On the frequency of N{}_{2}H{}^{+} and N{}_{2}D{}^{+}Based on observations made with the IRAM 30-m and the GBT 100-m. IRAM is supported by INSU/CNRS (France), MPG (Germany), and IGN (Spain). GBT is run by the National Radio Astronomy Observatory which is a facility of the National Science Foundation operated under cooperative agreement by Associated Universities, Inc.

On the frequency of NH and NDthanks: Based on observations made with the IRAM 30-m and the GBT 100-m. IRAM is supported by INSU/CNRS (France), MPG (Germany), and IGN (Spain). GBT is run by the National Radio Astronomy Observatory which is a facility of the National Science Foundation operated under cooperative agreement by Associated Universities, Inc.

L. Pagani LERMA & UMR8112 du CNRS, Observatoire de Paris, 61, Av. de l’Observatoire, 75014 Paris, France
laurent.pagani@obspm.fr, marie-lise.dubernet@obspm.fr
   F. Daniel LERMA & UMR8112 du CNRS, Observatoire de Paris, 61, Av. de l’Observatoire, 75014 Paris, France
laurent.pagani@obspm.fr, marie-lise.dubernet@obspm.fr Department of Molecular and Infrared Astrophysics ( DAMIR), Consejo Superior de Investigaciones CientíÞcas (CSIC ), C/ Serrano 121, 28006 Madrid, Spain
daniel@damir.iem.csic.es
   M.L. Dubernet LERMA & UMR8112 du CNRS, Observatoire de Paris, 61, Av. de l’Observatoire, 75014 Paris, France
laurent.pagani@obspm.fr, marie-lise.dubernet@obspm.fr
received : 11/7/2008; accepted : 10/11/2008
Key Words.:
Molecular data – ISM : kinematics and dynamics – ISM : lines and bands – Radio lines : ISM
offprints: L.Pagani
Abstract

Context:Dynamical studies of prestellar cores search for small velocity differences between different tracers. The highest radiation frequency precision is therefore required for each of these species.

Aims:We want to adjust the frequency of the first three rotational transitions of NH and ND and extrapolate to the next three transitions.

Methods:NH and ND are compared to NH the frequency of which is more accurately known and which has the advantage to be spatially coexistent with NH and ND in dark cloud cores. With lines among the narrowests, and NH and NH emitting region among the largests, L183 is a good candidate to compare these species.

Results:A correction of 10 kHz for the NH (J:1–0) transition has been found ( 0.03Êkm s) and similar corrections, from a few m s up to 0.05 km sare reported for the other transitions (NH (J:3–2) and ND(J:1–0), (J:2–1), and (J:3–2)) compared to previous astronomical determinations. Einstein spontaneous decay coefficients (A) are included.

Conclusions:

1 Introduction

In the quest for star forming cores, kinematic studies play a crucial role, trying to unveil slowly contracting cores or fast collapsing ones, depending upon which theory we rely upon or at what moment along the evolutionary track the prestellar core is standing. As already discussed by Lee et al. (1999), the accurate knowledge of every species line frequency is of the uttermost importance to track small systematic velocity gradients in molecular clouds. Therefore, because these velocity shifts can be as small as a few tens of m s, millimeter line transitions should be known with a precision of at least 10 and ideally 10. Some species are easily measured in the laboratory, especially stable species like CO, NH, etc,. Others are unstable and more difficult to measure (such as OH, HD,…). One possibility in the latter case, is to compare the transitions of the species of interest with the transitions of another well-known species in dark cloud cores where the lines are narrow enough to be accurately measured. However, the obvious difficulty is to be sure that the two species share the same volume of the cloud and undergo the same macroscopic velocity shifts. Even though, the line opacities might be a problem if too different in presence of a velocity gradient : the two coexistent species might then emphasize different parts of the cloud, depending on the depth for which their respective opacity reaches 1. A problem of opacity was indeed met in the comparison of CS with CCS made by Kuiper et al. (1996) in their attempt to measure the frequency of the CS lines as discussed in Pagani et al. (2001).

Caselli et al. (1995) performed such a measurement for NH, comparing NH (J:1–0) line emission to the CH (J : 2–1) line emission in L1512, confirming a sizeable difference between laboratory measurement and astronomical observations. Dore et al. (2004) expanding on a previous work by Gerin et al. (2001) also calculated and observed the ND (J:1–0) transition in L183, and extrapolated to the higher ND transitions, (giving slightly different values compared to Gerin et al. 2001, for the J:2–1 and J:3–2 transitions). They aligned their ND (J:1–0) observation onto their NH (J:1–0) towards the same source with the same telescope. The NH rotational constant was itself redetermined from a new evaluation of the NH (J:1–0) frequency from a comparison with CO (J:1–0) in the L1512 cloud (see Dore et al. 2004, for more details). This new value gave an offset of -4.2 kHz from their previous determination.

While the direct comparison of the ND and NH lines is presently the best option to choose because NH forcibly exists where ND exists, the hypothesis that CH is also present in the same volume as NH is more questionable because of differential depletion problems. Dore et al. (2004) also note that using CO has the problem of tracing different regions but hoped for a null velocity shift between the two tracers. We think that a better possibility exists to measure accurately the frequency of NH, namely by taking NH as the frequency reference. NH and NH are clearly coexistent species in depleted prestellar cores (e.g. Tafalla et al. 2002, 2004), having a common chemical origin and showing similar extents in most cores.

In this Note, we present a detailed comparison of NH with NH and ND in L183, checking that the measurable velocity shifts across the core are the same for all three species to convince ourselves of their coexistence and the absence of opacity effect on the velocity peak position. Schmid–Burgk et al. (2004) developed a similar strategy in their study of HCO and CO hyperfine structure (hereafter HFS) towards another dark cloud, L1512, with similar very narrow linewidths. With these comparisons in hand, we give all corrections for the 5 most currently observed transitions together with their Einstein spontaneous decay coefficients (A), determine the best fitting rotational constants and compute the expected frequencies for the next 3 rotational transitions (J:4–3, 5–4, 6–5).

2 Observations

Figure 1: NH (J:1–0) (left) and NH (1,1) (right) integrated intensity maps. The dotted lines AA and BB indicate the profiles along which the velocity gradients are traced in Figs. 2 & 3. Reference position : = 155408.5 = -2°52′48″

The whole elongated dense core of L183 (reference position : = 155408.5 = -2°52′48″) has now been fully mapped wih the IRAM 30-m telescope in a series of observations spanning several years from November 2003 to July 2007. The NH and ND (J:1–0) lines have been fully mapped while the NH (J:3–2), ND (J:2–1) and (J:3–2) lines have been mapped mostly towards the main core and its elongated ridge and partly towards the peak of the northern core (see Pagani et al. 2004, 2005). All observations have been performed in frequency–switch mode. For the (J:1–0) lines, the frequency sampling is 10 kHz, 10 or 20 kHz for the (J:2–1) and 40 kHz for the (J:3–2) lines, providing comparable velocity resolution for all lines in the range 30–50 m s. Spatial resolution ranges from 33″at 77 GHz to 9″at 279 GHz. For all lines, the spatial sampling is 12″ for the main prestellar core and 15″for the southern extension and for the northern prestellar core. We use Caselli et al. (1995) and Dore et al. (2004) frequencies for NH and ND transitions, respectively.

We performed observations of NH (1,1) and (2,2) inversion lines towards the whole core at the new Green Bank 100-m telescope (GBT) in November 2006 and March 2007 with velocity sampling of 20 m s and a typical T of 50 K, in frequency–switch mode. The angular resolution (35″) is close to that of the 30-m for the low-frequency (J:1–0) NDline. The spatial sampling is 24″ all over the source. We use the accurate measurement of Kukolich (1967) for NH (1,1), namely  = 23 694 495 487 (48) Hz which is an average estimated from the whole HFS (see also Hougen 1972, who revisited the NH and NH frequencies. The reported accuracy is higher but the NH (1,1) frequency remains basically unchanged, namely  = 23 694 495 481 22 Hz). For this frequency, the two strongest hyperfine components have the following frequency offsets :

(FF : 2, 2,) = 10 463 Hz

(FF : 2, 2,) = -15 196 Hz

Samples of these spectra (NH, ND and NH) are displayed in Pagani et al. (2007).

3 Spatial coexistence of ammonia and diazenylium

Figure 2: NH, ND (J:1–0) and NH (1,1) line of sight velocity along the AA cut (see Fig. 1). The NH data are displayed with the original frequency (uncorrected) and with a correction of -41 m s. The uncorrected ND (J:1–0) points are consistent with the uncorrected NH points despite the different opacities

Though depletion of molecules was predicted in the 70’s, it was only a few years after the publication of the Caselli et al. (1995) paper on the frequency of NH that depletion was actually discovered and traced (e.g. Willacy et al. 1998). Therefore the hypothesis made by Caselli et al. (1995) that CH and NH are spatially coexistent is probably refutable as it is clear now that such heavy Carbon carrier should be depleted in the same region as CO which is the region where NH appears. Indeed, the detection of ND in L1512 as a large fraction of NH (Roberts & Millar 2007) is a clear sign of heavy depletion of other molecules. Therefore the velocity coincidence between these two species is questionable.

Ammonia and diazenylium have the same chemical origin, starting from N and are well-known to be coexistent as discussed by e.g. Tafalla et al. (2002, 2004). This is in particular true in L183 as can be seen in Fig. 1 (but not for CH which has a much smaller extent, mostly concentrated towards the northern prestellar core as can be seen in Swade 1989). Interestingly, the velocity along the dense filament is constantly changing (Fig. 2), evoking a flow towards the prestellar cores and the cut perpendicular to the filament (marked BB′ in Fig. 1) is suggesting a rotation of the filament around its vertical axis (Fig. 3). NH (1,1), NH and ND (J:1–0) all trace exactly the same gradients and it seems therefore compulsory that their velocities be identical as there is no obvious possibility that the velocity gradients be exactly parallel but offset from each other, especially in the probable case of the cylinder rotation. With present NH (J:1–0) frequency as given by Caselli et al. (1995), there is indeed a clear offset with respect to the NH velocity gradient, close to 40 m s (and to 26 m s compared to the new value in Dore et al. 2004). Note also that Amano et al. (2005) have reinterpreted Caselli et al. (1995) observations along with new laboratory measurements but are therefore plagued by the velocity difference between NH and CH which appears to exist in view of the present discrepancy between NH and NH. Consequently, their best fit (#2 of their Table 2) is to be considered cautiously. Finally, the fact that ND velocity centroids are almost identical with those of NH indicates that the different opacities of the lines are not introducing any measurable bias here (though a very tiny shift is possibly visible in Fig. 3 where the ND displacement is symmetrically slightly less than the NH displacement).

In conclusion, the three species are spatially coexistent and trace the same velocities and one must adjust the frequencies of NH and ND to that of NH.

4 Frequency corrections

4.1 NH (J:1–0) correction

Frequency was measured using the MINIMIZE function in CLASS111http://www.iram.fr/IRAMFR/GILDAS with the HFS method for all species (for NH, the HFS method is similar to the internally built NH3(1,1) method). Because it is easier to deal with velocity offsets in CLASS, especially as we have to compare two species at different frequencies, the measurements have all been made in the velocity scale. Velocity differences are subsequently converted into frequency offsets using the approximate doppler shift formula (), being the velocity offset, c the celerity of light, and the corrected and original frequencies). The HFS method, in order to fit all the hyperfine components individually requires that we provide their list with their relative velocities and relative weights, these parameters not being adjusted during the fit. Therefore, we have used the detailed HFS provided by Caselli et al. (1995), Dore et al. (2004) and Kukolich (1967). Since an accurate determination of the hyperfine spectroscopic constants depends only slightly on the adopted rotational constants B and D222indeed, it can be noted that the HFS splitting is in first approximation identical for both NH and NDdespite a large, 20% variation in B rotational constant, we can safely use the previously determined ones. Doing so, our own determination for the relative velocity offsets between the hyperfine components in the J:1–0 line agree with Caselli et al. (1995) with a typical dispersion of 0.7 kHz. Though this is twice as much as the r.m.s. error on our frequency determination of each individual component ( 0.3 kHz), we find that using their offsets or ours, introduces a negligible difference of 0.13 kHz in the J:1–0 transition frequency determination, which is comparable to the r.m.s. error of the fit (0.12 kHz). We also did not find an improvement on the r.m.s. error of the fit itself. For the NH and ND transitions, the strongest hyperfine transition was given null velocity offset as it was also the strongest hyperfine transition frequency which was used to tune the receivers. The advantage of a complex and strong HFS is that it lowers the uncertainty on the velocity fit, compared to a single line estimate (fitting individually the NH J:1–0 lines with independent gaussians, gives errors between 0.85 and 1.2 m s instead of 0.38 m s with the global HFS fit for the reference spectrum).

Though the reference position has been observed often enough to get very high signal-to-noise ratios for most transitions, it seems more secure to measure the offset between NH and NH on all common positions (every other position in the central core, a few positions in the rest of the cloud) and measure the average difference. We have identified 65 common positions with sufficient signal-to-noise ratios and we have obtained the dispersion histogram of the velocity difference (Fig. 4). Fitting the histogram with a gaussian, we find a velocity difference of 40.8 m s with a dispersion  = 12.9 m s. This corresponds to a frequency correction of -13 4 kHz (or -8.8 kHz compared to Dore et al. 2004). For the reference position alone, the difference is also 40.8 m s with an error  = 0.56 m s (due to the very high signal to noise ratio obtained for both lines towards that position).

Figure 3: NH, ND (J:1–0) and NH (1,1) line of sight velocity along the BB cut (see Fig. 1). The NH data are displayed with the original frequency (uncorrected) and with a correction of -41 m s. The uncorrected ND (J:1–0) points are consistent with the uncorrected NH points despite the different opacities
Figure 4: NH (J:1–0) and NH (1,1) line of sight velocity difference histogram. The gaussian fit is centered on 40.8 m s with a dispersion  = 12.9 m s

4.2 NH (J:3–2) correction

For the NH (J:3–2) transition, only the reference position has been observed with a reasonably good signal-to-noise ratio (10). Therefore, we can only make a direct comparison for this position. The Jet Propulsion Laboratory (JPL) catalogue frequency for this line (279 511.701 0.05 MHz) is too vague to be useful for a precise velocity determination. The Cologne Database for Molecular Spectroscopy (CDMS) catalogue gives = 279 511.8577 MHz for the (FF: 4,5–3,4) strongest hyperfine component based on various works while Crapsi et al. (2005) give 279 511.863 MHz determined from the new rotational and centrifugal distorsion constants from Dore et al. (2004). These new values are respectively 26 and 31 kHz above our own determination.

4.3 ND corrections

For all three transitions of ND, we took advantage of the similar sampling with NH (J:1–0) to have a larger number of comparison points. We obtained 83, 73 and 51 comparison points with sufficient signal-to-noise ratio between NH (J:1–0) (using Caselli et al. 1995, frequency) and ND (J:1–0), (J:2–1), and (J:3–2) transitions respectively. The gaussian fit to each histogram yielded :

(J:1–0) : -5.1 m s ( = 10.5 m s)
(J:2–1) : 12.5 m s ( = 14.4 m s)
(J:3–2) : 18.4 m s ( = 8.1 m s)

The corresponding correction with respect to NH(1,1) is :

(J:1–0) : 35.7 m s or -9.2 ( 2.7) kHz
(J:2–1) : 53.3 m s or -27 ( 7.4) kHz
(J:3–2) : 59.2 m s or -49 ( 6.7) kHz

Direct comparison of the reference position with NH(1,1) spectrum yields :

(J:1–0) : 37.7 m s ( = 0.85 m s)
(J:2–1) : 47.7 m s ( = 0.92 m s)
(J:3–2) : 63.6 m s ( = 4.7 m s)

4.4 Rotational constants and Einstein–A coefficients

Except for the NH (J:3–2) line which has only one measurement, we have used the averaged comparisons for correcting the frequencies of all these transitions.

We have derived from these new frequencies the rotation (B) and centrifugal distortion (D) constants for NH and ND, using the hyperfine constants given by Caselli et al. (1995) and Dore et al. (2004) respectively. The error budget has been estimated by adding 1 to one of the frequency measurements and subtracting 1 to the other which we use to determine B and D, e.g. +2.7 kHz to the ND (J:1–0) line and -6.7 kHz for the ND (J:3–2) line. For the NH (J:3–2) transition, as we have only one measurement, we have taken the average of the 1 dispersion for all the other transition measurements as a probable dispersion for that measurement if we had had as many observations. We have found an average velocity dispersion of 11.5 m s which corresponds to 10.7 kHz at that frequency. The new constants are listed in Table 1. As expected from the fact that Amano et al. (2005) make use of the Caselli et al. (1995) frequency determination of NH and the related Dore et al. (2004) ND measurements, their rotational constants are different from ours by an amount directly related to the difference between CH and NH velocity determinations. The difference (5.3 kHz for B(NH) and 9.2 kHz for B(ND)) is significantly larger than the error estimate (conservatively given to be 2.5 and 1.7 kHz respectively for us and 1.3 and 1.2 kHz for Amano et al. 2005). It would be interesting to repeat Amano et al. (2005) analysis with our new frequency determinations to better secure these values.

Line strengths, from which Einstein–A coefficients are defined, are determined from the reduced transition matrix elements of the dipole moment operator:

(1)

where and are the wave–functions of the two levels involved in the radiative transition. In the case of hyperfine structures, the wave–functions can be defined according to an expansion on Hund’s case (b) wave–functions, the coefficients being determined by diagonalisation of the hyperfine Hamiltonian. In the case of NH and ND, the mixing of states is low so that a given hyperfine wave–function can be accuretaley defined as a pure Hund’s case (b) wave–function. Doing so, the line strengths can be expressed in a closed form (Gordy & Cook 1984), and for NH, the relevant expressions being given in Daniel et al. (2006). The Einstein–A coefficients are then given by:

(2)

(this is the same equation as in Daniel et al. 2006, but corrected for two typos)

The calculated line frequencies and A coefficients (the dipole moment – = 3.37 D – is taken from Botschwina 1984) are given in Tables 14 to 22 for all rotational transitions from (J:1–0) to (J:6–5) for both NH and ND. The frequency uncertainty is estimated by varying the rotational B and D constants by 1 .

Species B D
MHz MHz
NH 46586.8713(25) 0.08796(24)
ND 38554.7479(17) 0.06181(15)
Table 1: Rotation (B) and centrifugal distortion (D) constants for NH and ND. Errors in parentheses are given for the last two digits

5 Conclusions

  1. New, more accurate rotational constants and line frequencies are given along with the detailed Einstein spontaneous coefficients (A) for each of the hyperfine components.

  2. The main prestellar core LSR velocity is 2.3670 (0.0004) km s.

Acknowledgements.
We thank an anonymous referee for her/his critical reading which helped to improve the manuscript.

References

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\onltab

2

. Frequency A (MHz) (s) 1 1 0 0 1 1 1 1 2 0 1 2 1 1 2 0 1 1 1 1 1 0 1 0 1 1 1 0 1 2 1 1 1 0 1 1 1 2 2 0 1 1 1 2 2 0 1 2 1 2 3 0 1 2 1 2 1 0 1 2 1 2 1 0 1 1 1 2 1 0 1 0 1 0 1 0 1 1 1 0 1 0 1 2 1 0 1 0 1 0

Table 14: Hyperfine components and A Einstein spontaneous emission coefficients of the (J:1–0) transition of NH. The frequency uncertainty is 4.0 kHz for all hyperfine components. Summing A over all the hyperfine components with the same frequency always give the same total A, 3.628 10 s. 3.628(-5) means 3.628 10
\onltab

3

Frequency A
(MHz) (s)
2 2 2 1 2 1
2 2 2 1 2 3
2 2 1 1 2 1
2 2 3 1 2 3
2 2 2 1 2 2
2 1 1 1 0 1
2 2 3 1 2 2
2 1 2 1 0 1
2 2 1 1 2 2
2 1 0 1 0 1
2 2 2 1 1 1
2 3 3 1 2 3
2 2 2 1 1 2
2 2 1 1 1 1
2 3 3 1 2 2
2 3 2 1 2 1
2 2 3 1 1 2
2 2 1 1 1 2
2 3 4 1 2 3
2 3 2 1 2 3
2 2 1 1 1 0
2 3 2 1 2 2
2 1 1 1 2 1
2 1 2 1 2 1
2 1 2 1 2 3
2 1 0 1 2 1
2 1 1 1 2 2
2 1 2 1 2 2
2 1 1 1 1 1
2 1 1 1 1 2
2 1 2 1 1 1
2 1 2 1 1 2
2 1 1 1 1 0
2 1 0 1 1 1

Table 15: Hyperfine components and A Einstein spontaneous emission coefficients of the (J:2–1) transition of NH. The frequency uncertainty is 2.3 kHz for all hyperfine components.
\onltab

4

Frequency A
(MHz) (s)
3 3 3 2 3 2
3 3 3 2 3 4
3 3 2 2 3 2
3 3 4 2 3 4
3 3 3 2 3 3
3 3 4 2 3 3
3 3 2 2 3 3
3 2 2 2 1 2
3 2 2 2 1 1
3 2 1 2 1 0
3 4 4 2 3 4
3 3 3 2 2 3
3 2 3 2 1 2
3 2 1 2 1 2
3 3 3 2 2 2
3 3 2 2 2 1
3 3 4 2 2 3
3 4 3 2 3 2
3 4 4 2 3 3
3 2 1 2 1 1
3 4 5 2 3 4
3 3 2 2 2 3
3 4 3 2 3 4
3 2 2 2 3 2
3 3 2 2 2 2
3 2 3 2 3 2
3 4 3 2 3 3
3 2 3 2 3 4
3 2 1 2 3 2
3 2 2 2 3 3
3 2 3 2 3 3
3 2 2 2 2 1
3 2 2 2 2 3
3 2 2 2 2 2
3 2 3 2 2 3
3 2 1 2 2 1
3 2 3 2 2 2
3 2 1 2 2 2
Table 16: Hyperfine components and A Einstein spontaneous emission coefficients of the (J:3–2) transition of NH. The frequency uncertainty is 11 kHz for all hyperfine components.
\onltab

5

Frequency A
(MHz) (s)
4 4 4 3 4 3
4 4 4 3 4 5
4 4 3 3 4 3
4 4 5 3 4 5
4 4 4 3 4 4
4 4 5 3 4 4
4 4 3 3 4 4
4 3 3 3 2 3
4 5 5 3 4 5
4 4 4 3 3 4
4 3 3 3 2 2
4 3 2 3 2 1
4 3 4 3 2 3
4 3 3 3 4 3
4 3 2 3 2 3
4 4 4 3 3 3
4 4 3 3 3 2
4 4 5 3 3 4
4 5 4 3 4 3
4 5 5 3 4 4
4 4 3 3 3 4
4 5 6 3 4 5
4 5 4 3 4 5
4 3 2 3 2 2
4 3 4 3 4 3
4 4 3 3 3 3
4 3 4 3 4 5
4 3 2 3 4 3
4 3 3 3 4 4
4 5 4 3 4 4
4 3 4 3 4 4
4 3 3 3 3 2
4 3 3 3 3 4
4 3 3 3 3 3
4 3 4 3 3 4
4 3 2 3 3 2
4 3 4 3 3 3
4 3 2 3 3 3
Table 17: Hyperfine components and A Einstein spontaneous emission coefficients of the (J:4–3) transition of NH. The frequency uncertainty is 41 kHz for all hyperfine components.
\onltab

6

Frequency A
(MHz) (s)
5 5 5 4 5 4
5 5 5 4 5 6
5 5 4 4 5 4
5 5 6 4 5 6
5 5 5 4 5 5
5 5 4 4 5 5
5 5 6 4 5 5
5 4 4 4 3 4
5 5 5 4 4 5
5 6 6 4 5 6
5 4 4 4 5 4
5 4 3 4 3 2
5 4 4 4 3 3
5 4 5 4 3 4
5 4 3 4 3 4
5 5 5 4 4 4
5 5 4 4 4 3
5 5 4 4 4 5
5 5 6 4 4 5
5 6 5 4 5 4
5 6 6 4 5 5
5 6 5 4 5 6
5 6 7 4 5 6
5 4 5 4 5 4
5 4 3 4 5 4
5 4 4 4 5 5
5 4 5 4 5 6
5 4 3 4 3 3
5 5 4 4 4 4
5 6 5 4 5 5
5 4 5 4 5 5
5 4 4 4 4 3
5 4 4 4 4 5
5 4 4 4 4 4
5 4 3 4 4 3
5 4 5 4 4 5
5 4 5 4 4 4
5 4 3 4 4 4
Table 18: Hyperfine components and A Einstein spontaneous emission coefficients of the (J:5–4) transition of NH. The frequency uncertainty is 95 kHz for all hyperfine components.
\onltab

7

Frequency A
(MHz) (s)
6 6 6 5 6 7
6 6 6 5 6 5
6 6 5 5 6 5
6 6 7 5 6 7
6 6 6 5 6 6
6 6 5 5 6 6
6 6 7 5 6 6
6 5 5 5 4 5
6 6 6 5 5 6
6 7 7 5 6 7
6 5 5 5 6 5
6 5 4 5 4 3
6 5 5 5 4 4
6 5 4 5 4 5
6 5 6 5 4 5
6 6 5 5 5 6
6 6 5 5 5 4
6 6 6 5 5 5
6 7 6 5 6 7
6 6 7 5 5 6
6 7 6 5 6 5
6 7 7 5 6 6
6 7 8 5 6 7
6 5 4 5 6 5
6 5 5 5 6 6
6 5 6 5 6 7
6 5 6 5 6 5
6 5 4 5 4 4
6 6 5 5 5 5
6 7 6 5 6 6
6 5 6 5 6 6
6 5 5 5 5 6
6 5 5 5 5 4
6 5 5 5 5 5
6 5 4 5 5 4
6 5 6 5 5 6
6 5 4 5 5 5
6 5 6 5 5 5
Table 19: Hyperfine components and A Einstein spontaneous emission coefficients of the (J:6–5) transition of NH. The frequency uncertainty is 0.18 MHz for all hyperfine components.
\onltab

8

Frequency A
(MHz) (s)
1 1 0 0 1 1
1 1 2 0 1 2
1 1 2 0 1 1
1 1 1 0 1 1
1 1 1 0 1 0
1 1 1 0 1 2
1 2 2 0 1 1
1 2 2 0 1 2
1 2 3 0 1 2
1 2 1 0 1 0
1 2 1 0 1 2
1 2 1 0 1 1
1 0 1 0 1 2
1 0 1 0 1 0
1 0 1 0 1 1
Table 20: Hyperfine components and A Einstein spontaneous emission coefficients of the (J:1–0) transition of ND. The frequency uncertainty is 2.8 kHz for all hyperfine components.
\onltab

9

Frequency A
(MHz) (s)
2 2 2 1 2 1
2 2 2 1 2 3
2 2 1 1 2 1
2 2 3 1 2 3
2 2 2 1 2 2
2 1 1 1 0 1
2 2 3 1 2 2
2 1 2 1 0 1
2 2 1 1 2 2
2 1 0 1 0 1
2 2 2 1 1 1
2 3 3 1 2 3
2 2 2 1 1 2
2 2 1 1 1 1
2 3 3 1 2 2
2 3 2 1 2 1
2 2 3 1 1 2
2 3 4 1 2 3
2 2 1 1 1 2
2 3 2 1 2 3
2 2 1 1 1 0
2 3 2 1 2 2
2 1 1 1 2 1
2 1 2 1 2 1
2 1 2 1 2 3
2 1 0 1 2 1
2 1 1 1 2 2
2 1 2 1 2 2
2 1 1 1 1 1
2 1 1 1 1 2
2 1 2 1 1 1
2 1 2 1 1 2
2 1 1 1 1 0
2 1 0 1 1 1
Table 21: Hyperfine components and A Einstein spontaneous emission coefficients of the (J:2–1) transition of ND. The frequency uncertainty is 2.1 kHz for all hyperfine components
\onltab

10

Frequency A
(MHz) (s)
3 3 3 2 3 2
3 3 3 2 3 4
3 3 2 2 3 2
3 3 4 2 3 4
3 3 3 2 3 3
3 3 4 2 3 3
3 3 2 2 3 3
3 2 2 2 1 2
3 2 2 2 1 1
3 2 1 2 1 0
3 4 4 2 3 4
3 3 3 2 2 3
3 2 3 2 1 2
3 2 1 2 1 2
3 3 3 2 2 2
3 3 2 2 2 1
3 3 4 2 2 3
3 4 4 2 3 3
3 4 3 2 3 2
3 2 1 2 1 1
3 4 5 2 3 4
3 3 2 2 2 3
3 4 3 2 3 4
3 2 2 2 3 2
3 3 2 2 2 2
3 2 3 2 3 2
3 4 3 2 3 3
3 2 3 2 3 4
3 2 1 2 3 2
3 2 2 2 3 3
3 2 3 2 3 3
3 2 2 2 2 1
3 2 2 2 2 3
3 2 2 2 2 2
3 2 3 2 2 3
3 2 1 2 2 1
3 2 3 2 2 2
3 2 1 2 2 2
Table 22: Hyperfine components and A Einstein spontaneous emission coefficients of the (J:3–2) transition of ND. The frequency uncertainty is 6.2 kHz for all hyperfine components
\onltab

11

Frequency A
(MHz) (s)
4 4 4 3 4 3
4 4 4 3 4 5
4 4 3 3 4 3
4 4 5 3 4 5
4 4 4 3 4 4
4 4 5 3 4 4
4 4 3 3 4 4
4 3 3 3 2 3
4 5 5 3 4 5
4 4 4 3 3 4
4 3 3 3 2 2
4 3 2 3 2 1
4 3 4 3 2 3
4 3 3 3 4 3
4 4 4 3 3 3
4 3 2 3 2 3
4 4 3 3 3 2
4 4 5 3 3 4
4 5 4 3 4 3
4 5 5 3 4 4
4 5 6 3 4 5
4 4 3 3 3 4
4 5 4 3 4 5
4 3 2 3 2 2
4 3 4 3 4 3
4 4 3 3 3 3
4 3 4 3 4 5
4 3 2 3 4 3
4 3 3 3 4 4
4 5 4 3 4 4
4 3 4 3 4 4
4 3 3 3 3 2
4 3 3 3 3 4
4 3 3 3 3 3
4 3 4 3 3 4
4 3 2 3 3 2
4 3 4 3 3 3
4 3 2 3 3 3
Table 23: Hyperfine components and A Einstein spontaneous emission coefficients of the (J:4–3) transition of ND. The frequency uncertainty is 25 kHz for all hyperfine components
\onltab

12

Frequency A
(MHz) (s)
5 5 5 4 5 4
5 5 5 4 5 6
5 5 4 4 5 4
5 5 6 4 5 6
5 5 5 4 5 5
5