I. INTRODUCTION

The symmetry energy of asymmetric nuclear matter, especially neutron-rich matter, has been an important topic both in theory and experiment with the development of radioactive nuclear beam (RNB) facilities. The new or updated RNB facilities can provide nuclear beams with larger neutron-richness, which are supposed to promote the study of neutron-rich nuclear matter and extend the chart of nuclides to the more neutron-rich side. In heavy-ion collisions above the intermediate energy, nuclear matter from sub-saturation to supra-saturation densities can be produced. The symmetry energy of nuclear matter depends on density and temperature. In the past decade, though many probes have been proposed, there are still difficulties in measuring the symmetry energy of nuclear matter at different nuclear densities.

The isobaric yield ratio (IYR) method was first proposed by Huang et al. to extract the symmetry-energy coefficient for fragments with small isospins, which also have some temperature since they are measured in heavy-ion collisions [1, 2]. In the IYR, some terms deciding the fragment yield cancel out, which makes it useful to study the symmetry energy term of the fragment with a low temperature. The IYR method is extended to study the symmetry energy of fragment with larger neutron richness [3-6]. The results obtained by the IYR method are compared to a similar method, i.e., isoscaling, which extracts the symmetry energy of the colliding source, showing a large difference between them [7-10]. We studied the results of the IYR and isoscaling, and found that though the IYR and isoscaling methods both aim at the extraction of nuclear symmetry energy, the obtained results are for different nuclear matter. A new probe was constructed via IYR between two reactions, which is called the isobaric yield ratio difference (IBD), and the results of the IBD and isoscaling were found to be similar because they are both the probes for the chemical potential of neutrons and protons in the reactions [11, 12]. Moreover, the IBD probe is also sensitive to the density differences between the reactions [13, 14]. In this article, we will summarize the recent progress on the IBD method.

II. THEORETICAL DESCRIPTIONS

In thermodynamics theory, an equilibrium state is assumed for the reaction, which introduces the concept of temperature. The temperature affects the yield of fragment. With high temperature, light particles like neutrons and protons can escape from the binding of hot fragment. The formation time of the final fragments, which correspond to the measured ones in experiments, is called the "chemical freeze-out" (when the fragment ceases to emit light particles, like neutrons, protons or α, and the neutron and proton numbers do not change anymore). In general, the yield of fragment in heavy-ion collisions above the Fermi energy obeys an exponential law. For example, in the grand-canonical ensembles theory within the grand canonical limit, the yield of a fragment with mass numbers A and neutron-excess I (I = N–Z is the difference between the numbers of neutrons and protons) is written as [15, 16],

where T is the temperature of the system. F(I,A) is the free energy of the fragment which depends on T. μn(μp) is the chemical potential of the neutrons (protons) in the equilibrium system, which depends on the density and T of the systems. In the isoscaling method [17, 18], the difference between the μn(μp) of two reactions can be extracted from the isotopic (isotonic) ratio,

with indices 1 and 2 denoting the reactions. In Eq. (2), the T of the two reactions is assumed to be the same. For the chemical freeze-out fragments this can be true since they both have a very long evolution time. For the isotopic ratio,

Z does not change. The isotopic ratio shows scaling phenomena obeying the linear function. The fitted slope is defined as α (μ_n2 - μ_n1)/T. Similarly, for the isotonic ratio, N is a constant. The isotonic ratio also shows scaling phenomena obeying the linear function. The fitted slope is defined as β (μ_p2 - μ_p1)/T. α and β are used to extract the symmetry energy of the system. But the symmetry energy in the isoscaling method is an indirect result obtained from the yield of fragments.

Starting from Eq. (1), assuming the isobars differing 2 units in I have the same temperature, the isobaric yield ratio can be defined as,

To make cancellation of F, we adopt a reference reaction system with similar measurements as the isoscaling method does, and assume that the free energies of one fragment in the two reactions are the same. Defined as the difference between the IYRs of two reactions, the IBD is [11, 12],

The indices also denote the reaction systems. From the definition of the isoscaling parameters, the following correlation can be obtained,

Eq. (5) can also be written as,

The chemical potentials of neutrons and protons are related to the densities of neutrons (ρn) and protons (ρp) respectively, [14, 19],

which result in α = ln ρ_n2 - ln ρ_n1 = ln(ρ_n2/ρ_n1), and β = ln ρ_p2 - ln ρ_p1 = ln(ρ_p2/ρ_p1). We can also write the relationship between the IBD results and the densities of the reaction systems as [14],

Eqs. (9) and (10) are written according to Eqs. (5) and (7), respectively. The densities of protons can be determined by experimental measurement. Thus the relationships in Eqs. (9) and (10) can be used to determine the neutron density of the neutron-rich system. For example, if the reference reaction system is a neutron-proton symmetric one, in which ρ_n1 and ρ_p1 can be assumed to be the same and ρ_p2 is measurable, ρ_n2 can be determined from the IBD results. In Eq. (9), if ρ_p2 and ρ_p1 can be assumed to be the same (for isotopic projectiles) and ρ_p1 ρ_n1, ΔR(I + 2, I, A)₂₁ = ln (ρ_n2/ρ_p1). Similarly, in Eq. (10), the reference reaction is neutron-proton symmetric and ρ_p1 ρ_n1, by knowing ρ_p2, ΔR(I + 2, I, A)₂₁ = ln (ρ_n2/ρ_p2) and ρ_n2 can be determined from the IBD.

III. RESULTS AND DISCUSSION

Now we will show the IBD results obtained from the measured 140 A MeV ^{40, 48}Ca + ⁹Be and ^58,64Ni + ⁹Be projectile fragmentation reactions [20]. These reactions were measured by M. Mocko et al. at the National Superconducting Cyclotron Laboratory (NSCL) at Michigan State University. The measured fragments range from Z = 5 to Z_proj, which provides high quality data for the test of the IBD methods. The IBD results has been previously

In Fig. 1, the results of the IBD and isoscaling methods are re-plotted for comparison [11]. Four groups of reactions are analyzed, the ⁴⁸Ca/⁴⁰Ca + ⁹Be and ⁶⁴Ni/⁵⁸Ni + ⁹Be reactions of the isotopic projectiles, the ⁵⁸Ni/⁴⁰Ca + ⁹Be reactions with the projectiles both being neutron-proton symmetric, and the ⁴⁸Ca/⁶⁴Ni + ⁹Be reactions with the projectiles both being neutron-rich. The reactions using the relative neutron-proton symmetric projectile was set as the reference reaction denoted by index 1. Theoretically, the relationship between the IBD and isoscaling is described in Eq. (6). The fragment belongs to one isotopic chain and one isotonic chain, thus it can be related to α and β according to the respective isotopic and isotonic ratio fitting. For the isobaric chains with I from -1 to 2, the IBD and isoscaling results were found to be similar, except for some fragments with a large A. The IBD results were the same in the I = 0 and 1 isobaric chains for the ⁴⁸Ca/⁴⁰Ca reactions. The same phenomena can be found in the I = 1 and 2 isobaric chains for the ⁶⁴Ni/⁵⁸Ni reactions, and the I = -1 isobaric chains for the ⁵⁸Ni/⁴⁰Ca reactions.

Fig. 1.Full-screen View

(Color online) The comparison between the IBD and isoscaling results (a re-plotted figure from Ref. [11]). The circles and squares denote the IBD and isoscaling results, respectively. The lines are just for guiding the eyes to the plateaus in the IBD and isoscaling results.

It was also found that the height of the IBD-plateaus in the four groups were different, about 2, 1.4, 0.5 and 0.4, respectively [11]. And the lengths of the plateaus are also different. In the IBD of neutron-rich isobars, the plateau can disappear [11-14]. The IBD results denote the difference between the neutrons and protons for the two reactions according to Eqs. (9) and (10). If the IBD results change very small, it reflects that the difference between the densities of neutrons and protons for the reactions are small. The increasing IBD with A of the fragment indicates an enlarged difference between the densities of the neutrons and protons [21, 22]. Taking the IBD results for the ⁴⁸Ca/⁴⁰Ca reactions as an example, ρ_p1 and ρ_n1 for ⁴⁰Ca can be assumed to be the same, and assuming that ρ_p1 ρ_p2, ΔR(I + 2, I, A)₂₁ = ln(ρ_n2/ρ_p1). ρ_n2 of the ⁴⁸Ca reaction can be determined from the IBD results. For the ⁴⁸Ca/⁴⁰Ca reaction, $\Delta R{(I+\mathrm{2,}I\mathrm{,}A)}_{21}=\ln (\frac{{\rho }_{n2}}{{\rho }_{p1}})=2$ which results in $\frac{{\rho }_{n2}}{{\rho }_{p1}}=7.4$. It is very large compared to the isoscaling method. It should be noted that ⁴⁸Ca has a much larger neutron excess than ⁴⁰Ca, and α∼β, of which β cannot be omitted in Eq. (6). This makes Δln R(I+2, I, A)₂₁ = 2α, which results in the overestimation of $\frac{{\rho }_{n2}}{{\rho }_{p1}}$. Similar explanations can be found for the other three reaction groups. It should be noted that the densities determined by the IBD results are for the time of freeze-out, assuming thermal equilibrium. The relationship between the density of the projectile and the system at the freeze-out should be further studied in transport models.

The IBD-plateau was found to be sensitive to differences between the densities of the reaction systems. The asymmetry of these systems (N/Z)₂₁ are calculated and the results are plotted in Fig. 2. The asymmetry of the reaction systems (N/Z)₂₁ is defined as [(N_p+N_t)/(Z_p+Z_t)]₂/[(N_p+N_t)/(Z_p+Z_t)]₁. From high to low, (N/Z)₂₁ is in the order of ⁴⁸Ca/⁴⁰Ca, ⁴⁸Ca/⁵⁸Ni, ⁶⁴Ni/⁴⁰Ca, ⁶⁴Ni/⁵⁸Ni, ⁴⁸Ca/⁶⁴Ni, and ⁵⁸Ni/⁴⁰Ca. The (N/Z)₂₁ of the ⁴⁸Ca/⁶⁴Ni and ⁵⁸Ni/⁴⁰Ca reactions are very similar. In Fig. 3(a), the IBD-results for the I = 0 isobaric chain are re-plotted. From high to low, the IBD-plateau is in the order of ⁴⁸Ca/⁴⁰Ca, ⁴⁸Ca/⁵⁸Ni, ⁶⁴Ni/⁴⁰Ca, ⁶⁴Ni/⁵⁸Ni, ⁵⁸Ni/⁴⁰Ca, and ⁴⁸Ca/⁶⁴Ni. The IBD-plateaus for the ⁵⁸Ni/⁴⁰Ca and ⁴⁸Ca/⁶⁴Ni reactions are very similar. The order of the (N/Z)₂₁ and the IBD-plateau are consistent, except for the ⁵⁸Ni/⁴⁰Ca and ⁴⁸Ca/⁶⁴Ni reactions. The same phenomena can also found in the IBD results for the I = 1 isobaric chain in the reactions as plotted in Fig. 3(b).

Fig. 2.Full-screen View

(Color online) The asymmetry of the systems for obtaining the IBD results. The labels on the circles denote the reactions used.

Fig. 3.Full-screen View

(Color online) The IBD results for the I=0 (a) and I=1 (b) isobaric chain.

To test the sensitivity of the IBD-plateau to the density difference between the reactions, we calculate the fragment yield in the 80 A MeV ^38-52Ca + ¹²C reactions (only the even-A calcium isotopes are used) by using a modified statistical abrasion-ablation (SAA) model [6, 13, 14, 21-27]. The statistical abrasion-ablation model will not be described in this work. Readers can refer to Refs. [21-27]. The fragments produced in the ⁴⁰Ca + ¹²C reaction is set as reaction 1, which is the reference reaction. The IBD results for the isobaric chains from I=-1 to 2 are plotted in Fig. 3. Since the analyzed fragments are the prefragments which are determined by the neutron and proton densities, as well as the nucleus-nucleus reaction cross sections, the yields of prefragments are sensitive the density changes of the projectile [13]. It is found that, as the projectile (^XCa, with X denotes the mass number of projectile) becomes more neutron-rich, the IBD results increases regularly. The IBD obtained from the prefragments are sensitive to the difference between the neutron density of the ^XCa and ⁴⁰Ca reactions if we assume that the proton densities in ^XCa are the same or vary very little [13, 14].

We calculated the average values (<Δμ/T>) of the IBD-plateau for the isobars plotted in Fig. 4. The <Δμ/T> are plotted as a function of I in Fig. 5. It is shown that for the relative symmetry fragments, <Δμ/T> changes very small with I. In the inserted figure of Fig. 4, the difference between the neutron densities (Δ ln ρn) of ^XCa and ⁴⁰Ca is plotted. When r < 2 fm, Δ ln ρn are flat and change very small. It has been illustrated that the yield of fragments with small mass numbers are mostly produced in the central collisions [21, 22, 26, 27]. The flat distribution of Δ ln ρn between the core part of the calcium isotopes can account for the phenomena of the IBD-plateau. The enlarged Δ ln ρn with r (r > 2 fm for different calcium isotopes) can also account for the increasing of Δμ/T with A, as well as the disappearance of the IBD-plateau in the neutron-rich isobars [11-14].

Fig. 4.Full-screen View

(Color online) The IBD results for the prefragments calculated by using a modified statistical abrasion-ablation model.

Fig. 5.Full-screen View

(Color online) The averaged value of IBD-plateaus for the isobaric chains from I=-1 to 2. The inserted figure show the difference between the neutron density distributions of ^XCa and ⁴⁰Ca with Δ ln ρn = ln ρn (^XCa)- ln ρn (⁴⁰Ca).

Both from the experiment results and the calculated results by the statistical abrasion-ablation model [11-14], it is shown that the IBD probe is sensitive to the density difference between reactions. The IBD probe provides a new method to determine the neutron density in heavy-ion collisions. But as we have noted, the density is for the time of the chemical freeze-out, assuming there is a thermal equilibrium in the reaction. Actually, the process of the reaction is dynamic and changes with reaction time. The IBD results should be verified by a dynamic evolution process. Previously we have reported the IBD result for the ⁵⁸Ni + ¹²C reaction calculated by the anti-symmetric molecular dynamical (AMD) model to verify the sensitivity of the IBD to density [11]. In dynamical model calculations, the evolution of the IBD results with time can be studied. We have finished the calculations and the IBD analysis of the ^{58, 64}Ni + ¹²C reactions and hope to report the AMD results in near future to see the IBD for primary fragments and fragments after the decay process. Moreover, the Shannon information entropy uncertainty is introduced to explain the physical meaning of the IBD probe [28, 29]. In the dynamical model, the construct of fragments are important [30, 31], and at the same time the temperature for heavy fragments should be carefully treated [32-39]. It is also suggested that the IBD can be used to study the evolution of chemical potential and the density of the reactions. We will also study the evolution of the IBD results by using the AMD model.

IV. SUMMARY

In summary, the method of isobaric yield ratio difference is explained in this work. The IBD method is a more direct method than the isoscaling method to extract the symmetry energy or density of nuclear matter in heavy-ion collisions. It is proven that the IBD probe is sensitive to the difference between the densities of the reactions, and can be used to study the neutron density of the reactions.