Athanase Papadopoulos

NICOLAS-AUGUSTE TISSOT: A LINK BETWEEN CARTOGRAPHY AND QUASICONFORMAL THEORY

Keywords: AMS Mathematics Subject Classification: 01A55, 30C20, 53A05, 53A30, 91D20 Quasiconformal mapping, geographical map, sphere projection, Tissot indicatrix

published a series of papers on cartography in which he introduced a tool which became known later on, among geographers, under the name of the Tissot indicatrix. This tool was broadly used during the twentieth century in the theory and in the practical aspects of the drawing of geographical maps. The Tissot indicatrix is a graphical representation of a field of ellipses on a map that describes its distortion. Tissot studied extensively, from a mathematical viewpoint, the distortion of mappings from the sphere onto the Euclidean plane that are used in drawing geographical maps, and more generally he developed a theory for the distorsion of mappings between general surfaces. His ideas are at the heart of the work on quasiconformal mappings that was developed several decades after him by Grötzsch, Lavrentieff, Ahlfors and Teichmüller. Grötzsch mentions the work of Tissot and he uses the terminology related to his name (in particular, Grötzsch uses the Tissot indicatrix). Teichmüller mentions the name of Tissot in a historical section in one of his fundamental papers where he claims that quasiconformal mappings were used by geographers, but without giving any hint about the nature of Tissot's work. The name of Tissot is also missing from all the historical surveys on quasiconformal mappings. In the present paper, we report on this work of Tissot. We shall also mention some related works on cartography, on the differential geometry of surfaces, and on the theory of quasiconformal mappings. This will place Tissot's work in its proper context.

plane cannot preserve these three parameters at the same time, and they searched for a "best compromise." Making precise such a notion of compromise between the three distortions we mentioned depended on the practical use for which the map was intended.

Several mathematicians -some of them among the most prominent -were also geographers. One can mention Ptolemy, Euler, Lambert, Lagrange, and Gauss, and there are many others. Most of them were interested at the same time in the mathematical theory and the practical art of map drawing. Euler, at the Academy of Sciences of Saint Petersburg, besides being a mathematician, had the official status of a cartographer, and was one of the leaders of the huge project of drawing maps of the new Russian Empire. Motivated by cartography, he was led to study questions regarding conformal and almost-conformal mappings. In 1777, he published three memoirs on mappings from the sphere to the Euclidean plane: De repraesentatione superficiei sphaericae super plano (On the representation of spherical surfaces on a plane) [START_REF] Euler | De repraesentatione superficiei sphaericae super plano[END_REF], De proiectione geographica superficiei sphaericae (On the geographical projections of spherical surfaces) [START_REF] Euler | De proiectione geographica superficiei sphaericae[END_REF] and De proiectione geographica Deslisliana in mappa generali imperii russici usitata (On Delisle's geographic projection used in the general map of the Russian empire) [START_REF] Euler | De proiectione geographica Deslisliana in mappa generali imperii russici usitata[END_REF]. In the first memoir, [START_REF] Euler | De repraesentatione superficiei sphaericae super plano[END_REF], Euler examines several projections of the sphere, searching systematically for the partial differential equations that they satisfy. He recalls that there is no "perfect" mapping from the sphere onto a plane, and he considers the question of finding mappings with the least distortion. He highlights several kinds of properties that one may naturally ask for geographical maps: conformality; sending meridians to curves normal to a given axis and parallels to lines parallel to that axis; preservation of area (up to scale), etc. Most of all, Euler considered, like Lambert at about the same time, arbitrary mappings from the sphere onto a flat surface, and not only mappings obtained as a projection from a given point onto a plane, a cylinder or a cone, etc.

Lagrange, in his study of geographical maps (see [START_REF] Lagrange | 1779a. Sur la construction des cartes géographiques[END_REF] and [START_REF] Lagrange | Sur la construction des cartes géographiques[END_REF]), also considered general mappings from the sphere onto a flat surface. Like Euler and Lambert before him, he characterized angle-preserving general mappings analytically, and he applied his methods to the particular case of mappings from the sphere onto the plane that send meridians and parallels to circles. He writes that "these are the only curves which one can easily employ in the construction of geographic maps" ( [START_REF] Lagrange | 1779a. Sur la construction des cartes géographiques[END_REF] pp. 642). Lagrange introduced a function called the magnification ratio. The name given to this function clearly indicates its meaning. For conformal mappings, the magnification ratio depends only on the point on the sphere (and not on the direction at that point). Lagrange gave formulae for this parameter for general conformal mappings. We note right away that this notion played an important role in the later work of Chebyshev on the drawing of geographical maps: the latter noticed that the magnification ratio, under the condition that the distortion of the map is minimal in some appropriate sense, satisfies Laplace's equation, and he showed that under some natural conditions, the best angle-preserving geographical maps of a certain region having a reasonable size are those for which the magnification ratio function is constant on the frontier of the region. Chebyshev thus reduced the question of finding the best geographical map to a problem in potential theory, namely, solving the Laplace equation on a certain domain of the sphere for a function which is constant on the boundary of the domain. We refer the reader to the papers [START_REF] Chebyshev | a. Sur la construction des cartes géographiques[END_REF] and [START_REF] Chebyshev | Sur la construction des cartes géographiques. Discours prononcé le 8 février 1856 dans la séance solennelle de l'Université Impériale de Saint-Pétersbourg[END_REF] by Chebyshev (also included in his Collected works [START_REF] Chebyshev | OEuvres[END_REF]), and to the recent survey [START_REF] Papadopoulos | Euler and Chebyshev: From the sphere to the plane and backwards[END_REF] for some details on the subject. by several expeditions in the eighteenth century, which included especially French scientists, whose aim was to make precise measurements of the meridians near the poles.

It is also worth recalling that the motivation of several prominent mathematicians to study the differential geometry of surfaces came from geography. A famous case is Gauss, who declares explicitly in the preface of his paper [START_REF] Gauss | Allgemeine Auflösung der Aufgabe: die Theile einer gegebnen Fläche auf einer andern gegebnen Fläche so abzubilden, daß die Abbildung dem Abgebildeten in den kleinsten Theilen ähnlich wird[END_REF] that his aim is only to construct geographical maps and to study the general principles of geodesy for the task of land surveying. Gauss's paper [START_REF] Gauss | Allgemeine Auflösung der Aufgabe: die Theile einer gegebnen Fläche auf einer andern gegebnen Fläche so abzubilden, daß die Abbildung dem Abgebildeten in den kleinsten Theilen ähnlich wird[END_REF] is titled Allgemeine Auflösung der Aufgabe, die Teile einer gegebenen Fläche auf einer andern gegebenen Fläche so abzubilden dass die Abbildung dem Abgebildeten in den kleinisten Theilen ähnlich wird. (General solution of the problem: to represent the parts of a given surface on another so that the smallest parts of the representation shall be similar to the corresponding parts of the surface represented). This is an epoch-making paper, in which the author proves that every sufficiently small neighborhood of any point in an arbitrary real-analytic surface can be mapped conformally onto a subset of the Euclidean plane.

There are many other examples of mathematical works on the geometry of surfaces motivated by geography. We refer the reader to the recent survey [START_REF] Papadopoulos | Quasiconformal mappings, from Ptolemy's geography to the work of Teichmüller, to appear in: Uniformization, Riemann-Hilbert Correspondence[END_REF].

After differential geometry, we arrive at quasiconformal mappings. Teichmüller's paper [START_REF] Teichmüller | Extremale quasikonforme Abbildungen und quadratische Differentiale[END_REF] contains a section on the origin of these mappings in which he mentions the name of the French geographer Tissot. Grötzsch, in his paper [START_REF] Grötzsch | Über die Verzerrung bei nichtkonformen schlichten Abbildungen mehrfach zusammenhängender schlichter Bereiche[END_REF] mentions several times the name of Tissot and refers to his work. But the name of Tissot is never mentioned in any paper on the history of quasiconformal mappings, and it seems that the references to him by Grötzsch and by Teichmüller's remained unnoticed. Yet the work of Tissot, which is well known among geographers, and in particular in the drawing of geographical maps, is very closely related to quasiconformal mappings. Let us mention right away some of Tissot's important results.

(1) Tissot invented a device which provides a visual measure for the distortion of a geographical map. This is a field of infinitesimal ellipses which are the image of a field of infinitesimal circles on the region on the sphere which is represented. This field of infinitesimal ellipses is characterized by two parameters: their relative size, and the ratio of their two axes (the major axis divided by the minor axis). 2 These parameters represent respectively the local area distortion and the angle distortion of the mapping that is used to draw the geographical map. This device is known among geographers as the Tissot indicatrix. An example is given in Figure 1. From the differential geometric point of view, the Tissot indicatrix gives an information on the metric tensor of the metric obtained by pushing forward the metric of the sphere or the spheroid by the projection used. (2) Tissot noticed that for a given mapping between two surfaces, there is, at each point of the domain, a pair of orthogonal directions which are sent to a pair of orthogonal directions on the image surface. Unless the mapping is angle-preserving at some point, the pair of orthogonal directions is unique.

The orthogonal directions at the various points on the two surfaces define a pair of orthogonal foliations which are preserved by the mapping. We shall comment later on the uniqueness of the two fields of directions that are referred to in the above result. An important observation made by Tissot right at the beginning of his memoir [START_REF] Tissot | Mémoire sur la représentation des surfaces et les projections des cartes géographiques Paris[END_REF] (p. 1) is that the most appropriate mode of projection in order to build a geographical map depends on the shape of the region, that is, on the shape of its boundary as a subset of the sphere (or the spheroid). 2 The expression "infinitesimal circle" means here, as is usual in the theory of quasiconformal mappings, a circle on the tangent space at a point. In practice it is a circle on the surface which has a "tiny radius." These circles, on the domain surfaces, are all supposed to have the same small size, so that the collection of relative sizes of the image ellipses is a meaningful object. Tissot discovered that in order for the map to minimize the appropriately defined distortion, a certain function λ, defined by setting

dσ 2 = (1 + λ) 2 ds 2 ,
must be minimized in some sense. Here, ds and dσ are the line elements at the source and the target surfaces respectively. The minimality of λ may mean, for example, that the value of the gradient of its square must be the smallest possible. Darboux later on connected this work of Tissot with the work of Chebyshev that we mentioned above. We shall comment on this fact later in this paper.

(3) Tissot constructed mappings that realize the required properties of minimal distortion. Most of the mathematicians who preceded Tissot (Lagrange, Euler, etc.), and Chebyshev who was his contemporary, studied conformal maps between surfaces, with some distortion parameter to be minimized. In contrast, Tissot studied extensively distortion-minimizing non-conformal maps.

We shall elaborate on Tissot's ideas in § 3 below. Beyond the work of Tissot, a theory of "almost-conformal" mappings whose main object was to find mappings with least distortion between two surfaces had been inherent in the works of geographers for more than two thousand years. Geographical maps were used for various practical needs, including navigation, surveying for the purpose of collecting taxes, etc., but the theory behind these maps was an important element in the exact sciences.

It is always interesting how very simple ideas lead to elaborate theories.

On the works of Nicolas-Auguste Tissot

Nicolas-Auguste Tissot (1824-1897) was born (like Poincaré whom we shall mention very soon) in Nancy. In an article by A.-P. d'Avezac published in the Bulletin de la Société de Géographie [START_REF] Avezac | Coup d'oeil historique sur la projection des cartes de géographie (suite et fin), notice lue á la Société de géographie de Paris dans sa séance publique du 19 décembre 1862[END_REF], we read the following (p. 127)3 :

Mr. Nicolas-Auguste Tissot, an ancient capitaine du génie, and today assistant professor (répétiteur ) in geodesy at the École Polytechnique, presented to the Academy of Sciences a series of communications on geographical maps, whose outline is reproduced in the Comptes Rendus des Séances (Novembre 7, 1859, 49, pp. 673-676; March 5, 1860, pp. 474-476; and December 17, 1860, pp. 964-969). Their main goal is to determine the rule according to which the deformation is produced around each point, regardless of the system of representation; and to compare, from this point of view, the various systems that are used and proposed in the drawing of mappae mundi, and finally, to find, in this way, the best mode of projection for each particular country. In the investigation of this particular question, he outstripped Mr. Airy. 4In § 3, we shall describe the content of the Comptes Rendus notes referred to by d'Avezac.

Tissot defended his doctoral thesis on November 17, 1851, cf. [START_REF] Tissot | Mouvement d'un point matériel pesant sur une sphère[END_REF]. On the cover page of the thesis, the author is described as "M. A. Tissot, Ex-Capitaine du Génie." The work presented contains in fact two theses, 5 the first one on Mechanics, titled Mouvement d'un point matériel pesant sur la sphère (Motion of a material point moving on the sphere) and the second one on astronomy, titled Sur la détermination des orbites des planètes et des comètes (On the determination of the orbits of planets and comets). In his first thesis, Tissot studies the motion of a point on a sphere under the action of gravity. In the second thesis, Tissot describes various methods of determining distances between celestial bodies making use of works of Cauchy and of Lagrange on the determination of the orbits of celestial bodies. The techniques used in the two memoirs are those of differential geometry, differential equations and approximation theory. They are the same techniques that Tissot used later on in his work on cartography.

On the title page of Tissot's memoir [START_REF] Tissot | Mémoire sur la représentation des surfaces et les projections des cartes géographiques Paris[END_REF] (1881), the expression Examinateur à l' École Polytechnique follows his name. In his Éloge historique de Henri Poincaré [START_REF] Darboux | Éloge historique de Henri Poincaré, membre de l'Académie[END_REF], Darboux relates the following episode about Tissot, examining Poincaré at the entrance exam: 6 Besides his work on geographical maps, Tissot wrote several papers on elementary geometry; see e.g. [START_REF] Tissot | Formules relatives aux foyers des coniques[END_REF], [START_REF] Tissot | Démonstration nouvelle du théorème de Legendre sur les triangles sphériques dont les côtés sont très-petits relativement au rayon de la sphère[END_REF] and [START_REF] Tissot | Sur les hélices[END_REF]. We mention in this respect that several original, and usually consisted in the survey of some work in a field which is different from the candidate's field of expertise. The subject of this thesis was chosen by the faculty and communicated to the PhD student a few weeks before the defence. The Doctorat d' État disappeared in the 1990s. 6 Poincaré entered the École Polytechnique in 1873. It is not unusual that at such an examination, some students listen to the examination of other students. In fact, a student is given a question or a set of questions which he is asked to prepare while another student (who had already been given some time to prepare his questions) is explaining his solutions at the blackboard, in the same room.

prominent French mathematicians of the nineteenth and the beginning of the twentieth century were very much interested in such problems. The names include Serret, Catalan, Laguerre, Darboux, Hadamard and Lebesgue, and there are others.

Tissot's work on cartography

In the years 1856 and 1858, Tissot published several papers and Comptes Rendus notes on cartography in which he made a detailed analysis of the distortion of some known geographical maps (see [START_REF] Tissot | Sur la détermination des latitudes au moyen de la méthode de M. Babinet[END_REF], [START_REF] Tissot | Sur les altérations d'angles et de distances dans le développement modifié de Flamsteed[END_REF], [START_REF] Tissot | Sur le développement modifié de Flamsteed[END_REF]). He started developing his own theory in three Comptes Rendus notes [START_REF] Tissot | Sur les cartes géographiques[END_REF], [START_REF] Tissot | 1860a. Sur les cartes géographiques[END_REF] and [START_REF] Tissot | Sur les cartes géographiquesComptes[END_REF], written in the years 1859-1860.

In [START_REF] Tissot | Sur les cartes géographiques[END_REF], Tissot starts with the following observation. To construct a geographical map, or to represent one surface on another, one first chooses on each of the two surfaces a decomposition into infinitesimal parallelograms using two systems of curves (in reality, these two "systems of curves" are two transverse foliations), in such a way that the lines of the first system are sent to the lines on the second. In this way, points and their images are encoded as intersection points of corresponding lines.

Encoding points as intersections of two lines was widely used in projective geometry. At the practical level, it was used by engineering students in France in their study of "dessin industriel", by students in arts who were asked to magnify a drawing using graph paper, etc.

Based on the existence of this decomposition into nets of infinitesimal parallelograms, Tissot states the following:

Any representation from a surface onto another may be replaced, at each point, by an orthogonal projection made at an appropriate scale.

The proof is given in his later papers. Tissot then gives in [START_REF] Tissot | Sur les cartes géographiques[END_REF] the following principles that concern angle, distance, and area deformation. These principles play a very important role in the rest of his work.

1.-For any kind of representation of a surface onto another, there exists, at every point of the first surface, two perpendicular tangents, which are unique unless the angles are preserved at that point, such that the images of these two tangents are perpendicular on the second surface.

Tissot calls the tangents referred to in this statement principal tangents.

2.-The directions of the principal tangents are those at which the ratio of lengths of the corresponding infinitesimal elements attains its greatest and smallest values.

Tissot denotes these two values by a and b, a > b.

3.-In order to find the image of an infinitely small figure drawn in the tangent plane of the first surface, one uses the following procedure: Superpose the two tangent planes in such a way that the principal tangents coincide. Turn the first tangent plane by an angle whose cosine is b/a. Use the orthogonal projection given by the two decompositions by parallelograms on the two surfaces, then modify the image of the figure by applying in the second tangent plane a magnification of ratio a.

Tissot notes that the image of an infinitely small figure around a point on the first surface which is a circle of radius 1 centered at the given point is an ellipse whose major and minor axes are a and b. He then outlines a practical way to find the major and minor axes of these ellipses, and he provides formulae for them. This is the basis of the theory of what became known later as the Tissot indicatrix.

In the note [START_REF] Tissot | 1860a. Sur les cartes géographiques[END_REF], Tissot says that at any point of the Terrestrial globe, the angle which is deformed most by a geographical map is never the angle made by the meridian and the parallel at that point. Likewise, the directions which are stretched most, or contracted most, do not coincide with those of the meridian and the parallel unless the images of these directions are at right angles. He says that using the rules he gave in the previous note, one can easily compute the greatest alterations of angles and distances. This is the basis of the theory which is behind the maps that he will describe later on, in which he seeks for the most advantageous properties for what concerns distortion.

Tissot declares that the projection that he shall adopt depends on the particular country that has to be represented, in particular, its position with respect to the equator, its size, and the form of its contour. He then makes a list of 12 known projections or families of projections that are known, for which he gives formulae for the angle, distance, and area deformations. He compares the usefulness of these projections. The central question is again that of finding the best possible projection.

The third note, [START_REF] Tissot | Sur les cartes géographiquesComptes[END_REF], is more technical than the first two. Tissot mentions there the difficulties that arise in applying the theory to a country with a great area, like Russia. He gives several formulae for the projections of countries with a reasonable surface area, like France and Spain. These formulae will be discussed more thoroughly in his long memoir [START_REF] Tissot | Mémoire sur la représentation des surfaces et les projections des cartes géographiques Paris[END_REF] published several years later. In a note on pp. 2 of this memoir, Tissot declares that after he published the first of his Comptes Rendus notes on the subject, the statements that he gave there without proof were reproduced by A. Germain in his Traité des projections des cartes géographiques [START_REF] Germain | Traité des projections des cartes géographiques : Représentation plane de la sphère et du sphéroııde[END_REF] and by U. Dini in his memoir Sopra alcuni punti della teoria delle superfici [START_REF] Dini | Sopra alcuni punti della teoria delle superfici[END_REF]. He says that Germain and Dini gave their own proofs of these statements, which are nevertheless more complicated than those he had in mind and which he gives in the present memoir. He also declares that Dini showed that the whole theory of curvature of surfaces may be deduced from the general theory that Tissot developed. In fact, Dini applied this theory to the representation of a surface on a sphere, using Gauss's methods. Tissot also declares that his ideas were used in astronomy, by Hervé Faye, in his Cours d'astronomie de l' École polytechnique [START_REF] Faye | Cours d'astronomie de l' École Polytechnique[END_REF]. The texts of the two Comptes rendus notes [START_REF] Tissot | Sur les cartes géographiquesComptes[END_REF] and [START_REF] Tissot | Sur le développement modifié de Flamsteed[END_REF] of Tissot are reproduced in the treatise of Germain [START_REF] Germain | Traité des projections des cartes géographiques : Représentation plane de la sphère et du sphéroııde[END_REF].

Tissot developed his complete theory several years after his three Comptes Rendus notes, first in installments, in 1878-1880, (cf. Tissot [START_REF] Tissot | Mémoire sur la représentation des surfaces et les projections des cartes géographiques[END_REF]- [START_REF] Tissot | Mémoire sur la représentation des surfaces et les projections des cartes géographiques[END_REF]), and then in the long memoir [START_REF] Tissot | Mémoire sur la représentation des surfaces et les projections des cartes géographiques Paris[END_REF], which contains the work in [START_REF] Tissot | Mémoire sur la représentation des surfaces et les projections des cartes géographiques[END_REF]- [START_REF] Tissot | Mémoire sur la représentation des surfaces et les projections des cartes géographiques[END_REF] together with additional material.

The memoir [START_REF] Tissot | Mémoire sur la représentation des surfaces et les projections des cartes géographiques Paris[END_REF] is 337 pages long. It consists of a preamble and 4 chapters. We briefly review its content.

In the first chapter, Tissot studies the general properties of surfaces and the distortions of maps between them. He provides the proof of Statement 1 which we quoted above, asserting d the existence and uniqueness of the pairs of field of perpendicular lines that are sent to each other by the maps. He also gives the following complement:

We can vary in all possible manners the fields of perpendicular lines, obtaining an infinite number of decompositions into infinitesimal rectangles, each leading to a map between surfaces and having the property that these fields are the unique invariant orthogonal fields.

The first chapter contains, besides the proofs of the results announced in the Comptes Rendus notes, a description of the Tissot indicatrix, a detailed study of its properties, and a method of calculating the various deformations of angle, length and area using this indicatrix.

Practically, on a given geographical map, the Tissot indicatrix is a field of ellipses drawn at certain points (often located at intersections of the images of the parallels and the meridians that are drawn on the map). These ellipses illustrate the distortion at these points. See e.g. the maps drawn in the Album of map projections [START_REF] Snyder | An album of map projections[END_REF].

In terms of the differential geometry of surfaces, the Tissot indicatrix is a representation of the quadratic form that defines the metric tensor at each point of the geographic map. In the case of a conformal map, the Tissot indicatrix is simply a field of circles.

Tissot explains how one can draw the axes of the indicatrix. The two fileds of axes, when they are integrated over the surface, give the two orthogonal forming what which Tissot calls canevas. The development of the theory is an ingenious combination of elementary geometry, infinitesimal calculus, and differential equations. We recall that the results apply to maps between differentiable surfaces embedded in space, and not only between the sphere and the Euclidean plane. Tissot studies in particular the case of surfaces of revolution.

In the section starting at pp. 21 of the memoir, Tissot gives the formulae for the major and minor axes of the ellipses and its inclination for general maps between surfaces, in terms of the local parametrisations of these surfaces.

Tissot starts with two surfaces defined using orthogonal coordinate systems (x, y, z) and (x , y , z ) respectively, functions of two variables l and m. The coordinates are chosen so that the mapping between the surfaces is defined by equating the six coordinates with appropriate functions of l and m. He sets .

L
The lengths of two infinitesimal parallelograms on the two canevas that correspond to each other will be, for the first surface, Ldl and M dm, and on the second one, L dl and M dm. Setting h and k to be the ratios of the sides of these parallelograms and Θ and Θ their angles, we have Depending on the practical applications needed, the formulae for the major and minor axes and the inclination of the ellipses defining the Tissot indicatrix are derived from these formulae. We refer the reader to pages 22ff. of Tissot's memoir, where he considers a variety of special cases in which he derives precise formulae. For instance, in the case where the mapping is such that the ratios of lengths in the direction of the two linear elements have the same common value h, then he finds the following formulae for the major and minor axes of the ellipses:

h = L L , k = M M , cos Θ =
b = h cos Θ 2 cos Θ 2 and b = h sin Θ 2 sin Θ 2
Many other cases are considered. Tissot uses extensively power series expansion. We shall see below that Darboux, in Part II of his paper [START_REF] Darboux | Sur une méthode de Tissot relative à la construction des cartes géographiques[END_REF] on the work of Tissot, develops the same theory in the setting of Gauss's intrinsic geometry of surfaces. It is interesting to note that after he explains the methods of Tissot, Darboux says: "We can approach the same theory in a more rigorous way, where the power series developments intervene only at the end, and in more general conditions."

The second chapter concerns the practical applications of the theory developed in Chapter 1. It is dedicated to the answer to the question:

How to find the projection which is most appropriate to a given country?

In the introduction of this chapter, Tissot says that in the drawing of maps that are intended for the use in public services, and in particular in the services of the army, the most important question is the reproduction of angles. He says that for the map to be useful, as a topographical tool, the distortion of angles must be very small. He recalls that the distortion of distances varies from point to point, and he says that the supremum of this distortion must be reduced to a minimum. Another rule he mentions is that in working with geographical maps, formulae must be simple. He suggests equipping the map with a network of curves, using colors, which are loci where the distortion of length is constant. He shows that these curves will be in general algebraic of degree two, and in most cases they will be ellipses. Furthermore, for a given country that has to be represented, a point has to be chosen, called the central point, and the position of this point must be determined according to the required maximizing properties. The meridian and the parallel passing through this point will be called the mean meridian and the mean parallel. This chapter contains many formulae, concerning types of various projections. As an illustration, let us mention one of the formulae (p. 71), which is also contained in his Comptes Rendus note [START_REF] Tissot | Sur les cartes géographiquesComptes[END_REF] written 21 years before. It involves some notation.

Let l and m denote the latitude and longitude of a point on the surface of the Earth, r the radius of the terrestrial parallel at the latitude l and s the arc of meridian contained between the mean parallel and the parallel at latitude l. If x and y denote the coordinates of the corresponding point on the map with respect to a perpendicular system of axes, then the projection is given by the following formulae

x = s + 1 2 rm 2 sin l and y = rm(1 + 1 6 m 2 cos 2l).
The other formulae contain higher order terms.

Then follows a long development in which Tissot simplifies the formulae using power series expansions, with practical applications concerning the drawing of maps of France, Spain and other countries.

The last two chapters of the memoir [START_REF] Tissot | Mémoire sur la représentation des surfaces et les projections des cartes géographiques Paris[END_REF] are more computational than the first two. Tissot compares there the distortions of various projections.

Let us also mention the so-called Tissot projection used in cartography. It is described in Chapter 2 of the memoir [START_REF] Tissot | Mémoire sur la représentation des surfaces et les projections des cartes géographiques Paris[END_REF]. From the point of view of minimizing distortion, this map makes a compromise between conformal and equal area projection. The mathematical tools behind the definition are based again on power series expansions. The Tissot projection was used by the French army cartography department; see [START_REF] Snyder | Flattening the Earth: Two Thousand Years of Map Projections[END_REF]. It was also used in the cadastral survey of Egypt; cf. [START_REF] Lyons | The cadastral survey of Egypt 1892-1907[END_REF]. In fact, Tissot mentions in his memoir [START_REF] Tissot | Mémoire sur la représentation des surfaces et les projections des cartes géographiques Paris[END_REF] pp. 71, that his projection can be used for a country situated in the lune between two meridians which are not very distant from one another, and as an example he gives precisely Egypt.

Darboux was interested in the work of Tissot on geography, and in particular, in his projection described in Chapter 2 of his memoir [START_REF] Tissot | Mémoire sur la représentation des surfaces et les projections des cartes géographiques Paris[END_REF]. He wrote a paper on Tissot's work, [START_REF] Darboux | Sur une méthode de Tissot relative à la construction des cartes géographiques[END_REF]. In the introduction of that paper, Darboux recalls that using power series expansions, Tissot gave a new method for the representation of a given country. The region that has to be represented has to be small enough compared to the Earth. He also recalls that the series expansions reduce to a minimum amount the distortions of angles and distances, but he says that " [Tissot's] exposition appeared to me a little bit confused, and it seems to me that while we can stay in the same vein, we can follow the following method." He then provides another method to construct Tissot's projection. In this method, using a system of principal tangent lines, the linear element at a point of the surface is written as (1)

ds 2 = dx 2 + dy 2 + xdx R + ydy R 2 ,
where R and R are the principal radii of curvature at the given point. In this formula, the terms of the third order and more in x and y have been neglected. Then, one looks for power series,

α = f (x, y) and β = φ(x, y) such that dα 2 + dβ 2
coincides with ds 2 for terms up to a certain order (the highest possible). The condition that the map is a similarity at that point leads to an equation of the form

dα 2 + dβ 2 = (1 + λ) 2 ds 2
where λ is a function. Setting λ 0 to be the second order homogeneous component of λ, Darboux is led to the equation

λ 0 = x 2 + y 2 4RR + A(x 2 -y 2 ) + 2Bxy
where A and B are constants.

Darboux then says that Tissot determines A and B by an ingenious device. Let us admit for simplicity that the Earth is spherical, that is, let us take R = R = 1. Then Tissot's method amounts to searching, among all the conics with equation

λ = α 2 + β 2 4 + A(α 2 -β 2 ) + 2Bαβ = const.,
the one which fits better the form of the surface to be represented, and at the same time corresponds to the smallest value of λ, with some considerations to be taken into account, for instance, the mean value of the square of the gradient of λ is sought to be smallest possible. Darboux says that in the case where the curve is an ellipse, one obtains a "remarkable result,", namely, that the value of λ is maximal on the boundary of the ellipse, and the value of λ on this boundary is constant. Darboux (like Tissot before him) gives the value of this constant. Darboux notes that there is a relation with Chebyshev's theorem that we mentioned above, and he makes this relation explicit. In both theories, an integral has to be minimized, and if one takes the same degree of approximation, the two integrals are the same. Darboux then explains that conversely, the theorem of Chebyshev justifies the "rather elementary" reasoning of Tissot and his use of the set of conics. Indeed, by the theorem of Chebyshev, λ has to be constant on the boundary of the region to be represented and since, with the approximation made, it has to be constant on concentric conics, it is natural to choose among these conics the one which covers best the given region or, more precisely, its projection on the tangent plane, while having the smallest λ.

After that, Darboux extends the whole theory of Tissot to maps between surfaces in the setting of Gauss's theory, starting with arbitrary curvilinear coordinates with length element

ds 2 = Edu 2 + 2F dudv + Gdv 2
and using the theory of conformal representations. He makes the relation between the work of Tissot and the works of Gauss, Tchebyshev and Beltrami.

Since we mentioned the works of Lagrange and Chebyshev on cartography, let us note that Darboux wrote two papers which are directly motivated by these works. In the first paper, [START_REF] Darboux | Sur un problème posé par Lagrange[END_REF], Darboux gives a proof of a problem addressed by Lagrange in the paper [START_REF] Lagrange | 1779a. Sur la construction des cartes géographiques[END_REF] on cartography that we already mentioned. The question concerns a constant which Lagrange calls the "exponent of the projection." This question is reduced to the following problem in elementary geometry: Given three points on the sphere, can we draw a geographical map, with a given exponent, such that these three points are represented by three arbitrarily chosen points on the map? Lagrange, in his paper, says that a geometric solution seems very difficult, and that he did not try to find a solution using algebra. Darboux solves the problem in a geometric manner. He says that it is the recent progress in geometry that made this solution possible. In the second paper, [START_REF] Darboux | a. Sur la construction des cartes géographiques[END_REF], Darboux gives a detailed proof of the result of Chebyshev that we quoted above, saying that the most advantageous representation of a region of the sphere onto the Euclidean plane is the one where the magnification ratio is constant on the boundary of the surface to be represented. The proof he gives is the one outlined by Chebyshev, using potential theory. To make the relation with modern works, let us mention that Milnor also gave a proof of the theorem of Chebyshev, again following the latter's approach, see [START_REF] Milnor | A problem in cartography[END_REF]. It seems that Milnor was not aware of the work of Darboux on the same problem (he does not mention his name in his paper).

We end the discussion on geography by recommending to the interested reader the book [START_REF] Freeman | Portraits of the Earth: A Mathematician Looks at Maps[END_REF] by T. G. Feeman. It is addressed to mathematicians. It gives a nice exposition of the various distortions of a geographical map (angle, distance and area-distortions), with a review of several types of maps and an analysis of their distortion. The book also contains a section on the Tissot indicatrix.

Quasiconformal mappings

The theory of quasiconformal mappings appeared in complex analysis under various names, and with slighly different definitions. Presumably, the English name "quasiconformal mapping" is due to Ahlfors; cf. his comments in his Collected works edition. 7 Grötzsch, in his papers written at the end of the 1920s, used the expression "nichtkonformen Abbildungen" (non-conformal maps). He considered in [START_REF] Grötzsch | Über konforme Abbildung unendlich vielfach zusammenhängender schlichter Bereiche mit endlich vielen Häufungsrandkomponenten[END_REF] and in other papers he wrote in the same period the problem of finding the homeomorphism between two rectangles (the images of the vertices being fixed) that has the least deviation from conformality. He proved that in some reasonable sense of the expression " that has the least deviation from conformality" the solution is given by the natural linear map between these rectangles. According to Ahlfors [START_REF] Ahlfors | In Quasiconformal mappings and their applications[END_REF], the problem that Grötzsch solved in that paper was first considered as a mere curiosity, and the full strength of quasiconformal mappings and their use in the deformation theory of Riemann surfaces was first realized by Teichmüller. Let us quote Ahlfors from his 1954 paper [START_REF] Ahlfors | In Quasiconformal mappings and their applications[END_REF] pp. 156:

The very genesis of quasiconformal mappings was connected with the elementary extremal problem formulated by Grötzsch. Teichmüller was the first to extract a general principle: In a class of mappings it is required to find one whose maximal dilatation is a minimum. It is to be expected that the solution is unique, and that the extremal mapping is characterized by simple properties.

Ahlfors gives a brief summary of the early use of quasiconformal mappings in his 1964 paper [START_REF] Ahlfors | In Quasiconformal mappings and their applications[END_REF] (p. 153). In his 1978 ICM paper, he writes ([4] pp. 72): "Quasiconformal mappings might have remained a rather obscure and peripheral object of study if it had not been for Oswald Teichmüller." Quasiconformal mappings play a central role in the work of Teichmüller, who thoroughly developed the theory and made it at the basis of several research topics, including the theory of moduli of Riemann surfaces, value distribution theory of meromorphic functions, the type problem and the Bieberbach coefficient problem. We refer the reader to the surveys [START_REF] Ji | Historical development of Teichmüller Theory[END_REF], [START_REF] A'campo | On the early history of moduli and Teichmüller spaces[END_REF] and [START_REF] Papadopoulos | Quasiconformal mappings, from Ptolemy's geography to the work of Teichmüller, to appear in: Uniformization, Riemann-Hilbert Correspondence[END_REF] for more details.

Lavrentieff, around 1935, wrote two papers in French, [START_REF] Lavrentieff | a. Sur une classe de représentations continues[END_REF] and [START_REF] Lavrentieff | Sur une classe de représentations continues[END_REF], in which he introduced a class of mappings he called "fonctions presque analytiques" (almost analytic functions). The two papers contain a remarkable series of results in function theory and geometry that are based on quasiconformal mappings. Let us note that in Lavrentieff's papers, the dilatation of a mapping between surfaces, in the modern sense of quasiconformal theory, is highlighted, but it is not assumed to be uniformly bounded. We note by the way that the same holds in some of Teichmüller's papers, e.g. [START_REF] Teichmüller | Untersuchungen über konforme und quasikonforme Abbildungen[END_REF].

We mention some of the results of Lavrentieff, because they remain poorly known compared to those of the other founders of the theory of quasiconformal mappings, and because the notion (which we recall below) of quasiconformal mappings that he uses is very close in spirit to the work of Tissot.

In his paper [START_REF] Lavrentieff | Sur une classe de représentations continues[END_REF] (with an announcement of the results in the Comptes Rendus note [START_REF] Lavrentieff | a. Sur une classe de représentations continues[END_REF]), Lavrentieff gave a series of extremely interesting results on quasiconformal mappings. One of them ( §4 of [START_REF] Lavrentieff | a. Sur une classe de représentations continues[END_REF] and §3 of [START_REF] Lavrentieff | Sur une classe de représentations continues[END_REF]) is a generalization of the name, but I have also not been able to locate it elsewhere. Little did I know at the time what an important role quasiconformal mappings would come to play in my own work." In any case, the expression appears in print in Ahlfors' 1935 fundamental paper on covering surfaces [START_REF] Ahlfors | Zur Theorie der Überlagerungsflächen[END_REF].

Picard's theorem to the setting of quasiconformal mappings. Another one ( §5 of [START_REF] Lavrentieff | a. Sur une classe de représentations continues[END_REF] and §4 of [START_REF] Lavrentieff | Sur une classe de représentations continues[END_REF]) concerns a criterion, based on quasiconformal mappings, to find the type of a Riemann surface. There are several other results. We reproduce here Lavrentieff's definition of quasiconfomality to show how close it is to the ideas of Tissot.

Lavrentieff says that a function w = f (z) of a complex variable z in a domain D of the complex plane is almost analytic if it satisfies the following properties:

(1) f is single-valued and continuous on D.

(2) Except for a countable set of points z in D, the function f is an orientationpreserving local homeomorphism. (3) There exist two real functions p(z) ≥ 1 and θ(z) such that

• With the exception of points z in a set E consisting of a finite number of analytic arcs, p(z) is continuous, and θ(z) is continuous at all points z satisfying p(z) = 1. • In every domain ∆ which does not contain points of E and whose frontier is a simple analytic curve, p(z) is uniformly continuous, and if δ and its frontier do not contain points z satisfying p(z) = 1, θ is uniformly continuous on ∆. • Let z 0 be a point in ∆ which is not in E, and consider the ellipse E centered at z 0 , such that the angle between the major axis of E and the real axis of the plane is θ, and such that if a and b are the major and the minor axes of E, we have p(z 0 ) = a/b. Then, we have

lim a→0 f (z 1 ) -f (z 0 ) f (z 2 ) -f (z 0 ) = 1
where z 1 and z 2 are the points in E such that |f (z) -f (z 0 )| attains respectively its maximum and its minimum.

Lavrentieff calls the functions p(z) and θ(z) the characteristic functions of the almost analytic function f (z). Although he uses the same word as Tissot, it seems that Lavrentieff was not aware of his work. Lavrentieff notes that the problem of representing a 2-dimensional Riemannian manifold on a domain in the Euclidean plane is equivalent to the problem of constructing an almost-analytic function having the given characteristics p and θ. He gives several existence theorems that answer this question. We state his Theorem 3, pp. 414 of [START_REF] Lavrentieff | Sur une classe de représentations continues[END_REF]:

For any function p(z) ≥ 1 and θ(z) defined on the unit disc z ≤ 1 satisfying the conditions stated in the definition of an almost analytic function, one can construct an almost analytic function w = f (z) satisfying f (0) = 0, f (1) = 1 which realizes a conformal representation from the unit disc z ≤ 1 and the unit disc w ≤ 1 and which has p and θ as characteristic functions. This is one form of the fact which says (in modern terms) that an almost complex structure on a surface is integrable, which in turn is related to Gauss's existence of isothermal coordinates.

In a note (p. 408), Lavrentieff says that in the definition of almost analytic functions if one adds the requirement that the characteristic function p is bounded, then one obtains a class of functions which coincides with the ones considered by Grötzsch in [22] (1928). He also notes that he had already considered a special class of almost analytic functions in his ICM paper [28] (1928), in which he considers the question of constructing the Riemann Mapping Theorem by a sequence of explicit mappings obtained from the theory of partial differential equations, using a minimization principle. A similar application of quasiconformal mappings is mentioned by Teichmüller in the last part of his paper [START_REF] Teichmüller | Extremale quasikonforme Abbildungen und quadratische Differentiale[END_REF]. We refer the reader to the recent survey [START_REF] Papadopoulos | Quasiconformal mappings, from Ptolemy's geography to the work of Teichmüller, to appear in: Uniformization, Riemann-Hilbert Correspondence[END_REF].

Figure 1 .

 1 Figure 1. The Tissot indicatrix for the Cassini projection; from the Album of map projections [35] pp. 26.

This quote and the next one are my translations from the French.

Before questioning Poincaré, Mr. Tissot suspended the exam during 45 minutes: we thought it was the time he needed to prepare a sophisticated question. Mr. Tissot came back with a question of the Second Book of Geometry. Poincaré drew a formless circle, he marked the lines and the points indicated by the examiner, then, after wandering long enough in front of the blackboard, with his eyes fixed on the ground, he concluded loudly: "It all comes down to proving the equality AB = CD. This is a consequence of the theory of mutual polars, applied to the two lines." Mr. Tissot interrupted him: "Very good, Sir, but I want a more elementary solution." Poincaré started wandering again, this time not in front of the blackboard, but in front of the table of the examiner, facing him, almost unconscious of his acts; then suddenly he developed a trigonometric solution. Mr. Tissot objected: "I would like you to stay in elementary geometry." Almost immediately after that, the examiner of elementary geometry was given satisfaction. He warmly congratulated the examinee and announced that he deserves the highest grade.

The reference is to the British astronomer George Biddell Airy (1801-1892).

In France, the Doctorat d' État consisted of two theses; the second one was not necessarily

Ahlfors writes in[START_REF] Ahlfors | [END_REF], Vol. 1, pp. 213: "The truth is that I cannot recollect having invented

Acknowledgements.-I would like to thank Norbert A'Campo, Jeremy Gray, and François Laudenbach who read a preliminary version of this paper. The author acknowledges partial support of the French ANR (Agence Nationale de la Recherche) under the program "Géométrie de Finsler et applications."


Hartmann

J Khuyagbaatar
email: j.khuyagbaatar@gsi.de

A Yakushev

Ch E Düllmann

D Ackermann

L.-L Andersson

M Block

H Brand

J Even

U Forsberg

W Hartmann

R.-D Herzberg

F P Heßberger

J Hoffmann

A Hübner

E Jäger

J Jeppsson

B Kindler

J V Kratz

J Krier

N Kurz

B Lommel

M Maiti

S Minami

D Rudolph

J Runke

L G Sarmiento

M Schädel

B Schausten

J Steiner

T Torres De Heidenreich

J Uusitalo

N Wiehl

V Yakusheva

Nobel Symposium NS160 -Chemistry and Physics of Heavy and Superheavy Elements Fission in the landscape of heaviest elements: Some recent examples

Introduction

The stability, i.e. the existence of an atomic nucleus that consists of Z protons and N neutrons, strongly depends on the interplay between the attractive nuclear and repulsive Coulomb forces. In heavy nuclei the latter force eventually becomes dominant and spontaneous fission (SF) limits their stability [START_REF] Vandenbosch | Nuclear Fission[END_REF]. Some observables of the fission process were successfully explained within the liquid drop model, which also predicted that nuclei with too many protons cannot exist due to the absence of a (macroscopic) fission barrier (e.g. Z 112 [START_REF] Sobiczewski | [END_REF]). In the 1960's the shell-structure influence on the mass of the atomic nucleus has been investigated and accordingly the fission barrier was represented as a superposition of Nobel Symposium NS160 -Chemistry and Physics of Heavy and Superheavy Elements the liquid-drop and shell-correction energy terms [3,4]. Within these earlier versions of the macro-microscopic models the next spherical magic numbers Z = 114 and N = 184 have been predicted and nuclei at around these numbers are inferred to be surprisingly stable against spontaneous fission. A possible next magic proton number has several times alternatively been allocated to Z = 120 or 126 within modern self-consistent theoretical calculations [5,6].

To date many superheavy nuclei (SHN) with Z and N up to 118 and 176, respectively, have been synthesized. The heaviest ones have been produced in 48 Ca-induced fusion reactions with actinide targets and identified by their subsequent -decay chains, which always ended with fission of relatively neutron-rich unknown isotopes of elements down to Db [7].

One way for probing Z and A assignments of SHN is testing the systematics of observables like -decay energies and -decay (T ) and SF (T SF ) half-lives. Especially T SF of the relatively neutron-rich isotopes at the end of the Z = 114-118 chains can be compared with systematics based on the well known neutron-deficient ones, where shell-structure effects are well established.

The shell structure of the nucleus can strongly affect the stability against fission, primarily on its mass surface. Such an influence can be studied using the even-even nuclei as it has been observed for N = 152, where enhanced fission stabilities were found compared to neighbouring isotopes of Fm (Z = 100) and No (Z = 102) elements. Single and multiparticle configurations also hinder SF, and the strength of the hindrance depends strongly on the corresponding nuclear orbitals, e.g. quantum numbers associated with intrinsic angular momentum.

Present experimental techniques for the detection and identification of SHN are limited: they are sensitive for fissioning nuclei with half-lives roughly between tens of s up to a few hours at most. The identification of nuclei with T 1/2 outside these limits is challenging at best using standard experimental data acquisition (DAQ) methods. DAQ systems based on fast sampling ADCs have been in use for about a decade for detection of charged particles [8]. Recently, such a method has successfully been implemented at the gas-filled recoil separator TASCA and has been used for the first time in experiments dedicated to syntheses of SHN [9][10][11]. Its advantages for the detection of particles have already been described in Ref. [12]. In the present paper its benefits for the detection of SF-decaying nuclei will be demonstrated with examples of both short-lived ( 254-256 Rf [13]) and long-lived ( 266 Lr [11]) isotopes. Each case corresponds to a particular effect of the shell structure on the fission stability. Accumulated data and analyses procedures of fission activities will also be presented: (i) Spontaneous fission of even-even 254,256 Rf will demonstrate the influence of the shell structure on the nuclear mass. (ii) An influence of the single-particle configuration on the fission stability will be shown with even-odd 255 Rf. (iii) Fission hindrances due to multi-particle configurations will be demonstrated with even-even 254 Rf (K-isomeric state) and odd-odd 266 Lr.

Experimental setup

Since 2009 the gas-filled recoil separator TASCA is available for experiments aiming at SHN and has successfully been used for the studies of the elements flerovium (Fl, Z = 114) [14][15][16]), moscovium (Mc, Z = 115) [9], and Tennessine1 (Ts, Z = 117) [11]. TASCA consists of a deflecting dipole and a focusing quadrupole doublet and can be operated in Nobel Symposium NS160 -Chemistry and Physics of Heavy and Superheavy Elements different ion-optical modes [17], and with various fill gases [18], depending on experimental requirements.

Nuclei considered in this work were produced as evaporation residues (ERs) of excited compound nuclei formed in nuclear fusion reactions between 48 Ca and 50 Ti projectiles (beam) and solid 204,206,208 Pb [13] and 249 Bk [11,19] targets. Pulsed beams (pulse length: 5 ms, repetition frequency: 50 Hz) were accelerated by the UNIversal Linear ACcelerator (UNILAC) to energies around the Coulomb barrier. Typical target thicknesses were 0.5 mg/cm 2 . ERs leaving the target with relatively high charge states with broad distributions enter the gas-filled volume of TASCA, where charge-state equilibrating charge-exchange processes occur. After sufficiently many collisions the charge-state distribution becomes narrow at around an average one (often q < 10), before reaching the dipole magnetic field. Consequently, the trajectory of the ERs through TASCA can be defined by adjusting the dipole magnetic field to their magnetic rigidity [18,20]. For the present studies TASCA was filled with helium gas at 0.8 mbar pressure. At the focal plane of TASCA, ERs were implanted into a double-sided silicon strip detector (DSSSD) comprising 144 vertical (X) and 48 horizontal (Y ) strips on the front and back sides, respectively.

TASCA was significantly upgraded in 2011 by improving its performance including the assembling of a fast DAQ system for the implantation detectors [21]. All pre-amplifier signals from the Y -strips of the DSSSD were digitized by 60 MHz-sampling ADCs and stored in 50 s-long traces. Signals from the X-strips of were processed with standard analog electronics and peak-sensing ADCs with a dead-time of 35 s. These analog and digital branches, which independently initialized data storage, allowed determining the origin of events in two independent ways [11,12,21].

Experimental results

SF of 256 Rf, 255 Rf and 254 Rf

The isotope 256 Rf was produced in the 50 Ti( 208 Pb,2n) reaction. A typical energy spectrum of fission fragments detected in the DSSSD by the analog part of the DAQ is shown in Fig. 1a. These high-energetic events were in anti-coincidence with signals from the MWPC. Spatial and time correlation analysis between them and preceding evaporation residues (hereafter noted as ER-SF) allowed to determine the half-life of 256 Rf [see Fig. 2a] according to a method described in Ref. [22]. Traces of these fission events were stored as saturated signals in the digital part of the data. The saturation is due to the limited dynamic input range of the sampling ADCs. The saturation time can be used as a variable for the determination of the energy deposited in the DSSSD by the fission fragments in the digital data, as shown in Fig. 1b. The time distribution of fission events from decay of even-odd 255 Rf produced via 50 Ti( 206 Pb,1n) is shown in Fig. 2b. Due to the long half-life, 1.9(4) s, all fission traces were detected as single saturated signals similar to 256 Rf. 254 Rf was produced in the 50 Ti( 206 Pb,2n) reaction. Only a few events with saturated single signal, like the ones shown in Fig. 1a, were detected. They were found to be correlated with ERs within the short time as shown in Fig. 2c. The correlation time distribution of these events was terminated at about 50 s, because most of the fission events from 254 Rf were recorded as a second signal in ER-traces. Thus, these fission events were lost in the analog part of the data as being an individual fission-like event! One of them is exemplified in Fig. 1d. Energies of the first signal in such traces were corresponding to ER energies similar to that observed from the decay of 255,256 Rf. After extracting such short correlated ER-SF signals from the ER-traces, the full correlation time distribution was reconstructed; the corresponding halflife of 254 Rf was determined to T SF = 20(3) s. 254 Rf. Lines represent the calculated density distribution of the fission events on a logarithmic time scale according to Ref. [22]. The center of gravity of the distribution determines the lifetime ( = 1.443 • T 1/2 ) of the fissioning nucleus. Half-lives of the corresponding isotopes with uncertainties (in the bracket) and reactions in which they were produced are given. See text for details.

Nobel Symposium NS160 -Chemistry and Physics of Heavy and Superheavy Elements

Furthermore, in several traces of ER-like events a small signal was detected in between the ER and fission signals [23]. These signals with amplitudes not exceeding 500 keV were attributed to originate from the registration of electrons. Time differences between the ER and these electron-like signals are shown in Fig. 1e. They represent an activity with a half-life of 4(1) s. Therefore, all observed events containing ER-electron-SF signals were attributed to implantation of 254 Rf in an isomeric state associated with the emission of conversion electrons (CEs).

Nobel Symposium NS160 -Chemistry and Physics of Heavy and Superheavy Elements

Fission from 266 Lr and -decay assignment for 270 Db

The nucleus 266 Lr has not been produced directly so far. In 2012 at TASCA, this isotope was assigned to be produced in -decay chains associated with 294 117 [11]. Its safe observation was possible thanks to several experimental conditions, e.g., the use of a highly pixelized DSSSD [15], fast electronics [11], the UNILAC pulsed beam, and well-tuned TASCA [20] that allowed to assign a long-lived decay for the mother nucleus 270 Db in both -decay chains stemming from element 117.

As an example, energies of events measured in the pixel X=103 and Y =41 [11] without a coincident signal from the MWPC are shown in Fig. 3 as a function of time counting from the beginning of the experiment. Events occurring in the DSSSD during beam off periods are marked by additional horizontal lines. Within the energy range of 7-12 MeV where most of the known decays from heavy nuclei should occur, 55 beam-off events were detected. From the even-by-event analysis, 32 of them were identified as originating from known isotopes around 208 Pb produced in transfer reactions (see Fig. 3).

Alpha decays in the region "northeast" of 208 Pb occur with short half-lives. If the -particle energy is above 8 MeV and it originates from nuclei with Z ≈ 82-90 and A ≈ 210-230 then the corresponding ERs can mostly be found in the data within 1 s [24]. Therefore, depending on the counting rate of the individual DSSSD pixel, the predominant amount of beam-off events can be identified.

Several events were detected within a short time window (cf. Fig. 3, cluster). They were unambiguously attributed to originate from element Z = 117 based on the similarity to findings reported from the Dubna Gas-Filled Recoil Separator (DGFRS) [25][26][27]. Only one fission-like event was observed 5.1 hours after the last from the cluster-event. In between, four -like events with energies of 6.82, 7.89, 11.53 and 6.29 MeV were measured. Only the 7.89-MeV event could not be assigned to any known isotope. Furthermore, this was the only one event left without an assignment from in total seven events detected during the entire experiment within the energy range of 7.7-8.2 MeV, where a possible decay of 270 Db has been discussed in Ref. [27]. A similar situation has occurred in the second -decay chain, where the energy of the corresponding event was 7.90 MeV [11]. Therefore, these events were attributed to 270 Db and the fission events that terminate the decay chains to the decay of 266 Lr [11].

Discussion and summary

The influence of the shell structure on the spontaneous fission of heavy nuclei is well visible in systematics of T SF , as shown in Fig. 4. The results on isotopes presented in this work [11,18] are marked by solid symbols. Measured T SF of 254-256 Rf were in agreement with literature data [24]. Maxima of T SF of Fm (Z = 100) and No (Z = 102) isotopes observed at N = 152 show the enhanced fission stability due to the shell closure. However, even-even Rf (Z = 104) isotopes do not anymore show a well pronounced peak at N = 152 compared to lighter elements. This might be related to a weakened influence of the shell closure on the fission barrier. These nuclei are deformed, and thus the gaps between those nuclear orbitals responsible for the shell closure are strongly affected by interactions between the nucleons, i.e. both numbers of protons and neutrons. This is one of the distinct evidences of differences on the "magic" numbers (Z = 100, 108 and N = 152, 162) occurring in the deformed nuclei from those at, for example, Z = 82 and N = 126, where deformation is absent.

In case of nuclei with an unpaired proton and/or neutron (odd-A Rf, Lr and Db), T SF are significantly longer than in the neighbouring even-even ones, showing the effect of single and Nobel Symposium NS160 -Chemistry and Physics of Heavy and Superheavy Elements multi-particle configurations on the fission process. We note that experimentally observed SF branches from odd-A and/or odd-odd nuclei do not necessarily always correspond to the ground-state decay. Though T SF of odd-A Rf, Lr and Db isotopes show a maximum at N = 153 associated with the influence of both the N = 152 shell closure and single and/or multi-particle configurations. An increase in T SF towards N = 162 was observed in Rf-Hs, and in the case of Db a peak-like trend is observed at N = 163 similar to the situation N = 153, which indicates a shell closure. An intriguing finding is the assignment of the long-lived decay of 270 Db. From the experimental data analysis' point of view, two identical cases of 7.9 -MeV 's were clearly found to be member of the decay chains. However, in Ref. [28], the non-observation of analogous -branching in the odd-N neighbour 268 Db (N = 163) was pointed out, questioning the assignment of decay in 270 Db. According to Ref. [28] -decay properties of 268 Db could be expected to be similar to those of 270 Db. However, no decay was identified and assigned for 268 Db in several experiments performed at DFGRS [28], TASCA [9] and Berkeley Gas-filled separator (BGS) [29]. However, we note that the DGFRS and BGS experiments were significantly less sensitive for measuring long-lived decay because of the use of a DC beam where no (background-free) beam-off periods occur. The absence of an -decay branch in 268 Db would mean that it has a longer T compared to ≈ 1 hour of the neighbouring 270 Db (N = 165). Such cases are well known to occur in odd-odd and evenodd nuclei located above closed shells: for instance, T of 256 Lr (N = 153) is about eight times longer than 258 Lr (N = 155). In cases of spherically shaped nuclei such T differences between N = 127 and 129 isotopes of odd-Z elements vary up to 6•10 5 . Thus, it would not be surprising to expect a longer T and different -decay energy for 268 Db than 270 Db. For the future, more dedicated experiments to search for the long-lived decay of 268 Db and more data on 270 Db need to be performed. Their results will help to shed light on the shell structure around N = 162. We also note that in the case of odd-odd nuclei where both decay and SF are hindered, other radioactive decay modes ( decay and electron capture) can become preferable. Recently, statistical analyses of the -decay chains attributed to start at Z = 115 checking for the presence of such decays, have been carried out [31]. Their results show that . Theoretically calculated T SF of even-even Rf isotopes are also given as dashed lines [30]. Dashed-dotted lines represent the same calculation but multiplied by 10 3 , which simulates a hindrance factor due to an unpaired proton and/or neutron. Location of known and predicted neutron shell closures are marked by vertical dotted lines. different scenarios for the observed -decay chains can be proposed, assuming that one or more of the chain-members underwent electron capture decay.

Apparently, the identification of long-lived nuclei around N = 162 (expected also at N = 184) turned out to be a major task for further SHN studies. Suitable experimental techniques have to be implemented.

Similarly, very short-lived fissioning nuclei can also be studied in the future as demonstrated in the case of 254 Rf and its isomeric state. The latter observation made through electron detection without requesting a particular trigger was possible thanks to a fast DAQ, whose further advantage for the solution of fission related topics still needs to be explored. For instance, the observed ER-CE-SF sequence for 254 Rf indicates that an isomeric state is de-exciting into the ground state instead of direct fission. When such situation occurs in the strongly spontaneously fissioning nucleus then the most probable origin of the isomer is an excited state with high K-quantum number. K represents the projection of the total nuclear spin along the symmetry axis of the nucleus [32,33]. The same isomeric state in 254 Rf has recently been identified and half-lives are in agreement [34]. A two-quasi-neutron configuration ν7/2 + [624] ⊗ ν9/2 -[734] forming K = 8 has been suggested for this 4-s isomeric state in Ref. [34]. An additional 247-s isomeric state has also been observed. In the present experiment a threshold for the detection of signals in the DSSSD were not sensitive enough for measurements of electrons, thus the 247-s-isomeric decays reported in Ref. [34] could not be observed.

K-hindered fission stability might become an important issue for further studies of ultra short-lived nuclei with extreme numbers of Z and N . Very neutron-deficient nuclei with T SF much shorter than 1 s, which is approximately the flight-time of ERs through short in-flight separators for SHN, are just one example. Such an experimental scenario has been applied at TASCA for the hitherto unknown 252 Rf with a predicted T SF = 0.6 s [30]; these results will be presented elsewhere [23].

Figure 1 .Figure 2 .

 12 Figure 1. (a) Energy spectrum and (b) saturation times of traces versus energies of fission events from the decay of 256 Rf measured in the analog part of the data. (c)-(e) Examples of different types of traces containing fission events: (c) single-signal trace corresponding to 256 Rf. (d) Double-signal trace where the implantation signal (noted by ER) of 254 Rf was followed by fission. (e) Triple-signal trace where 254m Rf was implanted and after 2.3 s the isomeric state de-excited to the ground state which decays by SF after 9.4 s. The insert highlights the region of the electron signal.

Figure 3 .

 3 Figure 3. Energy versus time plot for events detected in pixel X=103 and Y =41 of the DSSSD where one of the -decay chains associated with 294 117 was observed[11]. The left panel shows the full time period of the experiment. The right panel shows a detailed view of the period the chain was observed. Events without a coincident signal from the MWPC and occurring during beam-off periods are marked by horizontal and vertical lines, respectively. Events with energies in the range of 7-12 MeV (an additional two low-energy events in the right panel), which were identified to originate from the products of transfer reactions, are marked by additional solid squares.

  Nobel Symposium NS160 -Chemistry and Physics of Heavy and Superheavy Elements Partial SF half-lives of even-even isotopes of the heaviest elements including odd-A Rf, Lr, and Db isotopes, as function of neutron number. Solid symbols mark experimental data provided in this paper, open symbols are from literature[24]

		10 14					152		1/2 T	(Db)	=	1/2 T	(Rf)	⋅	10	3
		10 11
		10 8
	SF (s)	10 5							Db
	T	10 2	Cm			Lr		Lr	Hs
		10 -1									Fl
		10 -7 10 -4	Cf	Fm	No	Rf	Sg	162	Cn	184
		136 140 144 148 152 156 160 164 168 172 176 180 184 188
							Neutron number
	Figure 4.

Here we adopt the suggested IUPAC names for elements 115 and 117.

We are grateful for GSI's ECR ion-source and UNILAC staff, and the Experimental Electronics department for their continuous support of the experiments.

Nobel Symposium NS160 -Chemistry and Physics of Heavy and Superheavy Elements