Mon. Not. R. Astron. Soc. 400, 1892–1896 (2009) doi:10.1111/j.1365-2966.2009.15589.x Formation of SiC by radiative association C. M. Andreazza,� R. M. Vichietti and E. P. Marinho Universidade Estadual Paulista, IGCE, DEMAC, Rua 10, 2527, CEP 13500-230, Rio Claro, SP, Brazil Accepted 2009 August 18. Received 2009 July 24; in original form 2009 February 5 ABSTRACT Rate coefficients for radiative association of silicon and carbon atoms to form silicon carbide molecule (SiC) are estimated. The radiative association of Si(3P) and C(3P) atoms mainly occurs through the C3� state followed by radiative decay to the X3� state. For the temperature range of 300–14 000 K, the rate coefficients slowly increase with temperature and they can be expressed by K(T ) = 2.038 × 10−17(T /300)−0.01263 exp(−136.73/T ) cm3 s−1. Key words: atomic data – atomic process – circumstellar matter – ISM: molecules. 1 IN T RO D U C T I O N The early condensation models for carbon-rich circumstellar re- gions performed by Gilman (1969) and Friedemann (1969) pre- dicted that solid SiC particles will be a significant constituent of the dust around carbon stars. The prediction of SiC dust grains was confirmed by the detection of a maximum of emission near 11 μm by Hackwell (1972) and Treffers & Cohen (1974) in the spectra of some carbon stars. The feature near 11 μm also appears in the spectra of some planetary nebulae (Aitken & Roche 1982), comets (Orofino, Blanco & Fonti 1994), and in the ejecta of novae (Gehrz et al. 1984). On the other hand, such a feature is not detected in Wolf–Rayet stars and in the ejecta of supernovae, both hydrogen- deficient objects. The absence of this feature in these environments is an intriguing puzzle. Additionally, solid SiC has never been identi- fied in interstellar medium (Whittet, Duley & Martin 1990). Kozasa et al. (1996) proposed that already in the stellar outflow the small SiC grains get surrounded by carbon mantles so that the 11.4 μm feature vanishes. Indeed, pre-solar SiC grains were isolated from primitive meteorites (Bernatowicz et al. 1987; Lewis et al. 1987). The isotopic compositions of these grains indicate that they origi- nated outside the Solar system. Dust condensation in atmospheres of carbon-rich stars is a plausible source of the majority of pre-solar SiC grains. However, the isotopic anomalies found in some pre- solar SiC X grains point to an origin in Type II supernovae ejecta, whereas novae and Wolf–Rayet stars have been suggested as stellar sources (Hoppe et al. 2000). The free SiC radical in gas phase was observed in the circum- stellar shell of the evolved carbon star IRC+10216 by Cernicharo et al. (1989) through observation of seven millimetre-wave rota- tional transitions in the ground electronic X3� state. It is not yet clear by which mechanism this molecule can be formed in a stellar outflow. The photodissociation of SiC2 by interstellar ultraviolet radiation field has been proposed as the formation mechanism of the SiC radical by Cernicharo et al. (1989). Therefore, the chem- �E-mail: carmenma@rc.unesp.br istry models do not explain the observed abundances of both SiC and SiC2 molecules (e.g. MacKay & Charnley 1999, and references therein). Silicon carbide has not been detected in the ejecta of Type II supernova SN 1987A. However, the SiC X grains found in mete- orites and the identification of SiO and CO molecules in the infrared spectra of the SN 1987A (e.g. Aitken et al. 1988; Spyromilio et al. 1988) suggest that SiC may exist in this environment. Unfortunately, the location of their emission bands is coincident with gaps in the observed spectra. The chemical processes producing the SiC molecule have been modelled by Cherchneff & Lilly (2008), but this species was not included among the dominant molecules produced in the ejecta. There are also studies on SiC condensation in Type II supernovae (Ebel & Grossman 2001; Deneault, Clayton & Heger 2003; Nozawa et al. 2003; Lodders 2006). However, the process by which this dust forms and survives in the ejecta of supernovae has not yet been explained. The formation of the SiC molecule is the first step in the process of SiC grain building. The expanding supernova interiors are hy- drogen free. In this environment, the dominant molecular formation process is the radiative association reaction (e.g. Gearhart, Wheeler & Swartz 1999). However, the radiative association rate coefficients of the SiC molecule are not known. Hence, using theoretical po- tential energy curves and transition dipole moments, we present estimates of the rate coefficients for radiative association of Si and C atoms (KSiC) to form the SiC molecule for temperatures in the 300–14 000 K range. 2 ME T H O D O F C A L C U L ATI O N A semiclassical description of the nuclear motion may be applied to collisions of massive reactants or strong stabilizing transitions (e.g. Zygelman & Dalgarno 1988; Latter & Black 1991; Dalgarno, Du & You 1990), which was first formulated correctly by Bates (1951). In this approximation, the cross-section integrated over all C© 2009 The Authors. Journal compilation C© 2009 RAS at Fundaà §Ã £o C oordenaà §Ã £o de A perfeià §oam ento de Pessoal de N à ­vel Superior on July 15, 2013 http://m nras.oxfordjournals.org/ D ow nloaded from http://mnras.oxfordjournals.org/ Radiative association 1893 impact parameter b for a given collision energy E is given by σ �uS = 4πg ( μ 2E )1/2 × ∫ ∞ 0 bdb ∫ ∞ rc A(r) dr{ 1 − [ V �uS (r)/E ] − (b2/r2) }1/2 , (1) in which μ is the reduced mass, g is the probability of approach along any particular molecular potential energy curve, rc is the distance of the closest approach, V�u is the molecular potential energy curve through which the colliding particles unite, and A(r) is the transition probability for the radiative decay. The rate coefficient for a Maxwellian distribution of particle ve- locities is given by K(T ) = ( 8 μπ )1/2( 1 kT )3/2 ∫ ∞ 0 E σ (E) exp ( − E kT ) dE. (2) Radiative association is most likely to occur in situations in which an excited state can be formed with potential minima lower than the dissociation limit, which is connected to a lower state of the molecule by an allowed electric dipole transition (Herbst & Bates 1988; Latter & Black 1991). The selection rules for allowed dipole transitions are �+ − �+, �− − �−, �S = 0, �� = 0, ±1. 3 SI L I C O N C A R B I D E (S I C ) The approach of ground state silicon (3P) and carbon (3P) atoms can occur through SiC molecular electronic states �, �(2), �+(2) and �−, each of multiplicities quintet, triplet and singlet. The ground electronic state of SiC is the X3� (Lutz & Ryan 1974), with a dissociation energy of 4.64 eV (Huber & Herzberg 1979). The first experimental observation of the SiC d1�1–b1� system was presented by Bernath et al. (1988). A detailed study of the pure rotational spectrum of SiC in the X3� ground state was reported by Cernicharo et al. (1989). The A3�−–X3� transition was observed by Brazier, O’ Brien & Bernath (1989) and the C3�–X3� system by Ebben, Drabbels & ter Meulen (1991). The A–X and C–X transition were also measured by Deo & Kawaguchi (2004) and Butenhoff & Rohlfing (1991), respectively. The first configuration interaction (CI) calculations on SiC have been performed by Lutz & Ryan (1974). They found that the ground state is a 3� state, and they also described two other excited states (1�+ and 3�−). Larsson (1986) have studied the B3�+ and C3� states using complete-active-space self-consistent field (CASSCF) and CI calculations. The dipole moment function of the X3� and A3�− states, radiative lifetimes, as well as electronic transition mo- ment functions of the A–X system have been reported by Langhoff & Bauschlicher (1990) at the multireference CI (MRCI) level. Trinder, Reinsch & Rosmus (1993) have studied transition moment functions for the C–X transition and radiative lifetimes for the C3� state by means of CASSCF calculations. The transition moment functions for the A–X, d–b, c–b and b–a systems have been calcu- lated by Sefyani & Schamps (1994) using the MRCI method. Other calculations on SiC, but only for low-lying potential energy curves were reported by Borin et al. (2005) using state-average CASSCF method. Recently, Pramanik & Das (2007) have determined partial radiative lifetimes and transition dipole moment functions for sev- eral allowed transitions based on multireference singles and doubles CI (MRDCI) calculations. Radiative association of Si and C atoms can occur following approach along the C3� state, which radiates to the ground X3� state of the SiC molecule. The probability of approach along the Figure 1. Potential energy curves for the lowest lying triplet states of SiC. C3� is 6/81. Potential energy curves of the X3� and C3� states have been modelled with Hulbert–Hirschfelder function (Hulbert & Hirschfelder 1941, hereafter HH), using relevant molecular con- stants extracted from spectroscopic studies of Ebben et al. (1991) and Butenhoff & Rohlfing (1991). These energy curves are depicted in Fig. 1, which are in excellent agreement with the ab initio po- tential energy curves reported by Larsson (1986) and Borin et al. (2005). The variation of the transition moment with internuclear distance (r) was taken from Trinder et al. (1993) in the 2.8–5.0 a0 range. At short range, the transition moments are fitted to the form Re(r) = ar + br2 (Dalgarno, Kirby & Stancil 1996), where a and b are found to be 0.94359 and −0.24629, respectively. At large separations, the transition moment falls off exponentially, but it is found to fit well to the function Re(r) = −124.24/r3 + 2032.60/r5. The C–X system has the broadly comparable transition dipole mo- ments among all the transitions studied. For temperatures ranging from 300 to 14 000 K, the radiative association of Si(3P) and C(3P) atoms through the C3� state of SiC, followed by a downward tran- sition to X3� state, is found to vary from 1.16 × 10−17 to 1.71 × 10−17 cm3 s−1 (Table 1). The rate coefficients for the radiative association of the SiC molecule were also calculated through A3�−–X3�, B3�+–X3�, D3�–X3� and C3�–A3�+ transitions. The probability of approach through the D3� state is 6/81 and for B3�+ and A3�− states is 3/81. The variation of the electronic transition moment (Re) with r for the A–X system has been taken from Langhoff & Bauschlicher (1990). The corresponding variation of Re with r for the B–X, D–X and C–A systems has been extracted from Pramanik & Das (2007). HH function has been used for construction of the potential energy curves. For A3� and B3�+, the molecular constants have been extracted from Deo & Kawaguchi (2004) and Larsson (1986), re- spectively. For D3� and F3�+ states, some spectroscopic constants (we xe, Be, αe) were still missing, and they were estimated from C© 2009 The Authors. Journal compilation C© 2009 RAS, MNRAS 400, 1892–1896 at Fundaà §Ã £o C oordenaà §Ã £o de A perfeià §oam ento de Pessoal de N à ­vel Superior on July 15, 2013 http://m nras.oxfordjournals.org/ D ow nloaded from http://mnras.oxfordjournals.org/ 1894 C. M. Andreazza, R. M. Vichietti and E. P. Marinho Table 1. Rate coefficients in cm3 s−1 for the SiC triplet–triplet transitions. T(K) C–X A–X B–X D–X C–A (10−17) (10−19) (10−19) (10−19) (10−19) 300 1.16 2.57 0.61 2.29 1.04 700 1.54 3.41 0.86 2.63 1.28 1000 1.66 3.72 0.99 2.74 1.40 1500 1.74 3.97 1.14 2.88 1.52 2000 1.76 4.07 1.25 3.04 1.61 3000 1.76 4.11 1.38 3.45 1.71 3500 1.75 4.11 1.42 3.71 1.75 4000 1.74 4.11 1.45 4.00 1.78 5000 1.72 4.10 1.48 4.61 1.83 6500 1.70 4.07 1.49 5.68 1.89 8500 1.68 4.05 1.49 7.30 1.98 10 500 1.68 4.06 1.47 9.10 2.07 12 500 1.70 4.10 1.45 11.05 2.18 14 000 1.71 4.24 1.43 12.59 2.23 the available data (T e, we, re) listed by Borin et al. (2005) using some well-known relationships between these constants (Herzberg 1950). The B3�+ state suffers an avoided crossing with the F3�+ state around 4.0 a0. The HH potentials are summarized in Fig. 1, which are in good agreement with the ab initio calculations (Borin et al. 2005). The rate coefficients through D–X transition are found to increase from 2.29 × 10−19 to 1.26 × 10−18 cm3 s−1 for the temperature range 300–14 000 K. The A–X state separation is rel- atively small, as well as the B–X and C–A transition moments. So, the total rate coefficients for association in these three channels are lower than 8.0 × 10−19 cm3 s−1 at any temperature as displayed in Table 1. As far as we know, there is no information about variation of Re with r for the C–B, D–C, F–X and F–C transitions. The probability of approach along the F3�+ state is 3/81. According to Pramanik & Das (2007), the transition probability of the C–B system is very low as well as the energy difference. Hence, this transition makes little contribution to radiative association. The transition frequency of the D–C and F–C systems is similar in magnitude, which is much smaller than the corresponding C–X transition. On the other hand, the F–X state separation is larger than those of the C–X transition. Even if the variation of Re with r for these transitions was the same as in the C–X, the contribution of the transitions D–C and F–C to the radiative association would be lower than 1.0 × 10−18 cm3 s−1 at any temperature. For F–X transition, the rate coefficients are lower than 8.2 × 10−18 cm3 s−1. The enhancing of the rate coefficient for radiative association through C–X transition relative to F–X transition as a result of the lower transition frequency is compensated by the greater probability of approach along the C3� state. Hence, these molecule-forming transitions are unlikely to alter our results significantly. There are no other accessible triplet states. Radiative association of silicon and carbon atoms is also possible through the 21�, d1�+, c1� and b1� states followed by radia- tive decay to the a1�+, b1�, b1� and a1�+ states, respectively. Potential energy curves were constructed through the HH function using molecular constants reported by Sefyani & Schamps (1994) and Borin et al. (2005) (see Fig. 2). The two 1�+ states exhibit an avoided crossing around 3.6 a0. The probability of approach along the 21�, b1� and c1� states is 2/81, and for a1�+, e1�− and d1�+ states is 1/81. The variation of Re with r for the singlet–singlet transitions has been taken from Sefyani & Schamps (1994) and Pramanik & Das (2007). The total rate coefficients for association Figure 2. Potential energy curves for the lowest lying singlet states of SiC. in these additional channels are lower than 7.0 × 10−19 cm3 s−1 at any temperature, as listed in Table 2. No information about transition dipole moments between 21�–e1�−, 21�–d1�+, 21�–c1�, 21�– b1�, e1�−–c1� and e1�−–b1� transitions are available. However, the state separations of theses transitions, as the probability of ap- proach along the singlet states, are lower than those of the C–X transitions. On account of this, the rate coefficients for radiative association through these transitions are expected to be very small. Six quintet states can be formed from ground state silicon and carbon atoms. Among then, the 5� state has an attractive potential energy curve (Borin et al. 2005). The second 5� state and the 5�− are repulsive in nature. Finally, the potential energy curves of the 5�+, 25�+ and 5� states exhibit a minima above the first atomic Table 2. Rate coefficients in cm3 s−1 for the SiC singlet-singlet transitions. T(K) d–a 2–a d–b c–b b–a (10−20) (10−21) (10−19) (10−20) (10−19) 300 3.48 0.71 1.30 1.11 2.05 700 4.02 0.86 1.67 1.70 2.55 1000 4.10 0.94 1.79 1.95 2.67 1500 4.09 1.06 1.89 2.18 2.73 2000 4.02 1.15 1.96 2.27 2.73 3000 3.87 1.34 2.00 2.31 2.69 3500 3.81 1.44 2.00 2.30 2.67 4000 3.78 1.57 2.00 2.29 2.66 5000 3.80 1.93 2.02 2.22 2.66 6500 4.03 2.86 2.04 2.14 2.69 8500 4.69 5.25 2.07 2.04 2.14 10 500 5.73 9.45 2.11 1.95 2.94 12 500 7.13 15.91 2.17 1.88 3.15 14 000 8.40 22.45 2.22 1.83 3.34 C© 2009 The Authors. Journal compilation C© 2009 RAS, MNRAS 400, 1892–1896 at Fundaà §Ã £o C oordenaà §Ã £o de A perfeià §oam ento de Pessoal de N à ­vel Superior on July 15, 2013 http://m nras.oxfordjournals.org/ D ow nloaded from http://mnras.oxfordjournals.org/ Radiative association 1895 Table 3. The total radiative association rate coefficients for Si-C. T K (K) (10−17 cm3 s−1) 300 1.26 700 1.67 1000 1.80 1500 1.89 2000 1.92 3000 1.93 3500 1.92 4000 1.91 5000 1.90 6500 1.88 8500 1.88 10 500 1.90 12 500 1.95 14 000 1.98 dissociation channel, to which they dissociated. Hence, the radiative association through the quintet states is negligible. The total rate coefficient in cm3 s−1, listed in Table 3, can be approximated within 4 per cent by means of the following equation, K(T ) = 2.038 × 10−17 ( T 300 )−0.01263 exp (−136.73 T ) . (3) Rate coefficients for SiC are plotted with other radiative asso- ciation rate coefficients in Fig. 3. The formation of SiC by radia- tive association tends to be more rapid than the formation of C2 (Andreazza & Singh 1997). On the other hand, the radiative asso- Figure 3. Rate coefficients for formation of SiS, SiC, SiO, CO, C2, CS and SO. ciation rate coefficients (KSiC) of SiC are smaller than those of SiO (Andreazza, Singh & Sanzovo 1995) and SiS (Andreazza & Marinho 2007). In addition, KSiC is greater than KCO (Singh et al. 1999) for T < 1600 K, and it is smaller than KCO for T ≥ 1600 K. As can be seen from Fig. 3, the radiative association-forming SiS has a reaction rate greater than that for SiO and SiC formation, resulting in large amounts of SiS compared to SiO and SiC. SiS has also greater rate coefficients than for CS (Andreazza & Singh 1997) and SO (Andreazza & Marinho 2005) and binds as much Si as there is S present. However, the condensation of FeS, CaS and MgS (Nozawa et al. 2003; Lodders 2006; Cherchneff & Lilly 2008) may remove S from the gas and allows more Si free, which might form SiC. 4 C O N C L U S I O N S We have calculated the rate coefficients for radiative association of the atoms Si and C to form the SiC molecule. The radiative association of Si(3P) and C(3P) atoms mainly occurs through the C3� state followed by radiative decay to the X3� state because of the highest dipole moments involved in this transition. 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