Nuclear Bubble Configuration in Heavy-Ion Collisions

Abstract

We study the effects of a bubble configuration in a nucleus on heavy-ion collisions at a few tens and hundreds A MeV. We first investigate the bubble structure of ${}^{206}$Hg using the relativistic continuum Hartree–Bogoliubov theory and then study the ${}^{206}$Hg + ${}^{208}$Pb and ${}^{206}$Hg+${}^{206}$Hg reactions using the DaeJeon–Boltzmann–Uehling–Uhlenbeck (DJBUU) transport model. To see the role of the bubble structure, we consider the maximum density of the produced nuclear matter, directed flow of neutrons and protons, and ${\pi }^{-}/{\pi }^{+}$ ratio. We observe that the maximum density is smaller with a bubble nucleus, and the directed flow of nucleons and ${\pi }^{-}/{\pi }^{+}$ ratio may depend on the bubble structure.

1. Introduction

The physics of dense matter, heavy-ion collisions, nuclear structures and compact stars are all intimately interrelated. Terrestrial dense matter with a large isospin asymmetry can be produced during heavy-ion collisions with rare isotope beams. To investigate the dense matter produced during heavy-ion collisions, we have to resort to nuclear transport simulations since the studies of dense matter in equilibrium do not directly lend themselves to experimental observation. We refer to [1] for a recent review of nuclear transport models.

The exotic nuclear property of nuclei such as a bubble structure is expected to affect the results of the nuclear transport simulations. For example, the effect of hypothetical nuclear bubble configurations on heavy-ion collisions has been studied by investigating the ${\pi }^{-}/{\pi }^{+}$ ratio at $E_{\mathrm{beam}}=400$ A MeV [2] and by exploring proton-induced reactions at $E_{\mathrm{beam}}=800$ A MeV [3] in the framework of the isospin-dependent Boltzmann–Uehling–Uhlenbeck (IBUU) transport model.

The bubble structure in a nucleus is characterized by a depleted central density, and its origin is closely related to a low occupation of the s state near the Fermi surface. There have been many theoretical investigations on bubble nuclei, for example see [4,5,6,7,8,9,10,11,12]. It is exciting to note that the first experimental evidence of the bubble configuration in the doubly magic ${}^{34}$Si nucleus was reported in 2017 [13].

In this work, we focus on a possible effect of the bubble structure in heavy-ion collisions at the energies relevant to the new generation of rare isotope beam facilities: Heavy Ion Research Facility in Lanzhou (HIRFL) [14], RIKEN Radioactive Ion Beam Factory (RIBF) [15], Rare isotope Accelerator complex for ON-line experiments (RAON) [16], Second Generation System On-Line Production of Radioactive Ions (SPIRAL2) [17], Facility for Rare Isotope Beams (FRIB) [18], etc. To this end, we study heavy-ion collisions with a realistic bubble nucleus at beam energies $E_{\mathrm{beam}}=50,100,200,300$ A MeV using the DJBUU transport code [19,20]. For the initialization of the spatial positions and momenta of nucleons in a bubble nucleus, we take the nucleon density profile calculated in the relativistic continuum Hartree–Bogoliubov (RCHB) theory [21,22,23], which provides a proper treatment of pairing correlations in the presence of the continuum. The RCHB theory was successfully applied to exotic nuclear property studies; it provided the first microscopic self-consistent description of halo structure in ${}^{11}$Li [24] and predicted the giant halos in light and medium-heavy nuclei [25,26,27]. It was also shown, in the framework of the RCHB, that the couplings between the bound states and the continuum through the pairing correlation extended the neutron drip-line further to the more neutron-rich side than in other mass models [28].

In this article, we discuss briefly bubble nuclei from the RCHB theory and the DJBUU in Section 2. Numerical results on the density distributions, the time evolution of the density at the collision, and the directed flow of neutrons and protons are described in Section 3. A summary and discussion are given in Section 4.

2. Bubble Structure in Heavy-Ion Collisions

For a theoretical investigation of the bubble structure in a nucleus we used the RCHB theory [21,22,23], and for the numerical simulations of heavy-ion collisions, we utilized the DJBUU model [19,20]. Now, we briefly describe our theoretical frameworks and the bubble structure in nuclei.

2.1. Bubble Nuclei in RCHB Theory

The RCHB theory has been developed for a spherical nucleus in coordinate space to investigate the impact of the continuum on exotic nuclear structures. The RCHB theory also provides an appropriate treatment of pairing correlations in the presence of the continuum through the Bogoliubov transformation.

The point-coupling Lagrangian density in the RCHB theory is given by [29]:

$$\begin{array}{ccc}\hfill \mathcal{L}& =& \hfill \overline{\psi }\left(i{\gamma }_{\mu }{\partial }^{\mu }-m\right)\psi -\frac{1}{2}{\alpha }_S\left(\overline{\psi }\psi \right)\left(\overline{\psi }\psi \right)-\frac{1}{2}{\alpha }_V\left(\overline{\psi }{\gamma }_{\mu }\psi \right)\left(\overline{\psi }{\gamma }^{\mu }\psi \right)\hfill \\ & & \hfill -\frac{1}{2}{\alpha }_{TV}\left(\overline{\psi }\overrightarrow{\tau }{\gamma }_{\mu }\psi \right)·\left(\overline{\psi }\overrightarrow{\tau }{\gamma }^{\mu }\psi \right)-\frac{1}{3}{\beta }_S{\left(\overline{\psi }\psi \right)}^3-\frac{1}{4}{\gamma }_S{\left(\overline{\psi }\psi \right)}^4-\frac{1}{4}{\gamma }_V{\left[\left(\overline{\psi }{\gamma }_{\mu }\psi \right)\left(\overline{\psi }{\gamma }^{\mu }\psi \right)\right]}^2\hfill \\ & & \hfill -\frac{1}{2}{\delta }_S{\partial }_{\nu }\left(\overline{\psi }\psi \right){\partial }^{\nu }\left(\overline{\psi }\psi \right)-\frac{1}{2}{\delta }_V{\partial }_{\nu }\left(\overline{\psi }{\gamma }_{\mu }\psi \right){\partial }^{\nu }\left(\overline{\psi }{\gamma }^{\mu }\psi \right)-\frac{1}{2}{\delta }_{TV}{\partial }_{\nu }\left(\overline{\psi }\overrightarrow{\tau }{\gamma }_{\mu }\psi \right)·{\partial }^{\nu }\left(\overline{\psi }\overrightarrow{\tau }{\gamma }_{\mu }\psi \right)\hfill \\ & & \hfill -\frac{1}{4}{F}^{\mu \nu }{F}_{\mu \nu }-e\frac{1-{\tau }_3}{2}\overline{\psi }{\gamma }^{\mu }\psi A_{\mu },\hfill \end{array}$$

(1)

where m is the nucleon mass and the subscripts $S,V$ and $TV$ stand for scalar, vector and isovector, respectively. The terms with ${\alpha }_V$, ${\beta }_S$ and ${\gamma }_S$ are introduced for the effects of medium dependence, and those with ${\delta }_S$, ${\delta }_V$ and ${\delta }_{TV}$ mimic the finite range effects. $A_{\mu }$ and $F_{\mu \nu }$ represent the vector field and field strength tensor of the electromagnetic field, respectively. For the numerical values of the coupling constants in the Lagrangian density, we adopt the PC-PK1 parametrization [29]. In the PC-PK1 parametrization, the coupling constants are fitted to the binding energies, charge radii and empirical pairing gaps of sixty selected spherical nuclei. 1 After using the mean-field approximation to the Lagrangian density in Equation (1), a Legendre transformation, a variational method and the Bogoliubov transformation, we obtain the relativistic Hartree–Bogoliubov Equation [21,22,23] to investigate nuclear properties.

In order to study the effect of the bubble structure, we define the depletion fraction factor DF that characterizes the depleted central density as follows:

$$\mathrm{DF}\equiv \frac{{\rho }_{\max }-{\rho }_{\mathrm{c}}}{{\rho }_{\max }}\times 100[\%],$$

(2)

where ${\rho }_c$ is the central density of the nucleus and ${\rho }_{\max }$ is the maximum density of the nucleus. In Figure 1, we show the nucleon density distribution of ${}^{206}$Hg obtained from the RCHB theory (with a bubble structure) and from the relativistic Thomas–Fermi (RTF) theory (without a bubble structure). With the RCHB theory, the DF factors for the total and proton densities are 13% and 18%, respectively.

2.2. DJBUU

To simulate heavy-ion collisions, we used the DJBUU transport code [19,20]. The DJBUU code adopts the widely used method for initialization, collisions and Pauli blocking. For the initialization in coordinate space, in addition to the Woods–Saxon parameterization, the nucleon density profile obtained from a RTF calculation can also be used in the DJBUU code. For the mean field potentials, the DJBUU code includes the relativistic mean fields (RMF) from [30] with scalar ($\sigma$) and vector ($\rho$, $\omega$) mesons. To address issues such as partial chiral symmetry restoration and chiral-invariant nucleon mass in heavy-ion collisions, the DJBUU code also contains an extended parity doublet model in [31,32].

In the present work, we used the mean field potential given in [30], where the symmetry energy at the saturation density is 30.5 MeV and the slope parameter is 84 MeV. This mean field potential has been, for instance, used for the Transport Model Evaluation Project (TMEP) [33,34,35]. As such, the potential is well-tested in transport model studies.

A distinctive feature of the DJBUU code is the profile function of the test particle, which is not the Gaussian type:

$$g\left(\mathbf{u}\right)=g\left(u\right)={\mathcal{N}}_{m,n}{(1-{(u/a)}^m)}^n\mathrm{for}0<u/a<1$$

(3)

where $u=\left|\mathbf{u}\right|$, ${\mathcal{N}}_{m,n}$ is the normalization constant and $m>1$ and $n>1$ are integers. This profile function vanishes smoothly at $u=a$ and is exactly integrable. The values of constants are $m=2$, $n=3$, $a_x=4.2\mathrm{fm}$ for the position profile and $a_p=s\hslash /a_x$ for the momentum profile with $s\approx 0.6$. As expected, choosing different values of the constants in our profile function affects our results. Especially, the width of the test particle, $a_x$, is crucial to initialize bubble nuclei.

The mean field potentials affect mainly the propagation of particles as they appear in the equations of motion. We remark here that since we are using the RCHB theory to initialize the nucleus, our propagation method using RMF can disrupt subtle initial structures such as the bubble if the nuclei are allowed to evolve before the collision. To minimize such disruptions, we started our simulations by placing the projectile and target nuclei almost touching; the distance between the centers was 16 fm.

3. Results

We compared three different systems to observe the effect of the bubble structure. The first system was ${}^{206}$Hg + ${}^{208}$Pb without bubbles for both nuclei. The second system was also ${}^{206}$Hg + ${}^{208}$Pb but the ${}^{206}$Hg nucleus had a bubble.

The last one was ${}^{206}$Hg+${}^{206}$Hg with bubbles for both nuclei. We simulated collisions at beam energies of 50, 100, 200 and 300 A MeV with the impact parameters 0, 3 and 6 fm. The number of test particles was 100 and ten events were simulated for each condition. We simulated until $t_{\mathrm{end}}=$ 250, 200, 120 and 100 fm/c for $E_{\mathrm{beam}}=$ 50, 100, 200 and 300 A MeV, respectively.

We used the nucleon density profile from the RCHB (RTF) theory to initialize the nuclei. With the RCHB theory, we obtained the DF of 12% for the nucleon density and 29% for the proton density. When we initialized a bubble nucleus, the resulting density distribution of the initialized nucleus could be different from the RCHB result due to the finite width of the test particles if we simply sampled the nucleon positions in the RCHB profile. Hence, in order to match the RCHB profile after taking into account the test particle profile, Equation (3), we sampled modified density profiles to initialize ${}^{206}$Hg with a bubble structure. The density distributions of ${}^{206}$Hg with the RCHB and RTF results are shown in Figure 2. In this figure, the blue lines are the modified nucleon density profiles that were used for the initialization of ${}^{206}$Hg with a bubble structure.

In Figure 3, we plotted the density contours in the reaction plane for three systems with $E_{\mathrm{beam}}=50$ A MeV and the impact parameter 3 fm. The initialized density distributions have clear differences between the bubble and nonbubble nuclei, especially near the center of the nuclei, and the differences can be also seen during the dynamical evolution.

Figure 4 shows the density profiles at the collision center at $E_{\mathrm{beam}}$ = 50, 100 and 200 A MeV with the impact parameter 3 fm. Comparing ${}^{206}$Hg (nonbubble) + ${}^{208}$Pb and ${}^{206}$Hg (bubble) + ${}^{208}$Pb reactions, we can see that the maximum density is smaller with a bubble structure in ${}^{206}$Hg, which is consistent with the observation made in Ref. [2] with hypothetically extreme bubble nuclei.

Since the momenta of the proton emitted from the collision center would be affected by the change of density, the directed flow $v_1$ of protons may be a good candidate for an observable that is sensitive to the existence of the bubbles. In Figure 5, the directed flow $v_1$ of free protons as a function of $p_t$ is summarized for ${}^{206}$Hg (nonbubble) + ${}^{208}$Pb and ${}^{206}$Hg (bubble) + ${}^{208}$Pb reactions. Here, $p_t$ is the transverse momentum. We regarded a proton in the effectively low density as a free proton and we picked only protons with negative rapidity to consider mainly the protons emitted from the Hg nucleus. We also present the directed flow $v_1$ of free neutrons in Figure 6. Our calculations suggest that there may exist some differences in the behaviors of $v_1$. However, considering the size of the error bars, a more extensive study is needed to conclude whether the directed flow of protons is a good probe to study the bubble structure of realistic nuclei in heavy-ion collisions. For example, one should check whether or not employing some other initial state models for initialization, parameter sets other than PC-PK1, mean field potentials or different transport models can also change $v_1$ and ${\pi }^{-}/{\pi }^{+}$ in a similar way.

As discussed in [2], the bubble structure in a nucleus can affect the ${\pi }^{-}/{\pi }^{+}$ ratio in heavy-ion collisions. In general, one can expect that the produced nuclear matter during the heavy-ion collisions with bubble nuclei has a different neutron and proton distribution compared to the one with nonbubble (or normal) nuclei. This will change the symmetry potential and affect the ${\pi }^{-}/{\pi }^{+}$ ratio. Furthermore, as shown in Figure 4, the maximum density of the produced nuclear matter during the heavy-ion collisions becomes smaller when bubble nuclei collide, which also affects the role of symmetry energy in the produced nuclear matter. Since it is a candidate to observe the effect of the initial isospin distribution, we conducted a study of the ${\pi }^{-}/{\pi }^{+}$ ratio as shown in

Table 1. ${\pi }^{-}$/${\pi }^{+}$ ratio at $E_{beam}=300$ A MeV.

b	${}^{206}$Hg + ${}^{208}$Pb	${}^{206}$Hg (Bubble) + ${}^{208}$Pb	${}^{206}$Hg (Bubble) + ${}^{206}$Hg (Bubble)
0 fm	3.18 (±0.11)	3.10 (±0.08)	3.30 (±0.10)
3 fm	3.18 (±0.11)	3.22 (±0.06)	3.23 (±0.07)
6 fm	3.32 (±0.10)	3.08 (±0.16)	3.37 (±0.17)

. Table 1 indicates that the ratio depends on the bubble structures of the colliding nuclei. However, considering the size of our current error bars, again a more thorough study is needed to reach a more definite conclusion.

4. Summary and Discussion

We investigated the possible effects of the nuclear bubble structure in heavy-ion collisions at a few tens and hundreds A MeV. We first calculated the bubble structure of ${}^{206}$Hg using the RCHB theory [21,22,23] and studied ${}^{206}$Hg + ${}^{208}$Pb and ${}^{206}$Hg + ${}^{206}$Hg reactions in the DJBUU code [19,20]. For the initialization of the bubble nucleus, we noticed that due to the finite width of the test particles we needed to modify nucleon density profiles slightly so that the actual density profiles could match the RCHB profiles.

We observed that the maximum density of the nuclear matter produced during the ${}^{206}$Hg + ${}^{208}$Pb reaction was smaller with a bubble structure in ${}^{206}$Hg, which was consistent with the observation made in Ref. [2] with a hypothetical bubble nucleus. We checked if the directed flow of nucleons and the ${\pi }^{-}/{\pi }^{+}$ ratio could serve as good probes to test the effects of the bubble structure in heavy-ion reactions. We did observe the expected differences in the directed flow and ${\pi }^{-}/{\pi }^{+}$ ratio in ${}^{206}$Hg (nonbubble) + ${}^{208}$Pb and ${}^{206}$Hg (bubble) + ${}^{208}$Pb reactions. However, in order to show clearly whether or not the directed flow and/or the charge pion ratio were good observables for the effect of the bubbles, we need to conduct more extensive studies. For example, one should check whether or not employing some other initial state models, parameter sets, mean field potentials or different transport models can also change $v_1$ and ${\pi }^{-}/{\pi }^{+}$ in a similar way. Furthermore, one may consider very light nuclei as a beam and/or target to see the effects of bubble structure in heavy-ion collisions more clearly.

One practical difficulty is that as discussed previously, due to the finite width of the test particles, there always exists a finite density contribution to the central density by nearby nucleons (test particles), which makes it difficult to initialize the bubble nuclei. For light nuclei, we observe that realizing a bubble nucleus is even more difficult since the test particles in a light nucleus tends to congregate near the center of the nucleus. One way to overcome this practical hindrance may be to use a very small width of the test particle and optimize the transport model to reasonably describe heavy-ion collisions. We leave that for our future studies.

Looking ahead, it will be interesting to investigate the effects of other exotic nuclear properties such as shape coexistence on heavy-ion collisions within our simulation framework.