Decay modes of highly excited nuclei

NUCLEAR PHYSICS AND INTERDISCIPLINARY RESEARCH

Decay modes of highly excited nuclei

Zhen-Fang Zhang，

De-Qing Fang，

Yu-Gang Ma

Nuclear Science and Techniques

Vol.29, No.6

Article number 78

Published in print 01 Jun 2018

Available online 25 Apr 2018

DOI：10.1007/s41365-018-0427-8

34100

The deexcitation of single excited ¹¹²Sn nuclei at T = 1–30 MeV is simulated using the isospin-dependent quantum molecular dynamics (IQMD) model and GEMINI model. The fragmentation mechanism, critical behavior, and kinematic characteristics are investigated within these two models. The results show that the IQMD model can be applied to the analysis of fragmentation processes, critical points, and slope temperature extraction. The results of IQMD are generally consistent with experimental 〈M_IMF〉- Z_bound data. However, GEMINI can reproduce the experimental data better than IQMD for isotopic distributions.

Excited nucleiIQMD modelGEMINI modelDeexcitation

1 Introduction

Nuclear fragmentation in heavy-ion collisions is one of the most challenging subjects. Initially, the proton-induced reactions were found to produce fragments of all mass numbers [1]. Subsequently, the multifragmentation phenomenon was discovered. This phenomenon can be described as many particles heavier than α being produced at the same time in a single collision at energies above 30 MeV/nucleon [2]. Since then, many experiments and theoretical works have been dedicated to exploring the reaction mechanism. However, the phenomenon is still not fully understood [3-11].

In intermediate-energy heavy-ion collision processes, at first, the nuclear system is compressed and heated. Then, the system breaks up into several fragments, including light particles, during the expansion stage. Many theoretical models have been proposed to describe this reaction scenario. A brief introduction to these models can be found in Ref. [12]. One may treat it as a statistical process, which means that before the system disintegrates into lightly charged particles, neutrons, and IMFs (intermediate mass fragments) simultaneously, statistical equilibrium is reached in some subsystems during the expansion. This kind of equilibrium can be maintained until each subsystem becomes deexcited and the whole system reaches the freeze-out stage [3]. The final products are the sum of the decay results from all the subsystems.

To further understand this physical process, the isospin-dependent quantum molecular dynamics model (IQMD) [13, 14] and the GEMINI statistical model [15] are adopted to investigate the decay process. The combination of IQMD and GEMINI is applied in many articles to obtain isotopic distributions which are close to experimental data. In this combination, the expansion dynamics are done by IQMD and GEMINI is just used for the decay of the fragments produced by IQMD. J. Hubele et al. [16] also emphasized that it is necessary to have a complete dynamical treatment of the decay process. To compare the decay processes between the two models, it is interesting to study the deexcitation of a single excited nucleus within IQMD and GEMINI. In this way, the influences of the parameters and approximations employed in these models could be reduced [17]. By comparing them with the experimental data, it can be learned how well the IQMD and GEMINI models can describe the deexcitation processes of excited nuclei.

This paper is organized as follows: a brief introduction to the IQMD and GEMINI models is given in Sect. 2. In Sect. 3, the calculated results and discussions are presented. Finally, the conclusions are given in Sect. 4.

2 Model description

2.1 Dynamical model: IQMD

The QMD model is a many-body theory that describes two-nuclei collisions from intermediate to relativistic energies. The IQMD model, which is based on QMD, has been successfully applied to heavy-ion collisions with consideration of the isospin degree-of-freedom for nucleons [18, 19]. The nucleons are represented by Gaussian wave packets with width L (L = 2.16 fm²). When the nucleons propagate in the mean field, they are governed by the following potential:

\begin{array}{l} U (ρ, τ_{z}) = & α (\frac{ρ}{ρ_{0}}) + β {(\frac{ρ}{ρ_{0}})}^{γ} + \frac{1}{2} (1 - τ_{z}) V_{c} \\ + C_{sym} \frac{(ρ_{n} - ρ_{p})}{ρ_{0}} τ_{z} + U^{Yuk} + U^{MDI}, \end{array}

(1)

where α = -390.1 MeV, β = 320.3 MeV, and γ = 1.14 are the momentum-dependent nuclear equation of state (EOS) parameters, with an incompressibility of K = 200 MeV [14]. ρ, ρ₀, ρ_n, and ρ_p are the total, normal, neutron, and proton densities, respectively. τz (= 1 for neutron or -1 for proton) is the zth component of the isospin degree-of-freedom. V_c is the Coulomb potential and C_sym is the strength of the symmetry energy, which equals 32 MeV. The last two parameters, U^Yuk and U^MDI, represent the Yukawa potential and the momentum dependent interaction, respectively.

In this work, a single excited nuclear source is simulated at different temperatures. Usually, the initial nucleus is in the state T = 0 (T is the temperature), corresponding to the ground state. The moment distribution is calculated with the Fermi gas approximation $P_{F}^{i} (\vec{r}) = ℏ {[3 π^{2} ρ_{i} (\vec{r})]}^{1 / 3}$ after its initial coordinate is sampled by the Monte Carlo method. ρi (i = n or p) is the local density of neutrons or protons, respectively. To study the states of nuclei with T > 0, we replace the previous moment distribution with the Fermi–Dirac distribution,

n (e_{k}) = \frac{g (e_{k})}{e^{\frac{e_{k} - μ}{T}} + 1},

(2)

where $e_{k} = \frac{p^{2}}{2 m}$ is the state kinetic energy. p and m represent the momentum and mass of the nucleon, respectively. $g (e_{k}) = \frac{V}{2 π^{2}} {(\frac{2 m}{ℏ^{2}})}^{\frac{3}{2}} \sqrt{e_{k}}$ is the state density. $V = \frac{4}{3} π r^{3}$ , with r=r₀A^1/3, is the volume of the source, and the source radius r is determined by the radius parameter r₀ and nuclear mass A. μ is the chemical potential, which is determined by the following integral equation:

\frac{1}{2 π^{2}} {(\frac{2 m}{ℏ^{2}})}^{\frac{3}{2}} \int_{0}^{\infty} \frac{\sqrt{e_{k}}}{e^{\frac{e_{k} - μ_{i}}{T}} + 1} d e_{k} = ρ_{i} .

(3)

In this paper, the simulated temperatures are from 1 MeV to 30 MeV and the excited nucleus evolves in the mean field without collisions after the initialization [20].

2.2 Statistical model: GEMINI

GEMINI is a Monte Carlo code written in 1986 that follows the decay of a compound nucleus through a series of sequential binary decays like light-particle evaporation and symmetric fission. All possible binary decays are performed until the resulting products cannot decay any further [15]. There are four basic parameters as the input: fragment mass number A, charge number Z, excitation energy E^*, and angular moment L. To study the same excited nuclear source, the excitation energy and angular moment of the hot nucleus are calculated in the IQMD when the initialization is just completed:

{\vec{L}}_{i} = {\vec{R}}_{i} \times {\vec{P}}_{i},

(4)

E^{*} = E_{bind}^{excited} - E_{bind}^{ground} .

(5)

${\vec{R}}_{i}$ and ${\vec{P}}_{i}$ are vectors of the coordinates and momentum, respectively, and i represents the ith nucleon of the nucleus [21]. $E_{bind}^{excited}$ is the binding energy of the excited nucleus and it can be calculated from the IQMD model. The $E_{bind}^{ground}$ is obtained from the nuclear data table [22]. The latest atomic mass evaluation can be seen in Ref. [23].

3 Results and discussion

In this section, the fragmentation mechanisms are first analyzed. Then, the critical behavior and slope temperature are compared between the two models. Finally, the comparison of the results between the calculated and experimental data is discussed. In this work, a nuclear source of normal density with 50 protons and 62 neutrons (¹¹²Sn) is chosen. The simulation in IQMD is calculated up to 200 fm/c and the number of simulated events is 1000 for each temperature point. The fragments are constructed by a coalescence model, in which nucleons with relative distances smaller than 3.5 fm and relative momenta smaller than 300 MeV/c are combined into one cluster [24].

3.1 Fragmentation mechanism

Before comparing the fragmentation phenomena of the two models, it has to be checked whether the time-length of 200 fm/c is sufficient for the nucleus decay in IQMD. This question is investigated by looking at the mean multiplicity and production rate of the IMFs, which is defined as the fragments with 3 ≤ Z ≤ 20.

From Fig. 1, it is found that the 〈M_IMF〉 remains almost the same after 100 fm/c at different temperatures. The production rates of the IMFs become almost zero after 100 fm/c. It indicates that the fragment formation is finished within 100 fm/c and the evolution time of 200 fm/c is enough in IQMD.

Figure 1:

Mean multiplicity of the IMFs (upper panels) and the production rate of the mean multiplicity of the IMFs (bottom panels) at 5, 15, and 25 MeV, respectively. The evolution time is up to 200 fm/c. As the fragment data is output every 10 fm/c in our calculation, the production rate is defined as the variation in the mean multiplicity of the IMFs every 10 fm/c. The production rate at ti is calculated by IMFti-IMFti-1(i=1,2,3...20). IMFti is the mean multiplicity of the IMFs at ti (ti = 0 fm/c, 10 fm/c, 20 fm/c...200 fm/c).

Figure 2 presents the mean multiplicity of the IMFs calculated by IQMD and GEMINI at different temperatures. They have very similar trends at lower temperatures, but the maximum value of the mean IMF multiplicity of IQMD is much bigger than that of GEMINI. The T-value corresponding to the maximum value of the mean IMF multiplicity is also higher for IQMD. In addition, nuclei in IQMD can keep the fragments’ structure, while they are almost completely disassembled into light particles in GEMINI at very high temperatures. The insert graph shows the correlation between the temperature and mean excitation energy calculated by Eq. (5). It is similar to a Fermi gas curve at low excitation energies and shows linear growth at high excitation energies. Thus, the excitation energies of the nuclei used as input for GEMINI at different temperatures are reasonable in this respect.

Figure 2:

Temperature dependence of the mean multiplicity of the IMFs for IQMD (circles) and GEMINI (triangles). The correlation between the excitation energy per nucleon and temperature is shown in the inset graph.

To conveniently compare the fragmentation mechanisms between the two models at different temperatures, three fragmentation phenomena, i.e., spallation, fission, and multifragmentation, are distinguished by the multiplicity M_H of the heavy fragments in a single event: M_H = 1 for spallation, M_H = 2 for fission, and M_H = 0 with at least one fragment of Z > 2 for multifragmentation. Fragments with Z > 20 are defined as heavy fragments [25]. It is well known that the binary decay modes that cause the particles evaporated one-by-one from the excited nuclei are adopted in the GEMINI code. Thus, the multifragmentation that is caused by chemical instability cannot be treated with GEMINI. Here, the multifragmentation only concerns the decay results for GEMINI.

In Fig. 3, the fractions of spallation, fission, and multifragmentation at different temperatures are presented for IQMD and GEMINI. Fission is a rare process that only occurs at lower temperatures. It is interesting to see the opposing trends of the spallation and multifragmentation below T = 10 MeV. There exists an apparent deviation between IQMD and GEMINI in the multifragmentation. The deviation corresponds to the discrepancy in the mean multiplicity of the IMFs at high temperatures, as we can see in Fig. 2. In general, the main fragmentation phenomena are similar in IQMD and GEMINI, which indicates that a statistical model and dynamical model may obtain comparable results for the decay process.

Figure 3:

Fractions of fragmentation phenomena (fission, spallation, and multifragmentation) at different temperatures. The triangles represent the results of GEMINI and the circles correspond to IQMD results.

3.2 Critical behavior

It is a significant topic in nuclear physics to explore the liquid-gas phase transition of finite nuclei that is associated with the nuclear equation of state and clarification of the mechanism when a heated nucleus disassembles [26, 27]. Many signatures have been applied to identify the critical region. The Fisher droplet model is extensively used in the study of multifragmentation and the extracted value of the power-law exponent is reasonable for the critical behavior analysis [28, 29]. We employed a power-law fit for the charge distribution in the range Z = 2–7 for IQMD and Z=1–7 for GEMINI. The effective Fisher-law parameter τ_eff is extracted from the following equation:

d N / d Z \sim Z^{- τ_{eff}} .

(6)

The upper and bottom panels of Fig. 4 show the charge distributions at T = 5, 15, and 25 MeV for GEMINI and IQMD, respectively. The charge distributions become steeper with increasing temperatures and the widths of the charge distributions of GEMINI are narrower than those of IQMD. The fitting curves are consistent with the IQMD data, but the fitting curves for GEMINI are not very good, and the errors of τ_eff are quite large, as presented in Fig. 5. Thus, the charge distributions of GEMINI are not well described by the power-law dependence.

Figure 4:

Charge distributions at T = 5, 15, and 25 MeV. The results of GEMINI are given in the upper panels. The same distributions by IQMD are shown in the bottom panels. The lines are the power-law fits for 2 ≤ Z ≤ 7 (IQMD) and 1 ≤ Z ≤ 7 (GEMINI).

Figure 5:

Extracted τ_eff parameters at different temperatures for GEMINI (triangles) and IQMD (dots).

The overall trend for τ_eff with temperature is that it first falls to a minimum at about 8 MeV for GEMINI and 10 MeV for IQMD and then rises slowly. The minimum values of τ_eff are 1.79 for GEMINI and 2.14 for IQMD, respectively. The Fisher droplet model predicted a reasonable critical exponent ~ 2.21 for the critical behavior [28]. The result of IQMD is closer to the critical exponent. It indicates that the critical behavior is better presented within IQMD model and a possible critical point around T = 10 MeV.

The occurrence of the largest fluctuation indicates that the system undergoes a continuous transition. Campi has suggested using the event-by-event distribution of the logarithm of the largest fragment’s size versus the normalized second moment. The normalized second moment is represented by S₂ with the heaviest fragment removed,

S_{2} = \frac{\sum_{Z_{i} \neq Z_{max}} Z_{i}^{2} \cdot n_{i} (Z_{i})}{\sum_{Z_{i} \neq Z_{max}} Z_{i} \cdot n_{i} (Z_{i})} .

(7)

Zi is the charge number of the ith fragment and ni(Zi) is the multiplicity of fragment Zi. Campi plots are presented in Fig. 6 and they are instructive for analyzing the critical behavior [30].

Figure 6:

Campi plots at different temperatures for GEMINI (upper panels) and IQMD (bottom panels).

There are three selected temperatures for each model. At low temperatures, the upper branch, which represents the liquid phase, is strongly dominant. At T=15 MeV, the lower Z_max, which represents the gas phase, is strongly dominant. The transition from the liquid-dominated branch to the vapor branch then occurs, which indicates that the region of maximal fluctuations is in the range of 5 to 15 MeV.

To further identify the critical point, we also introduce the quantity γ₂, which was suggested by Campi:

γ_{2} = \frac{M_{2} M_{0}}{M_{1}^{2}},

(8)

M_{k} = \sum_{Z_{i} \neq Z_{max}} Z_{i}^{k} \cdot n_{i} (Z_{i}) .

(9)

M₀, M₁, M₂ and Mk are the zeroth, first, second and kth moments of the charge distribution, respectively [31].

Figure 7 presents the temperature dependence of the average values of γ₂ obtained from the event-by-event analysis. There are well-defined peaks in γ₂ for both models, while the maximum value of γ₂ is larger for IQMD. The maximum values are reached at T = 6 MeV and 8 MeV for GEMINI and IQMD, respectively. They are defined as the critical points where the largest fluctuations in fragment sizes are obtained. However, the peak value of γ₂ is expected to reach at least 2 for the critical behavior. Although a maximum value of γ₂ lower than 2 has been observed in a light system with A = 36 [27], the value extracted by GEMINI in this work is suspect for such heavy system like ¹¹²Sn. Thus, the most probable critical region is T = 8–10 MeV in our calculation.

Figure 7:

Calculated values of γ₂ at different temperatures for GEMINI (the triangles) and IQMD (the dots).

3.3 Kinematic characteristic

Nuclear temperature is a concept derived from the compound nucleus definition[32]. The Coulomb [33,34], recoil [35], and collective effects [36,37] have been observed with the slope thermometer, whose value for the temperature is extracted from the distribution of the kinetic energy spectra. Fig. 8 presents the kinetic energy spectra of ²H at four temperatures for the two models and the fitting curves are calculated by the surface-emission-type method [33]:

Figure 8:

Kinetic energy distributions of ²H at T = 5, 10, 15, and 20 MeV. The left panels are the IQMD results and the right ones are the GEMINI results.

Y (E) \propto \frac{(E - E_{0})}{T_{slope}^{2}} \exp (- \frac{(E - E_{0})}{T_{slope}}),

(10)

where E₀ reflects the Coulomb repulsion.

As the temperature or excitation energy increases, the final kinetic energy distribution of ²H becomes broader, but the peak positions of the spectra that are close to the Coulomb barriers seem to be insensitive to the temperature. This phenomenon can be explained by the fact that the Coulomb barrier is dependent on the charge of the initial emitting source, which is fixed in the simulation [38, 39].

To further investigate the kinetic characteristics, the slope values of the center-of-mass kinetic energy spectra for H and He were extracted, which are shown in Fig. 9.

Figure 9:

Extracted slope temperatures for isotopes ¹H, ²H, ³H, ³He, and ⁴He, respectively. The upper and bottom panels correspond to the results of GEMINI and IQMD, respectively. The solid lines represent the initial given temperatures.

One can observe that the slope temperatures are mass dependent for the IQMD model [6, 40, 41]. The extracted T_slope values increase with increasing mass numbers for the same elements that are associated with the radial flow effect. Usually, the Coulomb effect can be observed by comparing the T_slope values extracted from two fragments with the same mass number but different charge numbers. However, there is no obvious discrepancy between ³H and ³He. It may be that the Coulomb effect is not so apparent for there is no compression process as in the collision reaction. The T_slope values of the neutron are also extracted, which are almost the same as those of the proton. Generally, the T_slope values of ⁴He are lower than ³He in the collision experiments for the abundant secondary emission of ⁴He from heavy fragments. The ⁴He production in different evolution times were checked and there were no obvious decreases in the momentum distributions, which indicates that this effect is limited in the simulation. The results of GEMINI show similar mass dependent trends, but deviations between the fitting curves and calculated data occur with increasing temperatures, which can be seen in Fig. 8. The kinetic energy distributions of ³He become rather dispersive at high temperatures, thus only several extracted values are given. In fact, the decay nucleus no longer acts like a thermal source for the increasing intensity of the sequential binary decay characteristic. Thus, the extracted T_slope values are more reliable in the lower temperature region for GEMINI. It is interesting to see that the T_slope values of ²H are closest to the given temperatures for both models.

3.4 Comparison with experimental data

The projectile fragmentation of ¹²⁴Sn-on-^natSn at 600 MeV/nucleon has been studied with the ALADIN forward spectrometer at the GSI Schwerionen Synchrotron [42]. To quantitatively compare it with the experimental 〈M_IMF〉 versus Z_bound projectile data, where Z_bound is defined as the sum of the charges of all fragments with Z ≥ 2 in a single event, a series of primary nuclear sources are mixed in the following way to simulate the excited residual nuclei produced in the experiment. First, the average mass of the equilibrated source $\bar{A}$ has the correlation with the excitation energy per nucleon Ex/A suggested by Botvina et al. [43]:

α \equiv \frac{\bar{A}}{A_{0}} = 1 - 0.001 (E_{x} / A) - 0.015 (E_{x} / A)^{2},

(11)

where A is assumed to have a Gaussian distribution around $\bar{A}$ and A₀ is the mass number of the projectile. The width of the distribution is decided by the following formula:

σ = σ_{0} A_{0} (1 + c_{0} \frac{ϵ^{2}}{ϵ_{max}^{2}})

(12)

ϵ = Ex/A and ϵ_max represents the maximum value of ϵ. The parameters ϵ_max, σ₀, and c₀ are 8.13 MeV, 0.07, and 2, respectively, in the simulation and the details can be seen in Refs. [42, 43]. Then, the distribution of the mass versus excitation energy is generated with the two equations above, as shown in Fig. 10.

Figure 10:

Mass number versus excitation energy distribution of different thermal sources calculated with Eqs. (11) and (12). The numbers of the sources are proportional to the areas of the squares.

Next, 20 primary nuclear sources are chosen with initial temperatures derived from the correlation of the excitation energy and temperature. As the correlation is not precise, integer temperature values are adopted. The specific mass numbers and mixed proportions of the nuclear sources are presented in Fig. 11. The charge numbers of the different sources are obtained with the N/Z ratio remaining the same as the projectile ¹²⁴Sn. The parameters of the nuclear sources are identical for the two models.

Figure 11:

Proportions and temperatures of different nuclear sources adopted to simulate the excited residual nuclei produced in a 600 MeV/nucleon ¹²⁴Sn-on-^natSn experiment. A is the mass number of the different nuclear sources. The solid squares represent the temperatures of the nuclear sources. The proportions of the different sources used for the simulation are presented with dots.

In Fig. 12, it can be seen that the simulated results of IQMD are generally consistent with the experimental data, while the results of GEMINI show apparent deviations. However, the rising and falling trends of the GEMINI data are close to the experimental data. Thus, it is reasonable continue exploring the isotopic distribution.

Figure 12:

Distributions of 〈M_IMF〉 versus Z_bound by IQMD and GEMINI compared with the experimental results in Ref. [42]. The stars represent experimental data. The circles and triangles are the calculated results with a series of mixed nuclear sources by IQMD and GEMINI, respectively.

As there is no impact parameter in the single nucleus decay process, the yields of the isotopes cannot be compared with the experimental cross sections. However, the peak position and shape of the distribution are important information. Thus, the relative yield is studied, where the isotope with the highest yield is set to 1.

In Fig. 13, the shapes of the isotopic distributions by GEMINI are very close to the experimental data, while those of IQMD are much wider. To quantitatively compare the parameters of the isotopic distributions between the two models, the average mass numbers and the standard deviations σ of the mass numbers for all isotopes in the events with 10 ≤ Z_bound ≤ 40 are presented in Fig. 14.

Figure 13:

Isotopic distributions of Z = 3–10. The yields are replaced by defined ratios where the highest yield is set to 1. The dots are experimental data integrated over 10 ≤ Z_bound ≤ 40. The circles and triangles are the calculated results of IQMD and GEMINI, respectively. The lines are for clarity.

Figure 14:

Average mass numbers 〈A〉 of the isotopes with different charge numbers Z in the events with 10 ≤ Z_bound ≤ 40 are plotted in the top panel and the corresponding standard deviations σ are presented in the bottom panel. The squares and solid triangles are the results of IQMD and GEMINI, respectively.

The average mass numbers of the isotopes for different charge numbers Z are almost the same for IQMD and GEMINI. The σ is much larger for IQMD, which means a broader isotopic distribution. It may be that the final fragments in the IQMD model do not completely reach the ground state while the experimental fragments are in the ground state. In the GEMINI model, extensive comparisons with data from a large range of compound-nucleus masses have been used to optimize the default parameters. The good overall agreement with the experimental data for the isotopic distributions in the calculations by GEMINI can be attributed to this.

4 Conclusion

The deexcitation processes of single excited ¹¹²Sn nuclei are simulated at T = 1 ~ 30 MeV with a dynamical model (IQMD) and a statistical model (GEMINI). The fragmentation mechanisms, critical behavior, and kinematic characteristics are investigated, and some results are compared with experimental data. It is found that IQMD and GEMINI give similar fragmentation phenomena in different temperature regions. However, the power-law fit and the extraction of γ₂ are more reasonable with the IQMD model. Moreover, the extracted slope temperatures of IQMD are more reasonable and closer to the original temperatures than those of GEMINI, which indicates a better kinematic description. The simulation results of IQMD with mixed thermal sources are consistent with experimental 〈M_IMF〉 - Z_bound data of the projectiles, but GEMINI can reproduce the widths of the experimental isotopic distributions better than IQMD.

In general, both IQMD and GEMINI could be used to explore the decay properties of excited nuclei like the fragment distributions or fragmentation mechanisms. However, the multfragmentation phenomenon, which is related to the chemical instability, cannot be treated with GEMINI. The critical phenomena and kinematic characteristics are also associated with dynamical behaviors. Thus, they might be better described within the IQMD model. However, GEMINI shows advantages in reproducing isotopic distributions.

Thus, the IQMD model can reproduce the decay properties of an excited nucleus rather well, and it is shown that the IQMD model can be used alone in most cases.

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