RIKEN Accel. Prog. Rep. 48 (2015)

Ⅱ-2. Nuclear Physics (Theory)

Pairing effect in thermal shape fluctuation model on the width ofgiant dipole resonance†
A.K. Rhine Kumar,∗1 P. Arumugam,∗1 and N. Dinh Dang∗2
The study of nuclear properties at high temperature,spin and isospin has gained much of interest in recent
times. Apart from these extremes, there are still some
unexplored regimes of hot nuclei. The properties of nuclei at very low temperatures and the phase transitions
associated with that belong to such area where conclusive experimental results are scarce. At such low temperatures, the shell (quantal) and pairing effects arequite active though being modified by thermal effects.
In hot nuclei, thermal fluctuations are expected to belarge since the nucleus is a tiny finite system. Thermal shape fluctuations and fluctuations in the pairing
field are the dominating fluctuations and have been sofar studied separately within different models. Bothof these fluctuations are expected to be present at lowtemperatures. However, the interplay between themhas not been investigated so far. The present work addresses this subject and we study the influence of thisinterplay on the experimental observables, namely thewidth of giant dipole resonance (GDR). The thermal
shape fluctuation model (TSFM), which is often usedby experimentalists, describes the increase of the GDR
width with temperature by averaging the GDR crosssection over all the quadrupole shapes. However thismodel is known to largely overestimate the GDR widthin open-shell nuclei at low temperatures. The successof a proper pairing approach1,2) suggests the necessityof including pairing correlations to cure this shortcoming of the TSFM. This is done in the present work.
We employ the thermal shape fluctuation modelbuilt on Nilsson-Strutinsky calculations3) with amacroscopic approach to GDR and examine the inclusion of the fluctuations in the pairing field. The nuclear shapes are related to the GDR observables usinga model comprising an anisotropic harmonic oscillator potential with separable dipole-dipole interaction.
In this formalism the GDR Hamiltonian can be written as H = Hosc + η D† D + χ P † P , where Hosc
stands for the anisotropic harmonic oscillator hamiltonian, the parameter η characterizes the isovector component of the neutron and proton average field and χ
denotes the strength of the pairing interaction. The
pairing interaction changes the oscillator frequencies[ωνosc (ν = x, y, z)] resulting in the new set of frequen2
+N ∆Ncies ων = ωνosc − χω P , where ω P = Z∆PZ+Nwith χ having the units of MeV−1

†∗1
∗2.

Alternatively,Condensed from the article in Phys. Rev. C. 90, 044308(2014)
Department of Physics, Indian Institute of Technology Roorkee, Uttarakhand
RIKEN Nishina Center

Fig. 1. (Color online) GDR width for 97 Tc, calculated
within the liquid drop model (LDM), without pairing,including BCS pairing and pairing fluctuations, as afunction of temperature in comparison with experimental data2) .

the role of pairing is to renormalize the dipole-dipole√
interaction strength such that, η = η0 − χ0 T ω P ,with χ0 having the units of MeV−5/2 . The effective
GDR cross-sections is calculated by averaging all thecross-sections obtained from thermal fluctuations ofquadrupole shapes by using the formula for the expectation value of an observable O as �O�β,γ,∆P ,∆N =OW (T, β, γ, ∆P , ∆N )D[α]/ W (T, β, γ, ∆P , ∆N )D[α]with W (T, β, γ, ∆P , ∆N ) = exp[−F (T ; β, γ, ∆P , ∆N )/T ],D[α] = β 4 | sin 3γ| dβ dγ d∆P d∆N . The total free energy (FT OT ) at a fixed deformation is
calculated using the expression FT OT = ELDM + p,n δF . The
liquid-drop energy (ELDM ) is calculated by summingup the Coulomb and surface energies corresponding toa triaxially deformed shape defined by the deformation parameters β and γ. The shell correction (δF )is obtained with exact temperature dependence using
the single-particle energies given by the triaxial Nilsson
model.
The results of our calculations for 97 Tc, 120 Sn and208
Pb demonstrate that the TSFM can be quite successful if the shell effects (with explicit temperaturedependence) and the pairing ones are properly incorporated in the free energy (See Fig. 1).
References1) N. Dinh Dang, A. Arima, Phys. Rev C 68, 044303(2003); N. Dinh Dang, N. Quang Hung, Phys. Rev C
86, 044333 (2012).
2) B. Dey, et al, Phys. Lett. B 731, 92 (2014).
3) P. Arumugam, G. Shanmugam, S. K. Patra, Phys. Rev.
C 69, 054313 (2004).

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