The effects of pre-equilibrium emission and secondary decay on the determination of freeze-out volume at intermediate energies

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Hui-Xiao Duan, Dong-Hai Zhang, Fan Zhang and Hai-Shun Wu. The effects of pre-equilibrium emission and secondary decay on the determination of freeze-out volume at intermediate energies[J]. Chinese Physics C. doi: 10.1088/1674-1137/abec39
Hui-Xiao Duan, Dong-Hai Zhang, Fan Zhang and Hai-Shun Wu. The effects of pre-equilibrium emission and secondary decay on the determination of freeze-out volume at intermediate energies[J]. Chinese Physics C.  doi: 10.1088/1674-1137/abec39 shu
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The effects of pre-equilibrium emission and secondary decay on the determination of freeze-out volume at intermediate energies

  • 1. Key Laboratory of Magnetic Molecules and Magnetic Information Materials Ministry of Education, School of Chemistry and Material Science, Shanxi Normal University, Linfen 041004, China
  • 2. Institute of Modern Physics, Shanxi Normal University, Linfen 041004, China
  • 3. Department of Electronic Information and Physics, Changzhi University, Changzhi 046011, China

Abstract: The effects of pre-equilibrium emission and secondary decay on the determination of freeze-out volume are investigated by using the isospin-dependent quantum molecular dynamics model accompanied by the statistical decay model GEMINI. Small-mass projectiles and large-mass targets with central collisions are studied at intermediate energies. It is found that the proton yields of pre-equilibrium emission are smaller than those of secondary decay. However, the determination of freeze-out volume from the proton yields is more easily affected by pre-equilibrium emission. It is also found that the percent of proton yields in freeze-out stage is approximately 50%.

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    I.   INTRODUCTION
    • Statistical equilibrium is a basic theoretical assumption in many theoretical studies [1-5]. In these studies, the primary focus is on phase transition [1, 2] and nuclear multifragmentation [3-5]. Many experimental studies have been devoted to testing this theoretical assumption [6-12]. These studies were motivated by the good agreements obtained between various observable trends related to the asymptotically resulting fragments and various multifragmentation models. In the framework of these multifragmentation models, a freezing out process is very important. For example, both statistical and chemical equilibrium are assumed between produced fragments in a statistical multifragmentation model (SMM) [3]. A hot source with mass and charge ($ A_{0} $, $ Z_{0} $) at temperature T expands to a freeze-out volume. Fragments are not allowed to overlap each other, and they are placed into a volume V (freeze-out volume). The source size, its excitation energy, and its volume are basic quantities of the statistical models. To obtain these data, one must carry out indirect evaluations via comparisons between experimental data and statistical model predictions. However, the correlation between experimental data and bulk properties is far from simple. When the system reaches the freeze-out stage, the primary fragments may be excited. Understanding of the multifragmentation phenomenon is difficult due to the decay of the primary fragments. The detected fragments are cold remnant fragments. To facilitate comparison with experimental data, the SMM not only needs the information of the excited sources, but also needs the information of the primary fragments. In fact, different primary configurations could lead to the same final results, because of compensatory effects between the primary and secondary emission mechanisms [13]. The difficulty of determining the freeze-out volume is reflected in the different values. Different freeze-out volumes have been obtained in many studies, ranging from 2.5$ V_{0} $ to 9$ V_{0} $ [14-16], where $ V_{0} $ is the volume corresponding to normal nuclear matter density.

      Heavy-ion collisions are the only way to study the properties of hot nuclei [17]. In such collisions, the dynamical process can be divided into three stages. (i) The system driven by intensive interactions between nucleons evolves toward thermalization. Fast particles leave the system. The time interval of this stage is approximately a few tens of fm/c. The particle emission of this stage is pre-equilibrium emission. (ii) The hot nuclear residue expands and breaks up into hot primary fragments. The produced fragments are in the freeze-out stage. (iii) The primary fragments will de-excite by emitting particles and gamma rays to the final ground states.

      In the present work, the focus is on the freeze-out volume. To determine the freeze-out volume, experimental studies are extremely valuable. The extraction of volume from the measured yields of particles is discussed, and, using the yields and quantum fluctuations of light charged particles (Z $ \leqslant $ 2 LCP), the temperature and density are studied [18]. However, the sources of LCPs are complex. The LCPs, which are measured by experiments, can originate from several sources: (i) pre-equilibrium emission, (ii) the composite excited system at the freeze-out stage, and (iii) sequential decay of excited fragments. LCPs cannot come from an approximate single system. Therefore, the effects of pre-equilibrium emission and sequential decay on the determination of freeze-out volume are studied herein.

    II.   MODEL AND METHODS
    • In this work, an attempt is made to study the influence of pre-equilibrium emission and secondary decay on the determination of freeze-out volume via the isospin-dependent quantum molecular-dynamics ($ {\rm{IQMD}} $) model incorporating the statistical decay model GEMINI [19-21]. In the present model, each nucleon is represented by a coherent state of a Gaussian wave packet

      $ \phi_{i}({{r}}_{i},t) = \frac{1}{(2\pi L)^{3/4}}{\rm e}^{\textstyle-\frac{[{{r}}_{i}-{{r}}_{i0}(t)]^{2}} {4L}}{\rm e}^{\textstyle{\rm i}\frac{{{r}}_{i}\cdot {{p}}_{i0}(t)}{\hbar}}, $

      (1)

      where ${{r}}_{i0}$ and ${{p}}_{i0}$ are the average values of the position and momentum of the ith nucleon, and L is related to the extension of the wave packet. L equals $ \sigma_{r}^{2} $, where $ \sigma_{r} $ is the width of wave packet. The width of wave packet affects the stability of nuclei at their ground state and the charge distribution of fragments in heavy-ion collisions. If the width of wave packet is smaller than 1 fm, the “spurious” emission number of nucleons sharply increase with the decrease of the width of wave packet. Too large wave-packet width causes the central densities to be obviously higher than the normal density [22]. When the width of wave packet is 1.1 fm, the stable nucleus can be produced. The corresponding value of L is 1.21 $ {\rm{fm}}^{2} $. The total N-body wave function is assumed to be the direct product of these coherent states. Through a Wigner transformation of the wave function, the one-body phase-space distribution function for N-distinguishable particles is given by

      $ f({{r}},{{p}},t) = \sum\limits_{i = 1}^{n}\frac{1}{(\pi\hbar)^{3}} {\rm e}^{\textstyle-\frac{[{{r}}-{{r}}_{i0}(t)]^{2}} {2L}}{\rm e}^{\textstyle-\frac{[{{p}}-{{p}}_{i0}(t)]^{2}\cdot 2L}{\hbar^{2}}}. $

      (2)

      The time evolutions of the nucleons in the system under the self-consistently generated mean field are governed by Hamiltonian equations of motion

      $ \dot{{{r}}}_{i0} = \nabla_{{{p}}_{i0}}H, \dot{{{p}}}_{i0} = -\nabla_{{{r}}_{i0}}H, $

      (3)

      where the Hamiltonian H is expressed as

      $ H = E_{\rm kin}+U_{\rm Coul}+\int V(\rho){\rm d}r. $

      (4)

      Here, the first term $E_{\rm kin}$ is the kinetic energy, the second term $U_{\rm Coul}$ the Coulomb potential energy, and the third term the local nuclear potential energy. Each term of the local potential energy-density function $ V(\rho) $ in the present work is

      $ V_{\rm sky} = \frac{\alpha}{2}\frac{\rho^{2}}{\rho_{0}} +\frac{\beta}{\gamma+1}\frac{\rho^{\gamma+1}}{\rho_{0}^{\gamma}}, $

      (5)

      $ V_{\rm sur} = \frac{g_{\rm sur}}{2}\frac{(\nabla\rho)^{2}}{\rho_{0}} +\frac{g_{\rm sur}^{\rm iso}}{2}\frac{(\nabla\rho_{n}-\nabla\rho_{p})^{2}}{\rho_{0}}, $

      (6)

      $ V_{\rm mdi} = g_{\tau}\frac{\rho^{8/3}}{\rho^{5/3}_{0}}. $

      (7)

      Here, $V_{\rm sky}$, which includes the two-body interaction term and the three-body interaction term, describes the saturation properties of nuclear matter. $V_{\rm sur}$ is the surface term that describes the surface of finite nuclei. $V_{\rm mdi}$ is the momentum-dependent interaction term. The symmetry potential energy-density functional $V_{\rm sym}$ is

      $ V_{\rm sym} = \frac{C_{\rm sym}}{2}\frac{(\rho_{n}-\rho_{p})^{2}} {\rho_{0}}. $

      (8)

      The parameters used in this study are $ \alpha $ = −168.40 MeV, $ \beta $ = 115.90 MeV, $ \gamma $ = 1.50, $g_{\rm sur}$ = 92.13 MeV $ {\rm{fm}}^{2} $, $g_{\rm sur}^{\rm iso}$ = -6.97 MeV $ {\rm{fm}}^{2} $, $C_{\rm sym}$ = 38.13 MeV, and $ g_{\tau} $ = 0.40 MeV. The corresponding compressibility is 271 MeV [23]. The fragments are identified by a minimum spanning-tree algorithm. The nucleons with relative distance $ R_{0} $ $ \leqslant $ 3.5 fm and momentum $ P_{0} $ $ \leqslant $ 250 MeV/c belong to a fragment.

      In this study, the dynamical description is used not only for the excitation stage but also for intermediate-mass-fragment (IMF) emission. After the excitation stage, the time evolution in the IQMD code continues until the excitation energy of the heaviest hot fragment decreases to a certain value $ E_{ \rm{stop}} $ in each event. If the excitation energy is lower than $ E_{ \rm{stop}} $ [21], the IQMD calculation stops and the charge, mass, excitation energy and momentum of each hot fragment are recorded. The outputs of the IQMD code are the hot fragments. To obtain the cold fragments, emission of light particles from hot fragments is performed using the statistical code GEMINI.

      To study the freeze-out volume, the central collisions of small-mass projectiles and large-mass targets are used to produce hot nuclei. For such reaction systems, there are enough nucleons in the overlap volume to experience enough collisions needed for hot-nuclei thermalization [24]. To reduce the effects of mass range of the hot nuclei on proton production, a narrow mass number range of the hot nuclei is required to be 190 $ \leqslant $ A $ \leqslant $ 200. The selection method of the hot nuclei is the same as that given in Ref. [25]. It is worth noting that using the hot nuclei with mass number range 190 $ \leqslant $ A $ \leqslant $ 200 is only to satisfy the requirement of event number. In this work, using the reaction system $ ^{36} {\rm{Ar}} $ + $ ^{197} {\rm{Au}} $ with beam energies 50, 60 and 70 MeV/u, the hot nuclei have a wide mass number range (approximately 160-230). If one selects another mass number range, one has to calculate more events because of low production of the hot nuclei.

      Using the hot nuclei, the freeze-out temperatures can be calculated by the isotope-yield-ratio method of Albergo $ et $ $ al. $ [26, 27]. The corresponding expression is

      $ T_{\rm BeLi} = 11.3\;{\rm{MeV}}/ {\ln}\left(1.8\frac{Y_{^{9}\rm Be}/Y_{^{8}\rm Li}} {Y_{^{7}\rm Be}/Y_{^{6}\rm Li}}\right). $

      (9)

      Using Eq. (9), one can only study the apparent temperature (${T}_{ \rm{app}}$). Cold fragments are used to calculate ${T}_{ \rm{app}}$. However, the primary fragments are normally excited at the freeze-out stage. To calculate the freeze-out temperature (${T}_{0}$), one can connect ${T}_{0}$ and ${T}_{ \rm{app}}$ by a linear approximation ${T}_{0}$ = $1.2 {T}_{ \rm{app}}$ [8].

      At the freeze-out stage, protons (p), neutrons (n), tritium, etc., follow Fermi statistics, while deuterium, $ \alpha $, etc., should follow Bose statistics. Employing distributions of particles, the temperature and density of the nuclear system have been studied [28, 29]. In this work, only protons that are abundantly produced in the collisions are studied. In the freeze-out stage, the density of protons can be determined via Fermi distribution

      $ \rho_{Fp} = \frac{4\pi(2m)^{3/2}}{h^{3}}\int_{0}^{\infty} \frac{ \varepsilon^{1/2}{\rm d}\varepsilon}{{\rm e}^{\textstyle\frac{\varepsilon-\mu}{T}}+1}, $

      (10)

      where m is the mass of the protons.

      The multiplicity for a proton can be given by

      $ N = \frac{4\pi V (2m)^{3/2}}{h^{3}}\int_{0}^{\infty} \frac{ \varepsilon^{1/2}{\rm d}\varepsilon}{{\rm e}^{\textstyle\frac{\varepsilon-\mu}{T}}+1}, $

      (11)

      $ \langle(\Delta N)^{2}\rangle = T\bigg(\frac{\partial N}{\partial \mu} \bigg)_{T,V}. $

      (12)

      Substituting Eq. (11) into Eq. (12), one obtains

      $ \langle (\Delta N)^{2} \rangle = \frac{4\pi V (2m)^{3/2}}{h^{3}}\int_{0}^{\infty} \frac{{\rm e}^{\textstyle\frac{\varepsilon-\mu}{T}}\varepsilon^{\textstyle\frac{1}{2}} {\rm d}\varepsilon} {\left({\rm e}^{\textstyle\frac{\varepsilon-\mu}{T}}+1\right)^{2}}. $

      (13)

      The multiplicity fluctuation for a proton (MFp) can be given by [30]

      $ \frac{\langle(\triangle N)^{2}\rangle}{N} = \frac{\displaystyle\int_{0}^{\infty}\varepsilon^{\textstyle\frac{1}{2}}{\rm d}\varepsilon \dfrac{{\rm e}^{\textstyle\frac{\varepsilon-\mu}{T}}}{\left({\rm e}^{\textstyle\frac{\varepsilon-\mu}{T}}+1\right)^{2}}} {\displaystyle\int_{0}^{\infty}\varepsilon^{\textstyle\frac{1}{2}}{\rm d}\varepsilon \dfrac{1}{{\rm e}^{\textstyle\frac{\varepsilon-\mu}{T}}+1}}. $

      (14)

      The MFp values can be calculated by IQMD code or measured by experiment. Using the MFp values, the integral variable $ \mu $ can be solved numerically by Eq. (14). The freeze-out temperature can be calculated by Eq. (9). Substituting $ \mu $ and freeze-out temperature into Eq. (10), one can obtain the density of protons at freeze-out. The freeze-out volume V can be calculated by $ N/\rho_{Fp} $, where N is the proton average multiplicity at freeze-out.

    III.   RESULTS AND DISCUSSION
    • To calculate freeze-out volume, one should bypass the effects of pre-equilibrium emission and sequential decay. To define equilibrium and freeze-out moment, the time evolution of the quadrupole momentum and IMFs, i.e., fragments with Z $ \geqslant $ 3 are shown in Figs. 1(a) and 1(b), respectively. They are calculated for the system $ ^{36} {\rm{Ar}} $ $ + $ $ ^{197} {\rm{Au}} $ at 50 MeV/u bombarding energy and center collisions. The quadrupole moment of the momentum distribution is given by

      Figure 1.  (a) Time evolution of quadrupole momentum for maximum mass cluster and (b) IMF multiplicity for reaction system.

      $ Q_{p} = \int(2p_{z}^{2}-p_{x}^{2}-p_{y}^{2}) f({{r}},{{p}},t){\rm d}{{r}}{\rm d}{{p}}. $

      (15)

      The quadrupole moment of the momentum will be calculated by all nucleons that belong to the largest cluster in the center of mass of the largest cluster.

      From Fig. 1(a), it can be seen that the quadrupole increases quickly at 10 fm/c. At this moment, projectile and target are in contact with each other. During approximately 80 fm/c, the quadrupole recovers to zero again. The momentum of nucleons reaches an isotropic distribution at 100 fm/c [24]. Thus, the protons emitted before 100 fm/c comprise pre-equilibrium emission. With the change of reaction time, the hot nuclei expand and break into primary fragments. The hot nuclei reach the freeze-out stage, and the freeze-out moment can be estimated from the time evolution of the multiplicity of IMFs. It can be seen that the multiplicity of IMFs ends its variation at approximately 400 fm/c. The protons produced after 400 fm/c constitute secondary decay. Therefore, in this paper, protons are divided into four parts: (i) pre-equilibrium emission (PEp), (ii) protons produced in freeze-out stage (FOp), (iii) secondary decay (SDp), and (iv) PEp+FOp+SDp (TOTAL).

      In the following discussion, the focus is on the moderate excitation energy range (6-8 MeV/nucleon). To produce moderate excitation hot nuclei, three beam energies are selected, i.e., 50, 60, and 70 MeV/u. The reaction system is $ ^{36} {\rm{Ar}} $ + $ ^{197} {\rm{Au}} $. At low excitation energy, evaporation will be the main de-excitation process. The protons come from the surface of the hot nuclei, not from the freeze-out volume. At moderate excitation energy (approximately 7 MeV/nucleon), an IMF has a peak value [9]. The hot nuclear system is fully broken. The nucleus breaks into pieces, large fragments representing the liquid and very small ones representing the vapor. The Fermi-gas approach is well justified for this weakly interacting system. More importantly, the nuclear liquid-gas phase transition may occur at moderate excitation energy [10, 31]. Figure 2 shows the proton yield of different stages for different excitation energies. It can be seen that the proton multiplicity of pre-equilibrium emission is approximately 4. Most of the protons are produced at freeze-out and via the secondary decay process. The production of secondary decay is approximately 10. Thus, most of the protons are produced by a de-excitation process. It can also be seen from Fig. 2 that the proton production exhibits a slow increase with increasing excitation energy.

      Figure 2.  (color online) Proton yield in different stages (PEp, FOp, SDp and TOTAL) as a function of excitation energy.

      In heavy-ion collisions, the hot nuclei will de-excite by disintegrating. The de-excitation process can be light-particle (Z $ \leqslant $ 2) evaporation or IMF (Z $ \geqslant $ 3) emission. The competition of the two modes determines the de-excitation process. The fragment charge distribution reflects the de-excitation process of hot nuclei. If the excitation energy is low, the charge distribution has a "U"-shaped characteristic corresponding to evaporation event. When the excitation energy is high, the charge distribution has a rapidly decreasing charge distribution, which is a corresponding vaporization event [11]. Figure 3 shows the charge distribution as a function of the charge number Z of the fragments. The experimental data are taken from Ref. [12]. The square symbols show the calculated values. The behavior of the calculated charge distribution is generally in agreement with the data. Therefore, our calculation can appropriately describe the de-excitation process of the hot nuclei. The main difference in the calculations compared with the experimental data is overestimation of the proton yields. However, for the following discussion, the relative yields of proton at different stages are more important. The main focus of this study is on the effects of proton relative yield.

      Figure 3.  (color online) Charge distribution N(Z) in central $ ^{197} {\rm{Au}} $ + $ ^{197} {\rm{Au}} $ collisions at 35 MeV/u. Experimental data are taken from Ref. [12].

      The freeze-out temperatures are shown in Fig. 4(a). Calculation points are plotted for 0.2-MeV/nucleon-wide bins in excitation energy per nucleon. In Fig. 4(b), the MFp versus excitation energy per nucleon is plotted. The proton yield of secondary decay is approximately 3 times that of pre-equilibrium emission (see Fig. 2). However, the normalized fluctuations are more easily affected by pre-equilibrium emission. Compared to a pre-equilibrium emission process, a secondary decay process is more complex. There are many de-excitation routes for secondary decay and therefore the competition among the different de-excitation routes increases the fluctuation of proton production.

      Figure 4.  (color online) (a) Freeze-out temperatures and (b) multiplicity fluctuation for proton vs excitation energy per nucleon $ E^{*}/A $.

      Freeze-out volume versus excitation energy is plotted in Fig. 5. The freeze-out volume can be calculated by four groups of protons (FOp+PEp, FOp, FOp+SDp, and TOTAL). The freeze-out volume calculated by FOp is approximately 2 times that of FOp+PEp. The freeze-out volumes are almost same between TOTAL and FOp+ PEp. The difference in freeze-out volume between FOp and FOp+SDp is smaller than that between FOp and FOp+PEp.

      Figure 5.  (color online) Freeze-out volume calculated by protons produced at different stage as a function of excitation energy.

      The study of freeze-out volume is helpful to better understand the multifragmentation process. The multifragmentation can provide a possibility for investigating the nuclear liquid-gas transition. To obtain the freeze-out volume and understand the freeze-out concept, experimental study is indispensable. However, the particles detected by experiment include the information of pre-equilibrium and secondary decay. The determination of freeze-out information is affected by the interference of pre-equilibrium and secondary decay. Therefore, it is necessary to study the effects of pre-equilibrium emission and secondary decay on the determination of freeze-out information. In this work, the freeze-out volume is studied by the multiplicity fluctuation for a proton. Calculations show that pre-equilibrium emission and secondary decay will affect the determination of freeze-out information. Using the quantum fluctuations of proton to study freeze-out volume, the effect of pre-equilibrium emission is more obvious. However, the present results that are calculated by the IQMD model depend on the model parameters. Using different model parameters may obtain different results. Therefore, the effects of different model parameters on the determination of freeze-out volume should be studied in the future.

    IV.   CONCLUSIONS
    • A study of freeze-out volume from the yields of protons emitted in heavy-ion collisions is presented in this paper. The study of freeze-out volume is very important for understanding the multifragmentation process. The multifragmentation opens a possibility for investigating the liquid-gas coexistence region. To obtain the freeze-out volume and understand the freeze-out concept, experimental study is indispensable. However, the particles which are detected by experiment include the information of pre-equilibrium emission and secondary decay. The determination of freeze-out information is affected by the interference of pre-equilibrium and secondary decay.

      Owing to the effects of pre-equilibrium emission and secondary decay, the percentage of protons in the freeze-out stage is only approximately 50%. In the de-excitation process, the proton yield produced by secondary decay is approximately 3 times that of pre-equilibrium emission. However, the normalized fluctuations are more easily affected by pre-equilibrium emission because the secondary decay process is more complex. The competition among the different de-excitation routes increases the fluctuation of proton production. Therefore, using the multiplicity fluctuation of proton to study freeze-out volume, one should pay more attention to pre-equilibrium emission.

      It should be stressed that the present results are based on specific IQMD model parameters. If one uses different model parameters, one may obtain the different results. Therefore, these effects should be studied in the future.

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