Polarization Transfer Rates by Isotropic Collisions between Astrophysical SiO Molecule and Electrons

Abstract

We are interested in quantum calculations of polarization transfer (PT) rates due to collisions of the SiO molecule with the electrons. We determine the inelastic PT rates associated to the transitions: X ${}^1{\mathrm{\Sigma }}^{+}$→${}^3\mathrm{\Pi }$; X ${}^1{\mathrm{\Sigma }}^{+}$→${}^3{\mathrm{\Sigma }}^{+}$; X ${}^1{\mathrm{\Sigma }}^{+}$→${}^3\Delta$; X ${}^1{\mathrm{\Sigma }}^{+}$→${}^3{\mathrm{\Sigma }}^{-}$. In addition, we calculate the elastic PT rates due to rotational transitions inside the electronic state X ${}^1{\mathrm{\Sigma }}^{+}$ which are related to observed astronomical SiO MASERs. Our PT rates are obtained through linear combination of excitation rates previously calculated for SiO-electron collisions. The calculations are performed on a collision energy grid large enough to ensure converged state-to-state rates for temperatures ranging from 1000 to 10,000 K for inelastic rates and from 5 to 5000 K for elastic rates. The dependence of the inelastic rates on temperatures is obtained analytically and given in useful form.

1. Introduction

Interpretation of the intensity profiles of molecular lines observed in interstellar and circumstellar environments provides a wealth of knowledge about chemical abundances and physical conditions in the astrophysical media. Furthermore, interpretation of polarization profiles, if observable, allows to obtain important information such as magnetic fields of the astrophysical media which are inaccessible by solely interpreting intensity profiles. Let us recall that abundances and magnetic field values are good indicators of the spectral class of the late type stars. In addition, stellar magnetic field is an essential ingredient to advance our understanding of the evolution and dynamics of the stars.

The SiO molecule is considered to be one of the most abundant molecules present in astrophysical media such as atmospheres of cool stars and circumstellar environments. Microwave Amplification by Stimulated Emission of Radiation (MASER) of the SiO molecule is an interesting parameter to be exploited since its polarization has been observed especially in circumstellar environments (see [1,2]). The MASER is a result of SiO-states population inversions, a physical situation in which the population of the upper state is constantly larger than that of the lower state. SiO MASER emissions have been used to obtain radial velocities for emitting sources in galactic center regions [3,4].

SiO polarization can be due to the Zeeman effect of magnetic fields and radiative anisotropic pumping/scattering (see [1,2]). Very Long Baseline Array (VLBA) observations in circumstellar environments have revealed the existence of linear polarization of the SiO MASER transitions (e.g., [5]). Theoretical interpretations suggest that polarization of the SiO MASERs are due to the presence of a strong magnetic field of ∼10 G (see [5]). However, Desmurs et al. (2000) [6] noted that this magnetic field appears to be unreasonably strong in circumstellar environments. In fact, Asensio Ramos et al. (2005) [2] adopted a theoretical model based on the radiative anisotropic scattering and on the Hanle effect occurring for critical magnetic fields of ∼10${}^{-2}$ G to fit the observations. Nevertheless, they have not taken into consideration the effects of collisions with electrons nor with hydrogen/helium atoms on the SiO MASER polarization.

Investigations of the polarized SiO MASER require determination of all possible rates of the processes intervening at the time of its formation. One of these processes is the electron-SiO collisions for which the polarization transfer (PT) rates are unknown. More comprehensive work in this direction based on precise spectropolarimetric observations combined with careful theoretical analysis based on the scattering physics might lead to a better understanding of the magnetism of the circumstellar environments.

Derouich (2006) [7] provided quantum depolarization rates of the SiO MASER lines by collisions with hydrogen atoms but effects of collisions with electrons on the polarization of the SiO levels are yet to be investigated. Our goal in the present work is to evaluate the (de)excitation and PT rates of the SiO molecule due to collisions with electrons. A fully quantum collisional approach is adopted to calculate the rate coefficients for SiO+e collisions in the temperature range 1000–10,000 K for inelastic collisions involving two different electronic states and ranging from 5 to 5000 K for elastic collisions occurring inside the ground electronic state of SiO. Astrophysical implications of our results are briefly highlighted.

2. Definitions and Numerical Calculations

We focus in this work on the following inelastic transitions: X ${}^1{\mathrm{\Sigma }}^{+}$→${}^3\mathrm{\Pi }$, X ${}^1{\mathrm{\Sigma }}^{+}$→${}^3{\mathrm{\Sigma }}^{+}$, X ${}^1{\mathrm{\Sigma }}^{+}$→${}^3\Delta$ and X ${}^1{\mathrm{\Sigma }}^{+}$→${}^3{\mathrm{\Sigma }}^{-}$. The states X${}^1{\mathrm{\Sigma }}^{+}$, ${}^3\mathrm{\Pi }$, ${}^3\Delta$, ${}^3{\mathrm{\Sigma }}^{-}$ and ${}^3{\mathrm{\Sigma }}^{+}$ are assumed to be polarized which implies that populations of Zeeman sub-levels are different and there exist quantum interferences between these sub-levels (e.g., [8,9]). Isotropic collisions tend to equalize populations of different sublevels and to decrease the coherence between them which means that the polarization can be completely or partially destroyed by these collisions. In astrophysical applications aiming to study the polarized light, one needs to describe the average state of the polarized emitting molecule within the framework of the density matrix formalism (e.g., [10]). Quantum description of the molecular levels is achieved by using the formalism of the density matrix $\rho$ (e.g., [10,11]).

Collisional molecular (de)polarization matrix and matrices corresponding to other processes intervening in the formation of the polarization should be expressed on the irreducible tensorial operators (ITO) basis $T_q^k$ (e.g., [7,9,10]). In the tensorial basis, k represents the tensorial order and q quantifies the coherences between the levels within the polarized atomic state (e.g., [10]). In the case where collisions are isotropic, all collisional transfer of polarization rates are q-independent. Therefore, one should compute the PT rates only for q=0 and, then, use these rates to determine the collisional effect on all ${\rho }_q^k$ elements whatever is the q-value.

In the framework of the coupling schemes explained, for instance, by Harrison et al. (2012) [12] and Corey and Smith (1985) [13], and by including the infinite order sudden (IOS) approximation, one has (see [14]):

$$\begin{array}{c}\hfill {\sigma }_{IOS}^k(e_{i}Nj\to e_f{N}^{\prime }{j}^{\prime },E)=\sum _{K_j}{(-1)}^{k+j^{\prime }+K_j+j+1}(2j+1)(2j^{\prime }+1)\\ \hfill (2N+1)(2N^{\prime }+1){\left\{\begin{array}{ccc}N& N^{\prime }& K_j\\ j^{\prime }& j& S_m\end{array}\right\}}^2\times \left\{\begin{array}{ccc}j& j^{\prime }& K_j\\ j^{\prime }& j& k\end{array}\right\}\\ \hfill {\left(\begin{array}{ccc}{N}^{\prime }& N& K\\ 0& 0& 0\end{array}\right)}^2{\sigma }_{IOS}(e_{i}0\to e_{f}K,E),\end{array}$$

(1)

where ${\sigma }_{IOS}^k(e_{i}Nj\to e_f{N}^{\prime }{j}^{\prime },E)$ are the IOS-PT cross sections (see, e.g., [15]) with:

–

E being the collision energy;

–

$S_m$ is the SiO spin;

–

N is the rotational momentum of the SiO;

–

$j=S_m+N$,

–

$K_j$ quantifies the angular momentum exchange between the molecular states at the time of the collision;

–

$e_i$ is the SiO lower electronic state and $e_f$ is the upper electronic state. In this work, $e_i$ = X ${}^1{\mathrm{\Sigma }}^{+}$ and $e_f$ represents the ${}^3\mathrm{\Pi }$, ${}^3\Delta$, ${}^3{\mathrm{\Sigma }}^{-}$ and ${}^3{\mathrm{\Sigma }}^{+}$ states, respectively.

We assume that ${\sigma }_{IOS}(e_{i}0\to e_f{K}_j)$ is sufficient to give the value of the excitation from one electronic state to another electronic state without accounting for vibrational and rotational states. This corresponds to making the assumption: ${\sigma }_{IOS}(e_{i}0\to e_{f}K)\simeq \sigma (e_i\to e_f)$. The accuracy of this approximation can be justified by the fact that the energy spacing between electronic levels $\Delta E_{el}$ is much larger than the energy difference $\Delta E_v$ between the vibrational levels which is, in turn, clearly larger than the energy difference $\Delta E_R$ corresponding to the molecular rotations. In fact, $\Delta E_R$ =$Bj(j+1)$ where B is the rotational constant which is around 0.7 cm${}^{-1}$ (see [16]). Thus, for j = 10, $\Delta E_R$≃ 80 cm${}^{-1}$. On the other hand, according to the values given by Varambhia et al. (2009) [17], $\Delta E_v$ is roughly around 1200 cm${}^{-1}$. The energy separations $\Delta E_{el}$, for the transitions considered in this work, are about 3,000,000 cm${}^{-1}$[17]. In conclusion, one can easily check that $\Delta E_{el}>>\Delta E_{el}>>\Delta E_{el}$.

Varambhia et al. (2009) [17] studied SiO+e collisions and obtained excitation cross sections $\sigma (e_i\to e_f)$ from the SiO ground state $X{}^1{\mathrm{\Sigma }}^{+}$ to the SiO excited states ${}^3\mathrm{\Pi }$, ${}^3\Delta$, ${}^3{\mathrm{\Sigma }}^{-}$ and ${}^3{\mathrm{\Sigma }}^{+}$. Fully quantum cross sections of Varambhia et al. (2009) [17] are computed for kinetic collision energies up to 10 eV. This allows us to determine the PT rates $C^k(e_{i}Nj\to e_f{N}^{\prime }{j}^{\prime },T)$ for temperatures up to 10,000 K. Rates $C^k(e_{i}Nj\to e_f{N}^{\prime }{j}^{\prime },T)$ are determined via an integration over Maxwellian distribution of velocities of the quantity [$n_e$×v×${\sigma }_{IOS}^k(e_{i}Nj\to e_f{N}^{\prime }{j}^{\prime })$], where $n_e$ is the electrons density and v is the relative velocity (see, e.g., [18]).

To verify the extent to which the inelastic collisions with electrons can affect SiO polarization, one should take the representative case where the collisional rates reach their maximum values. It is to be noticed that dipolar transitions with $\Delta j=|j^{\prime }-j|$ = 1 correspond to the case where maximum values of the PT rates are achieved. Inelastic collisional PT rates from the molecular upper electronic state having $j^{\prime }$ = 5 to the lower electronic state with j = 4 are shown in Figure 1 as an example. These PT rates are associated to transitions from the ground state $X{}^1{\mathrm{\Sigma }}^{+}$ to the SiO excited states ${}^3\mathrm{\Pi }$, ${}^3\Delta$, ${}^3{\mathrm{\Sigma }}^{-}$ and ${}^3{\mathrm{\Sigma }}^{+}$ for temperatures ranging from 1000 to 10,000 K.

Collisional rates with $k=0$ are directly related to the intensity, while the rates with $k=2$ are directly connected to the linear polarization which is formed by anisotropic scattering and is of interest in the present paper. We express all rates in a compact analytical form characterized by the coefficients $a^i$ tabulated in

Table 1. Coefficients $a^0$, $a^1$, $a^2$, $a^3$ and $a^4$ of Equation (2).

	k	${\mathit{a}}^0$	${\mathit{a}}^1$	${\mathit{a}}^2$	${\mathit{a}}^3$	${\mathit{a}}^4$
X ${}^1{\mathrm{\Sigma }}^{+}$
→${}^3\Pi$
	0	−1.5428 $\times {10}^{-11}$	1.7377 $\times {10}^{-14}$	−4.1598 $\times {10}^{-18}$	−2.5952 $\times {10}^{-22}$	1.0604 $\times {10}^{-25}$
	1	−1.1882 $\times {10}^{-11}$	1.3382 $\times {10}^{-14}$	−3.2035 $\times {10}^{-18}$	−1.9986 $\times {10}^{-22}$	0.8166 $\times {10}^{-25}$
	2	−0.9006 $\times {10}^{-11}$	1.0144 $\times {10}^{-14}$	−2.4282 $\times {10}^{-18}$	−1.5149 $\times {10}^{-22}$	0.619 $\times {10}^{-25}$
X ${}^1{\mathrm{\Sigma }}^{+}$
→${}^3{\mathrm{\Sigma }}^{+}$
	0	7.1625 $\times {10}^{-13}$	−1.202 $\times {10}^{-15}$	6.5227 $\times {10}^{-19}$	−1.4116 $\times {10}^{-22}$	1.0631 $\times {10}^{-26}$
	1	5.5158 $\times {10}^{-13}$	−0.9257 $\times {10}^{-15}$	5.0231 $\times {10}^{-19}$	−1.0871 $\times {10}^{-22}$	0.8187 $\times {10}^{-26}$
	2	4.1809 $\times {10}^{-13}$	−0.7017 $\times {10}^{-15}$	3.8075 $\times {10}^{-19}$	−0.8240 $\times {10}^{-22}$	0.62055 $\times {10}^{-26}$
X ${}^1{\mathrm{\Sigma }}^{+}$
→${}^3\Delta$
	0	9.8049 $\times {10}^{-13}$	−1.7041 $\times {10}^{-15}$	9.5997 $\times {10}^{-19}$	−2.1515 $\times {10}^{-22}$	1.6715 $\times {10}^{-26}$
	1	7.5508 $\times {10}^{-13}$	−1.3124 $\times {10}^{-15}$	7.3928 $\times {10}^{-19}$	−1.6569 $\times {10}^{-22}$	1.2873 $\times {10}^{-26}$
	2	5.7234 $\times {10}^{-13}$	−0.9947 $\times {10}^{-15}$	5.6036 $\times {10}^{-19}$	−1.2559 $\times {10}^{-22}$	0.9757 $\times {10}^{-26}$
X ${}^1{\mathrm{\Sigma }}^{+}$
→${}^3{\mathrm{\Sigma }}^{-}$
	0	2.8107 $\times {10}^{-13}$	−4.5924 $\times {10}^{-16}$	2.416 $\times {10}^{-19}$	−5.0649 $\times {10}^{-23}$	3.6956 $\times {10}^{-27}$
	1	2.1646 $\times {10}^{-13}$	−3.5367 $\times {10}^{-16}$	1.8606 $\times {10}^{-19}$	−3.9005 $\times {10}^{-23}$	2.846 $\times {10}^{-27}$
	2	1.6407 $\times {10}^{-13}$	−2.6807 $\times {10}^{-16}$	1.4103 $\times {10}^{-19}$	−2.9565 $\times {10}^{-23}$	2.1572 $\times {10}^{-27}$

:

$$\begin{array}{c}\hfill C^k(e_{i}Nj\to e_f{N}^{\prime }{j}^{\prime })=n_e\sum _{i=0}^4{a}^i{T}^i,\end{array}$$

(2)

where $C^k(e_{i}Nj\to e_f{N}^{\prime }{j}^{\prime })$ are in s${}^{-1}$, $n_e$ in cm${}^{-3}$ and T in K. The correlation coefficients evaluating the precision on the calculation of the analytical forms of Equation (2) are more than 0.99. These relationships are useful for astrophysical applications and can be efficiently included in the numerical codes simulating the line formation.

In the case of isotropic collisions, $C^k(e_{i}Nj\to e_f{N}^{\prime }{j}^{\prime })$ can be derived from the detailed balance relation:

$$\begin{array}{c}\hfill C^k(e_f{N}^{\prime }{j}^{\prime }\to e_{i}Nj,T)=\frac{2j+1}{2j^{\prime }+1}exp\left(\frac{E_j-E_{j^{\prime }}}{k_{B}T}\right)\\ \hfill C^k(e_{i}Nj\to e_f{N}^{\prime }{j}^{\prime },T),\end{array}$$

(3)

where $E_j$ is the energy of the level (j) and $k_B$ is the Boltzmann constant.

In Section 4, we use values of the $C^k(e_{i}Nj\to e_f{N}^{\prime }{j}^{\prime },T)$ rates to briefly investigate the effect of the inelastic collisions on the polarization of SiO MASER emission.

3. Rotational Rates

In the case of elastic collisions, the excited electrons move from one rotational level to another without leaving the electronic level. All excitations happen inside the ground electronic level X ${}^1{\mathrm{\Sigma }}^{+}$. In our case, the excitation energies are smaller than the threshold corresponding to the energy E = 4.6 eV of the first excited level 1${}^3\mathrm{\Pi }$ (e.g., Varambhia et al. 2009 [17]). SiO 43 GHz MASER lines resulted from the transitions between the j-rotational levels inside the electronic state X ${}^1{\mathrm{\Sigma }}^{+}$. The polarization of these lines can be affected by collisions with hydrogen atoms (see [7]) and with electrons considered in the present paper. In fact, we make use of the excitation rates given by Varambhia et al. (2009) [17] to obtain the population transfer rates $D^{k^{\prime }}(j\to j^{\prime })$. Varambhia et al. (2009) [17] provided analytical relationships giving all collisional excitation rates $C(j\to j^{\prime })$ for $j\le 8$ where the temperature is 5 K $\le T\le$ 5000 K.

In order to use the $C(j\to j^{\prime })$ rates in the calculations of the PT rates $D^k(j\to j^{\prime })$, we proceed in two steps. First, we start by calculating the population transfer rates $D^{k=0}(j\to j^{\prime })$ using the relation (see, e.g., [7]):

$$\begin{array}{c}\hfill D^{k=0}(j\to j^{\prime })=\sqrt{\frac{2J+1}{2j^{\prime }+1}}C(j\to j^{\prime })\end{array}$$

(4)

By adopting the assumptions explained in details by Derouich et al. (2019) [19], one can use the relation:

$$\begin{array}{ccc}\hfill D^k(j\to j^{\prime })& =& {(-1)}^{k-k^{\prime }}\frac{\left\{\begin{array}{ccc}j& j& k\\ j^{\prime }& j^{\prime }& g\end{array}\right\}}{\left\{\begin{array}{ccc}j& j& k^{\prime }\\ j^{\prime }& j^{\prime }& g\end{array}\right\}}{D}^{k^{\prime }}(j\to j^{\prime }),\hfill \end{array}$$

(5)

to obtain the $D^k(j\to j^{\prime })$ through the values of $D^{k=0}(j\to j^{\prime })$ where $g=1$ in the case of collisions with electrons and $k^{\prime }=0$ (see [10,19]).

Results are given in Figure 2 for dipolar transitions where $\Delta j$ = 1 inside the ground electronic state X ${}^1{\mathrm{\Sigma }}^{+}$. We notice that rotational transitions with $\Delta j$> 1 are not permitted because of the dipolar radiative selection rules. As it can be concluded from Varambhia et al. (2009) [17], collisional rates with $\Delta j$> 1 are very weak if compared to the rates inferred from dipolarly permitted transitions.

4. Astrophysical Implications

To estimate the (de)polarizing effect of isotropic collisions between astrophysical SiO and electrons, one should compare the PT rates $D^{k=2}(j\to j^{\prime })$ inside the SiO ground state ($X^1\mathrm{\Sigma }$) to the Einstein coefficient A for spontaneous emission of MASER SiO lines.

As it can be seen in Figure 2, PT rates per unit volume are about 10${}^{-6}$–10${}^{-5}$ cm${}^{-3}$ s${}^{-1}$ meaning that they are more than four orders of magnitude larger than those of collisions with hydrogen.1 On the other hand, inelastic collisions studied in the previous section (see Figure ) give very small collision rates (∼ 10${}^{-10}$ cm${}^{-3}$ s${}^{-1}$) compared to elastic collisional rates. According to Tipping and Chackerian (1981) [20], Einstein coefficient A is ranging between 10${}^{-4}$ to 10${}^{-6}$ s${}^{-1}$ for some observed MASER transitions $(v,j+1)$→$(v,j)$ (v denotes the vibrational level). For electron density $n_e\sim 1$ cm${}^{-3}$, $D^{k=2}$ will be in the same range as A and thus the effect of the collisions with electrons on the polarization of the SiO MASER lines will be important. On the contrary for $n_e<<1$, $D^{k=2}$ is smaller than A and the MASER polarization becomes insensitive to the collisions with electrons. This procedure of determining the electron density by interpreting the polarization profiles has been in use in the solar context for a long time now, see, for example, Bommier et al. (1986) [21]. The limiting case is where the electron density $n_e>>1$ corresponds to a complete destruction of the MASER polarization, i.e., the rates $D^2$ dominate the radiative rates. In this sense, the MASER polarization could be used to determine the electron density in the interstellar and circumstellar environments where the MASER lines are formed.

Note that our rates are obtained by unit of electron density since, in the framework of the impact approximation, rates are proportional to the perturber density. To realistically determine the role of the (de)polarizing collisions during the formation of the SiO lines, one has to take into account perturber’s density values and solve the polarized radiative transfer problem according to suitable astrophysical model.

5. Conclusions and Perspectives

Within the circumstellar environment and the interstellar medium, accurate study of SiO lines requires precise values of collisional rate coefficients. In this context, we study collisions between SiO molecules and electrons in the framework of a quantum approach. We provide excitation inelastic collision rates for the transitions X ${}^1{\mathrm{\Sigma }}^{+}$→${}^3\mathrm{\Pi }$, X ${}^1{\mathrm{\Sigma }}^{+}$→${}^3{\mathrm{\Sigma }}^{+}$, X ${}^1{\mathrm{\Sigma }}^{+}$→${}^3\Delta$ and X ${}^1{\mathrm{\Sigma }}^{+}$→${}^3{\mathrm{\Sigma }}^{-}$ as well as excitation elastic collision rates for the rotational levels of the X ${}^1{\mathrm{\Sigma }}^{+}$ state. In addition, by using the tensorial basis, we provide the PT rates due to collisions between SiO and electrons. These PT rates are essential for studying the SiO polarization. To the best of our knowledge, neither theoretical nor experimental PT rates are available to compare. However, we notice that our work is based on the quantum results of Varambhia et al. (2009) [17] who adopted an accurate and up-to-date ab initio R-matrix method. Then, our PT rates should be sufficiently accurate to analyze the observations. In the future, our results might be confirmed by adopting a more accurate quantum close-coupling method. We briefly discuss the implications of our results in the modeling of the MASER SiO lines.