Nucleus giant resonances from an improved isospin-dependent Boltzmann-Uehling-Uhlenbeck transport approach
aa r X i v : . [ nu c l - t h ] J u l Nucleus giant resonances from an improved isospin-dependentBoltzmann-Uehling-Uhlenbeck transport approach
Jun Xu ∗
1, 2 and Wen-Tao Qin
2, 3 Shanghai Advanced Research Institute, Chinese Academy of Sciences, Shanghai 201210, China Shanghai Institute of Applied Physics, Chinese Academy of Sciences, Shanghai 201800, China University of Chinese Academy of Sciences, Beijing 100049, China (Dated: July 23, 2020)We have studied the isoscalar giant quadruple resonance (ISGQR) and the isovector giantdipole resonance (IVGDR) in
Pb based on an improved isospin-dependent Boltzmann-Uehling-Uhlenbeck transport approach using an improved isospin- and momentum-dependent interaction.With the isoscalar nucleon effective mass and the nucleon-nucleon cross section which reproducesrespectively the excitation energy and the width of the ISGQR strength function, the slope pa-rameter of the symmetry energy and the neutron-proton effective mass splitting are constrainedrespectively within 36 < L <
62 MeV and 0 . δ < ( m ∗ n − m ∗ p ) /m < . δ , by comparing theresulting centroid energy of the IVGDR and the electric dipole polarizability with the experimentaldata. It is found that nucleon-nucleon collisions have considerable effects on the resulting electricdipole polarizability, which needs to be measured more accurately in order to pin down isovectornuclear interactions. I. INTRODUCTION
Understanding the microscopic nuclear interaction aswell as the nuclear matter equation of state (EOS) isone of the main goals of nuclear physics. Thanks to thegreat efforts made by pioneer nuclear physicists, so farthe uncertainties mainly exist in the isospin-dependentpart of the EOS, i.e., the nuclear symmetry energy E sym ,whose density dependence is generally characterized bythe slope parameter L around the saturation density. Inthe microscopic level, the exchange contribution of thefinite-range part of the effective nuclear interaction leadsto the momentum-dependent nuclear potential, which isrelated to the nuclear matter EOS. The nucleon effec-tive mass characterizing the momentum dependence ofthe nuclear potential can be different for neutrons andprotons in the isospin asymmetric nuclear matter. Theisospin splitting of the neutron and proton effective mass m ∗ n − m ∗ p is also related to the symmetry energy throughthe Hugenholtz-Van Hove theorem [1, 2]. Both the sym-metry energy and the neutron-proton effective mass split-ting have important ramifications in nuclear astrophysics,nuclear reactions induced by neutron-rich nuclei, and nu-clear structures. Reviews on the symmetry energy can befound in Refs. [3–7], and a recent review on the neutron-proton effective mass splitting can be found in Ref. [8].Observables of finite nuclei are important probes ofnuclear interactions in nuclear medium at subsaturationdensities. Both the isoscalar and isovector excitations offinite nuclei are good probes for the corresponding chan-nels of nuclear interactions and EOSs (see, e.g., Ref. [9]).The pygmy dipole resonance (PDR) and the IVGDR aretypical isovector excitations in nuclei and good probesof isovector nuclear interactions. The former represents ∗ [email protected] the oscillation of the neutron skin against the nucleus in-ert core, while the later is an oscillation mode in whichneutrons and protons move collectively relative to eachother. The strength function of the PDR generally peaksat lower excitation energies compared to that of theIVGDR [10, 11], while both are sensitive to the symmetryenergy which prevents the center-of-masses of neutronsand protons from being away from each other. Typi-cally, various studies have shown that the centroid energyand the electric dipole polarizability extracted from thestrength function of the IVGDR are found to be goodprobes of the symmetry energy [12–20]. On the otherhand, it is intuitively expected that the frequency of thecollective oscillation is sensitive to not only the bulk en-ergy but also to the microscopic nuclear interaction char-acterized by the nucleon effective mass. Fortunately, theisoscalar nucleon effective mass can be extracted from theexcitation energy of the ISGQR [21–28], with the help ofthe available experimental results from α -nucleus scat-terings [29–31]. For a given isoscalar nucleon effectivemass, more recent studies have shown that the centroidenergy and the electric dipole polarizability can be usedto extract the nuclear symmetry energy and the neutron-proton effective mass splitting simultaneously [26, 27].Nuclei giant resonances can be studied by both the ran-dom phase approximation (RPA) method and transportapproaches. Despite the succusses of the RPA method,the width of the strength function is generally miss-ing, unless higher-order contributions [32], such as theparticle-vibration coupling [33, 34], are taken into ac-count. The Boltzmann transport approach, which haspreviously been used to extract the EOS and symme-try energy at both subsaturation and suprasaturationdensities from heavy-ion collisions (see, e.g., Refs. [35–37]), is based on the Boltzmann equation, with the col-lision term effectively containing higher-order contribu-tions when derived from the von Neumann equation withthe n-body density matrix [38, 39]. The collision termleads to the damping of the collective excitation, or equiv-alently, the width of the strength function [40, 41]. Re-producing correctly the width can be important in ob-taining accurately observables related to the moments ofthe strength function.In the present work, we study giant resonances in Pb using an improved isospin-dependent Boltzmann-Uehling-Uhlenbeck (IBUU) transport approach. An im-proved momentum-dependent interaction (ImMDI) isused in the transport approach based on the latticeHamiltonian framework. Ground-state initializations areachieved with different parameters used in the ImMDImodel, and collisions are also improved with the morerigourous energy conservation condition and better Pauliblockings. These theoretical details together with for-mulas related to nuclei giant resonances are discussed inSec. II. We first reproduce both the excitation energy ofthe ISGQR and its width for
Pb by using a properisoscalar nucleon effective mass and a constant isotropiccross section. The slope parameter of the symmetry en-ergy and the neutron-proton effective mass splitting arethen extracted from the centroid energy of the IVGDRand the electric dipole polarizability for
Pb. Theseresults are discussed in Sec. III, and a summary is givenin Sec. IV.
II. THEORETICAL FRAMEWORKA. Effective nuclear interactions
The potential energy density of the ImMDI model,which can be obtained from an effective two-body inter-action with a zero-range density-dependent term and afinite-range Yukawa-type term based on the Hartree-Fockcalculation [42], has the following form in the asymmet-ric nuclear matter with isospin asymmetry δ and nucleonnumber density ρ [43, 44] V ImMDI ( ρ, δ ) = A u ρ n ρ p ρ + A l ρ ( ρ n + ρ p ) + Bσ + 1 ρ σ +1 ρ σ × (1 − xδ ) + 1 ρ X τ,τ ′ C τ,τ ′ × Z Z d pd p ′ f τ ( ~r, ~p ) f τ ′ ( ~r, ~p ′ )1 + ( ~p − ~p ′ ) / Λ . (1)In the above, ρ n and ρ p are number densities of neutronsand protons, respectively, ρ is the saturation density, δ = ( ρ n − ρ p ) /ρ is the isospin asymmetry, and f τ ( ~r, ~p ) isthe phase-space distribution function, with τ = 1( −
1) forneutrons (protons) being the isospin index. The single-particle mean-field potential for a nucleon with momen-tum ~p and isospin τ in the asymmetric nuclear matterwith isospin asymmetry δ and nucleon number density ρ can be obtained from Eq. (1) through the variational principle as U ImMDI τ ( ρ, δ, ~p ) = A u ρ − τ ρ + A l ρ τ ρ + B (cid:18) ρρ (cid:19) σ (1 − xδ ) − τ x Bσ + 1 ρ σ − ρ σ δρ − τ + 2 C τ,τ ρ Z d p ′ f τ ( ~r, ~p ′ )1 + ( ~p − ~p ′ ) / Λ + 2 C τ, − τ ρ Z d p ′ f − τ ( ~r, ~p ′ )1 + ( ~p − ~p ′ ) / Λ , (2)where the four parameters A u , A l , C τ,τ , and C τ, − τ canbe expressed as [44] A l ( x, y ) = A + y + x Bσ + 1 , (3) A u ( x, y ) = A − y − x Bσ + 1 , (4) C τ,τ ( y ) = C l − yp f Λ ln[(4 p f + Λ ) / Λ ] , (5) C τ, − τ ( y ) = C u + 2 yp f Λ ln[(4 p f + Λ ) / Λ ] . (6)In the above, p f = ~ (3 π ρ / / is the nucleon Fermimomentum in the symmetric nuclear matter at thesaturation density. The isovector parameters x and y are introduced to mimic the density dependence ofthe symmetry energy, i.e., the slope parameter L =3 ρ ( dE sym /dρ ) ρ = ρ , and the momentum dependence ofthe symmetry potential or the neutron-proton effectivemass splitting. The values of the parameters A , C u , C l , B , σ , and Λ are adjusted to reproduce the empiri-cal nuclear matter properties, i.e., the saturation density ρ , the binding energy E ( ρ ) at the saturation density,the incompressibility K , the symmetry energy E sym ( ρ )at the saturation density, the isoscalar potential U , ∞ atthe saturation density and at infinitely large momentum,and the isoscalar nucleon effective mass m ∗ s at the sat-uration density and at the Fermi momentum. The non-relativistic k-mass in the present study is defined as m ∗ n ( p ) m = (cid:18) mp ∂U n ( p ) ∂p (cid:19) − , (7)where m is the bare nucleon mass. The isoscalar nu-cleon effective mass is the same as the neutron or theproton effective mass in the symmetric nuclear matter,while the neutron-proton effective mass splitting in theisospin asymmetric nuclear matter with isospin asymme-try δ is related to the isoscalar ( m ∗ s ) and isovector ( m ∗ v )nucleon effective mass through the following relation tothe first-order of δ expansion m ∗ n − m ∗ p ≈ m ∗ s m ∗ v ( m ∗ s − m ∗ v ) δ. (8)Note that m ∗ s and m ∗ v generally depend on both the nu-cleon momentum and the density of the nuclear matter,but are usually represented by their values at the satu-ration density and at the Fermi momentum, indicated as m ∗ s and m ∗ v in the present manuscript. (b) m * s / m U ( M e V ) m *s0 = 0.7m m *s0 = 0.8m m *s0 = 0.9m (a) p (MeV/c) FIG. 1: (Color online) Momentum dependence of the isoscalarpotential (a) and the isoscalar nucleon effective mass (b) inthe nuclear matter at ρ = 0 . − . Figure 1 displays the isoscalar potential and theisoscalar nucleon effective mass as a function of the nu-cleon momentum in the nuclear matter at ρ = 0 . − ,i.e., the average density of a nucleus. All the values ofthe parameters A , C u , C l , B , σ , and Λ need to beadjusted, in order to get different m ∗ s but the same ρ , E ( ρ ), K , E sym ( ρ ), and U , ∞ , as listed in Table I. Theisoscalar potential is larger (smaller) below (above) theFermi momentum (about 225 MeV at ρ = 0 . − ) fora larger m ∗ s , while it is the same at the Fermi momen-tum for different m ∗ s by the model construction. Sincethe potential below the Fermi momentum is expected todominate the dynamics of nuclei resonances, a larger m ∗ s gives an overall less attractive potential. The isoscalarnucleon effective mass generally increases with increasingnucleon momentum, and its value in the nuclear matterat subsaturation densities is larger than m ∗ s .Figure 2 displays the symmetry potential [ U sym =( U n − U p ) / δ ] and the relative neutron-proton effectivemass splitting as a function of the nucleon momentumin the nuclear matter at ρ = 0 . − and δ = 0 .
2, aswell as the density dependence of the symmetry energy,by setting m s = 0 . m and other isoscalar parametersas listed in Table I. Adjusting the x parameter changesthe momentum-independent part of the symmetry poten-tial and the density dependence of the symmetry energy,while the neutron-proton effective mass splitting remainsunaffected. It is seen that a larger symmetry energy atsubsaturation densities corresponds to a stronger sym-metry potential in this case. Adjusting the y param-eter alone changes both the momentum dependence ofthe symmetry potential and the density dependence ofthe symmetry energy [44]. By adjusting both values of x and y , it is possible to get very similar symmetry energiesbut different symmetry potentials and neutron-proton ef- TABLE I: Values of parameters and some physics quantitiesfor ImMDI, with ρ the saturation density, E ( ρ ) the bind-ing energy at the saturation density, K the incompressibility, U ∞ the isoscalar potential in the nuclear matter at the satu-ration density and at infinitely large nucleon momentum, and E sym ( ρ ) the symmetry energy at the saturation density. A (MeV) -66.963 92.144 100.466 B (MeV) 141.963 167.144 175.466 C u (MeV) -99.70 -92.34 -87.52 C l (MeV) -60.49 -52.34 -47.19 σ p f ) 2.424 3.401 5.369 ρ (fm − ) 0.16 0.16 0.16 E ( ρ ) (MeV) -16 -16 -16 K (MeV) 230 230 230 U ∞ (MeV) 75 75 75 m ∗ s ( m ) 0.7 0.8 0.9 E sym ( ρ ) (MeV) 32.5 32.5 32.5 U sy m ( M e V ) ( m * n - m * p ) / m x=0.3, y=-200MeV x=0.5, y=-200MeV x=0.1, y=-200MeV E sy m ( M e V ) p (MeV/c) p (MeV/c) x=0.3, y=-200MeV x=0.15, y=-350MeV x=0.45, y=-50MeV / (a) (b) (c)(d) (e) (f) FIG. 2: (Color online) Momentum dependence of the symme-try potential [(a), (d)] and the relative neutron-proton effec-tive mass splitting [(b), (e)] in the isospin asymmetric nuclearmatter at ρ = 0 . − and δ = 0 .
2, as well as the densitydependence of the symmetry energy [(c), (f)], from differentparameter values of x and y . fective mass splittings. Again, since the low-momentumpart dominates the dynamics in the simulation of nu-clei resonances, a larger neutron-proton effective masssplitting generally leads to an overall stronger symme-try potential. The corresponding slope parameters L ofthe symmetry energy and the isovector nucleon effectivemasses m ∗ v from these x and y values are listed in Ta-ble II.Besides the bulk ImMDI interaction, we have alsoincorporated the density gradient interaction and theCoulomb interaction. The potential energy contribution TABLE II: Values of x and y parameters for ImMDI and thecorresponding slope parameters L of the symmetry energyand the isovector nucleon effective masses m ∗ v . x y (MeV) -200 -200 -200 -350 -50 L (MeV) 53 66 40 54 53 m ∗ v ( m ) 0.79 0.79 0.79 0.73 0.86 of the density gradient interaction is V grad = G S ∇ ρ ) − G V ∇ ( ρ n − ρ p )] , (9)where G S and G V are the isoscalar and the isovectordensity gradient coefficients, respectively. Although theFock contribution of the finite-range term in the ImMDIinteraction leads to the density-dependent density gra- dient coefficients in the density-matrix expansion frame-work [42], these coefficients are generally very differentfrom the empirical values. In the present work we adopt G S = 132 MeV fm and G V = 5 MeV fm as in Ref. [45].The potential energy contribution of the Coulomb inter-action is V coul ( ~r ) = e Z ρ p ( ~r ) ρ p ( ~r ′ ) | ~r − ~r ′ | d r ′ − e (cid:20) ρ p ( ~r ) π (cid:21) / , (10)with the first term representing the direct contributionand the second term being the exchange contribution. B. An improved isospin-dependentBoltzmann-Uehling-Uhlenbeck transport approach
The IBUU transport model originating from Ref. [46] basically solves numerically the isospin-dependent BUUequation ∂ ˜ f τ ( ~p ) ∂t + ∇ p U τ · ∇ r ˜ f τ ( ~p ) − ∇ r U τ · ∇ p ˜ f τ ( ~p ) = − ( d −
12 ) Z d p (2 π ) d p ′ (2 π ) d p ′ (2 π ) dσ τ,τ d Ω v rel × [ ˜ f τ ( ~p ) ˜ f τ ( ~p )(1 − ˜ f τ ( ~p ′ ))(1 − ˜ f τ ( ~p ′ )) − ˜ f τ ( ~p ′ ) ˜ f τ ( ~p ′ )(1 − ˜ f τ ( ~p ))(1 − ˜ f τ ( ~p ))] × (2 π ) δ (3) ( ~p + ~p − ~p ′ − ~p ′ ) − d Z d p (2 π ) d p ′ (2 π ) d p ′ (2 π ) dσ τ, − τ d Ω v rel × [ ˜ f τ ( ~p ) ˜ f − τ ( ~p )(1 − ˜ f τ ( ~p ′ ))(1 − ˜ f − τ ( ~p ′ )) − ˜ f τ ( ~p ′ ) ˜ f − τ ( ~p ′ )(1 − ˜ f τ ( ~p ))(1 − ˜ f − τ ( ~p ))] × (2 π ) δ (3) ( ~p + ~p − ~p ′ − ~p ′ ) . (11)In the above, ˜ f is the occupation probability with 1 − ˜ f representing the Pauli blocking effect, dσd Ω is the nucleon-nucleon differential cross section, and v rel is the relative velocity of the two nucleons before the collision. The relationbetween the phase-space distribution function f and the occupation probability ˜ f is f = d ˜ f , with d = 2 being thespin degeneracy.The left-hand side of the above BUU equation de-scribes the time evolution of the phase-space distributionfunction f τ ( ~r, ~p ) in the mean-field potential, and this canbe approximately realized by solving the canonical equa-tions of motion for test particles [46, 47]. In this ap-proach, the phase-space distribution f τ ( ~r, ~p ) as well asthe local density can be obtained by averaging N T P par-allel collision events, i.e., f τ ( ~r, ~p ) = 1 N T P AN TP X i ∈ τ h ( ~r − ~r i ) δ ( ~p − ~p i ) , (12) ρ τ ( ~r ) = 1 N T P AN TP X i ∈ τ h ( ~r − ~r i ) , (13)where h is a smooth function in coordinate space, and A is the number of real particles, with each representedby N T P test particles. The form of the smooth function h is taken from that in the lattice Hamiltonian frame- work [48], i.e., the phase-space distribution function f L and the density ρ L at the sites of a three-dimensionalcubic lattice are expressed as f L,τ ( ~r α , ~p ) = AN TP X i ∈ τ S ( ~r α − ~r i ) δ ( ~p − ~p i ) , (14) ρ L,τ ( ~r α ) = AN TP X i ∈ τ S ( ~r α − ~r i ) . (15)In the above, α is the site index, ~r α is the position ofthe site α , and S is the shape function describing thecontribution of a test particle at ~r i to the value of thequantity at ~r α , i.e., S ( ~r ) = 1 N T P ( nl ) g ( x ) g ( y ) g ( z ) (16)with g ( q ) = ( nl − | q | )Θ( nl − | q | ) . (17) l is the lattice spacing, n determines the range of S , andΘ is the Heaviside function. We adopt the values of l = 1fm and n = 2 in the present study.After using the above smooth function for f L,τ ( ~r α , ~p )and ρ L ( ~r α ), the Hamiltonian of the system can be ex-pressed as H = AN TP X i q ~p i + m + N T P e V , (18)with the total potential energy expressed as e V = l X α ( V ImMDI α + V grad α + V coul α ) , (19)where V ImMDI α = A u ρ L,n ( ~r α ) ρ L,p ( ~r α ) ρ + A l ρ [ ρ L,n ( ~r α )+ ρ L,p ( ~r α )] + Bσ + 1 ρ σ +1 L ( ~r α ) ρ σ [1 − xδ L ( ~r α )] + 1 ρ × X i,j X τ i ,τ j C τ i ,τ j S ( ~r α − ~r i ) S ( ~r α − ~r j )1 + ( ~p i − ~p j ) / Λ , (20) V grad α = G S ∇ ρ L ( ~r α )] − G V {∇ [ ρ L,n ( ~r α ) − ρ L,p ( ~r α )] } , (21) V coul α = e l X α ′ ρ L,p ( ~r α ) ρ L,p ( ~r α ′ ) | ~r α − ~r α ′ | − e (cid:20) ρ L,p ( ~r α ) π (cid:21) / − e l X α ′ X i ∈ p S ( ~r α − ~r i ) S ( ~r α ′ − ~r i ) | ~r α − ~r α ′ | (22)are the corresponding contributions of the ImMDI in-teraction, the density gradient interaction, and theCoulomb interaction, respectively. δ L ( ~r α ) = [ ρ L,n ( ~r α ) − ρ L,p ( ~r α )] / [ ρ L,n ( ~r α ) + ρ L,p ( ~r α )] is the isospin asymmetryat ~r α with ρ L,n ( ~r α ) and ρ L,p ( ~r α ) being respectively thenumber density of neutrons and protons there, and thethird term in Eq. (22) subtracts the self contribution ofthe Coulomb interaction from the same proton due toits finite size in the lattice Hamiltonian framework. Thecanonical equations of motion for the i th test particlefrom the above Hamiltonian can thus be written as d~r i dt = ∂H∂~p i = ~p i p ~p i + m + N T P ∂ e V∂~p i , (23) d~p i dt = − ∂H∂~r i = − N T P ∂ e V∂~r i . (24)Further improvements have been incorporated into theIBUU transport approach. The coordinates of initialneutrons and protons are sampled uniformly within asphere of the radius R n and R p respectively. The initialmomenta are sampled within the local isospin-dependent Fermi sphere. The values of R n and R p are adjusted toreproduce the minimum total energy of the system cal-culated according to Eq. (18), so that the ground stateof the system can be achieved as in Ref. [48]. In addi-tion, a special treatment is applied in nucleon-nucleoncollisions in order to guarantee that the energy conserva-tion condition is satisfied in each collision within numer-ical errors even with the momentum-dependent poten-tial, and this is detailed in Appendix A. We have alsoimproved the Pauli blocking treatment by calculatingthe isospin-dependent occupation probability in the lo-cal frame rather than in the collisional frame, and this,together with the previous interpolation method, helpsto enhance the Pauli blocking rate. C. Nuclei giant resonances
In the present study, we mainly focus on the ISGQRand the IVGDR in
Pb. Their corresponding operatorscan be written respectively asˆ Q ISGQR = 1 A A X i =1 r π (2ˆ z i − ˆ x i − ˆ y i ) , (25)ˆ Q IVGDR = NA Z X i =1 ˆ z i − ZA N X i =1 ˆ z i , (26)where N , Z , and A are respectively the neutron, proton,and nucleon numbers in a nucleus. In the linear responseregion, the oscillation frequency of the nucleus resonanceis independent of the way the nucleus is initially excited.For the ISGQR, nucleons in the nucleus are initially ex-cited as x i → x i /λ, y i → y i /λ, z i → z i λ , (27)( p x ) i → ( p x ) i λ, ( p y ) i → ( p y ) i λ, ( p z ) i → ( p z ) i /λ , (28)where λ = 1 . ~r i → ~r i + η ∂q ( ~r i , ~p i ) ∂~p i , (29) ~p i → ~p i − η ∂q ( ~r i , ~p i ) ∂~r i , (30)where η = 25 MeV/c is the small perturbation constant,and q IVGDR ( ~r i , ~p i ) = ( NA z i (protons) − ZA z i (neutrons) , (31)can be obtained from Eq. (26).With the time evolution of the corresponding moment Q ( t ) from IBUU transport simulations, the strengthfunction of the IVGDR can be obtained from S ( E ) = − πη Z ∞ dtQ ( t ) sin( Et ) . (32)By calculating the moments of the strength function m k = Z ∞ dEE k S ( E ) , (33)one can compare the transport simulation results withthe available experimental data. For example, the cen-troid energy E − and the electric dipole polarizability α D can be obtained respectively from E − = p m /m − , (34) α D = 2 e m − . (35) III. RESULTS AND DISCUSSIONS
In the present study, we reproduce both the excitationenergy and the decay width of the ISGQR in
Pb mea-sured experimentally, by adjusting the isoscalar nucleoneffective mass m ∗ s and a constant and isotropic nucleon-nucleon scattering cross section. Using the same m ∗ s andthe cross section, we further constrained the symmetryenergy and the neutron-proton effective mass splittingusing the centroid energy and the electric dipole polariz-ability extracted from the IVGDR in Pb, by compar-ing results from the IBUU transport approach with theexperimental data. We use several IBUU runs for eachscenario, and 200 test particles are used for each run. Thestatistical errors are calculated based on results of differ-ent IBUU runs. The moments of the ISGQR and theIVGDR are calculated from binded nucleons with theirlocal densities higher than ρ / A. Isoscalar giant quadruple resonance
With the initial
Pb nucleus excited according toEqs. (27) and (28), the time evolutions of the ISGQRmoment using different nucleon-nucleon cross sectionsare compared in Fig. 3(a), by using the parameters with m ∗ s = 0 . m as listed in Table I. It is obviously seen thata larger nucleon-nucleon cross section leads to a strongerdamping of the ISGQR oscillation, since more attemptedand successful nucleon-nucleon collisions occur. Even inthe Vlasov calculation with σ = 0 mb, the oscillationmode damps very slowly due to the Landau dampingmechanism. On the other hand, the oscillation frequencyis seen to be not much affected by the nucleon-nucleoncross section. It is interesting to see that the momentdoes not return to zero especially with larger cross sec-tions. From the observation, the ISGQR moment gener-ally shows a periodical oscillation behavior with an ex-ponential decay, so it can be fitted with the followingfunction [41] Q ISGQR ( t ) = a sin[ b ( t − t )] exp( − ct ) + d, (36)where a represents the oscillation magnitude, b representsthe oscillation frequency, t represents the initial oscilla-tion phase, c represents the decay width, and d represents some possible average displacement. The resulting decaywidths Γ ∼ c for different nucleon-nucleon cross sectionsare shown in Fig. 3(b). The larger decay width from thelarger cross section is intuitively understandable. Evenin the Vlasov scenario, the decay width is non-zero. Inthe present study, we invoke the experimental results ofthe ISGQR extracted in Ref. [49], where the decay widthis 3 . ± . σ = 40 mb reproduces this decay width rea-sonably well, and the collision effect is seen to be similarto that from the particle-vibration coupling [25]. ( M e V ) Q I S GQ R ( f m ) t (fm/c) = 0mb = 30mb = 40mb = 50mb (a) (b) (mb) FIG. 3: (Color online) Time evolution of the ISGQR moment(a) and the decay width of the ISGQR (b) from differentnucleon-nucleon cross sections. The experimentally measuredwidth [49] is plotted as a band for comparison.
Using different isoscalar nucleon effective masses m ∗ s ,the time evolutions of the ISGQR moment are comparedin Fig. 4(a), where the nucleon-nucleon cross section σ = 40 mb is used in each scenario. The different os-cillation frequencies from different m ∗ s can already beseen from the time evolution of the ISGQR moment.Fitting the ISGQR moment with Eq. (36), the exci-tation energies E x ∼ b from different m ∗ s are shownin Fig. 4(b). It is seen that a larger m ∗ s leads to asmaller E x . This is understandable from Fig. 1, sincea smaller m ∗ s leads to a more attractive isoscalar poten-tial below the Fermi momentum, serving as a strongerrestoring force of the ISGQR and increasing the oscil-lation frequency. The experimental measured excitationenergy E x = 10 . ± . m ∗ s = 0 . m . B. Isovector giant dipole resonance
Using the same nucleon-nucleon cross section σ = 40mb and the initial excitation as Eqs. (29), (30), and (31),we have stimulated the IVGDR in Pb, and the timeevolutions of the moment from different scenarios are dis-played in Fig. 5(a) and Fig. 5(c). The periodic oscillation E x ( M e V ) Q I S GQ R ( f m ) t (fm/c) m *s0 =0.7m m *s0 =0.8m m *s0 =0.9m (a) (b) m *s0 /m FIG. 4: (Color online) Time evolution of the ISGQR mo-ment (a) and the excitation energy of the ISGQR (b) fromdifferent isoscalar nucleon effective masses. The experimen-tally measured excitation energy [49] is plotted as a band forcomparison. and decay behavior of the IVGDR moment in all scenar-ios can be fitted with the following form Q IVGDR ( t ) = a sin( bt ) exp( − ct ) . (37)The advantage of the fitting is that the same oscillationbehavior is extrapolated to infinity time and the integralin Eq. (32) can be carried out analytically, i.e., S ( E ) = ac πη (cid:20) c + ( b + E ) − c + ( b − E ) (cid:21) . (38)The resulting strength functions from different scenariosare shown in Fig. 5(b) and Fig. 5(d). The different x and y values corresponds to different symmetry energiesand neutron-proton effective mass splittings, essentiallydifferent symmetry potentials, as shown in Fig. 2 andTable II. Results with different symmetry energies butthe same neutron-proton effective mass splitting are thuscompared in the upper panels of Fig. 5, while results withthe same symmetry energy but different neutron-protoneffective mass splittings are compared in the lower panelof Fig. 5. The effects can all be understood from the low-momentum part of the symmetry potential, which is thedominating restoring force of the IVGDR. Since the cases( x = 0 . , y = − x = 0 . , y = − x = 0 . , y = − x = 0 . , y = − -20-1001020 x=0.5,y=-200MeV x=0.1,y=-200MeV S ( E ) ( f m / M e V ) Q I V G DR ( f m ) x=0.3,y=-200MeV x=0.3,y=-200MeV,vlasov x=0.3,y=-200MeV x=0.3,y=-200MeV,vlasov x=0.15,y=-350MeV x=0.45,y=-50MeV t (fm/c) (a) (b)(c) (d) E (MeV)
FIG. 5: (Color online) Time evolution of the IVGDR moment(left) and the strength function of the IVGDR (right) fromdifferent scenarios.
With the analytical formula of the strength functionEq. (38), the moments as well as other observables canalso be expressed analytically as m − = − ab η ( b + c ) , (39) m = − ab η , (40) E − = p b + c , (41) α D = − e abη ( b + c ) . (42)Figure 6 displays the resulting centroid energies E − andthe electric dipole polarizabilities α D for the correspond-ing scenarios as in Fig. 5. The experimental results of E − = 13 .
46 MeV from photoabsorption reactions [51],and α D = 19 . ± . , which is measured from pho-toabsorption cross sections as well as polarized proton in-elastic scatterings [50] and further corrected by subtract-ing the contribution of quasideuteron excitations [18], arealso plotted for comparison. It is seen that the electricdipole polarizability can generally be reproduced with theparameterization adopted here, while the centroid energygives a very stringent constraints on the x and y param-eters. Comparing the results with and without nucleon-nucleon collisions, it is seen that the centroid energies arevery similar within statistical errors, while a considerableeffect on the electric dipole polarizability is observed, asa result of the different shapes of the strength functionshown in Fig. 5.The favored x and y values can be obtained by com-paring the resulting E − and α D with the experimentaldata, in the way as shown in Fig. 6. Using the sameisoscalar parameterization with m ∗ s = 0 . m as shown inTable I, the favored x and y values can be mapped inthe two-dimensional plane of the slope parameter L ofthe symmetry energy and the isovector nucleon effectivemass m ∗ v , as displayed in Fig. 7. The anticorrelationrelation between L and m ∗ v is observed. It is seen that x=0.45y=-50MeV x=0.15y=-350MeVx=0.5y=-200MeVx=0.1y=-200MeV x=0.3y=-200MeVvlasov D ( f m ) E -1 (MeV) x=0.3y=-200MeV FIG. 6: (Color online) The resulting centroid energies E − and the electric dipole polarizability α D from different sce-narios compared with the experimental results [18, 50, 51]shown as bands. the favored values of L and m ∗ v are within an area ofabout 36 < L <
62 MeV and 0 . < m ∗ v /m < . . δ < ( m ∗ n − m ∗ p ) /m < . δ at the saturation density and at the Fermi momentum.The constraint on L further narrows down the recent con-straint of L = 58 . ± .
35 40 45 50 55 60 650.720.740.760.780.800.820.840.86 favored disfavored m * v / m L (MeV)
FIG. 7: (Color online) Favored and disfavored values of theslope parameter L of the symmetry energy and the isovectornucleon effective mass m ∗ v from the experimental data of E − and α D . IV. SUMMARY
Based on an improved isospin-dependent Boltzmann-Uehling-Uhlenbeck transport approach and using an im-proved isospin- and momentum-dependent interaction, we have studied the isoscalar giant quadrupole reso-nance (ISGQR) and the isovector giant dipole resonance(IVGDR) in
Pb. The width of the strength functionand the excitation energy of the ISGQR are reproducedrespectively by choosing a proper nucleon-nucleon crosssection σ = 40 mb and isoscalar nucleon effective mass m ∗ s = 0 . m . With the same σ and m s , we have fur-ther constrained the slope parameter L of the symmetryenergy and the isovector nucleon effective mass m ∗ v , bycomparing the resulting centroid energy and the electricdipole polarizability, extracted from the strength func-tion of the IVGDR, with the corresponding experimentaldata. The isoscalar potential and the symmetry potentialbelow the Fermi momentum dominate the restoring forceof the ISGQR and IVGDR. Incorporating the nucleon-nucleon collisions leads to almost the same peak energyof the strength function but broads it by damping thecollective oscillation of the IVGDR, and thus has consid-erable effects on the resulting electric dipole polarizabil-ity. The favored values of L and m ∗ v are within an areaof about 36 < L <
62 MeV and 0 . < m ∗ v /m < . . δ < ( m ∗ n − m ∗ p ) /m < . δ . Although the exper-imental measured centroid energy of the IVGDR givesa stringent constraint on L and m ∗ v , further efforts onmeasuring more accurately the electric dipole polarizabil-ity is encouraged to pin down nuclear interactions in theisovector channel. Acknowledgments
This work was supported by the National Natural Sci-ence Foundation of China under Grant No. 11922514.Helpful discussions with Rui Wang are acknowledged.
Appendix A: Energy conservation innucleon-nucleon collisions with amomentum-dependent potential
Although the Bertsch’s prescription [46] conservesthe energy in each nucleon-nucleon (NN) collision infree space, this is not the case in the presence ofthe momentum-dependent potential. This is becausethe contribution of the momentum-dependent part inEq. (20) generally changes after a NN collision, due totheir different final nucleon momenta compared to thosebefore the NN collision. As a remedy, we modified theBertsch’s prescription in the following way.The collision between nucleon 1 and nucleon 2 hap-pens in their center-of-mass (C.M.) frame. In the originalBertsch’s prescription, the momentum in the C.M. framechanges its direction while keeping its magnitude after asuccessful NN collision, and their final momenta ~p , andkinetic energies E , = q ~p , + m are from the Lorentztransformation back to the collisional frame according to ~p , = γ ( ± ~p CM + ~βE CM ) , (A1) E , = γ ( E CM ∓ ~β · ~p CM ) . (A2)In the above, E CM = p ~p + m is the kinetic energyin the C.M. frame with ~p CM being the momentum af-ter the collision, γ = 1 / p − β is the Lorentz factorwith ~β being the velocity of the C.M. frame with re-spect to the collisional frame. The upper (lower) signsin the above equations are for nucleon 1(2). As men-tioned before, this prescription conserves the total mo-mentum and kinetic energy. In order to conserve boththe total momentum and total energy in the presence ofthe momentum-dependent potential, we modify the pre-scription by changing the magnitudes of ~p CM and ~β whilekeeping their direction, i.e., ~p ′ CM = c ~p CM and ~β ′ = c ~β where c and c are constants to be determined, and theLorentz transformation from the C.M. frame back to thecollisional frame is now expressed as ~p ′ , = γ ′ ( ± ~p ′ CM + ~β ′ E ′ CM ) , (A3) E ′ , = γ ′ ( E ′ CM ∓ ~β ′ · ~p ′ CM ) , (A4) with γ ′ = 1 / p − β ′ . 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