Damien Ehrhardt

Area, Space and Culture Reflections about 'Travelling Concepts' in

The notions of "cultural area / aires culturelles" and "cultural space" are "travelling concepts" as Mieke Bal calls it [START_REF] Bal | Travelling Concepts in the Humanities. A Rough Guide[END_REF] . They travel between cultures, disciplines (anthropology, geography, history, geo-strategy...), and time. To understand their history, it is necessary to refer:

-more generally to the concept of culture, which can be defined in many ways. Its sphere varies with the language (there are plenty of meanings of the only notion of culture in English to what you can add those of analog expressions in other languages); -more specifically, notions such as Kulturkreis, cultural landscape, ethnoscape, third space, Kulturraum… that have been developed in some disciplines or beyond disciplines, in various linguistic areas at different times. In this paper, I will consider only some French, German and English concepts. The aim of my paper is not to draw up a detailed history of all of these notions, but to place some of them in their historical and political contexts in order to provide a critical approach of cultural area / aires culturelles / cultural space in the contemporary world.

Aires, Spaces and Cultures

Before turning to historical and intercultural considerations, I will begin to cross the definitions of the French words aire and espace with these of culture. I based my precedent research on the meaning of these terms in French. Aire (in latin: area) designates a lot of significations:

-the floor where grain is beaten, -the flat surface of the rock where the eagle make his nest, -the delimited grounds reserved for a certain type of activity (aire de jeu). It is similar to the meaning of area as a bounded portion of a surface, a region (playing area), -the surface of a planar region, a solid or something else and its measures in square units 2 (it is also similar to the definition of area) Contrary to the definition of area in English, the notion of aire concerns neither open, sunken places, nor fields of experience, activity or knowledge. The stability suggested by the word aire came up again in the notion of aire continentale. This expression designates a vast region of the Earth's crust which remains stable over geological time. Beyond the notion of stability, the previous frame of the nest evokes the protective and perhaps confined nature of this surface.

The French word espace (from latin: spatium) is very closed to space. It designates:

-an indefinite extent that surrounds and contains all objects, -its representation, -in mathematics, a set of points (vectors), with a structure, -the three dimensional geometry in space, as opposed to the plan, -the volume occupied by something, -the environment affected for a particular purpose (advertising space), -the extent in which the stars move 3 . But the notion of espace or space is much wider in scope than the concept of aire. It can be used to designate imaginary and chimerical ideas (imaginary spaces), time (a space of time), and emptiness (Newton's absolute space, the space between two words). Within his "triplicity of space", Henri Lefebvre crossed different meanings of the term: the spatial practice of a society, the representations of space (l'espace conçu) and the spaces of representation (l'espace vécu) [START_REF] Lefebvre | La production de l'espace[END_REF] .

As we saw, aire and space seems to be quite opposite. That means: when a theorist chooses the word aire/area or space, the meaning may be very different. In order to situate different theories between these two poles, I propose to elaborate two opposite ideal types (M. Weber): It would be interesting to bring intercultural theories in relation with these concepts of culture. In this sense, the opposition between material and spiritual culture is crucial. Indeed, there is a similarity between the intellectual-aesthetic culture and the anthropological one. Both are based on a canonization process, linked to many values: aesthetics, ethics, religion… The situation of different theories of interculturality in a multipolar space à la Bourdieu is useful to prevent bipolarity (aire vs space, material vs spiritual culture). This space opposes, on the horizontal axis, the poles of aire and space; on the vertical axis that of "material" and "spiritual culture". All concepts will be situated between these poles, within each of the following four quadrants:

AIRE ESPACE/SPACE two-dimensional three-dimensional
-A, bottom right: cultural spaces taking into account material culture; -B, bottom left: cultural areas taking into account material culture; -C, top left: cultural areas based almost exclusively on spiritual culture; -D, top right: cultural spaces based almost exclusively on spiritual culture.

Quadrant A: older notions of Kulturkreis and Cultural Landscape

Theorized by scientists as Friedrich Ratzel, Leo Frobenius or Fritz Graebner, the concept of Kulturkreis (Cultural Circle) appeared in the late 19 th century. According to them, different cultures would not develop uniformly, but have gone through successive stages, during which they were able to transform themselves. A limited number of Cultural Circles may grow from a cultural center in various places at different time. Ratzel's anthropo-geography, for example, links human societies to natural environments. The societies, and more generally the states, are compared to living organisms that grow before declining. According to him, the development of the population depends on his environment: when people are well adapted to a country, they tend to invade another one. Ratzel concludes that people are constantly moving and that they should be expansive when they are sufficiently strong. Borders are seen as the expression of an organic movement, which leads to justify the recovery of neighboring areas. Ratzel's ideas, including the notion of Lebensraum 5 , were instrumentalized by the Nazis. This had discredited space studies for a while after WWII. But the new visions of Michel Foucault [START_REF] Foucault | Des espaces autres[END_REF] and Henri Lefebvre 7 lead, after the spatial turn, to the rehabilitation of this domain of research. The notion of Kulturkreis is linked essentially with material culture and is closer to the conception of space than those of aire, as the organic and dynamic character of the Cultural Circles witnesses.

Influenced by the German geographer Johann Sölch 8 , the US-geographer Carl Sauer, founder of the Berkeley School, developed since 1925 the concept of cultural landscape [START_REF] Sauer | The Morphologie of Landscape[END_REF] . Applied to human activity in its natural environment, this notion takes in account the relationships between groups, cultures and natural environment 10 . Far away from rigidly defined areas as will do some of his students and other theorists from quadrants B and C, Sauer is particularly interested in material culture, natural environment, and in how these could be jeopardized [START_REF] Claval | Des aires culturelles aux réseaux culturels[END_REF] .

Quadrant B : the concepts of aires de civilisation (Mauss & Braudel)

According to Marcel Mauss, in 1929, "a civilization is both an area and a form" [START_REF] Mauss | Les civilisations. Eléments et formes[END_REF] . "The form of a civilization is the sum total of the specific appearances taken by the ideas, the practices and the products common or more or less common to a certain number of given societies". The "areas of civilization" (aires de civilization) include societies whose representations, practices and products form a common heritage. A civilization has "always its breakpoints, its limits, its core and its periphery. The description and definition of these areas are a key work in the history […]." 13 Mauss believed that civilizations can be circumscribed not only by their capacity of borrowing from other cultures and their will to expansion, but also by the internal resistances of the societies that compose them. However, he notices a trend towards a kind of standardization in the process that we call nowadays globalization, largely due to the impact of science and cinema. Among the facts of civilization, Mauss mentions very different elements like industries, arts, science, local flavors, layout of ground... Therefore, his notion of civilization is linked to material culture. The fact that civilizations tend to isolate themselves refers further to the notion of aire than to those of cultural space.

Mauss has already highlighted the temporal dimension of civilizations. Braudel insists further on this point: for him, civilizations are perennial and their history is part of the "longterm", since they survive even revolutions. Cultural traits can show a great diversity: shape and material of houses, languages and dialects, culinary tastes, agriculture (he differentiates the civilizations of wheat, maize and rice)… Braudel therefore takes account of material culture, but he also determines cultural areas which remain relatively stable on long-term scales:

-European civilizations: America and the other Europe (Muscovy, Russia, USSR), -Non-European civilizations: Islam and the Muslim world, the black continent and the Far-East.

Quadrant C: the clash of civilizations (Huntington)

The clash of civilization is a notion proposed by the political scientist Samuel P. Huntington, who was an expert of the US National Security Council during the Carter administration. His notion flourished in the context of economic globalization following the fall of the Berlin Wall [START_REF] Huntington | The Clash of Civilizations?[END_REF] . According to him, the borders of the world are less politically determined but related to cultural areas that prophesy the increasing of the clash of civilizations. Huntington distinguished nine major civilizations: "Western" (Europe, North America, Australia and New Zealand), "Latin American", "African" (Sub-Saharan Africa), "Islamic", "Chinese" (including Korea, Vietnam and the Philippines), "Hindu" (India and Sri Lanka), "Orthodox" (Russia and partly in Eastern Europe and South-East), "Buddhist" (Tibet, Mongolia and Southeast Asia) and Japanese [START_REF]Le monde à l'époque du choc des civilisations après 1990[END_REF] . The reunified Western world fades in comparison with other regions which affirm their cultural independence and reject "Western intoxication": Huntington emphasizes the rise of civilizations that he describes as "Chinese" and "Islamic" 16 . The names given to his cultural areas show the importance of religions, especially those that are not "Western". Their renewal would be the most powerful manifestation of anti-westernism in non-Western societies 17 . The gap between the West, now reunited, and other civilizations has been masked before the fall of the Berlin Wall, by the antagonism issued from the Cold War. New alliances between civilizations in our complex world are motivated by common interests or enemies 18 . Much of the conflict would be located along the fault line between "Islamic" civilization and other areas. This theory develops precisely in the context of the events of 11 September 2001 and the Iraq War in 2003. If Huntington's theories continue to play a major role from a geo-strategic point of view, they remain highly contested, particularly because of the essentialist vision of civilizations, regarded as homogeneous entities back to themselves. In reality, cultural areas interact always, even in times of war. And even the Middle East conflict, essentially related to religion and culture, depends on a complex set of factors, including the legacy of colonialism [START_REF] Lüsebrink | Interkulturelle Kommunikation[END_REF] . Among many critics of Huntington's theory, we can mention those of geographer Yves Lacoste. He provides an arrangement of the world into five groups: the West, the Islamic world, India, China, and the islands and peninsulas of Western Pacific [START_REF] Lacoste | Tout sauf la fin de l'histoire[END_REF] . Unlike the civilizations of Huntington, Lacoste sets allow interpenetration of areas and the internal diversity of these. So he says that the clashes occur rarely in the fault lines between civilizations but more frequently within the same cultural areas [START_REF]Le choc des civilisations, vrai ou faux débat?[END_REF] . Far away from material culture, Huntington refers primarily to the religious character of non-Western civilizations. In addition, his essentialist vision remains close to the ideal type of aire. The model of Lacoste, however, admitting blurred and shifting boundaries, is more oriented towards quadrant D that brings together the latest concepts of ethnoscape (Appadurai) and third space (Bhabha).

Quadrant D: the postcolonial deconstruction of cultural areas (Appadurai, Bhabha)

Since 1989, the world has seen the clash of civilizations gaining ground. At the same time, area studies have been questioned. According to Arjun Appadurai, "thus left-wing critics of area studies, much influenced by the important work of Edward Said on orientalism, have been joined by free-marketeers and advocates of liberalization, who are impatient with what they deride as the narrowness and history fetish of area-studies experts". But Said, like the author of Orientalism, has often been regarded as the initiator of postcolonial studies. To what extent, concepts such as third space or ethnoscape, at the foundation of postcolonial theory, participate in the deconstruction of cultural areas? Homi K. Bhabha has much developed the concept of "third space" which is situated beyond the bipolarity: "existence" of nation / alterity of other nations. This space, those of DissemiNation (i.e. cleaved and hybrid Nation), is "internally marked by the discourses of minorities, the heterogeneous histories of contending peoples, antagonistic authorities and tense locations of cultural difference." [START_REF] Bhabha | Les lieux de la culture. Une théorie postcoloniale[END_REF] The third space applies therefore to a performative utterance, far away from the material sense of culture. Furthermore, because of its hybrid and heterogeneous character, the third space refers much more to the notion of space than to that of area.

In the current globalization, we are caught in cultural flows of different dimensions, depending on the adopted point of view. In this regard, Arjun Appadurai identifies five dimensions of global cultural flows that can be termed: ethnoscapes, mediascapes, technoscapes, financescapes, and ideoscapes 23 . These flows draw in our imagination fluid and irregular shaped landscapes. At the post-colonial era, Appadurai want to break with the structural opposition between center and periphery. Borrowing from the idea of rhizomechannel [START_REF] Cf | 1. Introduction : Rhizome[END_REF] , he focuses on the flux of globalization and the experience of migration resulting from it. Appadurai defines the concept of ethnoscape, which interests us here in the first place, as follows:

"By ethnoscape I mean the landscape of persons who constitute the shifting world in which we live: tourists, immigrants, refugees, exiles, guest workers, and other moving groups and individuals constitute an essential feature of the world and appear to affect the politics of (and between) nations to a hitherto unprecedented degree. This is not to say that there are no relatively stable communities and networks of kinship, friendship, work, and leisure, as well as birth, residence, and other filial forms. But it is to say that the warp of these stabilities is everywhere shot through with the woof of human motion, as more persons and groups deal with the realities of having to move or the fantasies of wanting to move." [START_REF] Appadurai | Après le colonialisme[END_REF] The notion of ethnoscape refers to images and landscapes, which are shared into the identification process of diasporic communities. The construction of this kind of transnational identities is based on very different elements, like nostalgic memories or new communication technologies. The ethnoscape concerning at first the creation of transnational identities in the global city is very far from material culture. Because of its rhizomic and diasporic nature, this notion contributes to the deconstruction of cultural areas and heralds a new form of transnational space. Appadurai considers the area studies tradition as a "double-edged sword" 26 . This tradition has been a "tiny refuge for the serious study of foreign languages, alternatives worldviews, and large-scale perspectives on sociocultural change outside Europe and the USA". Area studies have been "one of the few serious counterweights to the tireless tendency to marginalize huge parts of the world in the American academy and in American society more generally." 27 Appadurai regrets however that the area-studies tradition is probably too comfortable with: "its own maps of the world, too secure in its own expert practices, and too insensitive to transnational processes both today and in the past. […] What does need to be recognized, if the area-studies tradition is to be revitalized, is that locality itself is a historical product and that the histories through which location emerge are eventually subject to the dynamics of the global." 28 A path between aire and space, material and spiritual culture It is possible to locate graphically into the multi-space the various models discussed above, depending on their proximity to the poles of aire and space, and them of spiritual and material culture:

The fact that the newer models are moving away from material culture is most certainly due to the global dissemination of innovation and agricultural products. Thus, the tools or technologies cease to be specific to a natural environment or a cultural area. On the other hand, with the standardization of technologies and constructions (international airports or industrial areas), the diversity of landscapes tends to disappear. It also adds the importance of religions, whose visibility is increased after the fall of the Berlin Wall, in a world whose borders are less political than civilizational.

How to explain the transition from space to aire and then the return to space? After World War II, the concept of time has been preferred to that of space. This is due especially to the instrumentalization of the latter notion by Nazi propaganda. The term Raum has hardly been the subject of studies in German-speaking countries, with the exception of its use by authors living at the turn of the 19 th and 20 th Centuries as Georg Simmel or Walter Benjamin. The appearance of the concept of cultural area or aire culturelle is at least linked to the two following factors:

-the emergence of area studies which are much developed in the USA mainly for geostrategic reasons, notably under the impetus of the National Defense Education Act (1958); -the construction of a sustainable European civilization after the horrors of the two world wars. This civilization, based on the spirit of the Enlightenment, would lie beyond national and cultural differences. If European civilization is seen as a universal model in the 1960s, the situation changed in the 1990s. After the fall of the Berlin Wall, the worldview of a clash of civilizations emerges by neoconservatives in the USA. According to them, the reunited West (Europe, North America, and Australia) would be the target of other civilizations. With the end of "grand narratives" [START_REF] Lyotard | La condition postmoderne[END_REF] , post-modernity accords less emphasis to the temporal category. Meanwhile, the concept of space is reintroduced into international research since the mid-1980s. The end of the Cold War and the expansion of markets in the context of economic globalization have profoundly changed the representation of space and world mapping. Nowadays, global development depends not only on nations, but also on a constellation of relational dependency networks and international mutual relationship. Finally, the choice of models and their names also depends on the disciplinary culture of the theorists 30 . Indeed, the fact that Braudel's civilizations respond more to the ideal type of aire than those of space is largely related to his specialty, history, which was leading him to the question of long-time. It is quite different by authors whose theories are registered in the C-quadrant. Their specialty is in the field of geopolitics or international strategy. However, theorists oriented to cross-disciplinary research or cultural studies are located more towards the ideal type of space than those of aire. Thus, the A and D-quadrants fall within cultural sciences (Kulturwissenschaft/en) which designate:

-the cultural studies developed at the Centre for Contemporary Cultural Studies (CCCS) of the University of Birmingham, then in the USA and now all around the world; -the older concept of Kulturwissenschaft that emerged in Germany at the turn of the 19 th and 20 th century. Considered as too vague, Heinrich Rickert replaces the notion of Geisteswissenschaft by that of Kulturwissenschaft opposed to Naturwissenschaft. As nature is meaningless and valueless, a scientific approach to culture is necessary. If this first "cultural science", partly linked to neo-Kantianism, feeds upon the work of philosophers and theorists such as Georg Simmel, Max Weber or Ernst Cassirer, it has also been instrumentalized, like Ratzel's notion of Lebensraum. Because of these different approaches, it is essential, in order to reflect on cultural area and space, to bring together representatives not only from different disciplines (anthropology, history, geography, political science...) but also, beyond disciplines, from cultural studies in the wider sense of the term. Some perspectives... It is possible to envisage different perspectives. One solution is to rethink area studies by taking into account, f.e. overlays of areas and nomadology [START_REF] Cf | La recherche universitaire à l'heure de la mondialisation[END_REF] . Another solution is to move beyond cultural areas [START_REF]Au-delà des Area Studies -perspectives transculturelles et comparatistes is f.e[END_REF] . In this context, it would be interesting to establish a new theory by crossing area studies with other theories of interculturality as Empire or transfer studies [START_REF] Cf | Transfer studies : une introduction[END_REF] . The ideal positioning of this new theory would be the middle of the multipolar space previously defined, halfway between the poles "material" and "spiritual culture", but also between the ideal types of aire and space. If transfer studies, especially in its recent transformation, tend more towards the notion of space, the Empire studies, however, accommodate themselves rather with a return to the aire culturelle. The planetarism of Gayatri Spivak or Sonja Neef can be added to this [START_REF] Cf | Imperative zur Neuerfindung des Planeten / Imperatives to Re-Imagine the Planet[END_REF] . Nowadays the planet is seen as more than a natural space devoted exclusively to the market. This new point of view would mark a slight shift from spiritual culture back to material culture. Therefore, the center of gravity of this new theory could well be situated in the middle of the multipolar space.

The concept of intellectual-aesthetic culture (intellektuell-ästhetischer Kulturbegriff) related to Bildung and art. It derives from the traditional canon of great

		or more
	definite, delimited	indefinite, infinite
	multiple	united
	stable	variable
	close and confine	open
		writers, artists or
	composers, and their works that have played a central role. This notion is linked with an
	elitist conception of culture based on aesthetic, moral and ethical values, and opposed to
	popular or mass culture;
	2.

In my paper, aire and space are related to culture. Beyond the numerous significations of culture or Kultur, Hans-Jürgen Lüsebrink underlines three concepts:

1.

The concept of material or instrumental culture (materieller oder

	instrumenteller
	Kulturbegriff) that comes from the etymological sense of agricultura;
	3.

The concept of cultural anthropology (anthropologischer

  Kulturbegriff), which means all the collective representations, patterns of thought, and actions of a society.


B Jurado
email: jurado@cenbg.in2p3.fr

H Savajols

W Mittig

N A Orr

P Roussel-Chomaz

D Baiborodin

W N Catford

M Chartier

C E Demonchy

Z Dlouhý

A Gillibert

L Giot

A Khouaja

A Lépine-Szily

S Lukyanov

J Mrazek

Y E Penionzhkevich

S Pita

M Rousseau

A C Villari

Mass Measurements of Neutron-Rich Nuclei Near the N=20 and 28 Shell Closures

Mass measurements of very neutron-rich nuclei near the N=20 and 28 shell closures are presented. Seven masses have been determined for the first time and the precision of 36 masses has been significantly improved. These results are used to investigate the evolution of the odd-even staggering of binding energies with neutron number. Special attention is paid to the evolution of the N=28 shell closure as the neutron dripline is approached. Changes in shell structure are observed around N=28 for the P and S isotopes but not for Si. This may be interpreted as a persistence of the shell closure at N=28 or as the result of a very sudden onset in deformation at 42 Si.

The mass of a quantum mechanical system, such as the nucleus, is a fundamental quantity as it reflects the sum of all forces acting on it. Of particular interest are direct mass measurements far from the valley of stability (such as those described in this work) which permit tests of the reliability of nuclear mass models to be made and studies of the evolution of shell or subshell closures and correlations to be undertaken. However, the direct measurement of masses far from stability presents significant technical challenges. The principal one being the limited production cross sections. In addition, exotic nuclides are by definition very short lived. The development of a fast measurement technique is therefore imperative, but such technique must also be of sufficiently high resolution to make precision measurements. In the present work we describe how the time-of-flight technique using the SPEG spectrometer [1] at GANIL has been improved in order to determine the masses of very neutron-rich nuclei in the vicinity of N=20 and 28. The results are then used to study the evolution of the odd-even staggering (OES) of nuclear masses with neutron number. When representing nuclear masses as a function of the neutron number one observes that even-neutron-number nuclei are more strongly bound than their oddneutron-number neighbours. The OES originates from two fundamental physical mechanisms: the breaking of the mean-field spherical symmetry and pairing correlations [2]. Both mechanisms are strongly influenced by the number of neutrons. For instance, it is now well established that de-1 formation effects are responsible for the vanishing of the N=20 shell closure for the very neutronrich sodium and magnesium isotopes (see, for example, refs. [3][4][5]). Particle correlations occur mostly in a narrow zone of the phase space around the Fermi surface. In drip-line nuclei the Fermi surface is very close to the single-particle continuum. Consequently, the scattering of virtual pairs into the continuum has to be considered. The increase in neutron-pairing correlations as the neutron binding energy decreases seems to be a quite general result of calculations that include continuum effects (see for example refs. [6][7][8][9]). Our results also provide key information to investigate the evolution of the N=28 shell closure when approaching the neutron dripline. Deformation effects and the vanishing of the spin-orbit force could cause an erosion of this shell closure for very neutron-rich nuclei. Finally, our results will be used to check the predictive power of several mass formulas.

For direct mass measurements of exotic nuclei three main issues have to be faced: the production rates, the mass resolution and the systematic errors. A series of earlier mass measurements [10] (and refs. therein) has shown that this can be achieved at GANIL by combining a time-of-flight (TOF) technique with the highresolution energy-loss spectrometer SPEG. The very broad elemental and isotopic distributions resulting from heavy-ion projectile fragmentation reactions combined with fast in-flight magnetic selection allows the mapping of an entire region of the nuclear mass surface in a single measurement. Compared to other mass measurement methods, the availability at GANIL of very intense neutronrich beams (such as 48 Ca) together with the relatively high transmission rates through the SISSI device [11] and the alpha-spectrometer2 enables measurements to be made far from stability with reasonable yields. Nuclei with lifetimes as short as the flight time through the system of ∼ 1µs can be measured. The nuclei investigated here were produced by fragmentation of a 48 Ca beam of 4 µAe at 60.3•A MeV on a Ta target. The TOF technique requires the simultaneous determination of the masses of well-known nuclei for calibration. Consequently, apart from the neutronrich nuclei of interest, it is of great importance to measure simultaneously a broad range of reference nuclei. To achieve this, a Ta production target with three different thicknesses and two magnetic rigidities of the beam line and spectrometer were employed. In addition, a thin (25 µm) Be achromatic degrader was placed at the dispersive focal plane of the alpha spectrometer. In this manner, the light ions that caused saturation and pile-up in the detection system in previous measurements were eliminated. As a consequence, the 48 Ca beam intensity could be increased by more than one order of magnitude with respect to the previous experiment [10].

The principle of mass determination by means of the TOF technique relies on the relation between the magnetic rigidity, Bρ, and the velocity, v, of an ion of rest mass, m 0 , and charge, q, traversing an achromatic system: Bρ = γm 0 v/q, where γ is the Lorentz factor. A precise measurement of the magnetic rigidity and the velocity allows the ratio m 0-exp /q (where m 0-exp is the experimental value of the mass of the ion) to be deduced. Once the ion has been identified in A, q and Z, m 0-exp can be extracted. The velocity of the ions is obtained from the TOF measurement. Detailed descriptions of the technique may be found in [10] and refs. therein.

The atomic mass excesses are obtained by means of a multidimensional fit where the mass excess is expressed as a Taylor series development of the form (for

q = Z), M j (A, Z) = Z(m 0-exp /Z + α j 1 + α j 2 A Z ) + α j 3 A + α j 4 Z + f j 1 (AZ, A 2 , Z 2 , A 2 Z 2 , A 3 , ...) + f j 2 (∆E) The constant α j
1 and the first order terms serve to transform m 0-exp into the atomic mass excess. The function f j 1 , which is a linear combination of higher-order terms in A, Z, A/Z and AZ, and the energy-loss (∆E) dependent function f j 2 are required to correct for the systematic uncertainties associated with the technique. In contrast to previous experiments where no achromatic degrader was employed, higher order terms were required here to correct for the associated aberrations.

The coefficients of the fit α j i are obtained by minimizing the difference between the experimental and the adopted reference mass excesses of the 2003 atomic-mass evaluation [12]. The unknown masses are then determined using the coefficients and functions from the best fit. The uncertainties associated with such a determination arise not only from the statistical uncertainty, but also from the need inherent in the method to interpolate between and extrapole from the reference masses, as well as the systematic uncertainties which are a measure of the limiting precision of the measurement. In the present work a number of independent measurements 3 were made resulting in up to 5 independent mass determinations (denoted by the index "j") for many of the nuclei. Each measurement was analysed separately following the procedure outlined above. The uncertainty ∆M j (A, Z) for each measurement was determined from the combination in quadrature of the statistical and the systematic errors. The statistical error varied from a few tens of keV for nuclei situated close to the line of stability, to around 1 MeV for nuclei in the vicinity of the neutron dripline where the production rates were very low. The systematic error for each of the measurements was estimated to be 150 keV. The final mass excesses, M (A, Z), quoted in Table 1, are the weighted means of the masses derived from each of the independent measurements. In the present experiment several combinations of higher order terms allowed acceptable fits (as defined by the chi-squared per degree of freedom) to be obtained. The variations in the masses derived from these fits provided a measure 4 of the uncertainty arising from the extrapolations noted above. The uncertainty in the adopted mass excesses (Table 1) was thus derived as the combination in quadrature of that of the weighted mean and that determined for the extrapolation. The results of the analysis described above are listed 3 These corresponded to runs with different settings of the rigidty of beamline and spectrometer, together with a parallel chain of electronics. 4 Estimated by calculating the dispersion between the results of the acceptable fits M i (A, Z) and the final adopted mass excesses -∆Mext(A, Z)

= i (M (A, Z) -M i (A, Z)) 2 .
in Table 1, where seven new masses can be seen to have been determined: 23 N, 31 Ne, 35 Mg, 36 Mg, 42 Si, 44 P and 47 Cl. In addition, the precision of 36 masses has been considerably improved with respect to the compilation of ref. [12]. No statistically significant discrepancies between our results and the 2003 atomic-mass evaluation [12] are observed to occur.

The one-neutron separation energy S 1n is the most straightforward observable reflecting the OES of binding energies. It is defined as:

S 1n (A,Z) = [M(A-1,Z) -M(A,Z) + M n ]c 2
where M n is the neutron mass excess. In order to illustrate the odd-even staggering, figure 1 plots S 1n (N) as a function of S 1n (N-1) for four isotopic chains. Two diagonal lines can be clearly distinguished. The upper line corresponds to nuclei with an even neutron number (N) for the ordinate which are more strongly bound. The lower line corresponds to nuclei with an even neutron number (N-1) on the abscissa. The separation between the two lines reflects the intensity of the odd-even staggering. Indeed, it is easy to see that if S 1n (N)=const+δ for even N, and S 1n (N)=const for odd N, the points formed in a diagram x=S 1n (N-1) and y=S 1n (N) will lie on two lines, y = x -δ and y = x + δ. The separation between these two lines is then equal to δ• √ 2. The nuclei with the lowest S 1n are the most neutron-rich ones. This original diagram is therefore well suited to trace the dependence of the OES on binding energy. The OES staggering is often taken as an indicator of the strength of pairing correlations. As outlined above, most of the theories including continuum effects predict an increase in neutron pairing correlations with increasing N/Z. However, our results indicate that this does not translate to an increase in the OES. Instead, figure 1 shows an overall reduction of the OES when the binding energy decreases for Z=10 and 12. A strong attenuation of the OES close to the neutron dripline has been predicted for fluorine isotopes by [7]. As described in ref. [7], this does not imply a decrease in the pairing correlations as a decrease in the OES may result from the np-continuum coupling, even if pairing increases. Here, as noted above, we see such a reduction not only for Z = 10, 12 (figure 1) but also for the

This work

Ref. [ 1 Experimental atomic mass excesses (±uncertainties) in keV. In the last column the weighted mean of columns 2 and 3 is given. oxygen isotopes (not shown). For all the other elements the OES attenuation is less evident and we observe a stabilisation in the OES when approaching the neutron dripline. To illustrate this the Z=13 and 14 isotopic chains are also shown in figure 1, whereby the OES staggering gap is clearly apparent but for the lowest values of S 1n the separation between the two groups of data remains rather constant. This indicates that features other than correlations are important. For instance, the spin-isospin coupling, discussed in ref. [13], and thought to be responsible for the deformation effects around N=20, will introduce an effect dependent on the filling of shells. where N is the neutron number. The triangles are from the mass table [12], the diamonds and the squares represent the new and improved (respectively) results measured in the present work.

Several theoretical studies have investigated how to separate the pairing and the mean-field contributions to the OES. The authors of refs. [2,14] proposed to extract the pairing contribution to the OES from experimental data by using the three-point indicator:

∆ 3 (N)=(-1) N [M(N-1)+M(N+1)-2M(N)]c 2 /2.
It was demonstrated in ref. [2], that the indicator ∆ 3 evaluated for odd-N can be roughly associ-ated with the pairing effect, while the differences of ∆ 3 at adjacent even and odd values of N provide information related to the spacing between single-particle levels, that is, information related to the mean-field contribution. The values of ∆ 3 calculated using the experimental masses tabulated in ref. [12] and the results of the present work are displayed in figure 2 as a function of neutron number for four isotopic chains. Let us first consider the case of Ca isotopes (Z=20). As the difference between ∆ 3 evaluated at adjacent neutron numbers is sensitive to the single-particle energy differences, it peaks at the magic numbers N=20 and 28. The ∆ 3 is almost constant between N=20 and 28 because, for spherical nuclei, the single-particle energy is constant within a shell. At N=Z a strong jump in the ∆ 3 is observed for all nuclei, and can be attributed to the Wigner term (see, for example, ref. [15]). The Si isotopes (Z=14) show a very similar behaviour to that of Ca between N=20 and 27, which indicates a regular filling of the f 7/2 orbital. On the contrary, the behaviour of the P (Z=15) and S (Z=16) chains is different from that of Ca. A local maximum appears at N=26 rather than at N=28. Such an effect at N=26 was already observed in our previous measurement and was attributed to deformation [10]. Owing to the reduction in the uncertainties in the masses for the most neutron-rich P and S isotopes, the effect at N=26 and the vanishing of N=28 as a shell closure have become considerably more apparent than in ref. [10]. Moreover, the present data clearly show that there is no such effect at N=26 for the Si isotopes.

Apart from the effects of deformation, an erosion of the N=28 shell closure could also result from a decrease in the spin-orbit interaction far from stability [16]. The case of 42 Si is of particular interest as it could be stabilised against deformation by the Z=14 sub-shell closure. Interestingly, the theoretical predictions for this nucleus are rather contradictory. On the one hand, shell model calculations suggest that 42 Si has the characteristics of a doubly-magic nucleus such as 48 Ca [17]. The same conclusion was reached in ref. [18], where the deformed configuration was found to be located 1 MeV above the ground state, whereas in other N=28 nuclei (such as 40 Mg) the deformed intruder state is well below the closed shell. On the other hand, relativistic Hartree-Bogoliubov calculations predict a strong oblate deformed configuration for 42 Si [19], in agreement with relativistic mean field calculations with BCS pairing (RMF-BCS) [20] and with the results of the latest version of the Skyrme-Hartree-Fock-Bogoliubov (HFB-9) model [21]. While refs. [19] and [20] predict a rapid transition from prolate to oblate deformation at N=26, for HFB-9 [21] this sudden change in deformation arises at N=28, that is, when going from 41 Si to 42 Si. From the experimental point of view the situation is rather unclear as well. Grévy et al. have measured the β-decay half-lives of several Si isotopes [22]. The short half-life of 42 Si could only be reproduced theoretically assuming a strongly deformed configuration. The present mass measurements of the most neutron-rich Si isotopes are important in order to confirm the conclusions of [22], because the Q β influences to the fifth power the lifetime. Contrary to the conclusions of ref. [22], high-energy two-proton removal cross sections populating 42 Si from a 44 S beam were interpreted as a signature of the doubly magic character of 42 Si [23]. However, in a more recent article it was recognized that the two-proton knockout cross section populating 42 Si is not sensitive to the size of the N=28 gap [24]. The mass of 42 Si can clearly aid in shedding new light on this question. From the mass measured in the present work one-neutron separation energies of 5.03±0.69 MeV ( 42 Si) and 4.96±0.24 MeV ( 40 Si) were obtained. For the even Si-isotopes, the tabulated S 1n decreases by about 1 MeV when going to the next even isotope. One would then expect an S 1n of about 4 MeV for 42 Si. Our results thus suggest a possible increase in binding as would be expected from a shell closure. Figure 3 represents the microscopic energy as a function of neutron number for the same isotopic chains as in figure 2. The microscopic energy, a convenient quantity to inspect the presence of structure effects in nuclear masses, has been obtained by subtracting the macroscopic component (as given by the Finite Range Liquid Drop Model FRLDM [25]) from the experimental mass excess. The variation of the microscopic energy as a function of neutron number should exhibit deep minima at the shell closures. In figure 3 a strong change in slope at N=20 and near N=28 can be observed for Z=20. However, the Z=15 and Z=16 isotopes do not show such features at N=28 and present instead a discontinuity in the slope at N=26, as observed in our earlier work [10]. This discontinuity is absent for the Z=14 isotopes. The similar behaviour of the microscopic energy for Z=14 and 20 could be considered evidence of a shell closure at N=28, in agreement with the discussion of the OES above. However, in a recent γ-spectroscopy experiment [26] evidence for a low lying 2 + state in 42 Si was found, which is not compatible with such a picture. All these observations could be consistent, however, if deformation effects appear suddenly at N=28. Reliable mass formulas that permit extrapolations to the unknown regions of the nuclear chart are important for many domains, such as nuclear astrophysics, and many mass formulas have been developed in recent years [27]. Our results provide new data to test the predictive power of different mass formulas when approaching the neutron dripline. In figure 4 we have compared our experimental values of ∆ 3 for the Si isotopic chain with the HFB-9 mass formula [21], the Duflo-Zuker formula [28], and with the RMF-BCS model [20]. One can see that the best description is given by the Duflo-Zuker formula. This is also true for the other isotopic chains, in agreement with [27]. The HFB-9 model does, for example, not reproduce the shell closure at N=20 and, indeed, for N=21 and 22 it overpredicts the mass. For N >22 the differences between the HFB-9 mass formula and the data decrease. The values of ∆ 3 for odd-N derived from the RMF-BCS calculations are systematically lower than the experimental results, probably owing to a weak pairing strength. The predictive power of this model could, therefore, be greatly improved by utilizing a better parametrisation of the pairing. The rms value of the differences between the masses listed in table 1 and the predictions of the FRLDM is 2.721 MeV, for the Duflo-Zuker formula 0.777 MeV, for HFB-9 1.11 MeV, and 1.73 MeV for the RMF-BCS model.

Finally, our results have also been employed to derive a new shell-model parametrisation for the s-d shell Hamiltonians, called USDA and USDB. As a result, a better agreement with experiment for binding energies and excitation energies in the region of 24 O has been found, with a reduction of the rms deviation to around 0.13 MeV [29].

In conclusion, we have determined using a direct time-of-flight technique the masses of a broad range of neutron-rich nuclei situated in the vicinity of the N=20 and 28 shell closures. The masses of 7 nuclei have been measured for the first time and the precisions of 36 masses have been considerably improved, in many cases by more than a factor two. These results have been used to explore the evolution of the OES with neutron number, whereby we observe a clear reduction of the OES as the one-neutron separation energy decreases for the Z=8,10 and 12 isotopic chains. This effect needs to be carefully interpreted in order to establish the relative importance of pairing, spin-isospin coupling, and coupling to the continuum which may be needed to explain the observed behaviour. Our results also corroborate the changes in shell structure already observed for the P and S isotopes at the N=28 shell closure. In contrast, such effects are seen to be absent for the Si isotopes. This could indicate either the persistence of the N=28 shell closure for the Si isotopes, or it may reflect a very sudden change in deformation at N=28, as predicted by ref. [21]. To answer this question, 42 Si and neighbouring neutron-rich nuclei need to be further investigated. Finally, our results have been used to test and suggest improvements to various mass models as well as derive an improved shell model interaction.

Figure 1 .

 1 Figure1. S 1n (N) as a function of S 1n (N-1) where N is the neutron number. The triangles are from the mass table[12], the diamonds and the squares represent the new and improved (respectively) results measured in the present work.

Figure 2 .

 2 Figure 2. Experimental results for the three-point indicator ∆ 3 as function of neutron number. The same symbols are used as in figure 1. The dashed lines link the values of ∆ 3 for even-N nuclei and the full lines for odd-N nuclei. The vertical lines indicate the position of N=20 and 28.

Figure 3 .

 3 Figure 3. The microscopic correction as function of neutron number. The same symbols are used as in figure 1. The vertical lines indicate the position of N=20 and 28.

Figure 4 .

 4 Figure 4. Experimental values of ∆ 3 for the Z=14 isotopic chain in comparison with three different mass formulas (see text). The vertical lines indicate the position of N=20 and 28.

A feature arising from the strong forward focusing of the fragmentation products.

ACKNOWLEDGEMENTS

The authors would like to thank M. Yamagami for fruitful discussions and L. Gaudefroy for carefully reading the manuscript. The untiring efforts of the SPEG crew (J.-F. Libin and P. Gagnant) in preparing the experiment are, as always, gratefully acknowledged as is the effort of the GANIL cyclotron operations group in maintaining a high intensity 48 Ca beam and the SISSI device online.