M Bender

P Bonche

P.-H Heenen

Shape coexistence in neutron-deficient Kr isotopes: Constraints on the single-particle spectrum of self-consistent mean-field models from collective excitations

Keywords: numbers: 21.60.-n 21.60.Jz, 21.10.-k, 21.10.Re 21.10.Ky 27.50.+e

We discuss results obtained in the study of shape coexistence in the neutron-deficient 72-78 Kr isotopes. The method that we used is based on the mixing of axial mean-field configurations after their projection on particle-number and angular-momentum. The calculations are performed with a Skyrme interaction and a density-dependent pairing interaction. While our calculation reproduces qualitatively and quantitatively many of the global features of these nuclei, such as coexistence of prolate and oblate shapes, their strong mixing at low angular momentum, and the deformation of collective bands, the ordering of our calculated low-lying levels is at variance with experiment. We analyse the role of the single-particle spectrum of the underlying mean-field for the spectrum of collective excitations.

I. INTRODUCTION

In a mean-field framework [1], the presence of low-lying 0 + states in the spectrum of an even-even nucleus is usually interpreted as the manifestation of shape coexistence [2]: each of the 0 + states, including the ground state, corresponds to a mean-field configuration of a different shape. Nuclei around the neutron deficient Kr and Sr have very early been considered as among the most favorable ones for the presence of shape coexistence. First studies were performed for 72 Kr and a few neighboring odd nuclei with the help of the Nilsson-Strutinsky approach [3]. Detailed calculations have been carried out since then with an improved microscopic-macroscopic model [4] and with self-consistent mean-field models using non-relativistic Skyrme [5] and Gogny [6] interactions, as well as relativistic Lagrangians [7]. They confirm the presence of oblate and prolate minima in the deformation energy surface of some light Kr, Sr and Zr isotopes.

The existence of nearly-degenerate structures corresponding to different deformations raises immediately the question of their stability against dynamical effets beyond a mean-field approach. One can expect that the physical states result in fact from a mixing of states with spherical, prolate and oblate deformations. Such mixings were obtained from models with parameters specifically adjusted to the data: the proton-neutron Interacting Boson Model (IBA-2) [8], and a Bohr-Hamiltonian calculation built on a microscopic-macroscopic model [9].

An alternative to a mean-field description of shape co-existence is given by the shell model. However, the number of active particles and holes necessary to describe the neutron-deficient Kr isotopes is prohibitively large, and this mass region is out of reach of the conventional shell model. The problem is tractable in models performing a clever truncation of the configuration space as in the complex excited VAMPIR approach using a phenomenologically modified nuclear matter G matrix as residual interaction [10,11], or in the Shell-Model Monte Carlo (SMMC) method. A SMMC calculation using a schematic pairing+quadrupole interaction is presented in Ref. [12]. First experimental evidence for ground-state deformation in neutron-deficient Kr isotopes was found in Ref. [13]. Subsequent experiments revealed the rich structure of the low-lying excitation spectrum in these nuclei, with coexisting and mixed bands in 72-78 Kr [14,15,16]. Many more data on transition probabilities, both in and out of bands, have also now become available [17,18,19,20,21,22,23,24,25,26,27].

The description of such nuclei is a challenge for nuclear structure models. In general, their excitation spectrum varies very rapidly along a given isotopic line. For this reason, they constitute a testing ground for beyond-mean-field approaches built on Skyrme-Hartree-Fock [28,29]. In this paper, we present an application of an angular momentum projected generator coordinate method to the description of the low-energy excitations in 72-78 Kr. At present, our implementation of the method is limited to axial mean-field states which are invariant under spatial inversion and time reversal. Applications of the same method to shape coexistence in neutrondeficient Pb isotopes were presented in Refs. [30,31]. A good qualitative agreement with the data was obtained, in particular for the relative position of the coexisting bands which result from a mixing of oblate, spherical and prolate configurations. However, the mixing is extremely sensitive to many details of the model and on the effective interaction, preventing a detailed quantitative agreement with the data. Compared to the neutron-deficient Pb isotopes, the description of the neutron-deficient Kr isotopes looks simpler. The coexisting bands result from the mixing of only two kinds of structures, oblate and prolate. The number of spherical j shells which contribute to the shell structure is smaller than for Pb, and the shells are the same for neutrons and protons. It is therefore easier to relate the collective states to the underlying meanfield.

Our calculated values presented below have already been used for comparison in the experimental report of Refs. [24,25].

II. THE METHOD

The starting point of our method is a set of axial HF+BCS wave functions. They are generated by selfconsistent mean-field calculations on a three-dimensional mesh in coordinate space [5,32], with a constraint on a collective coordinate, the axial quadrupole moment Q 20 . In a spherical nuclear shell model picture, such mean-field states incorporate particle-particle correlations through pairing and many-particle many-hole correlations through nuclear deformation. As a result, the mean-field states break several symmetries of the exact many-body states. These symmetry violations make difficult the connection between mean-field results, expressed in the intrinsic frame of reference of the nucleus, and spectroscopic data, obtained in the laboratory frame of reference. This motivates the second step of our method, the restoration of the symmetries associated with particle numbers and rotation. Another ambiguity in the interpretation of mean-field results arises when the deformation energy varies slowly as a function of a shape degree of freedom, in particular when a potential energy surface presents several minima separated by a low barrier. In such a case, to assign a physical state to each minimum is not a well-founded approximation. This problem is eliminated by the third step of our method: for each angular momentum, states obtained by projecting mean-field configurations of different deformation are used as the generating functions of the generator coordinate method (GCM). The weight coefficients of the mixing are determined by varying the energy and by solving the Hill-Wheeler-Griffin equation [33]. The configuration mixing removes the contribution of vibrational excitations from the ground state wave function. It permits also to construct a spectrum of excited states. A detailed introduction to the method has been given in Ref [START_REF] Bender | Proceedings of ENAM'04[END_REF].

The same effective interaction is used to generate the mean-field states and to perform the configuration mixing: the Skyrme interaction SLy6 [36] in the particlehole channel and a density-dependent, zero-range interaction in the pairing channel. As required by the SLy6 parametrization, the full two-body center-of-mass correction is included in the variational equations to generate the mean-field and in the calculation of the projected GCM energies. The strength of the pairing interaction is the same as in previous studies of light and medium-mass nuclei, i.e. -1000 MeV fm 3 for protons and neutrons in connection with cutoffs above and below the Fermi energy, as defined in Ref. [37]. Similar implementations of angular-momentum projected GCM have been developed also for the non-relativistic Gogny interaction [START_REF] Egido | Extended Density Functionals in Nuclear Physics[END_REF] and for relativistic mean-field Lagrangians [38].

To define a dimensionless deformation parameter, we use the sharp edge liquid drop relation between the axial quadrupole moment

Q (i)
2 and a parameter β

(i) 2 β (i) 2 = 5 16π 4πQ (i) 2 3R 2 A , (1)
with R = 1.2 A 1/3 fm. The mass dependence of the quadrupole moment is partly removed in this parameter β 2 . It should not be compared with the multipole expansion parameters that are used in microscopic-macroscopic models whose origin is not the same and usually are significantly smaller.

With our method, B(E2, J → J ′ ) values for in-band and out-of-band transitions as well as spectroscopic multipole moments Q s (J) are determined directly in the laboratory frame of reference [39]. As we use the full model space of occupied states, we neither have to distinguish between valence particles and a core, nor is there a need for effective charges.

To make a comparison with other approaches easier, it is useful to define quantities similar to intrinsic frame deformations from the spectroscopic and transition moments. An intrinsic transition charge quadrupole moment Q (t) can be derived from the B(E2) values within the static rotor model

Q (t) (J) = 16π 5 B(E2, J → J -2) (J 0 2 0|J -2 0) 2 e 2 . (2)
In the same way, an intrinsic charge quadrupole moment can be related to the spectroscopic quadrupole moment Q c (J) via the relation

Q (s) (J) = (J + 1)(2J + 3) 3K 2 -J(J + 1) Q c (J). (3)
For axially-symmetric shapes, i.e. pure K = 0 states, this relation reduces to Q (s) (J) = -Q c (J) (2J + 3)/J, with a change of sign between Q (s) and Q c (J). Dimensionless deformation parameters β 2 can be determined from Q (t) and Q (s) , respectively, through an equation similar to Eq. ( 1), with A replaced by Z. Within a given band, Q (s) and Q (t) might still depend strongly on angular momentum.

Q (s) and Q (t) measure different properties: Q (s) depends on a single state, while Q (t) probes the geometry of the initial and final states whose wave functions can correspond to very different mixings of projected mean-field states. In general, these two moments take different values. Their near equality indicates that the assumptions of the static-rotor model are fulfilled, i.e. a well-deformed rotational band not mixed with other bands.

III. RESULTS

A. Potential landscapes

Figure 1 provides the Nilsson diagram of the selfconsistent single-particle energies for neutrons. The diagram for protons looks very similar, but shifted in energy. One can see two main differences with the singleparticle energies used in some other methods. In the Woods-Saxon and the Folded-Yukawa potentials used in the microscopic-macroscopic calculations of Ref. [4] and [40] respectively, as in the Woods-Saxon potential used to generate the single-particle spectrum for the SMMC calculations of Ref. [12], the p 3/2 and p 1/2 levels are closer to the g 9/2 level by approximately 2 MeV and the separation between the f 5/2 and the g 9/2 levels is larger. As a result, the p 3/2 and the f 5/2 levels are much closer and their order is even changed in the case of the latter two models. These modifications have a strong effect on the deformed gaps which may correspond to quite different deformations and vary in size. The spin-orbit splitting for the f and p levels in the three potentials are very similar to ours. The differences between the single particle spectra must thus be related to the relative position of states with different orbital angular momentum within FIG. 2: Mean-field (dotted) and J = 0 projected deformation energy curves (solid) for 72-78 Kr (see text). a given shell.

Figure 2 shows the mean-field and J = 0 projected energy curves for 72-78 Kr obtained with the SLy6 Skyrme parametrization. Throughout this paper, the projected energy is plotted as a function of the intrinsic β (i) 2 value of the mean-field states from which they are obtained. In our opinion, this quantity provides the most convenient and intuitive label that can be defined for all states, irrespective on the level of modelling. However, it should not be misinterpreted as an observable. With our method, one calculates transition and spectroscopic multipole moments in the laboratory frame, which can be directly compared to experimental data. However, the spectroscopic moments which characterize a state do not provide useful coordinates to plot potential energy curves as they scale with angular momentum and are even identically zero for J = 0. For projected states with J > 0 and large intrinsic deformation in nuclei with A larger than 100,

β (i)
2 is very close to the intrinsic deformation β (s) 2 determined from the laboratory frame quadrupole moment Q (s) through the static rotor model, Eq. ( 3). Note that any coordinate might be misleading for the GCM, as the metric is related to the inverse of the overlap matrix, which has no direct connection to any deformation.

The mean-field energy landscapes (dotted lines in Fig. 2) show that the energies of the four nuclei vary very slowly with deformation. Our calculations predict also a transition from a nucleus with coexisting prolate and oblate minima in 72 Kr to a soft, spherical, anharmonic vibrator in 78 Kr. The many shallow local minima and plateaus in the total energy curves can be directly related to the gaps in the Nilsson diagram in Fig. 1.

The two minima in the mean-field energy curve (dotted line) of 74 Kr reflect the N = 38 gaps in the Nilsson diagram at small oblate and large prolate deformations. For 76 Kr, the spherical mean-field minimum is related to the large spherical N = 40 subshell closure, while the shallow oblate and prolate structures correspond to two deformed N = 40 gaps in the Nilsson diagram. The prolate minimum has moved to smaller deformation than in 74 Kr, the deformed N = 38 gap being at larger deformations than the N = 40 gap. The spherical N = 40 subshell closure is strong enough to stabilize the spherical shape up to the N = 42 isotope 78 Kr.

As can be seen from the solid lines in Fig. 2, the energy landscapes are qualitatively modified by the projection on J = 0. Since a spherical mean-field state is already a J = 0 state, the energy gain by projecting it on angular momentum is zero while the projection of a deformed mean-field state always leads to an energy gain, which increases very rapidly at small deformation. In almost all spherical and soft nuclei, this creates minima at prolate and oblate deformations with

|β (i)
2 | values around 0.1. These states usually have a large overlap close to 1, which means they are not two different states, but represent the same one, which can be associated with a "correlated spherical state". In nuclei with shallow meanfield minima at small deformations as the Kr isotopes, the projection merges this near-spherical spherical minimum with the slightly oblate one into a broad structure. The prolate mean-field minimum being at larger deformation, two distinct structures appear on the prolate side in 74-78 Kr, a "spherical" one at a β 2 value around 0.15, and a well-deformed one at a β 2 value around 0.5.

The topology of the 72 Kr energy surface on the oblate side can hardly be explained by a single large oblate N = 36 gap in the Nilsson diagram. The mean-field and J = 0 minima at large oblate deformations seem to result from a delicate balance of shell effects from N = 34 to N = 40 which reduce the level density around the Fermi surface sufficiently to create a shallow oblate minimum and a very soft energy surface. The shallow minimum on the prolate side can be associated with two close N = 36 gaps.

The fact that the structures in the potential landscapes can be easily associated with gaps in a Nilsson diagram is an attractive feature of our approach which makes straightforward the connection with simpler models. This is an advantage compared to the interacting shell model, in which states are constructed as nonintuitive np-nh states in a spherical basis.

One can wonder whether it is meaningful to restrict the description of the light Kr isotopes to axial shapes. There have been a few explorations of the triaxial degree of freedom which can help to answer this question. Yamagami et al. [41] have calculated the potential landscape of 72 Kr along several shape degrees of freedom, including nonaxial octupole deformations, using the Skyrme SIII interaction. Their potential energy curve is similar to ours for purely quadrupole axial deformations. They found that the prolate and oblate energy minima are separated by a triaxial barrier of 500 keV, and that the oblate shapes are soft with respect to non-axial octupole deformations. Bonche et al. [5] have obtained rather large triaxial barriers for 74 Kr and 76 Kr, also using the SIII interaction. Let us also mention that Almehed and Walet [42] have selfconsistently determined a collective path in 72 Kr for different values of the angular momentum in a small model space using a schematic interaction. For J = 0, they obtain a purely axial path that connects the oblate ground state and the prolate minimum. For J = 2, the path that they obtain does not cross the spherical configuration and makes a small excursion through triaxiality, as can be expected from the projected energy curves. From these studies of triaxiality, one can conclude that in most models, the oblate and prolate minima found as a function of the axial quadrupole moment are true minima. The barrier between the minima in the γ degree of freedom may not be very high, but to restrict the model to axial symmetry can be expected to provide a valid first approximation.

B. Spectroscopy

We first focus on the three isotopes for which very complete sets of experimental data are available. This enables us to introduce our notations and all the ingredients which are necessary for a comparison between our results and the data. We will come back to the more prospective case of 72 Kr at the end.

74 Kr

In Fig. 3 are given the lowest projected GCM levels for J = 0 up to 8 together with the corresponding projected energy curves. The collective levels are plotted at the mean deformation β(i) 2 of the mean-field states from which they are constructed, defined as

β(i) 2 = β (i) 2 β (i) 2 g 2 J,k (β (i) 2 ), (4)
FIG. 3: (color online). Mean-field energy curve, angularmomentum projected energy curves and projected GCM for 74 Kr. The projected energy curves are plotted as a function of the intrinsic deformation of the mean-field they are projected from, the projected GCM levels at the average deformation β2, Eq. ( 4). The labels on the right-hand side give the total binding energy as calculated, the labels on the left-hand side the energy relative to the J = 0 GCM ground state.

which provides in many cases an intuitive picture of the band structure in a nucleus. Figure 4, however, illustrates the limits of the meaning of β2 . In this figure are shown the collective wave functions g J i for J = 0, 2 and 4. All low-lying 0 + states result from a strong mixing between prolate and oblate mean-field configurations. The values of β2 for the ground state and the second excited state reflect the dominance of oblate and prolate deformations. For the ground state 0 + state, the very small value of β2 does not mean that this state is nearly spherical, but that the weights of prolate and oblate shapes are nearly equal. For higher J values, the mixing between oblate and prolate configurations is less pronounced and the value of β2 represents better the structure of the states.

As could be seen in Fig. 2, the J = 0 projected energy curve for 74 Kr presents two nearly degenerate minima at oblate and large prolate deformations, and a third shallow excited minimum at small deformation. This structure is reflected in the spectrum of GCM states plotted in Fig. 3 and in the wave functions drawn in Fig. 4.

As can be seen in Fig. 4, the J = 0 GCM wave functions strongly mix projected oblate, near-spherical and prolate configurations. The near degeneracy of the oblate and prolate minima is lifted, the first excited 0 + state having an excitation energy of 0.49 MeV. The situation becomes simpler for larger angular momenta. The first 2 + level still strongly mixes prolate and oblate deformations and has a wave function rather similar to the first excited 0 + state. On the contrary, the second 2 + state is mainly constructed from oblate mean-field configurations projected on J = 2 with a negative mean deforma- tion larger in modulus than that of the ground state. The two first 4 + and 6 + states are dominated by only prolate or oblate deformations. One can thus expect that an approximation based on a fixed mean-field configuration, like the cranked mean-field methods, is justified beyond

J = 4.
The nodal structure of the collective wave function is more complicated than the structure that one would obtain in a one-dimensional potential well. It reflects the fact that the rotation of a deformed wave function around the y-axis generates wave functions which cannot be represented along a single axis: the exchange of the x and z axes generates prolate and oblate shapes for two different values of the angle γ. For this reason, the angularmomentum projected GCM wave functions is not onedimensional, even when only axial states are mixed [39].

74 Kr provides an excellent example that a comparison between the intrinsic deformation of a mean-field minimum and the transition quadrupole moment derived from a B(E2) value might be misleading. There are three reasons for that: (i) for light and medium-mass nuclei, the minima of the projected energy curve are usually at FIG. 5: Spectrum of states for 74 Kr as seen in experiment (left) and as obtained from our calculation (right). The arrows with numbers denote the B(E2, J → J -2) values for transitions between states with different J and B(E2, J → J) for transitions between states with the same J, both given in e 2 fm 4 . Experimental values for the B(E2) values are taken from [24,25]. The labels "oblate" and "prolate" given to the bands correspond to the main components of the collective wave functions. larger prolate and oblate deformations than the minima of the mean-field energy curve, (ii) the minima of the projected J = 0 and = 2 energy curves correspond to the projection of mean-field states with different deformation and the assumption of a static rotor is not valid at low angular momentum, and (iii) the wave functions of the low-lying states are quite broad and strongly mix oblate or prolate configurations; moreover, the mixing is different for the 0 states and 2 + states.

A schematic two-level-mixing model has been used by Becker et al. [15,43] to analyze the experimental data. In this model, the energies of "pure" prolate and oblate configurations are deduced from an extrapolation to low spins of the high spin parts of the bands. This is valid if configuration mixing decreases with spin, which seems reasonable in view of our results. Data are described well by assuming that the unperturbed levels are nearly degenerate with only a 20 keV energy difference. This leads to nearly equal contributions from the oblate and prolate unperturbed states. This result is consistent with our calculations, although the assumption that one has to mix only two configurations is too crude. The complicated structures that we obtain for the collective wave functions, of Fig. 4 cannot be reduced to the mixing of two or even three configurations with a well defined shape. It does not seem therefore possible to connect our results with a phenomenological two-level mixing model.

Theoretical and experimental excitation spectra and B(E2) values are compared in Fig. 5. As in the experiment, the theoretical levels are ordered in bands on the basis of their spectroscopic quadrupole moments and of the dominant E2 transitions. A low-lying γ band, which FIG. 6: (color online). Comparison of calculated and experimental Q (s) (J) and Q (t) (J → J -2) moments for states in the prolate and oblate bands, assuming that all states are purely axial. Note that -Q (t) is plotted for oblate states for better comparison with Q (s) . Experimental values are taken from [25].

has a 2 + band head at 1.74 MeV, has been omitted since K = 2 states are outside our model space. The B(E2) values are always given for transitions from J → J -2 or from J → J, although values found in the literature correspond sometimes to B(E2, J → J +2) values, which are by definition larger by a factor 5. Our calculation is only partly consistent with experiment. We reproduce the coexistence of oblate and prolate bands, strongly mixed at low angular momentum, but the lowest 0 + states do not have the right ordering. Our calculated ground state is predominantly oblate, but the prolate band becomes yrast already at J = 2. For higher spins the spectrum is too stretched out, a problem that has already been encountered in previous GCM studies [START_REF] Bender | Proceedings of ENAM'04[END_REF][START_REF] Egido | Extended Density Functionals in Nuclear Physics[END_REF].

The in-band transition probabilities are slightly overestimated for both bands, which hints at a slightly too large deformation of the dominating configurations in both bands. Nevertheless, all calculated values are within a factor two of the experimental ones. Out-ofband transitions between states at the bottom of the bands are large, in particular the experimental value of B(E2, 2 + pro → 0 + obl ) of 938 +110 -91 e 2 fm 4 given in [25], or 1120 ± 460 e 2 fm 4 given in [16]. Our predicted out-ofband B(E2) values become weak already at J = 4.

The theoretical spectroscopic quadrupole moments are compared with experiment in Fig. 6. These moments provide by their sign the only unambiguous way to assign a prolate or an oblate character to a band. To compare them with in-band B(E2) values as well, Q (s) and Q (t) values derived from the static rotor model assuming K = 0 are shown. Except for the oblate 2 + state, our calculated Q (s) and Q (t) values are very similar. Experimental values for Q (s) and Q (t) are also similar within the large error bars of Q (s) . Our calculation gives systematically larger deformation for both the prolate and oblate states as already discussed above. FIG. 7: (color online). The same as Fig. 3 for 76 Kr. FIG. 8: (color online). The same as Fig. 4 for 76 Kr.

76 Kr

The projected energy curves in 76 Kr, plotted in Fig. 7, are very similar to those obtained for 74 Kr, with coexisting prolate and oblate bands, except that the prolate minima at β 2 values around 0.45 are less pronounced. As a consequence, the ground state wave function is more FIG. 9: The same as Fig. 5 for 76 Kr. Experimental data are taken from [25].

dominated by oblate configurations than that of 74 Kr, as confirmed by the shape of the collective wave functions given in Fig. 7. This is at variance with experiment, which indicates a prolate ground state, see also Fig. 8. Again, we omit from Fig. 9 the experimental states assigned to a γ band with a 2 + band head at 1.222 MeV; see, e.g. [21,25]. The calculated prolate band head has a higher excitation energy than for 74 Kr. The prolate band becomes yrast at J = 4. Compared to 74 Kr, the increased "purity" of states within the prolate and oblate bands happens at higher angular momenta, which is reflected in the large out-of-band B(E2) values up to J = 4 in Fig. 9.

Figure 10 compares the intrinsic quadrupole moments. As for 74 Kr, our calculated values for Q (s) and Q (t) are very similar, with small deviations at low J. Notably, this is not the case for the experimental values when assuming K = 0, which might hint at a small admixture of triaxial shapes to the prolate band, which is outside the scope of our calculation. Our calculated Q (t) are very close to the FIG. 10: (color online). Same as Fig. 6, but for 76 Kr. Experimental values are taken from [25]. FIG. 11: (color online). Same as Fig. 3, but for 78 Kr. FIG. 12: (color online). Same as Fig. 4, but for 78 Kr.

experimental values for the prolate band, while the Q (t) of the 2 + state in the oblate band is underestimated.

FIG. 13: The same as Fig. 5, but for 78 Kr. Experimental values for the B(E2) values are taken from [26].

78 Kr

Among the Kr isotopes, 78 Kr has the lowest excitation energy for the first 2 + state, and the largest B(E2, 2 + 1 → 0 + 1 ) value, corresponding to a transition quadrupole deformation β (t) 2 of 0.35. The rotational band built on the ground state has been seen up to J = 24. Two additional bands are known at low excitation energy, a first usually interpreted as a γ band built on a low-lying 2 + state at 1.148 MeV, and a second one built on a 0 + state at 1.017 MeV, originally assumed to be oblate. According to the recent analysis of [26], the spectroscopic quadrupole moment of the 2 + state in this band is negative, which indicates that this band should, in fact, be prolate.

Looking to the energy curves of 78 Kr, the collective states obtained from the projected GCM, Fig. 11, and the collective wave functions, Fig. 12, one can attribute to the ground state a dominating oblate structure, and to the band head of the first excited band, located at 2 MeV, a well-deformed prolate shape. All low-lying states are strongly mixed, which leads to large out-of-band B(E2) values up to J = 6.

Data for transition moments are taken from a recent Coulomb excitation experiment [26]. Values for in-band transitions in the yrast band were published earlier in Refs. [20,27], with a B(E2; 2 + 1 → 0 + 2 ) value of 91(5) e 2 fm 4 in [17], compatible with the value of 130 (40) obtained in [26].

As can be seen in Figs. 13 and14, our values are close to the experimental data for the low-lying states in the prolate band. The slight increase of Q (t) with J that we obtain can be related to the gradual disappearance of the prolate minimum at β 2 values around 0.45 and to the deepening of the shoulder at β 2 values around 0.6, which becomes the prolate minimum at J ≥ 8, see Fig. 11. In contrast, the experimental Q (t) 's are constant or slightly decreasing with J, see also [20,27].

The large mixing between the prolate 0 + state with an-FIG. 14: (color online). Same as Fig. 6, but for 78 Kr. Experimental values are taken from [20,26].

other excited 0 + state considerably reduces the in-band B(E2; 2 + pro → 0 + pro ) value, as the E2 strength from the 2 + pro level is nearly equally distributed. Therefore, our B(E2; 2 + pro → 0 + pro ) probability is slightly smaller than the experimental B(E2; 2 + 1 → 0 + 1 ) from the corresponding experimental state. As a consequence of this strong mixing, the calculated Q (s) and Q (t) values differ more than for 74 Kr and 76 Kr, Q (s) being larger than Q (t) except for J = 4. Experimentally, Q (s) values are smaller than Q (t) ones [26].

72 Kr

Rotational bands at high spin in 72 Kr have been investigated extensively, mainly motivated by the quest for fingerprints of T = 0 proton-neutron pairing [44,45]. Much less is known about the structure of this nucleus at low excitation energy. Only one band and two states that cannot be grouped into bands have been observed at low spin [16]. The precise structure of the low-spin yrast states is not completely clear either, as the ground state is argued to be oblate, in parts by consistency with theoretical predictions, while at higher spin the yrast states are assumed to be prolate. It was also argued in Ref. [18] that the large Q (t) value obtained from the B(E2; 8 + → 6 + ) = 2090(780) e 2 fm 4 favors its interpretation as a transition between prolate states. 1 We obtain, however, a much larger B(E2) for the 8 + pro → 6 + pro transition than the experimental value, which is in fact consistent with our 8 + obl → 6 + obl value. In view of the systematic overestimation of the experimental B(E2)s at large spins given by our method for the heavier Kr isotopes, one cannot draw a conclusion on the nature of the 1 The authors of Ref. [18] use a definition of β 2 different from ours, so that the values cannot and should not be compared.

FIG. 15: (color online). Same as Fig. 3 for 72 Kr.

FIG. 16: (color online). The same as Fig. 4 for 72 Kr.

8 + state in 72 Kr. However, it is clear that these large B(E2) values do not exclude an oblate band as previously argued. The potential energy curves, together with the GCM spectra, are shown in Fig. 15 and the collective wave functions in Fig. 16. According to the usual interpretation of the low-lying 0 + states, our calculation repro-FIG. 17: The same as Fig. 5 for 72 Kr. There is no unambiguous assignment of the experimental yrast states into a prolate or oblate band (see text). Experimental data for the B(E2) values are taken from [18] (8 + → 6 + ) and [22] (2 + 1 → 0 + 1 ).

duces their order with a dominantly oblate ground state. In Fig. 17, the experimental and theoretical excitation energies and B(E2) values are compared. According to the usual interpretation of the two low-lying 0 + states, our calculation reproduces their order with a dominantly oblate ground state. The first excited 2 + is at about the same excitation energy as the experimental one, while the first excited 0 + is slightly too high. Our calculated oblate states are yrast up to J = 6. With the exception the low-lying 0 + states, which strongly mix oblate, spherical and prolate states, the mixing between the prolate and oblate configurations is less important than for the heavier Kr isotopes. Within the accuracy that can be expected from our model, our B(E2) value for the 2 + obl → 0 + obl transition is consistent with the experimental one, B(E2; 2 + 1 → 0 + 1 ) = 1000(130) e 2 fm 4 . 72 Kr is the only Kr isotope studied here for which we can assume to reproduce the ordering of the low-lying states.

C. E0 transitions and radii

The nuclear matrix element entering the monopole decay rate of a J π state to a state with the same spin and parity is given by [46]

ρ 2 E0 (J k ′ → J k ) = JM k|r 2 |JM k ′ R 2 2 , (5)
where R = 1.2A smaller for 78 Kr. The variation of the GCM values is much larger, but in view of their sensitivity to model details it is encouraging that our calculated ρ 2 E0 values are within the experimental error bars for 74 Kr and 78 Kr and underestimate the value for 72 Kr by only a factor two. The large underestimation for 76 Kr can be related to the much smaller mixing between the oblate and prolate 0 + states that we obtain for this isotope.

The reasonable description of the ρ 2 E0 states indicates that, with the exception of 76 Kr, we have about the right amount of mixing between the oblate and prolate 0 + states, which is independent of the relative placement of the energy levels. In contrast, the systematics of groundstate radii is not well described, as our calculations give the "wrong" ground state for 74-78 Kr. This can be seen from the right panel of Fig. 18, where the isotopic shifts of the mean-square charge radii

δr 2 (N ) = r 2 (N ) -r 2 (N = 50), (6)
are plotted. The Kr radii present an anomaly [48,49]: they are larger for 74-84 Kr than for the N = 50 isotope 86 Kr. The standard interpretation is that the ground states have an increasing admixture of prolate states of large deformation when going to more neutron-deficient nuclei, which overcompensates the volume effect of decreasing mass. As our calculations always predict dominantly oblate ground states, we obtain a negative isotopic shift for all for isotopes (triangles in the right panel of Fig. 18). Our calculated excited dominantly prolate 0 + states, by contrast, lead to positive isotopic shifts for all four isotopes. It is unclear how well the ground state of 86 Kr itself is described within our method, so one has to be careful with the interpretation of the absolute values of the isotopic shifts. However, the δr 2 data confirm once again that the prolate states should have a much larger contribution to the ground state. The relativistic mean-field calculations with the NL-SH interaction presented in Ref. [7] do reproduce the systematics of the isotopic shifts, which hints at the more realistic single-particle spectrum of this interaction in the f p shell.

IV. DISCUSSION AND CONCLUSIONS

Our results show that in medium-mass nuclei with coexisting shallow minima in soft potential energy landscapes, one has to be very cautious when comparing experimental data and the mean-field states corresponding to local minima. Projection on good angular momentum alters significantly the potential energy surfaces. After configuration mixing, the collective wave functions present a very large spreading and may extend over several mean-field minima. We also find rotational bands, for example in 78 Kr, which are not related to a minimum in the mean-field energy landscape, but to a plateau.

While our calculations reproduce many of the global features of the neutron-deficient Kr isotopes, there remain several discrepancies with experiment when looking in detail.

• The collective spectra are too spread out in excitation energy, which appears to be a general problem of GCM of angular-momentum projected axial mean-field states [START_REF] Bender | Proceedings of ENAM'04[END_REF][START_REF] Egido | Extended Density Functionals in Nuclear Physics[END_REF]. This problem could probably be cured by a larger variational space for the projected GCM, in particular by treating dynamically the pairing, by breaking axial symmetry or time-reversal invariance. Breaking axial symmetry will also permit the description of γ bands. Work in these various directions is underway.

• Our study of neutron-deficient Kr isotopes clearly points at a problem with the SLy effective interactions in the f p shell, related to the relative position of a few single-particle levels.

As already discussed in the literature, (see, e.g., the discussion in [12]), the spectroscopic properties of the neutron-deficient Kr region are extremely sensitive to the details of the shell structure. The number of levels around the Fermi energy is quite small which results in several shell gaps for every neutron and proton number from 34 to 42. Any change in the relative position of levels and of their ordering will modify the size and deformation of the gaps in Fig. 1.

Our results, and those of models with adjustable singleparticle spectra that perform better for Kr isotopes, suggest a few necessary modifications of the single-particle spectra obtained with the SLy6 Skyrme parametrization:

• The experimental evidence that there are no spherical 0 + states in the spectrum of 76 Kr shows that the N = 40 spherical shell gap is too strong. This is consistent with our results for the N ≈ Z ≈ 40 region obtained in a systematic study of mass and deformation where the ground state for 80 Zr was obtained spherical with a very pronounced shell effect [50], while experiment suggests that is a welldeformed rotor [51]. A g 9/2 level closer to the p 1/2 orbital would decrease the N = 40 spherical gap. It would also decrease the oblate N = 38 gap, and shift the prolate N = 38 gap to smaller deformation, probably reducing the deformation of 74 Kr.

• Taking the Wodds-Saxon single-particle level schemes of Fig. 17 in [40] as an example, a decrease of the separation between the f 5/2 and the p 3/2 levels should change the single-particle spectra at small prolate deformation: a gap at N = 38 that extends from oblate shapes to prolate shapes would replace the gap at N = 36 around β 2 = 0.2.

It can be expected that this would drive 74 Kr towards prolate shapes.

We have checked that all modern successful Skyrme interactions give single-particle spectra in the f p shell which are similar to those obtained with SLy6. This points to a deficiency of the standard Skyrme interaction in general and not to a difficulty intrinsic to SLy6. Other models as the relativistic mean-field model (RMF) might seem to work more satisfactorily in the f p shell and to predict potential landscapes in better qualitative agreement with the Kr data than ours [7]. However, those models do not describe correctly the neutron-deficient Pb region [52,53], where many Skyrme forces perform quite well [30,31].

Altogether, the present study confirms the conclusions drawn in [50] that the current functional form of the Skyrme energy functional is not yet flexible enough to cover the relevant aspects of nuclear structure with the same good quality for all regions of the nuclear chart.

The combined analysis of Figs. 1 and 2 suggests that the structure of collective states in the neutron-deficient Krypton isotopes provides a sensitive testing ground for future attempts to better effective interactions. It has been pointed out recently that a tensor interaction, absent in the existing standard mean-field models, introduces a particle-number dependence of single-particle spectra [START_REF] Otsuka | presentation at the workshop on "Récents développements sur le champ moyen et l'appariement[END_REF][START_REF] Brink | seminar at the IPN Orsay[END_REF][START_REF] Dobaczewski | talk at the 3rd ANL/MSU/INT/JINA RIA Theory Workshop[END_REF], that might resolve at least some of the deficiencies in the f p shell we encounter here, and that are known for other regions of the nuclear chart. Work in that direction is underway.

FIG. 1 :

 1 FIG.1: Nilsson plot of neutron single-particle energies with positive (solid lines) and negative (dotted lines) parity as a function of the intrinsic mass deformation β (i) 2 , as obtained for 74 Kr.

FIG. 4 :

 4 FIG. 4: (color online). Collective wave functions g (J ) kfor the lowest states with J = 0, 2, 4, and 6 for 74 Kr as a function of the intrinsic deformation of the mean-field states they are constructed from. The corresponding energies and average deformations β2 are plotted together with the projected energy surface.

  1/3 fm. Within a simple model[47], this matrix element can be related to the amount of mixing of configurations with different deformations in the physical states. As it is a non-diagonal matrix element, it is also very sensitive to the detailed structure of the initial and final states. Within the error bars, the experimental ρ 2 E0 values are very close for 72-76 Kr, and slightly Kr) and[17] ( 76-78 Kr). Right panel: isotopic shift of the mean-square charge radii for the lowest (dominantly) oblate 0 + state (triangles) and the lowest (dominantly) prolate 0 + state (squares). The ground state predicted by our calculation is always oblate.

	FIG. 18: (color online). Left panel: comparison of calculated values for ρ 2 E0 (0 + 2 → 0 + 1 ) with experiment. Experimental val-ues are taken from [16] ( 72-74

Acknowledgments

We thank E. Bouchez, E. Clément, A. Gade, A. Görgen and W. Korten for stimulating discussions on the available experimental data and for making their results available to us prior to publication. We also thank R. V. F. Janssens and C.J. Lister for interesting discussions. This work was supported by the Belgian Science Policy Office under contract PAI P5-07 and the US National Science Foundation under Grant No. PHY-0456903. MB acknowledges financial support from the European Com-mission. MB and PHH thank for the hospitality at the Institute for Nuclear Theory, Seattle, where part of this work was carried out.

Work performed in the framework of the ESNT (L


Jonas Lamy

Thibault Pelletier

Guillaume Lienemann

Benoît Magnin

Bertrand Kerautret

Nicolas Passat

Julien Finet

Antoine Vacavant

Preliminary Results with a New Annotation Tool for Liver Volume and Inner Vessels from DCE-MRI Data

Keywords: Annotation, Liver, Vessels, DCE-MRI

Context

Annotation plays a key role in the creation of reference datasets that are useful to evaluate medical image processing algorithms and to train learning based architectures [START_REF] Park | Annotated normal CT data of the abdomen for deep learning: Challenges and strategies for implementation[END_REF]. The efficiency of the datasets generation generally depends on the interface ergonomics that is proposed to medical experts. While the development of annotation interfaces is quite standard in 2D, its design in 3D requires more attention to handle precisely the interactions between 3D structures and photometric constraints. DCE-MRI (Dynamic Contrast Enhanced MRI) is a popular image modality choice for the diagnosis of liver cancer (e.g. for hepato-cellular carcinoma). To our best knowledge, no database of hepatic vessels annotation is publicly available for this image modality. Such database would be of great help for the design of vascular segmentation tools used in surgical planning. However, the manual annotation by experts of thin and low-contrasted structures is a long, tedious and error-prone task.

We present a plug-in for the 3D Slicer software aimed at speeding up hepatic vessels annotation. We have created a complete pipeline dedicated to segment the liver volume and inner vessels by means of different image processing tools (seeded region growing or centerline-based vessel reconstruction for instance). In an agile process, we have developed and tested iteratively our annotation tool, so that clinical practitioners may be able to handle it efficiently during clinical routine or for specific clinical research activities.

Methods

3D Slicer

The annotation tool is delivered as a plug-in for 3D Slicer [1,[START_REF] Kikinis | 3D Slicer: a platform for subjectspecific image analysis, visualization, and clinical support[END_REF]. This free and open-source software platform developed by Kitware Inc. already offers various features including 2D and 3D visualization, segmentation tools and handles most medical image formats. It is also designed to be highly modular making the integration of custom tools easier. 3D Slicer also includes an Extension manager where community-developed plug-ins can be shared publicly.

Our plug-in

The annotation process is divided into four steps:

• loading and managing medical imaging data;

• liver segmentation;

• vessels annotation;

• tumor segmentation; organized into dedicated tabs in the plug-in module (highlighted in blue in Fig. 1). The liver segmentation and vessels annotation tabs are highlighted in dotted lines respectively in Fig. 1-a and Fig. 1-b.

The modularity of 3D Slicer is sometimes a drawback especially with data properties and their visualization shared across two different modules. The first tab aims at unifying these properties in a single interface in order to avoid back-and-forth use of the modules. The second tab is the integration of the already existing segmentation tools of 3D Slicer. As depicted in Fig. 1-a, these allow voxel annotation using manual (paint, eraser, scissors, etc.) as well as automated tools (level-set tracing, seeded region-growing, thresholding, etc.). For the vessels annotation strategy of the third tab, we have chosen to use centerline annotations instead of voxel-based segmentation. The centerlines are defined as sets of keypoints connected by lines that can be edited manually or generated automatically between two end-points using the Vascular Modeling Toolkit Module (VMTK) [START_REF] Antiga | An image-based modeling framework for patient-specific computational hemodynamics[END_REF] (see Fig. 1-b). This automated step will be improved in the future by using algorithms to be designed in the context of the R-Vessel-X project (see acknowledgements). The last tab uses once again 3D Slicer segmentation tools for tumoral segmentation.

Finally, we added a split 2D/3D view to allow the expert to annotate and control his/her work at the same time. In particular, we observed that overlapping annotations in 3D with the MIP (Maximal Intensity Projection) view of the liver was an efficient way of detecting missed vessels.

Results and discussion

To assess the efficiency of the plug-in, we asked two radiologists trained to use 3D Slicer to segment a liver (whole liver and inner vessels) with and without the plug-in. In their clinical practice, they usually employ embedded image processing tools as the General Electric AW solution (Server 3.2) for such annotations. We have tested our plug-in with DCE-MRI acquisitions obtained from 6 "healthy" patients (i.e. not suffering from any hepatic disease) and 4 patients with liver cancer and cirrhosis. DCE-MRI exams were performed on a 1.5T SIGNA TM Artist (General Electric, Milwaukee, WI) with a phased array coil.

Tab. 1 and Tab. 2 present annotation times obtained with our plug-in vs. the conventional approach, for patients without and with cirrhosis and liver cancer respectively. These tables show that our plug-in allows a faster annotation of liver volume and inner vessels, for both patient populations. The General Electric AW software proposes less intuitive interaction for vessel labeling within DCE-MRI sequences, which is particularly critical for sub-branches beyond left and right portal veins for instance. the 3D Slicer framework. For instance, the estimation of orthogonal planes during the annotation process should be a relevant feature to improve the precision of the centerline positioning defined by the user [START_REF] Gr | Precise cross-section estimation on tubular organs[END_REF]. The development of new algorithms during our project or technological transfer of previous work [START_REF] Lebre | Automatic segmentation methods for liver and hepatic vessels from CT and MRI volumes, applied to the Couinaud scheme[END_REF] would also increase the pertinence of our plug-in with more automatic steps, particularly for vascular reconstruction.

Future works

Figure 1 :

 1 Figure 1: View of our software. (a) Whole liver segmentation. (b) Centerline annotation.

Table 2 :

 2 The next steps are the adaptation of the interface design to other advanced algorithms not yet included in Annotation time of the liver volume and inner vessels for patients suffering from cirrhosis and liver cancer, with standard approach (General Electric AW software) and with our plug-in.

	Segmentations	GE AW	Our plug-in
	Liver	10 ± 5 min	3 ± 2 min
	Vessels	30 ± 10 min	5 ± 3 min
	Total	40 ± 15 min	8 ± 5 min
	Table 1: Annotation time of the liver volume and inner
	vessels for healthy patients, with standard approach
	(General Electric AW software) and with our plug-in.
	Segmentations	GE AW	Our plug-in
	Liver	15 ± 10 min	4 ± 2 min
	Vessels	35 ± 15 min	7 ± 4 min
	Total	50 ± 25 min 11 ± 6 min

Acknowledgements

This work was funded by the French Agence Nationale de la Recherche (grant ANR-18-CE45-0018, project R-Vessel-X, http://tgi.ip.uca.fr/r-vessel-x).


G

rey wolves were reintroduced in the northern Rocky Mountains (NRM) in the mid-1990s, at about the same time wolves discretely came on their own to France, crossing the Alps from Italy. Twentyfive years later, about 1,800 wolves inhabit the NRM while over 500 wolves live in France. The number of wolf packs (72) in the French Alps is similar to that in the Greater Yellowstone Ecosystem.

The main difference, apart from the different landscapes, land tenures, and uses by humans, lies in the number of animals depredated in the two countries: a few hundred per year in the NRM; and more than 10,000 per year, mostly sheep, in France. Since 2008, yearly livestock killings due to wolves in France have grown linearly, reaching 12,515 in 2018.

Why the differences between NRM and France? What went wrong in France? Don't French wolves have access to wild ungulates as prey in the Alps? Aren't livestock protected in France?

Certainly, many species of wild prey are abundant in France, but those which escape from wolves adapt their behavior and become more difficult for wolves to catch, especially compared to livestock. Almost all concerned French farmers (about 3,000) have adopted nonlethal protection measures in the Alps: reinforced human presence, guard dogs, secure fences, and compulsory night pens.

During the first 20 years, farmers and hunters were not allowed to use lethal control. In 1992, France signed and implemented the EU Habitat Directive as the legal duty of conserving outstanding wildlife habitats and species, including the European grey wolf. For the past five years, farmers have been allowed to use lethal control, but only if they are suffering verified frequent attacks and request and receive temporary authorization to diverge from the law.

In France, wolves have settled into mosaics of cultivated and natural meadows, hedges, and forest undergrowth. In those locations, livestock are easy and predictable

Are Wolves Becoming Bolder?

Wolves hurt livestock and local communities in the United States and France. By Michel Meuret, Pierre-Louis Osty and Fred Provenza, Ph.D.s and rangeland scientists. Wolf attack in the French Alps. More than 10 ewes and lambs were killed in this single attack, yet very little meat was eaten. Attacks such as this are now common and wolves do not seem to be bothered by close proximity to human dwellings.

prey and humans do not represent any potential danger to them.

We contend, based on numerous studies, that the differences between NRM and France are due primarily to wolf management. Unlike in the NRM, as a species mostly protected from trapping and hunting in France, wolves have unlearned to associate livestock with humans and humans with danger.

Wolves are smart, opportunistic, and very adaptable. One sign of adaptation: half of successful wolf attacks now take place during the day in France, notwithstanding the constant presence of dogs and regular if not constant presence of humans.

Another sign: wolf attacks are still occurring on high mountain pastures, but there are ever more frequent attacks in valleys and plains. Attacks occur in virtually all seasons in regions without much snow, and in the immediate vicinity of farms, houses, towns, and frequented roads.

The toll on livestock and farmers is great, but that is only part of the story. The rest of the tale is embedded in the physical and emotional fabric of livestock, livestock breeders, herders, and local communities throughout Alps.

To the eyes of Antoine Le Gal, sheep breeder and experienced herder in Provence and the Alps, the costs of predation by wolves are far more than livestock losses:

"Herding in the presence of wolves is an extra workload, a stressful situation, but also a recurrent feeling of failure and guilt. This summer, my stress level was reinforced by the fact that I suffered attacks by particularly bold wolves. One evening, my sheep had been in the electrified night pen for an hour, it was dark, the female dog barked, I left the shepherd's cabin, there were two wolves 20 meters from the door and also 20 meters from the night pen. I went to take my rifle in the cabin, and I chased them for a quarter of an hour on the mountainside. I thought I scared them off. Well, not exactly. The next day, at the same time, they were back! I had to sleep, not in the main room, but in the kitchen, next to the door, and along with my rifle. Many of us, and each year more and many more of us, have to go through this kind of situation. Not only in the high mountains at summer, but also in the valleys and plains in autumn and spring.

"When we count dead bodies, eventually after taking our knife to finish off sheep that have been too seriously injured, what are we counting, actually? We count animals that, most often, we have given birth to. Which we accompanied, them and their mothers, every day. We knew them and they knew us too. Some of the bodies had names, in addition to the mandatory ID numbers. Some of them, when called, answered to their name. So they're not numbers, percentages of herds, or even, as I've heard, "legs on feet. " They are living, intelligent, active beings. Above all, we must-that is our joboffer them a decent life, or even if possible a pleasant and rich life. When we sometimes hear: 'What are breeders complaining about? They are refunded for the damage caused by wolves, ' it's absolutely revolting!"

Due to wolves, too many tasks and talents are required from farmers, thinks Sebastien Vedrines, a young sheep farmer on a milestone plateau in south-central France: "One morning, I discovered some guts that went over the thistles on the pasture. There was one of my young sheep with half of her belly hanging down, and her legs bleeding. And that's something to say! I know it's very hard for us to tell, but...if you want the wolf...okay, it's very beauti-

Great Pyrenees dogs on duty in the Southern French Alps. Dogs are on the alert for wolves 24 hours a day. They often risk their lives to protect the flock, as humans generally are not allowed to use lethal means of control. BELOW: A meeting of breeders, herders, scientists and National Park officials in the Cevennes mountains. Direct exchanges about livestock and wolves are essential for local, national and European authorities to better consider how to address wolf depredation.

ful, a wolf, you can make nice photos...but you also have to show to people the other side of the facts! "We can't work around the farm during the day, stay watchful against wolves on pastures at night, and have a family life. If, to be a shepherd, you have to become...masochist, insomniac, sniper, with a penchant for illegality...I lack all these qualities for sure! I'm willing to have one or two, try for a third, ha-ha [desperate laugh]...it's better for us to be single too, I forgot about that. No, we can't follow this!"

Jacqueline Dupenloup, the mayor of Saint-Alban-des-Villards, a small alpine village from the Northern French Alps confronted with wolves and their management, explains how predation impacts her valley much more widely than only livestock losses for farmers and herders:

"I am neither a livestock farmer, nor a shepherdess. I am the mayor of a small village in the Alps-Savoy, France-with responsibilities for managing the municipality, the local alpine landscape, and the local community. Here, more than 20 years ago the wolves came back. In our mountainous and narrow valley, sheep from three local farms are grazed from early May to late October on pastures made of mosaics of natural meadows and forest undergrowth. Last year, despite livestock guard dogs and fences, they suffered more than 15 attacks by wolves, sometimes a few meters from the village, with a total of nearly 90 mother sheep and lambs killed. At the end of June, local flocks are joined by other flocks that come by trucks from Provence to our high mountain pastures for the summer. They were also attacked by wolves several times.

"I can only repeat this sentence heard from a farm advisor or perhaps a shepherd: 'The wolf is unthinkable...' Indeed, the wolf is unthinkable. What a paradox: On a planet where people don't hesitate to plunder natural resources to enrich the wealthy, with consequences that include degraded biodiversity, we should admit that a super-predator and symbol for biodiversity calls into question millennia of shepherding, a practice that is among the most respectful of the environment on Earth.

"It would then be necessary to admit that herders, men and women, must now be equipped with guard dogs, as well as fences two meters high, electrified when possible, and should also become watchful, day and night. In France, as these practices have been proven to be poorly effective in preventing wolf predation, herders should be allowed to get a hunting license, learn to use rifles and become security guards instead of herders. What other profession is framed by such constraints? "Can we imagine the constraints and conflicts imposed by wolves on municipalities, on their landscapes, most of them public lands, when challenged to manage impossible cohabitations, between wolves and livestock, guard dogs and bikers or hikers, high fences and village inhabitants?

"Behind the iconic image of a wonderful wild large carnivore, reality imposes itself: in our country, wolves attack more and more often in the presence of humans, during daytime, sometimes in the outskirts of villages. They thwart the vigilance of guard dogs, and do not care much about electrified fences. In some cases, slaughter entire groups of animals, killing to kill, condemning farmers, herders, and even sometimes the mayor, to euthanize halfdevoured sheep or goats.

"In France, will our land-management policies shape the future only for urbanized areas with our alpine valleys intended to become merely pleasant recreational areas for city dwellers? And, if so, how can this role be played in alpine landscapes that have less plant and animal biodiversity and no more horizon for hikers, as the lack of herding due to wolves generates rapid and widespread encroachment of shrubs and trees?"

And so, people in France are at the stage Chance Gowan described in his article "Are Wolves Really the Problem" in the Summer 2019 issue of RANGE magazine: wolves have shifted their attention from other wild prey species to domestic livestock and they now show little fear of humans. As Gowan concludes, "Let's never forget that we also owe the people of rural America the opportunity to protect themselves, their families, their livestock and their livelihoods. " The same is true in France. As Jacqueline Dupenloup emphasizes, "For us, the wolf is unthinkable and untenable. "

  In the Alps, shepherds guide herds along daily grazing circuits that stimulate the animals' appetites for plant diversity, while enhancing biodiversity and producing lamb meat. This is a full-time job. The added task of repelling wolves is unbearable. BELOW: Wolf attack near Jacqueline Dupenloup's village of Savoy: 23 lambs and ewes were killed. This attack occurred the day after sheep arrived from the plain by truck. Before going to higher mountain pastures, they were first grazed near the village under the care of a shepherd and his guard dogs, in a pasture surrounded by electrified fence.

  field agronomist, raised on a cattle farm, devoted his career to studying the technical systems of livestock farming in relation to landscape dynamics. Fred Provenza, behavioral ecologist, directed an award-winning research group at Utah State Univ. His latest book, Nourishment: What animals can teach us about rediscovering our nutritional wisdom, was published by Chelsea Green in 2018. Contact: fred.provenza@aggiemail.usu.edu.

	Michel Meuret, animal ecology scientist, coor-
	dinates a researchers' network in France deal-
	ing with coadaptation between predators,
	humans, and other animals. Contact:
	michel.meuret@inra.fr. Pierre-Louis Osty,

n


Mireille Fares
email: fares@isir.upmc.com

Catherine Pelachaud
email: catherine.pelachaud@upmc.fr

Nicolas Obin
email: nicolas.obin@ircam.fr

Multimodal modeling of expressiveness for human-machine interaction

Keywords: multimodality, speech, gestures, prosody, intelligent embodied conversational agents

come   L'archive ouverte pluridisciplinaire

INTRODUCTION

Human-Human interaction inherently involves the communication through multiple channels. We employ several modalities, both sequentially and in parallel, to communicate in our daily life. The multimodal channels adopted in human communication are verbal and non-verbal [START_REF] Norris | Analyzing multimodal interaction: A methodological framework[END_REF]. Non-verbal are non-vocal signals that are sent by means of facial expressions, gaze, body postures, head or arm movements. The speech modality involves the speech content (words) and vocal signals such as speech prosody and acoustic signals. Both verbal and non-verbal modalities are essential to send and perceive new information. They both reflect our socio-emotional behavior which includes our psychological state and attitude. During a human-human interaction the alignment phenomena (tone of voice, speed of body movement) are signs of common understanding and engagement in the interaction [START_REF] Castellano | Expressive copying behavior for social agents: A perceptual analysis[END_REF][START_REF] Sidner | Where to Look: A Study of Human-Robot Engagement[END_REF]. Communicative modalities should be taken into consideration when developing Human Computer Interaction (HCI) applications because computers are becoming more integrated in our daily life. As a matter of fact, in the past decade, many HCI applications such as personal assistants, tutoring systems, reception agents, chatbots, smartphones and home assistants are being extremely used in our daily life. The rise of such applications are leading humans to interact with virtual agents.

A key problem in the design of virtual assistants is how to maintain user's engagement [START_REF] Bickmore | Maintaining engagement in long-term interventions with relational agents[END_REF][START_REF] Bickmore | Establishing and Maintaining Long-Term Human-Computer Relationships[END_REF] during the interaction so that the interaction lasts long and stays fluent. The main limitations concern are on the one hand the agent's weak expressiveness, and on the other hand the agent's weak adaptation of his behavior to the behavior of the user. As a matter of fact the agent's behavior is very monomodal and monotonous, rendering the interaction very short, and lacking in variety and interest. On the other hand, the agent does not adapt its behavior with respect to the interlocutor's behavior which decreases the engagement of the user in the interaction [START_REF] Castellano | Expressive copying behavior for social agents: A perceptual analysis[END_REF]. The behavior of the agent greatly impacts the user's attentive commitment, the duration of the interaction, as well as the user's understanding of the transmitted messages.

The present paper is part of a thesis work that aims to better understand and model the mechanisms that govern multimodal interaction (voice and gesture) between humans and Embodied Conversational Agents (ECA). We aim to develop an engaging conversational agent ECA with expressive gestures and prosody, capable of maintaining the attention of the interlocutor during the conversation, emphasizing important points, and making the interaction last longer by improving its quality. The ECA will be able to reinforce the interlocutor's engagement by adapting its behavior according to the behavior of the interlocutor. As a first step of our work, we present in this paper a State-of-the-Art review of how to design engaging virtual agents with expressive gestures and prosody. In the following sections, we discuss the different points that should be taken into account when designing an ECA capable of detecting the user's engagement level, maintaining and reinforcing it by displaying an appropriate behavior. We also discuss some of the existing models of the agent's behavior, based on different scenarios. An emphasis is placed on gestural and prosodic expressivity of the agent.

ENGAGING EMBODIED CONVERSATIONAL AGENTS

People's satisfaction and well-being result from their environment, the decision latitude available to them, as well as the social support that exist in their environment [START_REF] Bae Brandtzaeg | Enjoyment: lessons from Karasek[END_REF]. For instance, a good working life of employees consists of various activities in dynamic environments, the support of co-workers and management, as well as the decisions they are allowed to make. In the context of Human-Computer Interaction, the presence of repetitiveness and routine within a human-machine interaction, and the absence of unpredictability and variation will probably imply a passive and boring interaction. Personalising interaction is crucial to obtain engagement as well as long-term relationship development. For instance, in [START_REF] Adam | Hello Emily, how are you today?: personalised dialogue in a toy to engage children[END_REF] they propose a toy companion for children capable of personalizing the interaction based on personal knowledge about the child, in order to build a long-term relationship, and render the interaction with the child more engaging. Interactions that are not personalized, and that do not contain the experience of control will result in a stressful experience [START_REF] Bae Brandtzaeg | Enjoyment: lessons from Karasek[END_REF]. Engagement is "the process by which two (or more) participants establish, maintain and end their perceived connection during interactions they jointly undertake" as defined by Sidner et al [START_REF] Sidner | Where to look: A study of human-robot interaction[END_REF]. The design of Embodied Conversational Agents (ECA) capable of rendering the users engaged in the interaction, is essential and critical for Human Agent Interaction applications [START_REF] Clavel | Fostering User Engagement in Face-to-Face Human-Agent Interactions: A Survey[END_REF]. Various applications such as tutoring systems [START_REF] Mello | AutoTutor and affective autotutor: Learning by talking with cognitively and emotionally intelligent computers that talk back[END_REF], ambient assisting living [START_REF] Bickmore | Health dialog systems for patients and consumers[END_REF][START_REF] Griol | Modeling the User State for Context-Aware Spoken Interaction in Ambient Assisted Living[END_REF], and virtual museum agents [START_REF] Cafaro | First Impressions in Human-Agent Virtual Encounters[END_REF][START_REF] Campano | I like this painting too": When an ECA Shares Appreciations to Engage Users[END_REF][START_REF] Gockley | Designing robots for long-term social interaction[END_REF][START_REF] Kopp | A Conversational Agent as Museum Guide -Design and Evaluation of a Real-World Application[END_REF] convey the importance of maintaining the user's engagement during his/her interaction with the agent. User engagement is marked by nonverbal social behaviors at specific moments of the interaction: it can be feedback signals (to indicate being in phase with the interactant), a form of imitation (i.e. smiling for another smile or making the tone of the voice resemble that of the interactant), or signals synchronized with those of the interactant (management of speaking turns). In addition to that, the behavior of the agent should adapt to the multimodal behavior of the interlocutor [START_REF] Castellano | Expressive copying behavior for social agents: A perceptual analysis[END_REF][START_REF] Sidner | Where to Look: A Study of Human-Robot Engagement[END_REF]. In other words, it should be capable of adapting its behavior with respect to the behavior of the interlocutor, to strengthen the latter's engagement in interaction.

The two components underlying the engagement process are the attentional involvement and emotional involvement [START_REF] Peters | An exploration of user engagement in HCI[END_REF]. Shortterm engagement is needed for performing a specific task during the interaction, whereas long-term engagement is essential for longer periods of interactions [START_REF] Bickmore | Health dialog systems for patients and consumers[END_REF]. The development of an engaging ECA must take into account the socio-emotional behavior of users which are expressed by means of verbal or non-verbal signals [START_REF] Clavel | Fostering User Engagement in Face-to-Face Human-Agent Interactions: A Survey[END_REF]. A socio-emotional behavior includes user's social attitudes as well as their emotions. As Scherer defines it, a social attitude is "an affective style that spontaneously develops or is strategically employed in the interaction with a person" [START_REF] Scherer | What are emotions? And how can they be measured[END_REF]. An engaging agent detects the engagement level of its interlocutor and maintains it by displaying an appropriate socio-emotional behavior during the interaction. Some studies prefer to detect the user's level of engagement by analyzing his/her signals (such as the location of his/her face), instead of his/her socio-emotional behavior [START_REF] Bohus | Managing Human-Robot Engagement with Forecasts and[END_REF]. Other studies prefer to focus on acoustic features such as prosody, voice quality, and spectral features. Prosodic cues are also informative of the engagement level of the users, in the form of global cues (long term patterns), and local cues (short-term) [START_REF] Sidner | Where to look: A study of human-robot interaction[END_REF]. Furthermore, engagement can be reflected not only in vocal signals, but also in the speech of the interlocutor [START_REF] Sidner | Where to look: A study of human-robot interaction[END_REF]. User engagement cannot happen if the agent isn't expressive. In the following sections, we review the state of art of gestural expressivity and prosodic expressivity.

GESTURAL EXPRESSIVITY

An ECA with expressive gestures is capable of having a varied and coherent expression so that it can maintain the attention of the interlocutor. It is also capable of emphasizing important ideas, while leveraging the quality of the interaction, and lengthening its duration (to exceed one or two speaking turns). A key challenge in designing ECAs with expressive gestures, is to display the right gestures at the right time. For instance, the signification of a ring gesture can change according to other body movements as well as the utterance that are produced simultaneously [START_REF] Streeck | Elements of Meaning in Gesture[END_REF]. In addition to that, a ring gesture can also be used as a kinesic sign, to substitute speech [START_REF] Streeck | Elements of Meaning in Gesture[END_REF]. In this case, other body movement occurring at the same time can change its meaning. For instance, facial expressions of the person doing a ring gesture, can transform the meaning of this gesture: it can be a positive meaning such as "perfect" or "delicious", or a negative one such as "zero" or "worthlessness".

In a human-human interaction, gestures and speech are synchronized in one speaker, or even during an interaction with another speaker [START_REF] Mark L Knapp | Nonverbal communication in human interaction[END_REF]. A little change in one gesture will occur at the same time with the beginning of change in another behavior such as a phonological segment. For instance, stressed syllables are frequently in line with gestural strokes and even eye blinks [START_REF] Mark L Knapp | Nonverbal communication in human interaction[END_REF]. During an interaction, we tend to mimic each-other unconsciously, following the "Chameleon effect" by matching and mirroring each-other with our postures, facially, and vocally [START_REF] Mark L Knapp | Nonverbal communication in human interaction[END_REF]. This way, humans adapt their behavior passively and unintentionally. Gestures and body movements convey emotional intentions [START_REF] Dahl | Visual perception of expressiveness in musicians' body movements[END_REF]. For instance, in [START_REF] Castellano | Automated analysis of body movement in emotionally expressive piano performances[END_REF], they demonstrate that the quantity of motion of the upper body and the velocity of head are sensitive to positive emotional valence of the emotional expression. Developing human-like body language expressions in virtual agents and in robots enhance their expressiveness and improve their sociability.

Modeling agent's non-verbal behavior as well as its gestures' expressivity can be performed in multiple ways, based on different scenarios. In [START_REF] Cafaro | Exploiting evolutionary algorithms to model nonverbal reactions to conversational interruptions in user-agent interactions[END_REF], they designed a behavioral and computational model based on an evolutionary algorithm for generation behaviors following an interruption. They used an interactive genetic algorithm since it is capable of exploring the space of solutions. In [START_REF] Rodriguez | Spontaneous talking gestures using generative adversarial networks[END_REF], they use Generative Adversarial Networks (GAN) to generate gestures using upper body sequence of joint positions which are aligned with each utterance. They show that high expressivity can be achieved with a low number of degrees of freedom.

In [START_REF] Yoon | Robots learn social skills: End-to-end learning of co-speech gesture generation for humanoid robots[END_REF], they present a learning-based co-speech gesture generation system. Since the co-speech gesture generation problem consists of mapping a sequence of words to a sequence of human poses, this problem is close to the neural machine translation that uses sequence to sequence mapping. Therefore, they chose to use a neural network architecture that is composed of an encoder (for speech-text understanding) and a decoder (to produce a sequence of gestures/frame by frame poses). The model captures speech context by using a bidirectional recurrent neural network, and the results are sent to the decoder to produce gesture motions. They have used a recurrent neural network for decoding, with a soft attention mechanism so that the decoder focuses on specific words instead of the whole utterance when producing poses. The model produces different types of gestures: iconic, metaphoric, deictic, and beat gestures.

Humans' voice is the "mirror of the soul": it can reflect our emotional states and feelings. Our most personal experience can be expressed by the mean of speech, in multiple degree of variation, rendering each expression a unique act. Prosody refers to all suprasegmental aspects of speech [START_REF] Xu | Prosody, tone and intonation[END_REF]. It involves pitch, duration, amplitude, and voice quality that are used to perform lexical contrasts and convey meaning. As the articulatory functional principle states, speech is a system for transmitting communicative meanings that can be lexical, post-lexical, affective, and social ones. Nowadays, a key problem in the development of Text-to-Speech Systems (TTS) is that they only focus on the physiological (gender, age, intrinsic and co-intrinsic characteristics) speech generation as well as short-term variations of speech parameters (i.e the articulations) [START_REF] Obin | MeLos: Analysis and modelling of speech prosody and speaking style[END_REF]. Therefore, TTS systems are very monotonous, and prosody is a real concern since the generated prosodic parameters have a poor variability. Thus, it is very important to consider implementing expressive TTS systems when designing virtual agents or even physical robots. TTS systems used in robotics are very monotonous, lacking prosodic expressivity. Some of the prosodic tools that can express a speaker's affective and emotional state are non-lexical sounds. Non-lexical sounds are non-phonological sounds that are generated during the speech or outside the talk turn. These sounds are prosodically relevant, and they can convey the speaker's emotions, intentions, attitudes and mental state [START_REF] Ameka | Interjections: The universal yet neglected part of speech[END_REF][START_REF] Schröder | Experimental study of affect bursts[END_REF][START_REF] Schröder | Perception of non-verbal emotional listener feedback[END_REF]. There exist different types of nonlexical sounds such as interjections (i.e. Aie! Ah!), fillers (words used for pausing, i.e. um, okay, uh), grunts (guttural sounds made by humans/animals), and bursts (very brief, discrete, nonverbal expressions of affect in both face and voice as triggered by clearly identifiable events [START_REF] Eva | Affect bursts: dynamic patterns of facial expression[END_REF], i.e. Wow!). All these non-lexical sounds are pure prosodic tools and could be indicators of the emotional and affective state of the speaker. In [START_REF] Aubergé | The eee corpus: socio-affective" glue" cues in elderlyrobot interactions in a smart home with the emoz platform[END_REF], they show that these sounds given to a robot in a smart home create a strong "socio-affective glue" with elderly. "Glue" designates the process that is dynamically constructed during an interaction which is considered as an emerging global system whose interactants are not complete subsystems [START_REF] Aubergé | La glu socio-affective: enjeux et risques du robot «compagnon[END_REF]. This socio-affective glue allows speakers to actively build communicative channel during an interaction. Imitation can also produce a socio-affective "glue" in children language acquisition [START_REF] Tomasello | Understanding and sharing intentions: The origins of cultural cognition[END_REF]. Human-machine interactions can become as natural as human-human interactions by employing speech with prosodic expressiveness, that is, speech with emotional content [START_REF] Monzo | Voice quality modelling for expressive speech synthesis[END_REF]. The emotional content of speech can be conveyed by manipulating the voice quality as well as prosody parameters (f0, duration, and energy). Recent work suggest employing both quality voice and prosody parameters to improve the acoustic modelling of expressive speech. In the last few years, research has focused on the Harmonic plus Noise Model (HNM) to generate high quality and versality and perform speech transformation by modifying the speech prosody. For instance, humans can combine different voice pitch with different body motions to signal their desire to take the floor during interactions, and as a result, interlocutors react to their signals and therefore making the turn-taking mechanism happen [START_REF] Starkey | Some signals and rules for taking speaking turns in conversations[END_REF].

In a human-human interaction, speech and movement are rhythmically coordinated in syllables and even smaller units [START_REF] Mark L Knapp | Nonverbal communication in human interaction[END_REF]. For instance, humans can combine different voice pitch with different body motions to signal their desire to take the floor during interactions, and as a result, interlocutors react to their signals and therefore making the turn-taking mechanism happen [START_REF] Starkey | Some signals and rules for taking speaking turns in conversations[END_REF].

LIMITATIONS

The previous two sections summarized the latest works related to gestural and prosodic expressivity. Despite the great findings and conclusions that these works draw, further research efforts should be directed at the development of more expressive gestures and prosody. For instance, as we have previously discussed, in [START_REF] Castellano | Automated analysis of body movement in emotionally expressive piano performances[END_REF] they demonstrate that the quantity of head motion displayed by a pianist is sensitive to positive emotional valence of expression, but their analysis is based on one musician (pianist) playing one instrument (the piano). In addition to that, they didn't investigate all emotional expressive movements, and analyze various modalities of expressions that musicians can do while playing (i.e. gaze, posture, facial expressions, ...). In [START_REF] Rodriguez | Spontaneous talking gestures using generative adversarial networks[END_REF], they use GANs to generate gestures using upper body sequence of joint positions which are aligned with each utterance. The limitation of this study is that they generate gesture movements without taking into account the semantic meaning of the spoken text, nor considering emphasis on important words. Moreover, in [START_REF] Yoon | Robots learn social skills: End-to-end learning of co-speech gesture generation for humanoid robots[END_REF], they present a learning-based co-speech gesture generation system. This system makes repetitive and excessive gestures: the participants in the experiments noted that motions were "jerky", "fast" and "jump around from motion to motion". In addition to that, participants noted that gestures were faster than speech which makes the interaction looks unnatural. The system also lacks the personalization of robots' gestures: all robots make the same gestures when exposed to one speech context. There is no parameter to control the robots' expressiveness. Furthermore, prosody is not taken into consideration when generating co-speech gestures: the system lacks prosodic expressiveness.

OUR CONTRIBUTION

Our work consists in developing a multimodal model of expressiveness for human-agent interaction. We plan to model the gestural and prosodic expressiveness, while taking into consideration all the points discussed in the previous section. More specifically, we plan to develop a multimodal model to predict facial expressions based on previous head motion, facial expressions, text, and prosody. We will build a multimodal model and it will be implemented using a Sequence to Sequence encoder/decoder framework. This will also allow finding semantic and syntactic alignments. As mentioned in sections 4 and 5, human-agent interactions may become as natural as human-human interactions by employing speech with prosodic and gestural expressiveness. This is the reason why our model will do the prediction based on voice (voice prosody and voice quality) and gestural (facial expressions and head motions) input features. The input speech features that we will use in our model consist of prosodic features, voice quality features, as well as Mel Frequency features. The prosodic features that we will consider in our model are the energy, the fundamental frequency F0, as well F0 contour. As [START_REF] Monzo | Voice quality modelling for expressive speech synthesis[END_REF] suggests, the voice quality features that can be useful for expressiveness, and which we will use for our model are: Jitter, Shimmer, Harmonic-to-Noise Ratio (HNR), and the hammarberg index (Hamml). The gestural input features will be the action units of facial expressions as well as head motions. In addition to that, we will focus on the learning of a coherent multimodal behavior by using multimodal attention mechanisms applied to the synchronicity of the generated prosody and gestures. In later work, we will focus on rendering the agent's behavior coherently aligned with that of its interlocutor by using interactive and imitation learning.

ACKNOWLEDGMENTS

This work was performed within the Labex SMART (ANR-11-LABX-65) supported by French state funds managed by the ANR within the Investissements d'Avenir programme under reference ANR-11-IDEX-0004-02.