C Simenel

Ph Chomaz

Couplings between dipole and quadrupole vibrations in tin isotopes

We study the couplings between collective vibrations such as the isovector giant dipole and isoscalar giant quadrupole resonances in tin isotopes in the framework of the time-dependent Hartree-Fock theory with a Skyrme energy density functional. These couplings are a source of anharmonicity in the multiphonon spectrum. In particular, the residual interaction is known to couple the isovector giant dipole resonance with the isoscalar giant quadrupole resonance built on top of it, inducing a nonlinear evolution of the quadrupole moment after a dipole boost. This coupling also affects the dipole motion in a nucleus with a static or dynamical deformation induced by a quadrupole constraint or boost respectively. Three methods associated with these different manifestations of the coupling are proposed to extract the corresponding matrix elements of the residual interaction. Numerical applications of the different methods to 132 Sn are in good agreement with each other. Finally, several tin isotopes are considered to investigate the role of isospin and mass number on this coupling. A simple 1/A dependence of the residual matrix elements is found with no noticeable contribution from the isospin. This result is interpreted within the Goldhaber-Teller model.

I. INTRODUCTION

A particular interest in strongly interacting systems is their ability to present disorder or chaos, and, in the same excitation energy range, well-organized motion. Atomic nuclei are known to show both behaviors [START_REF] Bohr | Nuclear Structure[END_REF]. In particular, they exhibit a large variety of collective vibrations, also called giant resonances (GRs), with excitation energy usually above the particle emission threshold [START_REF] Harakeh | Giant Resonances: Fundamental High-Frequency Modes of Nuclear Excitations[END_REF]. The GRs are associated with anomalously large cross sections in some nuclear reactions.

Baldwin and Klaiber observed the isovector giant dipole resonance (GDR) in photofission of uranium nuclei [START_REF] Baldwin | [END_REF], interpreted as a vibration of neutrons against protons [4]. This GDR was investigated with several probes [START_REF] Harakeh | Giant Resonances: Fundamental High-Frequency Modes of Nuclear Excitations[END_REF] and was also observed on top of highly excited states, e.g., in hot nuclei [5]. The survival of ordered motion in hot nuclei, i.e., in a chaotic environment, is one of the most striking phenomena in nuclear physics. Other kinds of GR have been discovered, such as the isoscalar giant quadrupole resonance (GQR) associated with an oscillation of the shape between a prolate and an oblate deformation [6], and the isoscalar giant monopole resonance (GMR) corresponding to a breathing mode [7,8,9].

The GRs are usually associated with the first phonon of a small-amplitude harmonic motion. However, the proof of their vibrational nature came with the observation of their two-and three-phonon states [10,11,12]. Multiphonon studies also provided a good test of the harmonic picture. In particular, anharmonicity was found in an abnormally large excitation probability of these states, indicating that different phonon states couple because of the residual interaction [13,14]. Microscopic investigations, such as the random phase approximation (RPA) together with boson mapping techniques [15] and the nonlinear response to an external field in the time-dependent Hartree-Fock (TDHF) theory [16,17] showed, indeed, that strong couplings between GMR, GQR and GDR occur. In particular, a GMR or a GQR (resp. a GMR) can be excited on top of a GDR (resp. a GQR), leading to couplings between one-and two-phonon states. As a consequence, GRs cannot be described in a purely harmonic picture. Anharmonicities were also found to affect pygmy dipole resonances, though depending on the choice of the nuclear functional [18].

The goal of the present work is to get a deeper insight into the couplings between various GRs, which represents a first step toward understanding complexity and disorder in nuclei at high excitation energies. As an example, we focus on the coupling between isovector dipole and isoscalar quadrupole vibrations. A clear link between the linear dipole motion on a deformed state and the quadratic response of the quadrupole moment to an external dipole excitation (investigated in Ref. [16]) is made. The TDHF theory is used to compute the residual interaction coupling the one-phonon state of the GDR to the two-phonon state with a GQR built on top of the GDR. Applications to spherical tin isotopes are performed to investigate the role of the isospin degree of freedom and of the total number of nucleons on the coupling.

We present a schematic model describing couplings between GRs and their effect on onebody observables in Sec. II. The TDHF formalism and its application to nuclear vibrations are discussed in Sec. III. Numerical details on the 3-dimensional TDHF code are also given.

A detailed investigation of the couplings in 132 Sn is presented in Sec. IV, together with a more systematic analysis in tin isotopes. Finally, we conclude in Sec. V.

II. A SCHEMATIC MODEL FOR GR COUPLING

Let us illustrate the effect of couplings between vibrational modes within a simple schematic model introduced in Ref. [16]. We consider the Hamiltonian Ĥ = Ĥ0 + V [START_REF] Bohr | Nuclear Structure[END_REF] where Ĥ0 corresponds to the harmonic (HF+RPA) part and the residual interaction V couples collective modes. Eigenstates of Ĥ0 are one-and two-phonon states |ν and |νµ

with eigenenergies E ν = E 0 + hω ν and E νµ = E 0 + hω ν + hω µ , respectively, where ω µ,ν denote the collective frequencies and E 0 is the ground-state energy. In the following, h is omitted in the notation. Only the coupling between the two states |ν and |νµ is considered here.

The associated matrix element of the residual interaction is noted v µ = ν| V |νµ . Such couplings between one-and two-phonon states have been proven to be the most important one in nuclei [15]. Using first-order perturbation theory, the eigenvalues of Ĥ are those of Ĥ0 with eigenstates

|ν ≈ |ν -ε µ |νµ (2)
and

|νµ ≈ |νµ + ε µ |ν (3)
where ε µ = vµ ωµ .

The couplings are expected to affect the evolutions of expectation values of one-body observables such as the multipole moments Q ν (t) ≡ Qν (t). We investigate below three different manifestations of the couplings on these evolutions. They will be used in the next section to compute v µ from TDHF calculations in the case of coupling between giant dipole and quadrupole resonances.

A. Quadratic response

The effect of couplings in the quadratic response has been introduced in Ref. [16]. Highlights on the main steps are given here. At initial time, the ground state |0 of the system is excited by a boost with the one-body operator Qν

|Ψ(0) = exp(-ik ν Qν )|0 . (4)
Developing the exponential up to second order in the boost intensity k ν and considering an evolution under the Hamiltonian defined in Eq. ( 1), the state at time t reads at first order

in ε µ |Ψ(t) ≈ exp(-iE 0 t) 1 - k 2 ν q 2 ν 2 |0 -ik ν q ν e -iων t |ν -ε µ e -iωµt |νµ , (5)
where q ν = ν| Qν |0 is the transition amplitude that we assume to be real.

The expectation value of the one-body observable used in the boost exhibits oscillations.

Indeed, in case of no static deformation in the ground state, we have

Q ν (t) = -2k ν q 2 ν sin(ω ν t) + O(k 3 ν ). (6)
In particular, its amplitude increases linearly with the boost intensity in the small amplitude regime. In addition to this linear response, the coupling induces an oscillation of the other collective mode Q µ :

Q µ (t) ≈ 2k 2 ν q 2 ν q µ v µ ω µ [cos(ω µ t) -1] (7)
where we have assumed q µ = µ| Qµ |0 = µν| Qµ |ν . This oscillation is then quadratic in k ν and provides a first method to compute the residual interaction v µ , assuming the fact that a nonlinear theory, such as TDHF, is used to follow the expectation values of the one-body observables. We finally note that Q ν (t) and Q µ (t) have different frequencies and start in phase quadrature.

B. Linear response in an external static field

It is interesting to note that the coupling may also manifest itself in the linear response to the boost (4) if an external static field is added to the Hamiltonian (1)

Ĥ(λ) = Ĥ(0) + λ Qµ . (8)
We choose λ small enough to induce a linear static deformation defined as

Q 0 µ (λ) = 0(λ)| Qµ |0(λ) ≈ λ ∂Q 0 µ ∂λ λ=0 , (9)
where the ground state |0(λ) of Ĥ(λ) contains a contribution of the one-phonon state |µ :

|0(λ) ≈ |0 + λ 2q µ ∂Q 0 µ ∂λ λ=0 |µ . ( 10
)
The external potential modifies linearly the eigenenergies of the Hamiltonian and the frequency of the linear response to a boost (4) on |0(λ) follows

∂ω ν ∂λ λ=0 = v µ q µ ∂Q 0 µ ∂λ λ=0 , (11)
providing another direct way to extract the matrix element v µ of the residual interaction. We emphasize the fact that, here, the nonlinear response is not invoked and a RPA code allowing static deformation in the ground state would be sufficient to compute such couplings.

C. Response to two simultaneous excitations

We showed two manifestations of the coupling (i) in the quadratic response and (ii) in the linear response under a static constraint. Let us now introduce a third one where the response Q ν (t) is studied after a simultaneous double boost of Qµ and Qν :

|Ψ(0) = e -ikµ Qµ e -ikν Qν |0 . ( 12
)
The Qµ term modifies the response of Eq. ( 6) with an additional term

∆Q ν (t) = Qν (t) -Qν kµ=0 (t) = 4k ν k µ q 2 ν q µ v µ ω µ [1 -cos(ω µ t)] cos(ω ν t). (13)
It is convenient to write this evolution with the form

x(t) = Q ν (t) Q ν = sin ω ν t -β cos ω ν t + β 2 cos(ω µ + ω ν )t + β 2 cos(ω µ -ω ν )t (14)
where Q ν = -2k ν q 2 ν and β = 2k µ q µ v µ /ω µ . In fact, we can show that x(t) is a solution of the differential equation

ẍ ω 2 ν + 1 -2β ω µ ω ν sin ω µ t x + β ω 2 µ ω 3 ν ẋ cos ω µ t = 0 ( 15
)
if one keeps only the first-order terms in β. The two first terms of the left-hand side are equivalent to a Mathieu's equation. It is not surprising because the latter has been shown to qualitatively reproduce the preequilibrium dipole motion coupled to collective shape vibrations of the system in N/Z asymmetric fusions [19,20].

We see in Eq. ( 14) that the effect of the coupling produces vibrations at frequencies |ω ν ± ω µ |. By analogy to the standard response function related to the strength distribution [START_REF] Ring | The Nuclear Many-Body Problem[END_REF] we introduce the coupling response function

R c ν (ω) = -1 πk ν k µ ∞ 0 dt cos(ωt)∆Q ν (t) (16)
defined for ω ≥ 0. The latter can be used to investigate the coupling because it is linearly proportional to v µ :

R c ν (ω) ≈ q 2 ν q µ v µ ω µ [-2δ(ω ν -ω) + δ(ω ν + ω µ -ω) + δ(|ω ν -ω µ | -ω)] . (17)
Equation ( 17) provides then a third way to extract v µ .

Let us finally note that the contribution to the coupling response function at ω ν and those at |ω ν ± ω µ | have opposite signs in Eq. ( 17) and that the integral of the coupling response function is zero. It is interesting to note that this property is still valid at all order in k ν and k µ . To show it, let us recall that, in our schematic model, Qν |0 = q ν |ν and Q2

ν |0 = q 2 ν |0 , which implies e -ikν Qν |0 = c ν |0 -is ν |ν , where c ν = cos(k ν q ν ) and s ν = sin(k ν q ν ). The response Q ν (t)
following the double boost of Eq. ( 12) then becomes

Q ν (t) = -2c ν s ν q ν sin(ω ν t) + 4c ν c µ s ν s µ q ν v µ ω µ [1 -cos(ω µ t)] cos(ω ν t). (18)
We see that Eq. ( 14) is still valid if one replaces

Q ν by Q ′ ν = -2c ν s ν q ν and β by β ′ = 2c µ s µ v µ /ω µ .
Then, the ω-dependance of Eq. ( 17) is unchanged. As a consequence, the cancellation of the integral of the coupling response function defined in Eq. ( 16) is not limited to the small-amplitude regime.

III. THE TIME-DEPENDENT HARTREE-FOCK APPROACH A. Applications to nuclear vibrations

Coherent motion of fermions such as collective vibrations in nuclei can be modeled by time-dependent mean-field approaches like the TDHF theory proposed by Dirac [START_REF] Dirac | [END_REF]. Indeed, in its linearized version, TDHF is equivalent to the RPA which is the basic tool to understand the collective vibrations in terms of independent phonons.

As we saw in the previous section, giant resonance properties can be investigated by studying the response of the system to an external (collective) one-body field. In particular, time evolution of one-body (collective) observables, which can be computed using mean-field approximations, contain the necessary information to investigate the couplings between collective modes. Indeed, TDHF takes into account the effects of the residual interaction if the considered phenomenon can be observed in the time evolution of a one-body observable.

In particular, the nonlinear response in TDHF contains the couplings between one-and two-phonon states coming from the 3-particle 1-hole and 1-particle 3-hole residual interaction [16]. In that sense, it goes beyond the RPA, which is a harmonic picture and contains only 1-particle 1-hole residual interaction.

In its unrestricted form (i.e., with no constraint on spatial symmetry), TDHF authorizes all possible spatial forms of the nucleon wave functions, which is crucial because of both shell the effects and the wave dynamics. In addition, Landau spreading and evaporation damping are well accounted for [23]. However, it does not incorporate the dissipation from two-body mechanisms [24,25,26]. Inclusion of pairing correlations is possible within the time-dependent Hartree-Fock-Bogolyubov theory [27], but realistic applications in three dimensions are not yet achieved. Extension to theories going beyond the one-body limit such as extended TDHF [26], second RPA [28,29], time-dependent density matrix theory [30,31,32] or stochastic one-body transport theory [33] should be considered for realistic description of giant resonance properties [34].

Application of TDHF to nuclear dynamics has been possible thanks to the Skyrmetype effective interaction [35,36]. Early realistic TDHF codes have been applied to study collective vibrations in nuclei with simplified Skyrme interactions [37]. Recent increase of computational power allowed realistic TDHF description of giant resonances in 3 dimensions with full Skyrme energy density functional (EDF) [16,38,39,40,[START_REF] Simenel | Proceedings at the Joliot-Curie School[END_REF]. In particular, TDHF has been used to investigate nonlinear effects in nuclear vibrations [16,[START_REF] Reinhard | [END_REF].

B. Formalism

The TDHF equation can be written as a Liouville-Von Neumann equation

i ∂ ∂t ρ = [h[ρ], ρ] , ( 19
)
where ρ is the one-body density matrix of an independent particles state with elements

ρ(rsq, r ′ s ′ q ′ ) = A i=1 ϕ i (rsq)ϕ * i (r ′ s ′ q ′ ), ( 20
)
where A is the number of nucleons. The sum runs over all occupied single-particle wave functions ϕ i and r, s, and q denote the nucleon position, spin, and isospin respectively.

The Hartree-Fock single-particle Hamiltonian h[ρ] is related to the EDF, denoted by

E[ρ], through h[ρ](rsq, r ′ s ′ q ′ ) = δE[ρ] δρ(r ′ s ′ q ′ , rsq) . (21)

C. Numerical details

In this work, the TDHF equation ( 19) is solved iteratively in time on a spatial grid with a plane of symmetry using the tdhf3d code built by P. Bonche and coworkers [43] with the SLy4 parametrization of the Skyrme EDF [44]. The latter has been constrained on the pure neutron matter equation of state to improve the description of exotic nuclei.

For instance, together with a density-dependent zero-range pairing interaction, it allows a somewhat good reproduction of the isotopic shifts between proton and neutron mean-square radii in lead isotopes. It also improves the description of the isotopic evolution of the binding energies [44]. Defining the neutron drip line as the isotope for which the chemical potential vanishes, it is estimated to be around 176 Sn in the tin isotopic chain [45], though it might depend on the choice of the pairing interaction and on the inclusion of beyond mean-field correlations.

Good convergences of the quadrupole and dipole moment evolution is ensured with a lattice spacing ∆ r = 0.6 fm and a time step ∆ t = 5 × 10 -25 s. The size of the half-box where the single particle wave functions are evolved is 80 × 80 × 40 in mesh size unit ∆ r, unless otherwise specified.

IV. RESULTS

Let us now investigate the coupling between isovector dipole and isoscalar quadrupole vibrations in tin isotopes in the framework of the theoretical model presented in Sec. II where |ν ≡ |D and |µ ≡ |Q denote a GDR and a GQR phonon respectively. The isovector dipole moment is defined as

QD = NZ A ( Ẑn -Ẑp ) ( 22
)
where Ẑn (resp. Ẑp ) measures the neutron (resp. proton) average position on the z axis.

The isoscalar quadrupole moment reads

QQ = 5 16π A i=1 (2ẑ 2 i -x2 i -ŷ2 i ). ( 23
)
Their expectation value evolutions are computed using the tdhf3d code after different initial conditions as described below.

A. Nonlinear quadrupole motion induced by a dipole boost

We first investigate the quadratic response presented in Sec. II A in the 132 Sn nucleus.

Figure 1(a) shows the early time evolution of the dipole moment after a dipole boost according to Eq. ( 4) in the small-amplitude regime. The dipole moment follows asin function as indicated by Eq. ( 6). Extracting the frequency from the first minimum of D(t) leads to a GDR energy of ω D ≈ 15.2 MeV. This value is slightly lower than the maximum of the experimental GDR peak energy E max = 16.1 (7) MeV [46]. The same analysis in 120 Sn, in which almost all the dipole strength is located around the GDR energy, gives a value of ω

( 120 Sn) D ≈ 15.
3 MeV which is in good agreement with experimental data where a peak energy of E exp. GDR = 15.4 MeV has been obtained [47]. Note that, the extraction method of ω ν from the first extremum of Q ν (t) is, in first approximation, comparable to the ratio of the second over the first energy weighted moments of the strength function m 2 /m 1 [16].

We see in figure 1 residual interaction of Eq. ( 1) coupling the dipole and quadrupole vibrations. In particular, Q Q (t) starts in phase quadrature with Q D (t) and oscillates with a smaller frequency. These observations are in qualitative agreement with the quadratic response in Eq. (7).

To get a deeper insight into this coupling, we have computed the TDHF response for several dipole boost velocities k D . The first extrema of the dipole and quadrupole moments are reported in figure 2(a) and (b) respectively. Whereas the dipole amplitude is indeed linear in k D as expected from equation ( 6), indicating that these calculations are performed in the small-amplitude regime, the induced quadrupole motion is quadratic in k D , in agreement with Eq. (7).

To obtain a quantitative estimate of the coupling, we first extract the transition amplitude from a linear extrapolation of Q min D at k D → 0 in Fig. 2(a). According to Eq. ( 6), we get q D ≈ 6.98 fm. The same analysis with a quadrupole boost in the linear regime gives a transition amplitude q Q ≈ 61.4 fm 2 and a GQR energy of ω Q ≈ 13.0 MeV. Note that the same analysis in 120 Sn gives a GQR energy of ω ( 120 Sn) Q ≈ 13.3 MeV, in excellent agreement with the experimental value E exp. GQR = 13.24 ± 0.13 MeV [48]. These quantities, together with a quadratic extrapolation of the quadrupole maximum at k D → 0 in Fig. 2(b), give, according to Eq. ( 7), a matrix element of the residual interaction v

B. Dipole motion in a nucleus with a static quadrupole constraint

The formalism developed in Sec. II B, where the linear response is investigated in an external potential, cannot be directly applied to study the coupling between the dipole and quadrupole modes. The reason is that the external potential -λ QQ with the definition of Eq. ( 23) is not bound from below and its use in constrained HF calculations would lead to unphysical results. It is then necessary to consider another external potential such as

λ( QQ + κ λ QM ) (24)
where

QM = 1 √ 4π A i=1 r2 i ( 25
)
is the monopole moment and κ λ = √ 5/2 if λ ≥ 0 and -√ 5 if λ < 0. The expression (24) then reads Such an external field allows one to explore all quadrupole deformations from oblate (λ > 0) to prolate (λ < 0) shapes as shown in figure 3(a) where the ground state quadrupole deformation Q 0 Q of the constrained HF solution is plotted as a function of the Lagrange parameter λ. The quadrupole deformation is clearly linear in this perturbative regime and its slope at the origin is

3 5 16π λ A i=1      ẑ2 i if λ ≥ 0, -x 2 i -ŷ2 i if λ < 0. (26)
∂Q 0 Q ∂λ λ=0 ≈ -1260.4 fm 4 MeV -1 .
As discussed in Sec. II B, such a static deformation is expected to change the dipole frequency as compared to that of the GDR excited on the spherical ground state. In fact, the frequency of a dipole oscillation along the main quadrupole axis decreases (resp. increases) with a prolate (resp. oblate) deformation. This is indeed what we observe in figure 3(b) where the energy of the GDR is plotted as a function of λ. Note that, according to Eq. ( 11), this is consistent with the negative sign of the ratio v Q /q Q obtained in Sec. IV A.

We also observe in Fig. 3(b) that the evolution of this energy is linear both for λ > 0 and λ < 0, but the slopes are different in these two regimes. This is attributed to the presence of the monopole moment in the constraint (24). Indeed, the monopole vibration is also coupled to the dipole mode by a matrix element v M of the residual interaction [15,16]. According to Eq. ( 11), the dipole energy is expected to be modified as

ω D (λ) ≈ ω D (0) + λ v Q q Q ∂Q 0 Q ∂λ λ=0 + λκ λ v M q M ∂Q 0 M ∂λ λ=0 . ( 27
)
A compression of the nucleus increases the dipole frequency, which implies v M /q M < 0.

Because λκ λ ≥ 0 for all λ, the monopole and quadrupole moments have an opposite effect on ω D for λ < 0 and act in the same direction for λ > 0. This is indeed what we observe in Fig. 3(b) where the effect of the constraint almost cancels on the prolate side, whereas it strongly increases the GDR energy in the oblate one.

Finally, starting from Eq. ( 27), it is possible to isolate the coupling matrix element between the dipole and quadrupole modes

v Q = ∂Q 0 Q ∂λ -1 λ=0 q Q 3 ∂ω D ∂λ λ→0 - + 2 ∂ω D ∂λ λ→0 + . ( 28
)
Using the data extracted from Fig. 3 and the value of q Q obtained in Sec. IV A, we get v

Q ≈ -0.56 MeV. This result is in reasonable agreement with the one obtained with the quadratic response.

C. Response to a dipole+quadrupole boost

A third manifestation of the coupling between dipole and quadrupole motions occurs when both a dipole and a quadrupole boost are performed at initial time. We showed in Sec. II C that, in such a case, the dipole motion is affected by the quadrupole vibration. Such effect is not present in the linear response theory because the modifications are proportional to k D k Q . As can be seen in Eq. ( 6), there is no other quadratic term because the next-order terms affecting the dipole motion are in k 3 D and k 3 Q . The basic tool to study the effect of the coupling on the dipole motion is the coupling response function defined in Eq. ( 16). In principle, its calculation implies to follow the dipole moment over an infinite time. However, we use a filtering procedure to avoid numerical artifacts coming from the interaction of the nucleus with reflected nucleon wave functions because of the hard box boundary conditions [49]. We perform the calculations over 3000 iterations in time (450 fm/c). The dipole moment is multiplied by a filtering function exp -1 2 t τ 2 with τ = 100 fm/c [39]. This procedure induces an additional width of τ -1 ≈ 2 MeV. According to Eq. ( 17), this additional width is sufficiently small for the present discussion because the modes in the coupling response function are located at ω D -ω Q ≈ 2.2, ω D ≈ 15.2, and ω D + ω Q ≈ 28.2 MeV. However, the low-energy part of the spectrum, i.e., in the region of the ω Dω Q peak, is dependent on the choice of the filtering function within these numerical conditions. We checked with other filtering functions, e.g., a cosine instead of a Gaussian function, to confirm that the higher part of the spectrum (above ≈ 10 MeV) is not affected. In addition, the filtering function does not change the fact that the total integral of the coupling response function vanishes (see Sec. II C). The latter was found to be a solid numerical property of this function. Finally, we checked the convergence of the results presented in this section by comparing with calculations performed in a bigger box of 120 × 120 × 60 in mesh size unit ∆ r.

Figure 4 shows the coupling response function for the dipole motion following a quadrupole+dipole boost. We checked that, in the small-amplitude limit, the coupling response function is indeed independent of k Q and k D . As expected from Eq. ( 17), two peaks are present in this energy range at ω D and ω D + ω Q with opposite signs. Moreover, the integral of the positive peak at ω D is directly related to the coupling as

v Q = - ω Q 2q 2 D q Q R c D >0 dω R c D (ω). (29)
With the coefficients calculated in Sec. IV A, we obtain a coupling v (3) ≈ -0.68 MeV of the same order of magnitude than with the two previous methods.

Let us finally note that in the case of more complicated vibrations, e.g., the oscillations exhibit several frequencies, the coupling response function can be used for a more detailed investigation of the coupling. Indeed, it allows an analysis of the coupling effect at each energy whereas the two previous methods give only access to a weighted sum of the matrix elements of the residual interaction associated to each excited mode [16].

D. Evolution of the coupling with isospin and mass

We now repeat the study of the linear quadrupole motion induced by a dipole boost, described in Sec. IV A, to the tin isotopic chain. The choice of this method to investigate more systematically the coupling between dipole and quadrupole vibrations is motivated by its rather low computational time as compared to the two other methods. Our goal is to understand the evolution of v Q as a function of the isospin. To avoid any ambiguity coming from possible static deformation in the ground states, we focus on some of the tin isotopes that are spherical at the HF level: 100,106,114,120,132,140 Sn. These isotopes allow for an investigation of the coupling from the proton-rich to the neutron-rich side.

Let us first investigate the linear response to a quadrupole boost (Eq. ( 4)) to compute the energies ω Q and transition amplitudes q Q from the first minimum of the quadrupole moment (see Eq. ( 6)). These quantities are plotted in Fig. 5 as a function of the number of nucleons. The GQR energy is known to be proportional to A -1/3 [START_REF] Harakeh | Giant Resonances: Fundamental High-Frequency Modes of Nuclear Excitations[END_REF]. This is compatible with the TDHF results that are fitted by ω Q ≈ 65.5A -1/3 MeV. The evolution of the transition amplitude with A can be obtained from the energy weighted sum rule (EWSR)

for quadrupole vibrations which reads [21]

S 1 Q = α (E α -E 0 )| α| QQ |0 | 2 = h2 m 5 4π A r2 ≈ 14.3A 5/3 MeV.fm 4 (30)
where {|α } is an eigenbasis of Ĥ. In the last line of Eq. 30, we used the approximation of a constant density and a sharp surface which gives rL = 3 L+3 R L with R ≈ 1.2A 1/3 fm. If all the strength is located at the GQR energy, which is a somewhat good approximation for heavy nuclei [START_REF] Harakeh | Giant Resonances: Fundamental High-Frequency Modes of Nuclear Excitations[END_REF], then the EWSR reduces to

S 1 Q = ω Q q 2 Q ≈ 65.5A -1/3 q 2 Q . (31)
Equations ( 30) and (31) then lead to

q Q ≈ 0.466A fm 2 . ( 32
)
This linear dependence is plotted in Fig. 5(b) and reproduces well the TDHF results.

Let us now consider a dipole boost on these nuclei with a boost velocity k D = 0.01 fm -1 .

This value is small enough to generate a linear response of the dipole moment and a quadratic response of the induced quadrupole vibration in all considered isotopes. The GDR energy ω D is shown as a function of the number of nucleons in Fig. 6(a). It is compatible with the A -1/3 dependence expected in heavy nuclei [START_REF] Harakeh | Giant Resonances: Fundamental High-Frequency Modes of Nuclear Excitations[END_REF]. A fit of the TDHF results gives

ω D ≈ 76A -1/3 MeV. ( 33
)
Similarly to the quadrupole case, the dependence of the transition probability q 2 D can be obtained from the dipole EWSR

S 1 D = h2 2m (1 + κ) NZ A ( 34
)
where κ is the enhancement factor of the Thomas-Reiche-Kuhn (TKR) sum rule. Assuming all the strength in the GDR, i.e., q 2 D = S 1 D /ω D , we get from Eqs. ( 33) and ( 34)

q 2 D ≈ h2 2m (1 + κ) NZ 76A 2/3 (35)
In nuclear matter, the enhancement factor of the TKR sum rule is κ = 0.25 with the SLy4 parametrization [44]. This value clearly overestimates the transition probabilities (see dashed line in Fig. 6(b)), though the qualitative trend is in good agreement with the TDHF results. It is possible to compute κ in finite nuclei using [50]

κ = m 4h 2 A NZ [t 1 (2 + x 1 ) + t 2 (2 + x 2 )] dr 3 ρ n (r)ρ p (r) (36)
where the t i and x i are the usual Skyrme parameters. In the considered tin isotopes, κ is almost constant within the range 0.181-0.186 with no particular isospin or mass dependence.

This leads to a better agreement with the transition probabilities obtained with TDHF (see solid line in Fig. 6(b)), though a slight overestimation remains. The latter could be attributed to the fragmentation of the isovector dipole response. In this case, the dipole response reads

Q D (t) = -2k D i q 2 D i sin[(ω D + δω i )t]. (37)
In our calculations, the GDR properties (ω D and q D ) are extracted from the first minimum of the dipole response, which obeys to

Q min D -2k D = i q 2 D i cos π 2 δω i ω D ≡ q 2 D ≤ i q 2 D i . (38)
The last inequality implies that the TKR sum rule allows one only to compute the upper limit of q 2 D in our model. Finally, we investigate the coupling between the quadrupole and dipole vibrations from the quadratic response. We have shown in Sec. II A that, in the presence of a nonzero matrix element v Q of the residual interaction coupling the state |D to the state |DQ , a dipole boost is expected to generate an oscillation of the quadrupole moment. Before studying the evolution of v Q along the tin isotopic chain, it is mandatory to get a deeper insight into the mechanism responsible for this induced quadrupole excitation.

In a macroscopic approach, the isovector GDR is interpreted by a combination of the Steinwedel-Jensen model in which the total density is kept unchanged [51] and the Goldhaber-Teller model where proton and neutron fluids are incompressible [4]. It is obvious that the Steinwedel-Jensen model does not affect the quadrupole moment because any modification of the density of one isospin specie is exactly compensated by the other in every point of space. In the Goldhaber-Teller model, however, a displacement of the proton and neutron spheres in the opposite direction is considered. It produces a dipole moment

Q D = N Z
A X where X is the distance between their centers. This displacement also induces a prolate shape with a quadrupole moment quadratic in X. Indeed, assuming a displacement of a proton (resp. neutron) homogeneous sphere of density Zρ 0 /A (resp. Nρ 0 /A) by X p = -XN/A (resp. X n = XZ/A) produces a quadrupole moment

Q Q ∝ ZX 2 p + NX 2 n = NZ A X 2 . ( 39
)
Using Eqs.( 6) and (35), one gets Q Q ∝ NZ/A 1/3 . Finally, together with Eqs. ( 7), ( 32) and ( 35), the evolution of the coupling simply reads v Q ∝ 1/A. This is, indeed, in agreement with the TDHF results shown in Fig. 7. It is interesting to note that, in this simple approach, the coupling does not depend on the isospin of the nuclei, but only on their total number of nucleons. In fact, the decrease of the absolute strength of the coupling with the number of nucleons is attributed to the fact that these couplings are mediated by the surface [16]. One then expects less anharmonicities in heavy nuclei.

V. CONCLUSIONS

We have shown that the residual interaction is responsible for anharmonicities in nuclear vibrations using three different analyses of time evolutions of multipole moments. We investigated the coupling between one-and two-phonon states using a 3-dimensional TDHF code with a full Skyrme energy density functional. In particular, the excitation of a GDR couples to a GQR built on top of it, inducing a quadratic response of the quadrupole moment. The same coupling is responsible for the change of the GDR energy in static deformed states.

The latter could be investigated using deformed RPA codes. As a consequence, the dipole frequency is modulated in case of dynamical deformation, e.g., induced by a quadrupole boost. This last property, associated with a Fourier analysis, might be used to investigate couplings when more than one mode is excited with the same quantum numbers. We finally investigated these couplings with the quadratic response in several spherical tin isotopes. As a result, no dependence with isospin was found whereas an overall decrease of the coupling is obtained with increasing mass, showing that the couplings are mediated by the surface.

These observations are interpreted within the Goldhaber-Teller macroscopic model. These results indicate that no anharmonicity enhancement is expected in the standard giant resonances even for very exotic nuclei. However, couplings with exotic modes such as the pygmy dipole resonance should be investigated with the present method. The role of pairing and static deformation should be considered as well.

  FIG. 1: Time evolution of the dipole (a) and quadrupole (b) moments in 132 Sn after a dipole boost with an intensity k D = 0.01 fm -1 .

( 1 )

 1 Q ≈ -0.61 MeV.

FIG. 2 :

 2 FIG. 2: Circles: First minimum and maximum of the dipole (a) and quadrupole (b) moment evolution, respectively, in 132 Sn as a function of the dipole boost intensity k D . Dashed lines: linear and quadratic extrapolations at k D → 0 of the dipole (a) and quadrupole (b) amplitudes, respectively.

FIG. 3 :

 3 FIG. 3: (a) Static quadrupole moment from HF calculation (circles) under a quadrupole+monopole constraint (see text) as a function of the Lagrange parameter λ in 132 Sn. (b) TDHF energy of the GDR (circles) from the first minimum of the dipole moment after a dipole boost along the deformation axis with an intensity k D = 0.01 fm -1 . Dashed lines : linear extrapolations at λ → 0 ± of the quadrupole moment (a) and GDR energy (b).

FIG. 4 :

 4 FIG. 4: Coupling response function of the dipole moment after a dipole+quadrupole boost with intensities k D = 0.01 fm -1 and k Q = 0.001 fm -2 respectively.

FIG. 5 :

 5 FIG. 5: Evolution of (a) the GQR energy and (b) the transition amplitude as a function of the number of nucleons in tin isotopes from the TDHF linear response (circles). The line in panel (a) represents a A -1/3 fit of the TDHF results, whereas the line in panel (b) is obtained from considerations on the GQR energy weighted sum rule (see text).

FIG. 6 :

 6 FIG. 6: Evolution of (a) the GDR energy and (b) the transition probability as a function of the number of nucleons in tin isotopes from TDHF linear response (circles). The line in panel (a) represents a A -1/3 fit of the TDHF results. The lines in panel (b) is obtained from considerations on the GDR energy weighted sum rule (see text) with an enhancement factor of the Thomas-Reiche-Kuhn sum rule κ = 0.25 (dashed line) and κ = 0.183 (solid line).

FIG. 7 :

 7 FIG.7: Evolution of the coupling with mass number. The matrix element of the residual interaction is plotted as a function of 1/A. The line shows a linear fit of the TDHF results.

Acknowledgments

We thank P. Bonche for providing his TDHF code. We are also grateful to B. Avez and D. Lacroix for discussions and a careful reading of the article. Useful discussions with K. Bennaceur are also acknowledged. The calculations were performed at the Centre de Calcul Recherche et Technologie of the Commissariat à l' Énergie Atomique.


Samory Houzangbe
email: s.houzangbe@estia.fr

Marco Dias
email: marco.dias@gln.pt

Chloé Morel-Targosz

Nadine Couture
email: n.couture@estia.fr

Alexandre Pere-Laperne
email: alexandre.pere-laperne@1a3i.com

Comparing Unstructured 2D Files: An Interactive Combined Multiple Views Method

Keywords: Unstructured documents, Visual comparison, Information Visualization. CCS Concepts, Human-centered computing → Information visualization, HCI theory, concepts and models

An unstructured document is made of a collection of potentially important data that are difficult to process through a computer. In the context of exponential data growth, one of the main challenges for companies is to manage, develop and use unstructured data. The collaboration of Famolde and 1A3i, two Europeans companies, has led to the development of new ways to process and compare unstructured data in Famolde's industrial manufacturing process. However, an ergonomic study identified limitations and potential improvements with the solution developed. In this context, we explored the information visualization literature to propose a new prototype designed to facilitate the comparison and classification of information. A multiple views solution based on the initial juxtaposition comparison tool is enhanced with explicit information about the identified differences. Finally we propose experimentation protocol to validate our solution.

https://dl.acm.org/ccs/ccs_flat.cfm

Industrial context: comparing unstructured graphical 2D files

The RUD project is a two and a half year project that aims at developing and validating innovative integrated technologies capable of performing searches and comparing both unstructured and heterogeneous files, unstructured data from 2D files/3D models (see Figure 1). An unstructured document is a document that is not exploitable or difficult to exploit by a computer. Unstructured data can be displayed, printed, or exchanged. All the persons having knowledge of the domain can "understand" and exploit the content of this document once displayed. Internally the document does not contain any structured data directly processed by the machine. However, the expert can identify all the elements and understand the structure of the information.

The volume of digital data is growing exponentially, and it is expected that in a near future, all the company's data will be digital [START_REF] Reinsel | Data age 2025: The evolution of data to life-critical[END_REF][START_REF]IDG 2018 Annual White Paper[END_REF]. One of the key challenges for companies is to manage, develop and use unstructured data. For the past 30 years, lots of work focused on the restructuring of graphic data from paper media. However, the problems of restructuring information from paper and digital vector media are wholly different. There is no study, apart those of 1A3i, on restructuring vector graphics files including schematics, plans, or drawings [START_REF] Pere-Laperne | A)KDD for Structuring Destrured Documents[END_REF][START_REF] Pere-Laperne | La restructuration des documents graphiques destructurés[END_REF]. The RUD project addresses all these issues for the first time to answer market needs.

In manufacturing companies, 2D and 3D files are linked to a huge amount of data. These files are created by Computer Aided Design (CAD) tools and contain a considerable amount of important information about the company (its processes, products, etc.) which, if adequately structured, could be used and could provide very valuable knowledge. However, due to the current unavailability to transfer this data from one application to another, this available data is poorly or not reused today. In fact, it is impossible nowadays to transfer directly unstructured file's data, in a software, to use it.

For Famolde, nowadays, the analysis of versions of 2D drawings requires a dedicated effort, since the analysis is mostly developed in physical format. There is a visual comparison between two printed drawings, being invariably subject to human error, for various reasons and uncontrollable factors -tiredness, stress, pressure or lack of technical sensitivity to know how to look for the most minute details. Although changes between versions are mostly in the interest of those who produce them, technical knowledge and the perception of their feasibility can compromise a more agile analysis. It will compromise downstream work, when the detail should be assured in the preliminary phase. Additionally, it can cause unforeseen effort and costs, and consequently an unsustainable business model. Moreover, since it deals with industrial processes, the analysis of 2D drawings requires the participation of different technical people. Their approaches are vastly different depending on the different manufacturing steps. Thus, for Famolde, it is imperative to improve this process and provide the 2D drawing analyst with more reliable tools and mechanisms. It facilitates not only the entire analysis process, which can take under normal circumstances 1 or 2 days for each analysis, to a few hours of effort. Moreover, the results being more faithful to expectations, and allowing the analyst to have all the information supported to present an opinion of feasibility.

The software used to produce 2D CAD models keep them structured as long as they are open in their dedicated soft-ware. But once they are exported the files are considered unstructured. 1A3I, leader of the RUD project, developed a method that relies on the analysis of 2D sketches to compare and find differences between two documents containing unstructured data. It is much less expensive and more efficient to get the user to intervene throughout the process (and especially at the beginning) rather than at the end to validate or correct errors. Involving users reduces and even cancels the error rate for all the tasks [START_REF] Péré | Restructuring Unstructured Documents[END_REF]. In this way, a semi-automatic interface to structure unstructured data has been designed, placing the user at the heart of the extraction process and during all the restructuring phases. The solution developed by 1A3i offers to automatize that process. Indeed, by uploading two different files on the interface, the software is capable of comparing both files and highlighting the differences depending on selected parameters.

In this article we will focus on one specific innovative software tool enhancing 1A3i's solution: a tool to structure 2D data, allowing the comparison of unstructured and heterogeneous 2D files.

Existing 2D comparator

Designers and engineers who work with several versions of the same schema often need to know differences between tow versions. The prototype developed by 1A3i for the RUD project is able to find it, even through transformations (translation, rotation, scaling and elongation in one or two directions). But how to present, classify and store these differences? To ensure a comfortable use of the prototype, some ergonomic adjustments are needed.

A knowledge discovery database for destructured document Several strategies have already been implemented in the 1A3i's prototype. One consists in displaying the 2 compared files, one next to the other, and synchronizing zooms and scrolls on the 2 views (see Figure 3). This allows viewing the details on each part of the documents at the same time.

Thanks to a third view displaying the whole page in a reduced format, a flag indicates to the user the position of the zoomed part. With this "map-style view", the user keeps a global understanding of the file. Each of these views is on a different window so that it can be displayed on different screens: multiple screens environment enhances the capacity to display more information with more accuracy. 1A3i also uses several colors to underline differences between the 2 files: added elements, deleted elements, moved elements etc. Other interface dedicated to the comparison in the prototype of 1A3i displays both files in the same view, the one overlapping the other: the user can move a cursor to play with transparency ratio in order to distinguish each file and understand differences.

Some buttons allow to the user to sequentially iterate differences. There are several possible sort orders defined to iterate differences: depending on coordinates on the page (from left to right and top to bottom, etc.), depending on size of differences (from the biggest to the smallest, from the smallest to the largest) or depending on the contour of the drawing on which the difference appears. The user need to iterate differences because they want to inventory significant differences. Depending on their goal, the same difference can be significant or meaningless: the user are not interested in the same differences if they try to calculate a budget or if he tries to evaluate technical changes in the design. They are the only one who know how to distinguish significant and meaningless differences. This involvement of the human being in the process of restructuration comes from the (A)KDD method [START_REF] Pere-Laperne | A)KDD for Structuring Destrured Documents[END_REF][START_REF] Pere-Laperne | La restructuration des documents graphiques destructurés[END_REF] (see Figure 2). When it is identified, significant differences are flagged with a different color and the prototype of 1A3i can also build a report con-taining significant differences with screenshots and textual comments of the user. However, 1A3i consider that these strategies already applied in their prototype are effective but not enough to ensure a comfortable use of the prototype. One reason is that a lot of differences make the file less readable. Even with the use of colors and multiple views, a lot of differences in a small surface of the page can be confusing. Readability is essential to avoid errors.

Moreover, navigating a large list of differences can be tedious. Even if the "map-style view" indicates the position, the user must make an effort to understand it. Moreover, iteration implies an order. Even if the prototype allows choosing the contour or the difference to start the iteration, the user may not want to process differences in the sequence that the prototype proposes it. Iterate a list of differences is a repetitive task and it may be perceived as a boring task for the user. How can we represent and navigate the list of identified differences for a better and more efficient experience?

Ergonomic study

In order to develop a human centered solution to answer these concerns, an ergonomics analysis was performed to provide a diagnosis of the existing interfaces. Two methodologies were used : (1) ergonomic inspection of the software described previously (2) usability testing.

The ergonomic inspection method is the first step to identify user problems in software. It was carried out using Bastien and Scapin criterion [START_REF] Bastien | Ergonomic criteria for the evaluation of human-computer interfaces[END_REF]. Each interface was evaluated by comparing all the elements with the 8 criteria of Bastien and Scapin. Then, all usability problems identified was detailed in an inspection grid and illustrated with annotated screens. total of 69 points were identified during the analysis via the Bastien and Scapin criteria after the evaluation of all the interfaces of the software: 2 were considered blocking, 25 were considered critical, 30 were considered medium and 12 were considered minor. In order to validate this first analysis and to propose concrete suggestions to build the new solution, the user testing method was used in a complementary way.

The usability test method consists of evaluating the usability1 and user experience2 (UX) [START_REF] Lallemand | Méthodes de design UX: 30 méthodes fondamentales pour concevoir et évaluer les systèmes interactifs[END_REF]. It refers to evaluating a product or service (digital products, physical consumer, commercial products) by testing it with representative users. During a test, participants will try to complete typical tasks while observers watch, listen, and takes notes. The goal is to identify any usability problems, collect qualitative and quantitative data and determine the participant's satisfaction with the product.

To carry out these tests, 5 participants were recruited, 3 research engineers in additive manufacturing and 2 engineering school students. In total, 19 points were noted by the participants during the user tests. This number is much lower than the evaluation performed using the Bastien and Scapin criteria (69 points), because the scenario proposed for the usability test was not intended to evaluate all the interfaces of the software. Indeed, the tests focused only on the following workflow: the creation of a project with the synchronization of documents, the realization of a comparison (direct, transformation and by point) and the realization of a report. During the user tests, the participants were able to answer the System Usability Scale [START_REF] Brooke | SUS-A quick and dirty usability scale[END_REF]. This scale aims

Not intuitive Lack of guidance in the type of comparison No history

"it is sad that there is no history of the comparison" No trust in the "this resize bothers me since it is dependent point translation on mouse precision" to measure the perceived usability and general appreciation of a system. The score is between 0 and 100, but it is not a percentage. The average score obtained for the initial solution was 57/100. The interfaces were therefore evaluated as acceptable. This score is consistent with the feedback, knowing that the software is still in the prototype stage. To significantly improve the interfaces, modifications are to be expected The most critical usability issue was a lack of guidance when making comparisons. Users had real difficulty navigating through comparisons (see Table 1). Comparisons were not archived or classified according to their type (between direct comparison, transformation and point transformation). In order to guide the user, several suggestions were made including creating a tree structure to navigate through the comparisons and creating an annotation system to allow the user to identify each comparison.

We aim to establish the specifications for a functional interface for upstream efficiency. How can we efficiently browse a list of identified differences on a graphical files in order to correctly classify the major, minor and wrongly classified differences in two 2D graphical files?

Proposed prototype for visualization and exploration

A large body of visual tools have been used to facilitate comparison of multiple sets of data and information. They are essential to understanding, analyzing, and communicating data. The information visualization community has explored different solutions and proposed three main comparative designs: juxtaposition (showing different objects separately), superposition (overlaying objects in the same space) and explicit encoding of relationships [START_REF] Gleicher | Visual comparison for information visualization[END_REF]. The 2D comparator developed by 1A3i already implement the juxtaposition design to facilitate the comparison between tow files. They also analyze the documents and produce a list of identified differences between the two. To have a visual representation of the differences we want to propose the addition of a explicit encoding of relationships. This combination helps the user connect the juxtaposed views with the necessary additional information that will facilitate comparison. One interesting solution in this context consists in highlighting the explicit relationships on the juxtaposed views (for example zooming on the corresponding part of the juxtaposed views when selecting an explicit relation). To limit decontextualization, an explicit encoding of the representation may be shown next to a visualization of the objects.

The solution developed by 1A3i allows for the transformation of unstructured 2D graphical data to be restructured into tagged information. 1A3i's software produce a hierarchical representation in the form of marked-up files with groups, sub-groups and details in an XML-like format: SVG. XML schemas are hierarchical and as such can be represented as tree graph. Tree representation have been studied a lot in the field of information visualization [START_REF] Graham | A Survey of Multiple Tree Visualisation[END_REF]. The node-link layout is the most widespread, parent-child relations being represented by lines drawn between nodes, they usually follow a top-down orientation (see Figure 4). This type of representation is easily understandable and communicates structure immediately. The addition of explicit encodings can help the user link the juxtaposed information and gives context for the relations [START_REF] Gleicher | Visual comparison for information visualization[END_REF].

The goal is to study new ways for the user to exploit the results of the comparison tool in order to conduct their own comparison and refine the results. Taking into account the (A)KDD method, 1A3i's solution lets the operator decide for themselves if a difference is important or negligible. Providing new ways to quickly manipulate the data to highlight what the user is looking for will enhance the ability of the operator to compare and extract pertinent information. Following recommendation from the information visualization community, the multiple views solution already developed by 1A3i will be complemented with an explicit encoding of the identified differences between the juxtaposed views. The solution will consist in proposing a new way to select and manipulate sets of 2D subparts.

Conclusion and perspective

We want to propose to the users a new layer of information and new interactive ways to navigate and discriminate the differences. Adding an explicit encoding view, displaying the identified differences through a tree visualization could help the user navigate and visualize the construction of the document (see Figure 5). The nodes will be foldable to limit the complexity of the representation. When hovering a specific node, the corresponding data on the image will be highlighted. Each node will display if it contains differences that have been treated or not by the user. When a node is selected the users will be able to mark if the differences found in the node are significant or if they are to be discarded (in the same fashion the original 1A3i software).

We plan on realizing short experiences with different participants. The experiences will consist in comparing the solution developed by 1A3i to the one with extra layer of information we are developing. We'll get feedback from the users and measure experience and efficiency. We plan on measuring different data regarding the experience:

• Usability: is the system usable enough for the participants? Are they getting what they need from it and in what fashion?

• Accessibility: is the system accessible and understandable enough to be used without extensive training?

• Efficiency: is the system doing what it is supposed to do efficiently?

A combination of external users and Famolde engineer will be recruited to test this prototype. The tests will be conducted in a within-subject design study with the initial 1A3i tool to identify the enhancements provided by the addition of the new view.

Figure 1 :

 1 Figure 1: Diagram of the RUD project.

Figure 2 :

 2 Figure 2: The four steps the (A)KDD Method: 1)Structuring lines: Defining the type of lines and identify them based on their composition; 2) Structuring letters: Defining the font, size, style, direction and letters which associate one or several graphics; 3) Structuring words: Grouping together letters identified to create words, then text; 4)Structuring basic symbols: Defining these symbols and associating basic graphic items and text.

Figure 3 :

 3 Figure 3: Example of a visualization of two files being compared.

Figure 4 :

 4 Figure 4: Example of a tree graph.

Figure 5 :

 5 Figure 5: Tentative design of the prototype proposed. The top images represent the 2D files to be compared, the bottom tree is the hierarchy of the different parts in which differences have been identified.

Table 1 :

 1 Areas of improvement and comments from the user study.

Usability: the extent to which a product can be used by specified users to achieve specified goals with effectiveness, efficiency and satisfaction in a specified context of use". ISO 9241-11

[START_REF] Brooke | SUS-A quick and dirty usability scale[END_REF] User experience : a person's perceptions and responses that result from the use or anticipated use of a product, system or service" international standard on ergonomics of human system interaction, ISO 9241-210

Acknowledgements

We thank the Eureka Network for co-funding this project through the Eurostars-2 joint program and the European Union Horizon 2020 research and innovation program (2021-2023). We would also like to thank Patrick Reuters for his contribution during the conception of the tree visualization tool. We extend our thanks to all the partners that participated in this project, staff and users that participated in the study, who provided helpful comments. This research project was undertaken with the assistance of resources and services from the PEPSS platform, Human Factors for Interactive Technology.


Gabriela Gomes Fernandes

User's Perceived Needs for a Personalized Motivational Application for Running: Interviews With Future Users

Keywords: CCS Concepts:, Human-centered computing → Interaction design, User centered design, Participatory design User-centered design, mobile application, personalization, motivational technology

.

INTRODUCTION

Physical inactivity and a sedentary lifestyle are major risk factors for health problems. Reducing sedentary behaviors and practicing physical activities at all ages and in all life contexts is recommended for health reasons. The WHO's report on physical activity reveals gaps in implementing policies and interventions to increase physical activity (WHO, 2022). Personalized interventions seem more efficient than generic approaches when promoting people's engagement [START_REF] Oinas-Kukkonen | Persuasive Systems Design: Key Issues, Process Model, and System Features[END_REF]. Motivational technology has been gaining ground as an alternative when designing interventions to change users' behavior in healthcare [START_REF] Aubert | Personnalité, Motivation et Changement de Comportement : Application aux Personnes Atteintes de Diabète de type 2[END_REF] such as the self-management of chronic low back pain [START_REF] Debackere | DéMarche de Conception Participative d'une Application Mobile Motivationnelle Pour l'autogestion de La Lombalgie Chronique: Co-Design Process of a Motivational Mobile Application for the Self-Management of Chronic Low Back Pain[END_REF][START_REF] Debackere | Profiling of low back pain patients for the design of a tailored coaching application[END_REF]. User-Centered Design allows Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. Copyrights for components of this work owned by others than ACM must be honored. Abstracting with credit is permitted. To copy otherwise, or republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. Request permissions from permissions@acm.org. © 2024 Association for Computing Machinery. Manuscript submitted to ACM researchers to base design choices on concrete and objective factors. It consists of an iterative approach in which the results of each stage inform the subsequent phase. The first stage is the user research phase, identifying users' needs and expectations in their context. Then, the design and evaluation stages are carried out based on what was previously retrieved [START_REF] Hartson | The UX Book: Process and Guidelines for Ensuring a Quality User Experience[END_REF][START_REF] Nogier | UX Design et ergonomie des interfaces[END_REF]. Some factors may act as barriers or facilitators of physical activity behavior, and "knowing what facilitates and hinders physical activity behavior across domains is central for the development of actions for more active lifestyles".

In the literature, intra and interpersonal factors are considerably less explored than environmental conditions [START_REF] Garcia | Barriers and facilitators of domain-specific physical activity: a systematic review of reviews[END_REF]. In our study, we use a user-centered design approach to better understand the users' needs and expectations regarding motivational running mobile applications, as well as what helps or prevents them from maintaining running activities.

As a first step, we conducted individual interviews divided into narrative/life story interviewing to gather information from the subjective experience of individual participants and semi-structured interviewing with open questions [START_REF] Gubrium | The SAGE Handbook of Interview Research: The Complexity of the Craft[END_REF][START_REF] Magaldi | Semi-structured Interviews[END_REF].

METHODS

We recruited 34 participants; 10 were part of running clubs. We conducted video interviews via Teams (which generated the transcriptions automatically). All the participants gave informed consent, and this study was approved by the Ethics Committee of the University Paris-Saclay (#CER-Paris-Saclay-2023-037). The interviews were carried out by two investigators, both having access to the interview guide previously written based on existing literature [START_REF] Jones | Qualitative Research in Sport and Physical Activity[END_REF][START_REF] Allison M Morgan | A qualitative assessment of return to sport in collegiate athletes: Does gender matter? Cureus[END_REF][START_REF] Smith | Interviews: Qualitative interviewing in the sport and exercise sciences[END_REF].

We started with a narrative interview methodology to better understand people's experiences and behavior when running [START_REF] Anderson | Narrative interviewing[END_REF]. We asked participants to tell us about their last running session in the form of a story with as many details as they would like. No other constraints or specific directions were given at this point. Then, within a semi-directive approach, we used open-ended questions to ask the participants about their running habits, frequency of training, participation in running events, affiliation to groups/federations, and their goals and expectations concerning the sport.

The following block of questions was devoted to the motivational aspects and strategies adopted by the participants.

We explicitly asked about the approach they usually employ to ensure they will get a training session done, how they stay motivated in the long term, and the factors that work as barriers and may demotivate them. The last part of the interview was dedicated to the use of technology. We questioned whether participants already used running apps, if they had smartwatches, and what were their general impressions and impact of using said devices on their running activities. Then, we used more specific questions to guide the interview toward the discussion about existent features considered essential and features the participants did not have on their apps but believed could be valuable. These needs were explored within a motivational context. Participants were always encouraged to think about what would motivate them. We also invited them to think about their daily lives when asked to suggest the timing and frequency of receiving notifications. Finally, we asked if other topics adjacent to running per se -such as nutrition, hydration, or suggestion of stretches --should be present in a motivational app.

The Interviews were transcribed verbatim, yielding between 40 and 50 pages of single-spaced typed text each. The transcriptions are being content analyzed -with a deductive thematic analysis approach -using ATLAS/ti qualitative data analysis software package (Muhr,1997). ATLAS/ti is a flexible tool for constructing networks and is useful for qualitative data analysis.

PRELIMINARY RESULTS

The mean duration of the individual interviews was 58 minutes. From a preliminary analysis, several parameters emerged. Regarding the participants' aspirations, we were able to establish a distinction between two main types of goals: the first focused on health and aesthetics, and the second centered around performance and competition. Some terms frequently appear referring to the facilitators and barriers of running practice. Among the factors acting as barriers, the most cited were: the weather, cognitive fatigue, unplanned "last-minute" social invites, work or family-related obligations, injuries, and general time constraints. One of the most cited facilitators was seeing progress in their performance or health markers. Other important factors were: the social aspect, establishing short or long-term goals, and thinking about the well-being experienced after a running session. These facilitators and barriers are consistent with other studies in domains other than running [START_REF] Sutin | Sense of purpose in life and motivation, barriers, and engagement in physical activity and sedentary behavior: Test of a mediational model[END_REF]. When asked which features participants' considered crucial to a motivational running app, they all mentioned health and performance statistics, such as heart rate and variability, distance, mean pace, and time. Participants frequently considered social features motivating as long as they could have autonomy in choosing what to share and how to interact with other users. Needs in terms of personalization were evoked. A common complaint regarding the existing commercial running apps is the generic content. The timing of notifications and tips ought to be aligned with participants' objectives and the time of sessions for them to perceive the features as personal.