1. Introduction

The measured fragments in heavy-ion collisions have a lower temperature than the primary fragments which are formed in the hot source. The Albergo isotopic thermometer has been used to extract the temperature based on the yields of protons, neutrons, and some light isotopes ^[1]. Also, the isotopic thermometer has been used to extract the temperature of larger isotopes, such as the carbon isotopes ^[2,3] and intermediate-mass fragments (IMFs) ^[4,5]. Other methods employed to study this temperature in heavy-ion collisions include the thermal energy method ^[6], excitation energy method ^[7], momentum fluctuation method ^[8], the correlation of two-particle relative moment ^[9], and kinetic energy spectra of light particles ^[4]. Recently, the isobaric ratio method has been proposed to extract temperature for IMFs ^[10].

In the thermodynamic models, temperature is one part of the probes and cannot be separated easily. In the isobaric yield ratio method, the symmetry energy coefficient is studied by using its ratio to the temperature (a_sym/T) for neutron-rich nucleus ^[11]. In this article, an improved thermometer is proposed to extract T of the IMFs which is based on the IYRs ^[12,13].

2. Methods

Here, the canonical ensemble theory is adopted. Based on the grand-canonical limitation, the cross section σ(A, I) of a fragment has the form of ^[14]

where C and τ are constants; T is temperature, I = N− Z is the neutron-excess; μ_n(μ_p) is the chemical potential of neutron (proton), and F(A, I) is the free energy of a fragment, which can be parameterized as the T-dependent mass formula^[10,15-18].

The IYR is defined between the yield of isobars with I and I −2

where Δμ = μ_n−μ_p. Similarly, for isobars with I +2 and I, one has

Δμ/T for fragment changes very small, which has been shown in an isobaric ratio difference method^[19-25], Thus Δμ/T can be canceled out in the difference between isobaric yield ratios,

The residual free energy is defined as ΔF ≡ 2 F(A, I) − F(A, I−2) − F(A, I+2). If ΔF is known, T can be obtained. It has been proven that within the finite temperature range ΔF between two isobars can be replaced by that of the binding energy for the fragments ^[13]. Following the assumption in Refs. [12, 13], the residual free energy ΔF can be replaced by the residual binding energy ΔB = 2 B(A, I) − B(A, I−2) − B(A, I+2). From Eq. (4), the improved method to extract T based on the difference between IYRs (labeled as T_IB) is

where ΔlnR = lnR(A, I+2, I) − lnR (A, I, I−2) is defined for simplification. The binding energy in the AME12 will be adopted in the analysis ^[26].

3. Results and Discussion

The fragments in the 140A MeV ^{40, 48}Ca + ⁹Be (¹⁸¹Ta) and ^{58, 64}Ni + ⁹Be (¹⁸¹Ta) reaction are adopted to verify the T_IB method. They were measured by Mocko et al. at the National Superconducting Cyclotron laboratory, Michigan State University ^[27].

ΔB and ΔlnR will be discussed separately, beginning with the distributions of ΔB for fragments in the 140A MeV ^{40, 48}Ca + ⁹Be and ^{58, 64}Ni + ⁹Be reactions (Fig.1). For the I =1 fragments, ΔB increases almost monotonically with A; while for fragments of I = 3,5,7, ΔB staggers on the relative small A side. The staggering in ΔB becomes smaller for the A>33 fragments. ΔB for the I=9 fragments shows a small staggering but the staggering is more evident for the A>47 fragments.

Fig.1.Full-screen View

(color online) ΔB for the fragments in the 140A MeV ^{40, 48}Ca + ⁹Be (¹⁸¹Ta) and ^{58, 64}Ni + ⁹Be (¹⁸¹Ta) reactions.

Secondly, ΔlnR for related isobars in the 140A MeV ^{40, 48}Ca + ⁹Be (¹⁸¹Ta) and ^{58, 64}Ni + ⁹Be (¹⁸¹Ta) reactions, as plotted in Fig. 2. For the I =1 fragments, ΔlnR almost keeps constant on the small A side, but increases with A at A >40, with some staggering at A >30. For the fragments of I =3, 5, 7, an obvious staggering appears in ΔlnR on the small A side, but it staggers little when A is relative large. The target (⁹Be and ¹⁸¹Ta) shows very little influence on ΔlnR. In general, the distribution of ΔB and ΔlnR is similar in shape.

Fig.2.Full-screen View

(color online) ΔlnR for the fragments in the 140A MeV ^{40, 48}Ca + ⁹Be (¹⁸¹Ta) and ^{58, 64}Ni + ⁹Be (¹⁸¹Ta) reactions.

Finally, we use ΔB and ΔlnR in 140A MeV ^{40, 48}Ca+⁹Be (¹⁸¹Ta) and ^{58, 64}Ni+⁹Be (¹⁸¹Ta) reactions to calculate T_IB (Fig. 3). T_IB for the I =1 fragments is almost constant at 1.5 MeV. T_IB for I =3 staggers for small A fragments, but becomes small and constant at A>35. T_IB for I =5 staggers, too, in a small manner though. But the staggering becomes larger again for I =7 and I= 9 fragments. For most of the I >3 fragments, the T_IB values range from 0.6 MeV to 3.5 MeV, which agrees with the temperatures extracted by the IYR method ^[12,13]. Only for some fragments of very rich neutrons, the T_IB values are large. This agrees with the results in the canonical ensemble theory to estimate the mass of neutron-rich copper isotopes at T = 2.2 MeV ^[14]. Great difference can be seen between the T_IB values for the reactions using ⁹Be and ¹⁸¹Ta targets when I is large (I = 7 and 9), showing obvious target effect.

Fig.3.Full-screen View

(color online) T_IB for the fragments in the 140A MeV ^{40, 48}Ca + ⁹Be (¹⁸¹Ta)and ^{58, 64}Ni + ⁹Be (¹⁸¹Ta) reactions.

4. Summary

An improved isobaric ratio thermometer (T_IB) for intermediate mass fragments has been proposed based on the difference between IYRs, in which the residual free energy is replaced by the residual binding energy among these isobars. In contrast to the IYR thermometer, T_IB is directly obtained from the yields of fragments and avoids the fitting procedure, which serve as a direct probe of temperature. T_IB of the odd $I$ fragments in the 140 MeV ^{40, 48}Ca + ⁹Be (¹⁸¹Ta) and ^{58, 64}Ni + ⁹Be (¹⁸¹Ta) reactions has been obtained. The values of T_IB for most considered IMFs are low. There is some inference that for similar reactions with different asymmetries, T_IB can be assumed as the same, so the assumption is reasonable that the temperatures in two similar reactions are similar.