A COMPLETE SPECTROSCOPIC CHARACTERIZATION OF SO AND ITS ISOTOPOLOGUES UP TO THE TERAHERTZ DOMAIN

Since its first detection in the interstellar medium by means of rotational spectroscopy (Gottlieb & Ball 1973), sulfur monoxide, SO, has been observed in a wide variety of astrophysical environments (see Klaus et al. 1994, and references therein). This transient species is particularly abundant in star-forming regions characterized by high temperature and gas concentrations (Blake et al. 1987). It has been detected in the atmosphere of Venus (Na et al. 1990) and in Io, one of Jupiter's moons, where it is thought to be produced by volcanic and photochemical pathways (Lellouch 1996). In planetary atmospheres, the presence of the SO radical results from the photodecomposition of sulfur dioxide (Krasnopolsky 2012). SO is also an important reaction intermediate in combustion and Earth atmospheric chemistry due to its high reactivity and its role in reactions involving molecules containing nitrogen and oxygen (see Burkholder et al. 1987, and references therein).

SO was first identified in the laboratory in 1932 through the observation of its electronic transitions (Martin 1932). Since then, numerous electronic (Colin 1968), ro-vibrational (Kanamori et al. 1985; Burkholder et al. 1987), as well as pure rotational studies have been dedicated to this radical. The electric dipole moment of SO has also been determined by Stark measurements (1.55 D; Powell & Lide 1964). Pure rotational transitions of SO have been observed in the microwave (Lovas et al. 1992), the terahertz (THz), (Cazzoli et al. 1994), and far infrared (FIR) regions (Cheung et al. 1991; Martin-Drumel et al. 2013) in numerous vibrational states of the ground electronic state (Bogey et al. 1982). SO is also the first molecule for which pure rotational transitions have been detected in several excited electronic states (Saito 1970; Clark & De Lucia 1976; Endo et al. 1987; Bogey et al. 1997; Sanz et al. 2003). Since the sulfur atom possesses three relatively abundant isotopes (³²S: 95.02%, ³⁴S: 4.21%, ³³S: 0.75%), several studies have also been dedicated to the SO isotopologues (e.g., Tiemann 1982; Klaus et al. 1996; Sanz et al. 2003). Isotopically invariant parameters have been derived from combined fits of all isotopic species (Klaus et al. 1996) and Dunham coefficients have also been determined (Sanz et al. 2003).

Recently, pure rotational transitions of SO have been observed by means of synchrotron-based Fourier-transform FIR (FT-FIR) spectroscopy (Martin-Drumel et al. 2013). In this study, 102 transitions were recorded in the spectral range 1.3–2.8 THz with a frequency accuracy limited to 2 MHz. In the present study, we have re-investigated this range of the spectrum using a photomixing technique allowing a higher accuracy to be achieved (down to 10 kHz). We also have extended the observed transitions of both ³⁴SO and ³³SO (detected in natural abundance) toward higher frequencies. Improved effective molecular constants and isotopically invariant parameters have been derived from these new results. Frequency predictions obtained from this new isotopically invariant fit are accurate in the THz region and will permit further exploration of the SO (and its rare isotopologues) spectra in astronomical sources observed with the HIFI instrument (De Graauw et al. 2010) on board the Herschel satellite (Pilbratt et al. 2010; see HEXOS and CHESS Herschel guaranteed programs; Wang et al. 2011; Ceccarelli et al. 2010).

The continuous-wave terahertz (cw-THz) photomixing spectrometer used in this study has been described in detail elsewhere (Eliet et al. 2011; Hindle et al. 2011; Mouret et al. 2009). Briefly, the operation of the photomixing source is based on the spatial superposition of two extended cavity diode lasers (ECDLs; emitting around 780 nm) creating an optical beatnote, which is converted to free space THz radiation by a photomixer (ultrafast LTG-GaAs device). This spectrometer is tunable between 0.3 and 3.3 THz. A frequency comb (FC) generated by a frequency doubled erbium doped femtosecond fiber laser enables the achievement of the ECDLs' frequency stability of 1 MHz or better. The FC repetition rate, which separates each mode of the FC, is locked to a low phase-noise oven-controlled crystal oscillator, which is itself locked to the GPS timing signal (Spectracom, EC20S). This reference frequency provides an accuracy of 2 × 10⁻¹² over a period of 24 hr. Two phase-locked loops are used to lock the two ECDLs to the FC, accurately fixing the difference frequency; the accuracy in line frequency is thus only limited by the line profile. The use of a third ECDL ensures a continuous tunability up to 500 MHz, illustrated in the study of the SO radical, wherein a 510 MHz continuous portion of the ³²SO spectrum was recorded around 1 THz (Figure 1). This spectrum, showing a rotational triplet, has been recorded in about ten minutes using frequency steps of 400 kHz and a time constant of 100 ms. The periodic variations observed in the baseline of the recordings are caused by a weak Fabry–Perot effect established between the photomixer and the bolometer, here separated by 1.87 m. The photomixing technique has the greatest tuning range of any known coherent source in the THz region, and has a spectral purity smaller than the Doppler limit at room temperature.

Zoom In Zoom Out Reset image size

The THz beam is collimated through the cell and focused onto a silicon liquid-helium cooled detector by means of two off-axis parabolic mirrors. In this study, both amplitude modulation (AM) and frequency modulation (FM) modes of operation have been used. ΔJ = 1 transitions of ³²SO have been recorded in AM using a 263 Hz modulation frequency (Figure 1). ΔJ = 0 transitions of ³²SO (100 times weaker than ΔJ = 1) as well as lines of ³⁴SO and ³³SO have been recorded in FM with a modulation depth of 120 kHz and a rate of 273 Hz (Figure 2). A second harmonic scheme is employed in order to optimize the signal-to-noise ratio (S/N). Despite this rather slow modulation frequency, which was necessary due to the use of a "slow" Si-bolometer, an improvement in S/N by more than a factor 10 has been observed using the FM instead of the AM configuration.

Zoom In Zoom Out Reset image size

The SO radical was produced by a 50 W radiofrequency discharge using air in a cell containing a pure sulfur deposit. The air flow was ensured by a roughing pump. Absorption levels of up to 80% were obtained for the main isotopologue in its vibrational ground state. ³⁴SO and ³³SO isotopologues were observed in natural abundance. A discharge in a flow air and H₂S mixture yielded a lower SO concentration. SO₂ transitions in the ground and first excited vibrational state have been identified in the spectrum (Figures 1 and 2). Experimental conditions were adjusted for each transition in order to achieve the highest S/N possible in a reasonable acquisition time (pressure: 20–500 μbar; time constant: 0.5–2 s; steps: 100–600 kHz.) S/Ns up to 260, 70, and 40 have been obtained for ³²SO, ³⁴SO, and ³³SO, respectively.

In its fundamental X³Σ⁻ electronic state, SO is an intermediate between Hund's coupling schemes (a) and (b). However, by convention the rotational structure of the radical is described in the (b) formalism using the rotational quantum number N. The coupling between the electronic spin and the rotation of the molecule, by means of spin–spin (interaction between the magnetic moments associated with the two unpaired electrons under the effect of the rotation) and spin–rotation (interaction with Π excited electronic states) coupling, leads to a rotational fine structure of three components (for N > 0) associated with the quantum number J ($\boldsymbol{J}=\boldsymbol{N+S}$, S = 1) and separated by a few hundred GHz (a few wavenumbers).

Due to the wide range of observed frequencies (and thus of FWHMs) and S/Ns, we performed a line profile fit on all transitions to determine their center frequency and estimate their accuracy. Only lines presenting no more than 60% to 70% of absorption (when recorded in AM) were considered in the process to prevent saturating line shapes. Rotational transitions were fitted with a pseudo-Voigt profile of type (1 − s)G + sL (with G and L a Gaussian and Lorentzian function, respectively, and s the "shape" of the line):

$$\begin{eqnarray} y(\nu) &=& A \left\lbrace (1-s){\rm exp} \left[ -\ln 2 \left(\frac{\nu -B}{C}\right) ^2\right]\right. \nonumber\\ &&\left. +s\frac{1}{1+ [ (\nu -B)/C] ^2} \right\rbrace, \end{eqnarray} \tag{ 1 }$$

where A is the height of the line, B is the center frequency, and C is the FWHM. Compared to a classical Voigt profile, the second derivative of this analytical formula can be easily calculated in order to fit the transitions recorded in FM (second derivative shape). The accuracy of each line position δ(ν) was then estimated from the formula (Landman et al. 1982):

$$\begin{equation} \delta (\nu) = \frac{(C\Delta \nu)^{1/2}}{\rm S/N} \left\lbrace (1-s)\left(\frac{2}{\pi \ln 2}\right) ^{1/4} +s \left(\frac{32}{\pi }\right) ^{1/2}\right\rbrace, \end{equation} \tag{ 2 }$$

where Δν is the frequency step and S/N is the signal-to-noise ratio.

One-hundred five transitions of ³²SO in its vibrational ground state (16 ⩽ N'' ⩽ 58) have been recorded in the spectral region 0.731–2.511 THz with frequency accuracies Δ(ν) ranging from 10 to 800 kHz. 48 (17 ⩽ N'' ⩽ 32) and twenty-one (16 ⩽ N'' ⩽ 22) transitions of the ³⁴SO and ³³SO isotopologues in v = 0 have also been observed in natural abundance up to 1.388 THz and 978 GHz, respectively. The observed transitions are presented with their calculated uncertainty in Tables 1–3.

The following sections detail the two fitting processes performed in this work. We first performed an effective fit for each of the three isotopologues observed in this study. These fits provide the most accurate rotational parameters for the three molecules. However, due to the lack of predictive reliability of such effective fits (see an example in the next paragraph) we performed a global fit of all the data available on SO (all isotopologues and all vibrational states).

3.1. Effective Fit of ³²SO, ³⁴SO, and ³³SO

The effective Hamiltonian for SO in its X³Σ electronic ground state is:

$$\begin{eqnarray} &&\mathscr{H}_{\rm {eff}} = \mathscr{H}_{\rm {Rotation}} + \mathscr{H}_{\rm {Spin{-}Spin}} +\mathscr{H} _{\rm {Spin{-}Rotation}} \end{eqnarray} \tag{ 3 }$$

$$\begin{eqnarray} &=B\boldsymbol{N}^2+\frac{2}{3}\lambda \left(3S_z^2-\boldsymbol{S}^2\right)+\gamma \boldsymbol{N}\cdot \boldsymbol{S} \end{eqnarray} \tag{ 4 }$$

where B is the rotational constant, λ the spin–spin coupling constant, and γ the spin–rotation coupling constant, neglecting the centrifugal distortion. In the case of ³³SO, the non-null nuclear spin of the ³³S (I = 3/2) leads to a hyperfine structure of four components separated by a few tens of MHz. This effect could not be resolved in the present study.

The newly recorded transitions have been fitted using the SPFIT/SPCAT software (Pickett 1991) together with previously available data reported in the literature for ³²SO, ³³SO and ³⁴SO (see references in Table 4). Multiple experimental values for one transition were kept in the fit (weighted according to their respective accuracy) since they were all obtained by independent measurements. Reduced standard deviations (dimensionless) of 0.88, 0.82, and 0.85 have been obtained for the three fits. Results are compared with Ref. Martin-Drumel et al. (2013) in the case of ³²SO, and to a refit of previously available data for the two other isotopologues (Table 5). For each isotopologue, fitting the newly recorded transitions (only), with parameters fixed at their final values, resulted in rms values lower than 100 kHz, and a reduced standard deviation of 0.99, 0.75, and 0.84. This reflects the quality of our measurements and uncertainty estimation.

Table 4. References Used in the Effective and Isotopically Invariant Fits

Isotopologue	References
Effective fit
32S16O	Powell & Lide (1964); Winnewisser et al. (1964); Amano et al. (1967); Tiemann (1974); Clark & De Lucia (1976); Bogey et al. (1982);
	Tiemann (1982); Lovas et al. (1992); Cazzoli et al. (1994); Klaus et al. (1996); Bogey et al. (1997); Hansen et al. (1998);
	Martin-Drumel et al. (2013); Sanz et al. (2003)
33S16O	Amano et al. (1967); Klaus et al. (1996); Sanz et al. (2003)
34S16O	Amano et al. (1967); Tiemann (1974); Bogey et al. (1982); Tiemann (1982); Lovas et al. (1992); Klaus et al. (1996); Sanz et al. (2003)
Isotopically invariant fit (additional references)
32S16O	Kawaguchi et al. (1979); Wong et al. (1982); Kanamori et al. (1985); Burkholder et al. (1987)
34S16O	Burkholder et al. (1987); Hansen et al. (1998)
36S16O	Klaus et al. (1996)
32S17O	Klaus et al. (1996)
32S18O	Lovas et al. (1992); Tiemann (1974); Bogey et al. (1982); Tiemann (1982); Klaus et al. (1994, 1996); Burkholder et al. (1987)

Download table as:  ASCIITypeset image

Table 5. Ground State Molecular Constants (in MHz) of ³²SO, ³³SO, and ³⁴SO, and Comparison with Previous Values

	32SO		33SO		34SO
	This Work	Previous Valuesa	This Work	Previous Valuesb	This Work	Previous Valuesc
B	21 523.555 878 (79)	21 523.556 16 (37)	21 306.465 38 (86)	21 306.465 23 (97)	21 102.732 28 (31)	21 102.733 14 (82)
D × 103	33.915 261 (85)	33.915 56 (70)	33.238 7 (11)	33.239 6 (14)	32.600 73 (55)	32 606 4 (13)
H × 109	−6.974 (22)	−7.09 (17)			−6.06 (31)
λ	158 254.392 (09)	158 254.387 (13)	158 252.22 (14)	158 252.28 (15)	158 249.807 (24)	158 249.816 (27)
λD	0.306 259 (72)	0.306 53 (20)	0.303 65 (62)	0.303 9 (12)	0.300 63 (08)	0.300 28 (79)
λH × 106	0.478 (47)
γ	−168.304 3 (15)	−168.304 1 (40)	−166.622 (20)	−166.630 (21)	−164.994 8 (36)	−164.992 9 (65)
γD × 103	−0.525 45 (97)	−0.523 1 (85)	−0.358 (20)	−0.344 (35)	−0.511 6 (27)	−0.515 (22)
b			18.883 (19)	18.883 (19)
c			−96.34 (11)	−96.34 (11)
eQq			−15.453 1 (71)	−15.453 1 (71)
CI			0.015 1 (73)	0.014 8 (74)
Fitted lines	329	184	100	79	95	48
σd	0.88	0.64	0.82	0.82	0.85	0.95

Notes. Errors (1σ) are reported in parentheses in the unit of the last digit. ^aPrevious values from Martin-Drumel et al. (2013). ^bObtained by refitting data from Amano et al. (1967), Klaus et al. (1996), and Sanz et al. (2003). ^cObtained by refitting data from Amano et al. (1967), Tiemann (1974), Tiemann (1982), Bogey et al. (1982), Lovas et al. (1992), and Klaus et al. (1996). ^dReduced standard deviation (dimensionless).

Download table as:  ASCIITypeset image

The addition of new transitions with higher N values allows the determination of molecular parameters with higher accuracy. The error of the centrifugal distortion parameters is decreased by up to an order of magnitude. Furthermore, λ_H for ³²SO and H for ³⁴SO have been determined for the first time. In case of ³³SO, the hyperfine structure due to the ³³S nucleus has not been resolved, hence the hyperfine structure parameters b, c, eQq, and C_I are not better determined. The amelioration of the centrifugal distortion parameters is particularly visible when examining the dispersion on the frequency predictions for the ³²SO radical in Figure 3. This plot presents the difference between the frequencies of the lines observed in this study (cw-THz) and the calculated frequencies obtained with different parameter sets (obs. − calc.). The experimental uncertainty on the line frequency is plotted as error bars. The calculated frequencies based on the initial set of data (pure rotational data from the literature apart from Martin-Drumel et al. 2013) present the strongest discrepancy with the experimental values. This set of data contains pure rotational transitions up to 1.9 THz and the deviation of the prediction is visible, 2.5 MHz at 2.5 THz, while the error on the frequencies of the lines at 2.5 MHz were estimated to be of only 5 kHz. This illustrates the lack of reliability of effective fits regarding the predictions mentioned above.

Zoom In Zoom Out Reset image size

We also note that the addition of FT-FIR data in the fit allows a significant improvement of the prediction, despite the fact that these data have a limited frequency accuracy of 2 MHz. This is an illustration of the complementarity of synchrotron-based FT-FIR spectroscopy and cw-THz technique for the study of transient species (see also Martin-Drumel et al. 2012): the former allows a broad survey of the FIR region with a limited resolution (30 MHz), consequently allowing a fast and efficient recording at kHz resolution of the same transitions with the latter technique.

3.2. Isotopically Invariant Fit

In order to provide reliable frequency predictions of SO isotopologues—including rare isotopologues—in the THz range, we performed an isotopically invariant fit. This global fit includes all reported pure rotational and ro-vibrational transitions of SO isotopologues in all observed vibrational states using the fitting programs associated with the MADEX spectroscopic code (Cernicharo 2013). The fitting code was checked against the results of Klaus et al. (1996) and provided the same rotational constants. The Dunham expansions used to determine the matrix elements are those reported in Klaus et al. (1994), although the hyperfine structure has not been considered. For the sake of clarity, energy expressions are summarized below.

The rovibrational energy levels are expressed by an expansion involving the isotopically invariant parameters U_ij and the reduced mass μ (in a.u.):

$$\begin{equation} E_{\rm {Rotation}}(v,N) = \sum _{i,j=0}^{\infty }\mu ^{-(i/2+j)}U_{ij}(v + 1/2)^i \left[N(N+1)\right]^j. \end{equation} \tag{ 5 }$$

The spin–rotation (SR) and spin–spin (SS) interactions are expressed using the isotopically invariant parameters γ_ij and λ_ij:

$$\begin{eqnarray} &&E _{\rm {spin{-}rotation}} (v,J,N) = (1/2)\left[ J(J+1)-N(N+1)-2\right]\nonumber \\ &&\quad \times \sum _{i,j=0}^{\infty }\mu ^{-(i/2+j+1)}\gamma _{ij}(v+1/2)^i\left[ N(N+1)\right]^j \end{eqnarray} \tag{ 6 }$$

$$\begin{eqnarray} &&E_{\rm {Spin{-}Spin}}(v,J,N) = (2/3) \sqrt{30}(-1)^{2N+J+1}(2N+1) \nonumber \\ && \times \left\lbrace \begin{array}{@{}ccc@{}}J & 1 & N \\ 2 & N & 1\end{array} \right\rbrace \left(\begin{array}{@{}ccc@{}}N & 2 & N \\ 0 & 0 & 0\end{array} \right)\nonumber \\ && \times (1/2) \sum _{i,j=0}^{\infty }\mu ^{-(i/2+j)}\lambda _{ij}(v+1/2)^i \times 2 \left[ N(N+1)\right] ^j. \end{eqnarray} \tag{ 7 }$$

The breakdown of the Born–Oppenheimer approximation yields a slight modification of the previous equations. For instance, denoting the isotopes A and B for the two nuclei, the U₀₁ constant is defined as (Tiemann 1982):

$$\begin{equation} U_{01} = U_{01}^e + X_{01}^A \left(1-\frac{M_A^0}{M_A} \right) +X_{01}^B \left(1-\frac{M_B^0}{M_B} \right), \end{equation} \tag{ 8 }$$

where $X_{01}^A$ and $X_{01}^B$ carry the corrections to the Born–Oppenheimer approximation. M_A and M_B are the atomic masses of the considered isotopes while $M_A^0$ and $M_B^0$ are the atomic masses of the reference isotopes (here ³²S and ¹⁶O). Atomic masses used in our fits are taken from the AME2012 (Audi et al. 2012; Wang et al. 2012). Unlike the one used by Klaus et al. (1994), our program did not allow us to include hyperfine interactions. Consequently, for isotopologues containing the nuclei ³³S (spin I = 3/2) or ¹⁷O (spin I = 5/2), the observed hyperfine components have been averaged with a weight relative to their intensities (taken from the CDMS website; Müller et al. 2001), and the resulting frequencies have been included in the global fit.

Pure rotation and ro-vibration transitions of six isotopologues of SO from this work and the literature have been included in our fit. For the first time rotational and vibrational transitions have been merged in an isotopically invariant fit for the SO radical allowing the determination of the U_i0 constants. In total, 1766 transitions were included (see detailed references in Table 4): 1121 for ³²S¹⁶O with v = 0–33, 53 for ³³S¹⁶O with v = 0–8, 296 for ³⁴S¹⁶O with v = 0–11, 28 for ³⁶S¹⁶O with v = 0, 4 for ³²S¹⁷O with v = 0, and 264 for ³²S¹⁸O with v = 0–1.

A fit of all rotational and vibrational data using the uncertainties given by the different authors for the IR transitions produces a result in which the uncertainties for the rotational constants, U_ij(j ≠ 0), are slightly degraded. An examination of the different sets of IR lines (by looking at the individual reduced standard deviation of each set of data) suggests that the quoted uncertainties for these data are in some cases optimistic. Hence, we have multiplied these uncertainties by a factor of two and eliminated in the final fit all lines for which the observed minus calculated frequencies exceed 3σ. The complete linelist is presented in the machine-readable table. A short example is shown in Table 6. The resulting 43 isotopically invariant parameters are given in Table 7 (reduced standard deviation of the fit: 0.8997). For the sake of comparison with previous studies and to observe the influence of the newly recorded transitions, a global fit including only pure rotational transitions has also been performed. The results are given in column two and can be compared with the results of Klaus et al. (1996) presented in the third column. The uncertainties of all parameters are improved with our data by a factor of 2–5.

Table 6. Pure Rotation and Ro-vibration Transitions of SO Included in the Isotopically Invariant Fit

Sa	Ob	N'	J'	ν'	N''	J''	ν''	νobs	Unc.	Ref.
1	5	2	1	0	1	1	0	13043.8	0.20	1964JCP...41...1413c
1	5	0	1	0	1	0	0	30001.6	0.20	1964JCP...41...1413
1	5	3	2	0	2	2	0	36201.7	0.20	1964JCP...41...1413
1	5	1	2	0	0	1	0	62931.0	0.50	1964JCP...41...1413
1	5	4	3	0	3	3	0	66034.9	0.50	1964JCP...41...1413

Notes. Positive values indicate frequency in MHz, negative in cm⁻¹. ^a1 = ³²S; 2 = ³³S; 3 = ³⁴S; 4 = ³⁶S. ^b5 = ¹⁶O; 6 = ¹⁷O; 7 = ¹⁸O. ^cPowell & Lide (1964).

Only a portion of this table is shown here to demonstrate its form and content. A machine-readable version of the full table is available.

Download table as:  DataTypeset image

Table 7. Isotopically Invariant Parameters for the SO Radical

	Ro-vibrational	Rotational (This Work)	Rotationala	Units
U10	3757.49542(195)	⋅⋅⋅	⋅⋅⋅	cm−1
X$^S_{10}$	−0.02551(209)	⋅⋅⋅	⋅⋅⋅	cm−1
X$^O_{10}$	−0.017252(516)	⋅⋅⋅	⋅⋅⋅	cm−1
U20	−68.32167(622)	⋅⋅⋅	⋅⋅⋅	cm−1
U30	0.47556(818)	⋅⋅⋅	⋅⋅⋅	cm−1
U40	−0.04041(463)	⋅⋅⋅	⋅⋅⋅	cm−1
U50	0.005031(934)	⋅⋅⋅	⋅⋅⋅	cm−1
U01	230387.59070(231)	230387.59057(277)	230387.56040(520)	MHz amu
X$^S_{01}$	7.80287(568)	7.80306(679)	7.8170(230)	MHz amu
X$^O_{01}$	21.4882(454)	21.4880(542)	21.5320(270)	MHz amu
U11	−6001.6578(164)	−6001.6561(197)	−6001.4290(360)	MHz amu3/2
U21	25.7585(267)	25.7522(321)	25.4620(750)	MHz amu2
U31	−0.8254(175)	−0.8211(211)	−0.5910(560)	MHz amu5/2
U41	−0.01197(508)	−0.01296(611)	−0.1180(140)	MHz amu3
U51	−0.022075(633)	−0.021972(759)	⋅⋅⋅	MHz amu7/2
U61	0.0011700(256)	0.0011663(306)	⋅⋅⋅	MHz amu4
U02	−3.8543097(229)	−3.8543191(281)	−3.853730(130)	MHz amu2
U12	−0.004547(114)	−0.004482(144)	−0.008590(390)	MHz amu5/2
U22	−0.0011636(577)	−0.0011953(717)	⋅⋅⋅	MHz amu3
U03	−0.0000078489(791)	−0.0000078478(946)	−0.00001260(140)	MHz amu3
U13			0.0000299(46)	MHz amu7/2
U04	−0.000000001040(146)	−0.000000001041(174)	⋅⋅⋅	MHz amu4
γ00	−1787.4277(276)	−1787.4285(332)	−1787.4230(370)	MHz amu
γ10	−45.508(176)	−45.501(212)	−45.670(210)	MHz amu3/2
γ20	1.871(133)	1.857(160)	2.210(150)	MHz amu2
γ30	0.0986(216)	0.0995(260)	⋅⋅⋅	MHz amu5/2
γ01	−0.060460(184)	−0.060481(221)	−0.058950(400)	MHz amu2
γ11	0.00439(108)	0.00452(130)	⋅⋅⋅	MHz amu5/2
λ00	157795.4203(201)	157795.4208(241)	157795.5340(680)	MHz
X$^S_{00}$	0.299(229)	0.303(273)	−0.570(240)	MHz
X$^O_{00}$	−7.722(235)	−7.721(280)	−7.200(130)	MHz
λ10	2979.597(136)	2979.592(164)	2978.670(640)	MHz amu1/2
X$^S_{10}$	−2.697(160)	−2.702(191)	⋅⋅⋅	MHz amu1/2
λ20	112.732(159)	112.745(191)	114.50(160)	MHz amu
λ30	13.6225(953)	13.619(113)	12.00(130)	MHz amu3/2
λ40	−0.1057(327)	−0.1054(391)	0.780(310)	MHz amu2
λ50	0.22948(606)	0.22949(724)	⋅⋅⋅	MHz amu5/2
λ60	−0.018122(580)	−0.018125(692)	⋅⋅⋅	MHz amu3
λ70	0.0014205(229)	0.0014203(273)	⋅⋅⋅	MHz amu7/2
λ01	3.24222(125)	3.24213(151)	3.24430(200)	MHz amu
λ11	0.14226(791)	0.14287(954)	0.1490(120)	MHz amu3/2
λ21	0.03545(466)	0.03505(561)	0.03140(780)	MHz amu2
λ02	0.00005374(357)	0.00005379(426)	⋅⋅⋅	MHz amu2
σb	0.8997	0.8112	0.70

Notes. Errors (1σ) are reported in parentheses in the unit of the last digit. ^aConstants derived by Klaus et al. (1996). ^bReduced standard deviation.

Download table as:  ASCIITypeset image

Born–Oppenheimer Breakdown (BOB) coefficients have been derived from the isotopically invariant fit following the formula (for an atom A; Watson 1980):

$$\begin{equation} \Delta _{01}^A = -\frac{M_A^0}{m_e\left(U_{01} + X_{01}^A + X_{01}^B\right)} X_{01}^A. \end{equation} \tag{ 9 }$$

Table 8 presents these coefficients together with a comparison with Klaus et al. (1996). Δ₀₁ coefficients are slightly better determined while Δ₁₀ are determined for the first time.

Table 8. Born–Oppenheimer Breakdown (BOB) Coefficients Derived from the Isotopically Invariant Fit

Parameter	This Work	Klaus et al. (1996)
$\Delta _{01}^S$	−1.9725 (14)	−1.9772 (58)
$\Delta _{01}^O$	−2.7175 (57)	−2.7247 (34)
$\Delta _{10}^S$	0.395 (32)	⋅⋅⋅
$\Delta _{10}^O$	0.1338 (40)	⋅⋅⋅

Note. Errors (1σ) are reported in parentheses in the unit of the last digit.

Download table as:  ASCIITypeset image

Finally, molecular parameters for 12 isotopologues of SO in v = 0 to 5 (³²S¹⁶O, ³²S¹⁷O, ³²S¹⁸O, ³³S¹⁶O, ³³S¹⁷O, ³³S¹⁸O, ³⁴S¹⁶O, ³⁴S¹⁷O, ³⁴S¹⁸O, ³⁶S¹⁶O, ³⁶S¹⁷O, ³⁶S¹⁸O) have been derived from the isotopically invariant fit using a nonlinear least squares method. The obtained parameters are presented in Tables 9–12. The quality of the determination of the obtained parameters can be estimated since the error is dominated by the one of the dominant parameters (Y₀₁ for B₀, Y₀₂ for D₀, Y₀₃ for H₀,...). As an example, for ³²SO in v = 0 the error on B₀ (21 523.556 51 MHz, Table 9) should be of the order of 2 kHz (i.e., the error on Y₀₁; Table 7) for all isotopologues, which is the strength of the isotopically invariant fit. This can be verified by comparison with the B₀ value obtained using the effective fit for ³²SO (21 523.555 878 (79) MHz; Table 5). The two values differ by 0.6 kHz, attesting to the quality of the derived parameters. However, for ³²SO, ³³SO, and ³⁴SO in their vibrational ground state, the rotational constants of Table 5 are more accurate. For all other isotopologues and vibrational states, the parameters given in the previous tables will permit the prediction of reliable rotational frequencies in the THz domain.

Table 9. Derived Parameters (in MHz) for ³²SO in the Six Lowest Vibrational Levels of the Electronic Ground State

	v = 0	v = 1	v = 2	v = 3	v = 4	v = 5
Isotopologue 32S16O
B	21523.55651	21351.59505	21180.06610	21008.95451	20838.24344	20667.91389
D × 102	3.39162	3.39304	3.39465	3.39645	3.39845	3.40063
H × 109	−6.47734	−6.47734	−6.47734	−6.47734	−6.47734	−6.47734
L × 1014	−8.04693	−8.04693	−8.04693	−8.04693	−8.04693	−8.04693
γ	−168.30517	−169.57865	−170.81684	−172.01815	−173.18097	−174.30370
γD × 104	−5.26007	−5.14161	−5.02315	−4.90469	−4.78623	−4.66777
λ	158254.38332	159189.34399	160148.99572	161135.78421	162152.28832	163201.27051
λD × 101	3.06232	3.10943	3.16278	3.22236	3.28818	3.36023
λH × 107	4.73466	4.73466	4.73466	4.73466	4.73466	4.73466
Isotopologue 32S17O
B	20677.80911	20515.89547	20354.38135	20193.25309	20032.49559	19872.09182
D × 102	3.12977	3.13105	3.13250	3.13412	3.13591	3.13787
H × 109	−5.74192	−5.74192	−5.74192	−5.74192	−5.74192	−5.74192
L × 1014	−6.85242	−6.85242	−6.85242	−6.85242	−6.85242	−6.85242
γ	−161.66569	−162.86535	−164.03249	−165.16566	−166.26342	−167.32432
γD × 104	−4.85508	−4.74793	−4.64079	−4.53365	−4.42651	−4.31937
λ	158244.74695	159160.65402	160100.21102	161065.71645	162059.58924	163084.41473
λD × 101	2.94132	2.98556	3.03555	3.09130	3.15281	3.22006
λH × 107	4.36914	4.36914	4.36914	4.36914	4.36914	4.36914
Isotopologue 32S18O
B	19929.27880	19776.08755	19623.26769	19470.80683	19318.69126	19166.90561
D × 102	2.90683	2.90799	2.90930	2.91077	2.91239	2.91416
H × 109	−5.13947	−5.13947	−5.13947	−5.13947	−5.13947	−5.13947
L × 1014	−5.91098	−5.91098	−5.91098	−5.91098	−5.91098	−5.91098
γ	−155.79078	−156.92631	−158.03168	−159.10556	−160.14663	−161.15358
γD × 104	−4.51015	−4.41246	−4.31478	−4.21709	−4.11940	−4.02171
λ	158236.06300	159134.79727	160056.26376	161002.63355	161976.18753	162979.35856
λD × 101	2.83426	2.87602	2.92312	2.97556	3.03335	3.09648
λH × 107	4.05792	4.05792	4.05792	4.05792	4.05792	4.05792

Notes. Weighted sigma = 0.6791. Standard deviation = 8.8978 MHz.

Download table as:  ASCIITypeset image

Table 10. Derived Parameters (in MHz) for ³³SO in the Six Lowest Vibrational Levels of the Electronic Ground State

	v = 0	v = 1	v = 2	v = 3	v = 4	v = 5
Isotopologue 33S16O
B	21306.46510	21137.10196	20968.16273	20799.63267	20631.49538	20463.73240
D × 102	3.32341	3.32479	3.32635	3.32811	3.33005	3.33217
H × 109	−6.28291	−6.28291	−6.28291	−6.28291	−6.28291	−6.28291
L × 1014	−7.72650	−7.72650	−7.72650	−7.72650	−7.72650	−7.72650
γ	−166.60082	−167.85522	−169.07505	−170.25875	−171.40477	−172.51154
γD × 104	−5.15457	−5.03908	−4.92359	−4.80810	−4.69261	−4.57712
λ	158252.04078	159182.12015	160136.62293	161117.95670	162128.65888	163171.44597
λD × 101	3.03126	3.07763	3.13011	3.18869	3.25339	3.32420
λH × 107	4.63944	4.63944	4.63944	4.63944	4.63944	4.63944
Isotopologue 33S17O
B	20460.68961	20301.32252	20142.34669	19983.74883	19825.51426	19667.62643
D × 102	3.06426	3.06550	3.06691	3.06848	3.07022	3.07213
H × 109	−5.56258	−5.56258	−5.56258	−5.56258	−5.56258	−5.56258
L × 1014	−6.56855	−6.56855	−6.56855	−6.56855	−6.56855	−6.56855
γ	−159.96154	−161.14248	−162.29159	−163.40747	−164.48870	−165.53387
γD × 104	−4.75372	−4.64938	−4.54503	−4.44069	−4.33634	−4.23199
λ	158242.35753	159153.28989	160087.60564	161047.56603	162035.54963	163054.09712
λD × 101	2.91026	2.95378	3.00293	3.05771	3.11812	3.18417
λH × 107	4.27769	4.27769	4.27769	4.27769	4.27769	4.27769
Isotopologue 33S18O
B	19712.13399	19561.44258	19411.11459	19261.13798	19111.49943	18962.18404
D × 102	2.84370	2.84483	2.84611	2.84753	2.84911	2.85083
H × 109	−4.97297	−4.97297	−4.97297	−4.97297	−4.97297	−4.97297
L × 1014	−5.65704	−5.65704	−5.65704	−5.65704	−5.65704	−5.65704
γ	−154.08680	−155.20395	−156.29159	−157.34845	−158.37325	−159.36469
γD × 104	−4.41247	−4.31743	−4.22238	−4.12734	−4.03229	−3.93725
λ	158233.62960	159127.30151	160043.44012	160984.18000	161951.76266	162948.57746
λD × 101	2.80321	2.84426	2.89053	2.94203	2.99876	3.06072
λH × 107	3.96980	3.96980	3.96980	3.96980	3.96980	3.96980

Notes. rms = 8.8978 MHz. Reduced standard deviation = 0.6791.

Download table as:  ASCIITypeset image

Table 11. Derived Parameters (in MHz) for ³⁴SO Isotopologues in the Six Lowest Vibrational Levels of the Electronic Ground State

	v = 0	v = 1	v = 2	v = 3	v = 4	v = 5
Isotopologue 34S16O
B	21102.73177	20935.79500	20769.27415	20603.15484	20437.42111	20272.05496
D × 102	3.26002	3.26137	3.26289	3.26460	3.26649	3.26856
H × 109	−6.10403	−6.10403	−6.10403	−6.10403	−6.10403	−6.10403
L × 1014	−7.43459	−7.43459	−7.43459	−7.43459	−7.43459	−7.43459
γ	−165.00144	−166.23801	−167.44069	−168.60794	−169.73826	−170.83011
γD × 104	−5.05654	−4.94379	−4.83105	−4.71831	−4.60556	−4.49282
λ	158249.83181	159175.30919	160124.95906	161101.15343	162106.39109	163143.34570
λD × 101	3.00211	3.04778	3.09945	3.15712	3.22077	3.29042
λH × 107	4.55095	4.55095	4.55095	4.55095	4.55095	4.55095
Isotopologue 34S17O
B	20256.92979	20099.94025	19943.33428	19787.09895	19631.21996	19475.68122
D × 102	3.00340	3.00462	3.00599	3.00752	3.00921	3.01107
H × 109	−5.39771	−5.39771	−5.39771	−5.39771	−5.39771	−5.39771
L × 1014	−6.31026	−6.31026	−6.31026	−6.31026	−6.31026	−6.31026
γ	−158.36234	−159.52580	−160.65808	−161.75780	−162.82358	−163.85406
γD × 104	−4.65958	−4.55782	−4.45605	−4.35429	−4.25253	−4.15077
λ	158240.10389	159146.34526	160075.72015	161030.45505	162012.89091	163025.52682
λD × 101	2.88112	2.92396	2.97232	3.02620	3.08560	3.15052
λH × 107	4.19274	4.19274	4.19274	4.19274	4.19274	4.19274
Isotopologue 34S18O
B	19508.35030	19359.99236	19211.99045	19064.33282	18917.00655	18769.99714
D × 102	2.78509	2.78619	2.78743	2.78882	2.79035	2.79202
H × 109	−4.82003	−4.82003	−4.82003	−4.82003	−4.82003	−4.82003
L × 1014	−5.42626	−5.42626	−5.42626	−5.42626	−5.42626	−5.42626
γ	−152.48777	−153.58775	−154.65885	−155.69981	−156.70939	−157.68634
γD × 104	−4.32177	−4.22917	−4.13657	−4.04397	−3.95136	−3.85876
λ	158231.33403	159120.23139	160031.34651	160966.77998	161928.73653	162919.56506
λD × 101	2.77408	2.81446	2.85996	2.91058	2.96633	3.02719
λH × 107	3.88798	3.88798	3.88798	3.88798	3.88798	3.88798

Notes. rms = 8.8978 MHz. Reduced standard deviation = 0.6791.

Download table as:  ASCIITypeset image

Table 12. Derived Parameters (in MHz) for³⁶SO in the Six Lowest Vibrational Levels of the Electronic Ground State

	v = 0	v = 1	v = 2	v = 3	v = 4	v = 5
Isotopologue 36S16O
B	20727.98664	20565.48218	20403.37916	20241.66385	20080.32103	19919.33356
D × 102	3.14502	3.14631	3.14777	3.14940	3.15120	3.15317
H × 109	−5.78394	−5.78394	−5.78394	−5.78394	−5.78394	−5.78394
L × 1014	−6.91936	−6.91936	−6.91936	−6.91936	−6.91936	−6.91936
γ	−162.05980	−163.26380	−164.43512	−165.57230	−166.67389	−167.73844
γD × 104	−4.87867	−4.77087	−4.66308	−4.55528	−4.44749	−4.33969
λ	158245.74142	159162.69970	160103.36959	161070.05823	162065.19394	163091.37241
λD × 101	2.94850	2.99291	3.04310	3.09908	3.16083	3.22837
λH × 107	4.39043	4.39043	4.39043	4.39043	4.39043	4.39043
Isotopologue 36S17O
B	19882.13562	19729.48795	19577.20994	19425.28926	19273.71231	19122.46379
D × 102	2.89307	2.89423	2.89553	2.89699	2.89860	2.90036
H × 109	−5.10303	−5.10303	−5.10303	−5.10303	−5.10303	−5.10303
L × 1014	−5.85516	−5.85516	−5.85516	−5.85516	−5.85516	−5.85516
γ	−155.42104	−156.55257	−157.65409	−158.72427	−159.76181	−160.76539
γD × 104	−4.48887	−4.39176	−4.29464	−4.19753	−4.10042	−4.00331
λ	158235.92958	159133.48463	160053.71427	160998.78163	161970.95902	162972.66986
λD × 101	2.82752	2.86913	2.91605	2.96829	3.02584	3.08872
λH × 107	4.03872	4.03872	4.03872	4.03872	4.03872	4.03872
Isotopologue 36S18O
B	19133.51192	18989.41421	18845.65911	18702.23545	18559.13096	18416.33189
D × 102	2.67888	2.67992	2.68111	2.68242	2.68387	2.68546
H × 109	−4.54697	−4.54697	−4.54697	−4.54697	−4.54697	−4.54697
L × 1014	−5.02030	−5.02030	−5.02030	−5.02030	−5.02030	−5.02030
γ	−149.54677	−150.61541	−151.65628	−152.66820	−153.64997	−154.60040
γD × 104	−4.15740	−4.06919	−3.98098	−3.89277	−3.80456	−3.71635
λ	158227.08088	159107.13481	160008.94931	160934.56297	161886.11354	162865.87626
λD × 101	2.72049	2.75966	2.80376	2.85279	2.90674	2.96561
λH × 107	3.73972	3.73972	3.73972	3.73972	3.73972	3.73972

Notes. rms = 8.8978 MHz. Reduced standard deviation = 0.6791.

Download table as:  ASCIITypeset image

New pure rotational transitions of the SO radical and of its two most abundant isotopologues have been measured in the THz spectral region (up to 2.5 THz). The use of the cw-THz photomixing technique based on a FC allowed very high accuracy line frequency measurements. Reliable effective parameters have been obtained for these three isotopologues. We also performed a global mass independent fit including all pure rotational and ro-vibrational transitions reported in the literature on all observed isotopologues of SO.

Thanks to this new set of parameters it is now possible to predict with very high accuracy the frequencies of the ro-vibrational lines. This should enable the research of SO in the mid-IR using ground-based IR telescopes, space-based telescope archives (Infrared Space Observatory, Spitzer), and future space missions such as the James Webb Space Telescope. This set of parameters particularly well-adapted for the detection of SO lines in O-rich evolved stars or in molecular clouds in absorption against bright IR sources.

The authors gratefully acknowledge Caroline C. Womack for her helpful comments on the manuscript. The THz spectrometer was funded by the Communauté urbaine de Dunkerque, the Région Nord-Pas de Calais, the Ministére de l'éducation nationale, de l'enseignement supérieur et de la recherche, and European funds in the context of the IRENI (Institut de Recherche en ENvironnement Industriel) and InterregIVA "Cleantech" programs. J. Cernicharo thanks the Spanish MINECO program for funding support through grants CSD2009-00038 (CONSOLIDER program "ASTROMOL"), AYA2009-07304, and AYA2012-32032, and the European Research Council for funding under the European Union's Seventh Framework Program (FP/2007-2013) / ERC-2013-SyG, Grant Agreement No. 610256 NANOCOSMOS.