Fragment productions in DJBUU and SQMD: comparative study

Worldwide, there are many new rare isotope production facilities coming online in the foreseeable future. One of the new facilities under construction is RAON (Rare isotope Accelerator complex for ON-line experiments) by the Rare Isotope Science Project(RISP) of Institute for Basic Science (IBS) in Korea Tshoo:2013voa . RAON will provide an ample amount of exotic neutron- and proton-rich isotopes and enrich various advanced research fields: nuclear physics, astrophysics, and applications in medical, material, and biological science. For a brief summary of RAON physics, see Jeong:2018qvi .

The evolution of rare isotope (RI) science evolves through interplay between three basic ingredients: experiment, simulation and theory. To make a major progress in RI science, we must make use of all the three approaches. A good example is the nuclear symmetry energy. The nuclear symmetry energy originates from the difference in the proton and the neutron numbers in dense nuclear matter or nuclei and it is an important physical quantity in understanding the properties of finite nuclei such as the neutron-skin thickness of heavy nuclei, neutron star properties and the collapse of the core of a massive star. The nuclear symmetry energy may also play an important role in modeling the r-process nucleosynthesis. To study the symmetry energy in a terrestrial environment laboratory, we perform heavy ion collision experiments at RI facilities such as RAON. To connect experimentally observed quantities such as neutron to proton ratios are then compared with nuclear transport model calculations. The two transport models we are using in this study, DaeJeon Boltzmann-Uehling-Uhlenbeck (DJBUU) and Sindong Quantum Molecular Dynamics (SQMD), are general purpose models for low-to-intermediate energy heavy ion collisions but the primary thrust for developing them has been to support physics outcome at RAON.

In this work we compare features of BUU- and QMD-type transport models in general and specifically DJBUU and SQMD by studying fragment productions from the both models. For this we study fragment productions in ²⁰⁸Pb+⁴⁰Ca and ²⁰⁸Pb+⁴⁸Ca reactions at $50$ and $100$ AMeV with the impact parameters $b=0,~{}3,~{}6$ fm. Since the most direct comparison of different transport models is to compare primary fragments before using an afterburner in the simulations, we focus on the primary fragments from DJBUU and SQMD. For fragment production simulations in heavy ion reactions, we refer to Mocko:2008rj ; Gaitanos:2007mm ; Napolitani:2009pc ; Liu:2014poa .

Transport theory describes the dynamics of nuclear collisions at Fermi/intermediate energies semi-classically. The key quantity is the phase space density (coordinate and momentum distribution). Transport simulations in general take the following steps: initialization of the projectile and target nuclei, propagation of the (test) particles, hadron-hadron scatterings with Pauli blocking.

There are two complimentary transport model approaches. The first one is the BUU-type models whose goal is to calculate the one-body phase space distribution. Due to the test-particle ansatz and the mean-field potential, the BUU-type model is deterministic and contains no fluctuations except the numerical ones. The second one is the QMD-type models that are based on an N-body Hamiltonian. As the QMD models do not use the test particle method, they are better suited for event-by-event analysis of heavy ion collisions.

In the next section, we will briefly describe the DJBUU model and the SQMD model. For more details on these models we refer to DJBUU2016 ; Kim:2020sjy for DJBUU and Kim:2017hmt for SQMD. For a recent review of most of the transport models that are currently being used in low-to-medium energy heavy ion collisions, see TMEP:2022xjg .

In this section, we compare the “biggest fragments (BFs)” in DJBUU and SQMD simulations. Here, the BF means the fragment that has the largest mass number in each simulated event and the BFs means the aggregation of the BF from each simulation. As described in the previous section, DJBUU and SQMD use different methods to define the fragments.

We recall that DJBUU uses the test-particle method. In this method, each simulated event contains an ensemble of events. When calculating physical quantities such as baryon number densities, we effectively average over the ensemble in each time step. The BF in DJBUU means the largest cluster formed near the center of a collision at the end of the simulation time, and the BFs in our study is the collection of the BF from ten runs. While, the BF in SQMD is the fragment which has the largest mass number among fragments at the end of the simulation time, and the BFs are the aggregation of such a BF from ten-thousand runs.

We simulate ²⁰⁸Pb+⁴⁰Ca and ²⁰⁸Pb+⁴⁸Ca reactions at $50$ and $100$ AMeV with the impact parameters $b=0,3,6$ fm. Both in DJBUU and SQMD, the initialized projectile and target are placed $16$ fm apart from each other. Here, 16 fm is the distance between the center of the projectile and target. Then, we do the simulations until $t=300$ fm/c. In the case of DJBUU, we use one hundred test particles for each nucleon and perform ten runs. We perform ten thousand runs for the SQMD simulations and also check some of our results with twenty-thousand runs to confirm that the results are robust against the number of runs. Since we are interested only in the central region we plot up to $20$ fm in each of the transverse directions.

In Fig. 1, the density distributions of three different systems are shown up to $300$ fm/c simulation time; from the top to bottom, ²⁰⁸Pb+⁴⁰Ca at $E_{\rm beam}$ = 50 AMeV, ²⁰⁸Pb+⁴⁰Ca at $E_{\rm beam}$ = 100 AMeV and ²⁰⁸Pb+⁴⁸Ca at $E_{\rm beam}$ = 50 AMeV. We observe that the ²⁰⁸Pb+⁴⁸Ca and ²⁰⁸Pb+⁴⁰Ca reactions at $E_{\rm beam}$ = 100 AMeV show a similar trend and therefore we plot only the case with ⁴⁰Ca target at $E_{\rm beam}$ = 100 AMeV. For comparison, the results of DJBUU and SQMD are plotted alternatively, and contours for different impact parameters are displayed, $b$ = 0 fm in the first two rows, $3$ fm in the middle, $6$ fm in the last. Here, the colors denote baryon number density, and the density is averaged for whole simulation events.

It can be seen from Fig. 1 that when $E_{\rm beam}$ = 50 AMeV, the density has a maximum value near $t=40$ fm/c. When $E_{\rm beam}$ = 100 AMeV, it has the highest value near $t=30$fm/c, as expected. A significant difference between DJBUU and SQMD is the central density. Generally, the BUU-type models have higher stability than QMD-type models, which contributes to the difference. Also, the difference is partly attributed to the equation of state of nuclear matter adopted in DJBUU and SQMD.

In Table 1, the BFs from DJBUU and SQMD simulations are shown. All the BFs from the ten runs are presented for DJBUU; while the most abundantly produced three BFs are listed for SQMD. It can be seen from Table 1 that as the beam energy increases from $50$ AMeV to $100$ AMeV, the difference in the produced BFs between DJBUU and SQMD, especially when $b=0$ fm, becomes significant. This is partly because the difference in the equations of state adopted in DJBUU and SQMD is more pronounced at higher beam energies since the maximum density becomes larger as the bean energy increase. For example, the symmetry energy used in DJBUU and SQMD can be written as

where $\alpha_{I}$ is the isospin asymmetry defined by $\alpha_{I}=(\rho_{n}-\rho_{p})/(\rho_{n}+\rho_{p})$. In this study $\sigma=1$ and $\sigma=0.7$ are used for DJBUU Liu:2001iz and SQMD, respectively. We will discuss the symmetry energy more at the end of this section. Also, since the beam energy ($100$ AMeV) would be high enough to break nuclei into small pieces, the different definitions of the BF in DJBUU and SQMD can be more pronounced at the higher beam energy. It can be partly responsible for the difference in the BF distributions.

Another trend in Table 1 is that with a fixed beam energy the difference in the produced BFs between DJBUU and SQMD increases as the impact parameter decreases. This is related to the collision dynamics and the equation of state. At the central collisions ($b=0$ fm), almost all the nucleons participate in the collisions. Therefore the bulk of the system evolution depends sensitively on the equation of state. At the peripheral collisions ($b=6$ fm), however, substantial number of nucleons do not participate in the collision directly. Hence, the part of the system that can feel the equation of state is relatively smaller.

To investigate some features of the BFs produced in SQMD, we show the proton and neutron distributions of the BFs in SQMD for ²⁰⁸Pb+⁴⁰Ca and ²⁰⁸Pb+⁴⁸Ca reactions at $E_{\rm beam}$ = 100 AMeV in Fig. 2 and Fig. 3, respectively. From the left to the right, we plot the distributions with the impact parameters 0, 3 and 6 fm. The colors denotes the total number of the produced BF in ten-thousand simulation events. A trend we can observe in Fig. 2 and Fig. 3 is that for the central collisions ($b$ = 0 fm), the ranges of proton and neutron numbers are relatively wide, while those become narrow for the peripheral collisions ($b$ = 6 fm). One of the interesting results is that, even though the neutron number of the system is increased and the bombarding energy in the center of mass frame is decreased from the reaction ²⁰⁸Pb+⁴⁰Ca to ²⁰⁸Pb+⁴⁸Ca, the proton and neutron numbers of the BFs for ²⁰⁸Pb+⁴⁸Ca reactions are smaller than those for ²⁰⁸Pb+⁴⁰Ca reactions at $E_{\rm beam}$ = 100 AMeV. One way to understand this phenomenon is the role of the nuclear symmetry energy. This effect is more significant in the ²⁰⁸Pb+⁴⁸Ca reactions than ²⁰⁸Pb+⁴⁰Ca reactions. In a neutron-rich matter, the symmetry energy pushes out the neutrons and so disturbs the formation of large fragments. To check the effect of the symmetry energy, we calculate the baryon density and the isospin asymmetry $\alpha_{I}$ as a function of time at the collision center and show them in Fig. 4. Note that as in Eq. (6), in the expression of the energy of a system, the symmetry energy $E_{\rm sym}$ comes with the isospin asymmetry as $E_{\rm sym}\alpha_{I}^{2}$. As can be seen in the right panel of Fig. 4, the isospin asymmetry in the ²⁰⁸Pb+⁴⁸Ca reaction is overall bigger that the one in the ²⁰⁸Pb+⁴⁰Ca reaction. Therefore, we expect that the role of the symmetry energy is more pronounced in the ²⁰⁸Pb+⁴⁸Ca reaction, resulting in smaller BFs compared to the ²⁰⁸Pb+⁴⁰Ca reaction.

In this work, we have compared two transport model codes DJBUU and SQMD. Both models have been developed mainly for RAON physics where one of the main goals is to produce rare isotope. To this end, we have investigated the fragment productions in ²⁰⁸Pb+⁴⁰Ca and ²⁰⁸Pb+⁴⁸Ca reactions at $50$ and $100$ AMeV with the impact parameters $b=0,~{}3,~{}6$ fm, focusing on primary fragments from DJBUU and SQMD. We observed that overall the two models produce similar fragments. For example, though the fragments with ⁴⁸Ca are a bit smaller than those with ⁴⁰Ca, the produced fragments from the both models are in the range of ${\rm A}=162\sim 168$ with ${\rm Z}=70\sim 74$, where A is the atomic number and Z is the charge number. We found a noticeable difference between DJBUU and SQMD at $E_{\rm beam}=100$ AMeV with $b$ = 0 fm, which can be attributed to the difference in the equation of state used in the models and also to the difference in stability inherent to the BUU-type and QMD-type models. After some improvements such as including a surface term (time-dependent width of the nucleons) in DJBUU (SQMD), DJBUU and SQMD will play important roles in studying heavy ion reactions and related physics.