Élodie Muller
email: emuller@ipgp.fr

Magali Ader

Carine Chaduteau

Pierre Cartigny

Franck Baton

Pascal Philippot

The use of chromium reduction in the analysis of organic carbon and inorganic sulfur isotope compositions in Archean rocks

Keywords: carbon isotopes, decarbonation, limited sample, Precambrian

One of the most serious issues with deciphering the evolution of organisms and their biogeochemical environments from the ancient rock record is the difficulty in obtaining well-preserved samples. Although not much can be done to avoid diagenetic and metamorphic alteration when they have occurred, alteration due to weathering can be avoided by working on drill core samples. This implies however that the amount of sample is limited, which may in turn restrain the number of possible chemical and isotopic analyses that can be performed. In order to save sample we show here that the chemical protocol used for the sulfur sulfide extraction (for later sulfur isotope analyses) is also suitable to decarbonate samples (for later organic carbon isotope analyses). In the case of carbonated rocks, both sulfur sulfide extraction and decarbonation require high amounts of sample so that coupling them may save a significant amount of sample and time. In addition it allows both organic carbon (TOC and δ 13 C) and sulfur isotope ACCEPTED

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HIGHLIGHTS:

 One chemistry for both sulfur sulfide extraction and rock sample decarbonation  A new protocol that saves time and a significant amount of limited sample  Organic carbon and sulfur isotope composition analysis on the exact same powder

INTRODUCTION

As shown by [START_REF] Lowe | Tectonic controls on atmospheric, climatic, and biological evolution 3.5-2.4 Ga[END_REF], vanishingly few Archean rock units have survived to the present (less than 10 %; [START_REF] Garrels | Evolution of sedimentary rocks[END_REF] and outcrops have been affected by modern oxidative weathering. This surficial weathering can modify both the mineralogy and chemistry of the ancient rock record and thus represents an issue when investigating paleoredox conditions and early traces of life [START_REF] Hoashi | Primary haematite formation in an oxygenated sea 3.46 billion years ago[END_REF][START_REF] French | Reappraisal of hydrocarbon biomarkers in Archean rocks[END_REF]. In order to recover relatively well-preserved rock samples from below the present weathering profile

A C C E P T E D

M A N U S C R I P T 3 (below the groundwater table ~10-50 m) and to obtain continuous rock cores that retain soft or friable units that outcrop poorly at the surface, several scientific drilling programs have been carried out in Archean terrains in the past decade [START_REF] Kaufman | Late Archean biospheric oxygenation and atmospheric evolution[END_REF], Yamaguchi et al., 2009[START_REF] Philippot | Early traces of life investigations in drilling Archean hydrothermal and sedimentary rocks of the Pilbara Craton, Western Australia and Barberton Greenstone Belt, South Africa[END_REF]. The recovered cores have been extensively analyzed to investigate conditions at the surface of the Archean Earth: the composition, temperature, and redox state of the Archean ocean and atmosphere; the volcanic and sedimentary processes that operated early in Earth history; and above all, to search for pristine morphological or chemical traces of early life [START_REF] Mojzsis | million years ago[END_REF][START_REF] Philippot | Early Archean microorganisms preferred elemental sulfur[END_REF][START_REF] Schopf | Evidence of Archean life: stromatolites and microfossils[END_REF], Thomazo et al., 2009b, Bontognali et al., 2012).

Since the first description of recognizable Archean microfossils a few decades ago (e.g. [START_REF] Dunlop | Shallow-water sedimentation at North Pole, Pilbara Block[END_REF][START_REF] Buick | Stromatolite recognition in ancient rocks: an appraisal of irregularly laminated structures in an Early Archaean chert-barite unit from North Pole, Western Australia[END_REF], morphology-focused imaging techniques of fossillike objects [START_REF] Brasier | Questioning the evidence for Earth's oldest fossils[END_REF][START_REF] Schopf | Laser-Raman imagery of Earth's earliest fossils[END_REF][START_REF] Lepot | Microbially influenced formation of 2,724-million-year-old stromatolites[END_REF], Wacey et al., 2011) have failed to absolutely confirm their biogenic origin. This is mainly because several abiologic metamorphic and hydrothermal reactions have been identified that could produce kerogen-like polymers and graphite (e.g. Van Zuilen et al., 2002) and generate complex structures similar to microfossils (e.g. [START_REF] Garcia-Ruiz | Self-assembled silica-carbonate structures and detection of ancient microfossils[END_REF] and stromatolites (e.g. [START_REF] Grotzinger | An abiotic model for stromatolite morphogenesis[END_REF]. In view of these uncertainties and controversies, it is now recognized that studies of early life and associated environmental conditions depend on the multiplication of structural, isotopic and chemical tracers. Particularly, sulfur and carbon isotopes have been traditionally used to decipher ancient biogeochemical cycles (e.g., review

in Thomazo et al., 2009[START_REF] Farquhar | Connections between sulfur cycle evolution, sulfur isotopes, sediments, and base metal sulfide deposits[END_REF]. Indeed, C and S isotopes retain records of microbial metabolisms where secondary processes such as metamorphism and weathering have obliterated or erased evidence for early life such as microfossils or other biogenic remains.

A C C E P T E D M A N U S C R I P T

4

A prerequisite for measuring carbon isotopes of organic material in rocks (and sediments) is to eliminate carbonates which are widely occurring in various forms, i.e. mainly ankerite, dolomite and siderite in Archean rocks, since the isotopic value of the carbon atom of the carbonate would otherwise bias the isotopic signal. In the past, several methods dissolving carbonates by acid-treatment, thereby eliminating the carbonate carbon as carbon dioxide, have been described [START_REF] Froehlich | Analysis of organic carbon in marine sediments[END_REF], Weliky et al., 1983[START_REF] Hedges | Carbon and nitrogen determinations of carbonatecontaining solids[END_REF], the most commonly used being 10 % HCl followed by rinsing with deionized water [START_REF] Schubert | Effects of decarbonation treatments on δ 13 C values in marine sediments[END_REF]. In parallel, sulfur extraction for isotopic analysis of sulfides and sulfates is generally performed with a succession of different acidic attacks on sample powder.

Therefore, sample preparation for either bulk C or S isotopic analyses is time consuming and requires important and independent quantities of sample, which are limited when working on drill core samples.

In this study, we tested the chemistry used for sulfur extraction and based on acidic chromium reduction for decarbonation of sample powders in order to save both time and sample and most importantly to perform bulk organic carbon and sulfur isotope measurements in the exact same samples. To our knowledge, this chemistry has never been used for simple decarbonation, and our objective here is thus to test if it is suitable for the determination of the Total Organic Carbone (TOC) content and carbon isotope compositions (δ 13 C org ). Ader et al.

( 2009) showed that the even standard decarbonation might lead to an increase by up to 1.4 ‰ in δ 13 C org values especially for samples with low amount of organic carbon and without changing significantly the TOC contents. Given the hotter (around 190 °C) conditions of the acidic chromium chemistry compared to a standard decarbonation procedure, one may be concerned by its effects on the TOC and δ 13 C org values. In order to investigate such effects we compared the results obtained after the acidic chromium chemistry and the standard

SAMPLES

The samples studied here have been selected among some of the oldest best-preserved Archean rocks in the world. They stem from the Mapepe and Mendon formations on the Kaapvaal Craton (South Africa) and the Tumbiana and Kazput formations on the Pilbara Craton (Western Australia). These units, which have been deposited between 3.24 and 2.32

Ga, present various lithologies representative of the Archean eon. Geological details have been published in previous studies [START_REF] Philippot | Early traces of life investigations in drilling Archean hydrothermal and sedimentary rocks of the Pilbara Craton, Western Australia and Barberton Greenstone Belt, South Africa[END_REF], 2012, Thomazo et al., 2009[START_REF] Martindale | Sedimentology, chemostratigraphy, and stromatolites of lower Paleoproterozoic carbonates[END_REF], and a short summary of pertinent information is given here. The selected rock samples present low TOC contents between 0.01 % and 0.40 % and a broad range of carbonate concentrations between 0.07 % and 63.80 %, which are representative of the Archean rock record.

Mapepe and Mendon Formations

Mendon and Mapepe formations belong to the Barberton Greenstone Belt (South Africa), which consists of a succession of Archean supracrustal rocks surrounded and intruded by granitoids (Viljoen and Viljoen, 1969). The Mendon Formation is mainly composed of ultramafic volcanics, interpreted to reflect large mantle plume magmatic activity [START_REF] Lowe | Abiological origin of described stromatolites older than 3[END_REF], alternated with bedded black cherts, probably recording deep-water sedimentation during times of magmatic quiescence [START_REF] Hofmann | Silica alteration zones in the Barberton greenstone belt: a window into subseafloor processes 3.5-3.3 Ga ago[END_REF]. Radiometric age constraints on the tuffaceous band from the underlying Kromberg Formation provide a lower age limit of 3334±3 Ma [START_REF] Byerly | Prolonged magmatism and time constraints for sediment deposition in the early Archean Barberton greenstone belt: evidence from the Upper Onverwacht and Fig Tree groups[END_REF]. The unconformably overlying Mapepe Formation records shallow-water terrigenous and volcanoclastic sediments, interbedded with 3226±3 Ma (Kröner et al., 1991[START_REF] Byerly | Prolonged magmatism and time constraints for sediment deposition in the early Archean Barberton greenstone belt: evidence from the Upper Onverwacht and Fig Tree groups[END_REF]. Both formations experienced only low greenschist facies regional metamorphism with a maximum temperature of 300 °C [START_REF] Knauth | High Archaean climatic temperature inferred from oxygen isotope geochemistry of cherts in the 3.5 Ga Swaziland Supergroup[END_REF]Lowe, 2003, Tice et al., 2004) together with some deformation.

The samples analyzed in the present work were collected from a weathering-free drill core recovered by the Barberton Barite Drilling Project (BBDP; [START_REF] Philippot | Early traces of life investigations in drilling Archean hydrothermal and sedimentary rocks of the Pilbara Craton, Western Australia and Barberton Greenstone Belt, South Africa[END_REF][START_REF] Philippot | Variations in atmospheric sulphur chemistry on early Earth linked to volcanic activity[END_REF] at the so-called "barite syncline" locality. They have been selected as representative of the different lithologies forming the Mendon and Mapepe formations and include one basaltic komatiite and three carbonaceous black cherts from Mendon, together with two bedded barites, two ferruginous cherts, and four silicified terrigenous and volcanoclastic sediments from Mapepe (see [START_REF] Philippot | Variations in atmospheric sulphur chemistry on early Earth linked to volcanic activity[END_REF] for more details). Except for black cherts, the other lithologies present low organic carbon content, as deduced from a petrographic observation. Quartz, green chlorite, Fe-Mg-carbonate, barite and pyrite represent the main mineral assemblage.

Tumbiana Formation

The Tumbiana Formation belongs to the Hamersley Basin (Western Australia), which presents a well-preserved succession of volcanic and sedimentary rocks deposited between 2.77 and 2.41 Ga on the granitoid-greenstone basement of the Pilbara Craton (Thorne and Trendall, 2001). This formation consists of large-scale cross-bedded calcareous sandstone, stromatolitic and fenestrate carbonate, and micaceous sandstone interlayered with volcanic siltstone (Thorne andTrendall, 2001, Thomazo et al., 2009). The Tumbiana Formation was deposited at 2724±5 Ma, as indicated by zircon U-Pb dating (Blake et al., 2004), either in a shallow marine or lacustrine environment associated with influx of riverine freshwater derived from the continents [START_REF] Buick | The antiquity of oxygenic photosynthesis: evidence from stromatolites in sulphate-deficient Archaean lakes[END_REF], Thorne and Trendall, 2001, Bolhar and Van Kranendonk, 2007). The regional metamorphic temperature has been estimated < 300 °C

A C C E P T E D M A N U S C R I P T

7 using metamorphic mineral assemblages and the degree of organization of organic matter [START_REF] Lepot | Microbially influenced formation of 2,724-million-year-old stromatolites[END_REF].

The samples have been collected in the Pilbara Drilling Project (PDP1) diamond drill core, which recovered the upper section of the Tumbiana Formation at Meentheena. These include:

one siltstone, one laminated mudstone and two homogeneous mudstones. In all samples, organic matter is homogeneously disseminated. Quartz, green chlorite, carbonate (mainly calcite) and pyrite represent the main mineral assemblage throughout the drill core. The

Kazput Formation

The Kazput Formation consists of subtidal stromatolites, grainstones, and micrites deposited on a mixed carbonate-siliciclastic shelf and is lying within the upper part of Turee Creek Group in the southern Pilbara region of Western Australia [START_REF] Martindale | Sedimentology, chemostratigraphy, and stromatolites of lower Paleoproterozoic carbonates[END_REF]. The Turee Creek Group is though to be a continuous stratigraphic sedimentary section hosting both the Great Oxidation Event and possibly two Huronian glaciations [START_REF] Martindale | Sedimentology, chemostratigraphy, and stromatolites of lower Paleoproterozoic carbonates[END_REF]Mazumder, 2015). Geochronological constraints for the Turee Creek Group are provided by the 2449±3 Ma Woongarra Rhyolite at the top of the underlying Hamersley Group (Barley et al., 1997) and by the 2209±15 Ma Cheela Spring Basalt of the unconformably overlying lower Wyloo Group [START_REF] Martin | A pre-2.2Ga age for giant hematite ores of the Hamersley Province[END_REF].

The samples selected for this study come from the Turee Creek Drilling Project #3 (TCDP3) drill core and consist of four carbonated mudstones and one quartzite with low organic carbon content (as deduced from a petrographic observation) and mainly composed of quartz, carbonate, chlorite and pyrites.

METHODS

Twenty-one rock samples were powdered to <100 µm using a ring and puck mill. For fourteen samples, one single powder has been homogenized, processed using either the standard or the acid chromium method and analyzed for organic carbon content and isotope composition (Table 1). For seven additional samples, different initial powders have been used

for the two different decarbonation techniques (Table 2).

Standard sample decarbonation

This method is a common procedure widely used to remove various carbonates such as calcite, andesite, dolomite and siderite (e.g. Bernasconi et al., 1997, Wu et al., 1999[START_REF] Lorrain | Decarbonation and preservation method for the analysis of organic C and N contents and stable isotope ratios of low-carbonated suspended particulate material[END_REF], Thomazo et al., 2009). Sample powders (about 200 mg) were reacted with excess HCl (6 N) at room temperature in glass beakers during one night. After removing the supernatant, samples were acidified again with HCl (6 N) at 80 °C and agitated during two hours: this is to guarantee full sample decarbonation. After decantation, the residues were rinsed with deionized distilled water until neutral, centrifuged, and dried at 60 °C overnight.

During this step, acid-volatile sulfur (AVS) such as FeS or ZnS would react and release H 2 S, some being potentially re-oxidized into insoluble elemental sulfur [START_REF] Rice | The analysis of forms of sulfur in ancient sediments and sedimentary rocks: comments and cautions[END_REF]. Pyrite would however remain.

Combined sulfur sulfide extraction and sample decarbonation

The acidic CrCl 2 solution is extensively used for the extraction of reduced inorganic sulfur compounds in different types of geological samples, including modern sediments and ancient shales (Canfield et al., 1986, Fossing andJorgensen, 1989), acid-sulfate soils (Sullivan et al., 2000[START_REF] Burton | A simple and inexpensive chromium-reducible sulfur method for acid-sulfate soils[END_REF], and basaltic glasses [START_REF] Labidi | Determination of sulfur isotopes in glasses: A reappraisal of the MORB δ 34 S[END_REF]. It was first designed by Traube and Passarge (1916) for the reduction of (non-sulfur bearing) complex organic

A C C E P T E D M A N U S C R I P T

compounds (e.g. maleic, fumaric or cinnamic acids; [START_REF] Holmquist | Stereochemistry of the reduction of tertiary cyclohexyl halides by chromium (II) complexes: Theses, Dissertations[END_REF]. This chemistry was first introduced to the geochemical community by Zhabina and Volkov (1978), and later improved by [START_REF] Canfield | The use of chromium reduction in the analysis of reduced inorganic sulfur in sediments and shales[END_REF] and [START_REF] Gröger | Quantitative recovery of elemental sulfur and improved selectivity in a chromium-reducible sulfur distillation: Geostandards and Geoanalytical Research[END_REF] for sulfur extraction. Gröger et al. (2009, p. 21) have shown that CrCl 2 solution typically yields 99±1 % for both pyrite and elemental sulfur. In contrast, simple organic molecules are hardly reduced, with very low yields < 0.2 % (Gröger et al., 2009, p. 24). For more complex organic molecules, polysulfur bonds could be broken by the CrCl 2 solution. However, organic sulfur, to be quantitatively reduced, requires the use of Raney-nickel catalyst (e.g. [START_REF] Oduro | Multiple sulfur isotope analysis of volatile organic sulfur compounds and their sulfonium precursors in coastal marine environments[END_REF].

Originally, the CrCl 2 solution was prepared and injected using a so-called Jones reductor, which is somewhat complex and time-consuming (i.e. a glass-column packed with granulated Zn amalgamated for a few minutes with an acidic 2 % mercuric nitrate solution and washed with three column volumes of 0.5 N HCl; Zhabina and Volkov, 1978[START_REF] Canfield | The use of chromium reduction in the analysis of reduced inorganic sulfur in sediments and shales[END_REF], Tuttle et al., 1986, Fossing and Jorgensen, 1989). As mostly done worldwide, we prepared our CrCl 2 solution daily using a much simpler procedure, where chromic chloride hexahydrate associated with granulated Zn is acidified with HCl 0.6 N under continuous flow of N 2 . On reduction, the chromium solution changes from green to blue, reflecting the valence change from chromic (III) to chromous (II) ion.

Sulfur extraction is undertaken adding 20 mL of freshly prepared CrCl 2 solution (2.1 N) and 10 mL of concentrated HCl (6 N) to ~1g of sample powder in the digestion flask at ~100 °C.

As long as N 2 -gas bubbles through the solution, we did not face any sample powder aggregation that would prevent full decarbonation. In these hot, reducing and acidic conditions, reduced sulfur species (S 0 and S 2-) are quantitatively decomposed into H 2 S (sulfate remains stable). Then, the released H 2 S is transferred with a stream of N For sulfur isotope analysis, we generally estimate the sulfur content of the sample (with petrographic observations) to calculate the suitable quantity of initial powder needed to obtain ~ 4 mg of Ag 2 S. However, for samples with low organic carbon content, at least 1 g of initial powder is necessary to analyze organic carbon compositions. This can be an issue for samples with high sulfur content, for which the chemical extraction with ~ 1g of powder releases large amounts of H 2 S. In this case, the standard 10 mL of AgNO 3 quickly reaches the saturation and to prevent any H 2 S loss, the volume of AgNO 3 has to be multiplied by 5. It produces large quantities of Ag 2 S potentially heterogeneous that must be carefully homogenized before sulfur isotope analysis.

TOC and organic carbon isotope analysis

TOC content and δ 13 C org values were measured for carbonate-free residues obtained by both the standard and the new decarbonation method with a Flash EA1112 elemental analyzer coupled to a Thermo Finnigan DELTA plus XP isotope ratio mass spectrometer interfaced with a ConFlo IV interface at the stable isotope laboratory of the Institut de Physique du Globe de Paris (IPGP, Paris, France). Our results were normalized using three internal standards and an internal standard with 5 different amounts was used to estimate the concentration of C (wt%). Reproducibility of replicated standards is ± 0.1 ‰ for δ 13 C (1).

The δ 13 C value of internal standards was determined using a classic sealed tube combustion method (i.e. Dumas combustion) and isotope measurement on a Delta+XP mass spectrometer in Dual Inlet mode (see Ader et al., 1998 for a detailed description of the method). They are calibrated against the NBS-21 international standard and reported with the conventional δ notation relative to the V-PDB (Vienna Pee Dee Belemnite) standard. Reproducibility of the δ 13 C org and TOC measurements based on at least triplicate measurements of the samples is usually better than ±0.4 ‰ and ±0.01 wt%, respectively (1σ; Tables 1 and2).

RESULTS & DISCUSSION

TOC content and organic carbon isotope compositions for the 21 decarbonated samples from BBDP (South Africa), TCDP3 and PDP1 (Western Australia) drill cores are reported in Tables 1 and2. All the decarbonated samples present low TOC content between 0.008±0.001 and 0.87±0.04 wt% and cover a large range of δ 13 C org between -57.3±0.1 and -15.8±0.5 ‰.

Values measured by the two methods are in a range characteristic for organic material in Archean rocks (δ 13 C org = -26 ± 7 ‰ in average; e.g. [START_REF] Schidlowski | Carbon isotopes as biogeochemical recorders of life over 3.8 Ga of Earth history: evolution of a concept[END_REF] and reference therein).

Average reproducibility (1σ), estimated from replicate measurements of TOC and δ 13 C org after the acidic chromium decarbonation, is similar to that obtained after the standard method with ±0.014 wt% and ±0.41 ‰ against 0.010 wt% and 0.33 ‰ respectively, suggesting that no carbon addition (from reactant) or incomplete decarbonation occurred. In order to demonstrate that no carbon isotope fractionation is caused by the acidic chromium chemical protocol for decarbonation, we compared the results obtained after standard and acidic chromium decarbonation on the same powder and on different powders from the same samples (Figure 2).

First, starting from the same initial aliquot, we obtained similar results for both TOC and δ 13 C org values (δ 13 C Cr versus δ 13 C Std slope of 0.994 with R 2 =0.995; Figure 2), suggesting that the new chemical protocol for decarbonation does not induce significant carbon isotope fractionation (Figures 2 and3). Carbon-richest samples (TOC ≥ 0.3 wt%) show no significant difference in δ 13 C org with δ 13 C Cr -δ 13 C Std ≤ ±0.08 ‰, whereas carbon-poor samples (TOC ≤ 0.1 wt%) sometimes display differences reaching +1.2 ‰. No correlation between the carbonate content and the difference of carbon isotopic values has been observed (Figure 3), further confirming that decarbonation was complete. However, Ader et al. ( 2009) already observed similar deviations in δ 13 C org after successive standard decarbonation of samples with low TOC content <1 % (i.e. δ 13 C org increased by 0.74±0.5 ‰). Even though these variations are very small compared to the range of δ 13 C org observed, they are still significant compared to the external reproducibility and should be taken into account in future studies regardless of the method used. We also duplicated one sample of mudstone from the Tumbiana Formation previously analyzed by Thomazo et al. (2009). The good reproducibility of ±0.06 ‰ for δ 13 C org and ±0.02 % for TOC values estimated between the two studies (1σ; Figure 2) allows us to validate the new protocol implemented.

Secondly, we observed less consistency between δ 13 C org and TOC measurements obtained on different powders from the same sample, with differences between δ 13 C Cr and δ 13 C Std varying from -7 to +4 ‰ and differences in TOC from -0.01 to +0.2 wt% (grey dots in Figure 2).

These sometimes strong but non-systematic offsets are particularly striking when compared to the results obtained when the two procedures are undertaken on the same powder. These results point to rock sample heterogeneity at the centimeter scale and illustrate the importance of working on unique powder aliquots in particular when studying relations between different isotopic tracers. Therefore, our results demonstrate that the chromium reducing chemistry can be used for decarbonation without significant effects on TOC and δ 13 C org compared to the standard HCl decarbonation. This new protocol of decarbonation seems to be perfectly adapted for the analysis of organic matter in the ancient rock record that experienced at least low greenschist facies metamorphism.

A C C E P T E D M A N U S C R I P T 13

For sedimentary rocks having been exposed to lower thermal conditions, it remains to be demonstrated if this protocol can also be safely used because the high temperatures of the acid attack might induce the breakdown of soluble organic molecules from the condensed organic matter, which may artificially decrease the TOC and modify δ 13 C org . Likewise, modern sediments mostly contain immature organic matter and previous studies have already demonstrated that acidification, even at ambient temperature and lower acidity (down to 0.1 M HCl) might have a significant effect on their δ 13 C org values (from 0.12 to 1.77 ‰; [START_REF] Bunn | Effects of acid washing on stable isotope ratios of C and N in penaeid shrimp and seagrass: Implications for food-web studies using multiple stable isotopes[END_REF][START_REF] Jacob | Stable isotope food web studies: a case for standardized sample treatment[END_REF][START_REF] Jaschinski | Effects of acidification in multiple stable isotope analyses[END_REF][START_REF] Brodie | Evidence for bias in C and N concentrations and δ 13 C composition of terrestrial and aquatic organic materials due to pre-analysis acid preparation methods[END_REF]. In addition, it is highly unlikely that this chemistry, which is known to be a powerful reagent that can reduce some organic substrates, should be applicable for immature organic matter (see [START_REF] Hanson | Applications of chromium (II) salts in preparative organic chemistry[END_REF] for review).

CONCLUSIONS

Our results suggest that the acidic chromium solution can be used to eliminate inorganic carbon without significantly influencing the chemical composition of organic matter in rock samples of lower greenschist metamorphic grade or higher. This method is not only less sample consuming but also less time-consuming and ensures that bulk organic carbon and sulfur isotope measurements are performed on the exact same samples. Nevertheless, given the effects of simple HCl acidification on the δ 13 C org values of modern samples [START_REF] Brodie | Evidence for bias in C and N concentrations and δ 13 C composition of terrestrial and aquatic organic materials due to pre-analysis acid preparation methods[END_REF] as well as Neoproterozoic samples (Ader et al., 2009), we recommend that this new method should be tested before being applied to lower maturity samples.

A C C E P T E D M A N U S C R I P T 14

We acknowledge the financial support from the UnivEarths Labex program of Sorbonne Paris Cite (ANR 11-IDEX-00005-02). This is Institut de Physique du Globe de Paris contribution 3821.

  . Zircon ages constrain the Mapepe Formation between 3259±3 and

  δ 13 C org values of the Tumbiana samples studied here were previously determined by Thomazo et al. (2009).

  2 from the reaction flask to the sulfide trap filled with AgNO 3 (0.3 M) where it reacts to precipitate Ag 2 S (Trap 2 in Figure 1). A trap filled with distilled H 2 O (Trap 1 in Figure 1) is intercalated in the A C C E P T E D M A N U S C R I P T 10 transfer between to reaction flask and the Trap 2 to clean the gas from any acid vapors that have not been stopped by the condenser. Approximately one hour is needed until completion of the H 2 S transfer, as confirmed by the fact that no additional Ag 2 S would precipitate if the AgNO 3 trap was changed. At the end of the reaction the sample powder residue, which has been decarbonated by the hot acidic CrCl 2 solution, is rinsed with deionized distilled water until neutral, centrifuged, and then dried at 60 °C overnight.

FIGURE CAPTIONS Figure 1 :

 CAPTIONS1 FIGURE CAPTIONS

Figure 2 :

 2 Figure 2: Comparison of δ 13 C org values (A) and TOC content (B) obtained after standard HCl

Figure 3 :

 3 Figure 3: Variations of δ 13 C values (A) and TOC content (B) obtained by the two methods as

  Fig. 1

Table 1 :

 1 δ 13 C org and TOC measured after standard and new decarbonation of same initial sample powders. Carbonate content and δ 13 C carb have also been determined on the same samples. Underlined italic values are fromThomazo et al. (2009).

				ACCEPTED MANUSCRIPT
				Decarbonation by chromium solution	Standard decarbonation			Carbonate content
	Formation	Lithology	Sample	δ 13 C Cr (‰)	1σ	TOC Cr (wt%)	1σ	δ 13 C Std (‰)	1σ	TOC Std (wt%)	1σ	δ 13 C Carb (‰)	1σ	%Carb	1σ
	Mapepe	Ferruginous chert	BBDP_52.72	-19.10	1.42	0.020	0.005	-19.31	0.78	0.019	0.001	0.65	0.02	33.10	1.65
	Mapepe	Bedded barite BBDP_67.95	-26.09	0.39	0.009	0.002	-26.88	0.72	0.008	0.001	-1.51	0.00	0.250	0.003
	Mapepe	Felsic volcanics	BBDP_68.05	-23.66	0.22	0.015	0.001	-23.63	0.53	0.015	0.002	0.42	0.13	35.46	1.26
	Mapepe	Felsic volcanics	BBDP_68.42	-26.75	0.37	0.023	0.001	-26.50	0.15	0.025	0.003	1.10	0.06	32.61	0.26
	Mapepe	Bedded barite BBDP_78.18	-27.56	0.05	0.031	0.002	-26.59	0.10	0.042	0.004	-6.01	0.16	0.070	0.001
	Mendon	Laminated black chert	BBDP_95.41	-30.14	0.08	0.356	0.005	-30.22	0.08	0.398	0.002	-3.82	0.01	0.50	0.02
	Mendon	Laminated black chert	BBDP_114.38	-31.26	0.17	0.85	0.02	-31.22	0.15	0.87	0.04	-7.03	0.04	8.13	0.02
	Mendon	Laminated black chert	BBDP_139.8	-28.29	0.20	0.30	0.01	-28.43	0.10	0.249	0.005	-4.28	0.18	14.40	0.55
	Turee Creek	Laminated mudstone	TCDP3_100.52	-27.49	0.70	0.095	0.029	-27.09	0.11	0.106	0.004	0.36	0.40	27.33	0.03
	Turee Creek	Laminated mudstone	TCDP3_135.16	-27.87	0.67	0.16	0.06	-27.06	0.08	0.111	0.004	-0.10	0.40	19.68	0.03
	Turee Creek Mudstone	TCDP3_156.54	-29.04	0.51	0.136	0.036	-28.85	0.14	0.112	0.007	-5.55	0.40	9.56	0.03
	Turee Creek Mudstone	TCDP3_164.36	-29.83	0.48	0.130	0.062	-29.72	0.26	0.100	0.005
	Turee Creek Quartzite	TCDP3_183.7	-28.89	0.71	0.042	0.015	-27.68	0.25	0.012	0.001	-6.08	0.40	4.19	0.03
	Tumbiana	Homogeneous mudstone	PDP1_44.5	-55.82	0.24	0.433	0.006	-55.9	0.4	0.40	0.03	-7.7	0.4	25.80	0.03

Table 2 :

 2 δ 13 C org and TOC measured after standard and new decarbonation of different initial sample aliquots. Carbonate content and δ 13 C carb have also been determined on the same samples.

				Decarbonation by chromium solution	Standard decarbonation			Carbonate content
	Formation	Lithology	Sample	δ 13 C Cr (‰)	1σ	TOC Cr (wt%)	1σ	δ 13 C Std (‰)	1σ	TOC Std (wt%)	1σ	δ 13 C Carb (‰)	1σ	%Carb	1σ
	Tumbiana	Laminated mudstone	PDP1_88.2	-57.26	0.08	0.410	0.001	-52.70	0.40	0.21	0.03	-1.10	0.40	51.20	0.03
	Tumbiana	Siltstone	PDP1_68.9C	-51.55	0.10	0.290	0.002	-44.90	0.40	0.09	0.03	0.40	0.40	63.80	0.03
	Tumbiana	Homogeneous mudstone	PDP1_69.5	-49.84	0.27	0.135	0.002	-44.50	0.40	0.11	0.03	0.60	0.40	40.40	0.03
	Mapepe	Ferruginous chert	BBDP_39.75	-15.78	0.43	0.034	0.001	-18.46	0.16	0.026	0.001	-0.59	0.01	9.94	0.05
	Mapepe	Felsic volcanics	BBDP_45.99	-27.65	0.96	0.019	0.004	-31.53	0.31	0.015	0.000	-0.18	0.04	0.46	0.01
	Mapepe	Felsic volcanics	BBDP_56.66	-25.94	0.33	0.028	0.004	-30.22	0.41	0.038	0.002	0.67	0.18	19.84	0.43
	Mendon	Altered komatiite	BBDP_160.12	-28.50	0.31	0.077	0.024	-25.79	1.05	0.02	0.01	-1.06	0.06	15.64	0.54

ACKNOWLEDGMENTS

Van Kranendonk, M.J., and Mazumder, R., 2015, Two Paleoproterozoic glacio-eustatic cycles in the Turee Creek Group, Western Australia: Geological Society of America Bulletin no. 127, p. 596-607.

Van Kranendonk, M.J., Mazumder, R., Yamaguchi, K.E., Yamada, K., Ikehara, M., 2015, Sedimentology of the Paleoproterozoic Kungarra Formation, Turee Creek Group, Western


Synthesis of a [20]phenacene dodeca-ester by controlled condensation of seven naphthalene-based building blocks

From a conceptual point of view the thinnest graphene nanoribbons are ladder-type polycyclic aromatic compounds 1 composed of only one row of fused benzene rings, allowing two different very regular structures: polyacenes and polyphenacenes. Long acenes, made of linearly fused benzene rings, are very sensitive to oxidation due to minimal sextet stabilization, and are thus too unstable to be isolated if not functionalized by very specific substituents. 2 On the contrary, phenacenes are fully Fries-stabilized since their benzene rings are fused in a zig-zag way that allows the representation of a Kekulé formula where all hexagons are made of three double and three single bonds.

3 Short [3]-, [4]-, [START_REF] Jørgensen | Reviews about light-induced 6-electrocyclisations: a)[END_REF]-and [6]phenacenes are respectively known as phenanthrene, chrysene, picene and fulminene, and can be extracted from coal tar, 4 whereas their longer homologs have to be synthesized from smaller fragments. Mallory and coworkers have shown that photochemical 6-electrocyclisation 5 (or "photocyclisation" for simplicity) of stilbenoïd flexible precursors is an efficient approach for such syntheses, and applied it first to the formation of an alkyl-substituted [7]phenacene, by light-induced creation of a carbon-carbon bond between two condensed monofunctional phenanthrenes. 6 They subsequently adapted this strategy to the formation of the [11]phenacene 1 by double photocyclisation of a flexible precursor (Figure 1), obtained by condensation of one bifunctional and two monofunctional alkyl-substituted phenanthrene-based building blocks. 7 In these two cases, flexible aliphatic substituents are necessary to improve the solubility of the precursors and of the final polycyclic aromatic compounds.

For the last 5 years, our research group has focused on the development and the application of the glyoxylic Perkin reaction to the formation of large polycyclic aromatic compounds with a wide diversity of structures: Linear or macrocyclic, planar or twisted. 8 This Perkin condensation of arylglyoxylic acids with arylacetic acids, followed by same-pot esterification, leads to the formation of long flexible conjugated compounds composed of several aryl fragments linked by dialkyl maleate bridges. Therefore, besides its high efficiency, this method has the advantage of using only one active function, i.e. acetic or glyoxylic acid, for both the condensation reaction and the creation of a solubilizing flexible substituent, which significantly simplifies the synthesis of the building blocks. Contrary to alkyl substituents, these adjacent ester functions can later be transformed into other diesters or even into cyclic maleimides in order to modulate the solubility or the electronic properties of the final compounds. Following this strategy and inspired by Mallory's approach, we have recently reported the syntheses of tetracarboxysubstituted [8]-, [10]-, [12]-and [14]phenacenes 10 by double photocyclisation of flexible precursors obtained by double Perkin condensation of three naphthalene-and/or chrysenebased building blocks. The low solubility of the [14]phenacene 2 (Figure 1), obtained from three chrysene units, indicated the limits of this three-component approach concerning the size of the starting compounds. Using longer building blocks would increase the size of the final molecule, but it would also further decrease the density of solubilizing functions at its periphery and thus further decrease its solubility.

A first solution to this problem has been found by developing a protecting technique of the acetic acid function for the Perkin reaction.

11 By mono-saponification of arylacetic diesters, mono-protected bifunctional diacetic building blocks have been synthesized first, and their use in a multi-step condensation of five naphthalene-based building blocks has finally led to the formation of another [14]phenacene 3 (Figure 1) whose periphery is substituted by eight ester functions, and which was therefore found to be much more soluble than the previous tetra-ester [14]phenacene 2.

In this article, we present the completion of this synthetic strategy for the formation of long but still soluble phenacenes.

The development of complementary protecting techniques of glyoxylic acid functions for the Perkin reaction would allow the controlled condensation of more than five building blocks and could thus lead to longer soluble phenacenes, such as the [20]phenacene 4 (Figure 1) which is the result of the condensation of seven naphthalene units. First attempts of protection of the reactive keto group of the glyoxylic acid function, by usual formation of cyclic ketals and thioketals, were unsuccessful. Ketals appeared to be too unstable in the Perkin reaction conditions, whereas thioketals were not reactive enough to efficiently undergo deprotection once the condensation reaction was made. A counter-intuitive protecting method has been serendipitously found when performing, by mistake, a Perkin reaction with a glyoxylic ester instead of a glyoxlic acid, and then observing that the ester function remained quantitatively unreacted under these conditions. Although we cannot fully explain this mechanism, this discovery was particularly convenient due to the high similarity with the previously reported protection technique of arylacetic acids by formation of esters. 11 In full analogy with these previous results, we thereupon have developed the mono-protection of bifunctional aryldiglyoxylic acid compounds by formation of mono-acid mono-ester moieties, by saponification of a diester with only one equivalent of base, following an already welloptimized procedure.

Scheme 1: Synthesis of the asymmetric mono-acid mono-ester 6 by monosaponification of the diester 5.

Naphthylene-1,5-diglyoxylic diethyl ester 5 was treated in homogeneous conditions with one equivalent of potassium hydroxide in a mixture of ethanol and THF. After evaporation of the solvents and before neutralization, the solid crude mixture composed of the diacid 7, the desired mono-acid mono-ester 6 -both as potassium salts -and the remaining diester 5 was triturated in dichloromethane (DCM). The solid mixture of acids 7 and 6 was filtered off and evaporation of the filtrate recovering 28% of pure diester 5. The solid mixture containing the potassium salts of 6 and 7 was then acidified by trituration in aqueous HCl (2 M). DCM was added and a mixture containing mainly the diacid 7 and a small amount of 6 was filtered off. The organic phase of the filtrate was finally separated, dried and evaporated to afford 49% of pure mono-acid mono-ester 6 (Scheme 1). Very satisfyingly, the statistical character of this synthesis of an asymmetric mono-acid mono-ester 6 is largely compensated by a very easy and efficient separation, relying on solubility differences, of the three components of the obtained mixture. The [20]phenacene dodeca-ester 4 has thereupon been synthesized following a six-step convergent synthetic strategy from four different naphthalene-based building blocks (Scheme 2): monofunctional naphthyl-1-acetic acid 8, monoprotected bifunctional naphthylene-1,5-diacetic acid 9, bifunctional naphthylene-1,5-diglyoxylic acid 7 and its monoprotected homologue 6. Naphthyl-1-acetic acid 8 was first coupled with one equivalent of the mono-protected diglyoxylic compound 6, followed by in situ esterification of the central carboxylic functions. The resulting protected monofunctional dimer 10 was obtained with a yield of 61% and was then fully hydrolysed in basic conditions to afford the triacid 11 quantitatively. In parallel, naphthylene-1,5-diglyoxylic acid 7 was coupled with two equivalents of the mono-protected diacetic compound 9 and esterified in situ, with an excellent yield of 96%. Complete saponification of this doubly protected bifunctional trimer afforded the hexa-acid 13 with a yield of 86%. This bifunctional diacetic trimer 13 was finally coupled twice with the monofunctional glyoxylic dimer 11 by Perkin reaction, followed by in situ esterification of the twelve resulting carboxylic functions, yielding (53%) the linear heptameric flexible precursor 14 which is composed of seven naphthalene fragments linked by six dihexylmaleate bridges.

Long hexyl ester chains have been chosen to optimize the solubility of the dodeca-ester compounds. Finally, a sixfold photocyclisation triggered by UV light allowed the transformation of the flexible heptamer 14 into the targeted [20]phenacene dodeca-ester 4, with an isolated yield of 13% after ten days of reaction. This modest yield is partly due to the difficult separation of the fully condensed compound 4 from the mixture of incompletely cyclised intermediates. The optical absorption spectra of dilute chloroform solutions (Figure 2) show that the structurally regular [20]phenacene dodeca-ester 4 has an absorption maximum at 366 nm with close near-equal-intensity shoulder at 352 nm and a characteristic long-wavelength shoulder of about half-ofmaximum intensity at 399 nm, similar to the previously reported phenacene tetra-and octa-esters. The fluorescence spectrum, with maximum at 441 nm and a shoulder 462 nm, mirrors the shouldered main absorption peak. In comparison, the previously reported [14]phenacene octa-ester 3 shows two close main absorption peaks at 354 and 338 nm and a long-wavelength shoulder at 387 nm, and two close fluorescence peaks at 434 and 453 nm. Thus the absorption spectrum of 4, whilst very similar in shape, redshifts by about 12 nm with respect to 3, whilst the fluorescence red-shifts by about 8 nm. This slightly smaller Stokes shift in 4 compared to 3 points to a greater rigidity in 4 that restricts structural adjustments in the excited state.

Conclusions

In summary, ester functions have been serendipitously found to be convenient protecting groups of glyoxylic acid functions during Perkin condensations, and an efficient elaboration of appropriate naphthylene-1,5-diglyoxylic mono-acid monoesters 6 as mono-protected bifunctional building blocks has been developed. A resulting sequence of several Perkin condensations and deprotecting saponifications, in conjunction with complementary mono-protected diacetic building blocks, provided a simple and systematic access to a twelve-fold carboxy-substituted heptameric oligomer 14. This has subsequently been submitted to photochemical intramolecular cyclisations to form the fully condensed and well soluble [20]phenacene dodeca-ester 4, which is the longest phenacene so far reported. Although in this specific case the final photocyclisation step presented a modest yield, the two complementary protecting techniques developed for the Perkin strategy are both very efficient and widen the limits of this synthetic approach. Thus oligo-arylenemaleates of any size, and their corresponding nanoribbon-style cyclisation products, are now straightforwardly targetable, including macrocyclic species.

9Figure 1 :

 1 Figure 1: Substituted long phenacenes obtained by multiple photocyclisations.

Figure 2 :

 2 Figure 2: Normalised absorption (solid lines) and fluorescence (dotted lines) spectra of [14]phenacene 3 (grey) and [20]phenacene 4 (black) (4 μM in chloroform). Fluorescence spectra are recorded upon excitation at 350 nm.

protecting techniques for the Perkin reaction between arylacetic acids and arylglyoxylic acids efficiently allowed the multistep condensation of seven naphthalene fragments, linked by maleate bridges. Photocyclisation of this heptameric flexible precursor has led to a [20]phenacene dodeca-ester, the longest known phenacene to date.

	Guillaume Naulet,	a Stéphanie Huet-Exiga,	a Harald Bock	a and Fabien Durola*

a Complementary


Keywords: Nonlinear Optics, Spectroscopy, Sum-Frequency Generation, Difference-Frequency Generation, Second Harmonic Generation, Fitting Procedures

Experimental data from second-order nonlinear optical spectroscopies (SFG, DFG, SHG) provide parameters relevant to the physical chemistry of interfaces and thin films. We show that there are in general 2 N or 2 N-1 equivalent sets of parameters to fit an experimental curve comprising N resonant features, of vibrational or electronic origin for example. We provide the algorithm to calculate these sets, among which the most appropriate has to be selected. The main consequences deal with the existence of "ghost resonances", the need of a critical analysis of fit results and the procedure to search for better sets of parameters coherent with applied constraints.

Non uniqueness of parameters extracted from resonant second-order nonlinear optical spectroscopies

TITLE RUNNING HEAD Non uniqueness of parameters in NLO spectroscopies

For the past years, infrared-visible Sum-Frequency Generation (SFG) has become a widely used investigation technique for the chemical analysis of interfaces and surfaces. A lot of technical work has been done to improve the experimental set-ups dedicated to the production and detection of SFG photons.

In the nonlinear SFG process, two monochromatic light sources (frequencies ω 1 and ω 2 ) create a second-order polarization when interacting with a material. At the lowest-order (i.e. dipolar) approximation, this polarization is proportional to the product of the incoming electric fields through the second-order susceptibility tensor χ (2) , which accounts for the material's properties. The macroscopic polarization and susceptibility account for spatial averages of microscopic second-order dipole moments and hyperpolarisabilities (β), respectively. The secondorder polarization may oscillate at the sum (ω 1 +ω 2 , SFG) or the difference (|ω 1 -ω 2 |, DFG) of the incoming frequencies, acting as a coherent source of new beams at these frequencies. As such processes have a low cross section, they require the beams to have a high energy density, implying the use of short pulsed lasers. Infrared-visible SFG/DFG spectroscopy uses one beam in the visible range, usually with fixed frequency, which makes the detection of the few SFG photons easier, and a tunable (or broadband) one in the infrared (IR) frequency range. The SFG/DFG process becomes resonant (and therefore amplified) when the energy of the infrared beam matches that of an IR and Raman active vibrational transition of the material, making it a vibrational spectroscopy. The main advantage of the nonlinear vibrational spectroscopies as compared to the linear ones (e.g. IR absorption, Raman spectroscopy) lies in a fundamental property of second-order nonlinear optical (NLO) processes, which vanish within media possessing an inversion symmetry. The direct consequence is that SFG/DFG in centrosymmetric materials is only produced where the symmetry is broken, i.e. at the interface between two media. SFG/DFG spectroscopy is therefore intrinsically specific of interfaces. This has lead to the continuous spreading of this spectroscopic tool over time and its application to various kinds of interfaces ─ molecular monolayers adsorbed at the surface of liquids 1 , solids 2 and nanoparticles [3][4][5][6] ; the surface of liquids 7 ; thin molecular films 8 ─ under diverse environments, for example in vacuum 9 , in catalytic conditions 10 or under electrochemical control 11 .

Additional refinements include SFG imaging [12][13][14] , two-color and doubly resonant SFG 2,15,16 , and SFG near-field microscopy 17,18 .

Although IR-visible SFG is mostly used as a vibrational spectroscopy, resonances may occur with any of the three energies involved (i.e. IR, visible, SFG). Resonant second-harmonic generation (SHG) may also be handled within the same theoretical frame, as a special case of SFG.

The theoretical background for any resonant and nonresonant SFG/DFG process has been established in details in many books [START_REF] Bloembergen | Nonlinear Optics[END_REF][START_REF] Boyd | Nonlinear optics[END_REF] . The intensity of the generated beam is described by

        2 2 1 2 G I ω χ I ω I ω  (1)
where subscripts 1 and 2 account for the two incoming beams (degenerate in the SHG case), and

ω G stands for either ω 1 +ω 2 , |ω 1 -ω 2 |, 2ω 1 or 2ω 2 .
Depending on the polarizations of the three beams and the symmetries of the interface, the susceptibility term χ (2) comprises in general several tensor components.

Resonant processes are embedded into χ (2) and take the form of Lorentzian functions of the resonant frequencies [START_REF] Lin | [END_REF]22 . The Lorentzian description has been widely adopted as a consequence of the quantum perturbation calculation of the nonlinear susceptibility [START_REF] Boyd | Nonlinear optics[END_REF] , even if it does not account for the inhomogeneous broadening inside the sample. Doubly resonant processes give birth to products of such Lorentzian contributions 22,23 .

In a general way, the resonant susceptibility is

  2 1 N i R i= i i A χ = ω Ω ± iΓ   ( 2
)
where ω is the tunable experimental frequency involved in the resonance (i.e. ω G , ω 1 or ω 2 ).

Resonances are described by their frequencies Ω i , damping constants Γ i and amplitudes A i .

Coefficients A i are in general complex quantities:

i i i i A = Ae   (3)
The ± sign evidences the difference between SFG/SHG (+) and DFG (-) 24,25 . In the particular case of vibrational spectroscopy (ω = ω IR ), the summation runs over the N vibration modes of a molecule and the amplitudes involve the product of Raman and IR activities of vibration mode i 22 .

In the following, we focus on IR-visible SFG and DFG experiments, but the analysis below also applies to the other kinds of resonant second order generation processes. In an actual SFG/DFG experiment, additional effects interfere with the resonant (R) response of the sample. The most obvious one is the presence of a so-called nonresonant (NR) contribution, which appears essentially constant when ω scans the infrared frequency range. For vibrational IR-visible SFG/DFG, the nonresonant signal usually arises from the substrate supporting the molecules. Its amplitude and phase strongly depend on the nature of the substrate 26,27 , and on the frequencies and polarization combinations of the light beams. Therefore the general expression for χ (2) follows:

      2 2 2 0 1 N i i NR R i= i i A χ = χ + χ = χ e + ω Ω ± iΓ    (4)
Additional terms appearing in the calculation of the SFG intensity include Fresnel factors, local field contributions 28 , averaging of the molecular responses (β) into the macroscopic χ (2) 29 , surface versus bulk response 30 , breaking of the first-level approximations (e.g. higher order multipolar expansion 31 ). Most of these effects may be embedded into Equation (4) by using effective susceptibility components.

For the IR-resonant SFG/DFG case, the need for a distinct phase at the numerator of each Lorentzian function may be discussed. In the simple harmonic oscillator description, A i values are essentially real 22 . However, many authors consider the case where A i also must have a specific phase. It is possible to list several cases where complex amplitudes may be required:

-embedding of complex Fresnel factors in the effective amplitudes of the resonances 32,33 . As an example, in SFG experiments on an isotropic surface in the ppp polarization combination, the effective χ (2) is the sum of four components (i.e. zzz, xxz, xzx and zxx) 26 . Considering a vibration mode excited along the z-direction, Fresnel contributions in that case for parallel and perpendicular components of the electric fields contribute differently to xxz and zzz terms. In the case of substrates with complex refractive indices (e.g. a metal or a dielectric near an absorption edge), the phases of Fresnel terms differ; -the coupling of the SFG process to the localized surface plasmon of nanoparticles 34 or to surface plasmon polaritons in the visible 35 or infrared 36 ranges; -specific experimental geometries requiring to take into account the local field amplitudes and phases, as is the case in an ATR geometry [37][38][39] ; -embedding of doubly resonant effects in the amplitudes of the resonances 15 ; -influence of higher order terms in the multipolar expansion of the source fields of second-order nonlinear processes. Mostly magnetic and quadrupolar contributions have been considered in the literature [START_REF] Bloembergen | Nonlinear Optics[END_REF] . For example, resonant quadrupolar terms may be recast in effective second-order susceptibilities comprising Lorentzian terms 40 , the phases of which depend on the interference between several contributions involving various quadrupole matrix elements.

-a strong charge transfer between a metallic substrate and adsorbates, whose effect on different vibration modes creates phase shifts between them 41 ; -interference between reflected and refracted SF waves in thick structures leading to a sum of χ (2) components with different phases 42,43 ; -additionally, it is not rare to encounter situations for which fitting with real numerators is simply not possible [44][45][46] , even if it is difficult to discriminate between artifactual distortions of the experimental data (due to optical alignments, detection procedures, imperfect calibrations) and theoretical grounds justifying phase shifts between vibrational resonances, as listed above.

From the previous analysis, we deduce that we may write the most general expression for an effective resonant χ (2) as

    2 0 1 N i i= i A χ ω = χ + ω ω   (5)
where χ 0 is set real and positive, ω i stands for Ω i ± iΓ i , and phases Φ i in equation ( 3) are all distinct. This function contains the physical chemistry of the sample under study, which lies in the adjustable parameters. However, it is not directly accessible in an experiment, for which only the spectrum   2 2 ( ) χ  can be measured. In the following, we therefore distinguish the spectrum from its generator χ (2) (ω).

Most authors extract the chemical information by numerically fitting the experimental spectrum according to equations ( 1) and ( 5) to get the resonance parameters: frequencies, widths and amplitudes. Even if alternate methods exist 47 , curve fitting is universally used for IR-resonant SFG experiments, as it provides very fast results and does not require to dig deep into the theoretical concepts underlying the SFG process. As such, it is easily accessible to physical chemists who use SFG as a spectroscopic tool among others.

We point out two drawbacks of numerical fits according to equations ( 1) and ( 5):

-various choices of the initial guess for the fit parameters may lead to different final fits. For example, Sgura et al. 48 have shown that, in a nonlinear problem for which the best fit is sought by nonlinear least-square adjustment of parameters, it is possible to find several so-called numerical global minima of the objective function, whatever the value of the tolerance used. In other words, a nonlinear fit never ensures that the minimum found is the exact solution sought;

-even in the ideal case of a defect-free algorithm for the fitting procedure, the difference between a generator and its spectrum (i.e. the square modulus operation) raises the question of the singleness of the sets of fit parameters. As an illustration, Le Rille et al. 25 have shown that it is possible to find (and calculate) two sets of fit parameters (i.e. two distinct generators) which build exactly the same SFG spectrum made of one vibrational resonance and a non resonant background.

Therefore, it is reasonable that the growing number of users of SHG/SFG/DFG tools question the trust they may put in the results of their fits.

In this article, we show that care must indeed be taken when performing fits of experimental data according to Equations ( 1) and ( 5). The uniqueness of the fit parameters is not ensured, which may raise concerns when extracting chemical information from them. We therefore propose a method to calculate all the equivalent sets of fit parameters, which can be used after fitting to improve or even drastically change the fit results, in order to select the most appropriate parameters according to the physical chemistry of the material under study.

As demonstrated in the Appendix, there are, in the general case, for a given number of resonances N, 2 N equivalent generators, each described by its own set of parameters A i (i.e. modulus and phase), generating the same spectrum I(ω). In the case of vanishing χ 0 , there are still 2 N-1 such generators (and associated sets of parameters A i ). The Appendix also gives the guidelines for an algorithm calculating the different sets. For all sets of parameters, χ 0 and ω i remain the same.

This rather surprising result had already been demonstrated by Le Rille et al. 25 for N=1, which is the only case for which an analytical solution can be given using their method. We notice in the examples below that each amplitude à i distributes in the vicinity of two poles. Combining the ways to choose one of the poles for each coefficient leads to 2 N possibilities. This remark does not hold true in the case of a vanishing nonresonant contribution χ 0 .

The case where all resonances share a common phase is often used in the literature for the reasons exposed above. It represents a particular case of the more general situation where the dephasings between the resonances are set fixed. In such cases, there is only one free phase parameter (chosen as the phase of resonance 1). In this situations, it is reasonable to wonder whether, among the 2 N (resp. 2 N-1 ) sets of parameters, it is possible to find several ones which fulfil these additional constraints on the phases. The fact that the amplitudes A i have a fixed phase in the complex plane does not have any direct consequence on the values of roots J i of the polynomial J N (cf. Appendix).

In fact, the J i 's depend both on A i and ω i , so that a particular property of the A i alone has in general no influence on the generality of the problem. As a consequence, except accidental numerical combinations, there should be only one set of parameters with resonances in phase or having fixed dephasing. However, finding such a function by data fitting is not sufficient to ensure that the best fit has been found, as is illustrated in the second example below.

The most striking consequence of this theoretical analysis is the existence of ghost resonances.

Let's consider a generator function built of N resonances (Equation 5) and the associated spectrum (Equation 1). In addition to the 2 N -1 equivalent generators with N resonances, it is possible to build additional generators having the same spectrum with one (or more) extra resonances, all with nonvanishing amplitudes. The resonances artificially added to the initial set do not show up in the spectrum, whereas they exist as a first order pole in the new generators. This general property defines a ghost resonance. As a consequence, the generator may contain more resonances than the spectrum. In other words, seeing N resonating features on a spectrum only means that there are at least N resonances in the generator. This situation can be related to the case, well-known in SFG spectroscopy, of a nonresonant contribution χ 0 interacting with one resonance of width Γ, amplitude 2Γχ 0 and phase 270° (resp. 90° for DFG), which is mathematically invisible in the SFG signal as the spectrum will appear as a strict flat line. This theoretical case is very close to the experimental spectra from molecules adsorbed on gold and studied by IR-visible SFG in the blue-green region of the visible spectrum 26 . However, as illustrated in the first example below, the ghost resonances also exist without a nonresonant contribution, in which case the invisible resonance interferes destructively with the contributions of its neighbors. The phenomenon of ghost resonances comes from to the fact that, to the existing generator made of N modes with amplitudes chosen among the 2 N possibilities, it is possible to add a (N+1)th resonance with a vanishing amplitude and an arbitrary ω N+1 . One may then construct in total 2 N+1 equivalent generators. Among these, it is easy to show (cf. Appendix) that only the initial 2 N generators have a zero amplitude for the (N+1)th resonance, whereas the 2 N others have a nonvanishing amplitude for that resonance.

Turning to the analysis of experimental data, these results raise several issues. We suppose that a generator could be found by the fitting procedure, which accounts As an illustration of the ghost resonances issue, we take the example of an SFG experiment in which a strong nonresonant background from a gold substrate (either planar or nanoparticles) interferes with CH 3 vibration modes from adsorbates (e.g. an alkanethiol). We adapt the following from the example of Figure 6 and Table 2 in Ref 34, which studied dodecanethiol adsorbates on spherical gold nanoparticles. Due to the high signal-to-noise ratio in that experiment, it could have been possible to fit not only with the three CH 3 resonances but also with two CH 2 modes, as was done for the data of Figure 5. In other words, we can introduce two ghost peaks in the original fit parameters.

We therefore take as original parameters:

χ 0 = 1.13; Ã 1 = 1.98; Ã 2 = 1.17; Ã 3 = 2.14; Ã 4 = 0; Ã 5 = 0; Results are shown in Table 2, with the corresponding graph in Figure 1(b). We remark in that case that Set 1 is again very close to a realistic solution according to the above-mentioned criteria, as the phase span is very small (16.5°/12.5°). Even if the effect is less spectacular than in the previous case, it is worth noting that the amplitudes of the ghost resonances are not negligible at all, respectively around 0.6 and 0.2 times the amplitude of the smallest real resonance. Contrary to the previous case, the presence of the ghost resonances does not perturb too much the three other ones, both in amplitude and in phase. However, data interpretation is again greatly modified by their existence.

ω 1 = 2881cm -1 ; ω 2 = 2943cm -1 ; ω 3 = 2967cm -1 ; ω 4 = 2853cm -1 ; ω 5 = 2917cm -1 ; Γ 1 = Γ 2 = Γ 3 = Γ 4 = Γ 5 = 3.5cm -1 ; Φ 1 = Φ 2 = Φ 3 =
To go further in the use of the model presented here, we take another example from the literature enhancing strong interference between resonant and nonresonant contributions. It is in fact rather difficult to find references where the actual phases of the resonances, when relevant, are listed. It is the case in Ref 44, which presents SFG experimental spectra of a thiophenol monolayer on Ag(111), with three vibration modes of the adsorbates in the 1000cm -1 region. Neither the amplitudes and widths of the modes, nor the fit curve are presented in the paper, but it is possible to reconstruct an acceptable function from the known parameters (Figure 2). After a careful comparison between Ref 44 and 49, and with the help of the experimental and fit curves, it is possible to correct an error in Table I of Ref 44, where the phase of the resonance at 1074cm -1 should be -95° instead of -5°.

We therefore use for original parameters, with the notations of the present work:

χ 0 = 0.45; Ã 1 = 3; Ã 2 = 1; Ã 3 = 1.5;

ω 1 = 1003cm -1 ; ω 2 = 1025cm -1 ; ω 3 = 1074cm -1 ; Γ 1 = Γ 2 = 2.5cm -1 ; Γ 3 = 4cm -1 ; Φ 1 = 55° Φ 2 = 215°; Φ 3 = 275°
The nonresonant baseline is not flat, which makes fitting more difficult. The corresponding function is shown on Figure 2, exhibiting a good agreement with Figure 2 of Ref 44. We have calculated the seven other sets of parameters based on this one (Table 3).

The same observations as above about an organization of the amplitudes around two poles with significantly distinct values can be made. In this case, it is also interesting to look closer at sets 2 and 6. For these, the phases of the three resonances are close to the particular case of ideal and undisturbed harmonic vibrations, in which all phases are equal modulo 180°. We therefore used parameters of fit 2 and 6 as initial guesses to try and improve the fits by imposing such a condition on the phases and letting other parameters (Ã i , Γ i ) free. Set 6 gave the best results, with the following final parameters:

χ 0 = 0.446; Ã 1 = 2.889; Ã 2 = 2.477; Ã 3 = 2.211; Φ 1 = 74.5°; Φ 2 = Φ 3 = 254.5°. Γ 1 = 2.47cm -1 ; Γ 2 = 2.79cm -1 ; Γ 3 = 4.44cm -1 .
Set 2 on the contrary led to a bad quality fit for resonance 3. The curve corresponding to this alternate fit issued from set 6, drawn on Figure 2, shows that the quality of the fit is rather good.

The amplitudes of modes 2 and 3 exhibit strong variations as compared to the initial parameters. It is therefore possible to find a fit according to Equation ( 5) with purely real amplitudes. There are indeed discrepancies between the original and the new fit curves, especially for the shape of resonance 3, and we do not claim that the new fit is better than the previous one, in particular because the analysis was not performed on the original data. However, this example illustrates the difficult task of fitting SFG curves ab initio, that is without strong conditions imposed on the parameters. It shows that it seems possible to fit the experimental curve with in phase resonances, probably paying the price of a slightly lower quality fit, but within hypotheses which are much easier to justify. As above, the scientific content and conclusions drawn from both the amplitudes and the phases of the three vibration modes will indeed vary much from one fit to the other. Finding such an alternate fit was possible because the method developed here allowed to scan all the equivalent sets of parameters, only a small number of which could lead to satisfactory fitting functions.

As a conclusion, the data presented in this article illustrate the fact that it is not possible to consider a fit of SFG/DFG/SHG data as a unique solution. In the case of out of phase resonances, it is mathematically not true. We propose an algorithm to rapidly find the 2 N (resp. 2 N-1 for vanishing nonresonant contribution) alternate sets of parameters building up strictly the same intensity function as a given original set with N resonances. The existence of such alternate sets of parameters leads to the possibility of ghost resonances, which may completely blur the analysis of the experimental data and lead to erroneous conclusions on the structure and dynamics of the interface. Finally, we have illustrated some tricks hidden behind the concept of "best fit". The best fit is always defined within a given frame (fit algorithm, value of the tolerance, initial guesses, signal-to-noise ratio, model describing the process). In particular, considering that there are several local minima of the objective function to the fit problem, the initial guesses play an essential role.

Performing a full analysis of all the available alternate sets of parameters may help finding a more accurate and more compatible solution for a given fit, even in the rather classical case where all resonances are set in phase.

Due to the specificity of nonlinear optical spectroscopies, and in particular their high sensitivity to interfacial processes involving very low amounts of matter, it could appear interesting to use and analyze their experimental data in a self-sufficient manner, that is without the help of other more conventional experimental techniques. The results presented here prove that it is most of the times not possible, even after a careful analysis of the fit results. To discriminate between the various sets of parameters, it still seems advisable to establish data analysis on a firm ground with the help of other spectroscopies, for example IR absorption, Raman spectroscopy and HREELS in the case of IR-visible SFG/DFG. They will provide an image, even sometimes distorted as compared to SFG/DFG, of the processes at stake at the molecular level, and help to establish the fits of nonlinear spectroscopy data on a firmer ground. However, this is still not sufficient to prove that the result of the fitting procedure of the second-order nonlinear optical spectroscopies is correct. There is no universal rule to select the appropriate set of parameters among the multiple solutions. Anyway, it is interesting to note that there are some ways to discriminate between the various generators. For example, the comparison between SFG and DFG data 25 greatly helps selecting among several possible sets parameters, and get rid of ghost resonances. The use of phase-sensitive interference methods in nonlinear optics 50 may also give access to the phases of some of the complex quantities in χ (2) . Finally, the evolution of the IR-visible SFG/DFG/SHG spectra with a tunable parameter (e.g.

the polarizations of the light beams 51 , the electrochemical potential 11 or the visible wavelength 15 ) requires to fit series of experimental curves with coherent sets of parameters, usually helping to lift the ambiguity on their values. The black curve (squares) is the closest to the experimental data. The grey curves (circles) was created from an alternate set of fit parameters by imposing the three resonances in phase.

Appendix

Starting from Eq. ( 5) and the notations of the text, we write the generator:

  2 0 1 N i i= i A χ = χ + ω ω  
with χ 0 real and positive. All ω i are supposed distinct one from the others.

The 2N+1 adjustable parameters fitting the experimental spectrum are: χ 0 , A i , ω i . The question is: knowing one set of parameters, are there other sets (i.e. other generators) building the same spectrum, and how many ? We postulate the existence of such another set of 2N'+1 parameters χ' 0 , A'i, ω' i . In other words, we define two functions of the real variable ω

  0 1 N i i= i A J ω = χ + ω ω  
for which the values of χ 0 ≥ 0, A i , ω i are known, and

  ' 0 1 N i i= i A' K ω = χ' + ω ω'   , such as     2 2
J ω = K ω for any positive real number ω.

Considering the spectrum as a complex rational fraction with respect to variable ω, its partial fraction decomposition leads to  

2 2 0 0 0 1 1 1 1 1 1 N N N N j j i i i= j= i= j= i i j i j i A A J ω = χ + A χ + + A χ + ω ω ω ω ω ω ω ω                         (A1)
where the bars stand for complex conjugation. The last two sums in (A1) are complex conjugates of one another for a real ω. Parameters ω i having a fixed sign for their nonvanishing imaginary parts, they cannot be confused with their conjugates.

We define at this stage a ghost resonance i as one for which the coefficient of pole ω i in (A1)

vanishes. This implies that it is not visible in the spectrum. We first analyze the problem in the absence of ghost resonances in J and K.

The equality of the spectra     2 2

J ω = K ω must hold for any positive real number ω. The properties of the decomposition of complex rational fractions imply that  χ' 0 2 = χ 0 2 (uniqueness of quotient);

 the unprimed resonances correspond one to one to the primed ones (uniqueness of the partial fractions). This is equivalent to the fact that N equals N', and that a renumbering exists such that ω' i = ω i (i.e. Ω' i = Ω i and Γ' i = Γ i ).

Only A' i may thus differ from A i from one set of parameters to the other. The problem is therefore reduced as follows: considering

  0 1 N i i= i A J ω = χ + ω ω   ;   0 1 N i i= i A' K ω = χ + ω ω  
, and

    2 2
J ω = K ω for any positive real number ω, what are the values of A' i and how many possibilities are there to build such a K function ? 1) We first assume that χ 0 ≠ 0.

Writing

      0 N χ J ω J ω = D ω and       0 N χ K ω K ω = D ω with     1 N i i= D ω = ω ω  
, J N and K N are monic polynomial functions of degree N, they have exactly N complex roots J i and K i , respectively

    1 N N i i= J ω = ω J   and     1 N N i i= K ω = ω K   .
Let P(ω) be the greatest common divider of J N (ω) and K N (ω), so that J N (ω) = P(ω) A(ω) and

K N (ω) = P(ω) B(ω). Calling             N N J ω J ω A ω Q = = = K ω K ω B ω
we have 2 1 Q = for ω real and Q belongs to the unit circle of the complex plane.

The transformation ω i z = ω+i

 maps the real axis to the unit circle, its reciprocal is 1 1

+ z ω = i z  .
Therefore Q, as a function of z, is a rational fraction which maps the unit circle onto itself. As such, the roots of Q are exactly the images of its poles under the transformation 1 z z  . The reader may refer to Ref. 52 for more details.

For α i a root of Q, let us introduce the Blaschke factors B i 53 , defined by 1

i i i z α B = α z   , it is clear that i i R Q = ζ B    , with ζ constant, 1 ζ =
, where the product runs over R, the roots of Q.

Turning back to the ω variable gives 1 1

i i i i i i β R' i i ω β + α Q = ζ' B ,B = ,β = i α ω β      . The limit of Q as ω → ∞ shows that ζ' = 1. Therefore   i i β R' i ω β Q = ω β     
, which means that i β belong to the roots of A(ω) (i.e. to the J i 's)

and i β to that of B(ω).

The monic polynomials A(ω) and B(ω) having the same degree and no common roots, it implies that i β and i β describe exactly the roots of A and B, respectively. In other words, i i J β  and i i K β  , and B(ω) is the complex conjugate of A(ω) for real values of ω.

Knowing the N roots J i of polynomial J N , the N roots of K N are built by choosing N-m roots among the J i 's while the last m roots are the conjugates of the remaining J i 's, with m taking any value between 1 and N.

 

    1 N m m N i i i= i=N m K ω = ω J ω J       , 1 ≤ m ≤ N.
The number of different possibilities to build a polynomial K N reduces to the number of partitions into two sets of the set made of the N roots of J N . In the general case, for which all roots are distinct and not real, there are exactly 2 N ways to build K N polynomials. Consequently, the general number of different sets of parameters A i to build a unique spectrum is also 2 N .

As mentioned in the text, the case N=1, with two different sets of solutions, has been evidenced by Le Rille et al 25 . Using the present algorithm also makes it possible to analytically calculate the case N=2. For N>2, a numerical evaluation is necessary.

The reasoning presented here additionally gives an algorithm for such a numerical calculation of the 2 N sets of parameters once one of them is known, working with a fixed set of resonances ω i , following the successive steps:

-knowing J(ω), reduce it to common denominator D(ω);

-extract numerator and divide it by χ 0 to get J N (ω);

-numerically solve the polynomial J N to find the N roots;

-create the 2 N -1 K N polynomials -for each K N polynomial, the complex A' i are evaluated by partial fraction decomposition of

      0 N χ K ω K ω = D ω which leads to     0 1 N i i N i j j= j i χ K ω A' = ω ω    (A2 )
2) The case χ 0 = 0 is treated in the same way, apart from a few differences. For  

1 N i i= i A J ω = ω ω  
there is no absolute phase reference (contrary to the previous section, in which setting χ 0 real and positive fixes the phases). As a consequence, the A i are only determined modulo a phase shift, which would result in the same    

J transforms into       1 1 N N i i= J ω J ω = A D ω        
where J N-1 is a monic polynomial of order N-1.

Assuming that 1 N i i= A  does not vanish, the problem is solved in the same way as in the previous case, 1 N i i= A  playing the role of χ 0 and J N-1 that of J N . We define as above      

1 1 1 N N N i i i= i= i K ω A' K ω = = A ' ω ω D ω          
with the condition     2 2

J ω = K ω .

The asymptotic behavior of this equality for ω → +∞ implies that  real and positive (which fixes the phase for K) leads to   . The analysis over the N-1 roots of K N-1 (ω) is led in the same way as above for K N (ω), leading to the conclusion that there are 2 N-1 equivalent sets of parameters in that case.

If

1 N i i= A
 vanishes, the analysis is performed using the same scheme. In a general way, J(ω) may be written as the division of a polynomial by D(ω). One selects the nonvanishing ω p term of the numerator with highest order p (p ≤ N) to create the J p (ω) monic polynomial as above. Repeating the previous analysis shows that the number of equivalent sets of parameters becomes 2 p .

3) Both cases above have been analyzed in the absence of ghost resonances. Starting from a generator J(ω) with N poles and free from ghost resonances, it is easy to build generators with N+1 poles having the same spectrum. We add a (N+1)th term to the initial generator, with A N+1 = 0. For this particular case, the value of ω N+1 is arbitrary, but it will naturally be chosen among meaningful values deduced from the chemical physics properties of the sample under study.

Applying the algorithm above to this new generator, we create a total amount of 2 N+1 generators (resp. 2 N if χ 0 = 0). The condition A N+1 = 0 implies that the complex quantity ω N+1 belongs to the roots of polynomial J N+1 . Considering the way polynomials K N+1 are built, half of them also has ω N+1 as a root, the other half having 1 N ω  , and therefore not ω N+1 , as the corresponding root.

Consequently, from Eq. (A2) it follows that one half of the generators (labeled N 1 ) have A N+1 = 0, but for the other half A N+1 ≠ 0 (labeled N 2 ). For these latter, the resonance N+1 has a nonvanishing amplitude, even if it does not contribute to the spectrum, it is thus a ghost resonance. It must be stressed that the value of amplitude A N+1 depends on the choice of the parameter ω N+1 .

For generators in N 2 , we therefore have  

1 1 0 1 1 0 N j N j= N j A' K ω χ + ω ω       
, which in turn proves that pole ω N+1 disappears indeed from (A1), and allows to check the status of ghost resonance. We understand that there are in fact two ways for a resonance ω i not to appear in a spectrum (A1): as a consequence of either A i = 0 or 0 1 0

N j j= i j A χ + ω ω           
. In the former trivial case, the resonance does not belong to the poles of the generator; in the latter situation, resonance ω i is a true ghost resonance according to the definition given in the article. We can thus formulate a more accurate definition of a ghost resonance: for a given generator   0

1 N i i= i A J ω = χ + ω ω   , resonance ω i is a ghost if   0 i J ω = .

  Bertrand Busson* and Abderrahmane Tadjeddine Laboratoire de Chimie Physique; CNRS, Université Paris-Sud 11; Bâtiment 201 Porte 2; 91405 Orsay; France Email address: Bertrand.Busson@u-psud.fr; Phone: +33 1 69 15 32 75; Fax: +33 1 69 15 33 28 RECEIVED DATE (to be automatically inserted after your manuscript is accepted if required according to the journal that you are submitting your paper to)

  for the N resonances showing up on an experimental spectrum. It is then possible to create the 2 N -1 alternative generators of the same fitting function with N modes. It becomes necessary at that stage to find the one which carries the appropriate chemical information relevant to the case under study. As discussed below, complementary experiments coupled to a fine analysis of fit parameters seem mandatory for that purpose. In a second stage, one must deal with the ghost resonances issue, which may arise in two ways: there may be too many resonances in the generator (i.e. the generator includes ghost resonances) and/or there may be ghost resonances missing in the generator. In the first case, some of the resonances in the generator do not contribute to the spectrum, whereas in the second case the generator misses resonances which should be present even if they do not appear in the spectrum. As the ghost resonances have no effect on the experimental data, it is not possible to discriminate them using only one experimental spectrum. Finally, one must assess whether the ghost resonances found must be kept or discarded on the basis of physical chemistry grounds. Of course, in the case of IRvisible SFG for example, the resonances are usually taken among the IR-active vibration modes of the material. However, the IR and Raman activities may be strongly modified by the adsorption process (e.g. through charge transfer or molecular geometry modification) and the resonances by interactions inside the interface (e.g. by dipole-dipole coupling or Stark shift). These effects make the choice of the number of modes N and the guesses for the ω values of the ghost resonances rather uncertain. As a consequence, there are indeed exactly 2 N equivalent sets of parameters to construct a given fit function using exactly N modes, but rather an infinity if one considers an arbitrary nonvanishing number of ghost resonances.

no obvious reason why set 1

 1 could not represent an acceptable fit of experimental data with five resonances. Apart from the knowledge of the existence of the original set, only a critical analysis of the parameters coupled to a study of the experimental curves may lead to the rejection of set 1.Inversely, the experimental curve could have been fitted with the hypothesis of five vibration modes, and a fit close to set 1 obtained. In that case, applying the algorithm shows that there are eight sets with vanishing, or almost vanishing, amplitudes for modes 4 and 5, hinting at their potential status of ghost resonances. Naturally, the presence of CH 2 resonances in the SFG spectrum of an alkanethiol self-assembled film being often used as a sign of gauche defects and disorder in the monolayer 33 , the chemical interpretation of the data based on the original set of parameters severely differs from that based on set 1.The phases of the CH 2 resonances are very close to 270° and their amplitudes not far from the quantity 2Γχ 0 , which explains why they are transparent on the graph. The deviation from this perfect case represents the contribution of the neighboring resonances in the vanishing process. To test its amplitude, we performed the same calculation with a vanishing nonresonant contribution.

Figure 1 .

 1 Figure 1. (a) Simulated SFG spectrum made of three CH 3 resonances and a nonresonant contribution, according to parameters described in text (squares). (b) Simulated SFG spectrum made of the same CH 3 resonances without nonresonant contribution (circles). Continuous curves are the same mathematical functions, drawn using sets 1 in Tables 1 and 2 respectively, and comprising five resonances.

Figure 2 .

 2 Figure 2. Simulated SFG spectra based on the experimental curves and the fit parameters of Ref 43.

Figure

  Figure 2

2

 2 Figure 2

2 2 2 χ

 2 = J ω . To fix the phase in a symmetrical way, we impose 1 to be real and positive.

Then

  214.8°.The resonances relate, from 1 to 5, to symmetric CH 3 stretching, CH 3 Fermi resonance, asymmetric CH 3 stretching, CH 2 symmetric and CH 2 asymmetric stretching modes. The corresponding function is drawn on Figure1(a). In Table1, we report, among the 32 sets of parameters, the eight ones with nonvanishing CH 2 ghost resonances. The first result is that these ghost resonances have an amplitude around four times greater than the real CH 3 ones. It is not obvious from a simple look at the table that they simply do not show up at all in the spectrum.Secondly, we observe the above-mentioned repartition of amplitudes for resonances 1, 2 and 3 around two poles for each, with respective values very far apart, respectively around 3.5, 6.5 and 3 times greater. Among these eight sets, we can define criteria for acceptable sets as is done in Ref 32. As far as the phases are concerned, we expect a common phase for resonances 1, 2 and 3 on one side , and 4 and 5 on the other side, as they relate to the same chemical species. With these criteria, sets 2, 3, 4, 6 and 8 are not acceptable. For sets 5 and 7, the phase span remains acceptable (resp.

	35.8°/11.5°; 34.8°/2.8°), whereas it is very good for set 1 (16.5°/14.3°). As a consequence, there is

Table 1 .

 1 27 Tadjeddine, A.; Le Rille, A.; Pluchery, O.; Vidal, F.; Zheng, W. Q.; Peremans, A. Phys. Stat. Sets of equivalent parameters for curve (a) in Figure1. The original set is used to build the curve with a vanishing amplitude for resonances 4 and 5, the eight others have nonvanishing amplitudes for these resonances.

	Sol. A 1999, 175, 89-107.
	28 Zhuang, X.; Miranda, P. B.; Kim, D.; Shen, Y. R. Phys. Rev. B 1999, 59, 12632-12640.
	29 Hirose, C.; Akamatsu, N.; Domen, K. Appl. Spec. 1992, 46, 1051-1072.
	30 Han, S. H.; Ji, N.; Belkin, M. A.; Shen, Y. R. Phys. Rev. B 2002, 66, 165415.
	31 Guyot-Sionnest, P.; Shen, Y. R. Phys. Rev. B 1987, 35, 4420-4426.
	32 Tourillon, G.; Dreesen, L.; Volcke, C.; Sartenaer, Y.; Thiry, P. A.; Peremans, A.
	Nanotechnology 2007, 18, 415301.
	I SFG	NR	R	2	NR e	i	n q 1	IR	i a q e q i q
	Chem. Phys. Lett. 1999, 302, 528-532.
	36 Eliel, E. R.; van der Ham, E. W. M.; Vrehen, Q. H. F. Appl. Phys. B 1999, 68, 349-353.
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33 Himmelhaus, M.; Eisert, F.; Buck, M.; Grunze, M. J. Phys. Chem. B 2000, 104, 576-584. 34 Humbert, C.; Busson, B.; Abid, J. P.; Six, C.; Girault, H. H.; Tadjeddine, A. Electrochim. Acta 2005, 50, 3101-3110. We correct here the misprint in Eq.(1), which should appear as 2 . 35 Alieva, E. V.; Kuzik, L. A.; Yakovlev, V. A.; Knippels, G.; van der Meer, A. F. G.; Mattei, G. 40 Neipert, C.; Space, B.; Roney, A. B. J. Phys. Chem. C, 2007, 111, 8749-8756. 41 Vidal, F.; Busson, B.; Tadjeddine, A. Electrochim. Acta 2004, 49, 3637-3641.

Table 2 .

 2 Sets of equivalent parameters for curve (b) in Figure 1. The original set is used to build the curve with a vanishing amplitude for resonances 4 and 5, the four others have nonvanishing amplitudes for these resonances.

		Original Set 1 Set 2 Set 3 Set 4
	Ã 1	1.98	2.08	2.14	2.09	2.15
	Φ 1 (deg.) 0	0	0	0	0
	Ã 2	1.17	1.22	1.24	1.50	1.53
	Φ 2 (deg.) 0	-16.5 -42.2 13.9	-11.8
	Ã 3	2.14	2.17	2.18	2.41	2.43
	Φ 3 (deg.) 0	-8.4	-29.4 -40.0 -61.0
	Ã 4	0	0.72	0.73	0.72	0.73

Table 3 .

 3 The eight sets of equivalent parameters for the original curve in Figure3.

		Original Set 1 Set 2 Set 3 Set 4 Set 5 Set 6 Set 7
	Ã 1	3	5.11	3.05	5.20	3.00	5.12	3.05	5.20
	Φ 1 (deg.) 55	296.0 63.8	304.8 56.2	297.2 65.0	306.0
	Ã 2	1	1.05	2.52	2.65	1.00	1.05	2.52	2.65
	Φ 2 (deg.) 215	183.8 262.5 231.3 216.7 185.5 264.2 233.0
	Ã 3	1.5	1.52	1.51	1.53	2.17	2.19	2.18	2.21
	Φ 3 (deg.) 275	263.9 270.6 259.5 274.3 263.3 269.9 258.8

Table of Content

 of Graphic Table of Content CaptionA typical case of ghost resonances in SFG spectroscopy. Vibration modes 4 and 5 (CH 2 vibrations) belong to the nonlinear susceptibility whereas they disappear in the SFG spectrum. Left panel shows the five resonances (three CH 3 and two CH 2 modes) plotted separately and the non-resonant background (from a gold substrate). Right panel displays the corresponding SFG spectrum.

Acknowledgments

We are greatly indebted to J.C. Léger, Mathematics Department, Université Paris-Sud, Orsay, France for his great help in analyzing the mathematical part of this work. We also warmly thank E.R. Eliel, Leiden Institute of Physics, Netherlands, for providing us with Ref 49.