11institutetext: Dept. de Astrofísica Molecular, Instituto de Física
Fundamental (IFF-CSIC), C/ Serrano 121, 28006 Madrid, Spain.
11email<EMAIL_ADDRESS>22institutetext: Observatorio Astronómico
Nacional (OAN, IGN), C/ Alfonso XII, 3, 28014, Madrid, Spain. 33institutetext:
Centro de Desarrollos Tecnológicos, Observatorio de Yebes (IGN), 19141 Yebes,
Guadalajara, Spain.

# Discovery of thiofulminic acid with the QUIJOTE1 line survey: A study of the
isomers of HNCS and HNCO in TMC-1††thanks: Based on observations with the
Yebes 40m radio telescope (projects 19A003, 20A014, 20D023, 21A011, 21D005,
22A007, 22B029, and 23A024) and the IRAM 30m radiotelescope. The 40m
radiotelescope at Yebes Observatory is operated by the Spanish Geographic
Institute (IGN, Ministerio de Transportes y Movilidad Sostenible). IRAM is
supported by INSU/CNRS (France), MPG (Germany), and IGN (Spain).

J. Cernicharo 11 M. Agúndez 11 C. Cabezas 11 B. Tercero 2233 R. Fuentetaja 11
N. Marcelino 2233 and P. de Vicente 33

(Received: 9 December 2023; accepted: 9 January 2024)

We present the first detection of HCNS (thiofulminic acid) in space with the
QUIJOTE1 line survey in the direction of TMC-1. We performed a complete study
of the isomers of CHNS and CHNO, including NCO and NCS. The derived column
densities for HCNS, HNCS, and HSCN are (9.0$\pm$0.5)$\times$109,
(3.2$\pm$0.1)$\times$1011, and (8.3$\pm$0.4)$\times$1011 cm-2, respectively.
The HNCS/HSCN abundance ratio is 0.38. The abundance ratios HNCO/HNCS,
HCNO/HCNS, HOCN/HSCN, and NCO/NCS are 34$\pm$4, 8.3$\pm$0.7, 0.18$\pm$0.03,
and 0.78$\pm$0.07, respectively. These ratios cannot be correctly reproduced
by our gas-phase chemical models, which suggests that formation paths for
these species are missing, and/or that the adopted dissociative recombination
rates for their protonated precursors have to be revised. The isotopologues
H15NCO, DNCO, HN13CO, DCNO, H34SCN, and DSCN have also been detected with the
ultrasensitive QUIJOTE line survey.

###### Key Words.:

molecular data — line: identification — ISM: molecules — ISM: individual
(TMC-1) — astrochemistry

##  1 Introduction

Among the new species discovered in recent years with the QUIJOTE111Q-band
Ultrasensitive Inspection Journey to the Obscure TMC-1 Environment line survey
of TMC-1 (Cernicharo et al. 2021a, 2023a, 2023b, and references therein), it
is worth noting that 11 of them are S-bearing species: NCS, HCCS, H2CCS,
H2CCCS, C4S, HCSCN, HCSCCH, HC4S, HC3S+, HCCS+, and HSO (Cernicharo et al.
2021b, c, d; Cabezas et al. 2022a; Fuentetaja et al. 2022; Marcelino et al.
2023). The chemical models for S-bearing molecules in interstellar clouds
poorly reproduce the observed abundances and the results are strongly
dependent on the adopted depletion of sulfur (Vidal et al. 2017). Moreover,
many reactions involving S+ with neutrals, as well as radicals with S and CS,
are missing in the chemical networks and further study is required in order to
achieve better agreement between chemical models and observations of sulfur-
bearing species (Petrie 1996; Bulut et al. 2021). In addition to these new
species, other S-bearing molecules such as CS, SO, NS, NS+, HCS+, HSC, HCS,
CCS, C3S, C5S, H2CS, HSCN, and HNCS are present in TMC-1 (See Table 1 in
Cernicharo et al. 2021b). Despite the high depletion of sulphur in molecular
clouds, most of these species are the counterpart of the same molecular
structure but replacing sulphur with oxygen. Only C4O, HCCO+, and HC4O have
not yet been detected in TMC-1, or indeed in any other source. HNCS (Frerking
et al. 1979) and HSCN (Halfen et al. 2009) are isomers of the CHNS family, and
obvious counterparts of the isomers of CHNO. For the latter family, HNCO
(Snyder & Buhl 1972), HCNO (Marcelino et al. 2009), and HOCN (Brünken et al.
2009a; Marcelino et al. 2010) have been detected in space in a broad sample of
cold and warm molecular clouds.

The study of isomers of the same molecule provides clues as to the formation
mechanism of these species, as often they are produced from a common
precursor. The recent detection of five cyano derivatives of propene, which
have energies ranging from 0 to 19.6 kJ/mol (0-2357 K) but similar abundances,
indicates that kinetics, and not energetics, is dominating the production of
these different isomers (Cernicharo et al. 2022a). The most relevant isomers
in space are HCN and HNC, which are assumed to be produced from the
dissociative electronic recombination ($DR$) of HCNH+. In cold dark clouds,
these isomers have an abundance ratio of $\sim$1 in spite of the much higher
energy of HNC of 55 kJ/mol (Baulch et al. 2005; Mendes et al. 2012, and
references therein). Their chemistry in interstellar clouds was analysed in
detail by Loison et al. (2014).

Concerning the CHNS and CHNO isomers, it is assumed that the first are mainly
formed from $DR$ of H2NCS+ and HNCSH+ (Adande et al. 2010; Gronowski & Kolos
2014), and the second from $DR$ of H2NCO+ and HNCOH+ (Marcelino et al. 2010;
Quan et al. 2010). In the present work, we present the first detection of HCNS
(thiofulminic acid) in space (TMC-1) and a coherent and precise study of the
abundances of the isomers of CHNS and CHNO. The isotopologues H34SCN, DSCN,
H15NCO, HN13CO, DNCO, and DCNO are also detected in TMC-1. The derived
abundances for the CHNS and CHNO isomers, together with those of NCS and NCO,
are compared with the results of state-of-the-art chemical models for
S-bearing species.

captionObserved $Ju=3$ and $4$ ($K_{a}=0$) transitions of the isomers of HCNS.
The bottom panel shows the hyperfine components of the $J=5/2-3/2$ transition
of NCS in its ${}^{2}\Pi_{3/2}$ ground state. The abscissa corresponds to the
rest frequency. The ordinate is the antenna temperature corrected for
atmospheric and telescope losses, in mK. Blanked channels correspond to
negative features produced when folding the frequency-switched data. The red
line shows the computed synthetic spectra for these lines (see Sect. 3). The
derived line parameters are given in Table 3.

##  2 Observations

The observational data used in this work are part of QUIJOTE (Cernicharo et
al. 2021a), a spectral line survey of TMC-1 in the Q-band carried out with the
Yebes 40m telescope at the position $\alpha_{J2000}=4^{\rm h}41^{\rm
m}41.9^{\rm s}$ and
$\delta_{J2000}=+25^{\circ}41^{\prime}27.0^{\prime\prime}$, corresponding to
the cyanopolyyne peak (CP) in TMC-1. A detailed description of the system is
given by Tercero et al. (2021), and details on the QUIJOTE line survey are
provided by Cernicharo et al. (2021a, 2023a, 2023b). The data analysis
procedure is described by Cernicharo et al. (2022b). The total observing time
on source is 1202 hours and the measured sensitivity varies between 0.07 mK at
32 GHz and 0.2 mK at 49.5 GHz.

The main beam efficiency can be given across the Q-Band by $B_{\rm eff}$=0.797
exp[$-$($\nu$(GHz)/71.1)2]. The forward telescope efficiency is 0.95 and the
beam size at half power intensity is 54.4′′ and 36.4′′ at 32.4 and 48.4 GHz,
respectively.

In addition to the Q-band data, the isomers of HNCO and HNCS were observed in
the millimetre domain with the IRAM 30m telescope and consist of a 3mm line
survey that covers the 71.6-116 GHz domain (Cernicharo et al. 2012; Agúndez et
al. 2022; Cabezas et al. 2022a; Cernicharo et al. 2023b). The final 3mm line
survey has a sensitivity of 2-10 mK. However, at some selected frequencies,
the sensitivity is as low as 0.6 mK. The IRAM 30m beam varies between 34′′ and
21′′ at 72 GHz and 117 GHz, respectively, while the beam efficiency takes
values of 0.83 and 0.78 at the same frequencies (Beff= 0.871
exp[$-$($\nu$(GHz)/359)2]). The forward efficiency at 3mm is 0.95.

The absolute intensity calibration uncertainty is 10$\%$. However, the
relative calibration between lines within the QUIJOTE survey is certainly
better because all them are observed simultaneously and have the same
calibration uncertainties. The data were analysed with the GILDAS
package222http://www.iram.fr/IRAMFR/GILDAS.

##  3 Results

Line identification in this work was performed using the MADEX code
(Cernicharo 2012), and the CDMS and JPL catalogues (Müller et al. 2005;
Pickett et al. 1998). The references for the laboratory data used by MADEX in
its spectroscopic predictions are described below. The intensity scale used in
this study is the antenna temperature ($T_{A}^{*}$). Consequently, the
telescope parameters and source properties were used when modelling the
emission of the different species in order to produce synthetic spectra in
this temperature scale. The permanent dipolar moments of the molecular species
studied in this work are given in Table 1. In this work, we assumed a velocity
for the source relative to the local standard of rest of 5.83 km s-1
(Cernicharo et al. 2020). The source is assumed to be circular with a uniform
brightness temperature and a radius of 40′′ (Fossé et al. 2001). The procedure
to derive line parameters is described in Appendix A. The observed line
intensities were modelled using a local thermodynamical equilibrium hypothesis
(LTE), or a large velocity gradient approach (LVG). In the latter case, MADEX
uses the formalism described by Goldreich & Kwan (1974).

Table 1: Derived column densities for the isomers of HNCO and HNCS

Species | Trot | $\mu_{a}$ | $N^{A}$ | Notes
---|---|---|---|---
| (K) | (D) | (cm-2) |
HCNS | 7.0 | 3.85 | (9.0$\pm$0.5)$\times$1009 | a
HNCS | 6.0 | 1.64 | (3.2$\pm$0.1)$\times$1011 | b
HSCN | 5.2 | 3.33 | (8.3$\pm$0.4)$\times$1011 | c
HSNC | 5.2 | 3.13 |  $\leq$4$\times$1009 | a
NCS | 5.5 | 2.45 | (9.5$\pm$0.2)$\times$1011 | d
HNCO | 8.0 | 1.58 | (1.1$\pm$0.1)$\times$1013 | d
HCNO | 5.5 | 3.10 | (7.8$\pm$0.2)$\times$1010 | e
HOCN | 5.5 | 3.70 | (1.5$\pm$0.2)$\times$1011 | f
HONC | 5.5 | 3.13 |  $\leq$5$\times$1009 | g
NCO | 8.0 | 0.64 | (7.4$\pm$0.5)$\times$1011 | h
H2NCO+ | 5.5 | 4.13 | (1.8$\pm$0.4)$\times$1010 | i
H34SCN | 5.2 | 3.33 | (2.1$\pm$0.3)$\times$1010 | j
DSCN | 5.2 | 3.33 | (2.1$\pm$0.3)$\times$1010 | j
H15NCO | 8.0 | 1.58 | (3.3$\pm$0.3)$\times$1010 | j
HN13CO | 8.0 | 1.58 | (2.1$\pm$0.1)$\times$1011 | j
DNCO | 8.0 | 1.58 | (2.5$\pm$0.2)$\times$1011 | j
DCNO | 5.5 | 3.10 | (3.1$\pm$0.4)$\times$1009 | j
333 $A$$A$footnotetext: Column densities for the isomers of HCNS, together
with the most abundant S-bearing species, were reported by Cernicharo et al.
(2021b). The present estimations are $\sim$20% larger due to the recalibration
of the QUIJOTE data in 2023 (Cernicharo et al. 2023a). $a$$a$footnotetext:
$\mu_{a}$ from McGuire et al. (2016). $b$$b$footnotetext: $\mu_{a}$ from
Szalanski et al. (1978). $c$$c$footnotetext: $\mu_{a}$ from Woon & Herbst
(2009). $d$$d$footnotetext: $\mu_{a}$ from CDMS (Müller et al. 2005).
$e$$e$footnotetext: $\mu_{a}$ from Takashi et al. (1989). $f$$f$footnotetext:
$\mu_{a}$ from Brünken et al. (2009a). $g$$g$footnotetext: $\mu_{a}$ from
Mladenovic et al. (2009). $h$$h$footnotetext: $\mu_{a}$ from Saito & Amano
(1970). $i$$i$footnotetext: $\mu_{a}$ from Gupta et al. (2013).
$j$$j$footnotetext: Assumed identical to that of the main isotopologue.

###  3.1 The HCNS, HSCN, and HNCS isomers

The HCNS, HNCS, and HSCN isomers were observed with high spectral resolution
in the laboratory (Brünken et al. 2009b; McGuire et al. 2016). The frequencies
measured in these studies were fitted using the adapted Hamiltonian for each
species and the resulting rotational constants were implemented into the MADEX
code to predict transition frequencies and to model the observed emission line
profiles.

HNCS and HSCN are well-known molecular components of interstellar clouds
(Frerking et al. 1979; Halfen et al. 2009; Adande et al. 2010; Vastel et al.
2018). They harbour strong lines in our line survey. However, the isomer HCNS
has not been detected yet in space. It is the thio-form of HCNO, an isomer of
HNCO and a molecule detected in several interstellar clouds (Marcelino et al.
2009). Thanks to the sensitivity of QUIJOTE, we report in this work the first
detection of its two lines in the Q-band ($J=3-2$ and $J=4-3$) in space,
namely towards TMC-1. The lines are detected with a signal-to-noise ratio
(S/N) of $\sim$12 and match the predicted frequencies within 3 kHz (see Fig.
1). The derived line parameters are given in Table 3. The sensitivity of our
line survey at 3mm with the IRAM 30m telescope is not as high as that of
QUIJOTE. Nevertheless, the $J=6-5$ line of HCNS at 73815.849 MHz is detected
at a 3$\sigma$ level in the data (see Fig. 2). We also observed this line with
the Yebes 40m telescope using the W-band receiver described by Tercero et al.
(2021). Near the frequency of the HCNS $J=6-5$ transition, there is a line of
CCC13CH that is detected with good S/N with both telescopes. It has an
intensity of 10 mK and has been used to scale the data of the Yebes 40m and
the IRAM 30m telescopes. The two sets of data were merged, weighting them as
1/$\sigma^{2}$. The resulting data allow us to detect the $J=6-5$ line of HCNS
at 4$\sigma$. The results are shown in Fig. 2. We conclude that HCNS is
convincingly detected in TMC-1. A fit to the laboratory and astronomical
frequencies for this species provides the following new molecular constants:
$B$=6151.44367$\pm$0.00027 MHz, $D$=1.712$\pm$0.016 kHz, and
$eQq$=-0.7227$\pm$0.0018 MHz.

From the observed intensities of these three lines of HCNS, and using a line
profile fit method (Cernicharo et al. 2021d), we obtain Trot=7.0$\pm$0.5 K and
N(HCNS)=(9.0$\pm$0.5)$\times$109 cm-2. The synthetic spectra are shown in Fig.
1 and 2, and are in excellent agreement with the observations. No collisional
rates are available for HCNS, but with the aim to estimate the excitation
conditions that could be expected for this molecule in the LVG approximation,
we adopted the collisional rates of OCS with H2 calculated by Green & Chapman
(1978). OCS has a rotational constant similar to that of HCNS, and therefore
we consider that these rates provide a better approximation to those of HCNS
than the rates of HCNO (Naindouba et al.  2015), which has a rotational
constant twice that of HCNS. For the typical volume densities of TMC-1 of
1$\times$104 cm-3 (Agundez et al. 2023, Fuentetaja et al., in prep.), we
obtain that the excitation temperatures of the observed lines could vary
between 8.2 K for the $J=3-2$ line down to 6.2 K for the $J=6-5$ transition,
in good agreement with the derived averaged rotational temperature of 7 K.

The $J_{u}=3,4$ with $K_{a}=0$ transitions of HNCS and HSCN appear in our
QUIJOTE data as features exhibiting all their hyperfine components with a very
high S/N. These transitions are shown in Fig. 1. The corresponding $K_{a}=1$
lines of HNCS have upper energy levels of 66 ($J_{u}=3$) and 68 K ($J_{u}=4$)
and are not detected in our data. However, the corresponding energies for HSCN
are only at 16.9 and 19.2 K, respectively. We detected these latter; they are
shown in Fig. 3. Moreover, the $J_{u}=7,8$ transitions with $K_{a}$=0 of both
isomers were detected with the IRAM 30m telescope (see Fig. 3). The derived
line parameters for all these lines are given in Table 3. No collisional rates
are available for either of the two isomers. We used all observed lines using
the line profile fitting procedure (Cernicharo et al. 2021d) to derive their
rotational temperature and column densities. For HNCS, we obtain
Trot=6.0$\pm$0.5 K and N(HNCS)=(3.2$\pm$0.1)$\times$1011 cm-2. For HSCN, we
derived Trot=5.2$\pm$0.3 K and N(HSCN, $K_{a}$=0) = (6.9$\pm$0.2)$\times$1011
cm-2, and N(HSCN, $K_{a}$=1) = (1.4$\pm$0.3)$\times$1011 cm-2. The interladder
temperature between $K_{a}$=0 and 1 lines could be thermalised by collisions
at the kinetic temperature of the cloud. For an energy separation between
these ladders of 14.5 K, we derive a kinetic temperature for TMC-1 of
9.1$\pm$1.5 K, which is in excellent agreement with the result obtained by
Agundez et al. (2023).

The isomer HSNC has been characterised in the laboratory (McGuire et al. 2016)
and we searched for its lines in our survey. However, we only obtained an
upper limit of 4$\times$109 cm-2 for its column density. Of particular
interest for the chemistry of the CHNS isomers, it is worth deriving the
column density of NCS as it could participate in their formation. NCS was
discovered in TMC-1 with previous QUIJOTE line data (Cernicharo et al. 2021b).
The new data are shown in Fig. 1. Adopting a rotational temperature of 5.5 K,
the derived column density is (9.5$\pm$0.2)$\times$1011 cm-2. The NCS to HNCS
abundance ratio is 3.0$\pm$0.16.

Among the isomers of the CHNS family, the most abundant is HSCN. Its lines are
strong enough to allow a search for its isotopologues. We observed two
rotational transitions exhibiting hyperfine structure for H34SCN and DSCN (see
Fig. 4). Adopting the same rotational temperature as for the main
isotopologue, we derive the column densities given in Table 1. The 32S/34S
abundance ratio is 33$\pm$5, which agrees with previous determinations using
other molecular species (see e.g. Cernicharo et al. 2021b and Fuentetaja et
al. in prep., and references therein). The H/D abundance ratio is also
33$\pm$5 and represents a considerable single deuterium enhancement in HSCN,
in line with that observed in other molecules in TMC-1 (Cabezas et al. 2021,
2022b, and references therein).

###  3.2 The HNCO, HCNO, and HOCN isomers

HNCO was one of the first molecules detected in the early years of millimetre
radio astronomy by Snyder & Buhl (1972). Its isomer HCNO (fulminic acid) was
detected by Marcelino et al. (2009) towards B1, L1544, and L183, but not in
TMC-1. Another isomer, HOCN, was detected towards SgrB2, B1-b, L1544, L1527,
L183, and IRAS16293-2422 (Brünken et al. 2009a, 2010; Marcelino et al. 2010).
Therefore, these isomers have been detected in space in a broad sample of cold
and warm molecular clouds. The level energies, transition frequencies, and
intensities for HNCO and its isotopologues, as well as for HOCN, are from the
JPL and CDMS catalogues, respectively. HCNO is a quasi-linear molecule and its
rotational frequencies were obtained from a fit to available laboratory data
(Winnewisser & Winnewisser 1971). The three isomers and the isotopologues 15N,
13C, and D of HNCO have been detected with QUIJOTE in TMC-1. The data are
shown in Fig. 5 and the derived line parameters are given in Table 4. We note
that for HN13CO and DNCO, our measured frequencies have a significant lower
uncertainty than those in the entries of the JPL catalogue.

The HNCO lines are the strongest among those of all its isomers. We also
detected the $4_{04}-3_{03}$ and $5_{05}-4_{04}$ lines at 3mm with the IRAM
30m radio telescope. These lines are shown in Fig. 6. Collisional rates for
HNCO with He (T${}_{K}\geq$30 K; Green  1986) and H2 (T${}_{K}\geq$7 K;
Sahnoun et al. 2018) are available. For a volume density of 1-2$\times$104,
our LVG calculations using the $p$-H2 collisional rates provide excitation
temperatures for the $2_{0,2}-1_{0,1}$ line of HNCO of 8-9 K, decreasing to
$\sim$5 K for the $5_{05}-4_{04}$ transition. We therefore adopted a
rotational temperature of 8 K for HNCO and its isotopologues in the Q-band,
and of 4.8 K in the 3mm domain. The derived column densities are given in
Table 1 and the computed synthetic spectra are shown in red in Figures 5 and
6. The HNCO/HN13CO, HNCO/DNCO, and HNCO/H15NCO abundance ratios are 52$\pm$8,
44$\pm$8, and 333$\pm$60.

Table 2: Derived abundance ratios for the isomers of HNCO and HNCS in TMC-1

S-species | N(X)/N(HNCS) | O-species | N(X)/N(HNCO) | N(S)/N(O)
---|---|---|---|---
HCNS | 0.028$\pm$0.002 | HCNO | 0.0071$\pm$0.0008 | 0.12$\pm$0.01
HNCS | 1.0 | HNCO | 1.0 | 0.029$\pm$0.004
HSCN | 2.60$\pm$0.016 | HOCN | 0.014$\pm$0.003 | 5.52$\pm$0.91
HSNC |  $\leq$0.013 | HONC |  $\leq$0.0005 |
NCS | 3.00$\pm$0.16 | NCO | 0.067$\pm$0.011 | 1.28$\pm$0.11
NSa | 5.3$\pm$0.7 | NOb | 25$\pm$6 | 0.007$\pm$0.002
|  | H2NCO+ | 0.0016$\pm$0.0005 |
444 $a$$a$footnotetext: Column density of (1.7$\pm$0.2)$\times$1012 cm-2
(Cernicharo et al. 2018). $b$$b$footnotetext: Column density of
(2.7$\pm$0.5)$\times$1014 cm-2 (Gerin et al. 1993).

The isomers HCNO and HOCN have dipolar moments of 3.1 and 3.7 D, respectively
(see Table 1). For HCNO we used the LVG approximation using the collisional
rates of HCNO with He calculated by Naindouba et al.  (2015). For a density of
1-2$\times$104 cm-3, we obtain rotational temperatures in the range of 4.9-5.5
K. HCNO is detected for the first time in TMC-1. We also observed its $J=4-3$
transition at 3mm (see Fig. 3). The estimated column density for this isomer
of HNCO is (7.8$\pm$0.2)$\times$1010 cm-2. We also observed the $J=2-1$ of
DCNO in the Q-band (see Fig. 6). The derived HCNO/DCNO abundance ratio is
25$\pm$4\. For HOCN, we observed the $2_{0,2}-1_{0,1}$ with QUIJOTE and its
$4_{04}-3_{03}$ and $5_{05}-4_{04}$ lines at 3mm (see Fig. 6). All the
observed lines are well reproduced by the models with a rotational temperature
similar to that of HCNO ($T_{rot}$=5.5 K) as shown by the computed synthetic
spectra shown in Figures 5 and 6. The derived column densities for HCNO and
HOCN are given in Table 1. The fundamental transition of HOCN at 20.9 GHz was
tentatively observed in TMC-1 by Brünken et al. (2009b).

We detected the four strongest hyperfine components of the $J=7/2-5/2$
transition of NCO in its ${}^{2}\Pi_{3/2}$ ground state at 3mm. This species
was recently detected towards L483 by Marcelino et al. (2018) with an NCO/HNCO
abundance ratio of $\sim$0.2. Due to the modest dipole moment of this species
($\mu$=0.64 D; Saito & Amano 1970), we modelled the TMC-1 data adopting
Trot(NCO)=8 K. The derived column density is (7.4$\pm$0.5)$\times$1011 cm-2
and the NCO/HNCO abundance ratio is 0.067, which is three times lower than
towards L483. However, in TMC-1 the NCS/HNCS ratio is 45 times larger than the
NCO/HNCO one. We also searched for the isomer HONC (Mladenovic et al. 2009),
but only obtained upper limits (see Table 1). Finally, three lines of H2NCO+
were observed in the Q-band. These are shown in Fig. 7 and their line
parameters are given in Table 4. Adopting a rotational temperature of 5.5 K
and a $o$/$p$ abundance ratio of 3:1, the derived total column density for
this species is (1.8$\pm$0.4)$\times$1010 cm-2.

Figure 1: Abundances calculated with the chemical model for molecules of the
families of HNCO and HNCS. The peak calculated abundance of HCNS is around
10-15 relative to H2 and therefore lies outside the abundance range shown.
Horizontal dashed lines correspond to the abundances observed in TMC-1.

##  4 Discussion

The most energetically stable isomer of the CHNS family is HNCS, followed by
HSCN, HCNS, and HSNC at 3171, 17317, and 18120 K above, respectively
(Wierzejewska & Moc 2003; McGuire et al. 2016). The energy ordering in the
oxygen family is similar, with HNCO being the most stable isomer, and HOCN,
HCNO, and HONC lying at 12330, 34480, and 42235 K above, respectively
(Schuurman et al. 2003). In spite of the similarities in the energetics, the
CHNO and CHNS families show a very different behaviour in TMC-1. While HNCO is
far more abundant than HOCN, the abundance of HNCS is only half of that of
HSCN (see Table 2).

The chemistry of HNCS and its isomers has been discussed by Adande et al.
(2010), Gronowski & Kolos (2014), and Vidal et al. (2017). Here, we revisit
the subject by running a gas-phase chemical model of a cold dense cloud (see
details in Cernicharo et al. 2021b and Fuentetaja et al. 2022). The chemical
scheme of synthesis of HNCS and HSCN can be summarized as

$\rm XH^{+}+NCS\rightarrow HNCS^{+}+X,$ (1) $\rm HNCS^{+}+H_{2}\rightarrow
H_{2}NCS^{+}/HNCSH^{+}+H,$ (2) $\rm H_{2}NCS^{+}+e^{-}\rightarrow HNCS+H,$ (3)
$\rm HNCSH^{+}+e^{-}\rightarrow HNCS/HSCN+H,$ (4)

where the scheme begins with the protonation of NCS by abundant proton donors
XH+, such as HCO+, H3O+, and H${}_{3}^{+}$. This scheme, replacing sulfur by
oxygen, is the same one that describes the formation of the oxygen analogues
HNCO and HOCN (Marcelino et al. 2010; Quan et al. 2010; Lattanzi et al. 2012).
The chemical schema start with the radicals NCO and NCS, which are mostly
formed through neutral–neutral reactions. According to the chemical model, the
main reactions forming NCO are N + HCO and CN + O2, while NCS is mainly formed
through the analogous reactions N + HCS and CN + SO.

The abundances calculated with the chemical model are shown in Fig. 1. We
focus on times around a few 105 yr, when most molecules reach their peak
abundances. The abundances calculated for NCO and HNCO are in line with the
observed values. In the case of HOCN, the calculated abundance is close to
that of HNCO, while observations indicate that HOCN is much less abundant than
HNCO. It is likely that the dissociative recombination of HNCOH+ favours HNCO
over HOCN (the model assumes equal branching ratios) or that the reaction
between HNCO+ and H2 favours H2NCO+ over HNCOH+ (equal yields are assumed at
present). Investigation into these two reactions would allow us to understand
the high observed HNCO/HOCN ratio. Grain surface chemistry may also be
responsible for this. Indeed, the gas-grain model of Quan et al. (2010)
predicted HNCO to be somewhat more abundant than HCNO. The detection of the
precursors H2NCO+ and HNCOH+ would also be very useful to shed light on this.
While the H2NCO+ cation has been detected in interstellar clouds (Gupta et al.
2013; Marcelino et al. 2018), its isomer HNCOH+ is still awaiting detection in
space (laboratory frequencies are available from Mladenovic et al. 2009). Only
the H2NCO+ cation is detected in our TMC-1 data.

In the case of the sulfur analogues NCS, HNCS, and HSCN, the model severely
underestimates their abundances. It is likely that the model lacks efficient
routes to NCS, hampering the subsequent formation of HNCS and HSCN. A similar
situation was encountered in previous chemical models, with abundances
relative to H2 $\lesssim$ 10-12 for HNCS and HSCN (Adande et al. 2010;
Gronowski & Kolos 2014; Vidal et al. 2017). Apart from the general problem of
abundance underestimation, the fact that HSCN is observed to be more abundant
than HNCS, in contrast to the oxygen case, indicates that the branching ratios
of reactions (2) and (4) must be markedly different to those of the analogous
oxygen reactions. Identification of the precursor cations H2NCS+ and HNCSH+
would also help to constrain the chemistry of HNCS and HSCN, although their
rotational spectrum has not yet been characterised in the laboratory.

The third observed species of the family of HNCO isomers, namely HCNO, has a
different skeleton to the other two isomers and is therefore formed in a
different way. According to the chemical model, the neutral–neutral reaction
CH2 \+ NO (Marcelino et al. 2009) is the main source of HCNO. The analogous
molecule with sulfur, HCNS (the one reported here), is severely underestimated
by the chemical model. The analogous reaction CH2 \+ NS cannot account for
HCNS because in TMC-1, NS is 160 times less abundant than NO, while HCNS is
just 9 times less abundant than HCNO (see Table 2). Among the potential
missing routes to HCNS, we could consider the neutral–neutral reactions HCN +
SH, CN + H2S, N + H2CS, NH + HCS, and NH2 \+ CS. To our knowledge, these
reactions have not been studied either theoretically or experimentally, and
therefore we do not know whether or not they are fast at low temperatures and
produce HCNS as a product. Among these reactions, the most efficient by far is
N + H2CS, which would provide an HCNS abundance of the order of the observed
one assuming that it has a rate coefficient of the order of 1$\times$10-10 cm3
s-1 at low temperature.

We conclude that further experimental and theoretical work is needed to
understand the chemistry of the HNCS and HNCO isomers. In particular, the
enhancement of the abundance of HSCN relative to HNCS and HOCN, as well as the
behaviour of the NCS/HNCS and NCO/HNCO abundance ratios, are not well
reproduced by the present chemical models.

###### Acknowledgements.

We thank Ministerio de Ciencia e Innovación of Spain (MICIU) for funding
support through projects PID2019-106110GB-I00, and PID2019-106235GB-I00. We
also thank ERC for funding through grant ERC-2013-Syg-610256-NANOCOSMOS.

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##  Appendix A Line parameters

Line parameters for all observed transitions with the Yebes 40m and IRAM 30m
radio telescopes were derived by fitting a Gaussian line profile to them using
the GILDAS package. A velocity range of $\pm$20 km s-1 around each feature was
considered for the fit after a polynomial baseline was removed. Negative
features produced in the folding of the frequency switching data were blanked
before baseline removal.

Figure 2: Observed $J=6-5$ line of HCNS with the IRAM 30m (top panel) and
Yebes 40m (middle panel) radio telescopes. The data of the Yebes telescope
have been scaled to those of IRAM using the line $N=8-7$ $J=15/2-13/2$ of
CCC13CH at 73819.18 MHz that appears in both sets of data with a S/N of
$\sim$10 in the IRAM data and of $\sim$5 in the Yebes data. The merged
spectrum was obtained by averaging both sets of data weighted as
1/$\sigma^{2}$ and has a sensitivity of 1 mK. The derived line parameters for
this transition of HCNS are given in Table 3. The red line corresponds to the
computed synthetic spectra (see Sect. 3). The vertical blue line corresponds
to the measured frequency for a vLSR of 5.83 km s-1.  Figure 3: Observed lines
of HSCN and HNCS. The two upper rows show the observed lines of HSCN with
$K_{a}$=1. The two lower rows show the $7_{07}-6_{06}$ and $8_{08}-7_{07}$
lines of HSCN (third row) and HNCS (fourth row) observed at 3mm with the IRAM
30m radio telescope. The red line shows the computed synthetic spectra for
these lines (see Sect. 3). The derived line parameters are given in Table 3.

The derived line parameters for all observed lines of the CHNS and CHNO
families are given in Tables 3 and 4, respectively. The differences between
the measured frequencies and the predicted ones are always of a few kHz.
However, for some isotopologues, several hyperfine components are blended in
the laboratory data (measurements with uncertainties $\geq$ 30 kHz), while in
our observations the lines are well resolved and have an uncertainty of 10
kHz.

Figure 4: Observed $3_{03}-2_{02}$ and $4_{04}-3_{03}$ transitions of the
isotopologues H34SCN (top panels) and DSCN (bottom panels) of HSCN. The red
line shows the computed synthetic spectra for these lines (see Sect. 3). The
derived line parameters are given in Table 3.  Figure 5: Same as Fig. 1 but
for the $J=2_{02}-1_{01}$ transition of HNCO (isocyanic acid) and its
isotopologues HN13CO, H15NCO, and DCNO (upper and second raw panels). The
upper panel also shows a zoom to the data to exhibit the hyperfine lines of
HN13CO. The scale of the zoom is indicated in blue on the right-Y axis. The
third row shows the $J=2-1$ line of HCNO (the $J=4-3$ is shown in Fig. 3) and
the $J=2_{02}-1_{01}$ line of HOCN (cyanic acid). Finally, the bottom panels
show two hyperfine components of the $J$=7/2-5/2 line of NCO in its
${}^{2}\Pi_{3/2}$ ground state. The lambda-type doubling components, $e$ and
$f$, are indicated. The derived line parameters are given in Table 4. The
hyperfine structure was considered in the modelling of the lines (shown by the
red lines).  Figure 6: Observed lines of HNCO and HOCN at 3mm with the IRAM
30m radio telescope. The red line shows the synthetic spectra computed with
the models described in Sect. 3. The column densities are the same as those
derived from the lines observed in the Q-band. The bottom panels show the DNCO
$4_{04}-3_{03}$ and DCNO $J=2-1$ lines. The derived line parameters are given
in Table 4.  Figure 7: Observed $J_{u}=2$ lines of H2NCO+ towards TMC-1. The
red line shows the computed synthetic spectra for these lines (see Sect. 3).
The derived line parameters are given in Table 4.

The lines of HCNS, HNCS, HSCN, and NCS are shown in Figures 1 and 3. The
observations of the $J=6-5$ line of HCNS with the IRAM 30m and the Yebes 40m
radio telescopes, together with the resulting merged data, are shown in Fig.
2. Finally, the lines of the isotopologues H34SCN and DSCN are shown in Fig.
4. The lines of HNCO, HCNO, HOCN, and NCO are shown in Fig. 5 and those of
H2NCO+ in Fig. 7

Table 3: Observed line parameters for the isomers and isotopologues of HNCS

Species | $J_{u}-J_{l}$ |  $F_{u}-F_{l}$a |  $\nu_{rest}$ b |  $\int T_{A}^{*}dv$ c | vLSR |  $\Delta v$ d |  $T_{A}^{*}$ e | Notes
---|---|---|---|---|---|---|---|---
|  |  | (MHz) | (mK km s-1) | (km s-1) | (km s-1) | (mK) |
HCNS | 3-2 |  | 36908.475$\pm$0.010 | 0.95$\pm$0.06 | 5.83 | 1.04$\pm$0.09 | 0.85$\pm$0.07 |
| 4-3 |  | 49211.115$\pm$0.010 | 0.77$\pm$0.11 | 5.83 | 0.47$\pm$0.08 | 1.56$\pm$0.14 |
| 6-5 |  | 73815.835$\pm$0.020 | 1.34$\pm$0.39 | 5.83 | 0.38$\pm$0.10 | 3.40$\pm$0.90 |
HNCS | $3_{03}-2_{02}$ | 2-2 | 35186.557$\pm$0.010 | 0.19$\pm$0.06 | 5.83 | 0.63$\pm$0.10 | 0.29$\pm$0.07 |
|  | Main | 35187.089$\pm$0.010 | 5.05$\pm$0.06 | 5.83 | 0.99$\pm$0.09 | 4.82$\pm$0.07 |
|  | 3-3 | 35187.445$\pm$0.010 | 0.16$\pm$0.04 | 5.83 | 0.46$\pm$0.11 | 0.33$\pm$0.07 |
| $4_{04}-3_{03}$ | 3-3 | 46915.466$\pm$0.010 | 0.67$\pm$0.10 | 5.83 | 0.72$\pm$0.13 | 0.87$\pm$0.14 | A
|  | Main | 46915.976$\pm$0.010 | 5.98$\pm$0.11 | 5.83 | 0.68$\pm$0.05 | 8.19$\pm$0.14 |
|  | 4-4 | 46916.349$\pm$0.010 | 0.27$\pm$0.10 | 5.83 | 0.51$\pm$0.23 | 0.50$\pm$0.14 | A
| $7_{07}-6_{06}$ | Main | 82101.810$\pm$0.010 | 6.04$\pm$0.09 | 5.83 | 0.45$\pm$0.07 | 12.68$\pm$2.10 |
| $8_{08}-7_{07}$ | Main | 93830.036$\pm$0.010 | 3.29$\pm$0.16 | 5.83 | 0.53$\pm$0.04 | 5.83$\pm$0.60 |
HSCN | $3_{13}-2_{12}$ | 3-2 | 34227.919$\pm$0.020 | 0.14$\pm$0.06 | 5.83 | 0.62$\pm$0.12 | 0.21$\pm$0.09 |
| $3_{13}-2_{12}$ | Main | 34228.329$\pm$0.010 | 0.39$\pm$0.06 | 5.83 | 0.84$\pm$0.14 | 0.44$\pm$0.09 |
| $3_{03}-2_{02}$ | 3-3 | 34407.322$\pm$0.010 | 1.62$\pm$0.09 | 5.83 | 0.77$\pm$0.04 | 1.98$\pm$0.11 |
| $3_{03}-2_{02}$ | 2-1 | 34408.425$\pm$0.010 | 7.74$\pm$0.08 | 5.83 | 0.72$\pm$0.01 | 10.26$\pm$0.11 |
| $3_{03}-2_{02}$ | Main | 34408.660$\pm$0.010 | 28.58$\pm$0.09 | 5.83 | 0.90$\pm$0.01 | 29.82$\pm$0.11 |
| $3_{03}-2_{02}$ | 2-2 | 34410.448$\pm$0.010 | 1.38$\pm$0.09 | 5.83 | 0.67$\pm$0.04 | 1.93$\pm$0.11 |
| $3_{12}-2_{11}$ | 3-2 | 34587.252$\pm$0.010 | 0.27$\pm$0.05 | 5.83 | 1.02$\pm$0.21 | 0.24$\pm$0.06 |
| $3_{12}-2_{11}$ | Main | 34587.606$\pm$0.010 | 0.35$\pm$0.05 | 5.83 | 1.53$\pm$0.32 | 0.21$\pm$0.06 | B
| $4_{14}-3_{13}$ | 4-3 | 45637.236$\pm$0.010 | 0.28$\pm$0.07 | 5.83 | 0.45$\pm$0.14 | 0.57$\pm$0.14 |
| $4_{14}-3_{13}$ | Main | 45637.423$\pm$0.010 | 0.51$\pm$0.09 | 5.83 | 0.81$\pm$0.18 | 0.59$\pm$0.14 |
| $4_{04}-3_{03}$ | 4-4 | 45876.458$\pm$0.010 | 0.80$\pm$0.07 | 5.83 | 0.61$\pm$0.05 | 1.24$\pm$0.14 |
| $4_{04}-3_{03}$ | 3-2 | 45877.725$\pm$0.010 | 10.52$\pm$0.34 | 5.83 | 0.68$\pm$0.02 | 14.57$\pm$0.14 |
| $4_{04}-3_{03}$ | Main | 45877.828$\pm$0.010 | 27.29$\pm$0.34 | 5.83 | 0.65$\pm$0.01 | 39.75$\pm$0.14 |
| $4_{04}-3_{03}$ | 3-3 | 45878.621$\pm$0.010 | 0.39$\pm$0.08 | 5.83 | 0.88$\pm$0.08 | 0.41$\pm$0.14 |
| $4_{13}-3_{12}$ | 4-3 | 46116.344$\pm$0.020 | 0.18$\pm$0.04 | 5.83 | 0.60$\pm$0.00 | 0.30$\pm$0.13 | C
| $4_{13}-3_{12}$ | 3-2 | 46116.396$\pm$0.020 | 0.28$\pm$0.07 | 5.83 | 0.60$\pm$0.00 | 0.31$\pm$0.13 | C
| $4_{13}-3_{12}$ | 5-4 | 46116.492$\pm$0.020 | 0.36$\pm$0.07 | 5.83 | 0.60$\pm$0.00 | 0.60$\pm$0.13 | C
| $7_{07}-6_{06}$ | Main | 80283.160$\pm$0.010 | 19.98$\pm$0.44 | 5.83 | 0.61$\pm$0.02 | 28.89$\pm$0.64 |
| $8_{08}-7_{07}$ | Main | 80283.160$\pm$0.010 | 10.54$\pm$0.77 | 5.83 | 0.60$\pm$0.05 | 16.63$\pm$1.66 |
DSCN | $3_{13}-2_{12}$ | Main | 33258.268$\pm$0.010 | 0.24$\pm$0.07 | 5.83 | 0.72$\pm$0.21 | 0.32$\pm$0.09 |
| $3_{03}-2_{02}$ | Main | 33581.957$\pm$0.010 | 0.33$\pm$0.04 | 5.83 | 0.87$\pm$0.11 | 0.35$\pm$0.06 | D
| $3_{12}-2_{11}$ | Main | 33905.771$\pm$0.003 |  |  |  |  $\leq$0.3 |
| $4_{14}-3_{13}$ | Main | 44343.744$\pm$0.010 | 0.36$\pm$0.12 | 5.83 | 0.66$\pm$0.26 | 0.52$\pm$0.10 |
| $4_{04}-3_{03}$ | Main | 44774.598$\pm$0.010 | 0.84$\pm$0.13 | 5.83 | 0.78$\pm$0.14 | 1.01$\pm$0.13 |
| $4_{13}-3_{12}$ | Main | 45207.143$\pm$0.010 | 0.29$\pm$0.07 | 5.83 | 0.46$\pm$0.14 | 0.60$\pm$0.11 |
H34SCN | $3_{03}-2_{02}$ | 2-1 | 33642.089$\pm$0.020 | 0.39$\pm$0.07 | 5.83 | 0.91$\pm$0.17 | 0.40$\pm$0.07 |
| $3_{03}-2_{02}$ | Main | 33642.328$\pm$0.010 | 0.98$\pm$0.07 | 5.83 | 0.96$\pm$0.09 | 0.96$\pm$0.07 |
| $4_{04}-3_{03}$ | Main | 44856.085$\pm$0.020 | 0.64$\pm$0.05 | 5.83 | 0.57$\pm$0.06 | 1.06$\pm$0.11 |
NCS | 5/2-3/2 ef | 9/2-7/2 | 42699.184$\pm$0.010 | 2.25$\pm$0.07 | 5.83 | 0.67$\pm$0.02 | 3.14$\pm$0.10 |
| 5/2-3/2 ef | 7/2-4/2 | 42702.930$\pm$0.010 | 2.10$\pm$0.08 | 5.83 | 0.82$\pm$0.04 | 2.39$\pm$0.10 |
| 5/2-3/2 ef | 5/2-3/2 | 43705.851$\pm$0.010 | 1.61$\pm$0.08 | 5.83 | 0.81$\pm$0.05 | 1.88$\pm$0.10 |

555 $a$$a$footnotetext: Main refers to the three hyperfine components with the
strongest line intensities, $F_{u}-F_{l}$=$J+1$$\rightarrow$$J$,
$J$$\rightarrow$$J-1$, and $J-1$$\rightarrow$$J-2$, respectively. These are
unresolved but could produce a measurable line broadening. However, the model
to compute the synthetic spectrum of all CHNS isomers and isotopologues takes
into account all hyperfine components. $b$$b$footnotetext: Observed
frequencies for the transitions of the species studied in this work assuming a
vLSR of 5.83 km s-1 (see text). For lines for which only upper limits to the
intensity are obtained the predicted frequencies from the laboratory data are
given. $c$$c$footnotetext: Integrated line intensity in mK km s-1.
$d$$d$footnotetext: Line width at half intensity using a Gaussian fit in the
line profile (in km s-1). $e$$e$footnotetext: Antenna temperature (in mK).
Upper limits correspond to 3$\sigma$ values. $A$$A$footnotetext: These
hyperfine components appear too strong compared with the other transition of
the same molecule. They are probably blended with unknown lines as shown in
Fig. 1. $B$$B$footnotetext: Heavily blended line. Derived parameters are
uncertain (see Fig. 3). $C$$C$footnotetext: The hyperfine components are
blended but produce a well-defined line profile. The line width has been fixed
to 0.6 km s-1 (see Fig. 3). $D$$D$footnotetext: The main component is affected
by a line of HC4N at +8 MHz in the frequency switching data with a throw of 8
MHz, and by a line of c-C3HCCH at -10 MHz in the data with a throw of 10 MHz.
The corresponding line parameters are uncertain (see Fig. 4).

Table 4: Observed line parameters for the isomers and isotopologues of HNCO

Species | $J_{u}-J_{l}$ |  $F_{u}-F_{l}$a |  $\nu_{rest}$ b |  $\int T_{A}^{*}dv$ c | vLSR |  $\Delta v$ d |  $T_{A}^{*}$ e | Notes
---|---|---|---|---|---|---|---|---
|  |  | (MHz) | (mK km s-1) | (km s-1) | (km s-1) | (mK) |
HNCO | $2_{02}-1_{01}$ | 1-1 | 43962.015$\pm$0.010 | 21.16$\pm$0.10 | 5.83 | 0.63$\pm$0.01 | 31.17$\pm$0.14 |
|  | 1-2 | 43962.626$\pm$0.010 | 1.15$\pm$0.09 | 5.83 | 0.58$\pm$0.05 | 1.86$\pm$0.14 |
|  | 3-2 | 43963.013$\pm$0.010 | 174.41$\pm$0.01 | 5.83 | 0.72$\pm$0.01 | 229.15$\pm$0.14 | A
|  | 2-1 |  |  |  |  |  | A
|  | 1-0 | 43963.556$\pm$0.010 | 29.93$\pm$0.12 | 5.83 | 0.67$\pm$0.05 | 42.19$\pm$0.14 |
|  | 2-2 | 43963.659$\pm$0.010 | 18.87$\pm$0.11 | 5.83 | 0.59$\pm$0.05 | 29.71$\pm$0.14 |
| $4_{04}-3_{03}$ | 3-3 | 87924.350$\pm$0.010 | 10.42$\pm$1.68 | 5.83 | 0.58$\pm$0.10 | 16.89$\pm$2.02 |
|  | Main | 87925.238$\pm$0.010 | 307.62$\pm$1.58 | 5.83 | 0.63$\pm$0.01 | 525.21$\pm$2.02 |
|  | 4-4 | 87925.921$\pm$0.010 | 8.94$\pm$1.53 | 5.83 | 0.60$\pm$0.11 | 14.12$\pm$2.02 |
| $5_{05}-4_{04}$ | Main | 109905.734$\pm$0.010 | 175.38$\pm$1.31 | 5.83 | 0.51$\pm$0.01 | 324.74$\pm$3.67 |
HN13CO | $2_{02}-1_{01}$ | 3-2 | 43964.265$\pm$0.010 | 3.41$\pm$0.04 | 5.83 | 0.74$\pm$0.02 | 4.33$\pm$0.14 | A
|  | 2-1 |  |  |  |  |  | A
|  | 1-0 | 43964.807$\pm$0.010 | 0.58$\pm$0.03 | 5.83 | 0.57$\pm$0.04 | 0.94$\pm$0.14 |
|  | 2-2 | 43964.897$\pm$0.010 | 0.29$\pm$0.04 | 5.83 | 0.48$\pm$0.15 | 0.70$\pm$0.14 |
H15NCO | $2_{02}-1_{01}$ |  | 42646.445$\pm$0.010 | 0.73$\pm$0.09 | 5.83 | 0.59$\pm$0.09 | 1.15$\pm$0.10 |
DNCO | $2_{02}-1_{01}$ | 1-1 | 40785.534$\pm$0.010 | 0.54$\pm$0.10 | 5.83 | 0.58$\pm$0.11 | 0.87$\pm$0.07 |
|  | 3-2 | 40786.545$\pm$0.015 | 1.65$\pm$0.10 | 5.83 | 0.60$\pm$0.15 | 2.12$\pm$0.07 | B
|  | 2-1 | 40786.618$\pm$0.015 | 1.28$\pm$0.10 | 5.83 | 0.60$\pm$0.15 | 1.98$\pm$0.07 | B
|  | 1-0 | 40787.139$\pm$0.015 | 0.66$\pm$0.08 | 5.83 | 0.60$\pm$0.15 | 0.79$\pm$0.07 | B
|  | 2-2 | 40787.249$\pm$0.015 | 0.48$\pm$0.08 | 5.83 | 0.60$\pm$0.15 | 0.69$\pm$0.07 | B
| $4_{04}-3_{03}$ | Main | 81571.895$\pm$0.010 | 6.34$\pm$0.37 | 5.83 | 0.57$\pm$0.02 | 10.42$\pm$0.66 |
HCNO | 2-1 | Main | 45876.059$\pm$0.010 | 7.16$\pm$0.12 | 5.83 | 0.91$\pm$0.02 | 7.41$\pm$0.15 | C
| 4-3 | Main | 91751.350$\pm$0.010 | 2.11$\pm$0.60 | 5.83 | 0.30$\pm$0.10 | 6.60$\pm$1.68 | C,D
DCNO | 2-1 | Main | 41169.798$\pm$0.010 | 0.25$\pm$0.05 | 5.83 | 0.54$\pm$0.10 | 0.44$\pm$0.08 | C
HOCN | $2_{02}-1_{01}$ | 2-2 | 41949.966$\pm$0.010 | 1.54$\pm$0.09 | 5.83 | 0.61$\pm$0.05 | 2.36$\pm$0.15 |
|  | 1-0 | 41950.108$\pm$0.010 | 2.14$\pm$0.09 | 5.83 | 0.64$\pm$0.04 | 3.14$\pm$0.15 |
|  | 2-1 | 41950.848$\pm$0.010 | 5.22$\pm$0.25 | 5.83 | 0.70$\pm$0.15 | 6.79$\pm$0.15 | B
|  | 3-2 | 41950.902$\pm$0.010 | 6.97$\pm$0.25 | 5.83 | 0.70$\pm$0.15 | 9.99$\pm$0.15 | B
|  | 1-1 | 41952.292$\pm$0.010 | 1.78$\pm$0.20 | 5.83 | 0.69$\pm$0.06 | 2.45$\pm$0.15 |
| $4_{04}-3_{03}$ | Main | 83900.560$\pm$0.010 | 12.77$\pm$0.53 | 5.83 | 0.62$\pm$0.03 | 19.35$\pm$0.61 |
| $5_{05}-4_{04}$ | Main | 104874.696$\pm$0.010 | 4.44$\pm$0.77 | 5.83 | 0.32$\pm$0.06 | 13.11$\pm$2.53 |
NCO | 7/2-5/2e | 9/2-7/2 | 81404.306$\pm$0.015 | 2.08$\pm$0.26 | 5.83 | 0.67$\pm$ 0.10 | 2.92$\pm$0.53 |
| 7/2-5/2f | 9/2-7/2 | 81404.815$\pm$0.015 | 1.40$\pm$0.23 | 5.83 | 0.51$\pm$ 0.10 | 2.55$\pm$0.53 |
| 7/2-5/2e | 7/2-5/2 | 81413.143$\pm$0.020 | 0.81$\pm$0.25 | 5.83 | 0.47$\pm$ 0.14 | 1.62$\pm$0.65 |
| 7/2-5/2f | 7/2-5/2 | 81413.591$\pm$0.020 | 0.66$\pm$0.21 | 5.83 | 0.32$\pm$ 0.14 | 1.93$\pm$0.65 |

666 $a$$a$footnotetext: Main refers to the three hyperfine components with the
strongest line intensities, $F_{u}-F_{l}$=$J+1$$\rightarrow$$J$,
$J$$\rightarrow$$J-1$, and $J-1$$\rightarrow$$J-2$, respectively. These are
unresolved but could produce a measurable line broadening. However, the model
to compute the synthetic spectrum of all CHNO isomers and isotopologues takes
into account all hyperfine components. $b$$b$footnotetext: Predicted, or
observed, frequencies for the transitions of the species studied in this work
(see text). $c$$c$footnotetext: Integrated line intensity in mK km s-1.
$d$$d$footnotetext: Line width at half intensity using a Gaussian fit in the
line profile (in km s-1). $e$$e$footnotetext: Antenna temperature (in mK).
$A$$A$footnotetext: The $F=3-2$ and $F=2-1$ hyperfine componentes are blended
with a predicted separation between them of 40 kHz. The derived frequency
corresponds to that of the averaged value of both components.
$B$$B$footnotetext: The hyperfine components are blended. The line width has
been fixed. We estimate an uncertainty on this parameter of 0.15 km s-1.
$C$$C$footnotetext: The hyperfine components are blended. A single line has
been fitted (see Fig. 5). $D$$D$footnotetext: Blended with a U-line as shown
in Fig. 3 but line parameters can be still estimated.


# Meta Ordinal Weighting Net for Improving Lung Nodule Classification

###### Abstract

The progression of lung cancer implies the intrinsic ordinal relationship of
lung nodules at different stages—from _benign_ to _unsure_ then to
_malignant_. This problem can be solved by ordinal regression methods, which
is between classification and regression due to its ordinal label. However,
existing convolutional neural network (CNN)-based ordinal regression methods
only focus on modifying classification head based on a randomly sampled mini-
batch of data, ignoring the ordinal relationship resided in the data itself.
In this paper, we propose a Meta Ordinal Weighting Network (MOW-Net) to
explicitly align each training sample with a meta ordinal set (MOS) containing
a few samples from all classes. During the training process, the MOW-Net
learns a mapping from samples in MOS to the corresponding class-specific
weight. In addition, we further propose a meta cross-entropy (MCE) loss to
optimize the network in a meta-learning scheme. The experimental results
demonstrate that the MOW-Net achieves better accuracy than the state-of-the-
art ordinal regression methods, especially for the unsure class.

Index Terms—  Convolutional neural network, lung nodule classification, meta-
learning, ordinal regression

##  1 Introduction

Deep convolutional neural networks (CNNs) have recently achieved impressive
results in lung nodule classification based on computed tomography (CT) [1, 2,
3, 4, 5, 6, 7, 8]. However, this kind of method usually performs a binary
classification (malignant _vs._ benign) by omitting the unsure nodules—those
between benign and malignant—which is a great waste of medical data for
machine learning algorithms, especially for data-hungry deep learning methods.
Therefore, how to leverage unsure data to learn a robust model is crucial for
lung nodule classification.

To this end, the ordinal regression has been widely explored to utilize those
unsure nodules. The unsure data model (UDM) [9] was proposed to learn with
unsure lung nodules, and it regards this classification as an ordinal
regression problem, which optimizes the negative logarithm of cumulative
probabilities. However, the UDM has some additional parameters that need to be
carefully tuned. The neural stick-breaking (NSB) method calculates the
probabilities through the $C-1$ predicted classification bounds, where $C$ is
the number of classes [10]. The unimodal method makes each fully-connected
output follows a unimodal distribution such as Poisson or Binomial [11].
However, these methods do not guarantee the strict ordinal relationship.
Recently, the convolutional ordinal regression forests (CORFs) aim at solving
this problem through the combination of CNNs and random forests [12], which
have been shown effective for lung nodule classification in a meta learning
framework [13]. In summary, the existing methods do not explicitly leverage
the ordinal relationship resided in the data itself.

Fig. 1: The illustration of the MOW-Net. Each training sample is aligned with
a meta ordinal set which contains $k$ samples of each class. Forward: the
$\mathcal{L}_{\mathrm{CE}}^{i,c}$ denotes the cross-entropy loss of class $c$
w.r.t. the $i$-th training sample, and each $\mathcal{L}_{\mathrm{CE}}^{ic}$
is mapped to the corresponding weight through the specific MLP. Backward: the
two parts of parameters $\bm{\Theta}$ and $\bm{\Phi}$ are updated
alternatively based on the proposed MCE loss and the Meta loss, respectively.

In this paper, we assume that the ordinal relationship resides in not only the
label but also the data itself. Put differently, the ordinal relationship of
features from different classes also dominates the generalization ability of
the model. Therefore, we propose a meta ordinal weighting network (MOW-Net) to
learn the ordinal regression between each training sample and a set of samples
of all classes simultaneously, where this set is termed as meta ordinal set
(MOS). Here, the MOS implies the feature and semantic information of the
dataset, and acts as the meta knowledge for the training sample. As shown in
Fig. 1, each training sample relates to an MOS, and the meta samples in the
MOS are mapped to the corresponding weights representing the meta knowledge of
all classes. Furthermore, we propose a novel meta cross-entropy (MCE) loss for
the training of the MOW-Net; each term is weighted by the learned meta weight
as shown in Fig. 1. Different from the normal CE loss, the MCE loss indicates
that the training sample is guided by the meta weights learned from the MOS.
Hence, the MOW-Net is able to boost the classification performance and
generalization ability with the supervision from meta ordinal knowledge.
Moreover, our MOW-Net is able to reflect the difficulties of classification of
all classes in the dataset, which will be analyzed in Sec.3.4.

The main contributions of this work are summarized as follows. First, we
propose a meta ordinal weighting network (MOW-Net) for lung nodule
classification with unsure nodules. The MOW-Net contains a backbone network
for classification and a mapping branch for meta ordinal knowledge learning.
Second, we propose a meta cross-entropy (MCE) loss for the training of the
MOW-Net in a meta-learning scheme, which is based on a meta ordinal set (MOS)
that contains a few training samples from all classes and provides the meta
knowledge for the target training sample. Last, the experimental results
demonstrate the significant performance improvements in contrast to the state-
of-the-art ordinal regression methods. In addition, the changes in learned
meta weights reflect the difficulties of classifying each class.

##  2 Methodology

This section introduces the proposed meta ordinal set (MOS), meta cross-
entropy (MCE) loss, and the meta training algorithm, respectively.

###  2.1 Meta Ordinal Set (MOS)

We assume that the ordinal relationship resides in not only the label but also
the data itself. Therefore, we align each target training sample with an MOS
that contains $K$ samples from each class. The MOS for $i$-th training sample
is formally defined as follows:

$\displaystyle\mathcal{S}_{i}^{\mathrm{meta}}=\left\\{\bm{x}_{c}^{k}\right\\}_{c=1,\ldots,C,\
k=1,\ldots,K},$ (1)

where $C$ is the number of the classes.Note that the samples in the
$\mathcal{S}^{\mathrm{meta}}$ are not ordered, but the samples in one class
should go to the corresponding multi-layer perceptron (MLP) in Fig. 1. Then
the MPLs are able to learn the specific knowledge from each class. For a
target training sample $\bm{x}_{i}$, $\bm{x}_{c}^{k}(c\in\\{1,2,\ldots,C\\})$
is randomly sampled from the training set, and
$\bm{x}_{i}\notin\mathcal{S}_{i}^{meta}$.

###  2.2 Meta Cross-Entropy Loss

In order to enable the MOW-Net to absorb the meta knowledge provided by the
MOS, we propose an MCE loss to align the meta knowledge of each class to the
corresponding entropy term:

$\displaystyle\mathcal{L}_{\mathrm{MCE}}=-\sum_{c=1}^{C}\omega_{c}\log\hat{p}_{c},$
(2)

where $\hat{p}_{c}$ and $\omega_{c}$ are the prediction and the learned meta
weight of the $c$-th class, respectively. Note that the MCE loss implies no
ordinal regression tricks such as cumulative probabilities, it only holds the
correlation between the meta data and the predictions. Compared with the
conventional CE loss, the MCE loss enables the training samples to be
supervised by the corresponding meta data, hence, the learning of the MOW-Net
takes into account the meta ordinal knowledge resided in the data itself.

###  2.3 Training Algorithm

The MOW-Net is trained in the meta-learning scheme, which requires the second
derivatives [14, 15, 16, 17, 18], and two parts of the parameters,
$\bm{\Theta}$ and $\bm{\Phi}$, to be updated alternatively. We highlight that
the MOW-net offers a novel way of utilizing ordinal relationship encapsulated
within the data itself; however, the model is still the same as the one
trained with CE loss. What’s more, our MOW-Net does not modify the
classification head and can be adapted to various backbones.

Although the whole MOW-Net contains the two parts of parameters, $\bm{\Theta}$
and $\bm{\Phi}$, the trained model discards the MLPs ($\bm{\Phi}$) at the
inference stage. In other words, the MLPs are only involved in the meta
training phase to produce the class specific knowledge. Then the optimal
$\bm{\Theta}$ is learned by minimizing the following objective function:

$\displaystyle\bm{\Theta}^{*}(\bm{\Phi})$
$\displaystyle=\operatorname*{arg\,min}_{\bm{\Theta}}\mathcal{L}_{\mathrm{MCE}}(\bm{\Theta};\bm{\Phi})=\frac{1}{N}\sum_{i=1}^{N}\mathcal{L}_{\mathrm{MCE}}^{i}$
(3)
$\displaystyle=-\frac{1}{N}\sum_{i=1}^{N}\sum_{c=1}^{C}\underbrace{V_{c}(\mathcal{L}_{\mathrm{CE}}^{i,c}(\bm{\Theta});\bm{\Phi})}_{\omega_{c}}\cdot\log{\hat{p}}_{i,c}(\bm{\Theta}),$

where $\mathcal{L}_{\mathrm{CE}}^{i,c}$ denotes the conventional CE loss of
the $c$-th meta data with respect to the $i$-th training sample, and $V_{c}$
is the $c$-th MLP with $\mathcal{L}_{\mathrm{CE}}^{i,c}$ being the input (Fig.
1).

Following [14], $\bm{\Phi}$ can be updated through the following objective
function:

$\displaystyle\bm{\Phi}^{*}=\operatorname*{arg\,min}_{\bm{\Phi}}\frac{1}{M}\sum_{j=1}^{M}\sum_{c=1}^{C}\mathcal{L}_{\mathrm{CE}}^{j,c}(\bm{\Theta}^{*}(\bm{\Phi})),$
(4)

where $M=C\times K$ is the size of the MOS. Then we update these two parts of
parameters alternatively using meta-learning [14, 15]. First, we calculate the
derivative of $\bm{\Theta}$ through Eq. (3):

$\displaystyle\widehat{\bm{\Theta}}^{(t)}=\bm{\Theta}^{(t)}-\frac{\alpha}{N}\sum_{i=1}^{N}\sum_{c=1}^{C}\underbrace{V_{c}(\mathcal{L}_{\mathrm{CE}}^{i,c}(\bm{\Theta});\bm{\Phi})}_{\omega_{c}}\nabla_{\bm{\Theta}}\log\hat{p}_{i,c}(\bm{\Theta})|_{\bm{\Theta}^{(t)}}.$
(5)

Next, $\bm{\Phi}^{(t+1)}$ can be updated as follows:

$\displaystyle\bm{\Phi}^{(t+1)}=\bm{\Phi}^{(t)}-\beta\frac{1}{M}\sum_{j=1}^{M}\sum_{c=1}^{C}\nabla_{\bm{\Phi}}\mathcal{L}_{\mathrm{CE}}^{j,c}(\widehat{\bm{\Theta}}(\bm{\Phi}))|_{\bm{\Phi}^{(t)}}.$
(6)

Last, $\bm{\Theta}^{(t+1)}$ is updated based on the $\bm{\Phi}^{(t+1)}$:

$\displaystyle\bm{\Theta}^{(t+1)}=\bm{\Theta}^{(t)}-\alpha\frac{1}{N}\sum_{i=1}^{N}\nabla_{\bm{\Theta}}\mathcal{L}_{\mathrm{MCE}}^{i}(\bm{\Theta};\bm{\Phi}^{(t+1)})|_{\bm{\Theta}^{(t)}}.$
(7)

In the above equations, the superscript $(t)$ represents the $t$-th iteration
during training, $\alpha$ and $\beta$ are the learning rates for $\bm{\Theta}$
and $\bm{\Phi}$, respectively. The training algorithm is detailed in Algorithm
1. Note that after updating $\bm{\Phi}^{(t+1)}$ (line 8 in the Algorithm 1),
the MOW-Net obtains the meta knowledge through taking the normal CE loss of
the training sample as the input to all MLPs. Here, we can regard the updated
MPLs as the prior knowledge for each ordinal class.

To further analyze Eq. (6), we can obtain:

$\displaystyle\bm{\Phi}^{(t+1)}$ $\displaystyle=\bm{\Phi}^{(t)}+$ (8)
$\displaystyle\frac{\alpha\beta}{N}\sum_{i=1}^{N}(\frac{1}{M}\sum_{j=1}^{M}G_{ij})\frac{\partial{\sum_{c=1}^{C}V_{c}(\mathcal{L}_{\mathrm{CE}}^{i,c}(\bm{\Theta}^{(t)});\bm{\Phi})}}{\partial{\bm{\Phi}}}|_{\bm{\Phi}^{(t)}},$
$\displaystyle G_{i,j}$
$\displaystyle={\left[\frac{\partial{\sum_{c=1}^{C}\mathcal{L}_{\mathrm{CE}}^{j,c}(\widehat{\bm{\Theta}})}}{\partial{\widehat{\bm{\Theta}}}}|_{\widehat{\bm{\Theta}}^{(t)}}\right]}^{{}^{\textrm{T}}}\cdot\frac{\partial{\sum_{c=1}^{C}\log\hat{p}_{i,c}(\bm{\Theta})}}{\partial{\bm{\Theta}}}|_{\bm{\Theta}^{(t)}}.$
(9)

We can see that Eq. (9) denotes the derivative of the entropy of the training
sample (2nd term) is to approach the derivative of the MOS (1st term). This
implies that the learning of the backbone network is guided by the meta
knowledge. Please refer to http://hmshan.io/papers/mow-net-supp.pdf for the
detailed derivation.

Algorithm 1 Training Algorithm of MOW-Net.

1:Training data $\mathcal{S}^{\mathrm{tr}}$, randomly sampled MOS
$\mathcal{S}^{\mathrm{meta}}$.

2:The learned parameter $\bm{\Theta}^{(T-1)}$.

3:Initialize $\bm{\Theta}^{(0)}$ and $\bm{\Phi}^{(0)}$.

4:for $t=0$ to $T-1$ do

5: Sample a mini-batch of $\bm{x}_{i}$ from $\mathcal{S}^{\mathrm{tr}}$, and
align each $\bm{x}_{i}$ with a MOS $\mathcal{S}^{\mathrm{meta}}_{i}$.

6: Forward: input $\bm{x}_{i}$ and $S^{\mathrm{meta}}_{i}$ with
$\bm{\Theta}^{(t)}(\bm{\Phi}^{(t)})$ to obtain $\hat{p}_{i,c}$ and
$\omega_{i,c}^{meta}$ respectively.

7: Compute the first order derivative
$\widehat{\bm{\Theta}}^{(t)}(\bm{\Phi}^{(t)})$ by Eq. (5).

8: Forward: input $\mathcal{S}^{\mathrm{meta}}_{i}$ with
$\widehat{\bm{\Theta}}^{(t)}(\bm{\Phi}^{(t)})$, then obtain the meta CE losses
$\mathcal{L}_{\mathrm{CE}}^{i\\_meta}$.

9: Update $\bm{\Phi}^{(t+1)}$ by Eq. (6).

10: Forward: input $\bm{x}_{i}$ with $\bm{\Theta}^{(t)}(\bm{\Phi}^{(t+1)})$
and calculate the $\mathcal{L}_{\mathrm{CE}}^{i\\_tr}$; then input
$\mathcal{L}_{\mathrm{CE}}^{i\\_tr}$ into all the MLPs to obtain the
$\omega_{c}^{tr}$.

11: Update $\bm{\Theta}^{(t+1)}$ by Eq. (7).

12:end for

##  3 Experiments

In this section, we report the classification performance of our MOW-Net on
the dataset LIDC-IDRI [19].

###  3.1 Dataset

LIDC-IDRI is a publicly available dataset for low dose CT-based lung nodule
analysis, which includes 1,010 patients. Each nodule was rated on a scale of 1
to 5 by four thoracic radiologists, indicating an increased probability of
malignancy. In this paper, the ROI of each nodule was cropped at its annotated
center, with a square shape of a doubled equivalent diameter. An averaged
score of a nodule was used as ground-truth for the model training. All volumes
were resampled to have 1mm spacing (original spacing ranged from $0.6$mm to
$3.0$mm) in each dimension, and the cropped ROIs are of the size $32\times
32\times 32$. The averaged scores range from 1 to 5, and in our experiments,
we regard a nodule with a score between 2.5 and 3.5 as the unsure nodule;
benign and malignant nodules are those with scores lower than 2.5 and higher
than 3.5, respectively [9].

###  3.2 Implementation Details

We used the VGG-16 as the backbone network [20], and made the following
changes: 1) the input channel is 32 following [4]; 2) we only keep the first
seven convolutional layers due to a small size of the input, each followed by
the batch normalization (BN) and ReLU; and 3) the final classifier is a two-
layer perceptron that has 4096 neurons in hidden layer. We use 80% of data for
training and the remaining data for testing.

The hyperparameters for all experiments are set as follows: the learning rate
is 0.0001 and decayed by 0.1 for every 80 epochs; the mini-batch size is 16;
weight decay for Adam optimizer is 0.0001 [21]. The symbols P, R, and F1 in
our results stand for precision, recall, and F1 score, respectively [9].

Table 1: Results of classification on LIDC-IDRI dataset. Following [9], the values with underlines indicate the best results while less important in the clinical diagnosis. Method | Accuracy | Benign | Malignant | Unsure
---|---|---|---|---
P | R | F1 | P | R | F1 | P | R | F1
CE Loss | 0.517 | 0.538 | 0.668 | 0.596 | 0.562 | 0.495 | 0.526 | 0.456 | 0.360 | 0.402
Poisson [11] | 0.542 | 0.548 | 0.794 | 0.648 | 0.568 | 0.624 | 0.594 | 0.489 | 0.220 | 0.303
NSB [10] | 0.553 | 0.565 | 0.641 | 0.601 | 0.566 | 0.594 | 0.580 | 0.527 | 0.435 | 0.476
UDM [9] | 0.548 | 0.541 | 0.767 | 0.635 | 0.712 | 0.515 | 0.598 | 0.474 | 0.320 | 0.382
CORF [12] | 0.559 | 0.590 | 0.627 | 0.608 | 0.704 | 0.495 | 0.581 | 0.476 | 0.515 | 0.495
MOW-Net ($k=1$) | 0.629 | 0.752 | 0.489 | 0.592 | 0.558 | 0.851 | 0.675 | 0.600 | 0.675 | 0.635
MOW-Net ($k=5$) | 0.672 | 0.764 | 0.596 | 0.670 | 0.600 | 0.802 | 0.686 | 0.642 | 0.690 | 0.665
MOW-Net ($k=10$) | 0.687 | 0.768 | 0.623 | 0.688 | 0.668 | 0.705 | 0.686 | 0.606 | 0.792 | 0.687

Fig. 2: The visualization results on the testing set using $t$-SNE.

Fig. 3: The variations of the learned weights for all classes.

###  3.3 Classification Performance

In our experiments, we mainly focus on the precision of benign class, recall
of malignant and unsure classes [9]. We compared our MOW-Net with the state-
of-the-art ordinal regression methods and the normal CE loss. In Table 1, we
can see that the MOW-Net achieves the best accuracy by a large margin against
other methods. Specifically, the MOW-Net significantly improves the recall of
the unsure class by 0.28 over the previous best result. This is significant
for the clinical diagnosis since a higher recall of the unsure class can
encourage more follow-ups and reduce the probabilities of the nodules that are
misdiagnosed as malignant or benign. In addition, the precision of benign and
the recall of the malignant get a great improvement.

###  3.4 Analysis on Learned Weights

In order to further understand the weighting scheme of the MOW-Net, we plot
the variations of the weights in $\mathcal{L}_{\mathrm{MCE}}$ in Fig. 3. At
the beginning of the training, the weight for the unsure class is increasing
while the weight for the malignant class is decreasing, indicating that the
MOW-net focuses on classifying the unsure class from the other two classes,
and the malignant class is an easy-classified class. Then, at epoch 10, the
trends of these two weights become opposite. The curve of the benign
fluctuates slightly through the whole training process. At epoch 45, the
weights for all the three classes begin to converge. This indicates that the
model pays different attentions (weights) to different classes, and these
attentions affect the update of the backbone network. At the end of the
training, the model has similar sensitivities for each class.

Together with Fig. 2, the malignant samples are easier to be classified than
the other two classes at the beginning. At epoch 10, the unsure samples are
fused with other samples severely so that it has the highest weight.
Simultaneously, the malignant class performs worse than that at the beginning.
As the training continues, the weight for the malignant began to increase. At
epoch 45, the malignant samples are clustered again and the unsure samples are
more centralized than that of the previous epochs. At epoch 97, the model
achieves the best accuracy, and it is obvious that the samples are distributed
orderly, which demonstrates the effectiveness of the meta ordinal set.

###  3.5 Effects on the Size of MOS

The definition of the MOS in Eq. (1) shows that the parameter $K$ determines
the number of samples of each class. Here, we explore the effect of varied
$K$. Table 1 shows that when $K=10$, the MOW-Net obtained the best
performance. The performance of $K=5$ and $K=10$ is better than that of $K=1$,
which indicates that the more number in MOS, the better generalizability of
the model.

##  4 Conclusions

In this paper, we proposed an MOW-Net and the corresponding MOS to explore the
ordinal relationship resided in the data itself for lung nodule classification
in a meta-learning scheme. The experimental results empirically demonstrate a
significant improvement compared to existing methods. The visualization
results further confirm the effectiveness of the weighting scheme and the
learned ordinal relationship.

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