemi-ionization and chemi-recombination fluxes caused by the processes (1, 2) and (3, 4), i.e.,

$$\begin{eqnarray} I_{\rm ci}(n,T) &=& K_{\rm ci}(n,T)\; \cdot \; N_{n}N_{1},\nonumber\\ I_{\rm cr}(n,T) &=& K_{\rm cr}(n,T)\; \cdot \; N_{1}N_{i}N_{e}, \end{eqnarray} \tag{ 26 }$$

and I_i;ea(n, T), I_r;eei(n, T), and $I_{r;\rm ph}(n,T)$ be the fluxes caused by ionization and recombination processes (5), (6), and (7), i.e.,

$$\begin{eqnarray} && I_{i;\rm {ea}}(n,T) = K_{\rm {ea}}(n,T)\cdot N_{n}N_{e}, \nonumber\\ && I_{r;\rm {eei}}(n,T) = K_{\rm {eei}}(n,T) \cdot N_{i}N_{e}N_{e}, \nonumber\\ && I_{r;\rm ph}(n,T) = K_{\rm ph}(n,T)\cdot N_{i}N_{e}, \end{eqnarray} \tag{ 27 }$$

where N₁, N_n, N_i, and N_e are, respectively, the densities of the ground and excited states of a hydrogen atom, of ion H⁺, and of free electron in the considered plasma with given T.

Using these expressions, we will first calculate quantities F_i(n, T) given by

$$\begin{equation} F_{i}(n,T) = \frac{I_{\rm ci}(n,T)}{I_{i;\rm {ea}}(n,T)} = \frac{K_{\rm ci}(n,T)}{K_{\rm {ea}}(n,T)}\cdot {N_{1}}{N_{e}}, \end{equation} \tag{ 28 }$$

which characterize the relative efficiency of partial chemi-ionization processes (1, 2) together and the impact electron–atom ionization (5) in the considered plasma. The total chemi-ionization and recombination rate coefficients K_ci(n, T) are determined here by the method described in the previous section, and impact ionization rate coefficients K_ea(n, T) are taken from Vriens & Smeets (1980). In Figure 3, the behavior of the quantities F_i,ea(n, T) for 2 ⩽ n ⩽ 8 as functions of height h is shown, according to the data (N₁, N_e, and T) from Vernazza et al. (1981) for solar photosphere. One can see that the efficiency of the considered chemi-ionization processes in comparison with the electron–atom impact ionization is dominant for 2 ⩽n⩽ 6 and becomes comparable for n = 7 and 8.

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However, in order to compare the relative influence of the chemi-ionization processes (1) and (2) together to that of the impact electron–atom ionization process (5) on the whole block of the excited hydrogen atom states with 2 ⩽ n ⩽ 8, we will calculate quantity F_i,ea;2−8(T), given by

$$\begin{eqnarray} F_{i,\rm {ea};2-8}(T) &=& \frac{\sum \nolimits _{n=2}^{8} I_{\rm ci}(n,T)}{\sum \nolimits _{n=2}^{8} I_{i;\rm {ea}}(n,T)}\nonumber \\ & =& \frac{\sum \nolimits _{n=2}^{8} K_{\rm ci}(n,T)\cdot N_{n}}{\sum \nolimits _{n=2}^{8} K_{\rm {ea}}(n,T)\cdot N_{n}}\cdot {N_{1}}{N_{e}}, \end{eqnarray} \tag{ 29 }$$

which can reflect the influence of the existing populations of excited hydrogen atom states within a non-LTE model of the solar atmosphere. In Figure 4, the behavior of the quantity F_i,ea;2−8(T) as functions of height h is shown according to the same data from Vernazza et al. (1981). As one can see, the real influence of the chemi-ionization processes on the total populations of states with 2 ⩽ n ⩽ 8 remains dominant with respect to the concurrent electron–atom impact ionization processes almost in the whole photosphere (50 km ≲h≲ 750 km). This means that the chemi-ionization processes influence the radiative properties of the whole solar atmosphere in the optical region considerably.

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Then, in order to compare the relative influence of chemi-recombination processes (3) and (4) together and electron–electron–H⁺ ion recombination process (6) on the same block of excited hydrogen atom states with 2 ⩽ n ⩽ 8, we calculated quantity F_r,eei;2−8(T), given by

$$\begin{equation} F_{r,\rm {eei};2-8}(T) = \frac{\sum \nolimits _{n=2}^{8} I_{\rm cr}(n,T)}{\sum \nolimits _{n=2}^{8} I_{r;\rm {eei}}(n,T)} = \frac{\sum \nolimits _{n=2}^{8} K_{\rm cr}(n,T)}{\sum \nolimits _{n=2}^{8} K_{\rm {eei}}(n,T)}\cdot \frac{N_{1}}{N_{e}}, \end{equation} \tag{ 30 }$$

taking rate coefficients K_eei(n, T) also from Vriens & Smeets (1980). In Figure 5, the behavior of this quantity as a function of height h is shown. One can see that the considered chemi-recombination processes dominate with respect to the concurrent electron–electron–ion recombination processes within the region 100 km ≲h≲ 650 km. Consequently, the considered chemi-recombination processes are also very significant for the optical properties of the solar photosphere.

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Finally, we compared the relative influence of chemi-recombination processes (3) and (4) together and photo-recombination electron–H⁺ ion process (7), also within the block of the excited hydrogen atom states with 2 ⩽ n ⩽ 8. For that sake we calculated quantity F_r,ph;2−8(T), given by

$$\begin{equation} F_{r,\rm {ph};2-8}(T) = \frac{\sum \nolimits _{n=2}^{8} I_{\rm cr}(n,T)}{\sum \nolimits _{n=2}^{8} I_{r;\rm ph}(n,T)} = \frac{\sum \nolimits _{n=2}^{8} K_{\rm cr}(n,T)}{\sum \nolimits _{n=2}^{8} K_{\rm ph}(n,T)}\cdot N_{1}, \end{equation} \tag{ 31 }$$

taking rate coefficients K_ph(n, T) from Sobel'man (1979). This is necessary since in Mihajlov et al. (1997) only the states 4 ⩽ n ⩽ 8 were considered. Still, it was a natural expectation that the inclusion of states with n = 2 and 3 will increase the influence of photo-recombination electron–ion processes. The behavior of quantity F_r,ph;2−8(T) as a function of h is shown in Figure 6. One can see that here a domination of the chemi-recombination processes with 2 ⩽ n ⩽ 8 over the electron–ion photo-recombination processes is confirmed (although to a slightly lesser extent) in a significant part of the photosphere (−50 km ≲h≲ 600 km).

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