Nuclear matter parameters and their correlations

For the Skyrme effective interaction used in this work, the corresponding isospin asymmetric equation of state for cold nuclear matter is $\begin{array}{c}E/A=\frac{3{\hslash }^2}{10m}{\left(\frac{3{\pi }^2}{2}\rho \right)}^{2/3}\\ +\frac{\alpha }{2}\frac{\rho }{{\rho }_0}+\frac{\beta }{\eta +1}\frac{{\rho }^{\eta }}{{\rho }_0^{\eta }}+g_{\rho \tau }\frac{{\rho }^{5/3}}{{\rho }_0^{5/3}}+S(\rho ){\delta }^2\mathrm{,}\end{array}$ (9) where the density dependence of the symmetry energy S(ρ) is $\begin{array}{c}S(\rho )=\frac{{\hslash }^2}{6m}{\left(\frac{3{\pi }^2\rho }{2}\right)}^{2/3}+A_{\text{sym}}\frac{\rho }{{\rho }_0}\\ +B_{\text{sym}}{\left(\frac{\rho }{{\rho }_0}\right)}^{\eta }+C_{\text{sym}}(m_s^{*}\mathrm{,}{m}_v^{*}){\left(\frac{\rho }{{\rho }_0}\right)}^{5/3}.\end{array}$ (10) The terms gρτ in Eqs. (9) and C_sym in Eq. (10) originate from the energy density functional of the Skyrme-type momentum-dependent interaction, and its relationship to the standard Skyrme interaction can be found in Ref. [54]. The pressure in the nuclear fluid is calculated as follows: $P={\rho }^2\frac{\partial E/A(\rho \mathrm{,}\delta )}{\partial \rho }\mathrm{.}$ (11) The saturation density ρ₀ for symmetric nuclear matter is obtained using $P={\rho }_0^2\left(\frac{\text{d}}{\text{d}\rho }\frac{E}{A}(\rho \mathrm{,}\delta =0)\right){|}_{\rho ={\rho }_0}=0.$ (12) Correspondingly, the nuclear matter parameters at the saturation density were obtained. For example, the binding energy E₀ and the incompressibility K₀ are $E_0=E/A({\rho }_0)\mathrm{,}$ (13) $K_0=9{\rho }_0^2\frac{{\partial }^{2}E/A}{\partial {\rho }^2}{|}_{{\rho }_0}\mathrm{.}$ (14) The symmetry energy coefficient S₀ and slope of the symmetry energy L are $S_0=S({\rho }_0)\mathrm{,}$ (15) $L=3{\rho }_0\frac{\partial S(\rho )}{\partial \rho }{|}_{{\rho }_0}\mathrm{.}$ (16) The effective mass of neutron and proton is obtained from the neutron and proton potentials respectively, as follows: $\frac{m}{m_q^{*}}=1+\frac{m}{p}\frac{\partial V_q}{\partial p}\mathrm{,}q=\text{n}\mathrm{,}\text{p}\mathrm{,}$ (17) where Vq is the single-particle potential for a neutron or proton and the form of Vq can be found in Appendix 5. For the Skyrme interaction, the neutron and proton effective masses are $\frac{m}{m_q^{*}}=1+4m{C}_0\rho +4m{D}_0{\rho }_q\mathrm{.}$ (18) The isoscalar effective mass $m_s^{*}$ can be obtained at ρq=ρ/2 from Eq. (18), and the isovector effective mass $m_v^{*}$ can be obtained at ρq=0, which represents the neutron (proton) effective mass in pure proton (neutron) matter, as in Refs. [55, 22]. They are $\frac{m}{m_s^{*}}=1+4m\left(C_0+\frac{D_0}{2}\right)\rho \mathrm{,}$ (19) $\frac{m}{m_v^{*}}=1+4m{C}_0\rho \mathrm{.}$ (20) By using $m_s^{*}$ and $m_v^{*}$, the effective mass splitting $\Delta m_{\text{np}}^{*}=\left(m_{\text{n}}^{*}-m_{\text{p}}^{*}\right)/m$ can be expressed as $\Delta m_{\text{np}}^{*}=\frac{m_{\text{n}}^{*}-m_{\text{p}}^{*}}{m}=2\frac{m_s^{*}}{m}{\displaystyle \sum _{n=1}^{\infty }}{\left(\frac{m_s^{*}-m_v^{*}}{m_v^{*}}\right)}^{2n-1}{\delta }^{2n-1}\mathrm{,}$ (21) as in Ref. [22]. As described in Eq. (21), the exact value of $\Delta m_{\text{np}}^{*}=\left(m_{\text{n}}^{*}-m_{\text{p}}^{*}\right)/m$ depends on the expansion and the isospin asymmetry of the system, δ. To avoid dependence on the expansion and δ, we define the quantity $f_I=\frac{1}{2\delta }\left(\frac{m}{m_{\text{n}}^{*}}-\frac{m}{m_{\text{p}}^{*}}\right)=\frac{m}{m_s^{*}}-\frac{m}{m_v^{*}}$ (22) to describe the isospin effective mass splitting, which has the opposite sign to $\Delta m_{\text{np}}^{*}$.

Because the aforementioned nuclear matter parameters are obtained from the same energy density functional, one can expect correlations between them. For example, as expressed in Eq. (10), S(ρ) depends on the two-body, three-body, and momentum-dependent interaction terms. These three terms are correlated with E₀, K₀, and $m_s^{*}$ [55] and S₀, L, and $m_v^{*}$ [50]. The correlation strength depends on the effective set of Skyrme interaction parameters used [55].

To describe the correlation between different nuclear matter parameters with less bias, one can calculate the linear correlation coefficient CAB between the nuclear matter parameters A and B from the published parameter sets, which satisfy the current knowledge of the nuclear matter parameters [50]: $\begin{array}{c}200\text{ MeV}⩽K_0⩽280\text{ MeV}\mathrm{,}\\ 25\text{ MeV}⩽S_0⩽35\text{ MeV}\mathrm{,}\\ 30\text{ MeV}⩽L⩽120\text{ MeV}\mathrm{,}\\ 0.6⩽m_s^{*}/m⩽1.0,\\ -0.5⩽f_I⩽\mathrm{0.4.}\end{array}$ (23) The quantities A or B = $\left\{{\rho }_0\mathrm{,}{E}_0\mathrm{,}{K}_0\mathrm{,}{S}_0\mathrm{,}L\mathrm{,}{m}_s^{*}\mathrm{,}{m}_v^{*}\right\}$ and the correlation coefficient CAB are calculated as follows: $\begin{array}{c}{C}_{AB}=\frac{\text{cov}(A\mathrm{,}B)}{\sigma (A)\sigma (B)},\\ \text{cov}\left(A\mathrm{,}B\right)=\frac{1}{N-1}{\displaystyle \sum _i\left(A_i-\langle A\rangle \right)\left(B_i-\langle B\rangle \right)},\\ \sigma (X)=\sqrt{\frac{1}{N-1}{\displaystyle \sum _i{\left(X_i-\langle X\rangle \right)}^2}}\mathrm{,}X=A\mathrm{,}B\\ \langle X\rangle =\frac{1}{N}{\displaystyle \sum _i{X}_i}\mathrm{,}i=\mathrm{1,}\dots \mathrm{,}N,\end{array}$ (24) where $\text{cov}(A\mathrm{,}B)$ is the covariance between A and B, σ(X) is the standard deviation of X, and $\langle X\rangle$ denotes the average values obtained from N=119 standard Skyrme parameter sets, selected according to the criteria in Eq. (23).

The values of these parameters are listed in Table 1, and the correlation coefficients CAB are shown in Fig. 1. A positive value of CAB reflects a positive linear correlation, whereas a negative value indicates a negative linear correlation. Correlations exist between the different nuclear matter parameters. Specifically, the correlations between S₀ and ρ₀, L and S₀, $m_v^{*}\text{ and }{m}_s^{*}$, K₀ and ρ₀, and S₀ and E₀ are stronger than those of the other nuclear matter parameter pairs. The ‘strange’ correlation between ρ₀ and S₀ can be understood as follows: ρ₀ can be determined using Eq. (12), which is related to the parameters α, β, η, and gρτ, or to the nuclear matter parameters, as presented in Eq. (5) of Ref. [50]. These correlations indicate that obtaining tight constraints on the density dependence of the symmetry energy using HICs requires knowing information not only on S₀ and L but also on $m_s^{*}$ and $m_v^{*}$ (or the effective mass splitting).

Fig. 1Full-screen View

(Color online) Correlation coefficients between the different nuclear matter parameter pairs.

Symmetry potential

Based on Eq. (17), effective mass splitting is related to the symmetry potential, which plays an important role in HICs. The symmetry potential V_sym is also called the Lane potential, which equals the difference between the neutron and proton potentials: $\begin{array}{c}{V}_{\text{Lane}}\left(\rho \mathrm{,}p\right)=\frac{V_{\text{n}}-V_{\text{p}}}{2\delta }\\ =2A_{\text{sym}}\frac{\rho }{{\rho }_0}+2B_{\text{sym}}{\left(\frac{\rho }{{\rho }_0}\right)}^{\eta }\\ +{\hslash }^2{D}_0{\left(\frac{3{\pi }^2}{2}\rho \right)}^{2/3}\rho +D_0\rho p^2\\ =V_{\text{sym}}^{\text{loc}}+2D_{0}m\rho E_k\mathrm{,}\end{array}$ (25) where $V_{\text{sym}}^{\text{loc}}=2A_{\text{sym}}\frac{\rho }{{\rho }_0}+2B_{\text{sym}}{\left(\frac{\rho }{{\rho }_0}\right)}^{\eta }+{\hslash }^2{D}_0{\left(\frac{3{\pi }^2}{2}\rho \right)}^{2/3}\rho$ and Ek=p^2/2m.

To quantitatively understand the momentum and density dependence of V_Lane on HIC observables, we investigate V_Lane(ρ,p) for two typical Skyrme interaction parameter sets: SkM* and SLy4. These two Skyrme interaction parameter sets were selected for the following reasons: First, the incompressibility (K₀), symmetry energy coefficient (S₀), and isoscalar effective mass ($m_s^{*}$) should be within reasonable and commonly accepted ranges; that is, K₀=230± 20 MeV, S₀=32± 2 MeV, and $m_s^{*}/m=0.7\pm 0.1$. Second, the parameter sets have different signs of effective mass splitting: $\Delta m_{\text{np}}^{*}=(m_{\text{n}}^{*}-m_{\text{p}}^{*})/m>0$ or <0. The SLy4 set [55] has $\Delta m_{\text{np}}^{*}<0$ (or fI>0) in neutron-rich matter, and the slope of the symmetry energy L is 46 MeV. The set SkM* has $\Delta m_{\text{np}}^{*}>0$ (or fI<0) and L = 46 MeV. For convenience, the values of the nuclear matter parameters in SkM* and SLy4 are listed in Table 2.

In Fig. 2, we present V_Lane as a function of kinetic energy for cold nuclear matter with isospin asymmetry δ=0.2 at different densities. V_Lane increased (decreased) as the kinetic energy increased for $\delta m_{\text{np}}^{*}<0$ ($\delta m_{\text{np}}^{*}>0$). They influence the neutron to proton yield ratio Y(n)/Y(p) as a function of the kinetic energy in HICs according to the following relationship: $\begin{array}{c}\frac{Y(\text{n})}{Y(\text{p})}\propto \exp \left(\frac{{\mu }_{\text{n}}-{\mu }_{\text{p}}}{T}\right)\\ =\exp \left[\frac{2\left(V_{\text{sym}}^{\text{loc}}+2D_{0}m\rho E_{\text{k}}\right)\delta }{T}\right],\end{array}$ (26) where T is the temperature of the emitting source and μ_n and μ_p are the chemical potentials of neutrons and protons, respectively. The above relationship can be obtained using statistical and dynamic models [56-61]. Therefore, one can expect that the larger the Lane potential, the larger the neutron to proton yield ratios. Similar effects on the triton to ³He yield ratios are also expected [62]: $\begin{array}{c}\frac{\text{Y(t)}}{{\text{Y(}}^{\text{3}}\text{He)}}\propto \exp \left(\frac{{\mu }_{\text{t}}-{\mu }_{{}^3\text{He}}}{T}\right)\approx \exp \left(\frac{{\mu }_{\text{n}}-{\mu }_{\text{p}}}{T}\right)\\ =\exp \left[\frac{2\left(V_{\text{sym}}^{\text{loc}}+2D_{0}m\rho E_k\right)\delta }{T}\right].\end{array}$ (27) In addition, one can expect that the slopes of the Y(n)/Y(p) ratios with respect to Ek will differ from the effective mass splitting according to Eq. (26) and a similar behavior is also expected for Y(t)/Y(³He).

Fig. 2Full-screen View

(Color online) Lane potential V_Lane as functions of kinetic energy E_k at densities of ρ = 0.3 ρ₀, 0.8 ρ₀, and 1.2 ρ₀.

Y(n)/Y(p) and Y(t)/Y(³He)

To observe the effects of effective mass splitting on HIC observables such as Y(n)/Y(p) and Y(t)/Y(³He), we performed a simulation of the ⁸⁶Kr + ²⁰⁸ Pb system at beam energies from E_beam = 25A to 200A MeV. In the calculations, the impact parameter b = 1 fm and the number of events was 100,000. The dynamic evolution time is stopped at 400 fm/c.

The left panels of Fig. 3 show the Y(n)/Y(p) ratios as functions of the normalized nucleon center-of-mass energy $E_{\text{k}}/E_{\text{beam}}$. The errors of Y(n)/Y(p) are statistical uncertainties obtained using the error propagation formula from the errors of Y(n) and Y(p). By using $E_{\text{k}}/E_{\text{beam}}$, the shapes of Y(n)/Y(p) as a function of the kinetic energy can be compared and understood on a similar scale for different beam energies. The red lines correspond to the results obtained with SLy4 ($m_{\text{n}}^{*}<m_{\text{p}}^{*}$) and the blue lines correspond to SkM* ($m_{\text{n}}^{*}>m_{\text{p}}^{*}$). Our calculations show that the Y(n)/Y(p) ratios obtained with both SLy4 and SkM* decrease as the nucleon kinetic energy increases, owing to Coulomb effects. Furthermore, the Y(n)/Y(p) ratios obtained using SLy4 ($m_{\text{n}}^{*}<m_{\text{p}}^{*}$) are larger than those obtained using SkM* ($m_{\text{n}}^{*}>m_{\text{p}}^{*}$). At a beam energy of 200A MeV, a flatter Y(n)/Y(p) dependence on the nucleon kinetic energy was observed for SLy4. This is because SLy4 has stronger Lane potentials at high kinetic energies and enhanced neutron emission at high nucleon energies.

Fig. 3Full-screen View

(Color online) Yield ratios of Y(n)/Y(p) and Y(t)/Y(³He) as functions of the normalized nucleon center of mass energy $E_{\text{k}}/E_{\text{beam}}$ at beam energies of E_beam = 25A, 100A, and 200A MeV.

Specifically, the difference in Y(n)/Y(p) between SLy4 ($m_{\text{n}}^{*}<m_{\text{p}}^{*}$) and SkM*($m_{\text{n}}^{*}>m_{\text{p}}^{*}$) maintains a constant value with the nucleon kinetic energy at 25A MeV and increases with the nucleon kinetic energy at a beam energy of > 100A MeV. This can be understood from the Lane potentials shown in Fig. 2. At 25A MeV, the system is less compressed and excited than that at 100A or 200A MeV, and most of the emitted nucleons originate from the low-density region. The corresponding symmetry potentials for SLy4 and SkM* varied weakly as a function of kinetic energy (see Fig. 2(a)). Therefore, one can expect that the difference in Y(n)/Y(p) between SLy4 ($m_{\text{n}}^{*}<m_{\text{p}}^{*}$) and SkM* ($m_{\text{n}}^{*}>m_{\text{p}}^{*}$) is small and changes weakly as the kinetic energy increases. At a beam energy of >100A MeV, the system can be compressed to higher densities, where the magnitude of the splitting increases with the kinetic energy, as shown in Figs. 2(b) and 2(c).

The right panels of Fig. 3 show the Y(t)/Y(³He) ratios as functions of the normalized nucleon center-of-mass energy, that is, $E_{\text{k}}/E_{\text{beam}}$. Similar to Y(n)/Y(p), the Y(t)/Y(³He) ratios are also sensitive to effective mass splitting. This can also be explained using Eq. (27). At a beam energy of 200A MeV, the sensitivity of the Y(t)/Y(³He) ratios to the kinetic energy becomes weak, which may be due to cluster effects and stronger nonequilibrium effects than those at lower beam energies.

Furthermore, Fig. 3 also shows that the Y(n)/Y(p) ratio decreases exponentially with respect to $E_{\text{k}}/E_{\text{beam}}$ in the range $0.3⩽E_{\text{k}}/E_{\text{beam}}⩽1.0$. For t/³He ratios, a similar behavior can be observed in $0.2⩽E_{\text{k}}/E_{\text{beam}}⩽0.5$ since the kinetic energy per nucleon for the emitted tritons or ³He is approximately one-half of the beam energy. According to Eqs. (26) and (27), the exponentially decreasing behavior indicates that the emitted nucleons are in equilibrium in momentum space and can be described by the slopes of ln[Y(n)/Y(p)] or ln [Y(t)/Y(³He)], and the slopes of ln[Y(n)/Y(p)] and ln[Y(t)/Y(³He)] are directly related to D₀ as follows: $\begin{array}{c}{S}_{\text{n/p}}=\frac{\partial \ln \left[Y(\text{n})/Y(\text{p})\right]}{\partial E_{\text{k}}}=4D_{0}m\delta \rho /T,\\ S_{{\text{t/}}^3\text{He}}=\frac{\partial \ln \left[Y(\text{t})/Y\left({}^3\text{He}\right)\right]}{\partial E_{\text{k}}}=4D_{0}m\delta \rho /T.\end{array}$ (28) In the following analysis, we perform the linear fit of ln [Y(n)/Y(p)] and ln [Y(t)/Y(³He)]: $\ln \left[\frac{Y(\text{n})}{Y(\text{p})}\right]=S_{\text{n/p}}\frac{E_{\text{k}}}{E_{\text{beam}}}+b_0^{\text{n/p}}$ (29) in the range of $0.3⩽E_{\text{k}}/E_{\text{beam}}⩽1.0$ and $\ln \left[\frac{Y(\text{t})}{Y{(}^3\text{He})}\right]=S_{{\text{t/}}^{\text{3}}\text{He}}\frac{E_k}{E_{\text{beam}}}+b_0^{{\text{t/}}^{\text{3}}\text{He}}$ (30) in the range of $0.2⩽E_{\text{k}}/E_{\text{beam}}⩽0.5$ to obtain the slopes of S_n/p ($S_{\text{t}{/}^3\text{He}}$) and the intercepts of $b_0^{\text{n/p}}$ ($b_0^{\text{t}{/}^3\text{He}}$). To describe the goodness of linear fit of ln(Y(n)/Y(p)) and ln(Y(t)/Y(³He)), we present the coefficients of determination, R² [63] in Table 3.

Figure 4 presents S_n/p ($S_{{\text{t/}}^3\text{He}}$) and $b_0^{\text{n/p}}$ ($b_0^{t{/}^3\text{He}}$) as functions of the beam energy to determine the optimal energy for probing effective mass splitting. Panel (a) shows S_n/p and panel (c) shows $S_{{\text{t/}}^3\text{He}}$. Our calculations show that the values of S_n/p ($S_{{\text{t/}}^3\text{He}}$) obtained with SLy4 are higher than those obtained with SkM* except for the beam energy of 25A MeV. Specifically, the impact of effective mass splitting on S_n/p becomes evident at a beam energy of 200A MeV. For $S_{{\text{t/}}^3\text{He}}$, the impact of effective mass splitting is greatest at a beam energy of 100A MeV under the influence of the cluster formation mechanism. For the $b_0^{\text{n/p}}$, the calculations show that it weakly depends on the effective mass splitting, except for the value of $b_0^{\text{n/p}}$ at beam energies $E_{\text{beam}}=200A$ MeV. At this beam energy, the value of $b_0^{\text{n/p}}$ obtained using SkM* is larger than that obtained using SLy4. For $b_0^{t{/}^3\text{He}}$, the value obtained using SkM* was lower than that obtained using SLy4 at a beam energy of 25A MeV. At $E_{\text{beam}}>100A$ MeV, the values of $b_0^{t{/}^3\text{He}}$ obtained with SkM* were greater than those obtained with SLy4.

Fig. 4Full-screen View

(Color online) S_X and $b_0^X$ as functions of E_beam for SLy4 and SkM*. Upper panels are X = n/p and bottom panels are X = t/³He.