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The standard form of Skyrme effective interaction is[34-35]
where
$ t_{i}^{} $ ,$ x_{i}^{} $ $(i = 0,\, 1,\, 2, \,3)$ , the power$ \alpha $ of density dependence, and the spin-orbit strength$ W_{0}^{} $ are Skyrme parameters of the interaction;$ P_{\sigma}^{} = \frac{1}{2}\left(1+\sigma_{1}^{} \cdot \sigma_{2}^{} \right) $ is the spin exchange operator;$ {\boldsymbol{R}} = ({\boldsymbol{r}}_{1}^{}+{\boldsymbol{r}}_{2}^{}) / 2 $ is the center-of-mass coordinate;$ {\boldsymbol{r}} = {\boldsymbol{r}}_{1}^{}-\mathbf{r}_{2}^{} $ is the relative coordinate of the two particles;$ \overrightarrow{k} = {\rm{i}}(\overrightarrow{\nabla}_{1}^{}-\overrightarrow{\nabla}_{2}^{}) / 2 $ and$ \overleftarrow{k} = {\rm i}(\overleftarrow{\nabla}_{1}^{}-\overleftarrow{\nabla}_{2}^{}) / 2 $ are the relative momentum operators acting on the right and on the left, respectively.We begin with the Skyrme effective interaction and use mean-field Hartree-Fock approximation to obtain single-nucleon Hartree-Fork equation. In asymmetric nuclear matter with an asymmetry parameter
$ \beta = (\rho_{\rm{n}}^{}-\rho_{\rm{p}}^{})/(\rho_{\rm{n}}^{}+\rho_{\rm{p}}^{}) $ , the nucleon effective mass determined by the momentum dependence of single-nucleon potential is[36]where
$ q = n $ ,$ \omega_{q}^{} = 1 $ for neutrons and$ q = p $ ,$ \omega_{q}^{} = -1 $ for protons. The simplified notation isAnd the momentum-independent mean-field potential is
where the kinetic energy density
$ \tau_{q}^{}({\boldsymbol{r}}) $ satisfies$ \tau({\boldsymbol{r}}) = \tau_{\rm{p}}^{}({\boldsymbol{r}})+\tau_{\rm{n}}^{}({\boldsymbol{r}}) $ . -
Generally, in the Boltzmann-Uehling-Uhlenbeck (BUU) model, the mean-field potential is expressed as a function of the (continuous) phase-space distribution, which is resolved in terms of a (large) number of discrete test particles as[37]
where
$ A $ is the number of nucleons,$ \widetilde{N} $ is the test particle number per nucleon (200 in this work),$ {\boldsymbol{r}}_{i}^{} $ and$ {\boldsymbol{p}}_{i}^{} $ are the coordinates and momenta of the individual test particles. The available volume is divided into sufficiently small cubes of side$ l $ which can be considered as uniform nuclear matter. Under the local density approximation of nuclear matter, the gradient term of the density and spin-orbit term of mean-field potential vanish.Then the momentum-independent single-nucleon potential obtained from Eq. (4) is
and the momentum-dependent part is
The isospin- and momentum- depentent single-nucleon potential used in the BUU code is
where
$ q = n $ for neutrons and$ q = p $ for protons.For the BUU code, the kinetic energy density
$ \tau_{q}^{}({\boldsymbol{r}}) $ in the above Eq. (6) is derived with the phase-space distribution function$f({\boldsymbol{r}}, \,{\boldsymbol{p}})$ Therefore, we have
where the summation represents the sum of all the protons (
$ q = p $ ) or neutrons ($ q = n $ ) in the grid where they are located with coordinate$ {\boldsymbol{r}} $ .The symmetry energy can be obtained from the binding energy per nucleon
$ E/A $ [38]The initial density distributions of neutron and proton in nucleus are given by the Skyrme-Hartree-Fork calculations with corresponding Skyrme parameters. And we let the nucleon momentum distribution in the high-momentum-tail(HMT)[39-41]
and
The isospin-dependent baryon–baryon(BB) scattering cross section in medium
$ \sigma ^{\rm{medium}}_{\rm{BB}} $ is reduced compared with their free-space value$ \sigma ^{\rm free}_{\rm{BB}} $ by a factor of[42-43]where
$ \mu_{\rm{B B}}^{} $ and$ \mu_{\rm{B B}}^{*} $ are the reduced masses of the colliding baryon-pair in free space and medium, respectively. The nucleon effective mass in the factor$ R_{\rm {medium }}^{} $ is obtained using the Skyrme interaction, see Eq. (2). -
The neutron/proton double ratio has recently been used to study the effective mass splitting effect with NSCL experiments[33-44], and we also need to consider the choice of symmetry energy because it's still unclear, especially in high-density regions[45-48]. In this way, it is convincing that signals are mainly due to the neutron-proton effective mass splitting and not strictly depending on the choice of the stiffness of the symmetry energy. In this work, among the published 240 Skyrme interaction parameter sets in the Ref. [49], six Skyrme interactions predict the opposite choices of effective mass splitting for different stiffnesses of the symmetry energy (asy-hard, asy-sightly-soft, and asy-soft), as shown in Table 1. The corresponding two sets of Skyrme parameters selected with similar symmetry energy also predict very close saturation properties of nuclear matter, such as incompressibility
$ K_0^{} $ , the binding energy$ E_{0}^{} $ and isoscalar effective mass$ m^{*} $ , with the exception that the effective mass splitting is opposite. The saturation properties of nuclear matter are determined by standard Skyrme parameter sets with 9 parameters$ \{t_0^{},\, t_1^{},\, t_2^{},\, t_3^{}, \,x_0^{}, \,x_1^{},\, x_2^{}, \, x_3^{},\, \alpha \}$ . By the way, if we fix the symmetry energy coefficient$ S_{0}^{} $ , its slope$ L $ , and curvature$K_{\rm sym}^{}$ , the trend of the symmetry energy is roughly determined. The density dependence of nuclear symmetry energy with selected Skyrme parameters is shown in the left panel of Fig. 1. And the right panel displays the effective mass of neutron and proton as a function of density ($ \rho = \rho_0^{} $ ,$ 2\rho_0^{} $ ) and isospin-asymmetry with the KDE0v1 and NRAPR interactions, respectively. It shows that the effective mass of proton is slightly larger than that of neutron for the KDE0v1 interaction and the opposite for NRAPR.Stiffness Model $ m_{\rm{n-p}}^{*} $ $ K_{0}^{} $ $ S_{0}^{} $ $ L $ $ K_{\rm{sym}}^{} $ $ m^{*}/m $ $ m_{\rm{n}}^{*}/m $ $ m_{\rm{p}}^{*}/m $ $ \rho_{0}^{} $ $ E_{0}^{} $ hard SkI2[50] <0 240.93 33.37 104.33 70.69 0.68 0.662 0.703 0.158 −15.78 hard Gs[51] >0 237.29 31.13 93.31 14.07 0.78 0.807 0.757 0.158 −15.59 slightly soft KDE0v1[52] <0 227.54 34.58 54.69 −127.12 0.74 0.737 0.763 0.165 −16.23 slightly soft NRAPR[53] >0 225.65 32.78 59.63 −123.32 0.69 0.716 0.674 0.161 −15.85 soft BSk9[54] <0 231.32 30 38.29 −153.70 0.80 0.780 0.818 0.159 −15.92 soft SV-mas08[55] >0 233.13 30 40.15 −172.38 0.80 0.818 0.781 0.160 −15.90 Table 1. Corresponding saturation properties of nuclear matter with selected different Skyrme parameter sets. The effective mass of neutron and proton are obtained with isospin asymmetry
$ \delta = 0.2 $ for isospin asymmetric nuclear matter. All entries are in MeV, except for$ \; \rho_{0}^{} $ in fm−3 and the dimensionless effective mass ratios.Figure 1. Left panel: Density dependence of symmetry energy with different Skyrme parameter sets; Right panel: The effective mass of neutron and proton as a function of density at
$ \rho = \rho_0^{},\, 2\rho_0^{} $ and isospin-asymmetry for the parameters KDE0v1 (upper window) and NRAPR (lower window) in cold asymmetric nuclear matter. (color online)In cold asymmetric nuclear matter, the kinetic energy density of nucleon in the Eq. (6) can be written as
$ \tau_{q}^{} = \frac{3}{5} k_{F_{q}^{}}^{2} \rho_{q}^{} $ , and the symmetry potential can be obtained from the definition$ U_{\rm{sym}}^{} = (U_{\rm{n}}^{}-U_{\rm{p}}^{})/2\delta $ . Where$ U_{\rm{n}}^{} $ ($ U_{\rm{p}}^{} $ ) is the single-nucleon mean-field potential of neutron (proton). The symmetry potentials as a function of density and momentum at isospin asymmetry$ \delta = 0.2 $ with the previously selected Skyrme parameters in cold asymmetric nuclear matter are shown in Fig. 2. The left panel shows the negative effective mass splitting for different stiffnesses of the symmetry energy predicted by the SkI2, KDE0v1, and BSk9 interactions while the corresponding positive effective mass splitting predicted by the Gs, NRAPR, SV-mas08 interactions are shown in the right, respectively. One can see that for the case of$ m_{\rm{n}}^{*} < m_{\rm{p}}^{*} $ , the symmetry potential increases with momentum and increases more rapidly due to the squared term of momentum in the mean-field potential, while decreases for$ m_{\rm{n}}^{*} > m_{\rm{p}}^{*} $ case. Combining the Eq. (2), we can obtain the effective mass splittingFigure 2. The symmetry potential as a function of density and momentum with isospin asymmetry
$ \delta = 0.2 $ in cold asymmetric nuclear matter; Different color lines represent different densities ($ \rho = 0.5\rho_{0}^{} $ ,$ \rho = \rho_{0}^{} $ ,$ \rho = 1.5\rho_{0}^{} $ ,$ \rho = 2\rho_{0}^{} $ ). (color online)And combining the Eq. (8), we can obtain the momentum-dependent behavior of symmetry potential
From the Eq. (15) and Eq. (16), we can see the sign of the effective mass splitting is opposite to the momentum-dependent behavior of symmetry potential, and the sign is determined by the notation
$ \xi_{\nu}^{} $ . This is consistent with the previous analysis in Fig. 2. In this framework, the Skyrme force has a strong momentum-dependent behavior and it is also mentioned in Ref. [56].Due to the opposite choices of effective mass splitting corresponding to symmetry potentials' different momentum-dependent behaviors, the symmetry potential is almost positive for the case of
$ m_{\rm{n}}^{*} < m_{\rm{p}}^{*} $ , but may have a cross-over (from positive to negative) at higher momentum and density for$ m_{\rm{n}}^{*} > m_{\rm{p}}^{*} $ case. This distinction will be a key entry point for the analysis of the following results.
Probing the Effects of Density- and Momentum-dependent Potentials in Heavy Ion Collisions
doi: 10.11804/NuclPhysRev.39.2022033
- Received Date: 2022-03-15
- Rev Recd Date: 2022-04-10
- Publish Date: 2022-12-20
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Key words:
- heavy ion collision /
- Skyrme interaction /
- symmetry energy /
- nucleon effective mass splitting
Abstract: The single-nucleon mean-field potential, in-medium nucleon–nucleon cross-sections, and initial density distributions of nucleons are obtained from the Skyrme nucleon-nucleon effective interaction, which are self-consistently used in the Boltzmann-Uehling-Uhlenbeck(BUU) transport model. The
Citation: | Jianbo SHU, Wei ZUO. Probing the Effects of Density- and Momentum-dependent Potentials in Heavy Ion Collisions[J]. Nuclear Physics Review, 2022, 39(4): 446-453. doi: 10.11804/NuclPhysRev.39.2022033 |