1 Introduction

Temperature is importan in heavy-ion collisions, which increases rapidly with the colliding time to reach the maximum, and then decreases due to cooling down of the colliding system in expansion [1]. The methods for studying the temperature in heavy-ion collisions above the Fermi energy include the double isotopic ratio (Albergo thermometer) of the light particles/fragments [2], the thermal energy method [3], the excitation energy method [4], the momentum fluctuation method [5], and the slope temperature using the kinetic energy spectra of light particles [6]. Being mostly focused on the extraction of temperature from light particles, the methods for temperature are correlated to each other [1, 7]. Another method is the intermediate mass fragments (IMFs). Assuming the colliding system to be in an equilibrium state, and using the Albergo temperature to extract temperature, the isotopes of carbon and the heavier [8,9] were found to carry a much lower temperature than the light particles, because that the fragments and the light particles denote the different colliding times, and the light particles are emitted earlier than the fragments. The isobaric yield ratio (IYR) was used to determine the temperature for IMFs in the framework of the thermodynamics models, and different methods were used to approximate the free energy of IMFs [10-13]. This isobaric yield ratio method has been improved by adopting the difference between two IYRs and avoiding the fitting procedure to IYR, so as to obtain the temperature directly from the isobaric yields [14]. Liu et al developed an isobaric yield ratio method using the symmetry energy of primary fragment and the prefixed symmetry energy of the source in which the primary fragments are formed (the primary fragments means that they are formed in the colliding source, which are highly excited and will undergo the secondary decay to form the final fragments) [15-17]. The temperature of fragments is determined at the formation time in a self-consistent manner. In their works, the fragments in the ⁴⁰Ca + ⁴⁰Ca reactions were simulated using the anti-symmetrized molecular dynamics (AMD) model [15]. In Ref. [10], a similar method was used to determine the temperature for IMF by taking the symmetry-energy coefficient (a_sym) of ground state nucleus as the reference. Due to the strong mass dependence of a_sym in isobars, the temperature strongly depends on mass. It is interesting to see if the temperature determined by the method in Ref. [10] is similar to those in Refs. [15-17]. In this article, the IYR methods to determine the temperature are described in Sec. 2. The temperature obtained from the simulated fragments using the AMD model is discussed in Sec. 3. A summary of this article is given in Sec. 4.

2 Methods

In this work we call the nucleus produced in collisions as a fragment, which has temperature. And we call the nucleus in its ground state as nucleus to differ it from the fragment. The methods to obtain the ratio of a_sym/T for IMFs are similar to that developed by Huang et al [18]. In this article, the IYR method in Ref. [10] is adopted to obtain a_sym/T for the primary IMF in the framework of the modified Fisher model (MFM) [18-21],

where, a^*’_sym = 4a^’_sym and the prime denotes the fragment; R is the yield ratio between isobars differing 2 in I (I=N−Z); ΔE_C is the residue Coulomb-energy, which can be omitted [22]; Δ_I−2 and Δ_I are the mixing term of N and Z of the isobars.

a_sym of nucleus is obtained from the binding energy [27],

where, a^*_sym = 4a_sym, and B is the binding energy. T is determined from the ratio of Eq. (2) to Eq. (1),

Similar ratio form of Eq. (3) is used to determine T for the primary IMFs, with each nucleus having a different a^*_sym/A [10], while it is the same for the primary IMFs in the self-consistent fitting method and is prefixed in simulations [15].

We note that the coefficients of energy terms contributing to the free energy of fragment depend both on nuclear density (ρ) and temperature [18-21], i.e. a^*’_sym=a^*’_sym (ρ, T). It is shown that the ρ and T dependence of a^*’_sym/(AT) can be determined from the correlation between a^*’_sym/(AT) and T by introducing a T-dependent symmetry energy [10]. If the coefficients and temperature are free parameters in the modified Fisher model, they can be fitted from the fragment distributions [26].

3 Results and discussion

The ⁴⁰Ca + ⁴⁰Ca reactions at E = 35, 50, 80, 100, 140 and 300 MeV/u were simulated using the AMD model by Liu et al[15]. The yields of primary fragments were extracted and the temperature was investigated using the self-consistent fitting method. In this paper, the same yields of primary fragments are used to analyze the temperature according to Eq. (3). In the simulations and the primary fragment reorganizations in Ref.[15], we use three standard Gogny interactions [24] ranging from the soft g0 [24], the hard g0as [25] to the super-hard g0ass interaction [15, 25]. The impact parameter is set as b = 0 fm to simulate the central collisions. Over 10,000 events are simulated for different reactions and Gogny interactions. The fragments are recognized in the phase space at t = 300 fm/c with a coalescence radius R_c = 5 fm [15].

In Fig. 1, a^*’_sym/(AT) for IMF and a^*_sym/A for nucleus are plotted, with the fragments being A ≤ 30. The AME12 binding energy of nucleus was adopted to obtain a^*_sym/A [28]. Only the results for I = 1 fragments are shown. It can be seen that both a^*’_sym/(AT) and a^*_sym/A decrease with increasing A. The odd-even staggering is more obvious in a^*_sym/A than in a^*’_sym/(AT), indicating that the symmetry energy obtained from the primary fragment is smaller than that from the ground state nucleus. This is because that the temperature increase results in a decrease of symmetry energy [29]. The a^*’_sym/(AT) for IMF decreases in an order of the g0, g0as, and g0ass potentials, with the g0as and g0ass results being of a small difference.

Fig. 1Full-screen View

a_sym^*’/(AT) calculated by Eq. (1) for the primary IMF in the ⁴⁰Ca + ⁴⁰Ca reactions at 35, 50, 80, 100, 140 and 300 MeV/u by the AMD model with the g0, g0as and g0ass interactions, and a_sym^*/A for nucleus (squares) calculated by Eq. (2).

As shown in Fig. 2, T for IMFs calculated by Eq. (3) does not change much at 35 and 50 MeV/u, while it varies with A at E ≥ 80 MeV/u, especially for the IMFs of A > 20. In a reaction of certain incident-energy, T increases with the interaction in an order of g0, g0as, and g0ass, indicating that T increases with the hardness of interaction. In super-hard g0ass interaction, T for IMFs changes greatly with A at E ≥ 80 MeV/u. These phenomena reflect the change of a^*’_sym/(AT) for fragments in reactions of different reactions, induced by the change in yield of fragments. Our results are close to T obtained by the self-consistent fitting method (lines in Fig. 2) in Ref.[15]. Since the primary IMFs are all reorganized at t = 300 fm/c in the reactions of different incident energies, suggesting that in reaction of relative low incident energies, the primary IMF may easily reach thermal equilibrium. While the trend changes at higher incident energies, where the Pauli blocking is less effective and the symmetry potential is less important, thus leading to a shorter momentum relaxation time and a longer isospin relaxation time [23]. An A-dependence of temperature was found in measuring fragments in the 1A GeV ^124,136Xe + Pb reactions [10], and in other isobaric yield ratio methods with simulated primary fragments [15] or measured cold fragments [14]. The A-dependence of temperature has been explained as effects of the finite reaction system [15]. After the self-consistent fitting procedure, the temperature for the primary fragments is almost the same, while after modification by the secondary decay of the primary fragments, the temperature for the resultant cold fragments depends significantly on A. This is because that for the cold fragment, its free energy can be approximated by its binding energy at low temperatures.

Fig. 2Full-screen View

T calculated using Eq. (3) for I = 1 primary IMFs in ⁴⁰Ca + ⁴⁰Ca reactions simulated at different energies by the AMD model with the g0, g0as, and g0ass interactions. The line in each panel denotes the results of the self-consistent temperature in Ref. [15].

Figure 3 shows T for fragments of A = 15, 17 and 19 in reactions simulated using the g0, g0as, and g0ass interactions. Although staggering can be seen in the g0ass results generally increases with E, in an order of the g0, g0as, and g0ass, indicating that the harder the interaction the higher temperature of IMFs. This disagrees with the results in Ref.[15], where the temperature is insensitive to the interactions, while in our method T is sensitive to IYR or the cross section of fragment.

Fig. 3Full-screen View

Correlation between T and E of the reactions for the A = 15, 17, and 19 fragments with I = 1.

Staggering was found in T for reactions at around 100 MeV/u. Cross sections of the related isobars are plotted in Fig. 4. For I = −1 and I = 1 isobars, the interactions affect little the cross sections, while for I = 3 isobars, the interactions affect greatly the cross sections, with obvious staggering. The staggering in T, or a^*’_sym/(AT), is due to the staggering of cross sections of the I = 3 isobars, i.e. in the isobaric ratio method the temperature is vulnerable to the change of cross sections of fragments which are influenced by the incident energy, the interaction, and the nucleus-nucleus cross sections, etc.

Fig. 4Full-screen View

Cross sections of the simulated fragments in calculating temperature of the A = 15, 17 and 19 isobars shown in Fig. 3.

At last, it should be noted that, a^*_sym/(AT) for fragment is influenced by IYRs and the cross sections of fragments [19]. It has been shown that T for the measured fragments in the ^124,136Xe reactions are similar, indicating that the temperature tends to be similar in reactions of similar environments [10]. In fact, the results of a^*_sym/(AT) for fragments reflect the properties of the nuclear system at the time they are generated, which is reflected by different type of the Gogny interactions we used. One would like to expect that R_c affects the temperature in this isobaric method since it definitely affects the yield of fragment. We cannot perform the analysis for the primary fragments reorganized using different R_c due to difficulties in obtaining the data, but it can be assumed that the R_c effect on the temperature in this method shall be small. The reason is that the primary fragments are formed in similar colliding source, and the fragments can be viewed as the samples of the source. And the samples of different sizes shall deliver the similar information of the source.

One question frequently raised is whether the equilibrium state can be achieved in the transport model simulations. Liu et al proposed that the AMD simulations should have a freeze out stage, which is a common concept of statistical theory [15]. The Shannon information entropy theory has been introduced to study the complex evolution of heavy-ion collision, which can bridge the results in the transport models and thermal dynamics models [30, 31]. It will be interesting to study the temperature evolution in the simulated reactions by the AMD model using the information entropy method.

4 Conclusion

Temperature of the primary IMF in simulated ⁴⁰Ca + ⁴⁰Ca reactions at 35, 50, 80, 100, 140 and 300 MeV/u by the AMD model adopting the standard Gogny g0, g0as, and g0ass interactions [15] has been determined using the IYR method. The primary fragments are reorganized in the phase space at t = 300 fm/c with the coalescence radius R_c = 5.0 fm.

The a^*’_sym/(AT) for the primary fragments simulated by the AMD model is obtained from the difference of IYRs, and a^*_sym/A for nucleus is determined from the difference between the binding energies of isobars. T for the primary fragment is determined from the ratio of a^*’_sym/(AT) to a^*_sym/A. T increases with the incident energy and the interaction hardness. The temperature we obtained is close to those determined by self-consistent method. The data of primary fragments in the simulated reactions are useful.