Colin M Western

Luke Carter-Blatchford

Patrick Crozet

Amanda J Ross

Jérôme Morville

Dennis W Tokaryk

The spectrum of N 2 from 4500 to 15700 cm -1 revisited with PGOPHER

)) reported a line list from a discharge of molecular nitrogen from 4500 to 11000 cm -1 for this purpose. With a hollow-cathode discharge source, we have extended the experimental spectrum up to 15700 cm -1 , to include the range of Ti:sapphire lasers, since the density of N 2 lines is greater than atomic atlases in common use. Recognizing that experimental conditions can vary, we have also analysed the spectra (comprising several B ³Π g -A ³Σ u + , B' ³Σ u --B ³Π g , and W ³Δ u -B ³Π g N 2 bands) with standard Hamiltonians, so that any part of the discharge spectrum 4500 -15700 cm -1 can be simulated. Parameters are given to do this with the spectral simulation and analysis package PGOPHER. (C. Western, J. Quant. Spectrosc. . Transf., 186, 221 (2016)). The analysis also included high-level ab initio calculations of potential energy curves, transition moments and spin-orbit coupling constants and these were used in preparing the model, extending the potential range of applicability. The results are available in a variety of formats to suit possible applications, including the experimental spectrum in ASCII, a detailed line list with positions and Einstein A coefficients, and a PGOPHER input file to synthesize the spectrum at selectable temperature and resolution.

Rad

Introduction

The first positive system of the N 2 electronic transition appears readily in most discharges containing air or nitrogen. It occurs in emission spectra from the Earth's upper atmosphere, associated with phenomena including aurorae [START_REF] Ashrafi | Modelling of N 2 1 P emission rates in aurora using various cross sections for excitation[END_REF], sprites [START_REF] Luque | Modeling and analysis of N 2 (B 3 Π g ) and N 2 (C 3 Π u ) vibrational distributions in sprites[END_REF][START_REF] Bucsela | N 2 (B 3 Π g ) and N 2 + (A 2 Π u ) vibrational distributions observed in sprites[END_REF], and falling meteoroids, such as Leonid meteor showers [START_REF] Jenniskens | Search for the OH (X 2 Π) Meinel band emission in meteors as a tracer of mineral water in comets: Detection of N 2 + (A-X)[END_REF]. The transitions all involve the B 3 Π g state, with the strongest transitions to the A 3 Σ u + state and weaker contributions from transitions involving the B' ³Σ u -and W ³Δ u states. These electronic transitions between excited states have been extensively studied in the past, as have many other band systems of neutral and ionic forms of N 2 , but they do not figure in standard data bases for atmospheric absorption, such as HITRAN [START_REF] Gordon | The HITRAN2016 molecular spectroscopic database[END_REF]. The A 3 Σ u + state of N 2 is metastable, making it possible to detect B-A transitions in absorption in the laboratory, as well as in emission. The Lyon group became motivated to re-record the spectrum in the near-IR because they recognized contributions from the B-A system of 14 N 2 around 13000 cm -1 in an intracavity absorption experiment [START_REF] Rutkowski | Continuous Vernier filtering of an optical frequency comb for broadband cavity-enhanced molecular spectroscopy[END_REF],

where traces of air had leaked into a vacuum discharge cell. Although N 2 bands were of no spectroscopic interest in this context, they clearly had potential to provide independent calibration markers. Boesch and Reiners [START_REF] Boesch | Spectral line lists of a nitrogen gas discharge for wavelength calibration in the range 4500-11 000 cm -1[END_REF] recently pointed out that a (microwave) nitrogen discharge produces an extensive spectrum with many thousands of lines, and that spectral line lists from N 2 could provide useful wavelength calibration for high-resolution astrophysical spectrographs. To this end, they recorded the spectrum on a Fourier transform spectrometer, and reported transition wavenumbers and intensities, mostly without assignments, for lines in the 4500-11000 cm -1 region. Their work stopped short of the Ti:sapphire laser wavelengths, where electronic transitions provide signatures of first row transition metal hydrides and oxides in spectra of cool stars [START_REF] Bernath | Metal Hydrides in Astronomy[END_REF]. We have extended their N 2 discharge 'atlas', recording emission from a hollow cathode discharge up to 15700 cm -1 by Fourier transform spectrometry, so that the spectrum (in electronic format) could be matched to measurements from a different environment. We supply the raw spectrum, corrected for instrumental intensity response, as an ascii data file and as a simple peak list. However, because the conditions of production of N 2 will vary according to situation (type of discharge, pressure, temperature), comparisons will sometimes be more meaningful using a reference spectrum generated from reliable spectral parameters. The second objective of this work is therefore to present a global analysis of the transitions making up the N 2 atlases (taking all the unassigned lines of ref. [START_REF] Boesch | Spectral line lists of a nitrogen gas discharge for wavelength calibration in the range 4500-11 000 cm -1[END_REF], plus the new near-IR data), giving parameters from fits to what are now accepted as standard Hamiltonians for 3 Σ [9], 3 Π [10] and 3 Δ [11] states, with notation following IUPAC recommendations [START_REF] Hirota | Symbols for fine and hyperfine parameters[END_REF]. The line assignments are unchanged from the detailed investigations on triplet transitions in 14 N 2 carried out notably by groups in France in the 1970's and 1980's [START_REF] Effantin | Analysis of the (0-0), (1-0), and (2-0) bands of the B 3 Π g →A 3 Σ u + system of 14 N 2 and 15 N 2[END_REF][START_REF] Roux | High-resolution Fourier spectrometry of 14 N 2 infrared emission spectrum: Extensive analysis of the B 3 Π g -A 3 Σ u + system[END_REF][START_REF] Roux | High-resolution Fourier spectrometry of 14 N 2 infrared emission spectrum: Extensive analysis of the w 1 Δ u → a 1 Π g system[END_REF][START_REF] Effantin | High resolution Fourier spectrometry of 14 N 2 infrared emission spectrum: Analysis of the (2-0) band of the W 3 Δ u → B 3 Π g system[END_REF][START_REF] Roux | High-resolution Fourier spectroscopy of 14 N 2 and 15 N 2 infrared emission spectrum: Analysis of the B′ 3 Σ u --B 3 Π g system[END_REF][START_REF] Cerny | High-resolution Fourier spectrometry of 14 N 2 and 15 N 2 infrared emission spectrum: Extensive analysis of the W 3 Δ u -B 3 Π g system[END_REF][START_REF] Roux | Investigation of the rovibrational levels of the B 3 Π g state of 14 N 2 molecule above the dissociation limit N( 4 S)+N( 4 S) by Fourier transform spectrometry[END_REF], but the parameters derived here have higher precision in many cases. They are otherwise essentially identical to the earlier work once the definitions of the parameters are taken into account, so we do not discuss the spectroscopic parameters in detail. We do however examine A D and γ as our increased precision allows these to be determined separately; in most cases these are completely correlated [START_REF] Brown | Spin-orbit and spin-rotation coupling in doublet states of diatomic molecules[END_REF]. As well as conventional tables of constants, we supply the information as input for PGOPHER, a widely used program for simulating and fitting rotational, vibrational and electronic spectra [START_REF] Western | PGOPHER: A program for simulating rotational, vibrational and electronic spectra[END_REF][START_REF] Western | Automatic assignment and fitting of spectra with PGOPHER[END_REF][START_REF] Western | PGOPHER, a Program for Simulating Rotational, Vibrational and Electronic Spectra[END_REF].

Experiment

To complement the data published in ref. [START_REF] Boesch | Spectral line lists of a nitrogen gas discharge for wavelength calibration in the range 4500-11 000 cm -1[END_REF], we recorded N 2 emission from 9000 to 15700 cm -1 at Doppler-limited resolution, on a Fourier transform spectrometer. The discharge hollow-cathode head was adapted from a sputter source [START_REF] Vallon | Room-Temperature Metal-Hydride Discharge Source, with Observations on NiH and FeH[END_REF], and consists of a cylindrical hole bored through a water-cooled copper body. Dry N 2 enters this system via a mass flow regulator (40 standard cm 3 /min) through this aperture. The pressure inside the vacuum chamber was maintained around 1 mbar. The lower edge of the system has two chromium cathode plates, fixed either side of a 1.2 mm slit centred on the gas duct. The anode is a copper loop, held in place about 5 mm below the cathode. The entire source is mounted on a primary vacuum chamber, with viewing ports to send light to the spectrometer via a steering mirror and focusing lens. The discharge ran at 150 mA, 600 V for about 2½ hours to record a rotationally-resolved spectrum at an instrumental resolution of 0.023 cm -1 , averaging 80 scans. The spectrum was recorded using a Si-avalanche photodiode, which has a sharp peak in response around 960 nm. Raw intensities were therefore corrected using an instrumental intensity response function (deduced from the spectrum of a standard 100 W halogen bulb).

Typical full-width half-maximum linewidths were 0.04-0.05 cm -1 , and we estimate peak position uncertainties ~ 0.005 cm -1 for unblended lines of N 2 . As noted by Reiners and coworkers [START_REF] Boesch | Spectral line lists of a nitrogen gas discharge for wavelength calibration in the range 4500-11 000 cm -1[END_REF] systematic calibration errors in the wavenumber scale can arise if the apertures of the FT instrument are not correctly set, or if the optical injection is imperfect. Below 11000 cm -1 , we compared peaks in our spectrum with the carefully recalibrated data from Boesch & Reiners. At higher wavenumbers, we compared peaks of atomic Cr I, Cu I and N I lines with reference wavenumbers taken from the NIST atomic data base [START_REF] Kramida | NIST Atomic Spectra Database (ver. 5.1)[END_REF], and a selection of peaks from spectra reported in [START_REF] Roux | High-resolution Fourier spectrometry of 14 N 2 infrared emission spectrum: Extensive analysis of the B 3 Π g -A 3 Σ u + system[END_REF]. A linear correction was applied to the raw wavenumber scale of our FT spectrum to minimize the differences (scattered between -0.008 and +0.021 cm -1 ).

Most of the N I peaks measured on our spectrum were at slightly higher wavenumber than given in NIST tables (N I lines are quoted to 2 decimal places), whilst the N 2 lines around 15000 cm -1 were generally lower than those from the various spectra recorded at Laboratoire Aimé Cotton [START_REF] Roux | High-resolution Fourier spectrometry of 14 N 2 infrared emission spectrum: Extensive analysis of the B 3 Π g -A 3 Σ u + system[END_REF]. A final adjustment to the wavenumber scale of +0.001 cm -1 (at 9000 cm -1 ) to -0.004 cm -1 (at 15500 cm -1 ) was made to produce a self-consistent overall frequency scale for all the data, by ensuring that a selection of the stronger peaks measured above 11000 cm -1 matched wavenumbers calculated from constants derived from an initial fit to the Boesch and Reiners data alone. An overview of our spectrum is shown in Figure 1.

Figure 1. Overview of the Fourier transform emission spectrum of the first positive system of N 2 , identifying the strongest bands in the B ³Π g -A ³Σ u + system. Some N I atomic lines also appear. Intensities have been corrected for instrumental response.

Ab initio calculations

To assist in the simulation of the spectra, ab initio calculations for the four electronic states of interest were performed using MOLPRO 2010.1 [START_REF] Werner | Molpro: a general-purpose quantum chemistry program package[END_REF][START_REF] Werner | version 2010.1, a package of ab initio programs see[END_REF]. Note that the calculations were performed in D 2h symmetry, implying that the two components of the B 3 Π g and W 3 Δ u states have different symmetries, and both are calculated independently. Initially multiconfiguration self-consistent field (MCSCF) calculations [START_REF] Werner | A second-order multiconfiguration SCF procedure with optimum convergence[END_REF][START_REF] Knowles | An efficient second-order MC SCF method for long configuration expansions[END_REF] were performed, stateaveraged over the six component states, including the two components for the B 3 Π g and W 3 Δ u states. These were followed by multi-reference configuration interaction (MRCI) calculations [START_REF] Werner | An efficient internally contracted multiconfigurationreference configuration interaction method[END_REF][START_REF] Knowles | An efficient method for the evaluation of coupling coefficients in configuration interaction calculations[END_REF] on each of the four symmetries required based on the MCSCF calculations. The final energies used include the relaxed Davidson correction [START_REF] Werner | version 2010.1, a package of ab initio programs see[END_REF]. Transition dipole moments and spin-orbit matrix elements were also calculated, the later using the Breit-Pauli Hamiltonian. To investigate convergence with respect to basis set size and active space, a series of calculations were performed at a bond length, r, of 1.3 Å, close to the average equilibrium geometries of the four states; these are tabulated in Table S1. It is clear from the table that calculations are converged with respect to basis set and active space, with energies changing by < 40 cm -1 and transition dipole moments by < 2%. Calculations were thus performed using the default active space with an AV6Z basis at 72 points over the range of 0.8 -3 Å, with the outer limit determined by convergence problems. The calculated potential energy curves, transition moments and spin-orbit matrix elements are given in tables S2-S4 of the supplementary material. RKR curves are available for these states from Roux and Michaud [START_REF] Roux | Investigation of the rovibrational levels of the B 3 Π g state of 14 N 2 molecule above the dissociation limit N( 4 S)+N( 4 S) by Fourier transform spectrometry[END_REF][START_REF] Roux | Extension of the analysis of the B′ 3 Σ u -→ B 3 Π g and W 3 Δ u ⇌ B 3 Π g systems of the nitrogen molecule by Fourier transform spectrometry[END_REF], and the calculated potential energy curves are close to these, though there are some systematic differences, mainly with the RKR curves consistently shifted to slightly shorter bond length. The same calculations gave spin orbit matrix elements of ~75 cm -1 between the A 3 Σ + u and B' 3 Σ - u states, but these were ignored as they would only give an (approximately) r-independent shift of ~0.35 cm -1 . They will also give an equal and opposite contribution of ~0.18 cm -1 to λ for the two states, but this is clearly not a major contribution.

The potential energy curves were fitted to an extended Morse oscillator (EMO) function [START_REF] Roy | An accurate analytic potential function for ground-state N 2 from a direct-potential-fit analysis of spectroscopic data[END_REF][START_REF] Roy | betaFIT: A computer program to fit pointwise potentials to selected analytic functions[END_REF]:

(1)
where β EMO (r) is expressed as a polynomial in the Šurkus reduced variable, :

Note that the reference distance, r ref , in the Šurkus reduced variable is chosen for best fit, and is typically longer than r e . The number of terms in the β EMO (r) polynomial were chosen to give an overall residual < 2 cm -1 . A weighted fit was used, with points above the dissociation limit being given progressively smaller weights. For the B 3 Π g state the ab initio PE curve suggested a small maximum around 2.4 Å and the points around this region were also given a smaller weight. Details of the fits are given in table S2. Similar fits for the transition moments and spin-orbit coupling matrix elements are given in tables S3 and S4, though spline interpolation of the numerical values was used for the calculations below.

Data analysis and fitting with PGOPHER

The assignment and analysis of the N 2 spectra was straightforward, given the availability of previous analyses [START_REF] Effantin | Analysis of the (0-0), (1-0), and (2-0) bands of the B 3 Π g →A 3 Σ u + system of 14 N 2 and 15 N 2[END_REF][START_REF] Roux | High-resolution Fourier spectrometry of 14 N 2 infrared emission spectrum: Extensive analysis of the B 3 Π g -A 3 Σ u + system[END_REF]. For fitting, the Fourier transform (FT) emission spectrum described above was combined with the line list compilation given as supplementary data by Boesch and Reiners [START_REF] Boesch | Spectral line lists of a nitrogen gas discharge for wavelength calibration in the range 4500-11 000 cm -1[END_REF]. Their line list includes frequency and intensity pairs for each line, together with error estimates for both. Some lines are flagged as giving poor results in the line profile fitting process used to generate the line list, and these were discarded, but the remaining frequency and error estimates were used directly in our fits. The estimated errors in line positions were as low as 2×10 -6 cm -1 for strong, isolated lines with 3×10 -5 cm -1 typical for the stronger lines rising to 0.005 cm -1 for the weakest lines. An equivalent peak list was generated from the FT spectrum, extracting error estimates with algorithms included in the PGOPHER program. The errors in peak positions based on the quality of fits of individual peaks to Gaussians were ~2×10 -4 cm -1 for strong lines rising to 0.005 cm -1 for the weakest lines. Peaks with large uncertainties or with widths significantly larger than the average were discarded. Given the calibration procedure described in the experimental section we are confident that the absolute accuracy is better than 0.01 cm -1 even at the highest

( )( ) ( ) ( ) 2 EMO min exp 1 e e r r r D V - β - - + ( ) q r r y , ; ref ( ) ( ) ( ) ∑ = + - = = β 0 ref ref ref ref EMO , ;
where , ; i q q q q i i r r r r q r r y q r r y a r wavenumbers recorded. The fits below demonstrate the relative accuracy is much better than this, though not quite as good as the estimates based on fitting individual line profiles.

As discussed below, an important part of the assignment process involves comparison of calculated and observed intensities. The fluorescence intensities were calculated from the Einstein A coefficients for the transition (derived from the ab initio dipole moment and potential energy curves for the purpose of initial assignment), weighted by the Boltzmann factor for the upper state. The calculated area is then proportional to the number of photons emitted per second on the transition, requiring a correction factor proportional to frequency for the experimental areas, as these are proportional to emitted power. The Boesch and Reiners data gives peak width as well as intensity at the line centre, so the product of these two is taken to give the peak area. An additional correction is required, as their intensity calibration is done against a black body curve expressed as energy per wavelength, and the areas must by divided by frequency cubed to give a value proportional to the rate of emission of photons.

The automatic assignment feature of PGOPHER, described in ref. [START_REF] Western | Automatic assignment and fitting of spectra with PGOPHER[END_REF] was then used to assign these line lists. Given the previous analyses, an extensive search is not required and in practice most assignment was done by simply taking the line nearest to each calculated line, excluding obvious mis-assignments, fitting and repeating the process a couple of times with re-calculated positions of unassigned lines. Given the large number of lines assigned identifying mis-assignments requires computational assistance. The obvious mechanism of excluding lines with residuals above a certain threshold is rather crude, as it gives no assurance that lines are being excluded for good reasons. We have taken an alternative approach, based on implementing special intensity residuals plots in the PGOPHER program. This new feature offers various (user selectable) plots based on observed and calculated intensity, and allows transitions outside the expected intensity range to be easily discarded. A key feature of the method is that only transitions with unexpectedly high intensity are discarded, while transitions with observed intensity unexpectedly low give cause for concern.

Observed intensity too high simply corresponds to overlapping transitions and is frequently encountered because of the line density. If the fit is correct, all transitions above some intensity threshold should appear in the spectrum, and any absence suggests a mis-assignment or problem with the model. Figure 3 shows typical plots, in this case for the origin band of the B ³Π g -A ³Σ u + transition using data from [START_REF] Boesch | Spectral line lists of a nitrogen gas discharge for wavelength calibration in the range 4500-11 000 cm -1[END_REF]. Both plots show the log of the ratio of intensity of the experimental spectrum at the line centre to the calculated line strength, S, against the upper state energy. Note that we use the measured line position as line centre for assigned features, and take the associated intensity at that point on the spectrum, rather than attempt to use measured peak heights. Calculated line centres, and the intensity on the experimental spectrum, are used for unassigned (or missing) transitions. The intensity plots can then draw attention to missing, or unexpectedly weak, lines in the spectrum. The plots shown in Fig. 3 use the calculated line strength (which excludes the Boltzmann factor), rather than the overall calculated intensity, to give a ratio proportional to population, and so under ideal conditions a linear Boltzmann plot is expected. (The calculated line intensity can also be used for these plots, in which case a horizontal line is expected, though the relatively large scatter in measured intensities made the plots shown more convenient in the current study.) . Plots of the log of the ratio of experimental intensity at the line centre to calculated line strength against upper state energy for the origin band of the B ³Π g -A ³Σ u + transition using data from [START_REF] Boesch | Spectral line lists of a nitrogen gas discharge for wavelength calibration in the range 4500-11 000 cm -1[END_REF]. (a) includes all 978 transitions, and (b) only those 437 included in the final fit. Note that, as discussed in the text, there is a sharp lower limit at each energy. + andsymbols indicate e and f parity in the upper state.

The transitions were selected so that most of the observed/calculated intensity ratios were within ±10% of our final intensity model¸ with more latitude (up to 30%) given to the weak, high-J transitions. This pruned the line list quite significantly, and, for the band shown, only 45% of the transitions were retained. This may seem to be discarding much relevant information, but given the large number of lines assigned there are still sufficient observations for a good determination of the (up to 20) parameters for each band, particularly as the coverage of J, Ω and parity was even. It is important to be aware of this, as the discarded lines included some of the strongest, where a blend led to a poor fit to a Gaussian function or an anomalously high intensity, and such lines would typically be included in standard fits, though usually with reduced weights. Even after the selection by intensity, a few lines with large residuals remained (and were discarded) as, with our data quality, even a blend with a line of intensity 10% of the stronger line (and thus satisfying the intensity criterion) experiences a sufficiently large shift to be visible in the fit, and thus merits discarding. The intensity pruning was iterative, as a good intensity model is required for the final pruning, which was in turn generated as part of the modelling process; the intensities are discussed below. Combining line lists from our work and Boesch and Reiners, assignments were made for v = 0 -9 for the B ³Π g state, v = 0 -8 for the A ³Σ u + state, v = 0 -7 for the B' ³Σ u -state and v = 0 -12 for the W ³Δ u state. While the measurements here have significantly higher precision than the earlier work from Roux et al [START_REF] Roux | High-resolution Fourier spectrometry of 14 N 2 infrared emission spectrum: Extensive analysis of the B 3 Π g -A 3 Σ u + system[END_REF][START_REF] Cerny | High-resolution Fourier spectrometry of 14 N 2 and 15 N 2 infrared emission spectrum: Extensive analysis of the W 3 Δ u -B 3 Π g system[END_REF][START_REF] Roux | Investigation of the rovibrational levels of the B 3 Π g state of 14 N 2 molecule above the dissociation limit N( 4 S)+N( 4 S) by Fourier transform spectrometry[END_REF][START_REF] Roux | Extension of the analysis of the B′ 3 Σ u -→ B 3 Π g and W 3 Δ u ⇌ B 3 Π g systems of the nitrogen molecule by Fourier transform spectrometry[END_REF], the coverage in v is smaller, reflecting different discharge conditions. Overall most of the transitions in the spectra are modelled, with some atomic lines from Cr (the cathode) and N are also present (the latter easily recognizable from their hyperfine-broadened profiles), and some transitions from the A -X transition in N 2 + .

The Hamiltonian used to model this data is as standard as possible [START_REF] Hirota | Symbols for fine and hyperfine parameters[END_REF][START_REF] Western | PGOPHER: A program for simulating rotational, vibrational and electronic spectra[END_REF], and includes rotational, spin-orbit, spin-spin, spin-rotation and Λ-doubling terms, with centrifugal distortion included as necessary to fit the data:

rot SO SS SR LD H H H H H H = + + + + (3) 2 4 6 ˆˆr ot H B D H = - + N N N (4) 2 ˆ, 2 D SO z z z z A H AL S L S + ⎡ ⎤ = + ⎣ ⎦ N (5) ( ) ( ) ( ) 2 2 2 2 2 2 2 4 2 1 2 1 2 3 2 3 2 3 ˆˆˆˆˆˆ3 3 , 3 , SS z D z H z H S S S λ λ λ + + ⎡ ⎤ ⎡ ⎤ = - + - + - ⎣ ⎦ ⎣ ⎦ S S N S N (6) 2 1 2 ˆˆ, SR D H γ γ + ⎡ ⎤ = ⋅ + ⋅ ⎣ ⎦ N S N S N (7)
( ) ( ) ( ) 2 2 2 2 2 2 2 2 2 1 1 2 4 2 2 2 2 2 1 1 2 4 2 2 2 2 2 2 2 2 2 1 1 2 4
ˆˆˆˆ, ˆˆˆˆ, ˆˆˆˆ,

i i i i LD D i i i i D i i i i D

H o S e S e o S e S e p N S e N S e p N S e N S e

q N e N e q N e N e

- + - + + - + - + - - - - + + -- + + -- + - - - - + - + - + ⎡ ⎤ = + + + ⎣ ⎦ ⎡ ⎤ - + - + ⎣ ⎦ ⎡ ⎤ + + + + ⎣ ⎦ N N N ϕ ϕ ϕ ϕ ϕ ϕ ϕ ϕ ϕ ϕ ϕ ϕ (8)
Initially, a "traditional" fit to the data was performed, fitting each of the above parameters as required for each state. This gave good results, with 17967 transitions fitted to 298 parameters with an average unweighted error of 0.0023 cm -1 and no evidence of perturbations. See tables 1-4 for the parameters. Two points are worthy of comment. Firstly, the fit was a weighted fit, with weights for the observed frequencies derived from the fits of individual line profiles. The fit to the lines from reference [START_REF] Boesch | Spectral line lists of a nitrogen gas discharge for wavelength calibration in the range 4500-11 000 cm -1[END_REF] gave a weighted (dimensionless) average error of 7.1 and an unweighted average error of 0.0019 cm -1 , whereas the fit to our data gave a weighted average error of 3.3 and an unweighted average error of 0.0030 cm -1 . This implies our data is slightly less precise, which can be understood in terms of the greater Doppler width at higher frequency, and a slightly poorer signal to noise ratio. However, the weighted errors imply the uncertainty estimated (in both cases) from the fits to line profiles is too small, by a factor of 3 to 7. To balance the two data sets, the estimated errors from reference [START_REF] Boesch | Spectral line lists of a nitrogen gas discharge for wavelength calibration in the range 4500-11 000 cm -1[END_REF] were scaled up by a factor of 3, and the overall weighted error was then 2.7. The overall unweighted error was 0.0023 cm -1 though this is a slightly misleading figure given the wide range of weights used. For strong unblended lines the typical error was 2×10 -4 cm -1 for the data from reference [START_REF] Boesch | Spectral line lists of a nitrogen gas discharge for wavelength calibration in the range 4500-11 000 cm -1[END_REF] and 6×10 -4 cm -1 above 11000 cm -1 .

Table 1 Constants (/cm -1 ) for the A Given the precision of the fits, with residuals for the strongest lines down to 1/200 of the linewidth it is worth considering the effects of the unresolved hyperfine structure. For the B ³Π g -A ³Σ u + transition the hyperfine structure has been measured [START_REF] Forthomme | Hyperfine structures in the v = 1-0 vibrational band of the B-A transition of N 2[END_REF], and the structure of an individual rotational transition is typically spread over 0.003 to 0.015 cm -1 , depending on the branch and J. This is potentially of concern, given the typical Doppler linewidth of ~0.04 cm -1 , but tests with fitting a Doppler-broadened exact line profile to a simple Gaussian indicates the potential shifts are very small. Even for a transition with a large hyperfine splitting, such as s R 32 [START_REF] Ashrafi | Modelling of N 2 1 P emission rates in aurora using various cross sections for excitation[END_REF] where the components are spread over 0.014 cm -1 , the shift is only 10 -5 cm -1 , so this potential systematic error can be ignored.

The inclusion of both the centrifugal distortion of the spin-orbit coupling, A D and the spin-rotation, γ in the fit for the B ³Π g state also deserves special mention. It is well known that these two parameters are strongly correlated (see for example [START_REF] Brown | Spin-orbit and spin-rotation coupling in doublet states of diatomic molecules[END_REF] for a formal derivation), and normally only one of these is determined. However, the argument for this correlation is based on perturbation theory, and a numerical test based on exact calculations (given in the appendix) shows that the effects of these two parameters are not exactly identical, and our precision is just sufficient to determine both independently for selected states. For the origin band of the B ³Π g -A ³Σ u + transition, for which the best data is available, fitting to just A D or γ gives significantly worse fits, as shown in Table 5, with weighted residuals plotted in Figure 4. This test fit just included the data from reference [START_REF] Boesch | Spectral line lists of a nitrogen gas discharge for wavelength calibration in the range 4500-11 000 cm -1[END_REF],

437 lines in all with J up to 53.

Table 5. Results of alternative fits to the origin band of the B ³Π g -A ³Σ u + transition.

Fit Both γ = 0 A D = 0 There is still a strong correlation between the two parameters, as can be seen by the requirement, based on Watson's sensitivity parameter [START_REF] Watson | Rounding Errors in the Reporting of Least-Squares Parameters[END_REF], for several significant figures in the standard deviation where both are floated. The previous workers on the B ³Π g state [START_REF] Roux | High-resolution Fourier spectrometry of 14 N 2 infrared emission spectrum: Extensive analysis of the B 3 Π g -A 3 Σ u + system[END_REF] were aware of this issue, and constrained one or the other to zero, though their precision on these constants was much less, and in many cases they could determine neither. An additional consideration is that, as discussed more generally below, A D can be calculated directly from the ab initio A(r) curve, and this gives values ~2×10 -5 cm -1 , some 1/20 th of the observed values. This implies the major contribution to A D is not centrifugal distortion, and A D is an effective parameter. Looking at the other parameters, A D ~ -p, and a possible contribution to A D can easily be identified. p arises from the L-uncoupling interaction,

γ×10 4 /
𝐵 Π ! ! 𝐵𝐽 ! 𝐿 ! + 𝐵𝐽 ! 𝐿 ! Σ ! !
, with other 3 Σ g states, and if the same term is used to mix with other 3 Δ g states, a J-dependent contribution to the spin-orbit coupling is induced, with a similar J dependence to A D , and this is the most likely the major contributor to A D .

Comparing the results of this first fit with previous fits (tables [START_REF] Ashrafi | Modelling of N 2 1 P emission rates in aurora using various cross sections for excitation[END_REF][START_REF] Luque | Modeling and analysis of N 2 (B 3 Π g ) and N 2 (C 3 Π u ) vibrational distributions in sprites[END_REF][START_REF] Bucsela | N 2 (B 3 Π g ) and N 2 + (A 2 Π u ) vibrational distributions observed in sprites[END_REF][START_REF] Jenniskens | Search for the OH (X 2 Π) Meinel band emission in meteors as a tracer of mineral water in comets: Detection of N 2 + (A-X)[END_REF] indicates the parameters are essentially consistent. The Hamiltonians used are not identical, though the differences are minor, and mainly relate to differences in definitions of the parameters; see the footnotes to the table for the conversions used for comparison.

It is clear from these plots that all the constants show a smooth trend with v, and an alternative fit was tried, where constants were expressed as polynomials in v + ½, and the polynomial coefficients were fitted rather than constants for individual states. This approach worked well for the smaller constants, such as the spin-spin constant, λ, but required high order polynomials for origin, B and D making them of little value, so this was not pursued, particularly as it is not likely to give a good extrapolation to higher vibrational levels.

Instead, the ab initio potential energy curves described above were used as a basis for a global fit. For this the DUO program [START_REF] Yurchenko | Duo: A general program for calculating spectra of diatomic molecules[END_REF] was used, which allows coefficients in the potential energy expressions above to be fitted to observations. While DUO can fit directly to observed transitions, we chose to fit to rovibrational energy levels calculated by PGOPHER, as it was more convenient to fit the electronic states separately, and we also did not have line lists from some of the earlier work, but could calculate energy levels from published constants. The DUO calculations included much the same rotational terms in the Hamiltonian as the PGOPHER calculations, though the inclusion of the Schrödinger equation for vibrational motion meant that centrifugal distortion parameters were replaced by r dependence of the parameters. The specific parameters included, in addition to the potential, were λ(r) and γ(r) for all four states, A(r) and o(r) for the B 3 Π g and W 3 Δ u states and p(r) and q(r) for the B 3 Π g state. For most terms a linear dependence was assumed:

(9)
For A(r) the ab initio values were taken directly and cubic splines used for interpolation. The input to the fit included calculated levels for all the states measured here, with additional coverage from earlier work: v = 10 -21 of the B ³Π g state from [START_REF] Roux | High-resolution Fourier spectrometry of 14 N 2 infrared emission spectrum: Extensive analysis of the B 3 Π g -A 3 Σ u + system[END_REF][START_REF] Roux | Investigation of the rovibrational levels of the B 3 Π g state of 14 N 2 molecule above the dissociation limit N( 4 S)+N( 4 S) by Fourier transform spectrometry[END_REF], v = 9 -16 for the A ³Σ u + state from [START_REF] Roux | High-resolution Fourier spectrometry of 14 N 2 infrared emission spectrum: Extensive analysis of the B 3 Π g -A 3 Σ u + system[END_REF][START_REF] Roux | Investigation of the rovibrational levels of the B 3 Π g state of 14 N 2 molecule above the dissociation limit N( 4 S)+N( 4 S) by Fourier transform spectrometry[END_REF] and v = 8 -9 for the B' ³Σ u -state from [START_REF] Roux | Extension of the analysis of the B′ 3 Σ u -→ B 3 Π g and W 3 Δ u ⇌ B 3 Π g systems of the nitrogen molecule by Fourier transform spectrometry[END_REF].

Without adjustment, these calculations gave electronic state separations out by ~100 cm -1 , vibrational spacings out by ~10 cm -1 and rotational energies out by ~10 cm -1 at J = 30. The residuals showed smooth trends with v and J, and are excellent for what are essentially purely ab initio calculations. The residuals could all be reduced to < 2 cm -1 by floating just three coefficients in the potentials, specifically V min , D e and r e , with the exception of the B ³Π g state, where the particularly extensive range of vibrational levels required floating a 0 . The fitted coefficients are given in table S2. The residuals for all states still showed a systematic trend with J and v, but floating additional parameters in the potential energy curve did not improve the fit. We did not take this approach any further since it ignores some interactions between electronic states, which might be expected to introduce effects of this order of magnitude. Given the analytical form (1) and parameters chosen for floating, the resulting potentials retain the form of the ab initio potentials, having simply been subject to a uniform shift and scaling. We can therefore have some confidence that they can be used to predict properties for levels outside the observed range of v and J.

The adjustment to the B ³Π g potential energy curve is greater than the others, but the observed levels cover 80 % of the well depth.

The fine-structure constants were not varied as part of the fitting process above, as the residuals from varying these parameters were swamped by the systematic residuals from the rotational structure fit. The ab initio spin orbit constants, A, only required a uniform scaling of 1.003 (for the B ³Π g state) and 1.009 (for the W 3 Δ u state) for a good fit to the observed values. For the remaining fine-structure terms, λ, γ, o, p and q, the coefficients in equation [START_REF] Brown | Rotational Spectroscopy of Diatomic Molecules[END_REF] were determined by manual adjustment so that, when averaged over the vibrational wavefunctions as described below, the observed vibrational dependence of the constants was reproduced.

To generate the final rovibronic parameters, selected constants were then calculated from these refined potential energy curves. This was done using Le Roy's LEVEL program [START_REF] Roy | LEVEL: A computer program for solving the radial Schrödinger equation for bound and quasibound levels[END_REF]. LEVEL and DUO both solve the one-dimensional vibrational Schrödinger equation but

( ) ( ) ref 1 0 r r r - λ + λ = λ
LEVEL was more convenient for this phase; electron spin effects were ignored for these particular calculations. As well as vibrational energies, vibronic transition moments were calculated by integrating the ab initio transition moments over the vibrational wavefunctions calculated by LEVEL. Values for rotational and centrifugal distortion constants (B, D, H …) were derived by using LEVEL to calculate rotational energies (including the centrifugal barrier) for 21 different values of J in the range 0 to 55, and then fitting the energies for a particular vibrational level to the standard polynomial B v J(J+1) + D v J(J+1) + …. The number of terms was chosen to give a fit with an average error of < 10 -4 cm -1 . The values do not exactly match the B v and D v values given by LEVEL, which are only based on J = 0 wavefunctions, but correspond more closely to usual determinations of these parameters from experimental data, and avoid polynomial truncation problems. A similar approach was used to calculate the other parameters, including centrifugal distortion effects by averaging the fitted λ(r) over the vibrational wavefunctions for the same range of J values, and then fitting to a similar polynomial for each v:

λ v (J) = λ v + λ Dv J(J+1) + λ Hv J 2 (J+1) 2 + … (10)
In the light of the discussion above concerning A D and γ, the A v values were determined by applying the procedure above to A(r), but only including low J values so that A D from this source was constrained to zero. A D (r) was then treated as a separate interaction; in principle this is r dependent, but measurements indicated this was constant to our precision. A similar approach could be used to account for centrifugal distortion of the transition moments, but was not as initial calculations from the ab initio curves, and more detailed work by Mishra et al [START_REF] Mishra | Fouriertransform spectroscopic study of the rotational intensity distribution in the first positive B 3 Π g -A 3 Σ u + band system of the nitrogen molecule[END_REF], found these effects to be < 2% of the intensity, and thus below our likely error bars.

The parameters generated as above were used as a basis for our final set of recommended parameters, given in tables S5-S8. For practical use, some adjustment is required, given the limits to the potential energy fit. With no adjustment, our observations were reproduced with an overall weighted error of 8200 (0.7 cm -1 unweighted). Optimizing just the two dominant constants, the origin and B, for all states reduced the overall weighted error to 190, and the unweighted error (0.01 cm -1 ) to less than the linewidth. The largest contribution to the error was now A for the B ³Π g state, and optimising this parameter reduced the error to 11 (0.004 cm -1 unweighted). For our recommended values (tables S5-S8) we additionally optimised λ and γ for v = 0 -9 of the B ³Π g state, D for v = 0 and 1 of the A 3 Σ + u state and o and p for v = 0 and 1 of the B ³Π g state. (A particularly large range of J values have been measured for v = 0 and 1 for the A and B states.). This set of constants is almost as good as our traditional fit, with all parameters floated, giving a weighted error of 3.8 compared to 2.1 (0.0026 cm -1 compared with 0.0023 cm -1 ). Given the minor adjustment to the potential energy curves required, the predictions are likely to have predictive value outside the observed range of v and J.

The assignment process above requires a good intensity model, which we now discuss. The Boltzmann plots for individual bands, such as the one illustrated in Fig 3,all show curvature at high J, indicative of a non-equilibrium high temperature component. This could be fitted with populations modelled as the sum of two Boltzmann distributions, with two different temperatures. Fitting to a single strong band in each spectrum gave values that worked for all the bands in that spectrum. The values used for each spectrum are given in Table 6. We can also calculate the relative intensities of each band, given the ab initio transition moments and adjusted potentials described above. The transition dipole moment for each band is easily calculated by integrating the transition dipole moment function, µ(r) over the vibrational wavefunctions, and these are tabulated in tables S9-S11. These should predict all the relative intensities for each set of transitions involving a given upper level, and indeed do, within the ±10% spread for transitions within a given band. We found that we could account for the relative intensity between most bands with a separate vibrational temperature, given in Table 6. See Figure 5 for a plot of observed/calculated intensity ratio for the spectrum taken with the lowest power in [START_REF] Boesch | Spectral line lists of a nitrogen gas discharge for wavelength calibration in the range 4500-11 000 cm -1[END_REF]. The above process gives absolute absorption cross sections, and we can validate these for the B ³Π g -A ³Σ u + transition by comparing with measured overall lifetimes for vibrational levels of the B ³Π g state. The lifetime can be calculated by summing the Einstein A coefficients for a given upper state over all accessible lower states. (For the B ³Π g state this includes the X 1 Σ g + , B' ³Σ u -and W ³Δ u states, but the total rate to these is very small.) There are various measurements of these lifetimes; the most recent of these is by Eyler and Pipkin [START_REF] Eyler | Lifetime measurements of the B 3 Π g state of N 2 using laser excitation[END_REF] and, as shown in Table 7, their measured values are entirely consistent with our calculated values. It is clear from this that the agreement is very good, and indeed for simple small molecules the uncertainty in the theoretical transition moment is likely to be less than the experimental uncertainties in the lifetimes. (See for example reference [START_REF] Brooke | Line Strengths of Rovibrational and Rotational Transitions in the Ground State of OH[END_REF], where ab initio calculations are used as the basis for calculations of absolute transition intensities.) The intensities given in our final line list are thus likely to be accurate to a few %. Absolute intensities have also been calculated on a purely theoretical basis recently by Ni et al [START_REF] Ni | Ab initio calculations for the first-positive bands of N 2[END_REF][START_REF] Piper | Experimental determination of the Einstein coefficients for the N 2 (B-A) transition[END_REF]; their transition moments are essentially identical to ours, and the calculated results are consistent, though their tabulation of rotational line strengths is less useful as the spin splitting is ignored.

For the other states there are fewer values available for comparison. Some relatively old ab initio calculations are available [START_REF] Werner | Accurate ab initio calculations of radiative transition probabilities between the A 3 Σ u + , B 3 Π g , W 3 Δ u , B′ 3 Σ u -, and C 3 Π u states of N 2[END_REF], which are essentially consistent with ours (within 0.02 D) given they are not done to such a high level. Covey et al [START_REF] Covey | Transition probabilities for the W 3 Δ u -B 3 Π g system of molecular nitrogen[END_REF] 6. The tick marks at the top of the plot indicate transitions used in the fit; note the excluded transitions are also well modelled. This includes some strong transitions, and all the excluded transitions are blends.

Figure 6 shows the quality of our final fit, comparing a small part of the FTS and simulated spectra. This global fit encompasses all transitions between states A ³Σ u + , B ³Π g , B' ³Σ u -and W ³Δ u of the 14 N 2 isotopologue and is based on 17967 observed lines selected to avoid blends using the intensity criterion described above. The derived constants are thus more precise and also more standard than earlier partial contributions concerning B-A [START_REF] Effantin | Analysis of the (0-0), (1-0), and (2-0) bands of the B 3 Π g →A 3 Σ u + system of 14 N 2 and 15 N 2[END_REF][START_REF] Roux | High-resolution Fourier spectrometry of 14 N 2 infrared emission spectrum: Extensive analysis of the B 3 Π g -A 3 Σ u + system[END_REF], B'-B [START_REF] Roux | High-resolution Fourier spectroscopy of 14 N 2 and 15 N 2 infrared emission spectrum: Analysis of the B′ 3 Σ u --B 3 Π g system[END_REF], or W-B [START_REF] Cerny | High-resolution Fourier spectrometry of 14 N 2 and 15 N 2 infrared emission spectrum: Extensive analysis of the W 3 Δ u -B 3 Π g system[END_REF] systems; the new model allows a global description of the main features of the spectrum of N 2 between 4500 and 15700 cm -1 . Experimental line lists and molecular constants are provided as supplementary materials.

Applications

We have two illustrations of practical implementation of this N 2 atlas, based on recognition of features in the emission spectrum, without needing line assignments. The first is to calibrate an intracavity absorption spectrum, recorded around the peak of the gain curve of a mode-locked Ti:sapphire laser. The light source is a spectrally broad frequency comb, which is injected into a high finesse optical cavity (F c =17000), whose mirrors curtail the extent of emission emerging from the optical cavity (still of the order of 1000 cm -1 ). The highreflectivity mirrors produce a large effective pathlength of absorption for any species contained in the cavity, making this a high-sensitivity absorption experiment. An optical Vernier scheme is used to perform continuous optical filtering as the laser frequency comb is scanned across the absorption bands of the sample, analysing the full spectral range at GHz resolution on timescales of the order of 1 second [START_REF] Rutkowski | Continuous Vernier filtering of an optical frequency comb for broadband cavity-enhanced molecular spectroscopy[END_REF]. The optical frequency comb at ILM is not (yet) locked to any external frequency reference, so does not offer metrological precision.

Nevertheless, the arrangement combines fast recording times with enough sensitivity to observe absorption from discharge products, and is still an attractive tool for exploratory spectroscopy, except that there is an obvious need to connect the raw time-scale axis of detection (as the Vernier scans) to a reliable wavelength scale. By admitting a small air leak into a discharge fitted within the optical cavity, N 2 lines can supply the missing calibration.

Figure 8 shows a comparison between a record of N 2 B-A bands seen in absorption in this Vernier spectrometer, and the PGOPHER simulation, which has provided the wavenumber scale. The second practical implementation of this N 2 atlas is more general, since it offers an alternative to either hot (500 °C) iodine [START_REF] Gerstenkorn | Atlas du spectre d'absorption de la molécule d'iode 11000 -14000 cm-1[END_REF] or alkali vapours [START_REF] Ross | A Doppler-limited rubidium atlas in ascii format, 9500 -12 300cm -1[END_REF] to calibrate laser excitation spectra in the near IR (our particular concern is with Ti:sapphire-accessible wavelengths, 720 -1000 nm). Without access to a high-finesse cavity, only weak absorption signals would be expected for transitions from a (metastable) excited electronic state, but optogalvanic spectroscopy is sensitive enough to respond to resonant laser light. This has been demonstrated with cw lasers in the visible region of the B-A system [START_REF] Feldmann | Opto-galvanic spectroscopy of some molecules in discharges: NH 2 , NO 2 , H 2 and N 2[END_REF][START_REF] Sasikumar | High resolution optogalvanic spectrum of N 2 -rotational structure of (11, 7) band in the first positive system[END_REF], and in the near IR around 775 nm [START_REF] Bachir | Laser optogalvanic spectroscopy of N 2 from the A 3 Σ u + metastable state in a corona-excited supersonic expansion[END_REF]. Feldmann [START_REF] Feldmann | Opto-galvanic spectroscopy of some molecules in discharges: NH 2 , NO 2 , H 2 and N 2[END_REF] explored optogalvanic signals at modest resolution, tracking changes in the discharge voltage when laser light crossing the discharge tuned through molecular transitions from 570 -620 nm. He observed changes in sign, and a region of reduced response around 585 nm. Phase effects were also demonstrated at higher resolution in optogalvanic spectra recorded by Bachir et al. [START_REF] Bachir | Laser optogalvanic spectroscopy of N 2 from the A 3 Σ u + metastable state in a corona-excited supersonic expansion[END_REF] in a corona-excited supersonic expansion of argon seeded with N 2 . Figure 2 of their paper showed a 6 cm -1 spectral window where lines of the (2-0) band of the B-A system of N 2 were of opposite phase to transitions in the (8-7) hot bands. Ref. [START_REF] Bachir | Laser optogalvanic spectroscopy of N 2 from the A 3 Σ u + metastable state in a corona-excited supersonic expansion[END_REF] mentions that the relative phase of optogalvanic signals are sensitive to discharge conditions. To verify that these bands could really be used for secondary calibration purposes, we recorded a few optogalvanic signals around 780 nm at U. New Brunswick, using a home-made hollow cathode glass discharge cell operating at room temperature, and electric circuitry copied from the model developed by

Stoicheff and co-workers [START_REF] Dube | Optogalvanic spectrum of Ar excimers in a DC discharge with supersonic expansion[END_REF][START_REF] Pirali | Optogalvanic spectroscopy of the C" 5 Π ui -A′ 5 Σ g + electronic system of N 2[END_REF] to detect and amplify the weak variations of the resistivity of the discharge. A slow leak of air flowed through the system, maintaining a pressure ~0.5 Torr around the tungsten anode and stainless-steel cathode. The single-mode cw Ti:sapphire laser beam (output power around 450 mW from a CR 899 cavity) was mechanically modulated at 820 Hz so that signals could be extracted by means of a lock-in amplifier. Clear optogalvanic signals in the B-A system of N 2 were observed when the pressure was adjusted so that the mauve N 2 plasma just filled the hollow cathode. A stable discharge regime was achieved with a current of 15 mA, a DC voltage of about 2000 V and a 115 kΩ ballast resistance. Figure 9 shows a comparison between part of the FTS discharge emission spectrum and optogalvanic under the conditions described above. In the context of high-resolution spectroscopy, where individual lines can be distinguished, an N 2 discharge provides a stable and affordable reference spectrum with a high density of lines for calibration purposes. The upper trace in Fig. 9 assembles six individual scans of the Ti:sapphire laser. A single scan (typically < 1 cm -1 ) can easily fall between transitions listed in reference spectra from atomic lamps such as Th/Ar [START_REF] Kerber | The Spectrum of Th-Ar Hollow Cathode Lamps in the 691-5804 nm region: Establishing Wavelength Standards for the Calibration of Infrared Spectrographs[END_REF] or U [START_REF] Stephen | The Infrared Spectrum of Uranium Hollow Cathode Lamps from 850 nm to 4000 nm: Wavenumbers and Line Identifications from Fourier Transform Spectra[END_REF][START_REF] Palmer | An Atlas of Uranium Emission Intensities in a Hollow Cathode Discharge[END_REF] for the near IR, as illustrated in Figure 10. Mapping reference signals on to the richer N 2 spectrum makes it much easier to establish a reliable wavenumber scale.

We consider that molecular nitrogen could well be considered as a convenient reference, not only in the laboratory, but also for forthcoming high-resolution near IR telescope-based spectrographs, such as SPIRou [START_REF] Artigau | SPIRou: the near-infrared spectropolarimeter/high-precision velocimeter for the Canada-France-Hawaii telescope[END_REF].

Conclusion

The N 2 bands recorded in [START_REF] Boesch | Spectral line lists of a nitrogen gas discharge for wavelength calibration in the range 4500-11 000 cm -1[END_REF] have been analysed together with an FT emission spectrum covering 'Ti-sapphire' wavelengths using the PGOPHER program, providing constants for the A ³Σ u + , B ³Π g , B' ³Σ u -and W ³Δ u states, involved in the emission spectrum of the first positive system of nitrogen. These are essentially consistent with those reported previously in the 1980's, but more precise, and the Hamiltonian used is more standard.

The reference spectrum is available either as an ascii file (9000-15700 cm -1 ), or in parameterized format (PGOPHER). Molecular constants for the A, B, B' and W electronic states of N 2 obtained with PGOPHER describe the spectrum between 4500 and 15700 cm -1 , identify all observed transitions, and have allowed us to create a reference spectrum for desired conditions (temperature, resolution, lineshape). The global treatment here offers a closer representation of spectral features than earlier analyses. Similar work will be required for taking into account the other spectral features observed in the recorded FTS spectrum, such as those involving 15 N 2 and 14 N 2 + .

Supplementary Material:

Selected supplementary material for this paper has been deposited with the Journal, and a full set is available from the University of Bristol Research Data Repository, doi:10.5523/bris.1u7h3szdcpk5r2dfa1hao9o2u1. The files deposited with the Journal include a PGOPHER input file, the calibrated Fourier transform spectrum taken as part of this work, a sample calculated line list for the origin band of the B-X transition and supplementary tables S1 -S13, containing ab initio results and constants not given elsewhere. The additional files in the data repository include a complete line list, ab initio output files and input files to the fitting process.

Figure 2 .

 2 Figure 2. Ab initio potential energy curves for the four electronic states of N 2 , calculated as described in the text (plot a)), together with spin-orbit functions b), and transition dipole moments, c), as a function of internuclear distance, R.

Figure 3 (Figure 3

 33 Figure 3(b) includes only the 437 transitions included in the final fit, and is close to the expected linear form, though the plot shows a clear curvature arising from a non-Boltzmann population distribution discussed in more detail below. Figure3(a) includes all 978 calculated transitions with intensities more than 0.3% of the strongest, and shows many transitions stronger than calculated but, crucially, none weaker than calculated, implying all the discrepancies can be understood in terms of overlapping transitions.

Figure 4 .

 4 Figure 4. Plot of the weighted residuals for the three alternative fits given in table 5 to the origin band of the B ³Π g -A ³Σ u + transition in N 2 . The top trace corresponds to floating both γ and A D , the middle fixing γ at zero and the bottom fixing A D to zero. Note the different vertical scales in the three plots.

Figure 5 .

 5 Figure 5. Plot of observed/calculated intensities for the spectrum from [7] taken with the lowest power (25 W). The colours represent different upper electronic and vibrational states; the rotational temperature was modelled with two temperatures (576 and 1384 K) and a vibrational temperature of 8530 K. The W ³Δ u -B ³Π g 3-0 band has been excluded from the plot, as the intensity corrections are not available.

  estimated the strength of the B ³Π g -W ³Δ u transition by measuring the Einstein A coefficient of the 2 -0 band as 0.061 of that of the B ³Π g -A ³Σ u + 2 -4 band which is in good agreement with our calculated ratio of 0.081 considering that their estimate of uncertainty of up to ±50%. Lofthus and Kripenje [46] quote an estimate for B' state lifetimes for v = 0 -8 in the range 25 -52 µs, consistent with our calculated range of 17 -47 µs.

Figure 6 .

 6 Figure 6. Comparison between part of the FTS experimental spectrum (upper trace) and the PGOPHER simulation (lower trace) in the B-A (2-0) region. The populations were calculated using the parameters given in Table6. The tick marks at the top of the plot indicate transitions used in the fit; note the excluded transitions are also well modelled. This includes some strong transitions, and all the excluded transitions are blends.

Figure 7

 7 illustrates the relative importance of the B' ³Σ u --B ³Π g , W ³Δ u -B ³Π g and B ³Π g -A ³Σ u + systems to the N 2 emission spectrum discussed here, plotting emission rates per molecule as calculated for a rotational temperature of 500 K and vibrational temperature 5000 K.

Figure 7 .

 7 Figure 7. Overview of the contributions of the electronic systems B' ³Σ u --B ³Π g , W ³Δ u -B ³Π g and B ³Π g -A ³Σ u + to the first positive bands of N 2 . Note the different vertical scale for each system. Selected vibrational assignments are indicated on the diagram; * indicates emission from W to B rather then B to W.

Figure 8 .

 8 Figure 8. Comparison between Vernier spectrum (upper (blue) trace) and PGOPHER simulation of the 14 N 2 B-A (2-0) band (black trace). The discharge was operating with 0.5 Torr in pure N 2 , with 50 mA DC current; the simulation was performed setting T = 380 K, and a Gaussian FWHM of 0.18 cm -1 . The red diamond highlights an overlap of 3 lines (see Fig 7).

Figure 9 .

 9 Figure 9. Comparison between part of the FTS discharge emission spectrum (lower trace) and optogalvanic signals (upper trace). The three resolved lines around 12900 cm -1 correspond to the single peak also indicated by a (red) diamond in Figure 8.

Figure 10 .

 10 Figure 10. Comparison of a section of the N 2 discharge spectrum (upper trace, this work) and U atomic lines from the Uranium atlas [56] (frequently used for calibration of Ti:sapphire laser excitation spectra).

Table 2

 2 Constants (/cm -1 ) for the B 3 Π g state of N 2

							3 Σ +	u state of N 2
	v	Origin					λ			B			D×10	6			γ×10	3	λD×10	6	H×10	12
		This Work	a	Previous	b	This Work	Previous	b,c	This Work	Previous	b	This Work	Previous	b	This Work	Previous	b
	0	0	d			0 -1.328301(14) -1.32855(42) 1.445765671(219) 1.445774(6)	5.794165(384) 5.804(2)		-2.71836(55) -2.4(2)	3.690(59) -2.259(199)
	1	1432.907215(11)	1432.9066 -1.321342(11) -1.32165(42) 1.427459855(215) 1.427440(10) 5.834309(377) 5.827(5)		-2.69331(51) -2.5(2)	3.728(53) -2.892(197)
	2	2838.142474(15)	2838.1400 -1.314253(16) -1.31430(64) 1.409051702(218) 1.409044(17) 5.877523(383) 5.87(1)		-2.66775(59) -2.1(3)	3.743(66) -3.822(201)
	3	4215.637926(22)	4215.6347 -1.307003(20) -1.30770(42) 1.390528368(219) 1.390530(8)	5.924195(329) 5.935(4)		-2.64054(74) -2.3(2)	3.908(90) -5.000(146)
	4	5565.297562(30)	5565.2917 -1.299521(31) -1.30020(67) 1.371875068(283) 1.371905(14) 5.975698(524) 5.995(7)		-2.61286(95) -2.7(2)	3.91(12)	-6.158(288)
	5	6886.992792(41)	6886.9847 -1.291810(47) -1.29225(67) 1.353073432(365) 1.353090(9)	6.033523(721) 6.052(5)		-2.5804(13)	-2.3(4)	4.13(17)	-6.733(411)
	6	8180.556724(70)	8180.5487 -1.283961(85) -1.28400(54) 1.334101621(646) 1.334097(14) 6.09710(148)	6.115(9)		-2.5498(19)	-2.1(2)	4.08(28)	-9.216(946)
	7	9445.77848(11)	9445.7557 -1.27558(13)	-1.27500(64) 1.31493401(101)	1.314909(17) 6.17160(254)	6.160(8)		-2.5191(28)	-2.2(3)	4.18(42)	-9.62(179)
	8	10682.39534(17)	10682.3770 -1.26680(25)	-1.26735(75) 1.29554387(107)	1.295550(19) 6.2695(14)	6.29(1)		-2.4903(48)	-2.3(4)	3.95(80)	0	d
	9				11890.0380		-1.2575(15)		1.27597(3)			6.41(1)		-2.36(5)
	10				13068.4170		-1.2484(15)		1.25595(3)			6.51(2)		-2.24(12)
	11				14217.0100		-1.2390(15)		1.23555(5)			6.62(3)		-3.03(10)
	12				15335.2800		-1.2239(30)		1.21496(4)			6.89(2)		-2.96(8)
	13				16422.5320		-1.2183(15)		1.19370(4)			7.02(2)		-3.18(4)
	14				17478.0070		-1.2048(30)		1.17193(5)			7.25(2)		-3.20(8)
	15				18501.2760		-1.1904(30)		1.14940(7)			7.49(3)		-3.60(10)
	16				19490.4440		-1.1829(45)		1.12619(10)			7.82(4)		-3.57(12)
	Footnotes
	a Systematic error may be up to 0.01 cm -1 . Value for v = 0 is fixed at zero.
	b Ref [14] for v=0-8, [19]for v = 9-16
	c Value quoted is 3/2(ε+γ) from previous work
	d Fixed

Table 2

 2 continued. Constants (/cm -1 ) for the B 3 Π g state of N 2

	v	q×10	5			γ×10	3		AD×10	4			λD×10	7	γD×10	8	qD×10	9	H×10	12
		This Work	Previous	b,g	This Work	h	This Work	h	This Work This Work	This Work This Work
		8.4703(50) 8.52(10)		-3.62240(3797) -4.21312(3174) -6.73(72)	-3.653(93) 1.552(64)	1.130(194)
		8.268(12)	8.26(6)		-3.85913(6945) -4.38467(5736) -4.4(13)	-3.46(22)	1.14(16)	0	i
		7.975(21)	8.05(10)		-3.7324(1058)	-4.25834(8650) -7.9(22)	-4.29(38)	2.21(31)	0	i
		7.758(48)	7.90(20)		-3.4640(1895)	-4.0164(1530)	-14.3(40)	-3.35(85)	2.18(73)	0	i
		7.694(26)	7.60(30)		-3.5400(2306)	-4.0594(1841)	0	i		-2.6(11)	0	i	0	i
		7.402(39)	7.35(20)		-4.7047(3460)	-4.9589(2743)	0	i		0	i	0	i	0	i
		7.01(17)	7.29(30)		-3.1417(4796)	-3.7114(3747)	0	i		0	i	0	i	0	i
		7.05(16)	6.94(20)		-3.765(16)	-4.21312	i		0	i		0	i	0	i	0	i
		7.97(21)	6.99(20)		-3.900(25)	-4.21312	i		0	i		0	i	0	i	0	i
		8.5(13)	6.74(30)		-4.26(14)		-4.21312	i		0	i		0	i	0	i	0	i
				6.33(30)
				6.53(30)
				6.33(50)
				6.32(30)
				8.1(10)
				6.2(20)
				3.6(14)
				3.54(90)
				4.44(80)
				4.67(80)
				4.01(70)
				3.2(10)

Table 3

 3 Constants (/cm -1 ) for the W 3 Δ u state of N 2

	v	Origin				A			λ			B			D×10	6			γ×10	3
		This Work	a	Previous	b, c	This Work	Previous	b	This Work	Previous	b,d	This Work	Previous	b	This Work	Previous	b	This Work	Previous	b,e
	0	9619.60060(65)	9625.4504(32)	5.8174(12)	5.8141(13) 0.66597(61) 0.66975(60) 1.46180782(445) 1.461743(17) 5.6435(63)	5.608(12) -2.937(31)	-2.915(45)
	1	11101.13790(69) 11106.9165(17) 5.76283(90) 5.7568(14) 0.67042(60) 0.67395(60) 1.44473840(495) 1.444687(13) 5.6239(74)	5.616(7)		-3.005(24)	-2.970(40)
	2	12557.80477(48) 12563.5179(6)	5.71082(64) 5.7096(8)	0.67534(33) 0.67515(45) 1.42768803(318) 1.427643(10) 5.64479(447) 5.646(5)		-2.913(14)	-2.805(30)
	3	13989.73512(24) 13995.3789(7)	5.65683(39) 5.6542(13) 0.67848(27) 0.68040(45) 1.4106314(17)	1.410595(14) 5.6494(23)	5.665(7)		-2.877(11)	-3.035(40)
	4	15397.04969(45) 15402.6254(17) 5.60023(62) 5.6007(11) 0.68143(37) 0.68325(60) 1.39358996(324) 1.393536(17) 5.6767(50)	5.677(10) -2.850(18)	-2.940(38)
	5	16779.86826(38) 16785.3759(41) 5.54266(71) 5.5461(20) 0.68476(31) 0.68760(75) 1.37653387(196) 1.376489(19) 5.6935(21)	5.696(9)		-2.807(11)	-3.188(68)
	6	18138.28951(24) 18143.7291(27) 5.47991(42) 5.4785(21) 0.68894(22) 0.69075(90) 1.35947620(137) 1.359430(29) 5.7159(15)	5.715(23) -2.7953(87) -3.273(78)
	7	19472.40489(39) 19477.7802(16) 5.41716(59) 5.4357(22) 0.69256(30) 0.69255(75) 1.34240175(246) 1.34233(3)	5.7368(31)	5.74(2)		-2.767(13)	-2.625(75)
	8	20782.28719(36) 20787.5820	5.35270(53) 5.3427(36) 0.69595(25) 0.697(15)	1.32530630(216) 1.32522(6)	5.7585(28)	5.74(2)		-2.757(12)	-2.050(50)
	9	22067.99104(31) 22073.214	5.28740(57) 5.2676(40) 0.69973(25) 0.700(15)	1.30818187(173) 1.30815(6)	5.7866(19)	5.80(5)		-2.695(12)	-2.200(25)
	10 23329.54814(51) 23334.696	5.21759(77) 5.218(30)	0.70315(32) 0.702(15)	1.29101360(263) 1.29095(1)	5.81297(292) 5.8(1)		-2.678(13)	-2.300(25)
	11 24566.96640(50) 24572.035	5.14572(99) 5.138(50)	0.70514(39) 0.709(30)	1.27379929(248) 1.27384(1)	5.8511(27)	6.0(1)		-2.648(16)	-1.950(25)
	12 25780.22595(97) 25785.200	5.0712(17)	5.042(50)	0.70899(93) 0.706(45)	1.25652651(623) 1.25688(2)	5.9052(85)	5.7(2)		-2.669(39)	-2.0(5)
	Footnotes
	a Systematic error may be up to 0.01 cm -1 .
	b Ref [18] for v=0-6, [32]for v = 7-12
	c Value quoted is Origin -4B from previous work
	d Value quoted is 3/2ε from previous work
	e Value quoted is -4p from previous work

Table 6 .

 6 Effective temperatures (K) used in simulating spectra

	T 1	T 2	r a	T vib

Table 7

 7 Lifetimes (/µs) for v = 0 -12 for the B ³Π g state of N 2 .

	v	This work Eyler & Pipkin
				Error
		(Calculated)		Bar
	0	11.94
	1	9.63
	2	8.13
	3	7.08
	4	6.31
	5	5.73	5.87	0.21
	6	5.27	5.34	0.17
	7	4.91	5.05	0.16
	8	4.62	4.72	0.15
	9	4.39	4.41	0.15
	10	4.20	4.33	0.17
	11	4.06	4.19	0.17
	12	3.94	4.11	0.21

  Origin 0 2.171(21) × 10 -3 2.471(21) × 10 -3 5.788(29) × 10 -3 6.587(29) × 10 -3

	B	1.6	1.600000007(18) 1.600000007(18) 1.600000014(24) 1.600000014(25)
	A	40	39.995181(60)	39.994573(60)	39.995159(51)	39.994551(51)
	λ×10 3	0	-	-	-2.252(48)	-2.554(48)
	γ×10 4	-40 0	5.525(12)	0	5.5159(98)
	A D ×10 4 -4	-0.4810(10)	0	-0.48025(85)	0
	Average		0.00019	0.00020	0.00032	0.00033
	Error

Acknowledgments

This work was supported by the French INSU/PNPS (Programme National de Physique Stellaire) program. AJR acknowledges support from the Harrison-McCain foundation as visiting scientist at the University of New Brunswick in 2017. DWT thanks Joyce MacGregor for her assistance with the laser optogalvanic experiments, and the Natural Sciences and Engineering Research Council of Canada for financial support.

Appendix: Correlation between A D and γ

In Ref. [START_REF] Brown | Spin-orbit and spin-rotation coupling in doublet states of diatomic molecules[END_REF], Brown and Watson show that the parameters γ and A D produce indistinguishable contributions to the energy levels of diatomic molecules in 2 Λ (Λ > 1) states of diatomic molecules, and therefore only either one or the other of the two parameters can be used in the calculation of energy levels for such states. Their conclusion is subject to some constraints imposed by their approach, in which a zeroth-order Hamiltonian

is diagonalized to provide zeroth-order wavefunctions (0) n ψ , and the effects of a perturbing Hamiltonian: [START_REF] Hirota | Symbols for fine and hyperfine parameters[END_REF]are dealt with via first-order perturbation theory. Since this procedure involves taking expectation values

over the zeroth-order wavefunctions, some off-diagonal matrix elements of the form

ψ for m ≠ n are neglected; these arise from the N-dependence of each term in H 1 . Use of the eigenfunctions of H 0 as the basis for further calculations reduces the magnitude of the off-diagonal matrix elements, but since they are not eliminated, data of extensive scope and of high precision may be sensitive to their contributions to the energy when the full Hamiltonian is diagonalized. Under such circumstances the argument for the equivalence of γ and A Dv in 2 Λ states breaks down, and both parameters may be required to fit the data to experimental precision.

In triplet states such as in the B 3 Π g state of N 2 considered in this paper, Brown and

Watson's development should be extended to include the effects of the spin-spin interaction λ in H 1 (or H 0 ). To do this formally would require significant algebra, but it is straightforward to test for parameter redundancy numerically and this also allows the argument for the breakdown of the perturbation treatment to be quantified. To this end a series of fits were performed on a synthetic energy level list generated with parameters appropriate to B 3 Π g v = 0. Λ-doubling effects were ignored, and levels up to J = 50 were included. The fits were done for both 2 Π and 3 Π states. The "Input" column in the table below gives the parameters used to generate the energy level list; the subsequent columns give the results of fits with A D or γ constrained to zero. As can be seen from the table, a good fit can be obtained for both 2 Π and 3 Π with just one of A D or γ, but the fit is not perfect, with levels with J < 10 showing the largest errors. Note that the origin and λ are floated as both A D and γ have some small Jindependent matrix elements, and fixing the origin or λ gives average errors ten times larger.


Rodrigue Imad

Sébastien Houcke

Catherine Douillard

Sebastien Houcke

Blind frame synchronization on Gaussian channel

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

INTRODUCTION

In digital communication systems, one important step is Channel Coding. At the reception, we should be able to find the beginning of each codeword, so that the decoder can work properly. This is what we call Frame Synchronization. Classical frame synchronization is realized by adding to the transmitted sequence, some synchronization bits known by the receiver. The actual codes are very powerful and can decode sequences having several numbers of errors. To keep their performance, traditional synchronization methods should increase the length of the inserted training sequence, which reduces the spectral efficiency of the transmission. For this reason, we are interested in developing blind methods of frame synchronization. Note that when we say blind, we mean that no additional sequence is added to the coded one. We generalize in this paper a simple method of blind frame synchronization, initially developed for a Binary Symmetric Channel (Hard data), so it can work now on a Gaussian Channel (Soft data). The remaining part of this paper is organized as follows. Section 2 presents the Hard and Soft techniques of our blind frame synchronization method. For each technique, we find the theoretical probability of false synchronization, which is compared to our simulated results in the third section. A comparison between the performance of the two methods will also be done by synchronizing some LDPC and convolutional codes. Finally, section 4 concludes the work.

BLIND FRAME SYNCHRONIZATION METHOD

In this paper, we consider that the transmitter is sending a binary sequence of codewords. Suppose that a Binary Phase Shift Keying (BPSK) modulation is applied to codewords bits and that the propagation channel is an Additive White Gaussian Noise (AWGN) channel.

We will see later that our synchronization techniques are based on the use of the parity check matrix of the code. For a code of rate

R = n c -n r n c ,
it is often possible to find the corresponding parity check matrix H of size n r × n c , where n c represents the length of a codeword and n r the number of parity relations. Let s(r) be the syndrome of a received codeword r:

s(r) = rH T = eH T ,
where e is the corresponding error vector. At the synchronization position and in a noise free channel, it is clear that the syndrome of a given codeword is equal to zero. This is not the case when we are not synchronized. We will see in the next sections that our synchronization techniques are based on this simple idea. Let us define now the i th received sample by:

x(i) = b(i -t 0 ) + w(i), ( 1
)
where t 0 is a delay introduced by the propagation channel, b(i) = ±1 a modulated coded bit and w(i) a white gaussian noise.

The received sequence of N samples can be written as:

X = [x(1), . . . , x(N)].
The main target of frame synchronization is to find the position of a codeword in the received sequence. In other words, we have to estimate the delay t 0 . Without loss of generality, we assume that t 0 is always lower than n c .

Hard Synchronization

The method initially proposed in [START_REF] Houcke | Blind Frame Synchronization for Block Code[END_REF] For each block, let us calculate its corresponding syndrome, then we form the vector of syndromes S d :

S d = [S d (1), . . . , S d (Kn r )] T .
φ (d) = Kn r ∑ k=1 S d (k).
By repeating this procedure for n c consecutive values of d, the frame synchronization instant is estimated as the value of d minimizing φ :

t0 = argmin d=0,...,n c -1 φ (d).

Theoretical probability of false synchronization

Let P F be the probability of false synchronization:

P F = Pr φ (t 0 ) > min d=0,...,n c -1,d =t 0 φ (d) .
To be able to calculate this probability, we have to find the probability laws followed by φ (t 0 ) and φ (d).

At non-synchronized positions, the k th parity check equation has a probability 1/2 to be verified and therefore S d (k) is a Bernoulli random variable of parameter 1/2. On the other hand, at the synchronized position S t 0 (k) follows a Bernoulli law of parameter p k where p k is the probability that S t 0 (k) is equal to 1. It is the probability of having an odd number of errors in the u k bits verifying the k th parity equation.

p k = Pr(S t 0 (k) = 1) = u k -1 2 ∑ l=0 u k 2l + 1 p 2l+1 e (1 -p e ) u k -2l-1 , (2)
where u k is the number of ones in the k th line of H, p e the error probability of the equivalent binary symmetric channel, n k the number of combinations of k elements from n and n the nearest integer to n towards minus infinity. Let us assume now that the elements of the syndrome are independent. By assuming that u k is constant for all the rows of H, we have p k = p ∀k and therefore φ is a Binomial random variable:

φ (d) ∝ B(Kn r , 1/2) d = t 0 and φ (t 0 ) ∝ B(Kn r , p).
Finally, the probability of false synchronization is

P F = 1- Kn r -1 ∑ k=0 Kn r k p k (1-p) Kn r -k Kn r ∑ j=k+1 Kn r j 2 Kn r n c -1
.

Let us adapt now this method for a Gaussian Channel.

Soft Synchronization

The use of hard decision on the received sequence involves a loss of information. Therefore, it is logical to try to benefit from this information and the common way to do it is to calculate the LLR of the received bits. In our case, we are interested in the LLR of S d (k) [START_REF] Houcke | Blind Detection for block interleaved division multiple access[END_REF]. To simplify the notations, let us assume that the length of the synchronization window is n c bits. Each element of the syndrome being the sum (modulo 2) of u k bits:

S d (k) = W d h T k = u k ∑ i=1 z(d + k i ), (3)
where k i is the position of the i th non zero element in the k th line of H. The LLR of each element of the sum is:

L(z(d + k i )) = 2 σ 2 x(d + k i ), (4)
where σ 2 is the variance of the channel noise. An approximation of the LLR of a syndrome's element is given by [START_REF] Hagenauer | Iterative decoding of binary block and convolutional codes[END_REF]:

L(S d (k)) = (-1) u k +1 u k ∏ j=1 sign(L(z(d + k j ))) β (5) with β = min j=1,...,u k |L(z(d + k j ))|. (6)
The criterion that we consider now is:

φ L (d) = Kn r ∑ k=1 L(S d (k))
and the synchronization position is estimated by:

t0 = argmin d=0,...,n c -1 φ L (d).
Notice that L(z(k)) and x(k) are proportional. By assuming that the variance of the noise is constant during a transmission, we can replace L(z(d + k j )) by x(d + k j ) in ( 5) and ( 6), so that we do not need to know the variance of the noise to make our calculations.

L(S d (k)) is estimated by: L(S d (k)) = (-1) u k +1 u k ∏ j=1 sign(x(d +k j )) min j=1,...,u k |x(d +k j )|.
(7) Note that the term

(-1) u k +1 u k ∏ j=1 sign(x(d + k j ))
involved in (7) is the modulated value of S d (k). An estimation of φ L (d) is also obtained by:

φL (d) = Kn r ∑ k=1 L(S d (k)) = Kn r ∑ k=1 (2S d (k) -1)Y d (k), where Y d (k) = min j=1,...,u k |x(d + k j )| represents the reliabil- ity of S d (k).

Theoretical probability of false synchronization

This time we have continuous variables instead of discrete ones. By applying the Central Limit theorem, φL (d) follows a Gaussian law:

φL (d) ∝ N (Kn r m d , Kn r σ 2 d ),
where m d and σ 2 d represent respectively the mean and variance of L(S d (k)). For d = t 0 , we have:

Pr(S d (k) = 0) = Pr(S d (k) = 1) = 1/2,
whether we have noise or not and therefore S d (k) and Y d (k) are independent. This leads to:

m d = 0 and σ 2 d = E[Y d (k) 2 ].
Note that σ 2 d is estimated by simulations.

On the other hand, when d = t 0 , S t 0 (k) and Y t 0 (k) are not independent anymore because in this case, the sign of a syndrome depends only on the presence or not of an error. Therefore, m t 0 can be written as:

m t 0 = Pr(S t 0 (k) = 0)E L(S t 0 (k)) /S t 0 (k)=0 +Pr(S t 0 (k) = 1)E L(S t 0 (k)) /S t 0 (k)=1
where Pr S t 0 (k) = 1 = p is the same one computed in [START_REF] Houcke | Blind Frame Synchronization for Block Code[END_REF]. Let us find now

E L(S t 0 (k)) /S t 0 (k)=1 = E Y t 0 (k) /S t 0 (k)=1 .
Recall that a BPSK modulation is used (cf. (1)):

x(i) = b(i) + w(i)
where b(i) = ±1 is a modulated bit and w(i) the noise. Saying S t 0 (k) = 1 means that there is an odd number of errors in the u k bits z(t 0 + k i ) involved in [START_REF] Houcke | Blind Detection for block interleaved division multiple access[END_REF]. And because

z(k) = sign b(k) + w(k) + 1 2 ,
we can say that among the u k elements, there is an odd number of elements w(k) that, once added to b(k), changes its sign.

Notice that an error occurs when:

b(k) = +1 and w(k) < -1 or b(k) = -1 and w(k) > 1.
Furthermore, as

Pr(b(k) = 1) = Pr(b(k) = -1) = 1/2
and w(k) is a Gaussian variable of zero mean, we can say that

E[Y t 0 (k) /S t 0 (k)=1 ] = E[ min j=1,...,u k |1 + w(t 0 + k j )| /B 1 ],
where B 1 is the event: {Among u k noise samples, an odd number of these elements have them values lower than -1}. Doing the same procedure for the case S t 0 (k) = 0, we have:

E L(S t 0 (k)) /S t 0 (k)=0 = -E Y t 0 (k) /S t 0 (k)=0
and

E[Y t 0 (k) /S t 0 (k)=0 ] = E[ min j=1,...,u k |1 + w(t 0 + k j )| /B 2 ],
where B 2 is the event: {Among u k noise samples, none or an even number of these elements have them values lower than -1}.

Having this, the mean m t 0 of Lt 0 (k) is then equal to:

m t 0 = -(1 -p)E min j=1,...,u k |1 + w(t 0 + k j )| /B 2 +pE min j=1,...,u k |1 + w(t 0 + k j )| /B 1 .
Concerning the variance, we can easily show that: φL (t 0 ) ∝ N (Kn r m t 0 , Kn r σ 2 t 0 ). Finally, the probability of false synchronization is :

σ 2 t 0 = E[( min j=1,...,u k |x(t 0 + k j )|) 2 ] -m 2 t 0 , 0 0.5 1 1.5 2
P F = 1 -1 - 1 2 erfc -Kn r m t 0 2Kn r (σ 2 t 0 + σ 2 d ) n c -1
.

Simulation results

In order to estimate the performance of our synchronization algorithms, we estimated the probability of false synchronization. The evaluation of this probability is realized by Monte Carlo simulation: up to 1000000 realizations have been performed, where, for each realization, the noise, information bits and the delay of the channel were randomly chosen.

Consider two LDPC codes of length 511 bits and rate 0.7, that differ only by the composition of their parity check matrix: Code I (respectively II) has 4 (respectively 10) non zero elements on each line of its parity check matrix. By using a synchronization window of size 1 block (511 bits), we applied the two synchronization methods to these codes. Fig. 2 shows the probability of false synchronization versus the Signal to Noise Ratio (E b /N 0 ), for the Hard synchronization method. We notice on this figure that our theoretical and simulated results are almost the same. For code II, the theoretical results are a little bit better than the simulated ones because having 10 non zero elements in each line of the parity check matrix of this code decreases the number of independent syndrome elements. However, in our theoretical computations we considered that all the elements of a syndrome are independent, which explains this difference in performance.

Let us compare now the plotted curves for codes I and II. We notice that the probability of false synchronization increases when the number of non zero elements in the parity check matrix increases. Fig. 3 shows the theoretical and simulated results of Soft synchronization method applied to the previous LDPC codes. As in the case of Hard synchronization, the theoretical and simulated curves are very close to each other. Note that for Code I, for E b /N 0 greater than 1.5 dB, no error was found in the 1000000 realizations. Let us compare now the differences in performance between Hard and Soft synchronizations: Comparing Fig. 2 and3, it is clear that Soft synchronization has the best performance: For code I for example, there is a gap of around 0.62 dB between the two methods at a probability of false synchronization equals to 10 -3 . Let us apply now our methods to convolutional codes. For this type of codes, non zero elements in the parity check matrix are placed in a "staircase" form. For example, the parity check matrix of the convolutional code of generator polynomials (5,7) can be written as [START_REF] Johannesson | Fundamentals of Convolutional Coding[END_REF]:

H =            1 1 0 1 1 1 1 1 0 1 1 1 1 1 0 1 1 1 1 1 0 1 1 1 . . . . . .           
Therefore, for this code we might have:

S d (k) = S d-2n (k + n),
for k > 2 and n integer. This leads to a degradation in the performance of our synchronization methods. One simple solution to this problem is to interleave the bits in each codeword using a pseudo-random interleaver of size n c bits.

The convolutional codes to which we applied our synchronization methods have respectively the generator polynomials (5, 7), (23, 35) and (561, 753). These three codes have the same length (512 bits), the same rate (1/2) and constraint lengths equal to 3, 5 and 9 respectively. Figure 4 shows a comparison between the performance of the two methods after being applied to those convolutional codes. The same behavior obtained with LDPC codes is found here: Soft synchronization is always the best. By comparing the curves plotted for code (5, 7), for a probability of false synchronization equals to 2.10 -2 , we can see a gap of 0.7 dB between the two methods. Notice that when the constraint length of a code increases, the number of non zero elements in the parity check matrix increases also, which explains the degradation in the performance of our synchronization methods, as explained in [START_REF] Houcke | Blind Frame Synchronization for Block Code[END_REF]. Note also that if those three convolutional codes were decoded using a Maximum A Posteriori (MAP) decoder, a frame error rate of around 10 -2 would correspond to a E b /N 0 between 5 and 6 dB. However, Fig. 4 shows that for E b /N 0 greater than 4 dB, synchronizing codes (5,7) and (23,35) gave no error over the 1000000 realizations. Thus, our synchronization method is well adapted for convolutional codes.

Conclusion

In this paper, we have described a new method of blind frame synchronization that is based on the calculation of the sum of the LLR of a syndrome elements. Compared with the Hard technique, our Soft approach gives the best results for any Signal to Noise Ratio. We have also estimated the theoretical probabilities of false synchronization and the results were almost the same as the simulated ones. After these promising results, our target will be to apply this blind frame synchronization method to other types of codes, in particular turbo codes.

Figure 1 :

 1 Figure 1: Blind frame synchronization principle

Figure 1

 1 Figure 1 represents three different sequences of W d corresponding for d = 0, 1 and t 0 , where the size of W d is fixed to K = 3 blocks. Having S d , we evaluate φ (d), the sum of non-zero elements in S d :

Figure 2 :

 2 Figure 2: Hard synchronization of LDPC codes

Figure 3 :

 3 Figure 3: Soft synchronization of LDPC codes

Figure 4 :

 4 Figure 4: Hard and Soft synchronizations for convolutional codes

Code I -theor. Code I -simu. Code II -theor. Code II -simu. Code I -theor. Code I -simu. Code II -theor. Code II -simu.


David Gaudrie

Rodolphe Le Riche

Victor Picheny

Benoît Enaux

Vincent Herbert

Targeting well-balanced solutions in multi-objective bayesian optimization under a restricted budget

come

Targeting Well-Balanced Solutions in Multi-Objective Bayesian Optimization under a Restricted Budget David Gaudrie 1 Rodolphe le Riche 2 Victor Picheny 3 Benoît Enaux 4 Vincent Herbert 5

1 Groupe PSA, david.gaudrie@mpsa.com 2 CNRS, Mines Saint-Étienne, leriche@emse.fr 3 MIAT, Université de Toulouse, INRA, victor.picheny@inra.fr 4 Groupe PSA, benoit.enaux@mpsa.com 5 Groupe PSA, vincent.herbert@mpsa.com Key words: Gaussian Processes, Parsimonious Optimization, Computer Experiments, Preference-Based Optimization Multi-objective optimization aims at finding the Pareto set composed of all the best tradeoff solutions between several objectives. When dealing with expensive-to-evaluate black box functions, surrogate-based approaches, in the vein of EGO [START_REF] Jones | Efficient global optimization of expensive black-box functions[END_REF] have proven their effectiveness.

However, for extremely narrow budgets, and/or when the number of objectives is large, uncovering the entire Pareto set becomes out of reach even for these approaches. In this work, we restrict our search to well-chosen targeted parts of the Pareto set. This accelerates the search as only a subset of the objective space is considered. As an end-user would typically choose solutions with equilibrated trade-offs between objectives rather than ones favoring a single objective over the others, we will focus on the central part of the Pareto Front, which corresponds to the most well-balanced solutions.

First, we discuss how to define and estimate the center of the Pareto Front. That estimated point has to fairly represent the topology of the front, in spite of the parsimonious knowledge of the objective space. Then, three infill criteria which will guide the optimization, including the Expected Hypervolume Improvement [START_REF] Emmerich | Single-and multiobjective evolutionary optimization assisted by gaussian random field metamodels[END_REF] are studied and tailored through some of their hyperparameters to enable them to target.

To assess performance, a benchmark built from real-world airfoil aerodynamic data, with variable dimension and number of objectives is used. Compared with standard techniques, the proposed methodology leads to a faster and a more precise convergence towards the Pareto Front in the targeted region.