Jos -Alain Sahel
email: <j.sahel@gmail.com

Quality versus quantity: assessing individual research performance

Evaluating individual research performance is a complex task that ideally examines productivity, scientific impact, and research quality a task that metrics alone have been unable to achieve. In January 2011, the French Academy of Sciences published a report -on current bibliometric (citation metric) methods for evaluating individual researchers, as well as recommendations for the integration of quality assessment. Here, we draw on key issues raised by this report and comment on the suggestions for improving existing research evaluation practices.

BALANCING QUANTITY AND QUALITY

Evaluating individual scientific performance is an essential component of research assessment, and outcomes of such evaluations can play a key role in institutional research strategies, including funding schemes, hiring, firing, and promotions. However, there is little consensus and no internationally accepted standards by which to measure scientific performance objectively. Thus, the evaluation of individual researchers remains a notoriously difficult process with no standard solution. Marcus Tullius Cicero once wrote, Non enim " numero haec iudicantur, sed pondere ( ). Translation: The number does not matter, the quality does. In line with Cicero s outlook on " 1

' quality versus quantity, the French Academy of Sciences analyzed current bibliometric (citation metric) methods for evaluating individual researchers and made recommendations in January 2011 for the integration of quality assessment ( ). The essence of the report is 2 discussed in this Commentary.

Evaluation by experts in the field has been the primary means of assessing a researcher s performance, although it can be biased by ' subjective factors, such as conflicts of interest, disciplinary or local favoritism, insufficient competence in the research area, or superficial examination. To ensure objective evaluation by experts, a quantitative analytical tool known as bibliometry (science metrics or citation metrics) has been integrated gradually into evaluation processes ( ). Bibliometry started with the idea of an impact factor, which was Fig. 1 first mentioned in in 1955 ( ), and has evolved to weigh several aspects of published work, including journal impact factor, total Science 3 number of citations, average number of citations per paper, average number of citations per author, average number of citations per year, the number of authors per paper, Hirsch s -index, Egghe s -index, and the contemporary -index. ). Within the past decade, metrics have secured a foothold in the evaluation of individual, team, and institutional research because the use of such metrics appears to be easier and faster than the qualitative assessment by experts. Because of the ease of use of various metrics, however, bibliometry tends to be applied in excessive and even incorrect ways, especially when used as standalone analyses.

The French Academy of Sciences (FAS) is concerned that some of the current evaluation practices in particular, the uncritical use of -publication metrics might be inadequate for evaluating individual scientific performance. In its recent review ( ), the FAS addressed the --

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advantages and limitations of the main existing quantitative indicators, stressed that judging the quality of a scientific work in terms of conceptual and technological innovation of the research is essential, and reaffirmed its position about the decisive role that experts must play in research assessment ( , ). It also strongly recommended that additional criteria be taken into consideration when assessing 2 4

individual research performance. These criteria include teaching, mentoring, participation in collective tasks, and collaboration-building, in addition to quantitative parameters that are not measured by bibliometrics, such as number of patents, speaker invitations, international contracts, distinctions, and technology transfers. It appears that the best course of action will be a balanced combination of the qualitative (experts) and the quantitative (bibliometrics).

BIBLIOMETRICS: INDICATORS OR NOT?

Bibliometrics use mathematical and statistical methods to measure scientific output; thus, they provide a quantitative not a qualitative assessment of individual research performance. The most commonly used bibliometric indicators, as well as their strengths and weaknesses, are described below.

Impact factor

The impact factor, a major quantitative indicator of the quality and popularity of a journal, is defined by the median number of citations for a given period of the articles published in a journal. The impact factor of a journal is calculated by dividing the number of current-year citations by the source items published during the previous two years ( ). According to the FAS, the impact factor of journals 5

in which a researcher has published is a useful but highly controversial indicator of individual performance ( ). The most common issue is 2 variation among subject areas; in general, a basic science journal will have a higher average impact factor than journals in specialized or applied areas. Individual article quality within a journal is also not reflected by a journal s impact factor because citations for an individual ' paper can be much higher or lower than what might be expected on the basis of that journal s impact factor ( , , ). In addition, '

self-citations are not corrected for when calculating the impact factor ( ). On account of these limitations, the FAS considers the tendency 6

of certain researchers to organize their work and publication policy according to the journal in which they intend to publish their article to be a dangerous practice. In extreme situations, such journal-centric behavior can trigger scientific misconduct. The FAS notes that there has been an increase in the practice of using journal impact factors for the evaluation of an individual researcher for the purpose of career advancement in some European countries, such as France, and in certain disciplines, such as biology and medicine ( ). 2

Number of citations

The number of times an author has been cited is an important bibliometric indicator; however, it is a value that has several important limitations. First, citation number depends on the quality of the database used. Second, it does not consider where the author is located in the author list. Third, sometimes articles can have a considerable number of citations for reasons that might not relate to the quality or importance of the scientific content. Fourth, articles published in prestigious journals are privileged as compared with those with equal quality but published in journals of average notoriety. Fifth, depending on cultural issues, advantage can be given to citations of scientists from the same country, to scientists from other countries (in particular Americans, as often is the case in France), or to articles written in English rather than in French, for example ( ). For these cultural reasons, novel and important papers might attract little attention for 2 several years after their publication. Lastly, citation numbers also tend to be greater for review articles than for original research articles.

Self-citations do not reflect the impact of a publication and should therefore not be included in a citation analysis when this is intended to give an assessment of the scientific achievement of a scientist ( ). 8

New indicators ( -index, -index) h g

Recently, new bibliometric indicators borne out of databases indexing articles and their citations were introduced to address the needs of objectively evaluating individual researchers. In 2005, Jorge Hirsch proposed the -index as a tool for quantifying the scientific impact h of an individual researcher ( ). The -index of a scientist is the number of papers co-authored by the researcher with at least citations 9 h h each; for example, an -index of 20 means that an individual researcher has co-authored 20 papers that have each been cited at least 20 h times each. This index has the major advantage to measure simultaneously the scientist s productivity (number of papers published over ' years) with the cumulative impact of the scientist s output (the number of citations for each paper). Although the -index is preferable to ' h other standard single-number criteria (such as the total number of papers, total number of citations, or number of citations per paper), it has several disadvantages. First, it varies with scientific fields. As an example, -indices in the life sciences are much higher than in physics ( h 9

). Second, it favors senior researchers by never decreasing with age, even if an individual discontinues scientific research ( ). Third, 10 citation databases provide different -indexes as a result of differences in coverage, even when generated for the same author at the same h time ( , ). Fourth, the -index does not consider the context of the citations (such as negative findings or retracted works). Fifth, it is 11 [START_REF] Kulkarni | Comparisons of citations in Web of Science, Scopus, and Google Scholar for articles published in general medical journals[END_REF] h strongly affected by the total number of papers, which may underestimate scientists with short careers and scientists who have published only a few although notable papers. The -index also integrates every publication of an individual researcher, regardless of his or her role h in authorship, and does not distinguish articles of pathbreaking or influential scientific impact. Contemporary -index (referred to as h hc -index), as suggested by Sidiropoulos . ( ), takes into account the age of each article and weights recently published work more et al 10 heavily. As such, the -index may offer a fairer comparison between junior and senior academics than the regular -index ( ). hc h 13

The -index was introduced ( ) to distinguish quality, giving more weight to highly cited articles. The -index of a scientist is the g 14 g highest number of articles (a set of articles ordered in terms of decreasing citation counts) that together received or more citations; for g g 2 example, a -index of 20 means that 20 publications of a researcher have a total number of citations of at least 400. Egghe pointed out that g the -index value will always be higher than the -index value, making easier to differentiate the performance of authors. If Researcher A g h has published 10 articles, and each has received 4 citations, the researcher s -index is 4. If the Researcher B has also written 10 articles, ' h and 9 of them have received 4 citations each, the researcher s -index is also 4, regardless of how many citations the 10th article has ' h received. However, if the tenth article has received 20 citations the -index of the Researcher B would be 6; for 50 citations, the -index g g INSERM analysis emphasizes the fact that each indicator has its advantages and limitations, and care must be taken not to consider them alone as surrogate markers of team performance. Several indicators must be taken into account when evaluating the overall output of a

" " research team. The use of bibliometric indicators requires great vigilance; but, according to the INSERM experience, metrics enrich the evaluation committees debates about the scientific quality of team performance ( ). Research evaluation practices also vary by field and subfield owing in part to the large disparities across community sizes and the literature coverage provided by citation databases. As reviewed by the FAS, evaluation of individual researchers in the mechanical sciences, computing, and applied mathematics fields includes both the quality and the number of publications, as well as scientific awards and the number of invitations to speak at conferences, software, patents, and technology transfer agreements. Organization of scientific meetings and editorial responsibilities are also taken into consideration. The younger researchers are evaluated by experts during interviews and while they give seminars. In these fields, publication does not always play a leading role in transferring knowledge; thus, during a long professional career, metrics give rather weak and inaccurate estimation of research performance. Bibliometrics are therefore used only as a decision-making aid, but not as a main tool for evaluation.

In physics and its subfields, evaluation methods vary. In general, a combination of quantitative (number of publications, -index) and h qualitative measures (keynote and invited speeches, mentoring programs) plays a decisive role in the evaluation of senior scientists only.

In astrophysics, metrics are largely used for evaluation, recruiting, promotions, and funding allocations. In chemistry, the main bibliometric indicators ( -index, total number of citations, and number of citations per article) are taken into consideration when h discussing the careers of senior researchers (those with more than 10 to 12 years of research activity). In recruiting young researchers, experts interview the candidate to examine ability to present and discuss the subject matter proficiently; the individual s publication record ' is also considered. However, the national committees for chemistry of French scientific and university institutions Centre National de la individual s evaluation.

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In economics, evaluation by experts in the field plays the most important role for recruitments and promotions, but bibliometric indicators are used to help this decision-making. For the humanities and social sciences (philosophy, history, law, sociology, psychology, languages, political sciences, and art) and for mathematics, the existing databases do not cover these fields sufficiently. As a consequence, these fields are not able to properly use bibliometrics. In contrast, in biology and medicine the quantitative indicators in particular the journal s impact factor are widely used for evaluating individual researchers ( ).

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STRATEGIES AND RECOMMENDATIONS

The FAS acknowledged that bibliometrics could be a very useful evaluation tool when handled by experts in the field. According to its recommendations, the use of bibliometrics by monodisciplinary juries should be of nondecisive value; instead, the experts of these evaluation committees know the candidates well enough to compare more precisely and objectively the individual performance of each of them. In the case of pluridisciplinary (interdisciplinary) juries, bibliometrics can be successfully used, but only if the experts consider the differences between scientific fields and subfields (as mentioned above). For this purpose, the choice of indicators and the methodology to evaluate the full spectrum of research activity of a scientist should be initially validated. As emphasized by the FAS, bibliometrics should not be used for deciding which young scientists to recruit. In addition, the bibliometric set should be chosen depending on the purpose of the evaluation: recruitment, promotion, funding allocation, or distinction. Calculations should not be left to nonspecialists (such as administrators that could use the rapidly accessible data in a biased way) because the number of potential errors in judgement and assessment is too large. Frequent errors to be avoided include the homonyms, variations in the use of name initials, and the use of incomplete databases. It is important that the complete list of publications be checked by the researcher concerned. Researchers could even be asked to produce their own indicators (if provided with appropriate guidelines for calculation); these calculations should subsequently be approved. The evaluation process must be transparent and replicable, with clearly defined targets, context, and purpose of the assessment.

To improve the use of bibliometrics, a consensus has been reached by the FAS ( ) to perform a series of studies and to evaluate 2 various methodological approaches, including (i) retrospective studies to compare decisions made by experts and evaluating committees, (ix) studies on how bibliometrics affect the scientific behavior of the researchers; and (x) establishment of standards of good practice in the use of bibliometrics for analyzing individual research performance.

FIXING THE FLAWS

Assessing research performance is important for recognizing productivity, innovation, and novelty and plays a major role in academic appointment and promotion. However, the means of assessment namely, bibliometrics are often flawed. Bibliometrics have enormous

' 16 As 2 recruiting

 162 reported by the FAS, bibliometric practices vary considerably from country to country. A worldwide survey ( ) Nature 17emphasized that 70 of the interviewed academic scientists, department heads, and other administrators believe that bibliometrics are used % for recruitment and promotions, and 63 of them consider the use of these measures to be inadequate. Many Anglo-Saxon countries use % bibliometrics to evaluate performances of universities and research organizations, whereas for hiring and promotions, the curriculum vitae, interview process, and letters of recommendation count more than the bibliometric indicators ( ). In contrast, metrics are used for " " in Chinese and Asian universities in general, although movement toward the use of letters of recommendation is currently underway ( ). In France, an extensive use of publication metrics for individual and institutional evaluations has been noted in the 2 biomedical sciences ( ). 2

[

  Recherche Scientifique (CNRS) and Conseil National des Universit s (CNU), respectively usually avoid bibliometrics altogether for an é ]

  with results potentially obtained by bibliometrics; (ii) studies to refine the existing indicators and bibliometric standards; (iii) authorship clarification; (iv) development of standards for originality and innovation; (v) discussion on the citation discrepancies on the basis of geographical-or field-localism; (vi) monitoring the bibliometric indicators of outstanding researchers (a category reserved for those who have made important and lasting research contributions their specific field and who have obtained international recognition); (vii) examining the prospective values of the indicators for researchers that changed their field orientation with time; (viii) examining the indicators of researchers receiving great awards such as Nobel Prize, Fields Medal, and medals of notorious academies and institutions;

  the qualitative evaluation of individual researchers; however, none of the bibliometric indicators alone (or even considering a set of them) allow for an acceptable and well-balanced evaluation of the activity of a researcher. The use of bibliometrics should continue to evolve through in-depth discussion on what the metrics mean and how they can be best interpreted by experts in the given scientific field. des sciences . available at: http://www.academiesciences.fr/actualites/textes/recherche_08_07_09.pdf 2009 ;

Fig. 1

 1 Fig. 1 Can individual research performance be summarized by numbers? CREDIT: IMAGE COURTESY OF D. FRANGOV (FRANGOV DIMITAR PLAMENOV COMPANY)

Acknowledgements:

The author thanks K. Marazova (Institut de la Vision) for her major help in preparing this commentary and N. Haeffner-Cavaillon (INSERM) for critical reading and insights.

3 5 would be 9 ( ). Thus, one or several highly cited articles can influence the final -index of an individual researcher, thus highlighting 15 g the impact of authors.

CHOOSING AN INDICATOR

Bibliometry is easy to use because of its simple calculations. However, it is important to realize that the purely bibliometric approaches are inadequate because no indicator alone can summarize the quality of the scientific performance of a researcher. The use of a set of metrics (such as number of citations, -index, or -index) would give a more accurate estimation of the researcher s scientific h g ' impact. At the same time, metrics should not be made too complex because they can become a source of conceptual errors that are then difficult to identify. FAS discourages the use of metrics as a standalone evaluation tool, the use of only one bibliometric indicator, the use of the journal s impact factor to evaluate the quality of an article, neglecting the impact of the scientific field/sub-field, and ignoring author --For example, the articles of all teams received a number of citations irrespective of the journal in which they were published, with only low correlation between the journal impact factor and the number of times each publication was cited. No correlation was found between the journal impact factor and the individual publication citations, or between the Top 1 publications and the impact factor ( ).

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Khalid Rashdan

The essence of gesture is invisible

Keywords: The novelty of the gesture, abstract symbolic representation, mental imagery, the subject's intention, gesture control, aspects

of the gesture, the phenomenon of persistence.

There are six reasons for the invisibility of much of gesture:

1. The meaning of gesture 2. Abstract symbolic representation 3. Mental imagery 4. The intention of the subject 5. The predicted control of speed and position in gesture 6. The phenomenon of persistence

The meaning of gesture:

The pattern of brain responses related to observing the movement of others is far from being fixed and immutable. It depends largely on the nature of the actions observed. For example, a movement with a known meaning will not create the same response as a symbolic movement without meaning (Decety et al. 1997;Rumiati et al., 2005). Learning can take place only if the subject has precise knowledge of the result of his actions (Simonet, 1990, for a review).

Abstract symbolic representation:

The symbolic representation of abstract features is constructed by two factors:

a) The observation of gesture. b) The intensive practice of gesture.

The observation of a visual model allows the learner to draw on a symbolic representation of abstract features of the task (Bandura, 1969(Bandura, , 1977(Bandura, , 1986)).

Considerable research has therefore been undertaken to determine, both in children and in adults, the role of the observational learning of factors related to attention (McCullagh, 1986), motivation (Landers andLanders, 1973;Gould and Weiss, 1981), characteristics of the model (Landers, 1975;Ishikura and Inomata, 1995;Sambrook, 1998), verbalization (Weiss, 1983;Weiss and Klint, 1987) or mental imagery (Lejeune and al. 1994;Fery and Morizot, 2000).

The image is defined as "an evocation of the inner qualities of a perceptual object in the absence of that object" (Blanc-Garin, 1974, p. 539).

This link between image and sensation could suggest that the image is a mere residue of perception or, in other words, the image is built isomorphically to the observed image. This idea is to a large extent incorrect. The image is therefore defined here not as a mere residual perceptual trace, but as a "true deferred imitation" (Piaget and Inhelder, 1963, p. 70). The image is not a reflection but an operation.

When the subject observes a visual model, there is a whole occipito-temporo-parietal network involved:

1. Information first undergoes basic processing in the occipital visual areas.

2. A second transformation then takes place in the superior temporal sulcus (STS). This structure is involved in decoding biological motion (Perrett et al., 1985(Perrett et al., , 1989;;Oram and Perret, 1994;for a review, Allison et al., 2000). Furthermore, neuronal activities recorded in the STS are largely modulated by the perspective of the observer. If this perspective shifts, the cellular response changes (Perrett et al., 1989(Perrett et al., , 1991)). For example: if you're in a car and you draw a figure on the windshield, an outside observer will see this figure as a mirror image (a "b" will appear as a "d"). If the observer wants to reproduce your drawing "in place", he must, first, rebuild your route according to its original perspective.

b) Intensive practice of gesture:

In cognitive psychology, it is conventional to distinguish two main modes of movement control:

I.
A closed-loop control (retrospective): In this first mode, driving a trace stored in longterm memory determines the start of the movement which is then adjusted in real time through sensorimotor feedback. It thus applies to movements whose duration is long enough to accommodate these changes (greater than 100 ms).

II.

An open loop control (proactive):

in contrast, open loop control applies to very fast movements. It suggests that the movement is fully programmed in advance and therefore performed without reference to sensory cues.

The mode of movement control is essentially retroactive at the beginning of learning (P.

Zesiger 1995). During this stage, children use the sensory feedback (visual and kinesthetic) from their own hand movements. Through learning and automation, proactive monitoring of the movements will become dominant.

Mental imagery:

A visual model can have positive effects on performance when combined with mental imagery tasks (Christmas, 1980;Hall and Erffmeyer, 1983;Ziegler, 1987;Gray 1990;Lamirand and Rainey, 1994;Bucher 1993;Lejeune et al. 1996;McKenzie and Howe, 1997;Frey and Morizot, 2000).

Internalized representations evoked during imagery tasks are not visual but are essentially motor representations (Famose, 1976). In this context, mental repetition loses its status and becomes true sensory imitation (Piaget and Inhelder, 1963) or internalized achievement (Jeannerod 1994(Jeannerod , 2001)). This internalization of the action may lead to the specific solicitation of sensorimotor networks and act as practice or repetition without movement.

Three studies show the effect of imagery on learning:

1. A study by Li-Wei et al. (Li-Wei et al., 1992), conducted as part of learning forehand in table tennis, in children aged 7 to 10 years.

2. A recent study also on learning tennis (Fery and Morizot, 2000).

3. A study on learning table tennis (Lejeune et al., 1994).

These studies show the existence of an improvement when the demonstration was followed by a task of mental rehearsal.

Intention of the subject:

The neural substrate for imitative mechanisms is significantly different for the tasks of emulating an effect or of reproducing a movement (Koski et al., 2002).

Activations are different when the subject is instructed to watch the movement:

1. To observe.

To imitate.

3. To recognize the movement during a subsequent performance (Decety et al., 1997;Buccino et al. 2004b;Rumiati et al., 2005).

Fogassa and colleagues showed that the response of parietal cell mirrors was modulated by the purpose of the action (Fogassa et al., 2005). Thus, for example, when the experimenter grasped an object to bring it to his mouth or put it in a box, the pattern of neural response was significantly different.

Consistent with this result, an imaging technique (fMRI) has recently shown in humans that

mirror neurons in the frontal region responded differently when the same movement was observed in two different contexts (e.g. grabbing a cup in order to drink from it, or in order to change its position; lacoboni et al., 2005).

The predicted control of speed and position in gesture:

For decades, it has been accepted that most motor skills obeyed ballistic logic. This logic can be summarized as follows: once started, the motor program runs in "open loop" and it is impossible to amend its development. This hypothesis has been invalidated by a host of behavioral studies over the last two decades (for a review, Paillard, 1996;Prablanc et al. 2003;Desmurget andGrafton, 2000, 2003;Todorov, 2004).

The first research to establish the existence of a rapid control of gesture during its execution concerned visual feedback loops. (Paillard, 1980(Paillard, , 1996) ) noticed that capture of the foveal target was usually performed before or, in the case of large eccentricities, just after the start of the hand movement (Prablanc et al. 1979;Prablanc and Martin, 1992;Desmurget et al., 2001Desmurget et al., , 2004)).

This implies that the initial phase of rapid transit to the target member is performed under the control of peripheral vision, while the slow terminal phase is completed in central vision.

Paillard suggested that the peripheral retina could play a specific role in early control of the directional component of gesture.

In rapid pointing movements, the movement of the hand is initially prepared on the basis of retinal information devices. However, this information does not allow as precise an encoding of target position as foveal information (Prablanc et al., 1979, Abrams et al. 1990;Bock, 1993;Desmurget et al., 2005b). Work by Prablanc et al. showed that the original inaccuracy was corrected during the course of an arm movement, regardless of whether the subject can see his hand or not (Prablanc et al. 1986;Desmurget et al., 2005b).

The speed and position of the probable effector can not only be estimated immediately, but also predicted in advance (Desmurget and Grafton, 2000). This capability enables adjustment of the current response to environmental stress for even the fastest movements.

In recent work, Piselli et al. (Piselli et al., 2000) have shown that these adjustments are largely automatic and independent of voluntary mechanisms. This result, recently reproduced by Cressman et al. (Cressman et al., 2006), is not surprising if one considers the nature of analysis required to stop the action underway: it is necessary to perceive the movement (for example, a jump), to identify it consciously, to decide to block the response, and then to send an inhibitory control. We can conclude that the work mentioned above by Piselli et al. (Piselli et al., 2000) indicates that stopping the action takes more time to adjust it.

The phenomenon of persistence:

Two currents of motor learning:

1. Centripetal logic: the exploration of a wide range of responses is essential to the formation of motor skills (Schmidt, 1975(Schmidt, , 1976;;Wulf andShmidt, 1988, 1994;Wulf and Lee, 1993;for a review, Schmidt and Lee, 1999). We start from a multiplicity of experiences intertwined to develop, through abstraction, a general and generalizable rule.

2. Centrifugal logic: A stabilized practice, based only on the expression of a particular gesture, optimizes the extraction of the relevant skill (Lai et al., 2000;Wright and Shea, 2001;Whitacre andShea, 2000, 2002;Shea et al. 2001;for a review, Shea and Wulf, 2005). We start from a particular instance of "average" ability then go by progressive differentiation to a wide range of adaptive behaviors.

Centripetal approaches encourage adaptation functions, as opposed to centrifugal approaches which tend rather to annihilate the flexibilities of the system in order to ultimately facilitate the formation of rigid and compelling invariants (Summers, 1975;Shapiro, 1977;Ripoll, 1982;Weineck 1986;Dronkert and Verwey, 1996). Several studies illustrate the results of the inadequate practice of centrifugal logic. Studies by Zatsiorsky show how "The practice of exclusive and prolonged identical gymnastics movements is a serious impediment, even among high-level gymnasts, to the acquisition of new gestures" (quoted in Ripoll, 1982, p. 140). Other studies on learning the sprint suggest that the application of constant and identical drives results in the establishment of a genuine sign stereotype whose main characteristic is to make "more difficult, even impossible, a new speed development " (Weineck, 1986, p. 86).


P Defrance

J J Jureta

J Lecointre

X Urbain

Isotope effects in electron-impact dissociation of D 2 H +

Keywords: 34-80, 52-20 deuterium, hydrogen, isotope, molecular ion, electron-ion collision, ionization, excitation, dissociation, cross section

Absolute cross sections have been measured using the crossed electron-ion beams method for electron-impact dissociation of the D 2 H + molecular ion yielding H + , D + , HD + and + 2 D fragments. The collision energy ranges from a few eV up to 2.5 keV. Around the maximum, cross sections are found to be of similar amplitude: (3.25±0.08)×10 -17 cm 2 for H + , (3.56±0.15)×10 -17 cm 2 for HD + and (3.33±0.11)×10 -17 cm 2 for + 2 D ; except for D + for which the maximum cross section is (4.62±0.09)×10 -17 cm 2 . Individual contributions for dissociative excitation and for dissociative ionization are determined for each product. Close analysis of present data brings into evidence isotope effects in the fragmentation pattern of the D 2 H + target. Ejection of the lightest isotope is generally favored: for resonant dissociative excitation (H + over D + ), for dissociative excitation (molecular ions + 2 D and HD + , associated with ejection of H and D, respectively) and for dissociative ionization, but not for H + and D + produced via dissociative excitation. Present dissociative excitation cross sections for D 2 H + are found to be significantly lower than those measured for + 3 D , although those of DI agree well together.

Introduction

Dissociation of small polyatomic ions is an important process in the chemistry of planetary atmospheres, interstellar clouds [1] and various laboratory experiments [2]. Many deuterated molecules have been detected in the interstellar environment where the material is mainly in the molecular form. The observation of deuterated molecules appears to be easier than that of atomic deuterium itself. The cosmic ratio [HD]/[H 2 ] is detected to be in the order of 10 -5 , Confidential: not for distribution. Submitted to IOP Publishing for peer review 14 February 2011 D is abundant [5].

Breakup dynamics and the isotope effect in the dissociation of + 3 H and of + 3 D have been theoretically studied by Strasser et al [6]. Because of the D 3h symmetry, a small isotope effect is likely to be due to the difference in the width of the initial wave function for these two isotopologues. For H 2 D + and for D 2 H + , the D 3h symmetry is broken and only the C 2v symmetry remains. Based on the Born-Oppenheimer approximation, the potential energy surfaces of these heteronuclear targets (H 2 D + and D 2 H + ) are identical to that of the homonuclear targets ( + 3 H and + 3 D ). Rotational excitation of H 2 D + and of D 2 H + is lower than that of symmetric species because asymmetric isotopologues exhibit a non-vanishing permanent dipole moment (unlike + 3 H and + 3 D ), so they can radiatively cool down and reduce both vibrational and rotational internal energies [7].

The experimental results of Strasser et al [7] give evidence for an enhanced occurrence of linear dissociation geometries in the dissociative recombination (DR) of H 2 D + and of D 2 H + , revealing isotope effects. For H 2 D + , the D atom dominantly lies between the two hydrogen atoms (H-D-H), while no preference for the linear H-H-D (or D-H-H) geometry is observed. The situation is reversed for D 2 H + , in which case the linear D-D-H (or H-D-D) geometry is preferred compared to the linear D-H-D configuration. The force driving the breakup causes a larger acceleration when acting on the H atom than on the D atom. It influences the dissociation process by favoring the ejection of the lighter fragment. Weak enhancements of symmetric dissociation geometries were also found for + 3 H and + 3 D [6], but results were difficult to analyze because the three constituents are indistinguishable.

In the DR of H 2 D + , the relative fraction of the two two-body channels (H 2 +D and HD+H) is constant over the whole energy range [8]. However, in the DR of D 2 H + , Buhr et al [9] have recently found that the two-body (D 2 +H and HD+D) relative fraction is constant up to 0.2 eV but it strongly changes in the 1-10 eV energy range. The relative fraction 2(D 2 +H)/(HD+D) is expected to be 1 in the absence of any isotope effect, but it is measured to be 1.27±0.05 at relative electron-ion energies around 0 eV and it is found to increase to 3.7±0.5 at about 5 eV. This large isotope effect implies that, in the DR of D 2 H + , the formation of D 2 +H is more favorable than that of HD+D. It appears that the general tendency in DR reactions is to favor ejection of the lightest fragments and to fractionate into the heaviest deuterated molecules.

Buhr et al [9] estimated the population distributions of the vibrational levels for the molecular fragments D 2 (v) and HD(v). They were found to be similar for both channels and similar to the distributions obtained in the DR of + D target [10]. Results have been reported for formation of D + and + 2 D ions, from their respective appearance thresholds up to 2.5 keV. For both fragments, an overall good agreement is observed in the energy region below 30 eV between our results and those of storage ring facilities measured by Jensen et al [11] and by Le Padellec et al [12]. Appearance energies are measured to be (4.5±0.5) eV and (6.0±0.5) eV for D + and for + 2 D respectively. For D + , the cross section displays a broad peak in the low electron energy range, which is attributed to resonant dissociative excitation (RDE). The threshold measured at (14.0±1.0) eV is in good agreement with the vertical excitation energy calculated for direct dissociative excitation [13]. In order to obtain a more comprehensive understanding of the isotope effect on electron-ion collisions in the hydrogen family, we present here the experimental study of the electron-impact dissociation of the isotopologue D 2 H + . Results are reported for the production of the H + , D + , HD + and + 2 D fragment ions. Dissociation of this heteronuclear molecular target proceeds via ten possible reaction channels:

D 2 H + + e -H + + D 2 + e - (1)
H + + D + D + e - (2)
D + + HD + e - (3)
D + + D + H + e - (4)
HD + + D + e - (5)
+ 2 D + H + e - (6)
D 2 H + + e -HD + + D + + 2e -

H + + + 2 D + 2e - (7)
H + + D + + D + 2e - (8)
D + + D + + H + 2e - (9)
Dissociative excitation (DE) processes are represented by reactions (1-6) and the dissociative ionization (DI) ones by reactions (7)(8)(9)(10). In the present experiment, individual ionic fragments result from both DE and DI reactions. A specific procedure has been developed to separate the contributions and absolute cross sections are reported separately for DE and for DI, from their respective thresholds up to 2.5 keV [14].

Experimental method and apparatus

The animated crossed electron-ion beams method is applied [15]. The molecular ion beam of well-defined energy (a few keV) interacts at right angles with the electron beam whose energy is tuned from a few electron volts up to 2.5 keV. Product ions are separated from the primary ion beam by using a double focusing 90°magnetic analyzer. Product ions are further deflected by a 90°electrostatic spherical deflector and directed onto the channeltron detector [10].

The electron beam is swept across the ion beam in a linear motion at a constant speed u. The total number of events K produced during one complete electron beam movement is related to the measured cross section σ m by ( )

e i i m e i e i v v q e uK I I v v σ γ = + (11)
In this expression, γ is the detector efficiency, I e and I i , e and q i e, v e and v i , are the electron and ion beam current intensities, the charges and velocities of electrons and ions, respectively.

Assuming m i >> m e , the interaction energy E (eV) is given by:

( )

e e i i e i m E V q V V m = + - (12)
where V e and V i , m e and m i are the acceleration voltages and masses of electrons and target ions, respectively.

Due to the transfer of internal potential energy, dissociation fragments exhibit both broad velocity and broad angular distribution in the laboratory frame. The angular acceptance of the magnet analyzer allows the total transmission of the angular distribution of product fragments emitted at a given velocity v in the laboratory frame. In order to put the cross section on absolute scale, the velocity distribution is computed and the total cross section σ is obtained by integrating this distribution over the entire velocity range.

The sum of the kinetic energy released to the dissociation fragments is represented by ( )

E KER .
1 ( ) 1 ln b th E E a e c E E E σ = × - × × + × ( 14
)
where e is Euler's constant, a, b and c are fitting parameters and E th is the threshold energy.

The total uncertainty (90% confidence limit) for the absolute cross sections is estimated to be about 10%, at maximum, and that associated to the electron energy is estimated to be ±0.5 eV [14].

Results and discussion

Absolute total cross sections for the production of H + , D + , HD + and + ) that is found to be (2.13±0.06)×10 -17 cm 2 at 75 eV. For energies above 50 eV, the DI contribution is observed to be noticeably higher than that of DE.

The H + appearance energy at (5.5±0.5) eV fairly agrees with the energy threshold of reaction ( 1) (H + +H 2 +e -, E th = 4.4 eV, Table 1) and with the experimental value of (4.5±0.5) eV determined for dissociative excitation of + 3 D [10]. The cross section displays a broad peak in the region between 5 and 11 eV, centered on 8 eV, with a maximum value of 1.2×10 -17 cm².

The presence of this peak is attributed to resonant dissociative excitation (RDE) because it is observed below the threshold for direct dissociative excitation (DDE). Reaction (

(H + +D+D+e -, E th = 8.9 eV, Table 1) also contributes to RDE in the H + channel and its input is included in the present measurements. For a C 2v symmetry structure, the 2 A 1 excited electronic state of H 3 (dissociation limit: H 2 (X 1 g + Σ )+H(n=2)) crosses the X 1 A 1 fundamental electronic state of + 3 H (dissociation limit: H 2 (X 1 g + Σ )+H + ) at H 2 -H bond distance of 2.6 a 0 [16].

In the Franck-Condon region, the 2 A 1 potential energy surface of the excited H 3 neutral lays about 6 eV above that of the ground state of the + 3 H ion (X 1 A 1 , v= 0) [16]. The low experimental threshold for H + formation can be attributed to an initial capture of the electron into this excited state of H 3 which autoionizes resulting in the dissociation into H 2 +H + fragments. In addition to the vibrational population and to the resonant capture leading to the low RDE threshold, the large width of the Franck-Condon region associated with the excited vibrational levels must also be taken into account.

The cross section exhibits a second threshold at (11.0±1.0) eV, which is pointed out by the change of slope at this energy. This result is observed to be below the vertical excitation energy calculated for the dissociative excitation of + 3 H via the first excited triplet state 1 3 E' at 14.75 eV [13]. The signal measured for energies higher than this second threshold is attributed to non-resonant dissociative excitation. Excitation might also occur from excited vibrational levels populated within the 1 ' 1 X A ground state that would contribute to enlarge the Franck-Condon region and to lower the thresholds. The peak observed below 10 eV for H + is the signature of resonant dissociative excitation. As for predissociation, there is no evidence of its role in the present experiment, above 10 eV. The D + appearance energy (10.5 eV) is observed to be lower than the direct DE threshold (1 3 E', 14.75 eV) but the shoulder that is merely discernible around (15±1) eV is expected to correspond to the energy threshold of the DE process. A plausible assumption for the observed energy threshold is that excitation occurs from excited vibrational levels populated within the ground state.

HD + channel

Figure 2(c) shows results for the formation of the HD + fragment together with the DE contribution (HD + +D). Experimental conditions were favorable enough to allow DE and DI contributions to be obtained separately for HD + [14]. Figure 2(e) shows results for the corresponding dissociative ionization contribution (HD + +D + , reaction ( 7)). The HD + appearance energy threshold is measured to be (8.5±0.5) eV, which is higher than the expected one ( + 2 H +H+e -, E th = 6.2 eV). The DE contribution (reaction ( 5)) is measured to be noticeably higher than that of DI below 40 eV. The shape of the DE cross section is very sharp and its maximum is measured to be (3.12±0.20)×10 -17 cm 2 at 14 eV.

The dissociative ionization threshold is observed around (15.0±1.0) eV and the maximum cross section is found to be (2.03±0.04)×10 -17 cm 2 at 75 eV. Such energy is too low to be consistent with a threshold for a direct transition in the Franck-Condon (FC) region corresponding to a DI process and it is also too low to be attributed to the H + + + 2 H +2e - reaction (E th = 19.8 eV, Table 1). A distinctive shoulder is discernible at (35.0±1.0) eV, which is expected to correspond to the energy threshold of the DI process. From a full configuration interaction calculation, Gorfinkiel and Tennyson [20] determined the vertical ionization threshold of + 3 H to be 33.47 eV. D appearance energy threshold (8.5±0.5 eV) is the same as that of HD + formation and the shape of the DE cross section is sharp as well, its maximum is measured to be (2.99±0.10)×10 -17 cm 2 at 18 eV.

The threshold energy for the dissociative ionization process is observed to be (18.0±1.0) eV which is reasonably close to the predicted value (H + + + 2 H +2e -, E th = 19.8 eV).

The DI maximum cross section is found to be (2.03±0.04)×10 -17 cm 2 at 75 eV. As it is the case for the HD + fragment, the DI threshold energy is inconsistent with the threshold for a direct transition in the Franck-Condon region (33.47 eV). Nevertheless, a distinctive shoulder is also observed at (35.0±1.0) eV, corresponding to the energy threshold of the direct DI process. Present result indicates that there is no clear isotope effect above 30 eV. Carrington et al [17] studied the photodissociation of excited D 2 H + by recording H + and D + fragment separately.

They observed that predissociation results in the formation of H + rather than D + because the ratio of corresponding signals was equal to three. This result was later corroborated by the theoretical predictions of Pollak and Schlier [18]. Conversely, Badenhoop et al [19] demonstrated that the D This argument is supported by the classical potential scattering description [START_REF] Bransden | Physics of Atoms and Molecules[END_REF]: the trajectory of a particle (mass m) in the field of a central potential U(r) is confined to a plane and the total energy of the effective motion is a constant:

( )

2 2 2 1 2 2 dr L E m U r dt mr = + + ( 15
)
where r is the position of the particle, L is the angular momentum and the term L 2 /2mr 2 corresponds to the centrifugal potential barrier. Equation ( 15) can be solved for ( )

1/ 2 2 2 2 2 dr L E U r dt m mr = - + (16)
This can be inverted and integrated over r, to give the time length t of the motion:

( ) 2 2 2 2 m dr t E U r L mr = - + (17)
The motion is restricted to the allowed region determined by the range over which the argument of the square root is positive. For two isotopes particles (masses m 1 and m 2 )

performing the same path in the field of a given potential, it comes finally

2 2 1 1 t m t m = (18)
This purely classical reasoning indicates that, among isotopic species in the same kinematical conditions (potential and trajectory), the ejection of the lightest particle should be favored. This tendency is confirmed for RDE (H + over D + ); for the DE diatomic ions + 2 D and HD + , associated with ejection of H and D, respectively and for DI; but not for H + and D + produced via DE. is found to be 1.25±0.04. Both ratios are higher than unity, which indicates that the production of light ions is dominant, whatever the target may be. This effect is even more pronounced for The DE cross section for the atomic ions (H + +D + )/D 2 H + is the sum of the DE cross sections for H + or D + formation, its maximum is found to be (4.3±0.

Conclusion

Absolute cross sections for electron-impact dissociation of D 2 H + into H + , D + , HD + and + D fragments have been measured in the energy region from their respective thresholds to 2.5 keV in the crossed electron-ion beams experiment. The present results for D 2 H + are compared with those measured for + 3 D [10]. The analysis of the results brings into evidence significant isotope effects, which are supported by a purely classical reasoning. This reasoning indicates that, among isotopic species in identical kinematical conditions, the ejection of the lightest particle is favored. This tendency is first confirmed for the RDE process, which is observed in the low energy range (5-11eV) for H + /D 2 H + only and not for any of the other fragments.

Moreover, its contribution is found to be much smaller than the D + / + 3 D , observed in the previous experiment [10]. For DE, the tendency is confirmed for the molecular products +

3 H and + 3 D [ 6 ]

 336 . The deduced vibrational populations are wide, showing a bell-shape-like behavior (centered on v= 5-7) covering all possible vibrational states up to v= 15 or even v= 20. For D 2 H + , only the vibrational states within the electronic ground state of the fragments are energetically accessible in both channels: the available final states are D 2 (X 1 Σ g (v))+H(1s) with v= 0 to 20 and HD(X 1 Σ g (v))+D(1s) with v= 0 to 17, respectively. The shape of the distribution is independent of the isotope and the shift in the position of the most populated quantum state reflects the different vibrational level spacing of the species. In contrast to homonuclear + heteronuclear D 2 H + should have lower internal excitation because the latter possesses an electrical dipole moment, allowing a much faster radiative decay of the rotational excitation. Diatomic fragments formed in electronically excited states are not expected to remain bound because of strong predissociation and because of radiative decay to the 3 Σ u dissociative state. The decay of excited singlet and triplet states tends hence to contribute to three-body fragmentation. Electron-impact experiments on deuterated molecular ions are currently performed in our laboratory by means of a crossed electron-ion beams set-up. Absolute cross sections and kinetic-energy-release distributions have been published for dissociation of the homonuclear + 3

For + 2 D 3 D 2 D 3 H

 2323 , no isolated RDE peak can be clearly observed and the non-zero cross section below the direct DE threshold reflects the width of the Franck-Condon region of the + ground state. Dissociative excitation cross sections (σ DE ) as well as dissociative ionization cross sections (σ DI ) have been separately obtained for both fragments. Dissociative ionization thresholds are observed at (11.0±0.5) eV and at (12.0±0.5) eV, for D + and + production, respectively. Both thresholds are observed to be analogous, around 11.5 eV, which is too low to be consistent with a Franck-Condon transition. A shoulder is discernible in the cross section at about 30 eV and it is expected to correspond to the energy threshold of the DI process because it is close to the vertical ionization threshold of + (33.47 eV).

2 D are grouped in figure 1 ( 2 D 2 D 2 D 3 H 2 D 2 D

 21222322 figure 2(e). Energy thresholds for dissociation of + 3 H and of D 2 H + are listed in table 1 and absolute total cross sections are given in table 2, together with the associated total uncertainties. The following cross section ratios are presented in figure 2(f) in order to search for an apparent fingerprint of isotope effects: 2[H + +D 2 ]/[D + +HD] (DE for atomic ions), 2[ + 2 D

Figure 1

 1 Figure1and Table1

Figure 2 2 D+ 2 D 2 D

 2222 Figure 2(d) shows results for the formation of the + 2 D fragment together with the DE

Figure 2 3. 5 . 2 D

 252 Figure 2

Figure 3 3 3 D 3 D 2 D 2 D

 33322 Figure 3

the + 3 D 3 D 2 D (for the + 3 D 2 D

 33232 target than for the D 2 H + one.Results for electron-impact dissociation of D 2 H + are compared with those for the + D target[10], for DE and DI separately. The DE contributions to atomic ions and to diatomic ions production are compared in figure4(a) and 4(b), respectively. For DI, the cross sections for + are compared with those for D 2 H + in figure4(c). The corresponding cross section ratios figure 4(d) in order to examine the behavior of the two isotopologues. X + stands for the production of D + or + target), while Y + stands for the production of (H + +D + ) or (HD + + + 2 D ) (for the D 2 H + target). The following cross section ratios have been estimated: [D + ]/[H + +D + ] (DE for atomic ions), [ +

3 D 3 D 2 D 3 D

 3323 , respectively. The ratio of the DI cross sections is found to be close to 1 over the whole energy range, it is measured to be (1.07±0.05) on average.To summarize the comparison between the + and D 2 H + targets, the strongest isotope effect is observed for the dissociative excitation into the light ions ([D + ]/[H + +D + ], 2.7), followed by dissociative excitation into the heavy ions ([ + ]/[HD + + + 2 D ], 1.4) and finally no effect is observed for dissociative ionization (isotope ratio close to 1). The electronic potential energy curves are supposed to be identical for the + 3 D and D 2 H + targets under the Born-Oppenheimer approximation and neglecting relativistic effects.But the vibrational and rotational manifolds are supposed to be different for each isotopologues, which corresponds to isotope effects. In the case of D 2 H + , the symmetry is broken because of the mass difference of the H and D nuclei. It causes the center of mass to shift away from the center of charge, creating a dipole moment, which is absent in the homonuclear molecules. Present sets of cross sections allow the quantification of the nonnegligible isotope effects for D 2 H + and + which have to be related to these theoretical statements.

Figure 4

 4 Figure 4

2 D 3 D

 23 and HD + , associated with ejection of H and D, respectively, but not for H + and D + which are found to be of the same importance. Finally, the ejection of the lightest particle is also evident for DI. Present DE cross sections for D 2 H + are found to be significantly lower than those measured for + , although those for DI agree well together. Molecular dynamics treatments of these reactions should enlighten the present experimental results.

Figures and captionsFigure 1 . 2 D 2 )Figure 2 . 2 D 2 D 2 D 2 D

 12222222 Figures and captions

2 )Figure 3 . 3 D 3 D 2 D 2 )Figure 4 . 3 D

 23332243 Figure 3. Absolute total cross sections (DE+DI) versus the electron energy (a) for + + 3 D D ( ) and for ( )+ + + 2H +D D H (×),

  Above the ionization threshold, the spectrum becomes broader because of the Coulomb repulsion experienced by DI fragments. For the H + , D + , HD + and + 2 D fragments, the mean kinetic energies are measured to be 4.2, 4.7, 4.1 and 2.9 eV and 9.2, 6.4, 8.6 and 8.3 eV, for DE and for DI, respectively. The maximum corresponding uncertainty is of ±10%. The DI signal is isolated by fitting the broader part of the spectrum. Absolute values of DE cross sections σ DE are obtained by subtracting the DI contribution σ DI from the total absolute cross section σ for each fragment[14]. To help visualize dissociative contributions, corresponding cross sections are reproduced using linear combinations of the analytic expressions which are based upon a Bethe-Born form:

	By assuming the fragmentation of the target to be binary only and by applying the
	momentum conservation, this sum is given by
	E	KER	=	2 2 2 m w µ	(13)
	where w represents the fragment speed in the centre of mass frame. Ionic products basically
	form two velocity distributions whose shapes depend on the various E KERs involved. At low
	energies, only DE is observed and the spectrum is narrow, which corresponds to low E KERs .

  Figure 2(a) shows the results for the formation of the H + fragment together with the DE contribution (H + +D 2 , or H + +D+D). The DI contribution attributed to the H + channel is that of

	the + 2 D fragment (H + + + 2 D +2e

3.1. H + channel -, reaction (8)). Because of unfavorable signal-to-noise ratio it

has not been possible to obtain the DI contribution directly from the H + channel. The maximum DE cross section (reaction (1) or (

2

)) is measured to be (1.98±0.15)×10 -17 cm 2 at 35 eV, which is close to the maximum DI cross section (H + + + 2 D

Table 2 3.2. D + channel

 2 Figure 2(b) shows results for the formation of the D + fragment together with the DE contribution (D + +HD, or D + +D+H).As for H + , unfavorable signal-to-noise ratio prevented us from measuring directly the DI contribution for the D + channel so the DI contribution attributed to D + is actually that of HD + (HD + +D + +2e -, reaction(7)). The DI contribution is almost as broad as that of DE and the maximum DI cross section is found to be

	e -+ T +	T** F + + neutrals + e -	(RDE)
	where the T** state is located just above the corresponding dissociation limit. F stands for
	fragment (H + , + 2 H and isotopologues) and T stands for target ( + 3 H and isotopologues).
	Secondly, the DE process may occur via a direct transition to a T + * (d) dissociative excited
	state of the target (direct dissociation, DDE). Lastly, the DE process may also occur via a
	transition to a T + * (b) bound metastable excited state that is coupled with a T + * (d) dissociative
	excited state (predissociation, PDE):
	e -+ T +	T + * (b) + e -	T

(2.03±0.04)×10 -17 cm 2 at 75 eV. The DE contribution (reaction (3) or (

4

)), which is measured to be larger than that of DI over the whole energy range, is especially broad by comparison to the DE contributions measured for the three other fragments. It exhibits a plateau between 30 eV and 95 eV and its maximum cross section is measured to be (2.59±0.10)×10 -17 cm 2 at 75 eV.

The D + appearance energy threshold that is measured to be (10.5±0.5) eV almost coincides with the direct DE threshold for H + (11 eV), but it is 5 eV above the corresponding RDE threshold for (5.5 eV). It indicates that, for D 2 H + , no RDE contribution is detected in the D + channel, unlike what is observed for H + (within the experimental uncertainties). In the dissociation of + 3 D leading to the D + product, a RDE contribution is apparent in the 4 to 12 eV energy region

[10]

. There is a strong isotope effect in the dissociation of D 2 H + favoring the H + channel over the D + channel in the low energy range below 10 eV. Firstly, the DE process yielding the F + fragment may occur via the capture of the incident electron into a T** doubly excited autoionizing dissociative state (resonant dissociative excitation, RDE): + * (d) + e -F + + neutrals + e - (PDE)

  HD + +D + ], about 2) and finally no isotope effect is observed for dissociative excitation into the light ions (2[H + +D 2 ]/[D + +HD], about 1). In the low energy range (below 10 eV), a strong resonant process is observed for the H + channel only, indicating for a strong isotope effect favoring the H + channel over D + via RDE (see section 3.2). As for the ionization process, the corresponding potential energy curves are steep due to the Coulomb effect, so the ejection of the lightest fragment is

	+H]/[HD + +D] is measured to be (2.8±0.4) on average in the 30-1000 eV range. The isotope
	ratio of the DI cross sections 2[ + 2 D +H + ]/[HD + +D + ] is observed to be almost constant in the
	50-2500 eV, it is measured to be (2.1±0.1) on average. It underlines that an isotope effect promoted and ( + 2 D +H + , reaction (8)) dominates over (HD + +D + , reaction (7)), in agreement
	takes place above 50 eV, favoring the + 2 D channel over the HD + channel. Statistically the with the present observation.
	HD + +D + channel is 2 times more favorable than the + 2 D +H + channel. Therefore, if there were
	no isotope effect, the ratio 2[ + 2 D +H + ]/[HD + +D + ] would be equal to 1.
	To review the study of the isotope effect in the dissociation of D 2 H + in the 50-2500
	eV range: the strongest isotope effect is observed for the dissociative excitation into the heavy
	ions (2[ + 2 D +H]/[HD + +D], about 3), followed by dissociative ionization (2[ + 2 D
	+H + ]/[For + 3
	region, it is the lowest one leading to + 2 H with the dissociation limit + 2 H ( 2 X g + Σ )+H(2s). On
	the contrary, the curve associated to the 3 1 A state is almost flat and it corresponds to H +
	formation ( * 2 H +H + dissociation limit). Dissociation pathways are such that the potential
	energy decreases and approaches the minimum value associated with the dissociation limit,
	2 H + ions formed from + 2 D (v= 18, 24)+H 2 tend to dissociate in favor of following the principle of minimum energy. The large steepness of the 2 1 A state may explain
	the D + channel. They also demonstrated that the centrifugal barrier for D 2 H + D + +HD would for the relative importance of the ( + 2 D +H) channel and for the concomitant ejection of the
	always be lower than that for D 2 H + H + +D 2 , resulting in the dissociation favoring D + lightest H fragment (reaction (6) dominating over reaction (5)). Considering the lightest
	fragments. Present results do not support any of these predictions because the ratio of the fragment should be promoted, the (H + +D 2 ) channel (1) should take over the (D + +HD) channel
	cross sections 2[H + +D 2 ]/[D + +HD] is really close to 1, so that no channel is promoted. (3) but the present isotope ratio is close to unity. It appears from theoretical studies that a
	For DE yielding diatomic ions, the + 2 D channel is clearly favored over the HD + one,
	underlining another unambiguous isotope effect. The ratio of the cross sections 2[ + 2 D

H , Talbi and Saxon

[21]

have performed calculations on the C 2v geometries (isosceles triangle) as well as on the D 3h geometries (equilateral triangle). The electronic ground 1 ' 1 X A state is bound by 4.6 eV with respect to H 2 (X 1 Σ + )+H + (two-body asymptote) and by 9.3 eV with respect to H(1s)+H(1s)+H + (three-body asymptote). The first excited state 1 1 E in D 3h geometry is degenerate and it splits in 2

1

A and 3

1

A in C 2v geometry. The potential energy curve corresponding to the 2

1

A state is very steep in the Franck-Condon large majority of the channels, both in D 3h and C 2v geometries, yields H + in a way or another, so it becomes difficult to assert which electronic state is more likely to take part in the dissociation process. Nevertheless, the fact that the 3 1 A state is almost flat is coherent with the present observation.

  case because there is no RDE contribution in the D + formation from electronimpact dissociation of D 2 H + . In the 30-2500 eV energy range, the ratio of the cross sections for the atomic ions [D + ]/[H + +D + ] is seen to exhibit the less regular behavior of all the calculated ratios. It is estimated to be (2.7±0.4) on average. Irregularities affecting this ratio

	may be explained by the D + / + 3 D cross sections that are affected by larger uncertainties than
	those for (H + +D + )/D 2 H + . The DE cross section for the diatomic ions (HD + + + 2 D )/D 2 H + is the
	sum of the DE cross sections for HD + or + 2 D formation. The (HD + + + 2 D )/D 2 H + maximum
	cross section, (5.9±0.2)×10 -17 cm 2 at 16 eV, is in reasonable agreement with the maximum
	cross section for + 2 D / + 3 D , (6.3±0.5)×10 -17 cm 2 at 18 eV, which is in opposition to what is
	observed for the atomic ions. Above the maximum, i.e. above 20 eV, the cross sections for
	D 2 H + are observed to be lower than those for + 3 D . The ratio of the cross sections for the
	diatomic ions [ + 2 D ]/[HD + + + 2 D ] is observed to be almost constant in the 15-2500 eV range
	and it is found to be around (1.4±0.2) on average. The DI cross section for D 2 H + is the sum of
	the DI cross sections for HD + or + 2 D formation. Overall, the DI cross section for D 2 H + agrees
	very well with what is measured for + 3 D . The corresponding maxima are (4.16±0.05)×10 -17
	cm 2 and (4.3±0.2)×10 -17 cm 2 at 75 eV, for D 2 H + and +
	2)×10 -17 cm 2 at 35 eV. It
	is a factor of two lower than what is obtained for D + / + 3 D , (9.4±0.9)×10 -17 cm 2 at 55 eV. The
	cross sections for D 2 H + are found to be lower than those for + 3 D over the whole energy range.
	For D 2 H + , the resonant contribution is noticeable below 10 eV, but much less pronounced
	than in the + 3

D

Table 1 .

 1 Threshold energies (eV) for the dissociation of +

			1	Predicted (eV) 2	D 2 H +	Experimental (eV)
	RDE	H + +H 2	4.5 ± 0.5	4.4	H + +D 2	5.5 ± 0.5
					D + +HD	No
	RIP	H + +H+H -		8.1
	DE	H + +H+H		8.9	H + +D+D	11 ± 1
					D + +H+D	10.5 ± 0.5
			14 ± 1	14.75 (1 3 E') 3		15 ± 1
	RIP	+ 2 H +H -		5.8
	DE	+ 2 H +H	6.0 ± 0.5	6.2	HD + +D	8.5 ± 0.5
					+ 2 D +H	8.5 ± 0.5
	DI	H + + + 2 H	11.0 ± 0.5 12.0 ± 0.5	19.8	H + + + 2 D	18 ± 1 ( + 2 D )
					H + +D + +D
	DI				D + +HD +	15 ± 1 (HD + )
		H + +H + +H		22.5	D + +H + +D
				33.47 (FC) 4		35 ± 1 (HD + )
						35 ± 1 ( + 2 D )
				3 H and of D 2 H + .
	1 [10] ( + 3 D target), 2 [23], 3 [13], 4 [20]

Note: RIP (Resonant-Ion-Pair formation) and FC (Franck-Condon)

Table 2 :

 2 Absolute total cross sections (H + , D + , HD + and + 2 D , 10 -17 cm 2 ).

		H	+		D	+		HD	+	D	+ 2
	E (eV)	σ		∆σ	σ		∆σ	σ		∆σ	σ	∆σ
	5	-0.05	0.18
	6	0.46	0.17
	7	1.21	0.18
	8							0.00	0.30	-0.04	0.27
	9	0.96	0.17				0.18	0.26	0.21	0.06
	10				-0.07	0.11	0.87	0.35	0.66	0.09
	11	0.85	0.20	0.18	0.23	2.25	0.17	1.77	0.22
	12				0.65	0.17	2.80	0.26	2.03	0.14
	13	1.24	0.22	1.03	0.23
	14				1.18	0.24	3.12	0.20	2.61	0.29
	15	1.62	0.20	1.99	0.27
	16							3.38	0.18	2.88	0.18
	17	1.83	0.13	2.21	0.09
	18							3.50	0.18	3.01	0.10
	20							3.56	0.15	3.01	0.10
	21	2.26	0.12	3.00	0.16
	22							3.54	0.16	3.04	0.15
	25	2.54	0.15	3.39	0.09
	26							3.47	0.12	3.16	0.09
	30	2.84	0.11	3.94	0.11	3.47	0.12	3.09	0.08
	35	3.03	0.13	4.09	0.15	3.28	0.07
	40							3.19	0.12	3.20	0.12
	45	3.18	0.09	4.37	0.10	3.17	0.11	3.23	0.11
	55	3.21	0.12	4.51	0.08	3.01	0.10	3.33	0.11
	65	3.25	0.08	4.54	0.09	2.93	0.08	3.27	0.09
	75	3.20	0.11	4.62	0.09	2.89	0.08	3.17	0.06
	85	3.18	0.10	4.50	0.09	2.90	0.09	3.27	0.08
	95	3.16	0.06	4.47	0.16	2.80	0.08	3.15	0.04
	115	2.91	0.05	3.82	0.09	2.67	0.10	3.09	0.08
	135	2.83	0.08	3.65	0.09	2.39	0.07	2.87	0.08
	155	2.66	0.09	3.56	0.11	2.28	0.08	2.52	0.06
	195	2.27	0.08	3.14	0.12	1.92	0.06	2.23	0.08
	245	2.05	0.06	2.66	0.13	1.67	0.08	2.07	0.06
	295	1.74	0.07	2.35	0.09	1.47	0.08	1.81	0.06
	395	1.50	0.08	1.95	0.10	1.30	0.08	1.51	0.07
	495	1.32	0.07	1.63	0.09	1.06	0.07	1.30	0.05
	595	1.12	0.05	1.50	0.08	0.92	0.04	1.21	0.07
	795	0.90	0.06	1.14	0.08	0.82	0.06	0.96	0.05
	995	0.83	0.04	1.03	0.06	0.67	0.06	0.84	0.06
	1495	0.64	0.06	0.76	0.05	0.46	0.05	0.55	0.04
	1995	0.53	0.06	0.64	0.06	0.37	0.04	0.52	0.03
	2495	0.44	0.06	0.52	0.06	0.33	0.03	0.44	0.04

Acknowledgements

J. Lecointre acknowledges financial support provided by F.R.S.-FNRS (Belgium). X. Urbain is a Senior Research Associate of the F.R.S.-FNRS (Belgium). J. J. Jureta expresses his gratitude for the partial support of the project 141011 from the Ministry for Science and Technology of the Republic of Serbia. This work was supported by the Association Euratom-Belgian State. We thank the Forschungszentrum Jülich for the lending of the ECR ion source and all the staff members of the IMCN for their assistance in this experiment.


J Trømborg

J Scheibert

D S Amundsen

K Thøgersen

A Malthe-Sørenssen

Transition from static to kinetic friction: Insights from a 2D model

We describe a 2D spring-block model for the transition from static to kinetic friction at an elastic slider/rigid substrate interface obeying a minimalistic friction law (Amontons-Coulomb). By using realistic boundary conditions, a number of previously unexplained experimental results on precursory micro-slip fronts are successfully reproduced. From the analysis of the interfacial stresses, we derive a prediction for the evolution of the precursor length as a function of the applied loads, as well as an approximate relationship between microscopic and macroscopic friction coefficients. We show that the stress build-up due to both elastic loading and micro-slip-related relaxations depend only weakly on the underlying shear crack propagation dynamics. Conversely, crack speed depends strongly on both the instantaneous stresses and the friction coefficients, through a non-trivial scaling parameter.

Frictional interfaces are important in many areas of science and technology, including seismology [START_REF] Scholz | The Mechanics of Earthquakes and Faulting[END_REF], biology [START_REF] Urbakh | [END_REF]4] and nanomechanics [START_REF] Bhushan | Nanotribology and Nanomechanics[END_REF]. Whereas a satisfactory picture of the steady sliding regime of such interfaces has been developed during the last twenty years [START_REF] Persson | Sliding Friction[END_REF][START_REF] Baumberger | [END_REF][8], the dynamics of the transition from static to kinetic friction remains elusive. During the last decade, a renewed interest has grown in such transitions, due to experimental studies that directly measured the local dynamics of frictional interfaces [9][10][11][12][13]. They have shown that macroscopic sliding occurs only after shear crack-like micro-slip fronts have spanned the entire contact interface.

Experimentally, micro-slip front nucleation, propagation and arrest was shown to be controlled by the instantaneous stress field at the interface. Fronts nucleate preferentially at the trailing edge of the contact area [9,11,[14][15][16][17], an effect explained either by the enhanced shear stress near the loading point in side-driven systems [11,14,16,18] or by a friction-induced pressure asymmetry in top-driven systems [9,19]. Fronts can arise well below the macroscopic static friction threshold and arrest before the whole contact area has ruptured [14][15][16]. The length and number of these precursors depends on the precise way in which shear [14] and normal [16] forces are applied. Moreover, precursors are associated with significant changes in the spatial distribution of the real contact area [14], a quantity related to the local interfacial pressure. Finally, the propagation speed of micro-slip fronts, which covers a wide range [9][10][11]20], correlates with the local shear to normal stress ratio at nucleation [17].

Theoretically, some aspects of these observations have been studied using one-dimensional (1D) models. The conditions leading to a large range of front velocities were addressed using a 1D spring-block model with a timedependent friction law [18]. The role of an asymmetric normal loading on the length of precursors was considered using a 1D spring-block model with Amontons-Coulomb (A-C) friction and different normal forces ascribed to different blocks [16]. The dependence of the series of precursors on the friction-induced pressure asymmetry was described, for A-C friction, using a quasi-static 1D model [19]. In these three studies, the normal stress distribution was assumed to be uniform [18] or linear (either fixed [16] or friction-dependent [19]). Such assumptions impede quantitative comparison with experiments since determination of the actual stress field requires solution of the elastic problem for the two bodies in contact, including their precise geometry, elastic properties and boundary conditions, not only at the frictional interface but also on all their other boundaries. As a first step towards such complete description, we present a minimal 1+1D (along + orthogonal to interface) model for the transition from static to kinetic friction of an elastic slider on a rigid substrate. We show that this model, by enabling realistic boundary conditions, is sufficient to reproduce a series of still unexplained experimental observations. We used the 2D spring-block model sketched in Fig. 1(a). The slider has mass M and sizes L and H in the horizontal (x) and vertical (z) directions, respectively. It is divided along a square lattice into N = N x N z blocks of mass m = M/N . Blocks are coupled to their four nearest neighbours and their four next-nearest neigh- Typical loading curve, from static contact to macroscopic stick-slip, for setup 1 (h=5mm, L=140mm, FN =3500N). Green: Locations of slipping regions for all micro-slip event. 0 (1) corresponds to the trailing (leading) edge.

V a F N z x p 1 p 2 p 3 f 1 f 2 f 3 0 0.4 0.
bours by springs of equilibrium lengths l = L/(N x -1) = H/(N z -1) and √ 2l and stiffnesses k and k/2, respectively, giving an isotropic elastic model with Poisson's ratio 1/3. The force exerted on block i by block j is thus k ij (r ij -l ij ) ∆xij rij when blocks are connected, 0 otherwise, where x = (x, z), ∆x ij = x jx i , r ij = |∆x ij | and k ij and l ij are the stiffness and equilibrium length of the spring connecting blocks i and j. Block oscillations are damped using a viscous force η( ẋjẋi ) on the relative motion of connected blocks. We chose the coefficient η = √ 0.1km so that blocks are underdamped and eventtriggered oscillations die out well before the next event. All η satisfying these conditions gave similar results.

Boundary conditions were designed to model two different experiments described in the literature: [14] (setup 1) and [16] (setup 2). They differ in the way the top blocks are loaded. For setup 1, they are glued to a rigid rod of mass 75.6g, itself submitted to a normal force F N and coupled to a soft "spring mattress" of stiffness 0.4MN/m, the effect of which is modeled with a restoring torque proportional to the rod's tilt angle. For setup 2, the top blocks are submitted to a linear time-independent distribution of vertical forces FN Nx 1 + 2i-Nx-1 Nx-1 θ , where θ ∈ [-1, 1] controls the pressure asymmetry. In both setups, the bottom blocks lie on an elastic foundation of modulus k f = k, i.e. each block is submitted to a vertical force of amplitude

p i = k f |z i | if z i < 0 or 0 oth- erwise, where z i is the vertical displacement of block i.
All k f > k gave similar results, so that the substrate can be considered rigid compared to the slider. Both vertical boundaries are free, except for a horizontal driving force F T = K(V t -x h ) applied on the left-side block situated at height h above the interface, where x h is the x-displacement of this block. This models a pushing device of stiffness K driven at a small constant velocity V . The amplitudes f i of the friction forces on the bottom blocks follow the minimalistic local A-C friction law with static and kinetic friction coefficients µ s and µ k < µ s . If ẋi = 0, f i balances all horizontal forces on block i up to µ s p i , at which slip initiates; then f i = ±µ k p i when ẋi ≶ 0. The 2N equations of motion are solved simultaneously using a fourth order Runge-Kutta integrator on a uniform temporal grid of resolution ∆t.

Using this model, we simulated the transition from static to kinetic friction for the various loading configurations reported in [14] and [16]. Model parameters, chosen in accordance with experiments [21], are given in the caption of Fig. 2. Figure 1(b) (blue) shows a typical loading curve for setup 1, from static contact up to macroscopic stick-slip. As in 1D models, and in agreement with experimental results, we observe, well before macroscopic sliding, a series of partial force relaxation events, corresponding to precursors, which all nucleate at the trailing edge, and extend over increasing lengths L p (Fig. 1(b), green). We first focus on the dependence of L p on the applied tangential force F T just after relaxation. We simulated, for setup 1, different slider lengths L, pushing heights h and normal forces F N . The behaviour under different conditions differ by the number of precursors occurring along the transition: the increase in both L p and F T between precursors scales almost linearly with h for small h, so that larger L and smaller h yield more precursors. However, Fig. 2(a) shows that the results for all conditions can be collapsed on a single curve by plotting L p /L as a function of F T /F N . The same collapse was found in [14], with a very similar shape for the non-linear increase of L p /L with F T /F N . In particular, we reproduce the transition from a roughly linear increase up to L p /L ∼0.5 to a more rapid growth for longer precursors. We emphasize that a 1D model with homogeneous normal loading would produce a purely linear increase. In Fig. 2(b) we compare, for setup 2, the evolutions of L p /L as a function of F T /F N for three different linearly asymmetric normal loadings of the slider. Qualitatively, the lower the normal load on the trailing edge, the lower the threshold force required to nucleate precursors, and therefore the lower the tangential force F T necessary for the precursor to reach a given length L p , which explains the relative positions of the three curves in Fig. 2(b). Again, the simulated curves are in excellent agreement with the experimental results in Fig. 8 in [16], much better than the 1D simulation (Fig. 13 in [16]). These non-trivial results for both setups clearly demonstrate that, by enabling realistic boundary conditions, 2D models do offer significantly improved agreement with experiments.

We now use our simulation results to gain insight into how shear cracks nucleate, propagate and arrest at an interface obeying A-C friction. In particular, we will ex- amine the role played by the evolution of the shear and normal stress distributions at the interface. We emphasize that, in 2D, the latter is a result of the simulation, not an assumption, which is required in 1D. In Fig. 3(a) we show the typical stress evolution for setup 1. Initially (F T = 0) the normal stress p(x) is symmetric, with edge effects related to the flat punch geometry of the contact. The shear stress τ (x) is antisymmetric, due to friction-frustrated Poisson expansion. These stresses are in excellent agreement with those expected from contact mechanics and those measured in [17]. Application of a tangential force at the trailing edge modifies slightly the normal stress field and modifies significantly the shear stress field, with a large increase near the trailing edge over a distance of order h. The local slipping threshold τ (x) = µ s p(x) is therefore reached first near the trailing edge and a micro-slip front corresponding to the first precursor nucleates there. It then stops after propagation over a finite distance L p and the whole scenario is repeated until the leading edge is reached. Figure 3(a) illustrates this scenario for a typical precursor event.

We find that the shear stress τ (x) just after a precursor is always very close to µ k p(x) over the whole slipped length x ≤ L p . This shows that the arrest state of the interface is only weakly dependent of the static friction coefficient µ s , and is controlled primarily by the kinetic friction coefficient µ k . We emphasize that this behaviour is not specific to A-C friction, but remained true for slipweakening friction, provided the weakening distance is smaller than a few µm. Based on this robust behaviour, we propose the following procedure to predict, for given loading conditions and a given µ k , the non-linear evolution of L p with F T : We run the model for equal friction coefficients (µ s = µ k ). The transition to kinetic friction in this simplified model is smooth, with a continuously growing micro-slip region the length of which is shown as a function of the applied force in Fig. 2 (solid lines). For both setups and for all loading configurations, this curve is in good quantitative agreement with the curve for the length of precursors as a function of the arrest force. Such agreement is due to the fact that any arrest state in the full model is very similar to the state reached in the simplified model for the same force F T : (i) in the microslip region τ (x) = µ k p(x) and (ii) in the stuck region τ (x) arises from almost identical boundary conditions. Because internal dynamics are fast compared to changes in external loading, the prediction is V -independent and be obtained using simple equilibrium calculations, i.e. it does not require a complete dynamical simulation.

The very last precursor in the full model propagated over almost the entire interface and left a shear stress that was equal almost everywhere to µ k p, yielding a total shear force F T ≃ µ k F N . The last increment of tangential force required to trigger the first system-sized event brings the vicinity of the trailing edge to its threshold, whereas the shear stress on the rest of the interface is essentially unchanged, yielding a maximum total shear force only slightly above µ k F N (see Fig. 1(b)). This maximum force is classically interpreted as µ macro s F N with µ macro s the macroscopic static friction coefficient. Therefore our results suggest that, in side-driven systems, whatever the value of µ s , µ macro s ≃ µ k . The difference between the macro-and microscopic static friction coefficients, already discussed in recent 1D models [16,19], provides a possible explanation for the anomalously high values of µ s compared to µ macro s reported in [17]. Not only the shear stress τ but also the normal stress p is altered along the transition. Figure 3(b) shows p(x), normalized by the initial distribution (at F T = 0), after each of the successive precursors. The normal stress is found to be significantly decreased along the path of the precursor that just occurred, by up to 30% whereas, apart from edge effects, it is mainly unchanged in front of it. The normal stress then assumes a reproducible distribution in the macroscopic stick-slip regime (Fig. 3(b), inset). Recalling that, at normally loaded rough contact interfaces, normal stress is robustly found to be locally proportional to the area of real contact, the results of Fig. 3(b) show very good agreement with measurements of the real area of contact reported in [14] (Fig. 4a of [14]). Direct quantitative comparison is not possible, mainly because normal stress has a constant integral (normal force F N imposed) whereas the total real area of contact is not a conserved quantity, but typically decreases by 20% across the transition [14,[START_REF] Ben-David | [END_REF].

Let us now focus on the propagation dynamics of the simulated frictional shear cracks. Figure 4(a) shows the evolution of the local crack speed v c as a function of position x along the interface for a typical precursor event in setup 1. v c (x) decreases from around 2400m/s near the trailing edge to around 400m/s just before crack arrest and appears to be strongly correlated to the shear to nor- mal stress ratio τ (x)/p(x) (Fig. 4(a)). In fact, when v c is plotted as a function of τ /p, all points corresponding to all locations x along all successive events in a given simulation collapse on a single curve (Fig. 4(b), inset). Different geometries or loading conditions yield the very same curve but different friction coefficients yield different curves (Fig. 4(b), inset). All these curves can then be collapsed on the same master curve when v c (x) is plotted as a function of the non-trivial parameter τ (x)/p(x)-µ k µs-µ k , which represents the local distance to the slipping threshold and extends existing parameters [START_REF] Scholz | The Mechanics of Earthquakes and Faulting[END_REF]23] to spatial heterogenities in normal stress p(x). These results are a generalization, accounting for any value of the microscopic friction coefficients, of a similar collapse obtained for experimental data, from system-sized events only, using τ /p as a parameter (Fig. 3 in [17]). Our results also suggest that, if probed, precursors would follow the same experimental curve as system-sized events.

In surprising contrast with the excellent agreement found with experiments up to now, the master curve of Fig. 4(b) exhibits strong discrepancies with its experimental counterpart. The shape as well as the explored ranges of both v c and τ /p are different. In particular, our model does not produce very slow micro-slip fronts like those observed in various experiments [17,20]. Most likely this is because the minimalistic friction used here (A-C) lacks some time-dependent ingredient necessary to yield slow fronts, like those in [18].

All our results suggest two distinct levels of description of the transition from static to kinetic friction. First, a kinematic description of (i) the slow evolution of interfacial stresses between events and (ii) the stress condi-tions at crack nucleation and arrest. Second, a dynamic description of the fast propagation of micro-slip fronts along the interface. Crack dynamics appear to depend crucially, via the friction law, on the kinematic stresses at crack initiation. Conversely, the kinematic description was found essentially independent of the underlying dynamics: we could successfully reproduce all available experimental kinematic results, even with an unrealistic friction law. The key observation yielding this surprising success is that, at crack arrest, τ (x) = µ k p(x) over the whole slipped region. Since the other 2D boundary conditions on the slider could be accurately taken into account, our results strongly suggest that a very similar arrest condition holds in the experiments. We thus believe that any friction law leading to such arrest state will produce as good kinematic agreement as A-C friction, irrespective of the dynamic way of reaching this arrest state.

  FIG. 1. (a): Sketch of the 2D spring-block model. (b) Blue:Typical loading curve, from static contact to macroscopic stick-slip, for setup 1 (h=5mm, L=140mm, FN =3500N). Green: Locations of slipping regions for all micro-slip event. 0 (1) corresponds to the trailing (leading) edge.

F

  FIG. 2. Normalized precursor length Lp/L as a function of normalized shear force FT /FN . (a) Setup 1 for various configurations: L=140 or 200mm, h ranges between 2.5 and 15mm, FN =1750, 2700 or 3500N, H=75mm, K=4MN/m, M =75.6g, V =0.7875mm/s, l=2.5mm, k=13.5MN/m, µs=0.7, µ k =0.45, ∆t=0.2µs. (b) Setup 2 for θ=0.833 (+), 0 ( ) or -0.833 (▽), L=100mm, H=20mm, K=0.8MN/m, M =12g, V =0.45mm/s, FN =400N, h=2mm, l=1mm, k=9.375MN/m, µs=0.7, µ k =0.45, ∆t=0.09µs. Vertical dashes: µ k . Solid lines: prediction obtained with µs = µ k .

2 FIG. 3 .

 23 FIG. 3. Spatial distribution of stresses at three stages of the transition. Dotted lines: initial configuration (FT =0). Dashed (solid) lines: at crack initiation (arrest) for the 9th precursor shown in Fig. 1(b). Blue: τ . Black: µsp. Red: µ k p. Square (disk): nucleation (arrest) point. (b) Spatial distribution of p, normalized by the initial distribution, after each of the precursors shown in Fig. 1(b). Inset: Similar distribution for system-sized events during macroscopic stick-slip.

FIG. 4 .

 4 FIG. 4. (a) Local crack speed vc (solid line, spatial average on 5 neighbouring blocks) and local shear to normal stress ratio τ /p (dashed line) as a function of crack length x for the 11th precursor event shown in Fig. 1(b). (b) Inset: vc as a function of τ /p. Runs: 1 (left), 2 (center), 3 and 4 (right). Main: vc as a function ofτ /p-µ k µs-µ k . Runs: 1 (▽), 2 (⋄), 3 (△), 4 ( • ). The longitudinal (shear) wave speed is 1677m/s (968m/s). Run 1: L=140mm, FN =3500N, h=5mm, µs=0.5, µ k =0.2. Run 2: L=140mm, FN =3500N, h=5mm, µs=0.55, µ k =0.3. Run 3: L=140mm, FN =3500N, h=7.5mm, µs=0.7, µ k =0.45. Run 4: L=200mm, FN =2700N, h=10mm, µs=0.7, µ k =0.45.

  L=140mm, FN =3500N, h=5mm, µs=0.55, µ k =0.3. Run 3: L=140mm, FN =3500N, h=7.5mm, µs=0.7, µ k =0.45. Run 4: L=200mm, FN =2700N, h=10mm, µs=0.7, µ k =0.45.

	τ /p-µ k µs-µ k . Runs: 1 (▽), 2 (⋄), 3 (△), 4 ( • ). The longitudinal (shear) wave speed is 1677m/s (968m/s).
	Run 1: L=140mm, FN =3500N, h=5mm, µs=0.5, µ k =0.2.
	Run 2:

We thank J. L. Vinningland and M. Dabrowski for discussions, and G. Debrégeas and P. Meakin for comments on the manuscript. We acknowledge funding from the European Union (Marie Curie grant PIEF-GA-2009-237089). This paper was supported by a Center of Excellence grant to PGP from the Norwegian Research Council.