Giant dipole resonance in Sm isotopes within TDHF method
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Giant dipole resonance in Sm isotopes within TDHF method
A. Ait Ben Mennana and M. Oulne High Energy Physics and Astrophysics Laboratory, Department of Physics, Faculty of Sciences SEMLALIA, Cadi AyyadUniversity, P.O.B. 2390, Marrakesh, Morocco.Received: date / Revised version: date
Abstract.
In this work, we have studied the isovector giant dipole resonance (IVGDR) in even-even Smisotopes within time-dependent Hartree-Fock (TDHF) with four Skyrme forces SLy6, SVbas, SLy5 andUNEDF1. The approach we have followed is somewhat similar to the one we did in our previous work inthe region of Neodymium (Nd, Z=60) [Physica Scripta (2020)]. We have calculated the dipole strengthof − Sm, and compared with the available experimental data. An overall agreement between them isobtained. The dipole strength in neutron-deficient − Sm and in neutron-rich − Sm isotopes arepredicted. Shape phase transition as well as shape coexistence in Sm isotopes are also investigated in thelight of IVGDR. In addition, the correlation between the quadrupole deformation parameter β and thesplitting ∆E/ ¯ E m of the giant dipole resonance (GDR) spectra is studied. The results confirm that ∆E/ ¯ E m is proportional to quadrupole deformation β . Giant resonances (GRs) represent an excellent example of collective modes of many,if not all, particles in the nu-cleus [1]. GRs are of particular importance because they currently provide the most reliable information about the bulkbehavior of the nuclear many-body system. The so-called isovector giant diople resonance (IVGDR) is the oldest andbest known of giant resonances. This is due to high selectivity for isovector E in photo-absorption experiments. Severalattempts of theoretical description of GDR have been made using the liquid drop model. Among them, Goldhaber andTeller (GT) interpreted it as collective vibrations of the protons moving against the neutrons in the nucleus with thecentroid energy of the form E c ∝ A − / [2]. Somewhat later, Steinwedel and Jensen (SJ) interpreted it as a vibrationof proton fluid against neutron fluid with a fixed surface where the centroid energy has the form E c ∝ A − / [3]. Theexperimental data are adjusted by a combination of these two [4]: in light nuclei, the data follow the law A − / , whilethe dependence A − / becomes more and more dominant for increasing values of A. Since its first observation [5], it hasbeen much studied both experimentally (see for example Refs. [4,6–8]) and theoretically (see for example Refs. [9–13]).The GDR spectra of nucleus can predict its shape (spherical, prolate, oblate, triaxial). It has a single peak forheavier spherical nuclei while in light nuclei it is split into several fragments [1]. In deformed nuclei, the GDR strengthis split in two components corresponding to oscillations of neutrons versus protons along and perpendicular to thesymmetry axis [1, 14]. Several microscopic approaches have been employed to study GDRs in deformed nuclei suchas Separable Random-Phase-Approximation (SRPA) [11, 15], time-dependent Skyrme-Hartree-Fock method [10, 16],Relativistic Quasi-particle Random Phase Approximation (RQRPA) [17] and Extended Quantum Molecular Dynam-ics (EQMD) [18]. Experimentally, the GDR is induced by various ways such as photoabsorption [6, 7, 19] inelasticscattering [8, 20], γ -decay [21].The time-dependent Hartree-Fock (TDHF) [22] method has been employed in many works to investigate GRs innuclei. It provides a good approximation for GR. Early, TDHF calculations concentrated on giant monopole resonance(GMR) [23, 24] because they require only a spherical one-dimensional code. In the last few years with the increase incomputer power, large scale TDHF calculations become possible with no assumptions on the spatial symmetry of thesystem [10, 25, 26]. Such calculations are performed by codes using a fully three dimensional (3D) Cartesian grid incoordinate space [27]. a e-mail: [email protected] b e-mail: [email protected] a r X i v : . [ nu c l - t h ] J a n A. Ait Ben Mennana, M. Oulne: Giant dipole resonance in Sm isotopes within TDHF method
In our previous work [13], TDHF method provided an accurate description of the GDR in − Nd isotopes. FourSkyrme forces were used in this work. We obtained an overall agreement with experiment with slight advantage forSLy6 [28]. In this paper, we aim to study another even-even isotopic chain namely − Sm with four Skyrme forcesSLy6 [28], SLy5 [28], SVbas [29] and UNEDF1 [30]. The first three forces were used in our previous work [13] andgave acceptable results for GDR in Nd isotopes. The new Skyrme force UNEDF1 provided also satisfactory results in.Many previous experimental and theoretical works have studied the isotopic chain of Samarium Sm (Z = 62). Fromthe experimental point of view one can see for example Ref. [7]) and from the theoretical one Refs. [12,18]. Besides thestudy of GDR, many works (Refs. [12, 17, 31]) studied the so-called pygmy dipole resonance (PDR) which correspondto low-energy E strength in nuclei with a pronounced neutron exces. The pygmy mode is regarded as vibration ofthe weakly bound neutron skin of the neutron-rich nucleus against the isospin-symmetric core composed of neutronsand protons [32]. In Ref. [12], the authors studied PDR in some spherical nuclei such as Sm and deformed onessuch as − Sm. For spherical nuclei, they found a concentration of the E strength in low-energy between 8 and10 MeV, whereas for deformed nuclei the dipole strength is fragmented into low-energy states. They also showed thatthe nuclear deformation increases the low-lying strength E at E <
10 MeV. The PDR mode is out of our currentwork in which we aim at a description of the GDR which lie at a high excitation energy range of ∼ β , γ . In dynamic calculation, the ground-state of the nucleus is boosted by imposing a dipoleexcitation to obtain the GDR spectra and some of its properties (resonance energies, width).The paper is organized as follows: in Sec.2, we give a brief description of TDHF method and the GDR in deformednuclei. In Sec.3, we present details of the numerical calculations. Our results and discussion are presented in Sec.4.Finally, Sec.5 gives the summary. The time-dependent Hartree-Fock (TDHF) approximation has been extensively discussed in several references[34–36]. A brief introduction of the TDHF method is presented as follows.The TDHF is a self-consistent mean field (SCMF) theory which was proposed by Dirac in 1930 [22]. It generalizesthe static hartree-Fock (HF) and has been very successful in describing the dynamic properties of nuclei such as forexample, giant resonances [10, 23, 26, 37] and Heavy-ion collisions [25, 38].The TDHF equations are determined from the variation of Dirac action S ≡ S t ,t [ ψ ] = (cid:90) t t dt (cid:104) ψ ( t ) | (cid:18) i (cid:126) ddt − ˆ H (cid:19) | ψ ( t ) (cid:105) , (1)where | ψ (cid:105) is the Slater determinant, t and t define the time interval, where the action S is stationary between thefixed endpoints t and t , and ˆ H is the Hamiltonian of the system. The energy of the system is defined as E = (cid:104) ψ | ˆ H | ψ (cid:105) ,and we have (cid:104) ψ | ddt | ψ (cid:105) = N (cid:88) i =1 (cid:104) ϕ i | ddt | ϕ i (cid:105) , (2)where | ϕ i (cid:105) are the occupied single-particle states. The action S can be expressed as S = (cid:90) t t dt (cid:18) i (cid:126) N (cid:88) i =1 (cid:104) ϕ i | ddt | ϕ i (cid:105) − E [ ϕ i ] (cid:19) = (cid:90) t t dt (cid:18) i (cid:126) N (cid:88) i =1 (cid:90) dx ϕ ∗ i ( x, t ) ddt ϕ i ( x, t ) − E [ ϕ i ] (cid:19) (3) . Ait Ben Mennana, M. Oulne: Giant dipole resonance in Sm isotopes within TDHF method 3 The variation of the action S with respect to the wave functions ϕ ∗ i reads δSδϕ ∗ i ( x, t ) = 0 , (4)for each i = 1 ....N , t ≤ t ≤ t and for all x . More details can be found for example in Refs. [35, 39]. We finally getthe TDHF equation i (cid:126) ∂∂t ϕ i ( t ) = ˆ h [ ρ ( t )] ϕ i ( t ) for 1 ≤ i ≤ N . (5)where ˆ h is the single-particle Hartree-Fock Hamiltonian.The TDHF equations (5) are solved iteratively by a small time step ∆t during which we assume that the Hamiltonianremains constant. To conserve the total energy E, it is necessary to apply a symmetric algorithm by time reversal,and therefore to estimate the Hamiltonian at time t + ∆t to evolve the system between time t and t + ∆t [40, 41] | ϕ ( t + ∆t ) (cid:105) (cid:39) e − i ∆t (cid:126) ˆ h ( t + ∆t ) | ϕ ( t ) (cid:105) . (6) In deformed axially symmetric nuclei, one of the most spectacular properties of the GDR is its splitting intotwo components associated to vibrations of neutrons against protons along (K=0) and perpendicularly to (K=1) thesymmetry axis. Therefore, the GDR strength represents a superposition of two resonances with energies E i ∼ R − i ∼ A − / [3] where R is the nuclear radius, and even three resonances in the case of asymmetric nuclei. This splittinghas been observed experimentally [4, 7, 8, 19] and treated theoretically by different models [10–12]. For the axiallysymmetric prolate nuclei, the GDR spectra present two peaks where the low-energy E z corresponds to the oscillationsalong the major axis of symmetry and the high-energy E x = E y corresponds to the oscillations along transverse minoraxes of the nuclear ellipsoid, due to E ∼ R − . For an oblate nucleus, it is the opposite situation to the prolate case. Fortriaxial nuclei, the oscillations along three axes are different , i.e ., E x (cid:54) = E y (cid:54) = E z . For spherical nuclei, the vibrationsalong three axes degenerate and their energies coincide E x = E y = E z . In this work, the GDR in even-even − Sm isotopes has been studied by using the code Sky3D (v1.1) [27] .This code solves the HF as well as TDHF equations for Skyrme interactions [42]. Calculations were performed with fourSkyrme functional: SLy6 [28], SLy5 [28], SVbas [29], UNEDF1 [30]. These Skyrme forces are widely used for the groundstate properties (binding energies, radii...) and dynamics (as giant resonances) of nuclei including deformed ones. Inparticular they provide a reasonable description of the GDR: SLy6 [10, 11], SVbas [29], SLy5 [16] and UNEDF1 [30].The parameters set of these functionals used in this study is shown in Table 1.Table 1: Parameters ( t, x ) of the Skyrme forces used in this work.
Parameters UNEDF1 SVbas SLy6 SLy5 t (MeV.fm ) -2078.328 -1879.640 -2479.500 -2484.880 t (MeV.fm ) 239.401 313.749 -1762.880 483.130 t (MeV.fm ) 1574.243 112.676 -448.610 -549.400 t (MeV.fm σ ) 14263.646 12527.389 13673.000 13763.000 x x -5.078 -0.381 -0.465 -0.328 x -1.366 -2.823 -1.000 -1.000 x -0.161 0.123 1.355 1.267 σ (MeV.fm ) 76.736 124.633 122.000 126.000 A first step of calculation concerns a static calculation which allows to determine the ground state for a givennucleus. This state is obtained by solving the static HF + BCS equations (8) in a three-dimensional (3D) Cartesianmesh with a damped gradient iteration method on an equidistant grid and without symmetry restrictions [27].
A. Ait Ben Mennana, M. Oulne: Giant dipole resonance in Sm isotopes within TDHF method ˆ hψ i ( x ) = (cid:15) i ψ i ( x ) for i = 1 , ...., A, (7)where ˆ h is the single-particle Hamiltonien, and (cid:15) i is the single-particle energy of the state ψ i ( x ) with x = ( r , σ, τ ).We used a cubic box with size a = 24 fm and a grid spacing of ∆x = 1.00 fm in each direction. In SKY3D code [27],the static HF + BCS equations (7) are solved iteratively until a convergence is obtained , i.e ., when for example thesum of the single-particle energy fluctuations becomes less than a certain value determined at the beginning of thestatic calculation. In this study we take as a convergence value 10 − which is sufficient for heavy nuclei (for moredetails see Ref. [27]. The pairing is treated in the static calculation, which allows to calculate the pairing energy E pair = 14 (cid:88) q ∈{ p,n } V pair,q (cid:90) d r | ξ q | F ( r ) (8)where the pairing density ξ q reads [27] ξ ( r ) = (cid:88) α ∈{ p,n } (cid:88) s u α v α | ψ α ( r , s ) | (9)where v α , u α = (cid:112) − v α are the occupation and non-occupation amplitude of single-particle state ψ α , respectively,and the function F = 1 or F = 1 − ρ/ρ gives a pure δ -interaction (DI), also called volume pairing (VDI) where ρ → ∞ or density dependent δ -interaction (DDDI), respectively, while ρ = 0 .
16 fm − is the saturation density. V P,N represents the pairing strength which is obtained from the force definition in the SKY3D code [27].In dynamic calculations, the ground-state wave function obtained by the static calculations is excited by aninstantaneous initial dipole boost operator in order to put the nucleus in the dipole mode [10, 43, 44]. ϕ ( g.s ) α ( r ) −→ ϕ α ( r, t = 0) = exp( ib ˆ D ) ϕ ( g.s ) α ( r ) , (10)where ϕ ( g.s ) α ( r ) represents the ground-state of nucleus before the boost, b is the boost amplitude of the studied mode, and ˆ D the associated operator. In our case, ˆ D represents the isovector dipole operator defined asˆ D = N ZA (cid:18) Z Z (cid:88) p =1 z p − N N (cid:88) n =1 z n (cid:19) = N ZA (cid:18) R Z − R N (cid:19) , (11)where R Z (resp. R N ) measures the proton (resp. neutron) average position on the z axis.The spectral distribution of the isovector dipole strength is obtained by applying a boost (10) with a small valueof the amplitude of the boost b to stay well in the linear regime of the excitation. For a long enough time, the dipolemoment ˆ D = (cid:104) ψ ( t ) | ˆ D | ψ ( t ) (cid:105) is recorded along the dynamical evolution. Finally, the dipole strength S D ( ω ) can beobtained by performing the Fourier transform D ( ω ) of the signal ˆ D ( t ), defined as [45] S D ( ω ) = (cid:88) ν δ ( E − E ν ) (cid:12)(cid:12) (cid:104) ν | ˆ D | (cid:105) (cid:12)(cid:12) . (12)Some filtering is necessary to avoid artifacts in the spectra obtained by catting the signal at a certain final time,in order to the signal vanishes at the end of the simulation time. In practice we use windowing in the time domain bydamping the signal D ( t ) at the final time with cos (cid:0) πt T fin (cid:1) n [27]. D ( t ) −→ D fil = D ( t ) .cos (cid:18) πt T fin (cid:19) n , (13)where n represents the strength of filtering and T fin is the final time of the simulation. More details can be foundedin Refs. [27, 46] .In this work, all dynamic calculations were performed in a cubic space with 24 x 24 x 24 fm according to the threedirections (x, y, z) and a grid spacing of 1 fm. We chose nt= 4000 as a number of time steps to be run, and dt = 0.2fm/c is the time step, so T f = 800 fm/c is the final time of simulation. Pairing is frozen in the dynamic calculation, i.e ., the BCS occupation numbers are frozen at their initial values during time evolution. . Ait Ben Mennana, M. Oulne: Giant dipole resonance in Sm isotopes within TDHF method 5 In this section we present our numerical results of static calculations concerning some properties of the ground-state, and dynamic calculations concerning some properties of the GDR for − Sm nuclei.
The isotopic chain of Sm (Z=62) studied in this work displays a transition from spherical, when neutron numberN is close to magic number N = 82, to the axially deformed shapes when N increases or decreases [7,18,47,48]. Amongthe properties of the ground-state of nuclei, there are the deformation parameters β and γ which give an idea on theshape of the nucleus [45, 49]. These deformation parameters are defined as follows [27] β = (cid:113) a + 2 a , γ = atan (cid:18) √ a a (cid:19) (14) a m = 4 π Q m AR , R = 1 . A / ( f m ) , (15)where Q m is the quadrupole moment defined as Q m = (cid:90) ρ ( r ) r Y m ( θ, ϕ ) d r (16)The deformation parameters ( β , γ ) often called Bohr-Mottelson parameters are treated as a probe to select the ground-state of all nuclei in this article. Table 2 displays the numerical results obtained for the deformation parameters ( β , γ )based on Eq. (14) of − Sm isotopes with four Skyrme forces, including the available experimental data fromRef. [50] and the HFB calculations based on the D1S Gogny force [51] for comparison. Fig. 1 shows the variation of β as a function of neutrons number N.Table 2: The deformation parameters ( β , γ ) calculated with UNEDF1, SVbas, SLy6, and SLy5 are compared withthe experimental data are from Ref. [50], and data from Ref. [51]. Nuclei UNEDF1 SVbas SLy6 SLy5 HFB Gogny. [51] Exp. [50]
Sm (0.406; 0 . ◦ ) (0.398; 4 . ◦ ) (0.402; 8 . ◦ ) (0.401; 7 . ◦ ) (0.398; 8 . ◦ ) —– Sm (0.393; 0 . ◦ ) (0.377; 0 . ◦ ) (0.381; 0 . ◦ ) (0.381; 0 . ◦ ) (0.377; 0 . ◦ ) —– Sm (0.388; 0 . ◦ ) (0.374; 0 . ◦ ) (0.371; 0 . ◦ ) (0.382; 0 . ◦ ) (0.380; 0 . ◦ ) —– Sm (0.377; 0 . ◦ ) (0.399; 0 . ◦ ) (0.308; 14 . ◦ ) (0.314; 12 . ◦ ) (0.436; 0 . ◦ ) 0.366 Sm (0.260; 21 . ◦ ) (0.252; 22 . ◦ ) (0.261; 22 . ◦ ) (0.263; 21 . ◦ ) (0.252; 22 . ◦ ) 0.293 Sm (0.205; 27 . ◦ ) (0.207; 26 . ◦ ) (0.228; 25 . ◦ ) (0.227; 25 . ◦ ) (0.183; 25 . ◦ ) 0.208 Sm (0.026; 14 . ◦ ) (0.113; 0 . ◦ ) (0.181; 27 . ◦ ) (0.181; 27 . ◦ ) (0.147; 35 . ◦ ) —– Sm (0.000; 20 . ◦ ) (0.000; 14 . ◦ ) (0.001; 0 . ◦ ) (0.003; 17 . ◦ ) (0.000; 0 . ◦ ) —– Sm (0.001; 4 . ◦ ) (0.000; 8 . ◦ ) (0.000; 12 . ◦ ) (0.000; 1 . ◦ ) (0.000; 0 . ◦ ) 0.087 Sm (0.014; 58 . ◦ ) (0.052; 1 . ◦ ) (0.063; 0 . ◦ ) (0.064; 0 . ◦ ) (0.045; 2 . ◦ ) —– Sm (0.128; 0 . ◦ ) (0.151; 0 . ◦ ) (0.167; 3 . ◦ ) (0.162; 0 . ◦ ) (0.167; 0 . ◦ ) 0.142 Sm (0.211; 0 . ◦ ) (0.220; 0 . ◦ ) (0.225; 0 . ◦ ) (0.223; 0 . ◦ ) (0.204; 0 . ◦ ) 0.193 Sm (0.302; 0 . ◦ ) (0.306; 0 . ◦ ) (0.305; 0 . ◦ ) (0.302; 0 . ◦ ) (0.273; 0 . ◦ ) 0.306 Sm (0.335; 0 . ◦ ) (0.337; 0 . ◦ ) (0.341; 0 . ◦ ) (0.338; 0 . ◦ ) ((0.347; 0 . ◦ ) 0.341 Sm (0.349; 0 . ◦ ) (0.348; 0 . ◦ ) (0.350; 0 . ◦ ) (0.349; 0 . ◦ ) (0.336; 0 . ◦ ) —– Sm (0.357; 0 . ◦ ) (0.356; 0 . ◦ ) (0.362; 0 . ◦ ) (0.363; 0 . ◦ ) (0.351; 0 . ◦ ) —– Sm (0.361; 0 . ◦ ) (0.360; 0 . ◦ ) (0.368; 0 . ◦ ) (0.366; 0 . ◦ ) (0.361; 0 . ◦ ) —– Sm (0.365; 0 . ◦ ) (0.362; 0 . ◦ ) (0.369; 0 . ◦ ) (0.367; 0 . ◦ ) (0.360; 0 . ◦ ) —– Sm (0.367; 0 . ◦ ) (0.363; 0 . ◦ ) (0.373; 0 . ◦ ) (0.369; 0 . ◦ ) (0.360; 0 . ◦ ) —– From Fig.1, we can see the β values of our calculations are generally close to experimental ones [50]. On the otherhand, there is an agreement between our calculations and HFB theory based on the D1S Gogny force [51]. In thevicinity of the region where N = 82, the β values show minima ( β (cid:39)
0) as expected because all nuclei with the magicnumber N=82 are spherical. For the
Sm nucleus, we find different results between the four Skyrme forces in thisstudy. For the Skyrme forces SLy6 and SLy5,
Sm has a triaxial shape ( γ (cid:39) . ◦ ). It has a prolate shape for SV-bas( γ = 0 . ◦ ), and has an approximate spherical form for UNEDF1 force ( β (cid:39) . A. Ait Ben Mennana, M. Oulne: Giant dipole resonance in Sm isotopes within TDHF method Sm β Neutron number
SLy6SV-bas SLy5UNEDF1 HFB-GognyExp.
Fig. 1: The Quadrupole deformation parameter β of − Nd isotopes as function of their neutron number N. Theexperimental data are from Ref. [50].M¨oller et al. [52], based on the finite-range droplet model, predicted the ground state of
Sm nucleus to be triaxial( γ = 30 . ◦ ). In table 2, the ( β , γ ) values obtained in this work as well as those of HFB theory based on the D1S Gognyforce [51] and avialable experimental data [50] show a shape transition from spherical Sm (N=82) to deformed shapebelow and above the magic neutron number N=82. For − Sm isotopes below N = 82, the isotopic chains exhibita transition from prolate ( γ = 0 . ◦ ) to spherical shape ( β (cid:39) . . ◦ ≤ γ ≤ . ◦ )for − Sm isotopes, and for neutron number higher than N = 82, both the experimental and theoretical resultsshow that the prolate deformation increases gradually and then saturates at a value which closes to β (cid:39) − Sm nuclei
Based on the TDHF ground states for − Sm isotopes obtained in static calculations, we perform dynamiccalculation such as GDR in this work to obtain some of its properties as we will see later. D m ( t )The dipole moment D m ( t ) defined by Eq. (11) allows to predict the collective motions of nucleons along the threedirections x, y and z. The time evolution of D im ( t ) where i denotes x, y and z of Sm,
Sm and
Sm is plottedin Fig. 2. We note that the collective motion of nucleons in GDR is done generally along two axes. The oscillationfrequency ω i is related to the nuclear radius R i by ω i ∝ R − i where i ∈{ x,y,z } . Fig. 2(a) shows the time evolutionof dipole moment for Sm and
Sm. For the
Sm nucleus, the three components D xm ( t ), D ym ( t ) and D zm ( t ) areidentical , i.e ., the oscillation frequencies along the three axes are equal ( ω x = ω y = ω z ) which confirms that thisnucleus has a spherical shape as we predicted in static calculations ( β (cid:39) . Sm nucleus, the D xm ( t )and D ym ( t ) values are identical and differ from the values of D zm ( t ) , i.e ., the oscillation frequencies along the symmetryz-axis ω z are lower than that along the two other axes x and y which they are equal ω x = ω y . This confirms that Smhas a prolate shape because ω z < ω x = ω y [19] which is consistent with our static calculations ( γ = 0 . ◦ ). We pointout that we found almost the same situation for the prolate nuclei namely − Sm and − Sm. In Fig. 2(b), thevalues of the three components D im ( t ) are different from each other in the case of the Sm nucleus. We notice that . Ait Ben Mennana, M. Oulne: Giant dipole resonance in Sm isotopes within TDHF method 7 -0.6-0.4-0.2 0 0.2 0.4 0.6 A 144 154 Sm (a) SLy6 D m ( t ) x-axisy-axisz-axis -0.6-0.4-0.2 0 0.2 0.4 0.6 0 50 100 150 200 250 300A 138 Sm (b) SLy6 t(fm/c) x-axisy-axisz-axis
Fig. 2: The dipole moment D m ( t ) as function of the simulation time t(fm/c) calculated with the Skyrme force SLy6for Sm,
Sm and
Sm.the oscillation frequencies ω i along the three axes are different from each other ω x (cid:54) = ω y (cid:54) = ω z which confirms that thisnucleus has a triaxial shape as we predicted in static calculations ( γ (cid:39) . ◦ ). The same situation occurs for Sm.We note also that the time evolution of dipole moment D m ( t ) is almost the same for the others Skyrme forces (SLy5,UNEDF1, SVbas) with an exception for some nuclei as Sm. The periodicity of the three components D im ( t ) allowsthe excitation energies E i to be estimated for the oscillations along each of the three axes. For Sm, we obtain, for D xm ( t ), D ym ( t ) and D zm ( t ), the same period T (cid:39) E x = E y = E z (cid:39) E exp.GDR =15.3 ± Sm and
Sm nuclei with Skyrme force SLy6.Table 3: The excitation energies along the three axes for
Sm and
Sm with Sly6, obtained from the time evolutionof D im ( t ). Nuclei E x (MeV) E y (MeV) E z (MeV) Sm 14.75 16.52 13.40
Sm 15.62 15.62 11.90 A. Ait Ben Mennana, M. Oulne: Giant dipole resonance in Sm isotopes within TDHF method
The calculation of the Fourier transform of the isovector signal D(t) allows to obtain the GDR energy spectrum. Thespectral strength S(E) (12) is simply the imaginary part of the Fourier transform of D(t).Figs. 3 - 6 display the GDR spectra in − Sm isotopes calculated with the four Skyrme forces, compared withthe available experimental data [7]. It needs to be pointed out that the experimental data for Sm isotopes from A=128to A=142, and from A=152 to A=160, and
Sm are not yet available. The calculated GDR spectra in − Smisotopes together with the available experimental data [7] are shown in Fig.3. It can be seen that all four Skyrme forcesgive generally acceptable agreement with the experiment with a slight down-shift of the order of 0.5 MeV for SLy5,SLy6 in the case of the spherical nucleus
Sm and the weakly deformed − Sm nuclei , and slight up-shift ( ∼ − Sm nuclei , where all Skyrme forces producethe deformation splitting, in which rare-earth nuclei as Samarium (Sm) with neutron number N ≈
90 show an exampleof shape transitions [7, 10, 13]. For
Sm (N=82), its GDR strength has a single-humped shape. The vibrations alongthe three axes degenerate , i.e ., they are the same resonance energy E i ( E x = E y = E z ), which confirms that thisnucleus is spherical due to the relation E i ∝ R − i where i ∈{ x,y,z } [3]. For Sm and
Sm nuclei, the two resonancepeaks move away slightly from each other but the total GDR presents one peak, so they are also weakly deformednuclei with prolate shape. For
Sm and
Sm nuclei, the total GDR splits into two distinct peaks which confirmsthat these nuclei are strongly deformed with prolate shape since the oscillations along the major axis (K=0 mode) arecharacterized by lower frequencies than the oscillations perpendicular to this axis (K=1 mode) [14].The isotope
Sm for which we do not have experimental data, SLy6, Sly5 and SVbas give a weakly deformednucleus ( β (cid:39) β (cid:39) Sm. In order to verify the shapecoexistence [53, 54] in
Sm nucleus, we redid the static calculations several times from different initial deformationswith SLy6 force. In all cases, we obtained two minima (prolate and oblate) whose their properties are displayed inTable 4. We can see that the difference in energy between these two minima is around ∆ (B.E) (cid:39) Sm nucleus. According to the value of deformation parameter γ , thiscompetition of shape is between oblate ( γ = 60 ◦ ) and prolate ( γ = 0 ◦ ) shape, but the deformation is very weak( β (cid:39) .
05) in both cases. Fig.4 shows the calculated GDR spectra corresponding to two minima (prolate, oblate).It confirms this suggestion: the upper panel (Fig.4(a)) shows an oblate shape for
Sm due to oscillations alongthe shorter axis (K=0 mode) which are characterized by higher energies than the oscillations along the axis ( | K | =1mode) perpendicular to it, while the lower panel (Fig.4(b)) shows a prolate shape for this nucleus. In both cases, thedeformation splitting ∆ E between the two peaks is too small which confirms that this nucleus is very weakly deformed.Table 4: The ground-state properties of two minima for
Sm nucleus.
Properties Prolate minimum Oblate minimumBinding energy (B.E) -1999.73 MeV -1999.66 MeVRoot mean square (r.m.s) 4.970 fm 4.969 fmQuadrupole deformation β γ ◦ ◦ Fig.5 shows the GDR strength in neutron-deficient − Sm isotopes. We can see that the deformation decreasesgradually from the well deformed nucleus
Sm ( β (cid:39) Sm ( β (cid:39) i.e .,when the neutron number N increases and closes to the magic number N=82. We note that all Skyrme forces in thiswork give almost the same GDR spectra except for Sm. According to the GDR strength along the three axes, the
Sm nucleus is weakly triaxial with SLy6, SLy5 and SVbas whereas it has a prolate shape with UNEDF1. For the − Sm isotopes, all the four Skyrme forces predict a prolate shape for them. For
Sm, SVbas and UNEDF1predict a prolate shape, while SLy5 and SLy6 give a weak triaxial shape. For − Sm isotopes, we can see that theoscillations along the three axes correspond to different resonance energies E i ( E x (cid:54) = E y (cid:54) = E z ), which shows that thesenuclei are deformed with triaxial shape. The four Skyrme forces give different results for Sm as displayed in Fig.5.The SLy family (SLy5 and SLy6) predict a triaxial shape, SVbas predicts a prolate shape while UNEDF1 gives anapproximate spherical shape. For
Sm, all Skyrme forces predict a spherical shape where the GDR strengths alongthe three axes are identical , i.e ., ( E x = E y = E z ).Fig.6 shows the GDR strength in neutron-rich − Sm isotopes. We can see that all Skyrme forces providequite similar results. From
Sm (N=94) to
Sm (N=102), the deformation gradually gets broader, and their GDRsacquire a pronounced double-humped shape. Therefore, these nuclei are strongly deformed with prolate shape sincethe oscillations energies along the longer axis (z-axis) are lower than those of oscillations along the short axis (x and . Ait Ben Mennana, M. Oulne: Giant dipole resonance in Sm isotopes within TDHF method 9 Sm SLy6 β =0.341γ=0.0° Sm β =0.305γ=0.0° Sm β =0.225γ=0.0° Sm β =0.167γ=3.6° d i p o l e s s t r e n g t h ( a r b . un i t s ) Sm β =0.063γ=0.0° Sm β =0.000γ=12.7° E(MeV) Sm SLy5 β =0.338γ=0.0° Sm β =0.302γ=0.0° Sm β =0.223γ=0.0° Sm β =0.162γ=0.0° d i p o l e s s t r e n g t h ( a r b . un i t s ) Sm β =0.064γ=0.7° Sm β =0.000γ=1.5° E(MeV) Sm SV-bas β =0.337γ=0.0° Sm β =0.306γ=0.0° Sm β =0.220γ=0.0° Sm β =0.151γ=0.2° d i p o l e s s t r e n g t h ( a r b . un i t s ) Sm β =0.052γ=1.2° Sm β =0.000γ=8.5° E(MeV) Sm UNEDF1 β =0.335γ=0.0° Sm β =0.302γ=0.0° Sm β =0.211γ=0.0° Sm β =0.128γ=0.0° d i p o l e s s t r e n g t h ( a r b . un i t s ) Sm β =0.014γ=58.0° Sm β =0.001γ=4.0° E(MeV)
Fig. 3: (Color online) GDR spectra in the chain of − Sm calculated with SLy6, SLy5, SVbas and UNEDF1. Thesolid(red), dashed(green) and dotted-dashed(blue) lines denote the dipole strengths: total, along the long axis and theshort axis (multiplied by 2) respectively. The calculated strength total is compared with the experimental data [7]depicted by black solid squares. Sm (a) β =0.048γ=60.0° D i p o l e s s t r e n g t h ( a r b . un i t s ) tot|K|=1 modeK=0 mode Sm (b) β =0.063γ=0.0° E(MeV)
Fig. 4: (Color online) The calculated GDR spectra for
Sm with the Skyrme force SLy6. , y axes) , i.e ., E z < E x = E y .In order to compare the results between different Skyrme forces under consideration, we plot their GDR spectrainto one figure, together with experimental data. Fig.7 shows the GDR strength in Sm, and
Sm calculated bythe four Skyrme forces as well as the experimental data from Ref. [7]. It can be seen there is a dependence of the GDRspectra on various Skyrme forces. We note a small shift of the average peak position of ∼ Sm, the Skyrme force UNEDF1 reproduces well the shape and the peak among the fourSkyrme forces. The agreement is less perfect with other forces. The SLy5 and SLy6 forces give very similar results,the strength exhibits a slight downshift while a slight upshift with SVbas functional. For the deformed nucleus
Sm,there is an excellent agreement between the different functionals and the experiment, with a slight upshift for theK=1 mode for SVbas force. We can explain this dependence y the fact that it is linked to certain basic characteristicsand nuclear properties of the Skyrme forces as shown in Table 5. The isovector effective mass m ∗ /m is related tothe sum rule enhancement factor κ by m ∗ /m = 1 / (1 + κ ) [4], i.e ., the larger isovector effective mass corresponds tothe lighter value of the enhancement factor. We can easily see that the increase of the factor κ ( i.e ., low isovectoreffective mass m ∗ /m ) causes the GDR strength to shift towards the higher energy region, as indicated in Ref. [55] forthe GDR in Sm,
U and
No, and in Ref. [56] for
Yb. For example, the large collective shift in SVbas canbe related to a very high enhancement factor κ =0.4 compared to other Skyrme forces. In addition to the dependencewith the enhancement factor κ , Fig.7 also shows a connection between GDR energy and symmetry energy a sym . Thepeak energy of the GDR moves towards the higher energy region when a sym decreases, as pointed in Ref. [57] for theGDR in doubly magic Pb, and in our previous work for Nd isotopes [13]. . Ait Ben Mennana, M. Oulne: Giant dipole resonance in Sm isotopes within TDHF method 11 Sm SLy6 β =0.001γ=0.0° Sm β =0.181γ=27.7° Sm β =0.228γ=25.6° Sm β =0.261γ=22.4° d i p o l e s s t r e n g t h ( a r b . un i t s ) Sm β =0.308γ=14.8° Sm β =0.371γ=0.0° Sm β =0.381γ=0.0° Sm β =0.402γ=8.6° E(MeV) Sm SLy5 β =0.003γ=17.0° Sm β =0.181γ=27.6° Sm β =0.227γ=25.2° Sm β =0.263γ=21.9° d i p o l e s s t r e n g t h ( a r b . un i t s ) Sm β =0.314γ=12.2° Sm β =0.382γ=0.0° Sm β =0.381γ=0.0° Sm β =0.401γ=7.6° E(MeV) Sm SV-bas β =0.000γ=14.4° Sm β =0.113γ=0.0° Sm β =0.207γ=26.1° Sm β =0.252γ=22.5° d i p o l e s s t r e n g t h ( a r b . un i t s ) Sm β =0.399γ=0.0° Sm β =0.374γ=0.0° Sm β =0.377γ=0.0° Sm β =0.398γ=4.8° E(MeV) Sm UNEDF1 β =0.000γ=20.6° Sm β =0.026γ=14.8° Sm β =0.205γ=27.5° Sm β =0.260γ=21.3° d i p o l e s s t r e n g t h ( a r b . un i t s ) Sm β =0.377γ=0.0° Sm β =0.388γ=0.0° Sm β =0.393γ=0.0° Sm β =0.406γ=0.0° E(MeV)
Fig. 5: (Color online) GDR spectra in the isotopic chain − Sm calculated with SLy6, SLy5, SVbas and UNEDF1.The solid(red), dashed(green) and dotted-dashed(blue) lines denote the dipole strengths: total, along the long axis andthe short axis(multiplied by 2 except − Sm) respectively. The dotted (magenta) line denotes the strength alongthe third middle axis in the case of the triaxial nuclei − Sm. Sm SLy6 β =0.373γ=0.0° Sm β =0.369γ=0.0° Sm β =0.368γ=3.4° d i p o l e s s t r e n g t h ( a r b . un i t s ) Sm β =0.362γ=0.0° Sm β =0.350γ=0.0° E(MeV) Sm SLy5 β =0.369γ=0.0° Sm β =0.367γ=0.0° Sm β =0.366γ=0.0° d i p o l e s s t r e n g t h ( a r b . un i t s ) Sm β =0.363γ=0.0° Sm β =0.349γ=0.0° E(MeV) Sm SVbas β =0.363γ=0.0° Sm β =0.362γ=0.0° Sm β =0.360γ=0.0° d i p o l e s s t r e n g t h ( a r b . un i t s ) Sm β =0.356γ=0.0° Sm β =0.348γ=0.0° E(MeV) Sm UNEDF1 β =0.367γ=0.0° Sm β =0.365γ=0.0° Sm β =0.361γ=0.0° d i p o l e s s t r e n g t h ( a r b . un i t s ) Sm β =0.357γ=0.0° Sm β =0.349γ=0.0° E(MeV)
Fig. 6: (color online) The GDR spectra in the isotopic chain − Sm calculated with SLy6, SLy5, SVbas andUNEDF1. The solid(red), dashed(green) and dotted-dashed(magenta) lines denote the dipole strengths: total, alongthe long axis and the short axis(multiplied by 2) respectively. . Ait Ben Mennana, M. Oulne: Giant dipole resonance in Sm isotopes within TDHF method 13 Sm d i p o l e s s t r e n g t h ( a r b . un i t s ) σ γ ( m b ) UNEDF1SLy6SLy5SVbasExp. Sm E(MeV)
Fig. 7: (Color online) The calculated GDR spectra
Sm and
Sm with Skyrme forces UNEDF1, SLy6, SLy5 andSVbas for . the experimental data [7]are depicted by triangle.Table 5: The sum rule enhancement factor κ , isovector effective mass m ∗ /m = 1 / (1 + κ ), and symmetry energy a sym for the Skyrme forces under consideration. Forces m ∗ /m κ a sym ( MeV )SLy6 [28] 0.80 0.25 31.96SLy5 [28] 0.80 0.25 32.03UNEDF1 [30] (cid:39) ∆E and quadrupole deformation β As we mentioned above, the GDR strength splits into two peaks for deformed nuclei. Each peak corresponds toa resonance energy E i of GDR. We denoted by E and E the energies corresponding to K=0 and K=1 modesrespectively. The total resonance energy of giant resonance is defined by the formula [58] E m = (cid:82) + ∞ S ( E ) EdE (cid:82) + ∞ S ( E ) dE , (17)where S(E) (12) is the strength function of giant resonance. In Table 6 , the resonance energies E and E of − Smnuclei are presented, including the available experimental data from Ref. [7]. From this table, we can see an overallagreement between our results and the experimental data, with a slightly advantage for the Sly6 functional. Forinstance, the result of the semi-spherical
Sm gives E SLy GDR =15.05 MeV which is very close to E Exp.GDR =(15.30 ± Sm and
Sm, the results ( E , E ) with SLy6 are very close to those ofexperiment. Table 6: The resonance energy centroids E and E of − Sm corresponding to oscillation along the major axis(K=0) and the minor axis (K=1) respectively. The experimental data are from ref. [7].
UNEDF1 SVBas SLy5 SLy6 Exp. [6]Nuclei E E E E E E E E E E Sm Sm Sm Sm Sm Sm Sm Sm Sm ± Sm Sm ± Sm ± Sm ± ± Sm ± ± Sm Sm Sm Sm Sm Fig. 8 displays the resonance energies ( E , E ) evolution as function of the neutron number N from Sm (N=66)to
Sm (N=102). We can see for all the four Skyrme forces that the resonance energy E along the major axis (k=0mode) increases with the neutron number N ( i.e ., mass number A) until the region around N=82 (magic number) andthen trends to decreases. The opposite happens for the resonance energy E , i.e ., it decreases with the increasing ofN until N=82 , and then gradually increases. We can clearly see that the SLy6 reproduces the experimental data bestamong the four Skyrme forces. It was shown to provide a satisfying description of the GDR for spherical and deformednuclei [11, 59]. The SVbas functional gives somewhat high values of E and E among the other forces due to its largeenhancement factor κ ( κ =0.4) as we discussed above.In Fig. 9, we plotted the evolution of the GDR-splitting value ∆E = E − E as a function of the neutron numberN. It can be easily seen for all the four Skyrme forces, that the GDR splitting ∆ E decreases gradually with theincrease of N and then increases. It takes the minimum value ∆E =0 at N=82 (magic Number ) which corresponds tothe spherical nucleus Sm and achieves a maximum for strongly deformed nuclei as
Sm. Such a result confirmsthat the splitting of GDR is related to the deformation structure of nuclei.Since the GDR-splitting is caused by the deformation, it is possible to relate the nuclear deformation parameter β with the ratio ∆E/ ¯ E , where ¯ E is the mean resonance energy. Fig. 10 displays the correlation between the quadrupoledeformation β and ∆E/ ¯ E for − Sm nuclei calculated with the Skyrme forces under consideration. We can seefor all of the four Skyrme forces that there is an almost linear relationship between ∆E/ ¯ E and β , i.e ., ∆E/ ¯ E (cid:39) a.β , (18)where a is a parameter depending slightly on the Skyrme force. This fact confirms that the size of the GDR-splittingis proportional to the quadrupole deformation parameter β . The relation (18) was already studied in Refs. [13,17,60]. The isovector giant dipole resonance (IVGDR) has been investigated in the isotopic chain of Samarium (Sm). Thestudy covers even-even Sm isotopes from
Sm to
Sm. The investigations have been done within the frameworkof time dependent Hartree-Fock (TDHF) method based on the Skyrme functional. The calculations were performedwith Four Skyrme forces: SLy6, SLy5, SVbas and UNEDF1. In static calculations, some properties of ground statelike the deformation parameters ( β , γ ) have been calculated by using SKY3D code [27]. In dynamic calculations, thedipole moment D m (t) and the strength of GDR are calculated and compared with the available experimental data [7].The results obtained showed that TDHF method can reproduce the shape and the peak of the GDR spectra. Allfour Skyrme forces generally reproduce the average position of the GDR strength with a small shift depending on the . Ait Ben Mennana, M. Oulne: Giant dipole resonance in Sm isotopes within TDHF method 15 SLy6 SLy5 SV-bas UNEDF1 Exp. Sm E m ( M e V ) Neutron number N E E Fig. 8: (Color online) The peak positions E and E of GDR in − Sm along major axis (square symbol) andminor axis (circle symbol) respectively. The experimental data are depicted by black square ( E ) and circle ( E ). Sm Δ E ( M e V ) Neutron Number N
SLy6 SV-bas SLy5 UNEDF1
Fig. 9: (Color online) The GDR-splitting ∆E as a function of the neutron number N for − Sm nuclei calculatedwith SLy6, SVbas, SLy5 and UNEDF1.
SLy6 Δ E / E _ Deformation β Sm(128-164)y=0.688x-0.002
SLy5 Δ E / E _ Deformation β Sm(128-164)y=0.677x+0.005
UNEDF1 Δ E / E _ Deformation β Sm(128-164)y=0.693x+0.004
SVbas Δ E / E _ Deformation β Sm(128-164)y=0.706x+0.001
Fig. 10: (Color online) The correlation between the deformation parameter β and the ratio ∆E/ ¯ E . circles denote thedata in the Sm isotopes and lines are the fitting results.used Skyrme force. The agreement is better with the SLy6 force among these Skyrme forces. The GDR strengths in − Sm,
Sm and − Sm nuclei are also predicted in this work.Finally, some properties of GDR ( ¯ E , E , E , ∆E ) have been calculated with the four Skyrme forces. The resultswith SLy6 were very close to the experimental data compared to the other forces. A correlation between the ratio ∆E/ ¯ E and the quadrupole deformation parameter β was found. For all Skyrme forces, we have found the relation ∆E/ ¯ E = a.β + b with the value of b is negligible.In the light of the successful description of the GDR in deformed nuclei with the TDHF method, it was expectedthat this latter can also be applied for treating the shape coexistence as we predicted for Sm with the SLy6 force.
References
1. M. N. Harakeh and A. Woude, Giant Resonances: fundamental high-frequency modes of nuclear excitation, vol. . OxfordUniversity Press on Demand, (2001).2. M. Goldhaber and E. Teller Phys. Rev., vol. , p. 1046, (1948).3. J. Speth and A. van der Woude Reports on Progress in Physics, vol. , p. 719, (1981).4. B. L. Berman and S. Fultz Reviews of Modern Physics, vol. , p. 713, (1975).5. W. Bothe and W. Gentner Z.Phys, vol. , p. 236, (1937).6. P. Carlos, H. Beil, R. Bergere, A. Lepretre, and A. Veyssiere Nuclear Physics A, vol. , p. 437, (1971).7. P. Carlos, H. Beil, R. Bergere, A. Lepretre, A. De Miniac, and A. Veyssiere Nuclear Physics A, vol. , p. 171, (1974).8. L. Donaldson, C. Bertulani, J. Carter, and al. Physics Letters B, vol. , p. 133, (2018).9. K. Goeke and J. Speth Annual Review of Nuclear and Particle Science, vol. , p. 65, (1982).10. J. A. Maruhn, P. G. Reinhard, P. D. Stevenson, J. R. Stone, and M. R. Strayer Phys. Rev. C, vol. , p. 064328, (2005).11. W. Kleinig, V. O. Nesterenko, J. Kvasil, P.-G. Reinhard, and P. Vesely Phys. Rev. C, vol. , p. 044313, (2008).12. K. Yoshida and T. Nakatsukasa Phys. Rev. C, vol. , p. 021304, (2011).13. A. A. B. Mennana, Y. E. Bassem, and M. Oulne Physica Scripta, vol. , p. 065301, 2020.14. S. Josef, Electric and magnetic giant resonances in nuclei, vol. . World Scientific, (1991).. Ait Ben Mennana, M. Oulne: Giant dipole resonance in Sm isotopes within TDHF method 1715. V. Nesterenko, W. Kleinig, J. Kvasil, P. Vesely, and P.-G. Reinhard International Journal of Modern Physics E, vol. ,p. 624, (2007).16. S. Fracasso, E. B. Suckling, and P. Stevenson Physical Review C, vol. , p. 044303, (2012).17. D. P. n. Arteaga, E. Khan, and P. Ring Phys. Rev. C, vol. , p. 034311, (2009).18. S. S. Wang, Y. G. Ma, X. G. Cao, W. B. He, H. Y. Kong, and C. W. Ma Phys. Rev. C, vol. , p. 054615, (2017).19. V. M. Masur and L. M. Mel’nikova Physics of Particles and Nuclei, vol. , p. 923, (2006).20. E. Ramakrishnan, T. Baumann, and al. Physical review letters, vol. , p. 2025, (1996).21. J. Gundlach, K. Snover, J. Behr, and al. Physical review letters, vol. , p. 2523, (1990).22. P. A. M. Dirac Mathematical Proceedings of the Cambridge Philosophical Society, vol. , p. 376, (1930).23. J. B(cid:32)locki and H. Flocard Physics Letters B, vol. , p. 163, (1979).24. P. Chomaz, N. Van Giai, and S. Stringari Physics Letters B, vol. , p. 375, (1987).25. J. A. Maruhn, P.-G. Reinhard, P. D. Stevenson, and M. R. Strayer Phys. Rev. C, vol. , p. 027601, (2006).26. P. Stevenson, M. Strayer, J. Rikovska Stone, and W. Newton International Journal of Modern Physics E, vol. , p. 181,(2004).27. B. Schuetrumpf, P.-G. Reinhard, P. Stevenson, A. Umar, and J. Maruhn Computer Physics Communications, vol. ,p. 211, (2018).28. E. Chabanat, P. Bonche, P. Haensel, J. Meyer, and R. Schaeffer Nuclear Physics A, vol. , p. 231, (1998).29. P. Kl¨upfel, P.-G. Reinhard, T. J. B¨urvenich, and J. A. Maruhn Phys. Rev. C, vol. , p. 034310, (2009).30. M. Kortelainen, J. McDonnell, W. Nazarewicz, P.-G. Reinhard, J. Sarich, N. Schunck, M. V. Stoitsov, and S. M. Wild Phys.Rev. C, vol. , p. 024304, (2012).31. C. Tao, Y. Ma, G. Zhang, X. Cao, D. Fang, H. Wang, et al. Physical Review C, vol. , no. 1, p. 014621, 2013.32. N. Paar, D. Vretenar, E. Khan, and G. Colo Reports on Progress in Physics, vol. , no. 5, p. 691, 2007.33. J. W. Negele Reviews of Modern Physics, vol. , p. 913, (1982).34. Y. Engel, D. Brink, K. Goeke, S. Krieger, and D. Vautherin Nuclear Physics A, vol. , p. 215, (1975).35. A. Kerman and S. Koonin Annals of Physics, vol. , p. 332, (1976).36. S. E. Koonin, K. T. R. Davies, V. Maruhn-Rezwani, H. Feldmeier, S. J. Krieger, and J. W. Negele Phys. Rev. C, vol. ,p. 1359, (1977).37. P.-G. Reinhard, L. Guo, and J. Maruhn The European Physical Journal A, vol. , p. 19, (2007).38. C. Simenel and A. Umar Progress in Particle and Nuclear Physics, vol. , p. 19, (2018).39. C. Simenel The European Physical Journal A, vol. , p. 152, (2012).40. H. Flocard, S. E. Koonin, and M. S. Weiss Phys. Rev. C, vol. , p. 1682, (1978).41. P. Bonche, S. Koonin, and J. W. Negele Phys. Rev. C, vol. , p. 1226, (1976).42. T. Skyrme Nuclear Physics, vol. , p. 615, (1958).43. C. Simenel and P. Chomaz Phys. Rev. C, vol. , p. 064309, (2009).44. J. M. Broomfield and P. D. Stevenson Journal of Physics G: Nuclear and Particle Physics, vol. , p. 095102, (2008).45. P. Ring and P. Schuck, The nuclear many-body problem. Springer-Verlag, (1980).46. P.-G. Reinhard, P. D. Stevenson, D. Almehed, J. A. Maruhn, and M. R. Strayer Phys. Rev. E, vol. , p. 036709, (2006).47. J. Meng, W. Zhang, S. Zhou, H. Toki, and L. Geng The European Physical Journal A-Hadrons and Nuclei, vol. , p. 23,(2005).48. T. Naz, G. Bhat, S. Jehangir, S. Ahmad, and J. Sheikh Nuclear Physics A, vol. , p. 1, (2018).49. K. W. N. Takigawa, ”Fundamentals of Nuclear Physics”. Springer Japan, (2017).50. S. RAMAN, C. NESTOR, and P. TIKKANEN Atomic Data and Nuclear Data Tables, vol. , p. 1, (2001).51. J.-P. Delaroche, M. Girod, J. Libert, H. Goutte, S. Hilaire, S. P´eru, N. Pillet, and G. Bertsch Physical Review C, vol. ,p. 014303, (2010).52. P. M¨oller, R. Bengtsson, B. Carlsson, P. Olivius, T. Ichikawa, H. Sagawa, and A. Iwamoto Atomic Data and Nuclear DataTables, vol. , p. 758, (2008).53. J. Wood, K. Heyde, W. Nazarewicz, M. Huyse, and P. Van Duppen Physics reports, vol. , p. 101, (1992).54. K. Heyde and J. L. Wood Reviews of Modern Physics, vol. , p. 1467, (2011).55. V. Nesterenko, W. Kleinig, J. Kvasil, P. Vesely, P.-G. Reinhard, and D. Dolci Physical Review C, vol. , p. 064306, (2006).56. T. Oishi, M. Kortelainen, and N. Hinohara Phys. Rev. C, vol. , p. 034329, (2016).57. J. R. Stone and P.-G. Reinhard Progress in Particle and Nuclear Physics, vol. , p. 587, (2007).58. U. Garg and G. Col`o Progress in Particle and Nuclear Physics, vol. , p. 55, (2018).59. V. Nesterenko, W. Kleinig, J. Kvasil, P. Vesely, and P.-G. Reinhard International Journal of Modern Physics E, vol. ,p. 89, (2008).60. K. Okamoto Phys. Rev., vol.110