This fieldwork preliminary report is part of a project that has received funding from the European Research Council (ERC)

under the European Union's Horizon 2020 research and innovation programme (Grant agreement No. 726206).

TEAM COMPOSITION

Pr. Julien Loiseau (Aix-Marseille University) -team leader Pr. Bertrand Hirsch (Sorbonne University) Dr. Amélie Chekroun (Aix-Marseille University)

FIELDWORK OBJECTIVES

The main objective of the mission was to survey the area around the village of Igre Hariba (Tigray region, Kwiha Woreda) in order to find new evidence of ancient Muslim communities in Northern Ethiopia.

The area was chosen according to epigraphic evidence, Arabic inscriptions being major sources for the history of early Islam in Ethiopia. Arabic inscriptions have been found so far in four distinct locations in Ethiopia: Eastern Tigray, Tchertcher, South Amhara and Ifat. The stelae with Arabic inscriptions already found around Igre Hariba and Kwiha call for two comments:

1. These inscriptions all belong to Muslim gravestones and are dated from the 4 th to the 6 th Century A.H. / 10 th to 12 th Century C.E. They bear insofar witness of the presence of Muslim trade and communities in Tigray before and during the Zagwe period. The Muslim presence in Tigray might have been linked to the Dahlak Islands (today in Eritrea), considering that the name of Ḥafs b. ʿUmar al-Yamāmī is mentioned on stelae found in both places. It should be stressed that no other written document, neither Ge'ez nor Arabic, mentions the existence of a Muslim community in Tigray at that time.

2. The original location of the gravestones is not precisely known. According to N. Pearce (beginning of the 19 th Century), Wagar Ḥaribā / Igre Hariba was the main place of the area, where an important leader of the region, the brother of Walda Sellāśē, was settled. The same witness located the original place of the stelae near the village of "Quened". More than a century later, Carlo Conti Rossini did not recognize the site described by N. Pearce and, moreover, did not identify the original location of the stelae8 . Like his successors (the French archaeologists in the 1960's and W. Smidt in the 2000's), he witnessed only displaced and reused stelae, in a place named "Bilet". In the 1930's, according to Conti Rossini, the main locality of the area was hence Kwiha, the rise of which being probably linked to the construction of the road Asmara-Addis Ababa by the Italians and to the layout of a fountain still in use.

The two main objectives of the fieldwork were to identify the exact location of the Muslim graveyard from which the stelae were shifted and to understand the link between this Muslim community and the ancient trade route driving south from the Red sea shore and crossing the area. Identification of archaeological sites and their possible excavation will shed new light on the issue of islamization and the history of the earliest Muslim communities in Eastern Tigray.

OFFICIAL AUTHORIZATION

Producing the official letter and authorization by the ARCCH, signed by Ato Demerew Dagne (Ref. n os 08/R2-8-2/019 & 08/6h-8-2/020 -see Annex), we introduced ourselves to Ato Dawit Hailu, director of the Tourism and Culture Office of Tigray (Mekelle), on 7 March 2018, in order to start our fieldwork on 8 March. Ato Dawit Hailu valued our project, reminding the important and long-standing history of the region. Considering that we were only supposed to survey the area, and not to excavate, Ato Dawit Hailu decided to not assign us an expert from the office, provided that we introduce ourselves to the officials of the kebele.

STELAES IN MEKELLE CULTURAL AND TOURISM OFFICE MUSEUM

Ato Dawit Hailu gave us the authorization to visit the Museum of the Tourism and Culture Office of Tigray, where two stelae with Arabic inscriptions coming from Igre Hariba/Kwiha are preserved. They have been given to the Museum by Pr. Tekle Hagos (Addis Ababa University) in 1993 G.C. According to personal information given to us afterwards by Pr. Tekle Hagos, these two stelae were brought to him by people from "Old Kwiha" while he was surveying the area. No information about their original location is available, despite the fact that they belong to the same corpus than the stelae preserved in the National Museum of Ethiopia. The Arabic inscriptions of the two stelaes have never been published.

Mekelle Museum Stele 1

Fragment in basalt stone: H. 52 x W. 52 x Th. 6 cm. Only the last seven lines of the Arabic inscription are preserved.

Mekelle Museum Stele 2

Fragment in basalt stone: H. 13 x W. 27 x Th. 14 cm. Only the first four lines of the Arabic inscription are preserved. No evidence of an old settlement has been found during the survey of Igre Hariba. The few places indicated by local people (e.g. Kelkal, Gedel Kebatsani) did not provide any clue. Inhabitants were not informed of any archaeological finding in the area. But they identified the place name "Bilet", mentioned in the documentation since Conti Rossini's visit in 1937, as a collective field for grazing located two kilometers North of Igre Hariba, on the valley bottom, near the church of Kwiha Kirkos, outside Kwiha (see map above).

IGRE HARIBA: THE ISSUE OF TOPONYMY

Bilet seen from Igre Hariba

Igre Hariba seen from Bilet

This toponymic evidence is all the more important. According to the 19 th -Century testimony of Nathaniel Peirce, at a time when Igre Hariba was the main place of (and the name given to) the whole area, we named this fieldwork "Igre Hariba" and first surveyed the village known today as Igre Hariba. But the evidence we were looking for seemed to be located in the area called today "Kwiha", since the latter replaced the former as the main place (and the main name) of the area after the construction of the road in the 1930's. According to C. Conti Rossini, these ruins are the remnants of a rectangular building dating back to the early Middle Ages. In 1970, Francis Anfray suggested to compare the columns observed near Kwiha Kirkos with those of Däbrä Gärgiš (Ḥaramat), which he proposed to date from late Aksumite or early post-Aksumite times 10 . In 1977, Eric Godet also attributed the stones and columns from Kwiha to late Aksumite or post-Aksumite times [START_REF] Eric | Répertoire des sites pré-axoumites et axoumites du Tigré (Ethiopie)[END_REF] . Local traditions attribute the ruins to an ancient church founded by aṣe 'Amdä Ṣeyon and later destroyed by imam Aḥmad "Grāñ" in the 1530's. Archaeological research has not been carried out yet on the site.

KWIHA: ARCHAEOLOGICAL EVIDENCE AROUND KWIHA KIRKOS CHURCH

The survey of the surroundings of the present Kwiha Kirkos church revealed that the hill is, at least partly, an archaeological tell. Ruined walls were observed in three various locations:

3. in a path leading to orchards located bellow the tell: fragments of pottery and bones were also noticed, within 2,5 m. of archaeological stratigraphy. GPS: 13° 29 mn. 9 s. North / 39° 32 mn. 54 s. East. Altitude: 2120 m.

This archaeological evidence (walls, pottery, bones) observed all around the site strongly suggests that a permanent (urban?) settlement was located in the fields surrounding the present Kwiha Kirkos church built in the 1980's. It also suggests that the so-called Aksumite or post-Aksumite columns, known since Conti Rossini's first description, are not any more in their original location.

In light of such evidence, we propose to dig archaeological test pits in various locations of the site for a better understanding of this old settlement and the search for datation elements. Provided that the ARCCH gives its agreement, these first excavations could begin in December 2018.

KWIHA: THE STELAE FROM BILET

The present church of Kwiha Kerkos preserves a basalt stone bearing an Arabic inscription. Wolbert Smidt first witnessed the stone inside the church in 2000 and published its Arabic inscription according to a drawing he made on site12 .

According to W. Smidt, the inscription was believed to be written in Hebrew by the inhabitants, hence witnessing their "descent from ancient Israel", the reason why it was kept inside the church mäqdas. Local opinion about the stone has changed since Smidt's first observations: its Arabic script has been identified and its location changed inside the church. Moreover, according to oral tradition, a nun used the stone for cooking and the fire broke it: indeed, as we observed, part of the Arabic inscription has been lost. With the help of Ato Neguse Hagos, Ato Goitom Besrat, Ato Ephrem Kidane, Ato Kibrom Belay, and of inhabitants of the area, we found in four days of survey 15 stelae and 6 fragments of stelae bearing Arabic inscriptions. All these basalt stones were used to delineate paths and plots of land, with the exception of one fragment in limestone found on the ground (B02).

According to Ato Goitom Besrat's instructions, these 21 archaeological artefacts have been stored in Kwiha Kirkos church, waiting for their preservation by the Cultural and Tourism Office of Mekelle. On 12 March 2018, we informed Ato Gidey Gebre Egziabher, archaeologist at the Cultural and Tourism Office of Mekelle, of the discovery of the stelae and of their storage in Kwiha Kirkos church. Each stone received a number, paint on the side, for a safer storage and a better preservation. The list of the 21 stones is the following:

STELAE FROM BILET AT THE NATIONAL MUSEUM OF ETHIOPIA, ADDIS ABABA

The National Museum of Ethiopia (Addis Ababa) preserves within its historical collections stelae in basalt bearing Arabic inscriptions. On 15 March 2018, we get the authorization to study the stelae and take pictures of them. Historical collections were opened to us with the precious help of Ato Salomon Kebede.

RESEARCH PERSPECTIVES

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  On 8 and 9 March 2018, the fieldwork was dedicated to the survey of the area of Igre Hariba, a village located East of Kwiha (N 13°30.895'; W 39°30.615'; see Map 1:50.000 ETH 4, 1339 D1, 1 EMA 1997 "Kwiha").Megdelawit Maryam Church is supposed to be, according to the inhabitants of Igre Hariba, the oldest church of the area. Located on the North side of the main hill, the church is surrounded to the South by a graveyard. Ruins are present on the top of the hill: they might be the remains of the settlement of local leaders, like Ito Debbib, Walda Sellāśē's brother, who welcomed Nathaniel Pearce in the beginning of the 19 th century.The survey of Igre Hariba revealed very few evidence. A basalt stone adorned with a cross and the inscription "1801 EC" is present on a path leading to the village on the North side. It seems of recent production. GPS: 13° 28 mn 30 s. North / 39° 33 mn 60 s. East, altitude 2330 m.Two water reservoirs built in stones (out of three mentioned by the inhabitants) have been also observed: 1. the first on the East side of the hill of Megdelawit Maryam Church. GPS: 13° 28 mn 28 s. North / 39° 33 mn 58 s. East 2. the second on the West side of the same hill. GPS: 13° 28 mn 27 s. North / 39° 34 mn 18 s. East, altitude 2368 m. According to Igre Hariba inhabitants, both of them have been built during the reign of Haile Selassié(1930-1974): their modern datation is confirmed by material observations.

On 10

 10 March 2018, we introduced ourselves to the head of the kebele where Kwiha Kirkos and the field called "Bilet" are located. Ato Goitom Besrat stayed with us on behalf of the local administration during the whole time of the survey around Kwhiha Kirkos and Bilet: he was of great help and support. The church of Kwiha Kirkos (or Kwiha Tchärqos) was built in 1985-86 on the top of a hill at the Northern end of the town, on the Western side of the road: surrounding lands then became Church lands. Ruins are known on this location since their first description by Carlo Conti Rossini in 1937: about 30 broken columns and carved stones have been observed, along with occasional findings of pottery 9 .

  Basalt stone kept in Kwiha Kerkos church: H. 31 x W. 20 x Th. 20 cm. Only the first seven lines of the inscription are partly preserved.Ato Goitom Besrat introduced us to Ato Neguse Hagos, a farmer living in the vicinity of the Kwiha Kirkos church. Ato Neguse Hagos confirmed the location of the place name "Bilet", on the Eastern side of the road. He also informed us that a stone bearing an Arabic inscription was currently lying in the fields of Bilet, used with other stones to delineate paths and plots of land.Sample of stelae used in Bilet to delineate paths and plots of land A first stone was found on 9 March according to Ato Neguse Hagos' indications. GPS: 13° 29 mn. 17 s. North / 39° 32 mn. 56 s. East. Altitude: 2200 m.

  11 stelae belong to the corpus of Bilet: they have been brought to the Museum by French archaeologists in 1962 and their inscriptions published by M. Schneider in 1967 13 . The list (with the current reference number) is the following: RO -004 : H. 29-39 x W. 36 x Th. 12 cm. RO -006 : H. 40 x W. 29 x Th. 13-18 cm. RO -008 : H. 57 x W. 30-60 x Th. 20 cm. RO -0010 : H. 32 x W. 23 x Th. 17 cm. 13 M. SCHNEIDER, "Stèles funéraires arabes de Quiha", Annales d'Ethiopie 7 (1967), pp. 107-122

  +251 1r r54 00 44 Tel.No. +251 11 154 00 54 4.hh I Fax / +251 11 154 00 33 lot"l.+ I P.o.Boxl 13247

  Presence of Arabic Inscriptions in Eastern Tigray was first witnessed by the English traveller Nathaniel Pearce in the beginning of the 19 th century around Igre Hariba 1 (also Egär Hariba, Wegär Hariba, Wäger Hariba, Eger Hariba, Uogher Hariba, Wober Hariba -እገር፡ሓሪባ). • 4 by Carlo Conti Rossini in 1937. Their present location is unknown 3 . • 14 by French archaeologists in 1962 and 1966 (among which 11 are preserved in the National Museum of Ethiopia, Addis Ababa: see bellow) • 2 by Prof. Tekle Hagos (Addis Ababa University) in 1993. They are currently preserved in the museum of the Tourism and Culture Office of Tigray, Mekelle. • 1 by Dr. Wolbert Smidt in 2000 (preserved in the church of Kwiha Kirkos) C. PANSERA, "Quattro stele musulmane presso Uogher Haribà nell'Enderta (fine sec. IV Egira)", Studi etiopici racolti da C. Conti Rossini, Roma, Istituto per l'Oriente, 1945 pp. 3-7.

	The Arabic inscriptions of 15 stelae (out of 21) have been published and translated so far:
	• 2 by Costantino Pansera in 1945 4 , publication revised by Madeleine Schneider in
	2009 5
	• 12 by Madeleine Schneider in 1967 6
	• 1 by Wolbert Smidt in 2004 7
	1 N. PEARCE, The life and adventures of Nathaniel Pearce, written by himself, during a residence in Abyssinia,
	from 1810 to 1819, J. J. Halls ed., London, 1938, vol. 1, pp. 121-124.
	2 Carlo CONTI ROSSINI, "Necropoli musulmana ed antica chiesa cristiana presso Uogri Hariba nell'Enderta",
	Rivista di studi orientali 17 (1937-1939), pp. 399-408
	3 C. CONTI ROSSINI, ibid.

Between Pearce's journey and our survey in March 2018, 22 stelae bearing Arabic inscriptions have been witnessed around Igre Hariba and Kwiha: • 1 by the Italians in 1935-1936, which, according to Carlo Conti Rossini, was brought to Brescia (Italy) 2 . Its present location is unknown. 4 5 M. SCHNEIDER, "Des Yamāmī dans l'Enderta (Tigré)", Le Muséon 122 1-2 (2009), pp. 131-148. 6 M. SCHNEIDER, "Stèles funéraires arabes de Quiha", Annales d'Ethiopie 7 (1967), pp. 107-122 7 W. G. C. SMIDT, "Eine arabische Inschrift in Kwiḥa, Tigray", in Verena BÖLL, Denis NOSNITSIN, Thomas RAVE, Wolbert SMIDT, Evgenia SOKOLINSKAIA (eds.), Studia Aethiopica in Honour of Siegbert Uhlig on the Occasion of His 65th Birthday, Wiesbaden, Harrassowitz, 2004, pp. 259-268.

regard with these promising results, we will submit in 2018 to the ARCCH a new application to conduct excavations in the area of Igre Hariba/Kwiha.

  fieldwork of March 2018 was fulfilled under the supervision of the ARCCH (Addis Ababa) and of the Tourism and Culture Office of Tigray (Mekelle), with the academic support of the Institute of Ethiopian Studies and of the French Centre for Ethiopian Studies (Addis Ababa), and with the academic and financial support of Aix-Marseille University (France). It opens promising avenues of research.1. New archaeological evidence found around the present church of Kwiha Kirkos, in addition to the broken columns and carved stones known since the 1930's, suggests that a permanent (urban?) settlement developed there during the Middle Ages in link with the old trade route crossing the area from North to South.2. New epigraphic evidence found in reuse in the fields of Bilet, on the Eastern side of the present road exiting Kwiha from the North, increases the number of Arabic inscriptions coming from the area to 45 items. The corpus of Arabic inscriptions of Bilet is now the larger one known in Ethiopia.3. The identification of the exact location of the Muslim cemetery of Bilet opens for the first time the opportunity to study Muslim burials from the Middle Ages in Ethiopia. -4,ut,l,.t^l ô9rT.'1 urg hfi r\. [1l1oo\l1i fl Dr.Julien Loiseau îoLo,,l,-.t.î,T.î ÿ-Cÿ-C 0.$7 nhÀrl. h.lô ü(,q fi,,,l-rl..l0i,. ùÇt h.h.h h^lcl. B-15/2019 n:béh floo) hcl|ÿ,h"ï. fhrt4 (flci}Ê) r,1.T. d^l!t:tl À.ÿl:

	0h+rft.P Lg,/-qq, &,toht-il.?t, L'r.fi\h 0qûÀç +l;nr "7.Ln*c g+Cn .I'Ç+Ç Tn,+ q,1àÀ/?? Federal Democratic Republic of Ethiopia	+fr: Ref.No +7 Date '!	lt-. 1.,, ,	,*l	TLÜ
	Ministry of Culture & Tourism
	Authoriÿ of Research & Conservation of
	Cultural Heritage
	nq'Iàf' {l'l/^t, hÀ43 o,,'7oltu4'
	riliUri\ri .i:ô'rly" fi.t';
	oo21î 7çt
	'l-,ar?-1. Dr. JLrlien Loiseau 2" Romair, Mensan In Prof. Julien Loiseau, Aix-Marseille University Prof. Bertrand Hirsch, Sorbonne University 3. Dr. Beltrand Hirsch
	Dr. Amélie Chekroun, Aix-Marseille University 4" Dr. Anrelie Chekroun ô.ÿT. fn.$I. irQ4,T. ,#g)-hûllC 7rÇ ,P1T h4"Int.yu ,l2g-(: t\l;tf'l ùCii.e (luo1f\. hÀ^. .t-,In. ?nC4.'1 q^tu.! ot ÿ:(t hùà4,1p:l h,T.C.Àf .e.._r$ h.I&fÿ.C,,14,11t hÿal34't fl.Jl'r. Ô4hÇorî;frn. 4--tqd.4-hoL"C è?'c4-hÀrl. rü?"qDlrnüt. tll100. Ç h'| &^h
	^4.
	ANNEXES
	1. Official letter and authorization by the ARCCH Ref. n o 08/6h-8-2/020 (in Amharic)
	2. Official letter and authorization by the ARCCH Ref. n os 08/R2-8-2/019 (in English)

t Jtg& ÿT,cn r.1,T.f ÿ-cy'c l.ÿ ÿ: l1(1fl1'1' rtortof,! 4z,t,c 1412006 (l,,,l.llltt. tto(1/:1. hàl.4,l,oy "lÿo'l^l .l'R.Ct l.ÿÿ; ,l'rlq'{.;}À:: ['T1'T. 0..e."r. ]rQ4;1.:-

  : ameiie " CF{EKROUN@univ-amu. fr France Subject: Field Research Permit 2018 Field Season Dear Dr. Julien Loiseau The Cultural Heritage Directorate of the Authority for Research and Conservation of Cultural Heritage has received your application to conduct field research on Archaeological survey Around Igre Hariba, Tigray regional state, Ethiopia. lhe Directorate has caret-u1lr-evaluated \,oLlr perrrit application and approved your reqllest ro concir.rct field work fi-om 8tl'to i5tl'of March 2018.V/e kindly inform you that the newly approved Cultural Heritage Research Permit Directive no"1412006 will be effective as of last field season (2009 E.C.). Vy'e, therefore, would like ro inform you that Ardraeological survey around Igre Hariba permit is active for two consecutive years with annual renewal.

		+fc Ref.No rt ls -ü -2lt tç
	Federal Democratic Republic of Ethiopia Ministry of Culture & Tourism	4,'T Date '2 3 FrR 2org
	Authoriÿ of Research & Conservation of
	Cultural Heritage
	Juiien Loiseau
	Aix-Marseilie University
	We remain cooperative for the	project planned in2A18 field season.

Email

ROSSINI, "Necropoli musulmana ed antica chiesa cristiana", loc. cit.

ROSSINI, "Necropoli musulmana ed antica chiesa cristiana", loc. cit.

W. G. C. SMIDT, "Eine arabische Inschrift in Kwiḥa, Tigray", loc. cit. For a re-reading of its inscription, see also Frédéric BAUDEN, "Inscriptions arabes d'Ethiopie", Annales Islamologiques 45 (2011), pp. 285-306.


Yangang Ren

Max Mcgillen

Ibrahim Ouchen

Veronique Daële

Abdelwahid Mellouki
email: mellouki@cnrs-orleans.fr

Kinetic and product studies of the reactions of NO 3 with a series of unsaturated organic compounds

Keywords: rate coefficient, NO 3 radical, products, terpene, OVOC

published or not. The documents may come

Kinetic and product studies of the reactions of NO3 with a series of unsaturated organic compounds Yangang Ren, Max Mcgillen, Ibrahim Ouchen, Véronique Daële, Abdelwahid

Introduction

Although the hydroxyl radical (OH) is recognized as the dominant oxidant in the daytime, the NO 3 radical (NO 3 ) is known to be the main oxidant of VOCs during the night [START_REF] Atkinson | Kinetics and Mechanisms of the Gas-Phase Reactions of the NO 3 Radical with Organic Compounds[END_REF][START_REF] Wayne | The nitrate radical -Physics, Chemistry, and the Atmosphere[END_REF]. NO 3 is produced in the atmosphere via the reaction between O 3 and NO 2 , which can originate from combustion, biomass burning, lightning and bacterial sources [START_REF] Brown | Aircraft observations of daytime NO 3 and N 2 O 5 and their implications for tropospheric chemistry[END_REF][START_REF] Aldener | Reactivity and loss mechanisms of NO 3 and N 2 O 5 in a polluted marine environment: Results from in situ measurements during New England Air Quality Study[END_REF][START_REF] Brown | Nighttime radical observations and chemistry[END_REF]. During the night, NO 3 and NO 2 exist in thermal equilibrium with N 2 O 5 :

NO 2 + O 3 → NO 3 (R1) NO 3 + NO 2 + M N 2 O 5 + M (R2)
Oxidation by NO 3 is normally only significant during nighttime due to its rapid photolysis with a lifetime of ~5 s and the lifetime is significantly shorter in the presence of NO (in photochemical steady state with NO 2 ) during daytime (Finlayson-Pitts and Pitts 2000), however, it has been shown that NO 3 can be an important daytime oxidant within the canopy in forested environments [START_REF] Liebmann | Alkyl nitrates in the boreal forest: formation via the NO 3 -, OH-and O 3 -induced oxidation of biogenic volatile organic compounds and ambient lifetimes[END_REF]. Field measurements have reported NO 3 concentrations ranging from lower than the detection limit (10 pptv) to 800 pptv and N 2 O 5 concentrations could be as high as 10.9 ppbv dependent on local conditions [START_REF] Brown | Nighttime radical observations and chemistry[END_REF].

NO 3 is a major oxidant for a variety of VOCs and sulfur compounds, and is particularly reactive towards unsaturated biogenic VOCs (BVOC), including both isoprene

(2-methyl-1,3-butadiene) and the monoterpenes [START_REF] Wayne | The nitrate radical -Physics, Chemistry, and the Atmosphere[END_REF]. Isoprene is the most abundant non-methane hydrocarbon in the atmosphere with annual global emissions of 500

Tg [START_REF] Guenther | Estimates of global terrestrial isoprene emissions using MEGAN (Model of Emissions of Gases and Aerosols from Nature)[END_REF]). Reactions of NO 3 with isoprene and monoterpenes could produce both nitrated compounds and secondary organic aerosol (SOA) in variable yields, some model studies have suggested that these reactions may be responsible for a large fraction of regional and global SOA [START_REF] Griffin | Organic aerosol formation from the oxidation of biogenic hydrocarbons[END_REF][START_REF] Ng | Secondary organic aerosol (SOA) formation from reaction of isoprene with nitrate radicals (NO 3 )[END_REF][START_REF] Perring | A product study of the isoprene+NO 3 reaction[END_REF][START_REF] Rollins | Isoprene oxidation by nitrate radical: alkyl nitrate and secondary organic aerosol yields[END_REF]).

However, the nitrated compounds can be long-lived in the atmosphere acting as temporary reservoirs of odd nitrogen, which can then be transported downwind to remote regions where their decomposition, and subsequent NOx release, will influence the regional ozone production [START_REF] Atlas | Evidence for ≥C3 alkyl nitrates in rural and remote atmospheres[END_REF][START_REF] Buhr | Contribution of organic nitrates to the total reactive nitrogen budget at a rural eastern U.S. site[END_REF][START_REF] Aschmann | Products of the OH radical-initiated reactions of 2-propyl nitrate, 3-methyl-2-butyl nitrate and 3-methyl-2-pentyl nitrate[END_REF][START_REF] Suarez-Bertoa | Atmospheric Fate of a Series of Carbonyl Nitrates: Photolysis Frequencies and OH-Oxidation Rate Constants[END_REF].

To improve the representation of NO 3 oxidation of VOCs in tropospheric chemistry models, more kinetic studies of these reactions are needed. Especially since there are quite considerable discrepancies in measurements of NO 3 rate coefficients carried out in different laboratories. For example, even the reaction of NO 3 with isoprene, which has been studied extensively, remains quite uncertain, since reported rate constants span a range of a factor of 2 as shown in IUPAC [START_REF] Atkinson | Evaluated kinetic and photochemical data for atmospheric chemistry: Volume II – gas phase reactions of organic species[END_REF][START_REF] Ammann | Evaluated kinetic and photochemical data for atmospheric chemistry[END_REF]. In this work, gas-phase rate coefficient measurements were made for the following reactions:

Isoprene + NO 3 → products (R3) Methacrolein (MACR) +NO 3 → products (R4) Methyl vinyl ketone (MVK) + NO 3 → products (R5) Crotonaldehyde (CA) + NO 3 → products (R6) 2-carene + NO 3 → products (R7) 3-carene +NO 3 → products (R8)
In addition, the experimental rate coefficients obtained for the different VOCs are discussed and compared with regard to their chemical structures. Besides that, some exploratory work is conducted regarding the products formed from the reaction of 2-carene and 3-carene with NO 3 , using PTR-TOF-MS. To the best of our knowledge, this work provides the first mechanistic study of the reaction of 2-carene with the NO 3 radical.

Experimental methods

7300 L simulation chamber

Experiments were carried out in the 7300 L CNRS-ICARE indoor simulation chamber at ambient temperature (298±2 K) and pressure (760 Torr ± 5 torr). This chamber and the associated technique have been described in detail in a previous study [START_REF] Ren | Rate coefficients for the reactions of OH radical and ozone with a series of unsaturated esters[END_REF]) and is thus described only briefly here. The chamber was made of Teflon foil and surrounded by opaque protection panels. Two Teflon fans were utilized to promote rapid mixing of the reactants. Dry and purified air (RH<1 %) was used as a bath gas in experiments and also to clean the chamber after each experiment. Known volumes of studied hydrocarbons were injected into a Pyrex impinger and driven into the chamber with a stream of purified air.

Concentrations were calculated using the ideal gas law, based on the volume and density of the sample that was injected and the volume of the chamber. In order to compensate for sampling flows and leaks and to avoid any ingress of outside air, 5 L/min of purified air was added continuously during the experiments to maintain a slight overpressure. SF 6 was used to determine the dilution rate in the chamber, which ranged from the (7-8)×10 -6 s -1 .

A high-resolution proton transfer reaction time-of-flight mass spectrometer (IoniconAnalytik, PTR-TOF-MS 8000) with a hydronium ion (H 3 O + ) ion source was employed to monitor the loss of the organic compounds in the chamber. The mass resolution of this instrument is >5000M/δM. The pressure in the PTR-TOF-MS drift tube was maintained at 2.1 mbar and drift voltage of 400 V was used to obtain an electric field, E/N, of 98 Td. The sampling flow from the chamber to the PTR-TOF-MS was approximately 150 mL min -1 through a 1 m long 1/8 inch OD Teflon tube, heated to 60 °C. The mass spectral data were analyzed by PTR-TOF Data Analyzer software [START_REF] Müller | A new software tool for the analysis of high resolution PTR-TOF mass spectra[END_REF]. The detection sensitivities for the VOCs were derived from the slopes of the calibrations plots of the signal (in normalized counts per second, ncps) versus their corresponding mixing ratios (ppbv) as shown in Table 1. An in situ Fourier transform infrared spectrometry (FTIR, Nicolet 5700) coupled to a White-type multipass cell (143m optical path length) was also equipped to the simulation chamber, mainly for SF 6 detection. NO 3 radicals were generated from the thermal decomposition of N 2 O 5 .

Concentration ranges of the various chemicals of importance to these experiments are:

[alkene]= (0.8 -4.0) × 10 12 molecule cm -3 , [N 2 O 5 ] = (0.1 -1.2) × 10 11 molecule cm -3 , [NO 2 ]=(2 -5) × 10 11 molecule cm -3 , [NO 3 ] = (0.5 -4.9) × 10 10 molecule cm -3 . The typical experimental procedure consisted of (1) prior to injection of N 2 O 5 , the SF 6

Relative rate kinetic studies

and VOCs were introduced into the chamber and kept for one hour to quantify k d . The dilution rate was followed by monitoring SF 6 throughout the experiment period using the long-path Fourier transform infrared spectrometer equipped to the chamber (2) sufficient N 2 O 5 was continually introduced to the chamber from a stream of purified air, such that a measureable consumption of VOCs occurred on a reasonable timescale (30 mins to 1 hour).

The decays of both the studied VOCs and reference compounds were continuously monitored using PTR-TOF-MS.

Assuming that the VOCs and reference compounds were lost only by reaction with NO 3 and non-reactive loss, it can be shown that:

ln -k d *t = * (ln -k d *t) Eq.1
Where

Chemicals

The sources of chemicals used and their purities are: isoprene (99%), methacrolein (95%), methyl vinyl ketone (99%), crotonaldehyde (≥ 99.5%), 2-carene (97%), 3-carene (≥90%)

from Sigma-Aldrich. All the liquid compounds in this study were further purified by repeating freeze, pump, thaw cycles and fractional distillation before use.

Results and discussion

Rate constants of NO 3 reaction with studied VOCs

The rate coefficients for isoprene, methacrolein, methyl vinyl ketone, crotonaldehyde, 2-carene and 3-carene reaction with NO 3 radical were obtained in the 7300 L ICARE simulation chamber using references including cyclohexene, 2-methyl propanal, methyl methacrylate, methyl acrylate, 1,3-cyclohexadiene and α-cedrene. The concentration of VOCs ranged from 0.8 to 4.1× 10 12 molecule cm -3 as shown in (5.90±1.77)×10 -13 [START_REF] Calvert | The Mechanisms of Atmospheric Oxidation of the Alkenes[END_REF]. Unit in cm 3 molecule -1 s -1 . Eq.2

As shown in Table 2, good agreement between the rate coefficients using different reference compounds for methacrolein, crotonaldehyde, 2-carene and 3-carene suggests that individual experiments and the reference rate coefficients used demonstrate a high level of consistency.

The determined rate constants of studied VOCs were obtained from the weighted average of the k from each individual reference [START_REF] Ren | Rate coefficients for the reactions of OH radical and ozone with a series of unsaturated esters[END_REF], shown as k average in Table 2:

k average =(w 1 k 1 +w 2 k 2 + …)/( w 1 +w 2 +…)
, where w 1 =1/σ 1 , etc.

Eq.3

The error, σ average , was given by:

σ av =2*(1/ 1 +1/σ 2 + …) -0.5
Eq.4

Comparison with the literature

Table 3 summarizes the rate constants measured by this work with data from the literature for the reactions of the NO 3 radical with isoprene, methacrolein, methyl vinyl ketone, crotonaldehyde, 2-carene and 3-carene. As shown in Table 3, the kinetics of the isoprene reaction with NO 3 radical has been studied extensively using both absolute and relative methods. We recalculate the values for the studies of [START_REF] Atkinson | Kinetics of the gas-phase reactions of nitrate radicals with a series of dialkenes, cycloalkenes, and monoterpenes at 295 .+-. 1[END_REF], [START_REF] Barnes | Kinetics and products of the reactions of nitrate radical with monoalkenes, dialkenes, and monoterpenes[END_REF] and [START_REF] Stabel | Rate coefficients for the gas-phase reaction of isoprene with NO 3 and NO 2[END_REF] using more recent recommended reference values (trans-2-butene and 2-buten-1-ol) from IUPAC [START_REF] Atkinson | Evaluated kinetic and photochemical data for atmospheric chemistry: Volume II – gas phase reactions of organic species[END_REF]) and [START_REF] Zhao | Kinetics Studies of the Gas-Phase Reactions of NO 3 Radicals with Series of 1-Alkenes, Dienes, Cycloalkenes, Alkenols, and Alkenals[END_REF].

Although the rate coefficient obtained in this work is in good agreement with most of previous studies, a larger difference is observed compared with the work of [START_REF] Benter | Absolute rate coefficients for the reaction of NO3 radicals with simple dienes[END_REF], [START_REF] Barnes | Kinetics and products of the reactions of nitrate radical with monoalkenes, dialkenes, and monoterpenes[END_REF] and [START_REF] Ellermann | Absolute rate constants for the reaction of NO 3 radicals with a series of dienes at 295 K[END_REF]. With the exception of [START_REF] Barnes | Kinetics and products of the reactions of nitrate radical with monoalkenes, dialkenes, and monoterpenes[END_REF], both [START_REF] Benter | Absolute rate coefficients for the reaction of NO3 radicals with simple dienes[END_REF] and [START_REF] Ellermann | Absolute rate constants for the reaction of NO 3 radicals with a series of dienes at 295 K[END_REF] applied absolute method using DF-MS (discharge flow system and mass spectrometer and PR-A (pulse radiolysis combined with kinetic spectroscopy in the visible). As shown in Table 3, higher values were also observed for crotonaldehyde for two absolute determinations [START_REF] Cabañas | Night-time Atmospheric Loss Process for Unsaturated Aldehydes: Reaction with NO 3 Radicals[END_REF][START_REF] Salgado | Night-Time Atmospheric Fate of Acrolein and Crotonaldehyde[END_REF]) compared with the relative measurements [ [START_REF] Atkinson | Kinetics of the gas-phase reactions of NO 3 radicals with a series of alkynes, haloalkenes, and α,β-unsaturated aldehydes[END_REF][START_REF] Ullerstam | Reactions of acrolein, crotonaldehyde and pivalaldehyde with Cl atoms: structure-activity relationship and comparison with OH and NO 3 reactions[END_REF]) and this work]. As mentioned by [START_REF] Zhao | Kinetics Studies of the Gas-Phase Reactions of NO 3 Radicals with Series of 1-Alkenes, Dienes, Cycloalkenes, Alkenols, and Alkenals[END_REF][START_REF] Canosa-Mas | Reactivity of the nitrate radical towards alkynes and some other molecules[END_REF] and [START_REF] Wille | A discharge flow mass-spectrometric study of the reaction between the NO 3 radical and isoprene[END_REF], additional removal of VOCs may occur due to its side reactions initiated by F atoms / OH radicals or additional loss of NO 3 may result from its reaction with NO 2 , alkyl and peroxy radicals in these experiments. As a result, we recommend a weighted rate constant of isoprene + NO 3 of (6.6±0.8)×10 -13 cm 3 molecule -1 s -1 based on all the studies except [START_REF] Benter | Absolute rate coefficients for the reaction of NO3 radicals with simple dienes[END_REF], [START_REF] Barnes | Kinetics and products of the reactions of nitrate radical with monoalkenes, dialkenes, and monoterpenes[END_REF] and [START_REF] Ellermann | Absolute rate constants for the reaction of NO 3 radicals with a series of dienes at 295 K[END_REF], which is in good agreement with IUPAC [START_REF] Atkinson | Evaluated kinetic and photochemical data for atmospheric chemistry: Volume II – gas phase reactions of organic species[END_REF] value as (6.5±1.3)×10 -13 cm 3 molecule -1 s -1 at 298 K. The recommended value of crotonaldehyde + NO 3 is (5.7±1.2)×10 -15 cm 3 molecule -1 s -1 , only taking into account the relative rate measurements.

The rate constant of methacrolein + NO 3 has been measured mainly by the relative method [START_REF] Kwok | Product formation from the reaction of the NO 3 radical with isoprene and rate constants for the reactions of methacrolein and methyl vinyl ketone with the NO 3 radical[END_REF][START_REF] Chew | Kinetics of the gas-phase reactions of NO 3 radicals with a series of alcohols, glycol ethers, ethers and chloroalkenes[END_REF][START_REF] Canosa-Mas | A kinetic study of the reactions of NO 3 with methyl vinyl ketone, methacrolein, acrolein, methyl acrylate and methyl methacrylate[END_REF], only one study from [START_REF] Rudich | Rate Coefficients for Reactions of NO 3 with a Few Olefins and Oxygenated Olefins[END_REF] applied an absolute method and observed an upper limit of <8×10 -15 cm 3 molecule -1 s -1 . The value of this work agrees well with the studies using the relative method and hence we recommend a weighted average of (3.3±0.9)×10 -15 cm 3 molecule -1 s -1 , which is in good agreement with IUPAC value (3.4±0.6)×10 -15 cm 3 molecule -1 s -1 at 298 K. The rate constant of methyl vinyl ketone reaction with NO 3 radical in this work is in the range of upper limit value of [START_REF] Rudich | Rate Coefficients for Reactions of NO 3 with a Few Olefins and Oxygenated Olefins[END_REF], but lower than that of [START_REF] Canosa-Mas | A kinetic study of the reactions of NO 3 with methyl vinyl ketone, methacrolein, acrolein, methyl acrylate and methyl methacrylate[END_REF] and [START_REF] Kwok | Product formation from the reaction of the NO 3 radical with isoprene and rate constants for the reactions of methacrolein and methyl vinyl ketone with the NO 3 radical[END_REF] who used relative method. Hence we recommend a preferred value as (1.3±0.6)×10 -16 cm 3 molecule -1 s -1 obtained in this work.

As shown in Table 3, rate constants of the NO 3 reactions with both 2-carene and 3-carene have been measured using the relative method with 2-methyl-2-butene [START_REF] Atkinson | Kinetics of the gas-phase reactions of nitrate radicals with a series of dialkenes, cycloalkenes, and monoterpenes at 295 .+-. 1[END_REF][START_REF] Barnes | Kinetics and products of the reactions of nitrate radical with monoalkenes, dialkenes, and monoterpenes[END_REF][START_REF] Corchnoy | Kinetics of the gas-phase reactions of hydroxyl and nitrogen oxide (NO 3 ) radicals with 2-carene, 1,8-cineole, p-cymene, and terpinolene[END_REF]) and 2,3-dimethyl-2-butene [START_REF] Corchnoy | Kinetics of the gas-phase reactions of hydroxyl and nitrogen oxide (NO 3 ) radicals with 2-carene, 1,8-cineole, p-cymene, and terpinolene[END_REF]) as reference compounds. For the reaction of NO 3 + 3-carene, [START_REF] Barnes | Kinetics and products of the reactions of nitrate radical with monoalkenes, dialkenes, and monoterpenes[END_REF] measured a lower value than [START_REF] Atkinson | Kinetics of the gas-phase reactions of nitrate radicals with a series of dialkenes, cycloalkenes, and monoterpenes at 295 .+-. 1[END_REF], whereas a higher value was obtained in this work. Hence we recommended the weighted average as (8.7±0.5)×10 -12 cm 3 molecule -1 s -1 by taking into account of all three measurements. For the reaction of NO 3 + 2-carene, however [START_REF] Martínez | Absolute Rate Coefficients for the Gas-Phase Reactions of NO 3 Radical with a Series of Monoterpenes at T = 298 to 433 K[END_REF] measured the rate constant that was ~30% lower than other studies, we recommend the weighted average as (1.8±0.6)×10 -11 cm 3 molecule -1 s -1 by taking into account all the measurements of [START_REF] Corchnoy | Kinetics of the gas-phase reactions of hydroxyl and nitrogen oxide (NO 3 ) radicals with 2-carene, 1,8-cineole, p-cymene, and terpinolene[END_REF], [START_REF] Martínez | Absolute Rate Coefficients for the Gas-Phase Reactions of NO 3 Radical with a Series of Monoterpenes at T = 298 to 433 K[END_REF] and this work.

Reactivity of unsaturated VOCs toward to NO 3 radical

The reaction of NO 3 radicals with unsaturated VOCs can proceed by two main pathways:

addition to an unsaturated bond and H-atom abstraction. [START_REF] Canosa-Mas | Reactivity of the nitrate radical towards alkynes and some other molecules[END_REF] radical with isoprene, 2-carene and 3-carene proceed mainly through the addition on the C=C double bond. However, the data points of methacrolein, crotonaldehyde and methyl vinyl ketone fall close to the correlation line for abstraction reaction, even those which contain an olefinic bond. [START_REF] Martínez | Absolute Rate Coefficients for the Gas-Phase Reactions of NO 3 Radical with a Series of Monoterpenes at T = 298 to 433 K[END_REF] investigated the reaction of NO 3 with a series of monoterpenes and concluded that their room temperature rate constant depends on the number and the structure of alkyl substituents around the olefinic bond. In this work, the rate constants of 3-carene and 2-carene reaction with NO 3 radical were determined to be ≈16 and ≈30 times larger than isoprene, respectively. By analogy to their reaction with ozone (as shown in Table 4), the additional steric effects of 3-carene than 2-carene could be the reason for its lower rate constants than 3-carene. As discussed by [START_REF] Chen | Rate coefficients for the reaction of ozone with 2-and 3-carene[END_REF], the reduction in rate coefficient for 3-carene arises from two hypotheses: (1) the conformational constraints of a C6-ring system containing two sp 2 hybridized carbons could decrease its reactivity; (2) the bridgehead carbon is out of the plane of the C6-ring similar to α-pinene and camphene, which may cause additional steric effects. However, a higher donator inductive effect of the alkyl chain around the C=C bond should also be considered for the higher rate constant of 2-carene reaction with NO 3 radical than 3-carene.

Methacrolein and methyl vinyl ketone are the main products from the oxidation of isoprene in the atmosphere, which both contain one C=C and a C=O group. However, the rate coefficients of methacrolein and methyl vinyl ketone with NO 3 are at least one order magnitude lower than the reaction with their non-carbonyl analogues isobutene [(3.4±1.0)×10 -13 cm 3 molecule -1 s -1 , [START_REF] Calvert | The Mechanisms of Atmospheric Oxidation of the Alkenes[END_REF]] and propene [(9.50±5.50)×10 -15 cm 3 molecule -1 s -1 , IUPAC http://iupac.pole-ether.fr/#] with NO 3 , respectively. This indicates a strong electron-withdrawing effect of the C=O group in the α position, which strongly deactivates the NO 3 addition on the C=C bond [START_REF] Kerdouci | An Experimental Study of the Gas-Phase Reactions of NO 3 Radicals with a Series of Unsaturated Aldehydes: trans-2-Hexenal, trans-2-Heptenal, and trans-2-Octenal[END_REF] as well as for other electrophilic addition reactions such as ozonolysis [START_REF] Mcgillen | Structure-activity relationship (SAR) for the prediction of gas-phase ozonolysis rate coefficients: an extension towards heteroatomic unsaturated species[END_REF]. Conversely, the electron-donating inductive effect of the -CH 3 group attached to the C=C bond in methacrolein increases its reactivity relative to that of methyl vinyl ketone [START_REF] Canosa-Mas | A kinetic study of the reactions of NO 3 with methyl vinyl ketone, methacrolein, acrolein, methyl acrylate and methyl methacrylate[END_REF]. As shown in Table 3, the reaction of crotonaldehyde with NO 3 radical (5.7±1.2×10 -15 cm 3 molecule -1 s -1 ) is slightly faster than the reaction of methacrolein with the NO 3 radical (3.3±0.9×10 -15 cm 3 molecule -1 s -1 ). The structure of methacrolein and crotonaldehyde differs in that the -CH 3 in methacrolein is closer to the C=O group than in crotonaldehyde, which may decrease the electron-donating affection of -CH 3 to the C=C double bond.

Products investigation of 2-carene & 3-carene reaction with NO 3 radical and proposed reaction mechanism

The product formation from the reactions of 2-carene and 3-carene with the NO 3 radical were conducted in the 7300 L simulation chamber under ambient temperature and pressure. As shown in Figure 4a and 4c, the 2-carene or 3-carene were introduced into the chamber first, and kept for two hours to inspect the wall-loss/ dilution rate. Then N 2 O 5 was injected continually for 15 min twice during the experiment in order to initiate the reaction. The possible products were detected using PTR-TOF-MS. As shown in Figure 4 Several studies [START_REF] Griffin | Organic aerosol formation from the oxidation of biogenic hydrocarbons[END_REF][START_REF] Hallquist | Aerosol and Product Yields from NO 3 Radical-Initiated Oxidation of Selected Monoterpenes[END_REF][START_REF] Colville | The roles of individual oxidants in secondary organic aerosol formation from Δ3-carene: 1. gas-phase chemical mechanism[END_REF][START_REF] Fry | Secondary Organic Aerosol Formation and Organic Nitrate Yield from NO 3 Oxidation of Biogenic Hydrocarbons[END_REF] with NO 3 , which may relate to their different reactivity toward NO 3 radical as discussed in previous section, however, a difference in sensitivity for these species cannot be ruled out.

Hence, more studies should be conducted to better understand the reaction of NO 3 with 2-carene/3-carene in the atmosphere.

Conclusion and Atmospheric Implications

The first objective of our study was to establish accurate rate coefficients of the reaction of the NO 3 radical with a series of unsaturated VOCs in order to better define their atmospheric lifetimes. Recommended rate coefficients are provided for isoprene, 2-carene, 3-carene, methyl vinyl ketone (MVK), methacrolein (MACR) and crotonaldehyde (CA), as (6.6±0.8)×10 -13 , (1.8±0.6)×10 -11 , (8.7±0.5)×10 -12 , (1.24±1.04)×10 -16 , (3.3±0.9)×10 -15 and

(5.7±1.2)×10 -15 cm 3 molecule -1 s -1 , respectively at 295±2 K and 760 Torr. Hence, these lifetimes were estimated assuming an NO 3 concentration of 5 × 10 8 molecule cm -3 [START_REF] Atkinson | Kinetics and Mechanisms of the Gas-Phase Reactions of the NO 3 Radical with Organic Compounds[END_REF] ( [START_REF] Wingenter | Hydrocarbon and halocarbon measurements as photochemical and dynamical indicators of atmospheric hydroxyl, atomic chlorine, and vertical mixing obtained during Lagrangian flights[END_REF] and the rate constants from the literature [START_REF] Calvert | The Mechanisms of Atmospheric Oxidation of the Alkenes[END_REF][START_REF] Suh | Kinetic Studies of Isoprene Reactions Initiated by Chlorine Atom[END_REF][START_REF] Cabañas | Night-time Atmospheric Loss Process for Unsaturated Aldehydes: Reaction with NO 3 Radicals[END_REF][START_REF] Timerghazin | Kinetics of the gas-phase reaction of atomic chlorine with selected monoterpenes[END_REF][START_REF] Ullerstam | Reactions of acrolein, crotonaldehyde and pivalaldehyde with Cl atoms: structure-activity relationship and comparison with OH and NO 3 reactions[END_REF][START_REF] Sato | Kinetic measurements for the reactions of ozone with crotonaldehyde and its methyl derivatives and calculations of transition-state theory[END_REF][START_REF] Calvert | The Mechanisms of Atmospheric Oxidation of the Oxygenates[END_REF][START_REF] Chen | Rate coefficients for the reaction of ozone with 2-and 3-carene[END_REF][START_REF] Ren | Investigation of the reaction of ozone with isoprene, methacrolein and methyl vinyl ketone using the HELIOS Chamber[END_REF]. The calculated lifetimes are shown in Table 4. It is clear that the calculated tropospheric lifetimes of these VOCs with respect to the NO 3 reaction vary from a few minutes (isoprene, 2-Carene and 3-Carene) to few months (methyl vinyl ketone). The atmospheric lifetimes for studied VOCs due to reaction with OH radicals are roughly a few hours. Lifetimes with respect to ozone are longer, ranging from ∼1.7 hours to months. The reaction of NO 3 may be more important than other oxidants to for the overall loss of isoprene, 2-carene and 3-carene, assuming that the oxidant concentrations employed in these calculations are representative of typical tropospheric conditions. In addition, the reaction of NO 3 will contribute significantly at night when the NOx emissions are high. Based on our measurements, we expect the overall atmospheric lifetimes of the studied VOCs with respect to all these oxidants are few minutes to a few days using the following equation: = + + + .

The reactivity of studied VOCs toward NO 3 was also discussed. The experiments indicate that NO 3 radical reactions with all the studied unsaturated VOCs proceed through addition to the olefinic bond, however, it indicates that the introduction of a carbonyl group into unsaturated VOCs can deactivate the neighboring olefinic bond towards reaction with the NO 3 radical, which is to be expected since the presence of these electron-withdrawing substituents will reduce the electron density in the π orbitals of the alkenes, and will therefore reduce the rate coefficient of these electrophilic addition reactions. The different steric and inductive effects of the alkyl chain around the C=C bond were considered for the different rate constants of 2-carene and 3-carene reaction with NO 3 .

In addition, the product formation from the reaction of 2-carene and 3-carene with NO 3 radical were also investigated. Epoxide, nitrooxy ketone (C 10 H 16 O 4 N) and caronaldehyde (C 10 H 16 O 2 ) were identified for both reaction systems. In the atmosphere, the epoxide could react with OH and NO 3 radicals and the atmospheric fate of caronaldehyde is expected to be reaction with OH and photolysis [START_REF] Hallquist | Atmospheric Fate of Carbonyl Oxidation Products Originating from α-Pinene and Δ3-Carene: Determination of Rate of Reaction with OH and NO 3 Radicals, UV Absorption Cross Sections, and Vapor Pressures[END_REF]. Secondary organic aerosol (SOA) has been observed in the reaction of NO 3 with 3-carene in the simulation chamber with mass-based yields of up to 72 % [START_REF] Griffin | Organic aerosol formation from the oxidation of biogenic hydrocarbons[END_REF][START_REF] Hallquist | Aerosol and Product Yields from NO 3 Radical-Initiated Oxidation of Selected Monoterpenes[END_REF]) and organonitrates comprised 56% of the aerosol mass. Hence, more work on the identification of organonitrates SOA formation from reaction system of 2-carene and 3-carene with NO 3 radical would be desirable.

  Rate coefficients were determined by following the simultaneous decays of the studied VOCs and reference compounds from their reaction with NO 3 and other loss processes (wall loss, → wall loss/ dilution k d k 3-8 and k ref are the rate coefficient for the reactions of NO 3 with the studied VOCs including isoprene, mehacrolein, methyl vinyl ketone, crotonaldehyde, 2-carene, 3-carene and reference compounds (cyclohexene, methyl methacrylate, 2-methyl propanal, methyl acrylate, 1,3-cyclohexadiene, α-cedrene), respectively. k d is the non-reactive loss rate of studied VOCs and reference compounds.

N 2 O

 2 5 was synthesized in a vacuum line by the reaction of NO 2 with excess O 3 through reactions R1 and R2[START_REF] Zhou | Kinetics of the Reactions of NO 3 Radical with Methacrylate Esters[END_REF]). In the first stage of synthesis, NO 2 was produced from the reaction of NO and O 3 in a glass tube. The mixture of NO 2 and O 3 were then flushed into a bulb (1 L) where NO 3 and N 2 O 5 are formed. At the end of the synthesis, N 2 O 5 crystals were collected in a cold trap (190 K). N 2 O 5 crystals were then purified by trap-to-trap distillation under a flow of O 2 /O 3 , which could be kept for several weeks in a cold trap maintained 190 K.

Figure 1

 1 Figure 1 and 2 show straight lines of ln k d *t as a function of ln k d *t with zero

  [START_REF] Wayne | The nitrate radical -Physics, Chemistry, and the Atmosphere[END_REF] have shown that both the addition and H-atom abstraction rate coefficients of OH radical and NO 3 radical reaction with alkynes and aliphatic aldehydes correlate. D'Anna et al. (2001) updated these relationships for addition as: log(k NO3 ) = (3.43±0.24) × log(k OH ) + (22.7±2.5) and we updated the relationship for H-abstraction from alkanes based on the recent data of Zhou et al. (2019) giving: log(k NO3 ) = (1.78±0.57) × log(k OH ) + (4.26±6.59) as shown in Figure 3. The data points of isoprene, 2-carene and 3-carene, as shown in Figure 3, fall close to the correlation line for the addition reaction. This implies that the reaction of NO 3

  in the following equation: τ NO3 = . The atmospheric lifetimes of studied VOCs depending on their reaction with OH, O 3 and Cl were also calculated using the following oxidant concentrations: [OH] = 1 × 10 6 molecule cm -3 (Spivakovsky et al. 2000), [O 3 ] = 7 × 10 11 molecule cm -3 (Monks et al. 2009) and [Cl] = 1 × 10 4 molecule cm -3

Table 2

 2

	. The recommended rate

  , the m/z 153.127, 169.122 and 215.115 according to C 10 H 16 OH + , C 10 H 16 O 2 H + and C 10 H 16 O 4 NH + , respectively, were identified as the products. Figures 4b and 4d demonstrate the linearity of product formation vs the reactant consumption, indicating that the C 10 H 16 OH + , C 10 H 16 O 2 H + are primary products from the 2-carene/3-carene reaction with NO 3 radical.

Acknowledgements:

This work is supported by Labex Voltaire (ANR-10-LABX-100-01), ARD PIVOTS program (supported by the Centre-Val de Loire regional council), and the European Union's Horizon 2020 research and innovation programme through the EUROCHAMP-2020 Infrastructure Activity under grant agreement No. 730997 and the Marie Skłodowska Curie Actions Programme (690958-MARSU-RISE-2015). MRM thanks Le Studium for their support over

~ 0.3%, respectively, for 2-carene and 3-carene reaction with NO 3 radical. In this work, the m/z 153.127 was also detected by PTR-TOF-MS from the reaction system of 2-carene/3-carene with NO 3 and which was attributed to the epoxide, C 10 H 16 O. The epoxide formation from NO 3 addition to alkenes has been reported for several compounds, e.g. α-pinene, 2-butene, 2,3-Dimethyl-2-butene [START_REF] Benter | REMPI-MS and FTIR Study of NO 2 and Oxirane Formation in the Reactions of Unsaturated Hydrocarbons with NO 3 Radicals[END_REF], Wängberg et al. 1997), where formation yields generally range from 3±0.5% to 20±10% under atmospheric conditions. The epoxide formation yields from the reactions of 2-carene and 3-carene with NO 3 were estimated to be ~ 3.6% and ~10.6%, respectively, by assuming same detection sensitivity as 2-carene and 3-carene in PTR-TOF-MS. Some organic nitrates such as nitrooxy-ketone (C 10 H 16 O 4 N) were also detected in PTR-TOF-MS in m/z 215.115.

According to the product formation and a generalized reaction mechanism applicable to monoalkenes, dialkenes, and monoterpenes reactions with NO 3 radical [START_REF] Barnes | Kinetics and products of the reactions of nitrate radical with monoalkenes, dialkenes, and monoterpenes[END_REF]), as shown in Figure 5, we were able to propose a reaction mechanism for 2-carene/3-carene reaction with NO 3 radical as below: the reaction proceeds mainly through NO 3 addition to the C=C bond to form two nitrooxy alkyl radicals, which can either react with O 2 to form an RO 2 radical (nitrooxy-peroxy radical) or lose NO 2 to form an epoxide C 10 H 16 O via reactions R7a/ R8a or R7b/ R8b in Figure 5, respectively. Then the nitrooxy-peroxy radical (RO 2 ) can either react with NO 2 to form a nitrooxy-peroxy nitrate or other RO 2 radicals to form an RO radical (nitrooxy-alkoxy radical), hydroxy-nitrate and nitrooxy-ketone C 10 H 16 O 4 N through reactions R7c/ R8c or R7d/ R8d in Figure 5, respectively. As reported by [START_REF] Hallquist | Aerosol and Product Yields from NO 3 Radical-Initiated Oxidation of Selected Monoterpenes[END_REF], the nitrooxy-peroxy nitrate are quite unstable and can only be detected in the beginning of reaction, which could explain why it was not detected it in this work using low reactant concentrations. As shown in Figure 5, nitrooxy-ketone C 10 H 16 O 4 N could also be formed through the reaction R7e/ R8e, as RO radical (nitrooxy-alkoxy radical) reaction with O 2 .

However, the RO radical (nitrooxy-alkoxy radical) could also form caronaldehyde C 10 H 16 O 2 by losing NO 2 via reactions R7f/ R8f in Figure 5.

To our best knowledge, there are no previous product and mechanism studies for the reaction of 2-carene with NO 3 . It should be noted that, because of the lack of standards, all product formation yields from our work are based on the assumption of the same sensitivity of product peaks as 2-carene and 3-carene in PTR-TOF-MS. However, as shown in Figure 4, the part of this project.


Xie Rui-Xun

Larranaga Melvyn

Mompiou Frédéric

A general analytical method for interface normal determination in TEM

Keywords:

This paper presents a new analytical method to determine interface normals from a series of bright/dark field images taken from arbitrary orientations. This approach, based on a general geometrical model of interface projection, provides a generalized formulation of existing methods. It can treat an excessive number of inputs, i.e. orientation conditions. Given 6 or more sets of inputs, even with considerable experimental errors, we prove that this method is still very likely to yield results with satisfactory trueness. The robustness of the method can thus allow its implementation in problems dealing with a large amount of data. We show that this method can also be applied to determine 1D features or to check the planarity of microstructural features.

Introduction

Transmission electron microscopy (TEM) is a powerful tool to examine a variety of defects in materials, including point defects (zero dimension, 0D), dislocation lines (1D), interfaces (2D) and inclusions (3D). Since TEM images are 2D projections of the 3D space, the geometrical features of non-0D defects, e.g., their shapes, must be reconstructed from their projections on the observation plane. The identification of 3D inclusions, curved lines or curved planes is a cumbersome work [START_REF] Midgley | Electron tomography and holography in materials science[END_REF] [START_REF] Barnard | High-resolution three-dimensional imaging of dislocations[END_REF]. However, many 1D or 2D defects, e.g., interfacial dislocations or faceted interfaces, often have preferred orientations. They can be considered as straight lines or flat planes. Accurately determining their crystallographic orientations is fundamental to unravel the mechanism of associated microstructure evolution [START_REF] Zhang | Faceted interfaces: a key feature to quantitative understanding of transformation morphology[END_REF]. In this context, the problem is restricted to the line direction or the interface plane normal determination, using a technique called trace analysis [START_REF] Hirsch | Electron Microscopy of Thin Crystals[END_REF] [START_REF] Edington | Practical electron microscopy in materials science[END_REF].

The concept and methods of trace analysis using TEM were clearly introduced in the well-known book by Hirsch et al. [START_REF] Hirsch | Electron Microscopy of Thin Crystals[END_REF]. A line feature is always on the plane defined by its projection and the electron beam direction. Therefore it can be determined if observed from two directions. However, the projection of a planar feature usually contains the projection of two traces, the intersections of the interface with two foil surfaces, separated by a certain width.

An interface can be directly determined at its edge-on condition, where the projection width is zero and the interface ORCID(s):

RX Xie et al.: Preprint submitted to Elsevier normal lies in the viewing screen [START_REF] Hirsch | Electron Microscopy of Thin Crystals[END_REF]. Since an edge-on condition often carries non-negligible uncertainty, several modifications were later proposed to improve the accuracy, such as the single edge-on method [START_REF] Edington | Practical electron microscopy in materials science[END_REF] [START_REF] Sandvik | Characteristics of lath martensite: Part i. crystallographic and substructural features[END_REF], the double edgeon method [START_REF] Luo | A tem method for accurate measurement of habit plane (interface): double edge-on trace analysis[END_REF] [START_REF] Qiu | A tem study of the crystallography of austenite precipitates in a duplex stainless steel[END_REF], and the trace & edge-on method [START_REF] Meng | Precise determination of the irrational preferred interface orientation by tem[END_REF]. Despite their good accuracy, these methods are not easy to use, as finding an exact edge-on condition is usually time consuming or even impossible if it is out of the tilting range of the sample holder.

An edge-on condition is not always necessary, if the interface contains an additional sharp line feature, such as a straight dislocation line [START_REF] Sandvik | Characteristics of lath martensite: Part i. crystallographic and substructural features[END_REF][6] [START_REF] Liu | A new method for determining the normals to planar structures and their trace directions in transmission electron-microscopy[END_REF] or an intersection with another plane [START_REF] Li | Determination of surface crystallography of faceted nanoparticles using transmission electron microscopy imaging and diffraction modes[END_REF], using the so-called double-trace method.

The orientations of both trace and line feature in the plane can be measured by the trace analysis method of 1D features.

Then one can obtain the interface normal by making a cross product of the line and the trace direction. However, additional line features are not always present, which limits the application of this method.

When the condition for the double-trace method is not satisfied, the projection width can be used to calculate interface normals at arbitrary orientations. In this approach, the trace direction is assumed to lie on the screen plane at zero tilt. The interface normal can be determined, once the inclination angle between the foil surface and the interface is determined. Traditionally, this was done by measuring the foil thickness and the projection width of the interface at an orientation near to zero tilt [START_REF] Edington | Practical electron microscopy in materials science[END_REF]. Since this method contains considerable uncertainty, Zhang and Kelly [START_REF] Zhang | Determination of habit planes using trace widths in tem[END_REF] made an improvement by tilting the sample along the trace direction. But it is usually difficult to ensure the tilting axis exactly parallel with the trace direction. Qiu an Zhang [START_REF] Qiu | A simple and inclusive method to determine the habit plane in transmission electron microscope based on accurate measurement of foil thickness[END_REF] solved this problem on a single tilt holder by taking the angle between the trace and the tilting axis into consideration. In order to avoid ambiguous solutions, this approach still needs to track the trend of the projection width change during sample tilting. All these methods are based on the assumption that the upper and lower foil surfaces are both perpendicular to the electron beam direction at zero tilt. This may introduce systematic error when the foil has thickness variation or not flat.

Above methods are highly sensitive to experimental errors, since they rely on limited information -usually one or two sets of inputs (beam and projected trace directions, projection width). Particularly, the error could be greatly amplified by the cross product operation in double trace and double edge-on methods [START_REF] Meng | Precise determination of the irrational preferred interface orientation by tem[END_REF]. To improve the reliability, one may use excessive experimental data to calculate several solutions of the interface normal. The scattering of results can be plotted in a pole figure [START_REF] Meng | Precise determination of the irrational preferred interface orientation by tem[END_REF] [START_REF] Qiu | A simple and inclusive method to determine the habit plane in transmission electron microscope based on accurate measurement of foil thickness[END_REF], but the selection of the final result and the estimation of its uncertainty are largely based on the operator's experience.

In order to improve the accuracy of interface normal determination and simplify the TEM operation, we present here a close-form algorithm to optimize the result with multiple sets of inputs from arbitrary orientations. In this method, there is no specific requirements on the beam direction, nor conditions on the interface and the foil.

In Section 2, we will describe the methodology used to compute the interface normal. Section 3 will present experimental details, followed by Section 4 illustrating the applications of the present method. The results and method will finally be discussed in Section 5. The accuracy of the results will be adressed by an error analysis, and the method will be compared with other methods proposed in the literature. A generalization of our approach will be proposedat the end.

Geometrical model of interface projection

The approach presented below is a purely geometrical analysis of the orthogonal projection of planar features on the observation screen, without any requirement on the beam direction (projection direction) or the foil surface. It is based on the measurement of the interface width and apparent trace direction for different beam direction. The symbols used in the derivation and their meaning are listed in Table 1. A flat interface has two traces, the intersections with upper and lower foil surfaces. Firstly, let us assume the two foil surfaces to be parallel planes, resulting in two parallel traces separated by a certain width . This case is 1 If not pointed out, all the vector symbols noted in bold and italics are column vectors. where is the angle between trace and its projection p .

When the foil is bent or has a wedge shape, the assumption of parallel foil surfaces is no longer valid. Because of this, trace and trace are not parallel, in Figure 1 is no longer a parallelogram (Figure 1b). However, if a point on the interface can be tracked, it is still possible to resolve the relationship between and . As shown in Therefore, no matter two foil surfaces are parallel or not, the interface projection width is always given by

= cos cos (1)
Or alternatively using dot product:

= | ⋅ e | √ 1 -e ⋅ 2 (2)
In Eq. ( 2), and e can be measured, while and are unknown.

Vector can be determined from its measured projections p 's at different e 's, using:

e1 × p1 e2 × p2 ... em × p T = (3)
When = 2, Eq. ( 3) is solved by using the cross product = ( 1 × 1 ) × ( 2 × 2 ). When the number of input sets ≥ 3, it is an overdetermined homogeneous linear equation, and can to be solved by the least square method (LSM, see Appendix A).

The determination of allows to compute the non-linear part √ 1e ⋅ 2 in Eq. ( 2), turning Eq. ( 2) into a linear problem. However, the width of interface projection, , is always positive, while ⋅ ei could be negative. Thus sgn( ), the sign of ⋅ ei , is needed to remove the absolute sign in Eq. ( 2). Using the fact that ⋅ = 0, a synthetic formula incorporating the input measurements ( , ei ) can be derived:

⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ T e1 T e2
...

T en T ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ ( ) = ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ sgn(1) 1 √ 1 -e1 ⋅ 2 sgn(2) 2 √ 1 -e2 ⋅ 2 ... sgn( ) √ 1 -em ⋅ 2 0 ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ (4)
This equation can be solved by LSM, but the uncertainty of sgn( ) would result in multiple solutions of . When = 2, Eq. ( 4) is determined and two different solutions are expected. It is impossible to assess the actual one without additional information. When ≥ 3, Eq. ( 4) is overdetermined, and its residual error can be computed (see Appendix A). The uncertainty of sgn( ) will give 2 -1 solutions of . The one with the minimum residual error is the best solution, because an improper sgn( ) would result in significantly large residual error. The contribution to the residual error from each set of input ( , ei , p ) can be quantified by the respective element in the residual vector of Eq. ( 4) (see Appendix A). The element with the maximum absolute value and different sign from the others may indicate an abnormal input with significantly large deviation. One may drop this input and recalculate the result, which can reduce the residual error and improve the input consistency. Thus, with at least 3 different sets of experimental data ( , ei , p ), LSM will give the optimized solutions of , and .

The accuracy of the interface normal, which is the deviation between the determined value and the true value is an important concern. Unfortunately, the true value of is usually unknown, and needs to be estimated. The estimation of the true value is a range with a certain confidence, i.e., a confidence interval, which can represent the accuracy of the result, based on the internal consistency of inputs. In the present work, the 95% confidence interval of and that of can be calculated by the bootstrap method (see Appendix B) with Eq. ( 4).

A C++ implementation of the present algorithm can be found in Supplementary Materials.

Experimental details

The present method was tested in two cases. In both cases, bright field images were recorded at different orientation conditions in an FEI Tecnai G2 20 TEM operated at 200 kV. Observations were performed using a double tilt sample holder with the coordinate conventions as shown in Figure 2: the x-axis is parallel to the holder axis, and the y-axis is perpendicular to the x-axis and the beam direction. In the following, the subscripts H , P and S denote the indices in holder, projection screen, and sample coordinate system, respectively . In the holder coordinate system, the sample is firstly tilted about theaxis by angle , and then tilted about theaxis by angle .

The rotation angle from holder axis to the projection screen axis is about = axis. This accounts for the rotation of the image in the microscope column. Hence, the beam direction e perpendicular to the screen and the direction p in the observation plane can be calculated using right-handed rotation matrices Rx( ), Ry( ), and Rz( ), as shown below: where is the azimuth angle between the projected trace and the -axis on the viewing screen.

In the first experiment, a grain with two interfaces was investigated in a duplex stainless steel Fe-24.9Cr-7.0Ni-3.1Mo (wt%) sample prepared by the same procedure as reported in [START_REF] Du | In-situ tem study of dislocation emission associated with austenite growth[END_REF]. The images were taken in conditions where

= 90 • .
In the second case, a grain boundary with non-parallel traces was observed in an aluminum bicrystal with a misorientation close to a coincident Σ41 12.68

• < 0 0 1 > {5 4 0}.
The sample was first ground to 50 microns using SiC grain disks and then electro-polished to obtain electron transparency. The TEM foil was strained in-situ at c.a. 400 • C, as reported earlier in [? ] (ref missing). Plastic deformation leads to a complex microstructure of dislocations, resulting in significant bending of the wedge foil and possible deviation from the original orientation. Eq. ( 5) and Eq. ( 6) are still valid in this case, using here = 157 • for Eq. ( 6). The first step of the method is to determine . The e 's in holder coordinates were calculated with data in Table 2 and Figure 3, using Eq. ( 5) and Eq. ( 6), while the p 's of IntA and IntB are given by their azimuth angle with respect to the axis of Figures 3a∼g. The lattice coordinates of e 's and p 's were defined in austenite using the Kikuchi patterns shown in Figure 3, as listed in Table 2. Hence was determined by solving Eq. 3 in either coordinate system.

Application examples 4.1. Measuring two interfaces with a double-tilt holder

The second step is to substitute e 's, , and 's, into Eq. 4 to solve . By enumerating all the possibilities of sgn A ( ), and sgn B ( ), and its residual error in both conventions are solved by LSM. Among them, the one with the minimum residual error was chosen as the final result, whose sgn( ) is shown in Table 2. Then, the optimized results

were calculated, as shown by Table 3. The 's in holder coordinates were transformed to lattice coordinates using the transformation matrix determined from Figure 3e2 (see Supplementary materials).

The 's of IntA and IntB in holder coordinates are almost in the plane, indicating that the foil was almost perpendicular to the beam direction at zero tilt. For each interface, the real width ( ) and the normal ( ) calculated from different coordinate systems show a good consistency. This means the holder coordinate system can be used instead of the lattice coordinate system to simplify the process while reserving the accuracy. The correctness of is verified by tilting both interfaces to a near edge-on condition (Figure 3h), where the interface normal can be determined by e × p . Using the data of Figure 3h in Table 2, the normal of IntA is [0.562 0.592 -0.579], and that of IntB is [-0.594 -0.237 -0.766]. These results are inside the confidence range of the 's determined in Table 3. Hence, the present method is able to accurately determine multiple interfaces simultaneously, either in the holder or lattice coordinate system. for verification. The trace of the interface in holder coordinates deviates from the plane, confirming that the foil is bent. Through a similar process as the previous example, the optimized result in holder coordinates is calculated

Measuring a grain boundary with non-parallel traces

Discussion

Accuracy of

The use of the method presented here raises three questions concerning the accuracy of : (i) is the true value of the interface normal inside the bootstrap confidence interval; (ii) what is the main error source on interface normal determination; (iii) what is the optimal number of input datasets for an acceptable accuracy. To answer these questions, statistical analysis was performed using simulated erroneous data.

The simulated datasets were generated in holder coordinates, using a 100-nm-thick virtual sample with two parallel surfaces normal to . As the prerequisite, the true value of the interface normal must be known. Then, the and angles were randomly chosen in ±40 • range, and the true values of and were calculated. The dataset ( , , , )

was randomly varied to simulated the experimental error. The , and angles were randomly varied in ±1 • range, while the projection widths in ±5 nm range. With several erroneous datasets, the interface normal was determined, and the deviation from the true value, which is called trueness for clarity, was calculated.

In the first analysis, the interface normal was randomly chosen and determined with 6 sets of simulated data. This process was repeated 10000 times, and the true value was inside the bootstrap confidence interval for 9369 times. This agrees to the confidence level (95%) of the interval. The true value could be outside the confidence interval when the datasets ( , , , ) have relatively large systematic error, as this problem cannot be figured out from the consistency of the datasets.

In the second analysis, 4949 unit vectors evenly distributed on a hemisphere were used as known interface normals.

For each normal, 6 arbitrary conditions ( , , , ) were used to compute back an erroneous normal, and its trueness.

Because the trueness is dependent on the orientation conditions, it was averaged over 10000 arbitrary condition sets for each known normal.

The results are plotted in a pole figure in Figure 5a. It shows an symmetry due to the angular area covered by the sample tilt, limited by the dashed lines in Figure 5a. The average trueness of interface normals ranges between 0.65 • and 1.35 • , but most of the value is due to the error on as shown in Figure 5b, where the average trueness is computed when varying only . The ring-like shape of trueness in Figure 5a can be understood by inspecting how errors affect the value of cos in Eq. 4. Using ℎ = sin = √ 1 -( ⋅ ) 2 (Figure 1a inset) and Eq. 2, we can derive

cos = ⋅ e = ℎ √ 1 -e ⋅ 2 sin (7)
When the interface normal moves from the center to the edge in the pole figure, statistically, sin =

√ 1 -( ⋅ ) 2
will increase, making the value of cos more sensitive to the error of . In other words, largely inclined planes in the foil are less precisely determined, because they statistically present orientation conditions where the projection width is small. Meanwhile, the increase in makes the direction of less sensitive to the error of cos , as d = -d(cos )∕ sin . The two effects having opposite variations, this leads to a trueness distribution with a maximum value in a ring area inclined about 54 • with the foil normal (Figure 5b). In common cases, the projection of the interested interface is not very wide, which means that its normal is not located at the blue part in Figure 5a. Therefore, the error of mainly comes from the error of .

In the third analysis, the interface normal was fixed to [-0.547, 0.514, 0.661] H , an orientation with the worst trueness in Figure 5a, while the number of input datasets varied from from 3 to 15. The average of trueness were computed, again by randomly varying the condition sets over 10000 configurations. As shown in Figure 5c, the trueness of would reach 1.7 • with 6 sets of inputs. It would reach 1.5 • with 8 sets of inputs, but barely improves afterwards.

Based on the above analysis, 6 to 8 sets of inputs would be the optimal choice.

In practice, it is trivial to reach the data quality (±1 • , ±5nm) in this analysis, and hence to get the results with the corresponding accuracy. Moreover, data quality can even be improved by image measurements at higher magnification, and by using Kikuchi patterns.

Comparison with existing methods

The method proposed here relies on the measurements of the direction and width of interface projections, and the beam directions. It works both in lattice or holder coordinate systems. The double trace and edge-on methods can be considered as special cases of the present method in the lattice coordinate system. The geometry model used here offers a framework to understand the existing projection width methods, which, as shown below, used specific simplifications of the non-linear part √ 1e ⋅ 2 in Eq. 2. The summary of the comparison is listed in Table 4. [START_REF] Edington | Practical electron microscopy in materials science[END_REF] lattice Eq. (4) edge-on condition 1 double edge-on [START_REF] Luo | A tem method for accurate measurement of habit plane (interface): double edge-on trace analysis[END_REF] lattice Eq. (4) edge-on condition 2 trace & edge-on [START_REF] Meng | Precise determination of the irrational preferred interface orientation by tem[END_REF] lattice Eq. (3) and Eq. (4) edge-on condition 3 projection width [START_REF] Zhang | Determination of habit planes using trace widths in tem[END_REF] holder Eq. (4) tilting about the trace 2 projection width [START_REF] Qiu | A simple and inclusive method to determine the habit plane in transmission electron microscope based on accurate measurement of foil thickness[END_REF] holder Eq. [START_REF] Meng | Precise determination of the irrational preferred interface orientation by tem[END_REF] tilting axis in screen 2 this work either Eq. (3) and Eq. (4) none ≥ 3

In the double-trace method and edge-on methods, the interface normal is always derived by the cross product of two vectors. These two vectors can be two traces (double-trace), a projected trace and a beam direction (single edge-on), a trace and a beam direction (trace & edge-on), or two beam directions (double edge-on). Since the cross product is equivalent to LSM when solving a determined equation, these methods can all be treated as special cases of Eq. ( 4), with the traces determined by Eq. ( 3).

Existing projection width methods [START_REF] Zhang | Determination of habit planes using trace widths in tem[END_REF][14] treated the non-linear part √ 1e ⋅ 2 in Eq. 2 with different strategies. Zhang and Kelly [START_REF] Zhang | Determination of habit planes using trace widths in tem[END_REF] tilted the sample about the trace direction, resulting in e ⋅ = 0 (?) and = | ⋅ e |.

In their work, the sample thickness ℎ was used rather than . These two variables have a relationship, ℎ = cos (Figure 1a inset), which is valid only if the two foil surfaces are both perpendicular to the beam direction at zero tilt.

Qiu and Zhang [START_REF] Qiu | A simple and inclusive method to determine the habit plane in transmission electron microscope based on accurate measurement of foil thickness[END_REF] extended this work using the latter assumption, but they substituted √ 1e ⋅ 2 by taking the tilting axis of the sample holder as a reference vector. The derivation of this method using the present geometry model is given in Appendix C.

The limitation of the reference vector method is that this vector must be parallel to the tilting axis of the sample holder, otherwise the × ⋅ in Eq. 11 would not have a simple result like , making it too hard to be solved.

Therefore, the reference vector method cannot be applied to double-tilt holders, because the resulting tilting axis is neither constant, nor in the screen plane.

In addition, these two methods cannot determine the sign of 's without external help, as they only use two sets of inputs. As shown in this work, the use of at least 3 input sets enables the determination of the sign of 's straightforwardly.

Generalization of the present method

The present method can also be used as an enhanced method to determine the length and the direction of 1D features.

Let be the projected length of the 1D feature and be the original length of the 1D feature. Each measurement of a 1D feature projection ( ei , pi , ) will give

ei × pi ei T = 0 T (8)
Eq. 8 can be solved the same way as described above. The length vector of the 1D feature can thus be determined.

This approach may be used, for instance, to track accurately the distance between pining points on a dislocation line or the size of dislocation loops, as long as, 2 traceable features, such as intersection points between dislocation segments, can be detected. If 3 traceable points can be detected, for instance, to determine dislocation habit plane if at least 3 points (or 2 lines) can be tracked [START_REF] Naanani | Pure climb of [001] dislocations in tial at 850Âř c[END_REF], or to assess the planarity of 4 points (or 3 lines). (i'm not sure to understand this sentence) Since all algorithms proposed here employed LSM, they are equally robust against random errors and scalable on large datasets. Efficient LSM functions can be found in plenty of math libraries, such as Eigen, LAPACK and Intel ® MKL. Hence the present algorithms can be implemented as a real-time solver (as shown in Supplementary Material). They also have the potential to be integrated with automatic feature tracking in a tilting series, which could be a fast and efficient way to automatically measure microstructural features in a foil.

Conclusions

A new analytical method to determine interface normals with excessive inputs has been proposed. It is a robust algorithm based on a generalized geometrical model of interface projection, and it can also automatically deal with a large amount of data. The validity of the method was verified using experimental observations of interfaces in TEM.

This study proves the reliability and effectiveness of this method, regardless of the input orientation or foil surface configuration. Given 6 or a few more sets of inputs, even with considerable experimental errors, this method is still capable of yielding reliable results. It can also be extended to determine 1D features or to check the planarity of a set of features. The present method is compared with existing approaches, showing that many existing approaches can be treated as special cases of the present method, while the present method has only few constraints.

A. LSM for solving overdetermined linear equations

For an overdetermined = equation, e.g., Eq. 4 with ≥ 3, the least square solution of vector is + = + , where + is Moore-Penrose inverse, or pseudo-inverse, of matrix . The solution has the residual vector, = + -, and the residual error, ‖ + -‖ 2 . This method is applicable for most circumstances. However, when = , e.g., Eq. 3, this method will only give a trivial solution of = .

= is a typical problem of overdetermined homogeneous linear system, whose solution is the eigen-vector of T with the smallest eigen-value.

The residual error of the solution is the smallest eigen-value.

B. Bootstrap method for estimating confidence intervals

Bootstrap is a statistical technique to estimate the variation of statistics that are computed from a set of data [? ].

Here we use the bootstrap method to estimate the variation, i.e., confidence intervals, of and determined by Eq. 4. In Eq. 4, and sgn( ) are considered as constant parameters in the equation, as they have already been determined.

Thus, the data is composed of data points of ( , ). Then, the data is resampled with replacement to generate a resampled data set of size , with which the statistics * and * (star denoting resampled data) can be calculated.

This procedure is repeated 10000 times, the deviations between * and and the angle between * and can also be calculated. The 95% confidence interval of is given by the 95th percentile of the deviation of * . The 95% confidence interval of is given by the 95th percentile of the deviation angle of * , denoting a confidence cone around [? ].

C. The projection width method reported in [14]

Here we used the symbols and equations in present work to derive the equations reported in [START_REF] Qiu | A simple and inclusive method to determine the habit plane in transmission electron microscope based on accurate measurement of foil thickness[END_REF].

By replacing in Eq. 2 with × ∕‖ × ‖, and temporarily ignoring the absolute sign on e ⋅ , we get:

= ℎ e ⋅ √ 1 -( ⋅ ) 2 -× ⋅ e 2 (9)
By introducing , the angle between p and is expressed by:

cos = p ⋅ ‖ p ‖ = × ⋅ √ 1 -( ⋅ ) 2 -× ⋅ e 2 (10)
By substituting Eq. 10 into Eq. 9, we get Eq. 11. ℎ and have to be combined, for they both need solving.

e ⋅ (

ℎ × ⋅ ) = cos (11)
In the work of Qiu and Zhang, the sample is tilted about axis with the screen remains unchanged. The r , r , r vectors at zero tilt in their work are the basis vectors in the holder coordinates system. Therefore, the parameters should be set as e = [0, 0, 1] T H , = [0, -1, 0] T H , = Ry( )[ , , ] T H , and = Ry( )[0, 0, 1] T H . In addition, the trace direction was measured at zero tilt angle by 0 , which would directly give . Similar as Eq. ( 4), Eq. 12 is constructed using two sets of Eq. 11 and ⋅ = 0. The ± sign is caused by the uncertainty of signs, and should be eliminated by inspecting the trend of projection width change.

⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣
-sin 1 0 cos 1 -sin 2 0 cos 2 sin 0 -cos 0 0

⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ ℎ ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ = ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ 1 ∕ cos 1 ± 2 ∕ cos 2 0 ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ (12)
The solution of Eq. 12 is Eq. 13, the same as reported in their work.

⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ ℎ ℎ ∕ ℎ ∕ ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ = 1 sin( 2 -1 ) ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ 1 cos

Figure 1 :

 1 Figure 1: Geometrical description of interface projection. (a) Parallel traces case: The interface is projected on the viewing screen, giving the projection . The inset is the cross section view along the trace direction. (b) Non-parallel traces case: A traceable point on the interface and a trace are projected to the viewing screen, giving the projection .

Figure 1b ,

 1b Figure 1b, by tracking point on the interface , one can redefine as the distance between and the trace , intersection with one foil surface (orange plane), while the distance between the projected point and projected trace is measured as . In this case, it is crucial to measure at the same position each time. The area of interface triangle = | |∕2 and that of projection triangle = | |∕2 are still related by = cos .

Figure 2 :

 2 Figure 2: Illustration of the holder coordinate system when the sample is mounted in a double-tilt holder.

Figure 3

 3 Figure3shows a series of bright field (BF) images of a faceted austenite grain (A) in a ferrite matrix (F) taken at different orientations. Two FCC/BCC facets, IntA and IntB, intersect and form a well defined corner. In order to show the ability to determine multiple interfaces, normals of these two facets are determined simultaneously, using 6 sets of inputs among 8 different imaging conditions. The determination was performed both in the holder coordinate system and in the lattice coordinate system. The former is more convenient to use, as it only needs tilt angles of the sample holder, while the latter, which requires the analysis of Kikuchi patterns, usually has better accuracy. The normal of IntA is determined by Figures 3a∼f, and that of IntB by Figures3a ∼ e and 3g. Figure3h, a near edge-on condition of both IntA and IntB, was only used to check the results.

Figure 3 :

 3 Figure 3: BF images of the interfaces and the corresponding Kikuchi patterns taken in austenite at different tilt angles. The and tilt angles are reported in the bottom right. The projection widths and the azimuthal angle of both IntA and IntB interfaces are also indicated. Arrows in the Kikuchi patterns indicate the e × p directions of IntA and IntB.

Figure 4

 4 Figure 4 shows a series of BF images of an interface with non-parallel traces in an Al bicrystal sample taken at different tilt angles. The tracked point (pointed out by arrows) is defined by the position where the lower trace (dotted line) starts to deviate. Figures 4a∼e are used to solve the interface normal, while Figure 4f is a near edge-on condition

  as following: = [0.883, -0.348, 0.315] H , = [-0.473, -0.646, 0.600] H ∼ 2.4 • , and = 224 ± 12 nm. (what is 2.4 • ?) Using the transformation matrix determined from a Kikuchi pattern (Figure S1), the lattice coordinates of the interface normal is = [4.000 -4.176 0.144] with 5.3 • deviation from [4 -5 0]. Noting that the 95% confidence range is 2.4• , this deviation is considered to be the result of plastic deformation, rather than experimental errors. Using Eq. 5, the e at the near edge-on condition in Figure4fis [0.278, 0.536, 0.797] H , which is almost perpendicular to the interface normal with 0.1 • deviation. This edge-on condition confirms the result of . Therefore, the present method remains effective even on interfaces with non-parallel traces.

Figure 4 :

 4 Figure 4: BF images of the interface with non-parallel traces at different tilt angles. The and tilt angles are reported in the top left. The widths between the tracked point (indicated by arrows) and the trace (dash line), and the azimuth angles of the trace (dash line) are also indicated.

Figure 5 :

 5 Figure 5: Statistical analysis of the trueness of the results. (a) Distribution of the average trueness for interface normals chosen within the pole figure viewed from axis. , , and have random errors. The beam directions reached by sample tilting are limited by dashed lines. (b) Same as (a) when the error is restricted to . (c) The change of trueness with respect to the number of inputs for a fixed interface normal.

Table 1

 1 Definition of symbols1

	Symbol	Definition
	e	Reversed electron beam direction, unit vector
		Interface normal, unit vector
		Arbitrary vector in the screen plane, unit vector, e ⋅ = 0
		Foil surface normal, unit vector
		Trace direction, i.e., direction of the intersection between the interface and the
		foil surface, unit vector = × ∕‖ × ‖
	p	Projected direction of
	ℎ	Foil thickness
		Azimuth angle from axis of the screen to p
		Real interface width, i.e., real distance between traces
		Projected interface width, i.e., distance between projected traces
		Input number, index, 1 ≤ ≤
		Total number of inputs
	Rx( ), Ry( ), Rz( )	Right-handed rotation matrix with rotation angle about , , axis respectively
	sgn( )	A function that gives a value of ±1 respect to the sign of ⋅ ei

Table 2

 2 Directions in lattice coordinates from the Kikuchi patterns in Figure3and sgn( ) for solving Eq. (4)

	Figure	e		IntA p IntB p	sgn A ( ) sgn B ( )	B A	( • )	( • )
	3a	[-0.811 0.485 0.327]	[0.546 0.427 0.721] [0.249 0.792 -0.558]	-1 +1	72 38	-37.86	27.22
	3b	[-0.629 0.147 -0.763]	[-0.250 0.891 0.378] [0.772 0.239 -0.590]	+1 +1	33 175	35.23	25.19
	3c	[-0.107 0.628 -0.771]	[0.186 0.774 0.605] [0.963 -0.128 -0.238]	+1 +1	130 108	30.53	-27.39
	3d	[-0.281 0.951 0.130]	[0.485 0.023 0.874] [0.691 0.295 -0.660]	+1 -1	73 27	-29.90 -27.28
	3e	[-0.465 0.341 -0.817]	[-0.048 0.912 0.408] [0.885 0.157 -0.438]	+1 +1	81 149	35.48	4.56
	3f	[-0.677 0.684 0.272]	[0.597 0.295 0.746] [0.457 0.680 -0.574]	-1 not used	26 -	-37.32	6.65
	3g	[-0.654 0.704 0.277]	[0.612 0.277 0.741] [0.469 0.665 -0.582]	not used +1	-0	-38.01	4.51
	3h	[-0.592 0.776 0.219]	[0.578 0.220 0.786] [0.541 0.584 -0.601]	not used not used	--	-34.12	-0.66
	Table 3
	Results of trace directions and interface normals
	Coordinates Interface						Confidence range of ( • )	(nm)
	holder	IntA	[0.453, -0.891, 0.003] H	[-0.863, -0.432, 0.262] H	1.0	166 ± 5
					[0.567 0.588 -0.577]
		IntB	[0.776, 0.630, 0.037] H	[0.549, -0.670, 0.500] H	1.8	178 ± 6
					[-0.605 -0.208 -0.769]
	lattice	IntA	[-0.215 -0.581 -0.785]	[0.581 0.585 -0.567]	1.6	172 ± 8
		IntB	[-0.722 -0.253 0.644]	[-0.610 -0.240 -0.755]	1.6	182 ± 6

Table 4

 4

	A comparison between this work and previous methods
	Method	Coordinates	Formulas	Special requirements	Number of inputs
	double-trace [5]	lattice	Eq. (3)	another line feature	2
	single edge-on