Keywords: comparative study, labour market, advertisements for job offers, intermediaries, recruitment, IT workers. JEL-Code: J44, J41, D83, D4

The British labour market is distinguished by a stronger presence of job placement agencies than the French market. We analyse this difference by pointing out the diverse roles played by labour market intermediaries in each country. Theoretically, we elaborate a framework in order to point out that the role played by intermediaries is linked to the nature of job matching processes. Empirically, we compare two samples of advertisements for job offers found in the IT sector. In Great Britain the considerable involvement of specialised recruitment agencies, which contribute to making information transparent, facilitates standard job matching processes. On the contrary, the asymmetry of information, to the applicant's detriment, noticed in French job offers and the role of pre-selection played by the intermediaries are consistent with more specific matches.

Introduction

By introducing the hypothesis of imperfect information, [START_REF] Stigler | Information in the Labour Market[END_REF] helped to renew in-depth analysis of the labour market dynamics. Nevertheless, even today very few authors deal with the role of intermediaries within the recruiting process of the labour market, and this, in spite of the greater role of private employment agencies. In studies dedicated to different recruiting channels and their efficiency, these intermediaries are typically described as "black boxes"(formal vs. informal channels).

Furthermore, there are very few cross-national comparative studies that focus on the interface between the labour market dynamics and the structure of recruiting channels. This structure itself depends on different institutions that formalise recruitment practices in each country, for instance the importance of the public monopoly in job placement, the regulating of private agencies, the regulation within matters of discrimination, etc … In a previous work (Bessy et alii, 2001), we have applied such an approach within the framework of a comparison between France and Great Britain's labour markets. Our results indicate a higher level of worker mobility in Great Britain, as well as a more developed job placement structure. The French labour market is more often characterized by very important channels of informal job recruitment [START_REF] Marchal | L"équipement des relations sur les marchés du travail : Comparaison des méthodes de recrutement[END_REF]). If we focus on press job advertisements, which are typical of formal research channels, their use is three times more frequent in Great Britain than in France, and similarly the intervention of private employment agencies, is twice as important.

Historically, the prominent role for job placements within the British labour market precociously emerged from the political will of Beveridge to organise the labour market with a network of public labour exchanges (The Labour Exchanges Act -1909) which have always functioned simultaneously with a diversity of private agencies, but on different segments. In 1973, the Employment Agencies Act was introduced to provide a regulatory framework for employment agencies. This Act sought to regulate private agencies by a combination of licensing and statutory standards of conduct. The requirement for private agencies to license was revoked with effect from January 1995 by the Deregulation and Contracting Out Act 1994. Consequently, the number of private agencies has strongly increased [START_REF] Hasluck | The Market for Job Placements in the United Kingdom[END_REF]. In contrast, in France, the public monopoly in job placement, which was enacted in 1945 (Placement des travailleurs et contrôle de l'emploi ordinance) and is still officially in force, has always interfered with the rise of private activity. Nevertheless, prohibition primarily concerns recruitment activities by private agencies and not personnel consulting (including selection) or temporary work. For the past few years, the regulatory framework has become less constraining and the activities of French private agencies are more extensive but remain less developed and less specialised, in narrow market segments, than their British counterparts. Finally, one considers that the number of private recruiting agencies is 7000 in Great Britain and 1300 in France.

In this paper, we do not revisit the differences between the two structures of recruitment channels and their institutional factors. Instead, we provide an empirical verification of the diverse roles played by private British and French recruitment agencies; and to do so we used a unique empirical evidence consisting of press job advertisements. This kind of media can form a primary source of information on recruitment activities and an international data base at a relatively low cost. It is especially interesting to distinguish job vacancies published by recruitment agencies from those that are directly diffused by employers. In the first case, the agency publishes the job advertisement and manages the recruitment process. In the second case, the employer assumes responsibility for the whole recruitment process. Thus, in order to examine the differences between the two countries and the different role played by agencies, we have created a set of variables drawn from the coding used in job advertisements that specify certain pieces of information such as: recipient of candidates file (employer or agency), salary expectation, location, job description, required diplomas, required experience, personal qualities and means to contact the recruiter. These different kinds of variable can be analysed from the Rees approach (1966), which emphasises the role of the extensive and the intensive search for information.

The second section exposes the Rees approach used to elaborate our own framework in order to point out that the role played by intermediaries is linked to the nature of job matching processes. The third section presents the variables, developed from the coding of information contained within the job advertisements themselves, and our hypotheses regarding their interpretation. The fourth section presents our data set consisting of a merged sample of press job advertisements that were collected from the IT sector in France and Great Britain (see Annex). The characteristics of this sector, notably the standardisation of the technology, assure a minimum level of equivalents to highlight the differences specific to private recruitment agencies in each country. In the fifth section we present and comment our main results.

The labour market and its intermediaries.

Since the acknowledgement of the concept of imperfect information, economists see the labour market as a fog bank, in which employers and employees look for each other and meet each other with a given probability. On both sides, the challenge is to collect a maximum of information in order to minimise the undesired eventuality of an ex post low quality match. Therefore, improving upon the earliest job search models, studies on the subject focused on an approach involving methods or channels of research 1 . In this way, intermediaries have made their appearance in labour market analyses, linked to information networks. [START_REF] Rees | Labour Economics: Effects of more knowledge. Information in Labour Markets[END_REF] proposed such an approach distinguishing formal channels (employment agencies, newspapers, unions) from informal channels (professional and personal relations). In this context, a given agent (worker or employer) is able to increase his/her initial information on available partners in two ways: the extensive or the intensive margin. At first glance, formal channels prove more efficient for an extensive employment or candidature research, while informal channels prove more reliable when seeking additional information on given partners. This distinction is currently used in evaluating the efficiency of different channels or explaining the need to use one rather than another. In our work, we suggest to link the extensive / intensive opposition with the nature of the implicated match: standard vs. specific. 1 We better understand the reasons for leaving behind these models with the Osberg"s fish hatchery metaphor (1993) about job search, where people seem to be only interested in "big enough" fish to keep and not in the best strategy to catch the right one! A match has a standard nature if the prior expectation of its ex post quality is highly reliable, based on a knowledge of standard criteria linked to the partners involved (employee and position).

Consequently, an agent willing a standard match commitment would select the largest number of candidates or employment possibilities, using a minimal number of immediately evaluated criteria: the necessary research will be extensive. A match is considered to have a specific (or risky) nature when the prior expectation based on standard criteria is of poor quality. It is necessary, in this case, to anticipate the requisite characteristics specific to this relationship prior to actively entering an employment relationship: here, an intensive research is required. In fact, a look at the standard criteria of two potential partners may make them appear equivalent, and only an intensive research can provide specific information to distinguish one from the other. This distinction opens the way to a better appreciation of the role of intermediaries, going beyond their "black box" typification, by justifying the different work they carry out: "matchmaking" and evaluation. The more standard the matches are, the greater the probability that the intermediary will limit his/her role in "matchmaking" is (with an extensive research of candidates and jobs). Whereas, the more specific the matches are, the more important the task of evaluation will be (with an intensive research of information on each partner).

To our knowledge, no model is able to explain the task of intermediaries (matchmaking or evaluation) according to the nature of matches on the labour market 2 . Nevertheless, certain studies, starting with a transaction cost economics approach [START_REF] Williamson | The Economic Institution of Capitalism: Firms, Markets, Relational Contracting[END_REF], attempt to model this type of arbitrage. For example, Lesueur in his study (1997) tried to explain the type of intervention performed by French public agencies. Unfortunately, our data do not allow us to use this kind of approach. Indeed, we do not have any precise information on the specificity of the job that is advertised.

2 The two models we consider most important in illustrating the opposition between extensive and intensive modes of research, do not belong specifically to the labour economics. On the one side, [START_REF] Rubinstein | Middlemen[END_REF] consider a market where buyers and sellers meet each other through a time-consuming and stochastic process. Under these conditions, a private third agent enters the picture and, in taking advantage of the impatience of the first two, is able to exercise a profitable business activity as a "matchmaker". On the other side, Biglaiser"s model (1993) is affiliated with economic models of quality whereby adverse selection problems are central (and so is evaluation problem). It leads to a segmentation of the market: "high"quality goods pass through intermediaries (experts who had made a sunk investment), whereas "bad" quality ones are directly exchanged between sellers and buyers alone.

Classified job advertisements: a unique information source with regards to labour matching process

Empirical studies3 indicate that press job advertisements are an important recruitment channel on the labour market. They are costly for employers but provide inexpensive and easy resource materials to candidates. Compared to others channels, resorting to job advertisement is the most extensive research mode. So, according to the previous section, it should be associated with standard matches. We assume the empirical hypothesis that we are able to distinguish the job advertisements according to their more or less intensive characteristics in their information research. Therefore, we develop indicators based on information gathered in job advertisements and carry out a set of interpretative hypotheses.

Beyond the nature of the recipient of candidate"s application, we consider three types of information: the job conditions, the explicit selective criteria mentioned in the job vacancy and the means to be used by the applicant to contact the recruiter. The job conditions (wage, location and title) allow a first set of criteria which indicate the degree of transparency of the labour market. The second set of criteria (diploma, experience and personal skills) is traditionally called the employer"s hiring criteria. It constitutes an indicator of pre-selection intensity. Finally, from information about means of contacting, we draw inferences about the costs related to pre-selection.

Indicators of labour market transparency

Among the indicators of the degree of visible competition between enterprises, we consider the mentioning or non-mentioning of wage and location in the job advertisement. We have also added another indicator measuring the degree of explicitness in the text for the job description. Job openings, whose definitions are specific, are very difficult to compare from one firm to another. The candidate's research of information is more intensive and the wage is likely to be determined after bargaining with the recruiter. In this case, the advertisement is more expensive (because of a more detailed and longer text), but that may improve the expected match quality. In contrast, when the title of the job is in itself sufficient to define it, firms and applicants have common knowledge of the required tasks of the job; its nature is thus standard.

Finally, we assume that when wage and location are explicitly mentioned but the title of the job is given without further description of the tasks to achieve, then the job advertisement is linked to a standard match.

Selection criteria for candidates

The kind of matching process can also be characterized by the use of explicit criteria found within job advertisements, especially those making reference to experience, diplomas or personal qualities. These employer's hiring criteria allow the recruiter to limit the field of his/her extensive research of information. We assume that the multiplicity of hiring criteria provides a positive indicator of preselection intensity, and the research of information on each pre-selected candidate will be more intensive. When there are few hiring criteria, then the job advertisement may be linked to a standard match. Finally, let us note that the pre-selection intensity may also depend on the labour market conditions.

The costs of pre-selection

Information related to means of making contact with candidates (mailing, telephone, electronic mail…) provides information regarding the opportunities of interacting (or not) with recruiters before applying for a job. The motivation for this type of interaction is to reduce the number of candidates, thus allowing both sides to adjust to a more continuous recruiting process, which can reduce selection costs. We assume that a continuous process leads the recruiter to measure each candidate according to pre-set evaluation criteria and stop his/her research as soon as an applicant satisfies the criteria required. The recruiter does not compare one candidate with another in order to select the most suitable employee for the enterprise. The candidate having an average quality is not necessarily the best one for the job. Recruitment costs for standard short-term jobs are obviously reduced.

Other factors of recruiting costs are connected to the composition of a candidate's application, especially if CVs or application letters are required.

Finally, we assume that when a more interactive process of contact is proposed and the candidate's file required is not too substantial, then the job advertisement is linked to a standard match.

Recipient of candidate's application

The key variable in our analysis is the recipient of a candidate"s application: i.e. the employer or the agency. We assume that when an agency is the recipient, it will manage the whole process of recruitment (at least, the pre-selection phase). Our data allow us to distinguish specialised agencies4 in the IT sector from "general" agencies that operate in various sectors. We assume that specialised agencies, in addition to exhibiting economies of scale and scope, acquire an expertise within their field of recruitment, which can function to limit their recruiting costs. They can increase the transparency of the market by centralising information on wages and job quality. In this perspective, they favour standard matches.

With the help of these different interpretative hypotheses about the job advertisements content, we are now able to characterize them according to their link with a standard or specific match. We can also compare the different roles of private agencies in the British and French IT labour markets.

Construction of the sample and method of coding

To demonstrate evidence of the existence of national differences, it is necessary to ensure that a certain number of equivalents are present in the data set of press job advertisements. To assure equivalence in our study, we have chosen to restrict our sample to job advertisements related to the recruitment of IT workers. On the one hand, the standardisation of technology introduces a certain basis for comparison in contrast to other professions. On the other hand, both countries have witnessed a shortage in IT workers during the end of 1998 and the beginning of 1999, the period to which the data are related. We assume that these shortages existed in both countries. Finally, even though it is evolving very quickly, we consider this profession relatively well identified to carry out our comparison.

We assume that the nature of the medium leads to a particular mode of characterising the position available and the qualities of candidates. For this reason, we have been very selective in our study in choosing equivalent media in Great Britain and in France. We cannot guarantee the representativeness of our data set because we do not have precise information on the parent population5 . Nevertheless, in choosing a great diversity of media, we can control the differences between the types of media in both countries (see Annex).

Our sample is composed of 600 job advertisements extracted from twelve publications. Each country accounts for half of the sample. The media (five British, seven French) are either weeklies or weekly supplements to a daily published for a national readership. Five newspapers, out of which two are British, publish job openings linked directly to IT (qualified as specialised media); five others, two of which are British, address a general audience (qualified as general media); and one British and one French are edited by private associations subsidised either by private or public funds (which we call institutional media), and address all professions. Each of these three categories of media represents a third of the total sample, or 200 job advertisements. Whenever a publication classifies its job advertisements, they are laid out by heading or sub-heading in order to give each an equivalent weight.

When job advertisements are not classified, they are laid out at random. Table 1 shows the distribution of the job advertisements with regard to the type of medium, the national origin of the publications and its share in the sample. In this text, we undertake an analytical comparison based on sub-samples comprised uniquely of specialised and general media (400 job advertisements), because "institutional media" seem too specific to each country (see annex). Indeed, these types of media mainly advertised jobs for young graduate workers. It is also important to notice that the sub-samples of job advertisements are comparable in terms of skill levels. All the media we have selected in our reduced sample publish job vacancies mainly for highly qualified IT workers. The proportion of advertisements for professional workers (developers, engineers, analysts, experts, etc.) reaches 94.0 % for the British media and 85.0% for the French ones. The remainder corresponds to jobs for technical workers (low level programmers or people who do maintenance or assistance); there are no "routine data entry clerks" or "call centre staff".

Our method of coding differs from previous studies [START_REF] Rivard | L"espace de qualification des cadres[END_REF], Todd et al [1995]) 6 , which include intensive application for certain job advertisement items to the exclusion of others. Our method is more focused on using all aspects of job advertisement content in order to develop the indicators we have presented. The principle of our coding scheme is based on the presence or absence of information. In all, the coding was composed of 97 items separated into six groups: the medium, the job advertisement, the employer, the position, the candidate and the application process.

The comparative focus brought us to the selection of certain items.

Results

The results presented here bring together our first three indicators by crossing them with the recipients of candidates" application, i.e. agency or employer. In order to show the intermediaries" role in each country, relative to the employers, we proceed systematically to Chi-squared tests. 6 These two studies are carried out on the IT sector but with historical perspectives and focusing on training issue for the first and skills for the second. 7 Statistics from the French newspaper La Tribune (April 2000).

The British market's higher transparency

The most noticeable difference (see Table 3) between the two countries is probably the nearly systematic publication of information concerning wage in Great Britain (85.5% of the job advertisements) in contrast to a very low occurrence in France (12.5%). There is no significant difference pertaining to the type of recipient of candidates (agency or employer).

By publishing wages, enterprises get into the position of competition by attracting new candidates.

They confirm to the line of the market"s pricing, which they can influence, although published wages are always negotiable. In contrast, the non-disclosure of wages undermines market transparency and can weaken candidates" bargaining power in the decentralised process8 .

In Great Britain, the wage is more often accompanied by a mention of location (86.0% vs. 57.5% in France). In both countries, it is to notice that this piece of information is significantly more mentioned by agencies than by employers. French job advertisements are more explicit in defining the position itself than British job advertisements (75.3% as opposed to 58.5%), especially when the advertisement is published by a recruiting agency (89.3% vs. 68.8% for French Employers). This is opposite to British job advertisements (51.2% vs. 70.7%). This last difference (significant at 1% level) indicates the unique role played by specialised British agencies in the IT sector, as opposed to their French counterparts9 .

These specialised British agencies occupy for the most part a relational function between employers and candidates, both having common knowledge about the job and its tools. Only 42.0% of the job advertisements published by specialised agencies embody an explanation of the job"s content.

A stronger pre-selection by French intermediaries

As far as the different criteria for selecting candidates are concerned, the mention of a qualification introduces the main difference between the two countries (see Table 5). From the coding, we have ascertained 6 headings with reference to the type of education required by the advertisers for a position in the IT field. Obviously, the difficulty of analysis has come from the difference between the two countries with regard to the organisation of higher education and , we have therefore created for our study a new "diploma" variable including three categories: i) no diploma mentioned; ii) diploma in computer science (regrouping the British university qualifications in IT and the French qualifications in IT such as IUT, BTS, etc.); iii) general diploma (regrouping British graduates without a specific major, and the French bac + 2, 4 or 5 without further specification and les Grandes Ecoles). Among the candidate"s profiles designated by their diplomas, we contrast, then, between "specialists" and "generalists".

In Great Britain, 70.5% of the job advertisements outline the ideal candidate profile without necessarily specifying his/her education. On the contrary, in France, 66.5% of the profiles specify a diploma. The ideal IT worker, according to the job advertisements, is more often a generalist than a specialist (43.0% vs. 23.5%). In Great Britain, the tendency is reverse with a smaller proportional gap (10.4% of the job advertisements for specialists, compared to 12.8% for generalists). Where British job advertisements do not use a qualification as a discriminatory element in the selection process, the French job advertisements specify a degree level by default ("Bac + years of studies"). ***: significant at the 0,1 per cent level (probability under the null hypothesis of independence ) **: significant at the 1 per cent level, *: significant at the 5 per cent level.

An interpretation of this result can be the specificity of the French education system. Indeed, in a country where education is very centralised and where the value of a "diploma" is guaranteed by the State, this element gives a clear signal which is more frequently used by the recruiters. This interpretation can only be valid for initial education corresponding to French "Bac +2 years of undergraduate studies" level10 . As concerns superior level of education, we can assume that the quality of the signal given by the degree is the same in each country.

One can then wonder about an eventual "intermediary" effect on the editing of the ideal profile. So, on the one hand, it appears that 83.0% of the French employers using agencies will mention a degree, when 59.6% of those not using agencies, will not mention it (significant at 1% level). On the other hand, among British employers not using intermediaries, the mention of a degree occurs much more frequently: 40.0% compared to 23.2% in the reverse case (significant at 5%). The "intermediary" effect we just mentioned can be noted here but with opposite results across the two countries: French intermediaries tend to demand qualifications (more likely general education), whereas their British colleagues ignore this selection criterion at this stage of the recruitment process.

We approached the issue of a candidate"s experience in three ways. We distinguished experience acquired in a particular and identical field (sector, duty, type of business) from experience related to a specific technology. In this perspective, we can notice that the first type of experience we described could correspond to competency acquired in the world of business. That would allow us to take into account the cases in which the recruiters are looking for candidates who do not necessarily have IT experience. The second type would target IT competency. In other respects, we have also taken into consideration whether the job advertisement specifies a minimum amount of experience required or not, which provides another good indicator of the selective aspect of the job advertisement.

Of the three criteria above mentioned, the difference between the two countries is rather insignificant.

Nevertheless, French intermediaries mention more frequently a "job experience" or a minimum length of experience than French employers : respectively 79.7% as opposed to 50.4%, 52.5% as opposed to 24.1% (significant at 0.1% level). There are not such striking differences within the British job advertisements.

It may be noted that the simultaneous mention of experience and diploma is more frequent within French job advertisements than within British ones (34.0% as opposed to 25.0%). In the former, it does not depend on the type of recruiter. In contrast, this simultaneous mention is significantly less used (at 0.1% level) within the job advertisements published by British agencies than within those directly published by employers (13.6% as opposed to 45.3%).

For each job advertisement, we have pointed out the presence of a personal skill (or not) used to complete the ideal candidate"s profile (enthusiasm, curiosity, tenacity, relational qualities, etc.) and accounted of them numerically (see Table 4). There is no difference between the two countries. But again British agencies mention less frequently personal skills than British employers : 21.8% as opposed to 39.2% (significant at 0.1% level).

Thus, it appears that, depending on employers" practises, British intermediaries stress less the preselection of applicants than their French counterparts. According to our interpretation of pre-selection criteria, they make more standard matches.

Initial interactive pre-selection more customary in Great Britain

Generally speaking, pre-selection costs are lower in Great Britain than in France according to the indicators we have applied (see Table 5). Chi-squared test on the influence of type of recruiter (agency or employer); Val 2: value of the Chisquared statistic. ***: significant at the 0.1 per cent level (probability under the null hypothesis of independence ) **: significant at the 1 per cent level, *: significant at the 5 per cent level. Hence, 33.0% of French job advertisements mention conventional mail as the means of making contact in the recruiting process, whereas this figure is only 8.0% for British recruiters who prefer (most frequently) either conventional mail or telephone, fax or e-mail (50.4%), or solely interactive way (46.4%).

Moreover, whereas in French job advertisements, the way of contact does not distinguish the job advertisements in function to the type of recruiter, the difference is very striking in British job advertisements. British agencies favour more interactive ways of contact than employers. Due to this very significant statistical difference (at 0.1% level), we can deduce that the process of selection by British intermediaries is more continuous and less costly, which would be consistent with standard matches.

Other factors related to lower recruiting costs in Great Britain are observed through the composition of candidates" applications. The mention of a C.V. is more frequent in France than in Great-Britain (78.5% as opposed to 38.0). Moreover, British intermediaries make significantly less reference to a C.V. than employers (28.8% as opposed to 53.3%).

French recruiters ask for a covering letter (73.0% in France; 4.5% in Great Britain). This gap reflects the presence of the French school of graphology within the recruiting profession [START_REF] Marchal | L"équipement des relations sur les marchés du travail : Comparaison des méthodes de recrutement[END_REF] 11 .

Discussion

The overall observations converge on the idea that French intermediaries, which are not specialised agencies, make more severe and more costly pre-selection than their British counterparts. This could be an indicator of their "evaluative" orientation, whereas British intermediaries are more likely positioned in a relational one ("matchmaking") as specialists in IT recruiting.

These results can be interpreted depending on the nature of the activity of private agencies in each country (Bessy et alii, 2001). The typical French agency would more often be a recruitment consulting firm capable of providing other related consulting services (such as defining and parametring skills, training, etc.), but not other placement related activities. The agency"s aim is to find and place the right candidate within a long-term perspective (the client enterprise is more truthfully an internal labour market). This process tends to lead the agency to enlarge its operations of evaluation. The severity of the selective criteria found in their job advertisements may be due to the employers" demand related to their investment in using the agency"s services. In contrast, in Great Britain, the typical agency is a specialised one that answers quickly to firms" labour demand. A look at their job advertisements indicates they deal with permanent and temporary openings as well as with contracts for independent workers. This shows how specialised they are in the IT sector and more involved into customer contacts than skills evaluation.

The different role played by private agencies in the two countries is obviously linked to the specificity of the institutional environment in the matter of job placement structure. In Great Britain, the most important part of specialised agencies alters the conditions of the trade-off concerning the eventual recourse to an external intermediary. Indeed, this more competitive configuration reduces the bilateral dependence between employers and agencies.

These results seem coherent with a Transaction Cost Economics approach [START_REF] Williamson | The Economic Institution of Capitalism: Firms, Markets, Relational Contracting[END_REF]. But for further verification of this approach, we need data on the actual role played by the agencies in the recruitment process and on their degree of investment specificity. Indeed, we have made the assumption that the agencies manage the whole process of recruitment. In contrast, we can assume that agencies manage only the diffusion of the job advertisement and constitute a pool of applicants. In this perspective, the recourse to an agency can be explained by its complementary activities that supplement those of an employer 12 . Co-operation between the two entities is made possible because it allows the co-ordination of activities that corresponds to different production process phases. This complementary aspect can explain why, during phases of manpower shortages, employers turn to employment agencies, even when an in-house recruitment department is available. A good illustration of this configuration is given by the "contingency recruiting" procedures in which several agencies compete for delivering CVs to the employer but do not participate to the selection step. The agency is paid only if one of the applicants who has been presented by it, is actually recruited13 .

Our main difficulty is to assess the specificity of the job from information found in job advertisements.

Whether an explicit job description testifies to a larger employment specificity, and contrarily nonexplicit job advertisements signal a standardisation in which a job description is not necessary among specialists in the field, is a fragile hypothesis. Our work with coding is indeed dependent on the way job advertisements are edited, certain editors using more redundant information than others.

Finally, our last concern was the labour market situation in each country during our period of observation (the end of 1998-the beginning of 1999). We have assumed that IT workers shortages were identical in both countries. In contrast, if we assume that shortages are stronger in Great Britain than in France14 , it can explain the fact that British recruiters make less reference to qualification and, more generally, less selection than their French counterparts. Nevertheless, that does not explain the differences in job advertising between the agencies and the employers in each country.

Conclusion

In this paper we provide a framework in order to point out that the role played by private recruitment agencies within the labour market is linked to the nature of job matching processes. In a comparative study of the IT labour market in France and Great Britain, using a merged sample of press job advertisements, we show the differences between the two countries. In Great Britain, standard matches are linked to the involvement of specialised recruitment agencies in the IT field. They contribute to the transparency of information by the way they edit their job advertisements (systematically announcing wage levels and job location) and procedures for continuous interaction that are put in place to inform the candidate. In contrast, the degree of asymmetry of information (to the detriment of the candidate) is much higher in French job advertisements. In other respects, French intermediaries play a more pre-selective role than their British counterparts, in particular by focusing on the "general diploma" as a pre-selection criterion. That is coherent with more specific matches.

Our analysis of the dynamics of the IT labour market based on an exploitation of classified job advertisements would be improved by expanding the scope of the analysis to a more general study on the different recruitment channels in this market. Along the same lines, an extension of our work would be to apply the same type of analysis to other sectors in order to confirm the differences we have found between the two countries in the realm of private agencies" role and in reference to qualification within the job advertisements. Indeed, in our study we have assumed that the two labour markets are completely separate. Nevertheless, in the highest level jobs, the IT sector has a more international labour market than most industries, and maybe British specialist agencies recruit for vacancies in other countries. The development of "e-recruitment" provides a powerful support for the internationalisation of this activity. By widening the comparison to other sectors, we could test more precisely the influence of institutional environment of the national labour markets. The specificity of this French institutional medium is, in a certain way, duplicated by the British publication "Prospects Today", a journal from the Association of university placement offices. With this type of medium, the recipient of the candidate"s applications is more often the employer and the mention of job location is less frequent. These job advertisements target young graduates and that explains the common reference to a C.V., a letter of candidature, a degree notably in IT, but also the non-mention of some prior experience.

To make the best use of our data, we have characterized media by the employers" strategy to resort to job advertisements and some defining elements of the employers. It seemed to us that if announcers were carrying out a veritable campaign to seduce potential applicants, they would be using certain means, such as publicity slogans, identifying specifically the employer and the abundance of job offers. This is in particular the case with the British publication "Prospects Today".

We have categorised employers by the presence (or not) of advertisement content as far as they are concerned. Coding this kind of information required us to perform a type of interpretation, which weakened our analysis. Apart from the sector of the enterprises and their public or private status, their position with regards to the usage of IT, also seemed to us to be a determining factor. We have distinguished, on the one hand, those, not necessarily specialists, who make an IT infrastructure available to their employees, and on the other hand, those who are developers of IT materials, environments and software, as well as those actually working with these tools. We have thus gathered constructors, software editors, engineering and IT service enterprises and consulting agencies in the same category.

Beyond the employer"s identity, his/her specific activity gives information to candidates (see Table 7).

To apply as an IT "professional" to an enterprise, which seeks such worker, does not mean the same thing as wanting a position in an IT constructor or in a service and consulting IT enterprise. Both institutional media of our sample show a common policy: they publish in large part offers from IT specialised enterprises, which guarantees the latter with an important distribution at low cost.

Due to their specificity, both publications seem to blur the differences observed in the two countries and especially, in regard to the selection criteria contained in the job advertisements, Which introduces a bias into our sample. This specificity symbolises, in a way, the two national labour markets: the existence in Great Britain of structures to place students, and a job placement organism for executives in France. Nevertheless, it damages a concrete comparison of the job advertisements since the objectives of these institutional media differ in the two countries. Hence, that justifies the necessity to continue the analysis with a restricted sample.

  test on the influence of type of recruiter (agency or employer); Val 2: value of the Chisquared statistic.

Table 1 :

 1 The sample of job advertisements

		Frequency	%	Nationality
	Professional periodicals:
	"Computer Weekly"	50	8,3%	GB
	"Computing"	50	8,3%	GB
	"Informatiques Magazine"	30	5,0%	F
	"Le Monde Informatique"	32	5,3%	F
	"01 Informatique"	38	6,3%	F
	National daily
	Newspapers:	29	4,8%	GB
	"Inter//face"	35	5,8%	F
	"Le Figaro Économie"	30	5,0%	F
	"Le Marché du Travail"	35	5,8%	F
	"Le Monde Interactif"	71	11,8%	GB
	"The Guardian"
	Institutional periodicals:
	"Courrier Cadres"	100	16,6%	F
	"Prospects Today"	100	16,6%	GB
	Total	600	100,0%

Table 2 :

 2 The recipients of candidates (frequency, %)Table2shows that the distribution of the job advertisements among the types of the recipients of the candidate"s application (employer or middleman) is very different between the two countries. Thus, 29.5% of French job advertisements emanate from recruiting agencies, while the proportion for their British counterparts is 62.5% (see Table1). This distribution appears to approximate more closely the respective weight on the market of intermediaries in both countries. Even though statistics concerning the IT labour market are not available, this pattern could be applied regardless of the sector, i.e. 34% of published French job advertisements are published by intermediaries 7 compared to 85% of British job advertisements. The distribution of these job advertisements testifies to the most important role played by British private agencies. Where general intermediaries in both countries play nearly identical roles, specialised agencies in Great Britain have a more important role than in France (49.4% vs. 8.0%).

	Nationality	France	Great
	Recipients of candidates		Britain
	Employer	71.5	37.5
	Intermediary	29.5	62.5
	Private specialised agency	8.0	49.4
	Private non spec. Agency	21.5	13.1
	Total	100.0	100.0

Table 3 :

 3 Market transparency indicators (frequency, %)

	Nationality		France			Great Britain
	Make mention of	Overal	Agenc	Empl. Val 2 Overal	Agenc	Empl. Val 2
		l	y			l	y
	Wage	12.5	18.6	9.9	2.9	85.5	84.8	85.3	0.01
	Location	57.5	78.0	48.9 14.3**	86.0	91.2	77.3	7.5**
					*
	Job description	75.0	89.3	68.8	9.8**	58.5	51.2	70.7	7.3**
	Chi-squared test on the influence of type of recruiter (agency or employer); Val 2: value of the Chi-
	squared statistic.
	***: significant at the 0.1 per cent level (probability under the null hypothesis of independence )
	**: significant at the 1 per cent level,
	*: significant at the 5 per cent level.

Table 5 :

 5 Pre selection costs (frequency, %)

	Nationality		France			Great Britain
	Make mention of	Overal	Agenc	Empl. Val 2 Overal	Agenc	Empl. Val 2
		l	y			l	y
	Way of contacting
	Conventional mail solely	33.0	37.3	31.2		8.0	3.2	16.0
	Alternative mail/interactive	60.0	61.0	59.6		59.5	50.4	74.7
	way	7.0	1.7	9.2		32.5	46.4	9.3
	Interactive way solely	100.0	100.0	100.0	3.8	100.0	100.0	100.0 34.1**
	Total								*
	Application
	CV	78.5	84.5	75.9	1.9	38.0	28.8	53.3	12.0**
	Covering letter	73.0	79.7	70.2	1.9	4.5	3.2	6.7	*
									Not
									valid

  Nationale pour l'Emploi des Cadres)15 , which plays, in France, an important role in executive placement. For selection costs indicators, we observe a dependency link that needs to be related to APEC recruitment procedures (application by mail only and implicit requirement of a C.V. plus a covering letter). In contrast, criteria of selection are not significantly different across the three types of media.

		34	26	33	1,77	62	11	79	100,20**
									*
	IT graduate	44	42	42	0,10	22	27	10	9,66**
	Graduate (general)	22	32	25	2,71	16	62	11	75,75***
	No mention of	11	5	12	3,38	3	37	4	59,81***
	experience
	Job experience	61	67	57	2,14	70	18	69	70,90***
	Technical experience	72	78	78	1,31	90	60	88	34,31***
	Mention of duration	32	47	33	6,01*	30	11	41	23,19***
	No mention of	50	47	47	0,24	43	37	46	1,72
	personal skills
	More than two	24	29	26	0,65	32	36	25	2,89
	personal skills
	Chi-squared test on the influence of type of media; Val 2: value of the Chi-squared statistic.
	***: significant at the 0,1 per cent level (chi-square test, probability under the null hypothesis of
	independence )
	**: significant at the 1 per cent level,
	*: significant at the 5 per cent level.
	With regards to French job advertisements, among the transparency indicators within the labour

market, only the description of job"s tasks is not dependent on the medium. The differences in terms of indicating wage and job location are attached to job advertisements advertised by "Courrier Cadres", a publication by APEC (Agence

Table 7 :

 7 Advertising Strategies and employers" characteristics following media (frequency, %)

	Media		French supports			British supports
	Characteristics	General	Institut. Special. Val 2 General	Institut. Special. Val 2
		.			.
	Advertising Strategies
	Slogan	36	1	36	44,35***	18	37	38	11,87**
	Employer identity	84	25	83	99,04***	52	88	41	50,50***
	Only one job	51	49	52	0,911	64	17	47	46,30***
	Several identical jobs	29	43	16	17,59***	14	39	26	16,11***
	Several different jobs	20	8	32	18,00***	22	42	27	10,25**
	Employers
	IT conceptors/SSII	55	70	42		35	64	38
	IT users	41	22	53	33,5***	42	30	46	25,91***
	No precise	4	8	5		23	6	16
	Private firm	89	90	72		50	82	59
	Public firm	7	1	13	19,23***	23	6	16	24,50***
	No precise	4	9	15		27	12	25
	Manufacture	8	11	20		3	18	10
	Services	87	84	76	6,92	72	76	72
	No precise	5	5	4		25	6	18

See Holzer (1988),[START_REF] Jones | Job Search Methods, Intensity and Effects[END_REF] and[START_REF] Osberg | Fishing in Different Pools: Job-Search Strategies and Job-finding Success in Canada in the Early 1980s[END_REF] for the use by candidates of want ads;[START_REF] Russo | Firm's Recruitment Behaviour: an Empirical Investigation of the Use of Search Channels, the rate of Arrival of Applicants, and the Spatial Radius of Search[END_REF],[START_REF] Gorter | Employers" recruitment behaviour and vacancy duration: an empirical analysis for the Dutch labour market[END_REF] and[START_REF] Roper | Recruitment Methods and Vacancy Duration[END_REF] for the use of wanted ads by employers;[START_REF] Lindeboom | Matching Employers and Workers: an empirical Analysis on the Effectiveness of Search[END_REF] for an estimation of the matching function through wanted ads.

Within the frame of our coding, the specialisation of an agency is identified from the number of different ads it placed in a given medium.

According to available studies, the use of want ads in the recruitment process is twice more important in Great Britain than France[START_REF] Marchal | L"équipement des relations sur les marchés du travail : Comparaison des méthodes de recrutement[END_REF].

So, on the goods market,[START_REF] Wang | Bargaining versus posted-price selling[END_REF] analysesfor a given sellerthe arbitration between two popular selling methods -bargaining and posted price selling. He finds that the bigger the seller"s bargaining power, the more a selling negotiated price is being used. On the contrary, an augmentation of the competition increases the probability of using posted prices.

Notice that, in each country, the mention of a job description is independent on the nature of media. See Annex.

On the specificity of the French education system and on the role played by the French State in the certification of intermediary level of education, see Buechtmann and Verdier (1998).

Also, French recruiters ask for a photograph (24% of the cases), whereas this practice is non-existent in Great Britain.

See Lesueur (1997) who finds evidence in the works of[START_REF] Richardson | The Organisation of Industry[END_REF]

According to our observations on placement markets in each country, this practise is more developed in Great Britain than in France(XXX, 2001).

According to a rapport ofOCDE (2000), in 1999, IT workers shortages raised to 80 000 in Great Britain and to 25 000 in France. This difference is explained by a more voluntary education policy set out by the French State for training qualified IT workers.

This paper draws from a research funded by the French Commissariat Général du Plan. We would like to thank François Eymard-Duvernay and Emmanuelle

Annex: Job advertisements media and their characteristics in the large sample (n=600)

The nature of the medium modifies the definition of the jobs and the employee qualities sought. This link of dependency is statistically verified for each country by taking into account most of the variables considered in our analysis (see Tables 6 and7), even though, for certain variables, the null hypothesis of independence is accepted (Chi squared test). In this annex, we justify the choice of our sub-sample by eliminating the "institutional media".


N Combe

C Taillan

J Morillo

Unidimensional model of the ad-atom diffusion on a substrate submitted to a standing acoustic wave I. Derivation of the ad-atom motion equation

The effect of a standing acoustic wave on the diffusion of an ad-atom on a crystalline surface is theoretically studied. We used an unidimensional space model to study the ad-atom+substrate system. The dynamic equation of the ad-atom, a Generalized Langevin equation, is analytically derived from the full Hamiltonian of the ad-atom+substrate system submitted to the acoustic wave. A detailed analysis of each term of this equation, as well as of their properties, is presented. Special attention is devoted to the expression of the effective force induced by the wave on the ad-atom. It has essentially the same spatial and time dependences as its parent standing acoustic wave.

I. INTRODUCTION

While the semi-conductors industry extensively uses the lithography process to stamp the micro-devices at the nanoscale, research centers and laboratories have investigated the self-assembling properties of materials to avoid this expensive and time consuming process. Most strategies to self-assemble materials at the nanoscale, especially during the atomic deposition process of semiconductors benefit from the elastic properties or from the structure of the substrate: the Stranski-Krastanov growth mode relies on the competition between the surface and elastic energies to organize the 3D-growth; 1,2 buried dislocations networks in the substrate induce a periodic strain field at the substrate surface that drives the diffusion of ad-atoms; 3,4 and finally the use of patterned substrates(vicinal surfaces, holes or mesas) can create some preferential nucleation sites. [5][6][7][8] An alternative approach to self-assemble materials at the nano-scale, the dynamic substrate structuring effect has been recently proposed. 9 At the macroscopic scale, a sand bunch on a drum membrane excited at one of its eigenfrequencies self-structures by accumulating around the nodes or anti-nodes displacements of the membrane. 10 Transposing this concept at the nanoscale, we investigate the diffusion of an ad-atom on a crystalline substrate submitted to a standing acoustic wave (StAW). [START_REF]The abbreviation SAW is usually used to design a Surface Acoustic Wave, we thus introduce here a distinct abbreviation, StAW for Standing Acoustic Wave. 29 δi[END_REF] Molecular Dynamic simulations have evidenced that the StAW structures the diffusion of the ad-atom by encouraging its presence in the vicinity of the maximum displacements of the substrate. 9 The typical and relevant StAW wavelengths vary from few to hundreds of nanometers. Experimentally, the production of standing surface acoustic waves of a few hundred nanometers to microns wave lengths are nowadays available through the use of interdigital transducer 12,13 or optically excited nanopatterned surfaces, 14 whereas one does not know yet how to efficiently generate smaller wavelengths (few to tens nanometers) phonons.

In this study, we propose to analytically study the diffusion of a single ad-atom on a crystalline surface sub- The ad-atom (red) and substrate atoms (blue) are characterized by their coordinates x and xj (j ∈ {-N..N }) in the reference frame of the center of mass of the substrate. Note that, for clarity reasons, the ad-atom is not reported on the same horizontal line as the substrate atoms, but the model is unidimensional.

mitted to a StAW. The goal of this study is to establish the formalism and the dynamic equation that describes the diffusion of an ad-atom on a crystalline substrate submitted to a StAW. In section II, a generalized Langevin equation governing the ad-atom diffusion on a one-dimensional substrate is analytically derived from the Hamiltonian of the system (ad-atom+substrate). Sects. III, IV, V and VI detail the different terms involved in this generalized Langevin equation, as well as their properties.

II. AD-ATOM MOTION EQUATION

We consider the diffusion of an ad-atom on a crystalline substrate submitted to a StAW with a wave-vector in the x-direction. Since the ad-atom diffusion is expected to be mainly affected in the x direction, we specialize to a system with one degree of freedom. The extension to a 2D system to model a more complex StAW system (for instance, two StAWs with wave vectors in the x and y directions form a square lattice of nodes and anti-nodes) is straightforward, though analytical calculations may become tedious. Fig. 1 reports a sketch of the model under study. x and x -N , ..., x N respectively design the positions of the ad-atom and of the 2N + 1 substrate atoms in the reference frame of the center of mass of the substrate. Following the work of Zwanzig 15 and related works, [16][17][18] we start with the Hamiltonian of the isolated system (ad-atom+substrate):

H 0 = p 2 2m +Φ(x, x -N , ...., x N )+ N j=-N p 2 j 2m j +V sub (x -N , ..., x N ), (1)
where m, p and m j , p j are respectively the masses and momenta of the ad-atom and of the substrate atoms, V sub (x -N , ..., x N ) and Φ(x, x -N , ...., x N ) the potential enegies of the substrate-substrate and ad-atom-substrate interactions. At this point, the generation process of the StAW has not been yet introduced, this will be done later on.

The motion of the substrate atoms will be described in the harmonic approximation [START_REF] Kittel | Quantum theory of solids[END_REF] with the associated phonons of eigenvibration frequencies ω n , normal coordinates Q n and momenta Π n .

j p 2 j 2m j +V sub (x -N , ...., x N ) ≃ 1 2 n Π n Πn + ω 2 n Q n Qn , (2)
where over-bar quantities are complex conjugate quantities and where the potential origin has been fixed at the equilibrium positions, V sub (x 0 -N , ...., x 0 N ) = 0. In this equation and in all the following equations, unless otherwise stated, the summations over the substrate atoms j are from -N to N , and those over the normal modes n from -N to N excluding n = 0. Note that, within the harmonic approximation, for an isolated substrate there is no substrate dilation with temperature nor energy exchanges between the phonons.

The substrate atom displacements, u j = x j -x 0 j , around their equilibrium positions, x 0 j , are thus given by:

u j = 1 √ m j n e iknx 0 j Q n , (3)
where k n is the wave vector of the n normal mode. In Eqs.

(2) and (3) we have:

k -n = -k n , ω -n = ω n , Qn = Q -n , and Πn = Π -n . (4)
From Eqs. ( 2) and (3) and performing a developement of the potential Φ to first order in the u j 's:

Φ(x, x -N , ...., x N ) = Φ(x, x 0 -N , ...., x 0 N ) + j u j . ∂Φ ∂x j (x, x 0 -N , ...., x 0 N ) = Φ 0 (x) + 1 2 n Q n .Ψ n (x) + Qn . Ψn (x) (5)
where

Φ 0 (x) = Φ(x, x 0 -N , ...., x 0 N ), (6)
Ψ n (x) = j 1 √ m j e iknx 0 j ∂Φ ∂x j (x, x 0 -N , ...., x 0 N ). ( 7
)
The interaction of the ad-atom with the substrate has been separated in two contributions. Φ 0 (x), the first one, appears as an external static force field. It is due to the frozen equilibrated substrate inter-atomic periodic potential. The second one, represents the interaction of the ad-atom with the phonons Q n , i. e. with the moving substrate atoms around their equilibrium positions. Eq. ( 1) hence writes

H 0 = p 2 2m + Φ 0 (x) + 1 2 n Q n .Ψ n (x) + Qn . Ψn (x) + 1 2 n Π n Πn + ω 2 n Q n Qn , (8)
Note that the coupling between the substrate and the ad-atom is linear in the phonon variables and non-linear in the ad-atom variable, i.e. the reverse situation of the one studied by Cortes et al. 17 To model the presence of a StAW in Eq. ( 8), we add a forcing term with the same F amplitude on two specific normal variables of opposite wave vectors Q nex and Qnex (= Q -nex ). However, since our model does not consider any dissipation of the substrate vibration modes, we slightly detune the forcing frequency Ω nex = ω nex + δω nex from the eigenfrequency ω nex to avoid any resonance and subsequent divergence of the amplitude of the mode Q nex . These two modes will be equally excited and thus, from basic forced oscillation theory, [START_REF] Kittel | Mechanics[END_REF] one expects a forced oscillation substrate displacement field proportional to that of the parent standing wave:

u(x, t) = - 2F M ∆ 2 cos[Ω nex t] cos(k nex x + η), (9)
where M is the mass of the oscillator, η a phase depending on the initial conditions and with:

∆ 2 = Ω 2 nex -ω 2 nex = (ω nex + δω nex ) 2 -ω 2 nex . (10)
We thus consider the following Hamiltonian for the system (ad-atom+substrate submitted to a StAW):

H = p 2 2m + Φ 0 (x) + 1 2 n Q n .Ψ n (x) + Qn . Ψn (x) + 1 2 n Π n Πn + ω 2 n Q n Qn -(Q nex + Qnex )F cos[Ω nex t] (11)
Note that, in Eq. ( 11), the addition of the StAW term makes the Hamiltonian time-dependent. In addition, the work of the operator to induce the StAW (the last term of Eq. ( 11)) is not null on average and leads to a monotonous increase of the average energy of the system (ad-atom + substrate). This would be the case even taking into account all the nonlinear terms we have omitted in Eq. (11). We however assume that despite this monotonous increase of the energy, the temperature of the system remains constant, either by considering that the substrate is infinite and has the behavior of a thermostat, or by considering that the system is not totally isolated and coupled to an external thermostat.

The dynamic equations derived from Eq. ( 11) read: 21

dQ n dt = Π n (12a) dΠ n dt = -ω 2 n Q n -Ψn (x) + Λ n,nex F cos[Ω nex t] (12b) dx dt = p m (12c) dp dt = - dΦ 0 dx (x) - 1 2 n Q n dΨ n dx (x) + Qn d Ψn dx (x) (12d)
Where Λ i,j = δ i,j + δ i,-j with δ i,j the Kronecker symbol 29 . In Eq. (12b), -Ψn (x) is the force on the substrate normal mode n, induced by the ad-atom at position x. Solving Eq. (12a) and Eq. (12b) between t 0 and t, the normal substrate coordinates read:

Q n (t) = Q n (t 0 ) cos[ω n (t-t 0 )]+ Π n (t 0 ) ω n sin[ω n (t-t 0 )]- t t0 Ψn (x(t ′ )) sin[ω n (t -t ′ )] ω n dt ′ + t t0 Λ n,nex F cos[Ω nex t ′ ] sin[ω n (t -t ′ )] ω n dt ′ , (13)
where Q n (t 0 ) and Π n (t 0 ) are fixed by the initial conditions. An integration of the second integral and an integration by parts of the first one gives:

Q n (t) = C n (t 0 ) cos[ω n (t -t 0 )] + D n (t 0 ) sin[ω n (t -t 0 )] - Ψn (x(t)) ω 2 n + t t0 cos[ω n (t -t ′ )] ω 2 n dx dt (t ′ ) d Ψn dx (x(t ′ ))dt ′ -Λ n,nex F ∆ 2 cos[Ω nex t], (14)
with:

C n (t 0 ) = Q n (t 0 ) + Λ n,nex F ∆ 2 cos[Ω nex t 0 ] + Ψn (x(t 0 )) ω 2 n (15a) D n (t 0 ) = Π n (t 0 ) ω n -Λ n,nex F ∆ 2 Ω nex ω n sin[Ω nex t 0 ]. (15b)
From Eqs. ( 4) and (7) we have:

Cn = C -n and Dn = D -n (16)
Using Eq. (12c), (12d) and ( 14), we derive the generalized Langevin equation governing the ad-atom diffusion:

m d 2 x dt 2 = - dΦ eff dx (x) - t t0 γ(x(t), x(t ′ ), t -t ′ ) dx dt (t ′ )dt ′ + ξ(t) +F SAW (x, t). (17)
The left hand side term of Eq. ( 17) is the usual inertial term. On the right hand side, we distinguish four terms, which are successively:

• the force induced by the effective crystalline potential Φ eff (x), defined by:

Φ eff (x) = Φ 0 (x) - 1 2 n 1 ω 2 n Ψ n (x) Ψn (x). ( 18
)
The properties of this potential will be studied in Sect. VI.

• The friction term -t t0 γ(x(t), x(t ′ ), t -t ′ ) dx dt (t ′ )dt ′ that depends on the ad-atom velocity and on the memory kernel γ(x, x ′ , t -t ′ ):

γ(x, x ′ , t -t ′ ) = n cos(ω n (t -t ′ )) ω 2 n dΨ n dx (x) d Ψn dx (x ′ ). ( 19
)
The properties of γ(x, x ′ , t -t ′ ) will be studied in Sect. IV.

• the stochastic force 17,18 ξ(t):

ξ(t) = - n C n (t 0 ) cos[ω n (t -t 0 )] +D n (t 0 ) sin[ω n (t -t 0 )] dΨ n dx (x(t)). ( 20
)
This term depends on the initial conditions and adatom position and is a quickly varying force gener-ated by the substrate. The of this force will be described in Sect. V.

• The last term F SAW (x, t) is the effective force due to the applied forcing term at Ω nex , i.e. the force F SAW (x, t) induced by the StAW on the ad-atom through the substrate:

F SAW (x, t) = F ∆ 2 dΨ nex dx (x) + d Ψnex dx (x) cos[Ω nex t].
(21) This force will be detailed in Sect. III.

The three first forces, crystalline, friction and stochastic, exist even in the absence of the StAW excitation. They are the usual forces describing the dynamics of the atoms in a crystalline material. We have chosen to keep in F SAW (x, t) only the forced oscillation term at the pulsation Ω nex . All the other terms depending on F have been included in the stochastic force ξ(t). They correspond to the answers of the oscillators Q nex and Qnex to the initial conditions at t = t 0 . Since the normal modes of the substrate are undamped, these last terms are periodic and do not cancel. For damped oscillators, the terms depending on F in the stochastic force would correspond to a transient regime and would thus cancel, contrary to the forced oscillation term at the pulsation Ω nex .

III. THE STAW FORCE

To derive the expression of the force F SAW (x, t) induced by the StAW, we need to explicit the expression of Ψ nex (x) in Eq. (21). Since interaction potentials depend only on the relative position of the interacting particles, so do Φ 0 and Ψ n . Ψ n writes then:

Ψ n (x) = j 1 √ m j e iknx 0 j ∂Φ ∂x j (x -x 0 -N , ...., x -x 0 N ) = α n (x)e iknx , (22)
with α n (x) defined as:

α n (x) = j 1 √ m j e -ikn(x-x 0 j ) ∂Φ ∂x j (x -x 0 -N , ..., x -x 0 N ) (23
) Note that for an infinite crystal the α n (x) functions have the lattice periodicity. [START_REF]Bloch function character of the Ψn functions[END_REF] . In addition, ᾱn (x) = α -n (x) so that introducing the real α r n (x) = ℜ(α n (x)) and imaginary α i n (x) = ℑ(α n (x)) parts of α n (x), we have:

α r n (x) = α r -n (x) and α i n (x) = -α i -n (x), (24)
which leads to:

dΨ n dx + d Ψn dx = 2[g n (x) cos(k n x) + h n (x) sin(k n x)], (25) with
g n = dα r n dx -k n α i n h n = -(k n α r n + dα i n dx ). ( 26
)
where g n (x) and h n (x) have the lattice substrate periodicity. The F SAW (x, t) force writes then:

F SAW (x, t) = 2F ∆ 2 cos(Ω nex t) [g nex (x) cos(k nex x) + h nex (x) sin(k nex x)] (27)
The comparison of Eq. ( 27) to Eq. ( 9) shows that, as expected, the SAW force on the ad-atom, induced by the standing surface acoustic wave through the substrate, has the large scale spatial and time dependence of the corresponding standing wave. This dependence at 2π/k nex scale has been exhibited in Molecular Dynamic simulations 9 of ad-atom diffusing on a substrate submitted to a standing surface acoustic wave. However, at a finer scale, x smaller than the lattice parameter, this force experiences an amplitude and a phase modulation due to the presence of the crystalline potential through the functions g n (x) and h n (x). At this point, it is instructive to turn to a particular case by specifying the substrate and the interaction potential between the ad-atom and the substrate atoms, especially in order to get an explicit expression of the functions α n (x) and thus of g n (x) and h n (x). We assume that the substrate atoms have the same mass M and that the ad-atom interacts with each substrate atom through an attracting pair potential V pair (x -x i ) that cancels at infinity. We choose for V pair an exponential curve of extension σ (roughly the pair interaction range), i.e. a potential expression, that is physically meaningful and that allows the derivation of analytical calculations.

Φ(x, x 0 -N ...x 0 N ) = j V pair (x -x 0 j ) = - j V 0 e -|x-x 0 j | σ (28)
Where V 0 is the bonding energy. Note that minima of Φ correspond to atoms substrate positions. We have x 0 j = ja where a is the lattice spacing and j ∈ [-N, N ]. α n (x) then writes:

α n (x) = 1 √ M j e -ikn(x-x 0 j ) ∂Φ ∂x j (x -x 0 -N , ..., x -x 0 N ) = - 1 √ M j e -ikn(x-x 0 j ) dV pair dx (x -x 0 j ) = V 0 √ M j e -ikn(x-ja) d dx e -|x-ja| σ (29)
To take into account the discontinuties of the derivative of V pair at its minima, we define m 0 (x) and r(x) respectively, the quotient and the rest of the Euclidian division of x by a: x = m 0 a + r, with m 0 ∈ [-N, +N ] and 0 ≤ r(x) < a. m 0 (x) tells us in between which potential wells [m 0 a, (m 0 + 1)a] the ad-atom is and r(x) where it is exactly in between. Extending the size of the substrate to infinity (N → ∞) in Eq. ( 29), we obtain:

α n (x) = V 0 σ √ M   ∞ j=m0+1 e -ikn(x-ja) e x-ja σ - m0 j=-∞ e -ikn(x-ja) e -(x-ja) σ   (30) = V 0 σ √ M e -iknr+ r σ e a σ -ikna -1 - e -iknr-r σ 1 -e -ikna-a σ ( 31
)
α n (x) appears then as a function of r(x) only, which writes:

α n (r(x)) = V 0 σ √ M e ikna cosh( r σ ) -cosh( r-a σ ) cosh( a σ ) -cos(k n a) e -iknr
(32) From this expression of α n we deduce the following expressions for Ψ n , g n and h n :

Ψ n (x) = V 0 σ √ M e ikna cosh( r σ ) -cosh( r-a σ ) cosh( a σ ) -cos(k n a) e iknm0a , ( 33
)
g n (r) = V 0 σ 2 √ M [cosh( a σ ) -cos(k n a)] cos[(k n (r -a)] sinh( r σ ) -cos(k n r) sinh( r -a σ ) , (34)
h n (r) = V 0 σ 2 √ M [cosh( a σ ) -cos(k n a)] sin[(k n (r -a)] sinh( r σ ) -sin(k n r) sinh( r -a σ ) . (35)
Note that, since g n (x) and h n (x) in Eq. ( 27) have the lattice substrate periodicity, we have g n (x) = g n (m 0 a + r) = g n (r) and h n (x) = h n (m 0 a + r) = h n (r). One can easily verify that F SAW (Eq. ( 27)) is a continuous function of x, despite the discontinuity of the derivative of V pair . A more symetric expression can be obtained through the r = r ′ + a/2 translation, with now -a/2 ≤ r ′ ≤ a/2 (r ′ = 0 corresponds to the mid position between two successive potential wells, located at r ′ = ±a/2):

F SAW (x, r ′ (x), t) = F saw (r ′ ) cos(Ω nex t) cos[k nex (x -r ′ ) + ϕ 0 (r ′ )] = F saw (r ′ ) cos(Ω nex t) cos[k nex x + ϕ(r ′ )], (36)
with

F saw (r ′ ) = 2F 0 cos 2 k nex a 2 sinh 2 a 2σ cosh 2 r ′ σ + sin 2 k nex a 2 cosh 2 a 2σ sinh 2 r ′ σ 1/2 , = 2F 0 cos( k nex a 2 ) sinh( a 2σ ) 1 + 1 + tan 2 k nex a 2 coth 2 a 2σ sinh 2 r ′ σ 1/2 , (37) tan
(ϕ 0 (r ′ )) = tan k nex a 2 coth a 2σ tanh r ′ σ , (38)
F 0 = 2V 0 F ∆ 2 σ 2 √ M [cosh( a σ ) -cos(k nex a)] (39)
where F saw (r ′ ) and ϕ(r ′ ) = ϕ 0 (r ′ ) -k nex r ′ are respectively the amplitude and the phase of the large scale spatial dependence of F SAW (x, r ′ (x), t). Eq. (36) again the large scale spatial and time dependence of the SAW. This point is evidenced by evaluating the force at the substrate atoms positions, r ′ = ±a/2, and at the midway position between two successive potential wells, r ′ = 0: ] in Eq. ( 36)) and midle: interatomic potential (Eq. ( 28)), as a function of x/a, for kn ex a = 2π/15 and two values of σ/a: 0.5 (black) and 1.5 (red). Top and Bottom: envelop curve at the substrate atom positions (blue, Eq. ( 41)) and midway in between ( magenta, Eq. ( 40)). 36)) induced by the StAW and the interatomic potential (Eq. ( 28)) as a function of x/a for k nex a = 2π/15 and two values of σ/a: 0.5 and 1.5. The large scale spatial dependence in cos(k nex x) of the force F SAW (x, r ′ (x), t) is clearly evidenced, whereas the finer scale, between two successive potential wells exhibits the sinus hyperbolicbased dependence of the force evidenced in Eq. (37). As σ increases, the amplitude of the wells of the ad-atomsubstrate potential (Eq. ( 28)) and the amplitude of the variations of the force at both the large scale 2π/k nex and at the fine scale a decrease: indeed, if the interaction between the ad-atom and the substrate is less pronounced, the force induced by the wave on the ad-atom well will also be reduced on both fine and large spatial scales.

F SAW (x, r ′ = ±a/2, t) = F 0 sinh( a σ ) cos(k nex x) cos(Ω nex t) (40)
F SAW (x, r ′ = 0, t) = 2F 0 cos( k nex a 2 ) sinh( a 2σ ) cos(k nex x) cos(Ω nex t) (41)

IV. THE MEMORY KERNEL

Let's now study the memory kernel γ(x, x ′ , t-t ′ ) of the friction force (Eq. ( 19)) that depends on the α n functions through Ψ n (x) Eq. ( 22):

γ(x, x ′ , t -t ′ ) = n cos(ω n (t -t ′ )) ω 2 n dΨ n dx (x) d Ψn dx (x ′ )
Note that this memory kernel depends on the ad-atom position so that the dissipation term in Eq. ( 17) is nonlinear in the ad-atom variables. 15,[START_REF] Lindenberg | [END_REF] An explicit expression of γ is out of scope. However, since the α n functions are periodic functions of period the lattice parameter a, we can make an evaluation of the kernel without taking into account their spatial variations. They are then replaced in Eq. ( 19) by their mean value over the period a. This is equivalent to take into account only the first term, αn (0), of their Fourrier expansion:

γ(x, x ′ , t-t ′ ) ≈ n cos[ω n (t -t ′ )]e ikn(x-x ′ ) ω 2 n k 2 n αn (0)α n (0), (42) with αn
(0) = 1 a a 0 α n (x)dx. ( 43
)
Again using the particular inter-atomic potential (Eq. ( 28)) with Eq. ( 31) or (32) one gets:

αn (0) = 2ik n V 0 aσ √ M (k 2 n + 1/σ 2 ) ( 44
)
this approximation, the memory kernel reads:

γ(x -x ′ , t -t ′ ) ≈ 4V 2 0 a 2 M n k 4 n cos[ω n (t -t ′ )] ω 2 n σ 1 + k 2 n σ 2 2 e ikn(x-x ′ ) (45)
In the same spirit, we will use the Debye model, [START_REF] Ashcroft | Solid State Physics[END_REF] which is well adapted for simple monoatomic lattices at intermediate temperatures, to describe the phonon dispersion relation, ω n = c s k n , where c s is the speed of sound of the substrate; and change the discrete summation to an integral:

γ(x -x ′ , t -t ′ ) ≈ 4V 2 0 σ 2 a 2 M c 2 s kD -kD k 2 cos[c s k(t -t ′ )]e ik(x-x ′ ) [1 + k 2 σ 2 ] 2 g(k)dk, (46)
where k D = π a is the Debye wave number and g(k) = L/(2π) the density of states in the reciprocal space, with L = 2N a the size of the substrate. Moreover, since the function k 2 /[1 + k 2 σ 2 ] 2 is a peaked function centered at k = 0 of extension 1/σ, and considering that σ is generally larger than a, the limits of integration are extended to ∞. An integration by parts leads to the calculation of Fourier transform of Lorentzians and to the following approximated γ expression:

γ(x -x ′ , t -t ′ ) = LV 2 0 2c 2 s a 2 M σ H |x -x ′ + c s (t -t ′ )| σ + H |x -x ′ -c s (t -t ′ )| σ (47) with H(x) = (1 -x)e -x
The expression of the memory Kernel in Eq. ( 47) is an even function of x -x ′ and t -t ′ . The dependence on x -x ′ is a direct consequence of the elusion of the dependence of α n (x) on x (at the scale a) (see Sect. VI). We do not find for γ(x, x ′ , t -t ′ ) a simple exponentially decreasing function of |t -t ′ | as usually assumed in most textbooks [START_REF] Pottier | Physique statistique hors d'équilibre[END_REF] . However, we emphasize that the γ expression in Eq. ( 47) crucially depends on the interaction potential chosen (Eq. ( 28)) and that Eq. (47) provides a rather crude estimation of γ(x, x ′ , t -t ′ ): we have ignored the dependence of Ψ n on the length scale a and the extension of the integral Eq. ( 46) to infinity is a rough assumption (σ/a is not in general very large compared to 1). In addition, from Eq. ( 47) the correlation time appears to be of the order of σ/c s . Knowing that σ is of the order of magnitude of the lattice paramater, this correlation time is of the order of the inverse of the Debye frequency.

V. THE STOCHASTIC FORCE

In this section, we describe the properties of ξ(t), the stochastic force (Eq.( 20)). Since this force depends on the ad-atom position through the coupling term dΨn dx (x(t)), it represents multiplicative fluctuations. [START_REF] Lindenberg | [END_REF] Using Eqs. (15a) and (15b), it writes: This force partially results from the initial state of the substrate. In that sense, our system is completely deterministic. However, we have considered a quadratic approximation in Eq. ( 2) and a linear development of Φ in Eq. ( 5). In a real substrate, the non-linear terms can hold and/or exchange some energy with the normal substrate modes and in addition the substrate is never completely uncoupled to the experimental set-up. To take into account these exchanges of energy without explicitly describing them, we characterize the state of the substrate ( Q , Π) at t 0 using a probability distribution p( Q(t 0 ), Π(t 0 )), where Q and Π are vectors whose coordinates are the variables Q n and Π n . We suppose that the StAW forcing terms in Eq. ( 11) initially switched off are switched on at t 0 : the Hamiltonian describing our system at t < t 0 is thus given by Eq. ( 8).

ξ(t) = - n Q n (t 0 ) + Λ n,
Besides, if we want Eq. ( 17) to be regarded as a conventional Generalized Langevin equation, the quantity ξ(t) ought to have the properties that are expected for Langevin noise. Especially, its average is expected to cancel with respect to the probability dis-tribution p( Q(t 0 ), Π(t 0 )). [START_REF] Zwanzig | Nonequilibrium statistical mechanics[END_REF] In order to satisfy this last requirement, we choose the following expression for p( Q(t 0 ), Π(t 0 )):

p( Q(t 0 ), Π(t 0 )) = Z -1 e -βHs , (49)
where β = 1/(k B T ), k B the Boltzmann constant, T, the temperature of a surrounding thermostat that mimics the coupling of the system with the experimental set-up and H s given by:

H s ( Q, Π) = 1 2 n Π n Πn + ω 2 n Q n Qn + 1 2 n Q n .Ψ n (x(t 0 )) + Qn . Ψn (x(t 0 )) + 1 2 Q nex + Qnex F ω 2 nex ∆ 2 cos[Ω nex t 0 ]. ( 50
)
H s describes the coupling between the substrate and the ad-atom at position x(t 0 ) and contains a term derived from the StAW force to take into account the initial conditions imposed by the StAW on the Q n variables

FIG. 1 .

 1 FIG. 1. Schematic representation of the model under study.The ad-atom (red) and substrate atoms (blue) are characterized by their coordinates x and xj (j ∈ {-N..N }) in the reference frame of the center of mass of the substrate. Note that, for clarity reasons, the ad-atom is not reported on the same horizontal line as the substrate atoms, but the model is unidimensional.

5 FIG. 2 .

 52 FIG. 2. Top and bottom: maximum force induced by the StAW (t = 0[ 2π Ωn ex

Fig. 2

 2 Fig. 2 reports both the maximum force (t = 0[ 2π Ωn ex] in Eq. (36)) induced by the StAW and the interatomic potential (Eq. (28)) as a function of x/a for k nex a = 2π/15 and two values of σ/a: 0.5 and 1.5. The large scale spatial dependence in cos(k nex x) of the force F SAW (x, r ′ (x), t) is clearly evidenced, whereas the finer scale, between two successive potential wells exhibits the sinus hyperbolicbased dependence of the force evidenced in Eq. (37). As σ increases, the amplitude of the wells of the ad-atomsubstrate potential (Eq. (28)) and the amplitude of the variations of the force at both the large scale 2π/k nex and at the fine scale a decrease: indeed, if the interaction between the ad-atom and the substrate is less pronounced, the force induced by the wave on the ad-atom well will also be reduced on both fine and large spatial scales.

  nex F ∆ 2 cos[Ω nex t 0 ] +

						Ψn (x(t 0 )) ω 2 n	cos[ω n (t -t 0 )]
	+	Π n (t 0 ) ω n	-Λ n,nex	F ∆ 2	Ω nex ω n	sin[Ω nex t 0 ] sin[ω n (t -t 0 )]	dΨ n dx	(x(t)).	(48)

at t = t 0 The Hamiltonian H s is hence different from the H 0 one (Eq. ( 8)) of the system for t < t 0 i.e. the probability distribution p( Q(t 0 ), Π(t 0 )) corresponds to a non-equilibrium (macro-)state of the system described by H 0 coupled to a thermostat at temperature T. We will now establish the properties of the fluctuating force ξ(t) for the probability distribution Eq. (49).

The examination of Eqs. ( 48) and (50) reveals that the appropriate variables are:

With these variables H s and ξ(t) write:

From Eq. ( 49) and ( 52), variables Π n and R n appear as complex variables with centered Gaussian distributions of variance β -1 . Note however, that since Πn = Π -n and Rn = R -n , all these variables are not independent. One can easily re-write Eq. (52) using a set of 2N independent variables (R n , Π n ) with n > 0:

So that, for any two variables X and Y ∈ {ω n R n , Π n } (n > 0) , their mean values X are 0 and their covariances

From which we deduce the stochastic properties of ξ(t)

We recover in this last equation the fluctuationdissipation theorem: this result is especially independent of the precise expression of the potentials Φ and V sub in Eq. ( 1) as soon as this later can be approximated by Eq. ( 8). The same result has been demonstrated in a general frame by Zwanzig. 15 The non-null value of ξ(t) is related to the time-depend Hamiltonian (Eq.( 11)) and more precisely to the initial conditions that are imposed by abruptly switching on the StAW term at t 0 . The Hamiltonian H s Eq. ( 52) actually takes into account the initial conditions imposed by the StAW on the Q n variables but not on the Π n variables. As a consequence, the non-null value of ξ(t) is directly correlated to the initial conditions imposed on the Π n variables. To recover that the average value of the stochastic force cancels, we impose that Ω nex t 0 = 0[π]: this corresponds to switching on the StAW force at an extremum of the force.

VI. THE EFFECTIVE CRYSTALLINE POTENTIAL

The effective crystalline potential Φ eff (x) reads:

Using Eq. ( 22), ∆Φ 0 (x) writes:

∆Φ 0 (x) is then a periodic function of the lattice. ∆Φ 0 (x) physically corresponds to the modification of the potential seen by the ad-atom induced by the auto-coherent interaction between the substrate atoms and the ad-atom at position x. Such interaction also appears in the memory kernel. Actually, both terms ∆Φ 0 (x) and the memory kernel derive from the integration by parts of the third term of Eq. ( 13) leading to Eq. ( 14). The term ∆Φ 0 (x) derived from the third term of Eq. ( 14), corresponds to the static and instantenous modification of the substrate variables due to the presence of the ad-atom at position x, while the memory kernel derived from the fourth term of Eq. ( 14), corresponds to the retarded effects, i.e. how the past positions of the ad-atom influence the substrate positions at present. Both quantities ∆Φ 0 (x) and γ(x, x ′ , t -t ′ ) can be related introducing the function Θ(x, x ′ , t -t ′ ):

). An explicit expression of the spatial dependence of ∆Φ 0 (x) can be obtained using the particular inter-atomic potential (Eq. ( 28)), and the Ψ n expression of Eq. (33), in Eq. (58):

where the two sums are only numerical factors independant of x. We recover in Eq. ( 60) that ∆Φ 0 (x) is a periodic function of the lattice.

VII. CONCLUSION

We have studied the diffusion of an ad-atom on a substrate submitted to a StAW. We found that the ad-atom motion is governed by a Generalized Langevin equation:

We have characterized each of the terms involved in this equation and have given them their analytical expression and most of the time, an explicit expression. A keyresult is the expression of the force F SAW induced by the StAW as a function of x and t. F SAW essentially varies as cos(k nex x) cos(Ω nex t) where k nex and Ω nex are the spatial and angular frequencies of the StAW. However, a deeper analysis exhibits that this force also varies on the crystalline substrate lattice scale. The next paper of this series is devoted to the study of the solutions of the equation Eq. ( 61).