Ⅱ-2. Nuclear Physics (Theory)

RIKEN Accel. Prog. Rep. 53

S-shaped heat capacity in an odd-odd deformed nucleus††∗1∗2∗3∗4∗5∗6∗7100

080-40

Entropy-20212

At211
Po200Tl184
Re

-60
-80
0.4

0.8

1.2

1.6

Temperature (MeV)

2.0100- 72 -4000.0

Condensed from the article in Phys. Lett. B 789, 634 (2019)
Bankura University
Institute of Fundamental and Applied Sciences, Duy Tan University
Variable Energy Cyclotron Centre
Dept. of Phys., Barasat Govt. CollegeRIKEN Nishina Center
Department of Natural Science and Technology, University ofKhanh Hoa
Dept. of Phys., Raniganj Girls’ College6020-100

0.0

0.4

0.8

1.2

1.6

2.0

0.0

0.4

0.8

1.2

1.6

2.0

Temperature (MeV)

12010080

Heat Capacity

Average Energy (MeV)

A long-standing problem in nuclear physics is theexperimental observation of the pairing phase transition in atomic nuclei. Thermodynamic properties suchas superﬂuidity and pairing phase transition are wellestablished facts in inﬁnite nuclear matter.1) However,these properties digress from inﬁnite nuclear matter toﬁnite nuclei owing to statistical ﬂuctuations in the order parameter. Therefore, the gradual transition fromstrongly correlated paired states to unpaired ones inatomic nuclei may not be as evident as in inﬁnite matter.1) This induces a high degree of interest in the studyof nuclear thermodynamics, especially in the energy domain of neutron binding.
The study of thermodynamics of ﬁnite nuclei involves
the reliable extraction of experimental nuclear level density (NLD). The experimental determination of NLDs
has been diﬃcult in the past owing to the absence ofsuitable experimental techniques. Recently, the thermalproperties of diﬀerent nuclei were measured by the Oslogroup using a new technique, called the Oslo method,
and the S-shaped canonical heat capacity was extracted,
signalizing the pairing phase transition around the critical temperature Tc = 0.5–1.0 MeV.2,3) However, thismethod is limited to a low excitation energy E ∗ (belowthe particle threshold) and angular momentum J (onlya few multiples of ℏ). This limitation in E ∗ and J has
recently been overcome by measuring the NLD from theevaporated particle spectra in compound nuclear reactions.3) In addition, thus far, most of the thermal properties have been studied in nuclei where the shell eﬀect(δS) is very small (less than ∼1 MeV). The mass region
exhibiting a large shell eﬀect (δS ∼ 10 MeV) has rarely
been studied; and thus, it is very important to investigate how the thermodynamic properties behave becauseof the large eﬀect of nuclear shells. In this work, theNLDs and, consequently, the nuclear thermodynamic
quantities (TQs) for four diﬀerent nuclei, 184 Re, 200 Tl,
211
Po, and 212 At, are estimated to understand the shelland deformation eﬀects on the TQs.
Figure 1 shows the free energy F , average energy E,entropy S, and heat capacity C as functions of T . Fand E behave smoothly with varying T for all the nuclei. However, S and C show diﬀerent variations with T .
The diﬀerence in entropy appears primarily because ofFree Energy (MeV)

B. Dey,∗1 N. Quang Hung,∗2 D. Pandit,∗3 S. Bhattacharya,∗4 N. Dinh Dang,∗5 L. T. Quynh Huong,∗6D. Mondal,∗3 S. Mukhopadhyay,∗3 S. Pal,∗3 A. De,∗7 and S. R. Banerjee∗760
40208060
40
20000.0

0.4

0.8

1.2

1.6Temperature (MeV)

2.0

Temperature (MeV)

Fig. 1. TQs as functions of temperature obtained using theEP+IPM level densities for J = 12 ℏ.

the diﬀerent shell structures of the nuclei at low T since
they show a smooth variation at higher T (above T =1.6 MeV), where the shell structures are predicted to bemelted. Most noteworthy is the nature of the heat capacity of 184 Re, which is completely diﬀerent from thoseof the other three nuclei. There is a pronounced bump(S-shape) in the heat capacity of 184 Re at T ∼ 0.8 MeV,
which is close to the critical temperature Tc , where thepairing gap collapses, as predicted by the conventional
BCS theory. This S-shape in the heat capacity is interpreted as a ﬁngerprint for a pairing transition in nuclei.
The heat capacities of 211 Po, 212 At, and 200 Tl do notshow the pronounced S-shape structure, as expected forodd-odd and odd-even nuclei. In order to understandthe S-shaped heat capacity in such odd-odd deformed
nuclei, the exact neutron and proton pairing gaps havebeen calculated as functions of T , as discussed in detailin Ref. 4). It has been shown that S-shape of the heat
capacity observed in the odd-odd deformed 184 Re nucleus at J = 12 ℏ is only due to the change in pairing
gaps caused by the change in deformation.
The authors are very grateful to Dr. Pratap Roy,VECC, Kolkata for providing experimental neutron energy spectra.
References

1) A. Belic et al., Nucl. Phys. A 731, 381 (2004).
2) E. Melby et al., Phys. Rev. Lett. 83, 3150 (1999).
3) B. Dey et al., Phys. Rev. C 96, 054326 (2017) and references therein.
4) B. Dey et al., Phys. Lett. B 789, 634 (2019).