domain: chim sdv

text:
Yassin REFAHI 1 , Thibaut Viné 1 , Gabriel Paës 1 yassin.refahi@inrae.fr

Fractionation of AgroResources and Environment (FARE) laboratory, INRAE, University of Reims Champagne Ardenne, Reims, France

Understanding the underlying principles behind biochemical processes usually requires acquisition of detailed experimental data over the course of study. 4D (space + time) imaging has emerged as a powerful tool to capture quantitative information on the behaviour of the processes and their dynamics.

To process the collected data, image processing algorithms are needed to provide quantitative data to develop computational models to decipher and predict non-intuitive aspects of bio-chemical processes.

We developed an open-source user-friendly modular 4D image manipulation and analysis software platform, called BIOMODLAB to analyze 4D datasets (time-lapse sequences of 3D stacks like confocal, light-sheet microscopy images). BIOMODLAB integrates various components to visualize, segment, manipulate and analyse 4D datasets. Importantly, it also integrates several quantitative tools to extract cellular properties (volumes, cell-to-cell contact area, etc) and track their evolution over time. BIOMODLAB is developed using C++, Python, and PyQt5 programming languages. Its graphical user interface has an embedded Python interpreter which can be used to write scripts with access to the images and quantifications. BIOMODLAB was initially developed to study cell size and growth kinetics [START_REF] Willis | Cell size and growth regulation in the Arabidopsis thaliana apical stem cell niche[END_REF]. It was also used to analyse flower meristem morphogenesis to investigate the link between cellular growth and gene regulatory networks [START_REF] Refahi | A multiscale analysis of early flower development in Arabidopsis provides an integrated view of molecular regulation and growth control[END_REF]. BIOMODLAB is adapted and further developed to study lignocellulosic biomass deconstruction during enzymatic hydrolysis which can be seen as the opposite mechanism compared to growth [START_REF] Zoghlami | Multimodal characterization of acid-pretreated poplar reveals spectral and structural parameters strongly correlate with saccharification[END_REF]. In brief, BIOMODLAB provides a single interface integrating state-of-art image processing and computational tools to obtain accurate quantitative data required for analysis and modelling of biochemical processes at cellular and tissular scale.


teaching and research institutions in France or abroad, or from public or private research centers.

Comment on " Gain coefficient method for amplified spontaneous emission in thin waveguided film of a conjugated polymer " [APL 93, 163307 (2008)]

Sebastien Chenais, Sebastien Forget

To cite this version:

Sebastien Chenais, Sebastien Forget. Comment on " Gain coefficient method for amplified spontaneous emission in thin waveguided film of a conjugated polymer " [APL 93, 163307 (2008)]. 2008. hal- (I out is the edge emitted intensity; I is the incident pump intensity; g is the net gain; α is related to spontaneous emission), the authors claim that the gain g appearing in equation ( 1) depends on the excitation stripe length.

The letter contains several errors, misconceptions and confusing statements.

Firstly, it is obvious that the gain, as defined in eq.( 1), cannot depend on the length of the stripe, but only on the position within the stripe. The net gain g appearing in eq (1) is fundamentally a local parameter, defined by ( )

( ) η - ∆ σ r N r g em r r =
, where σ em is the emission cross section, ∆N the local population inversion density (in cm -3

), and η the losses (cm -1

). When the 1-dimensional approximation is valid, g is actually only dependant on the position z. As a consequence, in a transversely-pumped thin slab geometry used by the authors, the gainaveraged over the waveguide thicknessis related to the pump intensity (in W/cm²) and not to the total pump power, a misconception that leads to a supposed expression for the gain written as ( )

( ) I w z z b z g . . . 2 =
where z is the stripe length and w the stripe width. The introduction of b(z) is useless since g(z) has no particular reason to be linear with z. In contrast, the inhomogeneous pump profile (in general a truncated Gaussian along the stripe direction, with possible diffraction effects added by the slit) should be taken into account if the pump intensity cannot be made constant, as well as gain saturation (upon introduction of a saturation intensity), which may affect the shape of g(z). Examples of such analyses may be found in refs [2,3].

Secondly, the reason why the term I α of equation ( 1), accounting for spontaneous emission (SE), is replaced by an Arrhenius expression of the form

( ) 0 / - max 1 / α I I Ce I +
is not clearly justified, and is not physically reasonable. This expression involves that when the pump laser is off (I = 0), the emitted intensity should grow linearly, according to eq. ( 1), which has obviously no physical meaning. It is exact that spontaneous emission may not scale linearly any more with I, for instance when stimulated emission is strong enough to significantly deplete the excited state population [START_REF] Siegman | Lasers[END_REF].

This gain-saturation related effect is in general neglected since it corresponds to a regime where SE is negligible itself compared to stimulated emission. Whatever the new form proposed for this term, the linear dependence with the pump intensity I should be preserved at low I values.

Finally, the authors introduce four new independent unknown parameters (αI max , C, b and I 0 ) and obtain good fits, which is not very surprising. Besides the fact that I 0 is an ill-defined parameter, obtained from the Arrhenius expression, it is referred to as a "threshold intensity" (expressed in W/cm²) establishing the crossing from the SE to the Amplified SE regime. It is then, like the gain, a local parameter which cannot depend on the stripe length, as shown in figure 4.

At last, the authors use the 1-dimensional approximation (eq. 1) in limit cases where it is highly questionable. Since the width w of the stripe is given to be 0.02 cm, data obtained for a length L = 0.03 cm cannot be processed with a simple 1-D model without care.

In conclusion, the authors report the experimental evidence of an apparent variation of gain with the stripe length, but do not provide a clear framework for analysis. A thorough analysis of the Variable-Stripe Length technique for the study of ASE in thin films may be found in references [1,3,[START_REF] Lange | [END_REF].

Comment on « Gain coefficient method for amplified spontaneous emission in thin waveguided film of a conjugated polymer » [APL 93, 163307 (2008)]

	In a recent letter, Silvestre et al. address the interesting issue of finding a general technique to fit the
	experimental data obtained with the Variable Stripe Length technique, which is commonly used to
	measure gains in waveguide lasers [1]. Relying on the one-dimensional equation
	out dz dI	= α	I	+	g	( ) out I z	(1)
	Sébastien Chénais and Sébastien Forget
	Laboratoire de Physique des Lasers, UMR 7538 du CNRS, Institut Galilée, Université Paris-Nord, 99
	avenue Jean-Baptiste Clément, 93430 Villetaneuse, France.


Bochra Benmeziane

Jean-Yves Baudais
email: 2jean-yves.baudais@insa-rennes.fr

Stéphane Méric
email: 3stephane.meric@insa-rennes.fr

Kevin Cinglant
email: 4kevin.cinglant@zf.com

Comparison of ZF and MF filters through PSLR and ISLR assessment in automotive OFDM radar

Keywords: radar signal processing, matched filters, zero forcing, OFDM

Many filtering strategies can be applied to estimate target position along the range profile in orthogonal frequency-division multiplexing radar. Chief among these filters are the matched filter (MF) and the zero forcing (ZF) method. For each technique, this paper compares the impact on the peak to side lobe ratio (PSLR) and the integrated side lobe ratio (ISLR). The results show that ZF scores higher than MF with respect to PSLR and ISLR in low-noise environments.

I. INTRODUCTION

The number of vehicles equipped with radars has been steadily increasing in recent years which led to a higher risk of saturated band [START_REF] Bourdoux | Phenomenology of mutual interference of fmcw and pmcw automotive radars[END_REF]. It has then become urgent to investigate detection techniques that can mitigate interference such as Orthogonal Frequency Division Multiplexing (OFDM). OFDM is a multiplexing technique that uses orthogonal sub-carriers to transmit data [START_REF] Nguyen | Design and analysis of superresolution algorithm and signal separation technique for an OFDM-based MIMO radar[END_REF]. This technique can be exploited for radar detection. The estimation of the delay induced by a target range can be performed through a variety of ways exploiting the OFDM signal. Chief among these solutions are matched filter (MF) and zero forcing (ZF). The majority of the current radars operate on a frequency modulated continuous waveform (also known as FMCW). These radars use an MF to estimate the range. MF was also used in the first OFDM radars described in [START_REF] Sen | Adaptive design of OFDM radar signal with improved wideband ambiguity function[END_REF]. ZF as a range estimation technique was first introduced by [START_REF] Sturm | A novel approach to OFDM radar processing[END_REF]. The interest for this technique came from a desire to perform radar detection and inter-vehicular communication using the same OFDM signal. Both techniques can be used for radar detection. The peak to sidelobe ratio (PSLR) and the integrated sidelobe ratio (ISLR) are crucial in an automotive radar [START_REF] Van Thillo | Almost perfect auto-correlation sequences for binary phase-modulated continuous wave radar[END_REF]. A high sidelobe might result in a false alarm, causing the vehicle to brake for no reason. Meanwhile, a high general level of sidelobes would raise the threshold above peaks linked to low power echos (farther targets). This would result in an equally dangerous effect.

This paper compares the impact of each technique on the PSLR and the ISLR. Up to our knowledge, ZF and MF as range estimation techniques in OFDM radars have not been compared yet. This paper also highlights the link between ZF and MF for each metric and shows that ZF scores higher in low-noise environments. The rest of the paper is organized as follows. Section II describes the OFDM echo and each range estimation technique. The PSLR and ISLR metrics are then defined and estimated for each technique. The connection between the techniques for each metric is described in Section III. Simulations with respect to different levels of noise are presented in Section IV. Section V finally concludes the paper.

II. RANGE ESTIMATION IN OFDM RADARS

Exploiting an OFDM signal for range estimation has already been thoroughly explored [START_REF] Hakobyan | Orthogonal Frequency Division Multiplexing Multiple-Input Multiple-Output Automotive Radar with Novel Signal Processing Algorithms[END_REF], [START_REF] Sturm | An OFDM system concept for joint radar and communications operations[END_REF]. The OFDM signal uses orthogonal sub-carriers to transmit data through an inverse Fourier transform. The N orthogonal sub-carriers used in an OFDM frame of duration T are f n = n T , n ∈ [0, N -1], which guarantees the orthogonality of the sub-carriers. The emitted OFDM symbol is expressed by:

∀t ∈ [0, T + T g ], x(t) = 1 √ N N -1 n=0 a n e 2πfn(t-Tg) (1)
where a n is the nth emitted complex symbol that has been obtained through any modulation such as M-QAM. The duration T g of the cyclic prefix is needed by the receiving end for demodulation. The received echo contains distortions due to the environment. The delay due to the distance of the target is then estimated through two different techniques investigated in this paper. One of these techniques exploits the multiplication and convolution property of Fourier transforms to estimate the correlation function of the received echo. This technique is referred to as the MF technique. The other technique is the estimation of the channel through its impulse response. This technique is referred to as ZF.

In the following, the signal is assumed to be narrowband and the duration T is short enough that the Doppler variation within one OFDM symbol is considered to be negligible. The duration T g is properly sized to absorb the delay. The received and demodulated signal is then [START_REF] Sen | Adaptive design of OFDM radar signal with improved wideband ambiguity function[END_REF]:

Y (l) = a l e φ e -2π τ T l + W (l) (2)
where φ = -2πf c τ , f c is the carrier frequency, τ is the delay of the target, and W (l) is a white Gaussian noise, the variance of which is σ 2 w . As previously discussed, both techniques can provide a range profile estimation through different methods. For ZF, each Y (l) sample is divided by a l in order to estimate the transfer function of the channel. The impulse response is therefore:

χ ZF (i) = e φ √ N N -1 l=0 Y (l) a l e 2π il N = e φ √ N N -1 l=0 e 2π( iT N -τ ) l T + N -1 l=0 W (l) a l e 2π il N (3)
For the MF technique, Y (l) samples are multiplied by a l . The correlation function is therefore:

χ MF (i) = e φ √ N N -1 l=0 Y (l)a l e 2π il N (4) = e φ √ N N -1 l=0 |a l | 2 e 2π( iT N -τ ) l T + N -1 l=0 a l W (l)e 2π il N
with a l denoting the conjugate of a l . Each of these χ(i) functions represents a range profile, with peaks at the delays of the detected targets. If i τ = τ T N is the index of a peak, then:

χ ZF (i τ ) = e φ √ N + e φ √ N N -1 l=0 W (l) a l e 2π iτ l N (5) χ MF (i τ ) = e φ √ N N -1 l=0 |a l | 2 + N -1 l=0 a l W (l)e 2π iτ l N (6)
Note that in the following, i τ is assumed to be integer in order to simplify the analysis. In reality, τ N T is often non integer, which causes spectral leakage. This issue can be mitigated by applying a window [START_REF] Harris | On the use of windows for harmonic analysis with the discrete fourier transform[END_REF]. With the right window parameters, the analysis applies regardless.

MF was designed to obtain the highest peak SNR possible. Meanwhile, the division by the transmitted symbols a l in ZF provides range profiles that are independent from these symbols. This allows for the transmission of random symbols, making both inter-vehicle communications and radar detection possible with the same transmitted signal.

III. PSLR AND ISLR CALCULATION AND COMPARISON

Two metrics can be used to characterize the range ambiguity function of the automotive radar. The PSLR is the ratio of the main lobe, identified by χ(i τ ), to the highest sidelobe in a range profile. The ISLR is the ratio of the main lobe to the energy of the rest of the range profile. Both PSLR and ISLR are expressed by γ such as:

γ 2 = E[|χ(i τ )| 2 ] E θ {|χ(i)| 2 } i =iτ (7)
where θ({x i }) = max i x i for γ PSLR and θ({x i }) = i

x i for γ ISLR [START_REF] Lu | Parameter assessment for SAR image quality evaluation system[END_REF]. Note that in our case, the sampling frequency is equal to the bandwidth. This means that the main peak is one sample wide. Before we can proceed to compare these metrics for ZF and MF, we need to derive E[|χ(i)| 2 ] for both i τ and i = i τ .

With independent and identically distributed noise samples, the expectation of |χ(i)| 2 in (3) leads to:

E[|χ ZF (i)| 2 ] = |E[χ ZF (i)]| 2 + σ 2 1/a σ 2 w ( 8
)
where σ 2 1/a is the variance of 1/a. This result is obtained with N large enough such that the expectation can be approximated by the average:

σ 2 1/a = E 1 a 2 l = lim N →∞ 1 N N -1 i=0 1 a 2 l ( 9
)
The values σ 2 1/a depend on the modulation used and can be tabulated. With MF, the expectation of |χ(i)| 2 in (4) leads to:

E[|χ MF (i)| 2 ] = |E[χ MF (i)]| 2 + σ 2 a σ 2 w ( 10
)
where σ 2 a is the variance of a l , also dependent on the modulation used and (10) is obtained with N large enough.

The next step is to compute the squared expectation of the range profile for both i τ and i = i τ :

|E[χ ZF (i)]| 2 = 1 N N -1 l=0 e 2π( il N -τ ) 2 + σ 2 1/a σ 2 w ( 11
)
and

|E[χ MF (i)]| 2 = 1 N N -1 l=0 |a l | 2 e 2π( il N -τ ) 2 + σ 2 a σ 2 w (12)
In the main lobe, i = i τ , the squared expectation becomes:

|E[χ ZF (i τ )]| 2 = N + σ 2 1/a σ 2 w ( 13
)
and

|E[χ MF (i τ )]| 2 = N σ 4 a + σ 2 a σ 2 w (14)
Meanwhile, when i = i τ , let α l defined as

α l = |a l | 2 σ 2 a , hence ∀l, E[α l ] = 1 and E[|α l | 2 ] ≥ 1.
The connection between the squared expectations of ZF and MF for N large enough is:

N -1 l=0 α l e 2π( il N -τ ) 2 = N (E[α 2 l ] -1) + N -1 l=0 e 2π( il N -τ ) 2 (15)
Consequently, in the case of PSLR, the link between MF and ZF is expressed as:

N + σ 2 w σ 2 a γ 2 PSLR,MF - σ 2 w σ 2 a = max α 2 l -1 + N + σ 2 1/a σ 2 w γ 2 PSLR,ZF -σ 2 1/a σ 2 w ( 16
)
In low-noise environments, σ 2 w → 0, and since max In the case of ISLR, the link between MF and ZF is expressed as:

α 2 l ≥ E[α 2 l ] ≥ 1, (16) becomes: 1 γ 2 PSLR,MF = max α 2 l -1 N + 1 γ 2 PSLR,ZF ≥ 1 γ 2 PSLR,ZF (17)
N + σ 2 w σ 2 a γ ISLR,MF - σ 2 w σ 2 a = (N -1)(E[α 2 l ] -1) + N + σ 2 1/a σ 2 w γ ISLR,ZF -σ 2 1/a σ 2 w ( 18
)
and in low noise environments:

1 γ 2 ISLR,MF = N -1 N (E[α 2 l ] -1) + 1 γ 2 ISLR,ZF ≥ 1 γ 2 ISLR,ZF (19)
This shows that for both PSLR and ISLR, ZF performs better in low noise environments.

IV. RESULTS AND DISCUSSION

Simulations are performed using the OFDM radar range detection techniques ZF and MF. The OFDM radar system parameters are summarized in Table 1. Using these parameters, together with (3) and ( 4), the range profiles are estimated and compared in Fig. 1. This figure shows that in a low-noise environment, ZF shows significantly lower levels of sidelobes. However, in order to compare ZF and MF in terms of PSLR and ISLR, further simulations need to be performed in different environments. Using ( 7), ( 17) and ( 19), PSLR and ISLR are computed for each technique. The elements required for this computation are shown in Table 2. Fig. 2 show the comparison of ZF and MF with N = 1024 sub-carriers and using a 16-QAM modulation in different levels of SNR. The simulated plots are computed through [START_REF] Sturm | An OFDM system concept for joint radar and communications operations[END_REF], while the analytical ones are computed using (17) and (19). The plots show three main trends. In the high end of SNR, (17) and ( 19) are proven to be accurate, since ZF shows a higher level of PSLR and ISLR, while MF plateaus. For lower SNR, the noise is too high for (17) and (19) to apply. There, MF displays a slightly higher level or PSLR and ISLR. When the SNR < -30 dB, the noise level is too high for detection. The PSLR and ISLR for MF are plotted in terms of PSLR and ISLR for ZF in Fig. 3 and Fig. 4. The figures confirm the previous findings, by showing these metrics steeply escalating for MF before plateauing in low-noise environments.

The main reason that could explain this difference in low-noise environments is the fact that no matter how low the noise gets, the level of the sidelobes due to the correlation function remains unchanged. Whereas, the noise floor of an impulse response depends on the noise only.

The next simulations compare ZF and MF with N = 1024 sub-carriers and SNR= 20 dB for different orders of M-QAM modulation. The results are plotted in Fig. 5. The figure show that the difference between ZF and MF in their PSLR and ISLR remains virtually unchanged regardless of the order of the used M-QAM modulation and is consistent with the results The last simulation compares ZF and MF using 16-QAM in an environment with SNR = 20 dB and for different numbers of sub-carriers.

Once more, Fig. 6 shows the clear superiority of ZF over MF in the PSLR and ISLR departments regardless of the number of orthogonal sub-carriers in a low-noise environment. Hence, it has been proven that for a high SNR level, ZF as a range estimation technique performs better with PSLR and ISLR than MF regardless of the number of sub-carriers or the used modulation. This makes it a better candidate for radar detection in automotive.

V. CONCLUSION

In this paper, two metrics have been used which are PSLR and ISLR. These two metrics are crucial in an automotive radar. Range estimation techniques which are ZF and MF have been analytically compared on the basis of these two metrics. ZF has proven to be more advantageous in low-noise environments. Simulations have been performed and the superior performances of ZF, which is the considered solution for a joint radar and communication OFDM automotive systems, have been proven.

Fig. 1 .

 1 Fig. 1. Normalized range profile comparison.

Fig. 2 .

 2 Fig. 2. PSLR and ISLR comparison in different levels of SNR.

Fig. 3 .

 3 Fig. 3. PSLR comparison in different levels of SNR.

Fig. 4 .

 4 Fig. 4. ISLR comparison for different levels of SNR.

Fig. 5 .

 5 Fig. 5. PSLR and ISLR comparison for different M-QAM orders.

Fig. 6 .

 6 Fig. 6. PSLR and ISLR comparison for different numbers of subcarriers.

Table 1 .

 1 OFDM radar parameters.

	Parameter	Notation	Value
	Number in sub-carriers	N	1024
	Bandwidth	B	375 MHz
	Order of M-QAM modulation	M	16
	Signal to noise power ratio SNR =

Table 2 .

 2 QAM parameters.

	M	4 8	16	32	64	128	256	512	1024
	E[α 2 l ] max α 2 l	1 1.44 1.32 1.52 1.38 1 2.78 3.24 4.98 5.44	1.54 1.4 6.68 7.01 7.75 7.94 1.54 1.4


Holger S P Müller

Jean-Claude Guillemin

Frank Lewen

Stephan Schlemmer

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Keywords: rotational spectroscopy, submillimeter spectroscopy, interstellar molecule, cyclic molecule, centrifugal distortion, reduced Hamiltonian

We studied the rotational spectrum of oxirane in a sample of natural isotopic composition in selected regions between 158 GHz and 1093 GHz. Investigations of the isotopologs with one 13 C or one 18 O were the primary focus in order to facilitate searches for them in space. We also examined the main isotopic species mainly to look into the performance of Watson's A and S reductions both in an oblate and in a prolate representation. Even though oxirane is a rather asymmetric oblate rotor, the A reduction in the III l representation did not yield a satisfactory fit, as was observed frequently earlier for other molecules. The other three combinations yielded satisfactory fits of similar quality among each other; the A reduction in the I r representation required two parameters less than both S reduction fits.

Introduction

Oxirane, c-C 2 H 4 O, also know as ethylene oxide, oxacyclopropane, epoxyethane, and dimethylene oxide, is a molecule of astrochemical interest that was detected first toward the prolific high-mass Galactic center source Sagittarius (Sgr) B2(N) [START_REF] Dickens | Detection of Interstellar Ethylene Oxide (c-C 2 H 4 O)[END_REF]. It was found subsequently in several other high-mass star-forming regions [START_REF] Nummelin | Abundances of ethylene oxide and acetaldehyde in hot molecular cloud cores[END_REF][START_REF] Ikeda | Survey Observations of c-C 2 H 4 O and CH 3 CHO toward Massive Star-forming Regions[END_REF]. It was also observed in three rotationally cold, but kinetically warm, Galactic center sources [START_REF] Requena-Torres | The Galactic Center: The Largest Oxygen-bearing Organic Molecule Repository[END_REF] and more recently toward the prototypical low-mass protostellar source IRAS 16293-2422 [START_REF] Lykke | The ALMA-PILS survey: First detections of ethylene oxide, acetone and propanal toward the low-mass protostar IRAS 16293-2422[END_REF] and toward a prestellar core [START_REF] Bacmann | Cold and Yet Complex: Detection of Ethylene Oxide in a Prestellar Core[END_REF].

Observations of molecules containing 13 C are viewed as important diagnostic tools because the 12 C/ 13 C ratio in the interstellar medium (ISM) differs from the terrestrial value of 89 [START_REF] Berglund | Isotopic compositions of the elements 2009 (IUPAC Technical Report)[END_REF]. It is around 20 to 25 in the Galactic center region [START_REF] Halfen | The 12 C/ 13 C Ratio in Sgr B2(N): Constraints for Galactic Chemical Evolution and Isotopic Chemistry[END_REF][START_REF] Müller | Rotational spectroscopy of isotopic vinyl cyanide, H 2 CCHCN, in the laboratory and in space[END_REF][START_REF] Belloche | Exploring molecular complexity with ALMA (EMoCA): Deuterated complex organic molecules in Sagittarius B2(N2), Astron[END_REF][START_REF] Müller | Exploring molecular complexity with ALMA (EMoCA): Alkanethiols and alkanols in Sagittarius B2(N2), Astron[END_REF][START_REF] Jacob | First detection of 13 CH in the interstellar medium[END_REF], increases to about 68 in the Solar neighborhood and even further in the outskirts of the Milky Way [START_REF] Halfen | The 12 C/ 13 C Ratio in Sgr B2(N): Constraints for Galactic Chemical Evolution and Isotopic Chemistry[END_REF][START_REF] Jacob | First detection of 13 CH in the interstellar medium[END_REF][START_REF] Wilson | Abundances in the Interstellar Medium[END_REF][START_REF] Milam | The 12 C/ 13 C Isotope Gradient Derived from Millimeter Transitions of CN: The Case for Galactic Chemical Evolution[END_REF]. Even lower 12 C/ 13 C ratios were found in the envelopes of some late-type stars, such as K4-47 [START_REF] Schmidt | Exotic Carbon Chemistry in a Planetary Nebula: The Unusual Case of K4-47[END_REF][START_REF] Schmidt | Extreme 13 C, 15 N and 17 O isotopic enrichment in the young planetary nebula K4-47[END_REF].

Numerous detections of isotopologs containing 13 C were reported in recent years [START_REF] Halfen | The 12 C/ 13 C Ratio in Sgr B2(N): Constraints for Galactic Chemical Evolution and Isotopic Chemistry[END_REF][START_REF] Müller | Exploring molecular complexity with ALMA (EMoCA): Alkanethiols and alkanols in Sagittarius B2(N2), Astron[END_REF][START_REF] Margulès | Spectroscopic study and astronomical detection of doubly 13 C-substituted ethyl cyanide[END_REF][START_REF] Jørgensen | The ALMA-PILS survey: isotopic composition of oxygen-containing complex organic molecules toward IRAS 16293-2422B[END_REF][START_REF] Jacob | First detection of 13 CH in the interstellar medium[END_REF][START_REF] Cernicharo | Discovery of HC 4 NC in TMC-1: A study of the isomers of HC 3 N, HC 5 N, and HC 7 N[END_REF] and some even much earlier. Variations in the 12 C/ 13 C ratios within one source, such as in IRAS 16293-2422 [START_REF] Jørgensen | The ALMA-PILS survey: isotopic composition of oxygen-containing complex organic molecules toward IRAS 16293-2422B[END_REF], may provide clues on the formation pathways of these molecules. Some of these observations benefited greatly from recent or concomitant laboratory investigations of 13 C containing isotopologs, such as ethanol [START_REF] Bouchez | Millimetre-wave spectrum of anti-13 C 1 and 13 C 2 isotopologues of ethanol[END_REF], acetaldehyde [START_REF] Margulès | Millimeter and submillimeter wave spectra of mono-13 Cacetaldehydes[END_REF], or ethyl cyanide with two 13 C [START_REF] Margulès | Spectroscopic study and astronomical detection of doubly 13 C-substituted ethyl cyanide[END_REF]. Other recently studied isotopologs await detection in space. These include methylamine [START_REF] Motiyenko | Millimeter and submillimeter wave spectra of 13 C methylamine[END_REF], methyl isocyanate [START_REF] Kolesniková | A Comprehensive Spectral Rotational Analysis of the Interstellar Methyl Isocyanate CH 3 NCO[END_REF], methyl mercaptan [START_REF] Ilyushin | Rotational spectrum of isotopic methyl mercaptan, 13 CH 3 SH, in the laboratory and towards Sagittarius B2(N2)[END_REF], and cyclopropenone [START_REF] Müller | Rotational spectroscopy of isotopic cyclopropenone, c-H 2 C 3 O, and determination of its equilibrium structure[END_REF].

The relative abundance of 18 O is less favorable; the terrestrial 16 O/ 18 O ratio is almost exactly 500 [START_REF] Berglund | Isotopic compositions of the elements 2009 (IUPAC Technical Report)[END_REF], and the ratio in the Galactic center of ∼200 [START_REF] Müller | Exploring molecular complexity with ALMA (EMoCA): Alkanethiols and alkanols in Sagittarius B2(N2), Astron[END_REF][START_REF] Wilson | Abundances in the Interstellar Medium[END_REF][START_REF] Guesten | H 2 12 CO/H 2 13 CO ratios from molecular clouds near the galactic center[END_REF][START_REF] Gardner | First detection of the 18 O isotope of methanol[END_REF] is only slightly lower. Nevertheless, some complex organic molecules containing 18 O have been detected in recent years, among them methyl formate [START_REF] Tercero | Microwave and submillimeter spectroscopy and first ISM detection of 18 O-methyl formate[END_REF] and formamide [START_REF] Belloche | Re-exploring Molecular Complexity with ALMA (ReMoCA): interstellar detection of urea[END_REF]. Cyclopropenone is an example of a molecule detected in the ISM whose 18 O isotopolog was studied recently, but was not yet detected [START_REF] Müller | Rotational spectroscopy of isotopic cyclopropenone, c-H 2 C 3 O, and determination of its equilibrium structure[END_REF].

The first studies of the rotational spectrum of oxirane date back to the early days of microwave spectroscopy and include the determination of structural parameters and of its dipole moment through Stark effect measurements [START_REF] Cunningham | The Microwave Spectra, Structure, and Dipole Moments of Ethylene Oxide and Ethylene Sulfide[END_REF]. Its dipole moment was determined even earlier through investigations of its dielectric properties along with those of several other molecules [START_REF] Stuart | Über die Temperaturabhängigkeit der Dielektrizitätskonstanten von Gasen und Dämpfen[END_REF].

ACCEPTED MANUSCRIPT -CLEAN COPY

More accurate transition frequencies of several isotopic species were determined later in the microwave and lower millimeter-wave regions [START_REF] Hirose | Microwave spectra and r 0 , r s , and r m structures of ethylene oxide[END_REF][START_REF] Hirose | Laboratory Microwave Spectrum of Ethylene Oxide[END_REF][START_REF] Creswell | Centrifugal distortion and oxygen-17 quadrupole coupling in ethylene oxide[END_REF]. Extensive data pertaining to the main isotopolog were obtained more recently between 260 and 360 GHz [START_REF] Pan | The Millimeter-and Submillimeter-Wave Spectrum of Ethylene Oxide (c-C 2 H 4 O)[END_REF] and between 15 and 73 cm -1 [START_REF] Medcraft | The Far-infrared Rotational Spectrum of Ethylene Oxide[END_REF]. Results of a millimeterwave and far-infrared study on c-C 2 H 3 DO were reported very recently [START_REF] Albert | The Gigahertz and Terahertz spectrum of monodeutero-oxirane (c-C 2 H 3 DO)[END_REF]. Unlabeled atoms in formulae designate 12 C and 16 O here and in the following. We also mention a high-resolution IR study [START_REF] Flaud | First high-resolution analysis of the ν 15 , ν 12 , ν 5 , ν 10 and ν 2 bands of oxirane[END_REF] and intensity measurements on c-C 2 H 4 O [START_REF] Kwabia Tchana | Absolute line intensities for oxirane in the 11.4 µm spectral region[END_REF].

Our present work describes the investigations of c-13 CCH 4 O and c-C 2 H 4 18 O employing samples of oxirane in natural isotopic composition to facilitate their detection in space. We also measured transition frequencies of the main isotopic species mainly to test the performances of Watson's S and A reductions of the rotational Hamiltonian in combination with an oblate and a prolate representation.

Experimental details

Our measurements were carried out at room temperature using two different spectrometers. Pyrex glass cells with an inner diameter of 100 mm were employed. Both spectrometer systems used VDI frequency multipliers driven by Rohde & Schwarz SMF 100A microwave synthesizers as sources. Schottky diode detectors were utilized below 250 GHz, whereas a closed cycle liquid Hecooled InSb bolometer (QMC Instruments Ltd) was applied above 340 GHz. Frequency modulation was used throughout. The demodulation at 2 f causes an isolated line to appear close to a second derivative of a Gaussian, as can be seen in Fig. 1.

A 5 m long double pass cell equipped with Teflon lenses was used for measurements between 158 and 250 GHz. The pressures were about 1 Pa. Further information on this spectrometer is available elsewhere [START_REF] Martin-Drumel | Millimeter-and submillimeter-wave spectroscopy of disulfur dioxide, OSSO[END_REF]. We achieved frequency accuracies of 5 kHz for this spectrometer in a study of 2-cyanobutane [START_REF] Müller | Rotational Spectroscopy of the Lowest Energy Conformer of 2-Cyanobutane[END_REF], a molecule that displays a much richer rotational spectrum. Measurements between 340 and 505 GHz and between 758 and 1093 GHz were carried out employing a 5 m long single pass cell at pressures of ∼2 Pa. A pressure slightly below 1 Pa was chosen for measurements of medium strong lines of the main isotopolog. Our studies on isotopic formaldehyde [START_REF] Müller | Submillimeter spectroscopy of H 2 C 17 O and a revisit of the rotational spectra of H 2 C 18 O and H 2 C 16 O[END_REF] or thioformaldehyde [START_REF] Müller | Laboratory spectroscopic study of isotopic thioformaldehyde, H 2 CS, and determination of its equilibrium structure[END_REF] demonstrate that accuracies of 10 kHz can be reached quite easily for very symmetric lines with good signal-to-noise ratio up to 1.5 THz. We assigned uncertainties of mostly 5 or 10 kHz for the best lines up to ACCEPTED MANUSCRIPT -CLEAN COPY 50 kHz for less symmetric lines or lines fairly close to other lines.

Results

Spectroscopic properties of oxirane

Oxirane is a very asymmetric rotor of the oblate type with κ = (2B -A -C)/(A -C) equals 0.4093, 0.3557, and 0.6389 for c-C 2 H 4 O, c-13 CCH 4 O, and c-C 2 H 4 18 O, respectively. The main isotopolog as well as the one with 18 O have C 2v symmetry, and the four equivalent H atoms lead to para and ortho spin-statistics with a 3 : 5 intensity ratio, see Fig. 1. The para and ortho levels are described by K a + K c being odd and even, respectively. These spin-statistics are absent in c-13 CCH 4 O because the molecule has only C S symmetry.

The dipole moment of 1.90 D of c-C 2 H 4 O is along the b-axis [START_REF] Cunningham | The Microwave Spectra, Structure, and Dipole Moments of Ethylene Oxide and Ethylene Sulfide[END_REF]. This value includes a small increase of 0.02 D caused by the difference of the OCS reference value used in [START_REF] Cunningham | The Microwave Spectra, Structure, and Dipole Moments of Ethylene Oxide and Ethylene Sulfide[END_REF] compared to more modern values [START_REF] Tanaka | Dipole moment function of carbonyl sulfide from analysis of precise dipole moments and infrared intensities[END_REF][START_REF] Lahaye | CO 2 laser saturation Stark spectra and global Stark analysis of carbonyl sulfide[END_REF]. The lower symmetry of c-13 CCH 4 O is associated with a small a-dipole moment component of ∼0.17 D.

The rotational spectrum of oxirane is sparse at the level of the strongest lines because the rotational parameters are fairly large. However, the spectrum is much richer at the level of the weakest transitions in the line list which are about 4.5 orders of magnitude weaker, leading to somewhat increased accidental overlap of lines.

Observed spectrum and determination of spectro

- scopic parameters of c-C 2 H 4 O
The initial calculation of the rotational spectrum of c-C 2 H 4 O was based on the second version of the Cologne Database for Molecular spectroscopy, CDMS, [START_REF] Müller | The Cologne Database for Molecular Spectroscopy[END_REF][START_REF] Müller | The Cologne Database for Molecular Spectroscopy, CDMS: a useful tool for astronomers and spectroscopists[END_REF][START_REF] Endres | The Cologne Database for Molecular Spectroscopy, CDMS, in the Virtual Atomic and Molecular Data Centre, VAMDC[END_REF] entry of the oxirane main isotopic species from July 2012, which, in turn, is based on the FIR study from that year [START_REF] Medcraft | The Far-infrared Rotational Spectrum of Ethylene Oxide[END_REF]. This study employed Watson's A reduced Hamiltonian in the prolate I r representation, as was done in the first version of the c-C 2 H 4 O CDMS catalog entry of March 2000, on which the FIR study built. The same combination of reduction and representation was used in an IR study [START_REF] Flaud | First high-resolution analysis of the ν 15 , ν 12 , ν 5 , ν 10 and ν 2 bands of oxirane[END_REF], whereas Pan et al. employed Watson's A reduction in the oblate III l representation using 25 spectroscopic parameters in their fit [START_REF] Pan | The Millimeter-and Submillimeter-Wave Spectrum of Ethylene Oxide (c-C 2 H 4 O)[END_REF]. The initial CDMS fit, in contrast, used only 14 spectroscopic parameters, and the FIR study as well as the second CDMS fit used only one additional spectroscopic parameter.

We scrutinized the existing line lists in order to approach the best possible spectroscopic parameters as closely as possible. The FIR transition frequencies had uncertainties of 0.00020 cm -1 assigned initially [START_REF] Medcraft | The Far-infrared Rotational Spectrum of Ethylene Oxide[END_REF]. These uncertainties were judged rather conservatively given their rms value was 0.00011 cm -1 in the fit of the FIR study [START_REF] Medcraft | The Far-infrared Rotational Spectrum of Ethylene Oxide[END_REF]. In addition, one pair of transitions, J = 64 -63, K c = 62 -61 at 61.468827 cm -1 , was omitted from our fit because of a large residual of ∼0.0012 cm -1 between measured and calculated frequencies, and one pair of transitions, J = 61 -60, K c = 61 -60 supposedly at 57.435516 cm -1 , was reassigned to a much stronger transition (37 26,12 -36 25,11 ) much closer to the calculated position. The FIR rms was reduced to 0.00009 cm -1 after these two corrections, and we assigned 0.00010 cm -1 as uncertainties to the FIR transition frequencies. The assignments of four more transition frequencies were extended eventually because more individual transitions contributed substantially to each measured line than was indicated in the initial line list [START_REF] Medcraft | The Far-infrared Rotational Spectrum of Ethylene Oxide[END_REF]; further details are available in our line list which is part of the Supplementary Material. These corrections, however, had only very small effects on the fit. It is important in this context to emphasize that in the case of two or more overlapping transitions, our fitting program SPFIT only judges the intensity-weighted average position of all overlapping transitions.

One set of MW transition frequencies [START_REF] Creswell | Centrifugal distortion and oxygen-17 quadrupole coupling in ethylene oxide[END_REF] was reportedly accurate to 0.01 MHz, but several lines displayed larger residuals between calculated and measured frequencies. In fact, the rms of this data set was very close to 0.02 MHz, and we attributed this value to this set of transition frequencies. Uncertainties of 0.05 MHz and 0.15 MHz, respectively, were employed for the remaining datasets [START_REF] Hirose | Laboratory Microwave Spectrum of Ethylene Oxide[END_REF][START_REF] Pan | The Millimeter-and Submillimeter-Wave Spectrum of Ethylene Oxide (c-C 2 H 4 O)[END_REF], as was done earlier [START_REF] Medcraft | The Far-infrared Rotational Spectrum of Ethylene Oxide[END_REF].

Spectroscopic parameters were determined for this as well as various intermediate line lists using the A reduction in the I r representation and employing the S reduction in the III l and I r representations. The A reduction in the III l representation was only used for the final line list as this combination of reduction and representation appeared to require many more spectroscopic parameters than the other combinations. Pickett's SPFIT program [START_REF] Pickett | The fitting and prediction of vibration-rotation spectra with spin interactions[END_REF] was applied for all fitting, and the SPCAT program was utilized for calculations of all rotational spectra.

We recorded individual transitions or pairs thereof if transitions happened to be close in frequency throughout our study. Series in particular combinations of K a and K c were followed in all instances. Asymmetry doublets with significant, but not complete splitting were avoided as their positions are usually determinable with less accuracy compared to well collapsed or well re- P KK J × 10 15 -204.5 [START_REF] Hirose | Laboratory Microwave Spectrum of Ethylene Oxide[END_REF] P K J × 10 15 30.7 [START_REF] Belloche | Exploring molecular complexity with ALMA (EMoCA): Deuterated complex organic molecules in Sagittarius B2(N2), Astron[END_REF] P JK × 10 15 -33.9 [START_REF] Müller | Rotational spectroscopy of isotopic vinyl cyanide, H 2 CCHCN, in the laboratory and in space[END_REF] p K × 10 15 90.7 [START_REF] Kolesniková | A Comprehensive Spectral Rotational Analysis of the Interstellar Methyl Isocyanate CH 3 NCO[END_REF] p KK J × 10 15 ACCEPTED MANUSCRIPT -CLEAN COPY solved asymmetry doublets. The strongest transitions of the main isotopolog were avoided because their uncertainties were already small such that their impact in the fit would be small. Moreover, these transitions are affected by opacity issues. The 340-506 GHz region was studied extensively with 842 transitions in the final line list. Additional measurements were made between 160-250 GHz and 764-1046 GHz, where 61 and 188 further transitions, respectively, were recorded and retained in the final line list. The number of different frequencies is 727; 728 transitions represent unresolved asymmetry doublets. The maximum quantum numbers J, K a , and K c of our new transition frequencies are 55, 43, and 52, respectively.

The newly recorded transition frequencies deviated by modest amounts from the calculations based on the previous FIR study [START_REF] Medcraft | The Far-infrared Rotational Spectrum of Ethylene Oxide[END_REF], the largest deviation was 4.32 MHz. The deviations were largest for transitions with high values of J and with K c = J. The rms of the new lines fit with the old parameters was 580 kHz, which corresponded to 38 times the experimental uncertainties on average. The signed deviation was 15 kHz, which means that deviations to higher frequencies were only marginally more important than deviations to lower frequencies.

We test the need for additional spectroscopic parameters usually by adding one parameter at a time and evaluating which of the new parameters improves the fit the most. This procedure ensures that the parameter set is about as small as possible and that it is fairly unique. The procedure works usually very well for prolate rotors, though correlation may in some cases require sets of two parameters to be tested. Fitting oblate type rotors was often less straightforward. In all combinations of reductions and representations we tested sets of two parameters to fit the c-C 2 H 4 O transition frequencies; we even tested sets of three and four parameters in the case of the A reduction in the III l representation.

The previous FIR study [START_REF] Medcraft | The Far-infrared Rotational Spectrum of Ethylene Oxide[END_REF] employed 15 parameters using the A reduction in the I r representation; these were a nearly complete set of parameters up to sixth order, except Φ J , plus l K . Our corresponding new fit consists of seven more parameters and is described as a nearly complete set of parameters up to eighth order, except L J and l J , as summarized in Table 1. The weighted rms (or rms error) of the fit is 0.946 with modest scatter for the individual data sets: 1.087 and 1.057 for 19 [START_REF] Creswell | Centrifugal distortion and oxygen-17 quadrupole coupling in ethylene oxide[END_REF] and 40 microwave lines [START_REF] Hirose | Laboratory Microwave Spectrum of Ethylene Oxide[END_REF], respectively; 1.000 for 607 millimeter and submillimeter transitions [START_REF] Pan | The Millimeter-and Submillimeter-Wave Spectrum of Ethylene Oxide (c-C 2 H 4 O)[END_REF]; 0.872 for 1185 FIR transitions [START_REF] Medcraft | The Far-infrared Rotational Spectrum of Ethylene Oxide[END_REF]; and 0.957 for our 1091 transitions with an rms of 20.3 kHz.

The quality of the fits utilizing the S reduction in the III l or the I r representation were similar, but required two parameters more in most fits; the difference was in some intermediate fits larger, in few smaller. The final spectroscopic parameters are also listed in Table 1. We tested the predictive power of the three parameter sets occasionally and found that there was no persistent preference for either set.

Obtaining a somewhat satisfactory fit applying the A reduction in the III l representation with the final line list was challenging. A complete set of parameters up to the eighth order plus several decic parameters yielded a fit that had an rms error more than 20% worse than any of the other fits; the parameters of this fit are also given in Table 1. We tested all reasonable parameters individually, all or nearly all combinations of two of these along with several combinations of three and even some four parameter combinations. These improved the rms error only marginally, ∼1.12 was the best we were able to achieve. This modest refinement was discarded as it came at the expense of many more spectroscopic parameters. 18 O, respectively. The uncertainties were reportedly 10 kHz, a value that was used in early fits, but later increased to 20 kHz because of the rms of these data, as in the case of the main isotopolog, see section 3.2. Data reported by Hirose [START_REF] Hirose | Microwave spectra and r 0 , r s , and r m structures of ethylene oxide[END_REF] were not considered initially even though slightly more transition frequencies were reported extending to slightly higher quantum numbers. The reasons were the larger uncertainties and in part large residuals between measured and calculated frequencies already in the publication, especially for c-C 2 H 4 18 O. The number of transition frequencies was small for both isotopologs. Therefore, the spectroscopic parameters of the main isotopic species in the A reduction and I r representation from the previous FIR study [START_REF] Medcraft | The Far-infrared Rotational Spectrum of Ethylene Oxide[END_REF] were taken as starting values for both isotopic species. The rotational parameters were adjusted in a first step. Through several trial fits, δ K and δ J were determined as the parameters which improved the rms error of the c-13 CCH 4 O fit the most. The corresponding parameters for c-C 2 H 4 18 O were also δ K and δ J and additionally ∆ JK . Minor adjustments of some further parameters, as frequently done [START_REF] Müller | Rotational spectroscopy of isotopic cyclopropenone, c-H 2 C 3 O, and determination of its equilibrium structure[END_REF][START_REF] Müller | Submillimeter spectroscopy of H 2 C 17 O and a revisit of the rotational spectra of H 2 C 18 O and H 2 C 16 O[END_REF][START_REF] Müller | Laboratory spectroscopic study of isotopic thioformaldehyde, H 2 CS, and determination of its equilibrium structure[END_REF], improved the fit only slightly.

Observed spectrum and determination of spectroscopic parameters of c-

We started our investigations of the two minor isotopic species in the 340-505 GHz region followed by 158-250 GHz and 758-1092 GHz, as for the main isotopic species. There was no caution required to avoid strong transitions of these isotopologs as their abundances in natural isotopic composition are lower than those of the main species by factors of ∼45 (there are two structural identical C atoms in the molecule) and ∼500, respectively. The quality of the extrapolations to submillimeter wavelengths was unclear because of the limited data set. Moreover, only one R-branch transition frequency was reported for either isotopolog, and this was the J = 1 -0 transition. Since patterns of two transitions are easier to recognize than individual transitions, we searched for transitions with small, but resolved asymmetry splitting. In the case of c-13 CCH 4 O, these were transitions with K a = J and J = 9, 8, and 7 as well as transitions having K c = J -4 and J = 14, 13, and 12. Both types of transitions were also searched for in the case of c-C 2 H 4 18 O, but with slightly different values of J because of the very different κ value, see section 3.1, namely 10 to 7 for the nearly prolate paired transitions and 11 to 9 for the nearly oblate paired transitions plus the oblated paired J = 14 in this series. After almost all of these transitions were found to be unblended, the spectroscopic parameters of both isotopologs were updated, and more transitions with similar combinations of quantum numbers were recorded. Subsequently, further series were sought, including large series of Q-branch transitions until there were no more transitions that had enough intensity to be recorded and a calculated uncertainty at least of order of the achievable measurement accuracy.

The final line lists consisted of 553 and 274 transitions between 340-505 GHz, 211 and 186 transitions between 158-250 GHz, and 341 and 207 transitions between 758-1092 GHz for the isotopologs containing one 13 C and one 18 O, respectively. 680 of the 1111 and 468 of the 667 transitions correspond to unresolved asymmetry doublets for c-13 CCH 4 O and c-C 2 H 4 18 O, respectively, resulting in 771 and 433 different transition frequencies. The maximum quantum numbers J, K a , and K c of our new transition frequencies are 45, 34, and 39 for c-13 CCH 4 O and 38, 30, and 38 for c-C 2 H 4 18 O. As the A reduction in the I r representation required the least number of spectroscopic parameters to fit the transition frequencies of the main isotopic species, we only tried this combination of reduction and representation. Additional parameters for the main isotopolog were subsequently transferred as fixed parameters to the fits of the minor isotopologs. Higher order distortion parameters of these, which were kept fixed in the fit, were adjusted if lower order parameters differed markedly from those of the main isotopic species. The changes were applied by evaluating trends in related parameters, such as ∆ K , Φ K , and L K .

Our new transition frequencies of the two minor isotopic species deviated much more from the initial calculations than in the case of the main isotopolog because of the much smaller initial line lists for both minor isotopologs which resulted in substantially less reliable spectroscopic parameters. The rms of the new lines fit with the initial parameters is 29.9 MHz, the average shift is 17.3 MHz, and the rms corresponds to more than 2500 times the experimental uncertainties on average for c-13 CCH 4 O. In the case of c-C 2 H 4 18 O, that rms is 63.4 MHz, the average shift is -14.5 MHz, and the rms corresponds to almost 3450 times the experimental uncertainties on average. Spectroscopic parameters derived from the main isotopolog or new spectroscopic parameters were tested as was done for the main isotopic species. That is, we searched for the parameter whose floating or inclusion improved the fit the most in each fitting round. An additional prerequisite to keep a new parameter in the fit or to keep a previously parameter floated was that the parameter was determined sufficiently well, meaning that its uncertainty should be less than one fifth of its magnitude.

In the final fitting round, we tested the transition frequencies reported by Hirose [START_REF] Hirose | Microwave spectra and r 0 , r s , and r m structures of ethylene oxide[END_REF]; 15 and 18 lines were included with uncertainties of 0.05 MHz for c-13 CCH 4 O and c-C 2 H 4 18 O, respectively, and three respectively four lines were omitted because of large residuals. The final sets of spectroscopic parameters are given in Table 2.

The weighted rms (or rms errors) of the fits are 0.907 and 0.906 for the isotopologs containing one 13 C and one 18 O, respectively. The rms errors for 11 lines from Creswell and Schwendemann [START_REF] Creswell | Centrifugal distortion and oxygen-17 quadrupole coupling in ethylene oxide[END_REF], 15 lines from Hirose [START_REF] Hirose | Microwave spectra and r 0 , r s , and r m structures of ethylene oxide[END_REF], and 1111 lines from our present study are 0.898, 0.805, and 0.909, respectively for c-13 CCH 4 O; the rms of our lines is 22.1 kHz. The corresponding values for 16 lines from Creswell and Schwendemann [START_REF] Creswell | Centrifugal distortion and oxygen-17 quadrupole coupling in ethylene oxide[END_REF], 18 lines from Hirose [START_REF] Hirose | Microwave spectra and r 0 , r s , and r m structures of ethylene oxide[END_REF], and 667 lines from our present study are 1.056, 0.978, and 0.897, respectively; the rms of our lines is 24.4 kHz.

Discussion

The main interest for our study of the oxirane main isotopolog was the test of various combinations of reduction and representation. We point out that we did not have any program at our disposal to try out the II r or II l representation, but these are very rarely used in published fits anyhow. A rather large set of spectroscopic parameters was needed to obtain a somewhat satisfactory fit in the III l representation using the A reduction, similar to results from a millimeter and submillimeter study [START_REF] Pan | The Millimeter-and Submillimeter-Wave Spectrum of Ethylene Oxide (c-C 2 H 4 O)[END_REF]. This combination of reduction and representation would be considered by many spectroscopists to be the natural choice as they commonly use the A reduction for traditional reasons and as oxirane is an asymmetric top of the oblate type that is with κ = 0.4093 quite far from the symmetric limit.

Considerably fewer parameters, however, are required in the III l representation using the S reduction, as can be seen in Table 1, emphasizing again the greater versatility of the S reduction compared to the A reduction if the representation is used that fits to the asymmetry type of the molecule. We may, of course, also use a different representation. The S reduction in the I r representation yields a fit of similar quality and with the same number of parameters as the S reduction in the III l representation. It is interesting to note that the A reduction in the I r representation requires two parameters fewer still to produce a fit of about the same quality. Thus, while the S reduction is the prime choice if one only considers the III l representation, the A reduction in the I r representation is preferable if all these four combinations are considered.

The poor performance of the A reduction in an oblate representation (III l and III r differ only in some signs of off-diagonal distortion parameters) is not unique to oxirane, but is at least somewhat more widespread. Yamada and Klee [START_REF] Yamada | Pure Rotational Spectrum of H 2 S in the Far-Infrared Region Measured by FTIR Spectroscopy[END_REF] carried out an FIR study on H 2 S and fit their data together with rotational data of microwave accuracy, employing full parameter sets of eighth order and diagonal decic parameters. Although H 2 S is an asymmetric rotor of the oblate type (κ = 0.5234), they encountered convergence problems using the A reduction in the III r representation. A satisfactory fit resulted from a fit in which the S reduction in the same representation was applied. The authors also tried fits in the I r representation. In this case, the A reduction gave a fit that was noticeably worse than the S and III r combination, whereas the S reduction (in I r ) yielded the best fit.

Another example is the lowest energy conformer of 2-cyanobutane, which is a very asymmetric rotor of the oblate type κ = 0.1404 [START_REF] Müller | Rotational Spectroscopy of the Lowest Energy Conformer of 2-Cyanobutane[END_REF]. The combination of A reduction and III l representation required the most parameters; the best result was achieved with the S reduction in the III l representation while both reductions in the I r representation required an intermediate number of parameters.

In a study of SO 2 isotopologs containing one and two 18 O, asymmetric top molecules somewhat close to the ACCEPTED MANUSCRIPT -CLEAN COPY prolate limit, Margulès et al. [START_REF] Margulès | Millimeterwave and submillimeterwave spectra of sulfur dioxide 32 S 16 O 18 O and 32 S 18 O 2 , centrifugal distortion analysis and equilibrium structure[END_REF] reemphasized Watson's recommendation to favor the S reduction over the A reduction because of the smaller correlation coefficients [START_REF] Watson | Aspects of quartic and sextic centrifugal effects on rotational energy levels[END_REF]. The authors added that the S reduction may be a better choice even in cases in which more spectroscopic parameters are needed because of the more favorable condition numbers [START_REF] Margulès | Millimeterwave and submillimeterwave spectra of sulfur dioxide 32 S 16 O 18 O and 32 S 18 O 2 , centrifugal distortion analysis and equilibrium structure[END_REF]. The condition number is the ratio of the largest singular value of the Jacobian matrix over the smallest and indicates the degree of ill-conditioning; a large value may indicate that the determined spectroscopic parameters are not reliable. Among the condition numbers of the fits of the main isotopolog, the value is very large, almost 15000, only in the case of the III l representation and the A reduction; the values are much smaller and fairly similar, 150, 91, and 92, for the combinations III l & S, I r & A, and I r & S, respectively. Therefore, our preference of I r & A is supported from the viewpoint of the condition number.

Substitution of one 12 C with 13 C or of one 16 O with 18 O reduces the rotational parameters by modest amount, except for B of the c-C 2 H 4 18 O isotopolog. Substitution of an atom on the symmetry axis does not affect the equilibrium rotational parameter in the Born-Oppenheimer approximation. The corresponding ground state rotational parameter of a heavy-atom substitution is frequently slightly larger because of effects of anharmonicity, as is observed in the case of oxirane. The quartic distortion parameters of both minor isotopic species are quite close to those of the main species, those of the 13 C species more so than those of the 18 O, which is commensurate with the changes in rotational parameters. The situation is more complex among the higher order parameters, in part possibly because fewer parameters were varied as a consequence of the smaller data sets extended to lower quantum numbers. Some parameters of c-13 CCH 4 O display large changes compared to the main isotopolog, possibly as a consequence of the different symmetry. The Φ J value of the main species appears to be accidentally very small in magnitude compared with almost all of the remaining sextic distortion parameters. The Φ J value of c-13 CCH 4 O is larger in magnitude and has an opposite sign; it was even necessary to fit the related L J for this isotopolog. On the other hand, φ J is much smaller in magnitude than that of the main species. The values of some of the octic distortion parameters will depend on the values of parameters that were kept fixed in the fit.

We were interested if c-13 CCH 4 O, as the more promising of the two minor isotopologs, could be found in existing astronomical data. The 12 C/ 13 C ratio is particularly favorable in the Galactic center. Among its sources, Sagittarius (Sgr) B2 is among the most promi-nent sources in the whole Galaxy. A search toward Sgr B2(N1S) and Sgr B2(N2) in the ReMoCA data, a molecular line survey at 3 mm with the Atacama Large Millimeter Array [START_REF] Belloche | Re-exploring Molecular Complexity with ALMA (ReMoCA): interstellar detection of urea[END_REF], was unsuccessful. Assuming a 12 C/ 13 C ratio of 20 [START_REF] Müller | Rotational spectroscopy of isotopic vinyl cyanide, H 2 CCHCN, in the laboratory and in space[END_REF][START_REF] Belloche | Exploring molecular complexity with ALMA (EMoCA): Deuterated complex organic molecules in Sagittarius B2(N2), Astron[END_REF][START_REF] Müller | Exploring molecular complexity with ALMA (EMoCA): Alkanethiols and alkanols in Sagittarius B2(N2), Astron[END_REF], all lines of c-13 CCH 4 O were heavily blended with other, stronger lines or did not exceed the 3σ average noise level of the survey (A. Belloche, private communication, 2021).

Conclusion

We have obtained greatly improved spectroscopic parameters for the c-13 CCH 4 O and c-C 2 H 4 18 O isotopologs which should be sufficiently accurate for all radio astronomical observations. The investigations into the performance of various combinations of representations and reductions in fitting an extended data set of the oxirane main isotopolog revealed once more the poor performance of the A reduction in an oblate representation, suggesting that these combinations should be avoided. In the more general case of oblate-as well as prolatetype asymmetric rotors we conclude that the S reduction should be preferred for fits employing a natural choice of representation. Other choices are only recommended if all common combinations of representation and reduction were tested.

Concerning c-13 CCH 4 O in space, it appears as if dedicated searches are required to detect this isotopolog in the interstellar medium.

Note added in proof

After acceptance of our manuscript, we became aware of an interesting article on dimethylsulfoxide [START_REF] Margulès | Choice of the reduction and of the representation in centrifugal distortion analysis: A case study of dimethylsulfoxide[END_REF] that also discusses reductions and representations. The findings are in agreement with the findings in publications discussed in our article.

CRediT authorship contribution statement

Figure 1 :

 1 Figure 1: Sections of the rotational spectrum of oxirane displaying the presence of 3 : 5 para to ortho spin-statistics for c-C 2 H 4 O (a) and c-C 2 H 4 18 O (c) as well as the absence in c-13 CCH 4 O (b).

a

  Numbers in parentheses are one standard deviation in units of the least significant figures. b Dimensionless. c See Section 3.2 for additional details on the numbers of transitions and the maximum quantum numbers. d Weighted standard deviation of the fit.

a

  Watson's A reduction was used in the I r representation. Numbers in parentheses are one standard deviation in units of the least significant figures. Spectroscopic parameters without uncertainties were evaluated from the main isotopic species and kept fixed in the fits, see section 3.3. b Dimensionless. c See Section 3.3 for additional details on the numbers of transitions and the maximum quantum numbers. d Weighted standard deviation of the fit.

  Holger S.P. Müller: Conceptualization, Investigation, Methodology, Formal analysis, Validation, Data curation, Writing -Original Draft, Writingreview & editing. Jean-Claude Guillemin: Resources, Writingreview & editing. Frank Lewen: Resources, Writingreview & editing. Stephan Schlemmer: Funding acquisition, Resources, Writingreview & editing.

Table 1 :

 1 Spectroscopic parameters a (MHz) of the main isotopolog of oxirane employing Watson's S and A reduction of the rotational Hamiltonian both in the oblate representation III l and in the prolate representation I r .

		S reduction			A reduction
	Parameter	III l	I r	I r	III l	Parameter
	A	25483.88987 (11) 25483.88751 (10)	25483.86273 (9)	25483.88386 (10) A
	B	22120.83638 (9)	22120.82714 (9)	22120.87303 (7)	22120.84394 (9)	B
	C	14097.83590 (11) 14097.85210 (11)	14097.82581 (8)	14097.83400 (16) C
	D K × 10 3	26.07490 (13)	2.95932 (23)	27.58930 (22)	27.63421 (33) ∆ K × 10 3
	D JK × 10 3	-68.65174 (30)	50.46597 (26)	20.91041 (21)	-70.50171 (68) ∆ JK × 10 3
	D J × 10 3	50.83976 (24)	15.75678 (24)	20.68240 (14)	51.14486 (26) ∆ J × 10 6
	d 1 × 10 3	9.02067 (5)	-6.21001 (6)	18.10824 (9)	3.39130 (43) δ K × 10 3
	d 2 × 10 3	-0.15233 (2)	-2.46284 (1)	6.21005 (2)	-9.01957 (5)	δ J × 10 3
	H K × 10 9	-191.41 (11)	2712.32 (23)	2.2415 (2)	-10.8512 (10)	Φ K × 10 6
	H K J × 10 9	373.61 (21)	-3535.95 (28)	-2.6387 (3)	16.0605 (16)	Φ K J × 10 6
	H JK × 10 9	-198.52 (24)	1096.22 (23)	624.58 (18)	-5321.27 (97)	Φ JK × 10 9
	H J × 10 9	17.92 (19)	-46.58 (18)	-1.797 (69)	110.91 (22)	Φ J × 10 9
	h 1 × 10 9	-8.211 (44)	5.268 (45)	-0.3278 (1)	-18.0197 (18)	φ K × 10 6
	h 2 × 10 9	46.364 (24)	22.470 (14)	277.30 (10)	1546.63 (67)	φ JK × 10 9
	h 3 × 10 9	-49.529 (4)	-6.868 (3)	-1.730 (7)	-57.773 (50)	φ J × 10 9
	L K × 10 12	-3.325 (25)	-14.450 (79)	-0.62 (9)	-132.5 (38)	L K × 10 12
	L KK J × 10 12	8.202 (86)	26.454 (117)	-6.79 (14)	177.5 (61)	L KK J × 10 12
	L JK × 10 12	-8.000 (98)	-13.601 (102)	8.18 (11)	-33.5 (23)	L JK × 10 12
	L JJK × 10 12	4.200 (57)	-1.218 (58)	-3.00 (5)	-9.36 (40)	L JJK × 10 12
	L J × 10 12	-1.083 (49)	0.469 (44)		-1.31 (6)	L J × 10 12
	l 1 × 10 15	479.6 (119)	117.3 (99)	-13.00 (4)	-207.8 (48)	l K × 10 12
	l 2 × 10 15	-72.7 (81)	-297.1 (36)	8.07 (5)	-110.5 (14)	l K J × 10 12
	l 3 × 10 15	87.5 (25)	-147.8 (15)	-2.42 (3)	4.43 (26)	l JK × 10 12
	l 4 × 10 15	-11.9 (4)	31.7 (2)		0.561 (14)	l J × 10 12
					-187.5 (9)	P K × 10 15
					361.2 (30)

  Only 11 and 15 transitions were reported by Creswell and Schwendeman for c-13 CCH 4 O and c-C 2 H 4

13

CCH 4 O and c-C 2 H 4 18 O

Table 2 :

 2 Spectroscopic parameters a (MHz) of the oxirane isotopologs containing one13 C or one18 O.

	Parameter	c-13 CCH 4 O	c-C 2 H 4	18 O
	A	25291.99778 (17) 23992.43450 (20)
	B	21597.97459 (13) 22121.17197 (14)
	C	13825.77925 (14) 13628.24006 (9)
	∆ K × 10 3	27.62104 (47)	24.65010 (70)
	∆ JK × 10 3	20.47198 (31)	17.45625 (69)
	∆ J × 10 6	20.03716 (31)	20.48033 (8)
	δ K × 10 3	18.29406 (30)	16.34518 (30)
	δ J × 10 3	6.05179 (5)	6.31363 (4)
	Φ K × 10 6	1.9510 (4)	2.0748 (8)
	Φ K J × 10 6	-2.2614 (6)	-2.4735 (9)
	Φ JK × 10 6	0.5141 (2)	0.5999 (6)
	Φ J × 10 9	4.97 (30)	-1.9
	φ K × 10 6	-0.2208 (6)	-0.3603 (3)
	φ JK × 10 6	0.2292 (5)	0.2704 (3)
	φ J × 10 9	-0.251 (29)	-1.6
	L K × 10 12	-0.51	-0.56
	L KK J × 10 12	-7.42 (37)	-6.56
	L JK × 10 12	7.00	7.40
	L JJK × 10 12	-2.40	-2.48
	L J × 10 12	-1.158 (97)
	l K × 10 12	-12.66 (35)	-10.1
	l K J × 10 12	9.45 (45)	7.0
	l JK × 10 12

Acknowledgments

It is our great pleasure to dedicate this work to the memory of J.K.G. Watson for his many and invaluable contributions to the theory of high-resolution spectroscopy. We will also keep him in mind for his great sense of humor and for his Scottish accent.

We thank Arnaud Belloche for communicating results of his search for c- 13 CCH 4 O in astronomical data. The work in Köln was supported by the Deutsche Forschungsgemeinschaft through the collaborative research center SFB 956 (project ID 184018867) project B3 and through the Gerätezentrum SCHL 341/15-1 ("Cologne Center for Terahertz Spectroscopy"). J.-C. G. acknowledges support by the Centre National d'Etudes Spatiales (CNES; grant number 4500065585) and by the Programme National Physique et Chimie du Milieu Interstellaire (PCMI) of CNRS/INSU with INC/INP co-funded by CEA and CNES. Our research benefited from NASA's Astrophysics Data System (ADS).

Declaration of competing interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Appendix A. Supplementary Material

Supplementary data associated with this article can be found, in the online version, at https://doi.org/101016/j.jms. ...