# Temperature dependence of the volume and surface contributions to the nuclear symmetry energy within the coherent density fluctuation model

###### Abstract

The temperature dependence of the volume and surface components of the nuclear symmetry energy (NSE) and their ratio is investigated in the framework of the local density approximation (LDA). The results of these quantities for finite nuclei are obtained within the coherent density fluctuation model (CDFM). The CDFM weight function is obtained using the temperature-dependent proton and neutron densities calculated through the HFBTHO code that solves the nuclear Skyrme-Hartree-Fock-Bogoliubov problem by using the cylindrical transformed deformed harmonic-oscillator basis. We present and discuss the values of the volume and surface contributions to the NSE and their ratio obtained for the Ni, Sn, and Pb isotopic chains around double-magic Ni, Sn, and Pb nuclei. The results for the -dependence of the considered quantities are compared with estimations made previously for zero temperature showing the behavior of the NSE components and their ratio, as well as with the available experimental data. The sensitivity of the results on various forms of the density dependence of the symmetry energy is studied. We confirm the existence of “kinks” of these quantities as functions of the mass number at MeV for the double closed-shell nuclei Ni and Sn and the lack of “kinks” for the Pb isotopes, as well as the disappearance of these kinks as the temperature increases.

###### pacs:

21.60.Jz, 21.65.Ef, 21.10.Gv## I Introduction

The study of the nuclear symmetry energy and particularly, its density () and temperature () dependence is an important task in nuclear physics. This quantity is related to the energy connected with the conversion of the isospin asymmetric nuclear matter (ANM) into a symmetric one. It is an important ingredient of the nuclear equation of state (EOS) in a wide range of densities and temperatures (see, e.g., NSE2014 (); Lattimer2007 (); Li2008 ()). Using approaches like the local-density approximation Agrawal2014 (); Samaddar2007 (); Samaddar2008 (); De2012 () and the coherent density fluctuation model Gaidarov2011 (); Gaidarov2012 (); Gaidarov2014 (), the knowledge of EOS can give information about the properties of finite systems. As noted in Bao2017 () information about NSE from laboratory experiments can be obtained from quantities that are sensitive to it, such as static properties, nuclear excitations, collective motions, heavy-ion reactions and others. One can add also interesting phenomena including the supernova explosions, properties of neutron stars and rare isotopes, frequencies and strain amplitudes of gravitational waves from both isolated pulsars and collisions involving neutron stars that depend strongly on the EOS of neutron-rich nuclear matter. Here we would like to mention a broad range of works devoted to the study of -dependence of single-particle properties of nuclear as well as neutron matter (e.g., Agrawal2014 (); Moustakidis2007 (); Sammarruca2008 (); De2014 (); Zhang2014 (); Lee2010 (); BLi2006 (); Mekjian2005 (); Xu2007 ()), and of -dependence of NSE (e.g., Li2006 (); Piekarewicz2009 (); Vidana2009 (); Sammarruca2009 (); Dan2010 ()). Among the important studies of the -dependence of hot finite nuclear systems we will note Ref. Brack74 () on thermal Hartree-Fock (HF) calculations, the semiclassical approaches based on the microscopic Skyrme-HF formalism Brack85 (), the Thomas-Fermi (TF) approximation Suraud87 (), the inclusion of the continuum effects in HF calculations at finite temperature Bonche (), the refined TF approach De96 () and extended TF (ETF) method Brack84 (). We mention also the Hartree-Fock-Bogoliubov (HFB) models Khan2007 (); Sandulescu2004 (); Monrozeau2007 (); Yuksel2014 (), as well as the relativistic TF approach with different relativistic mean-field nuclear forces Zhang2014 (). Here we should also note the studies of the role of short-range and tensor nucleon-nucleon correlations that change considerably the kinetic and potential energy contributions to the NSE (e.g. NSE2014 (); Bao2017 ()).

The method of the CDFM Ant80 (); AHP () allowed us to make the transition from nuclear matter to finite nuclei in the studies of the NSE for spherical Gaidarov2011 () and deformed Gaidarov2012 () nuclei, as well as for Mg isotopes Gaidarov2014 () using the Brueckner energy-density functional (EDF) of ANM Brueckner68 (); Brueckner69 ().

In our previous work Antonov2017 () we used a similar method to investigate the -dependence of the NSE for isotopic chains of even-even Ni, Sn, and Pb nuclei following the LDA Agrawal2014 (); Samaddar2007 (); Samaddar2008 (); De2012 () and using instead of the Brueckner EDF, the Skyrme EDF with SkM* and SLy4 forces. The -dependent local densities and kinetic energy densities were calculated within a self-consistent Skyrme HFB method using the cylindrical transformed deformed harmonic-oscillator basis (HFBTHO) Stoitsov2013 (); Stoitsov2005 () with the same forces. For comparison, the kinetic energy density was calculated also by the TF expression (for low temperatures) up to term Lee2010 (). In addition, we studied the -dependence of NSE for Pb using the ETF method Brack85 (); Brack84 () and the rigorous density functional approach Stoitsov87 () to obtain -dependent local density distributions.

In his pioneering work Feenberg47 () Feenberg in 1947 showed that the surface energy should contain a symmetry energy contribution due to the failure of nuclear saturation at the edge of the nucleus and that the volume saturation energy also has a symmetry energy term. In later works Cameron57 (); Bethe71 (); Green58 (); Myers66 () were given analyses of the volume and surface components of the NSE, as well as of their ratio. We should also mention some works in which the volume and surface components to NSE and their ratio are explicitly related to the neutron-skin thickness Dan2003 (); Dan2004 (); Dan2014 (); Dan2009 (); Dan2011 (); Tsang2012 (); Tsang2009 (); Ono2004 (); Dan2006 (); Warda2010 (); Centelles2010 (). Agrawal et al. Agrawal2014 () pointed out the substantial change in the NSE coefficients for finite nuclei with temperature in comparison with the case of nuclear matter. Lee and Mekjian Lee2010 () emphasized the bigger sensitivity of the surface component of NSE to the temperature in respect to the volume energy term. We note also studies of the volume and surface NSE, e.g., in Refs. Steiner2005 (); Danielewicz (); Diep2007 (); Kolomietz (); Nikolov2011 (); Jiang2012 (); Ma2012 (), including those by analyzing the experimental nuclear binding energies Jiang2012 (); Ma2012 ().

The values of the temperature-dependent symmetry energy coefficients for sixty nine spherical and non-spherical nuclei with mass and charge numbers, respectively, have been evaluated in Ref. De2012a () in the subtracted finite-temperature TF framework by using two different energy-density functionals. A substantial temperature dependence of the surface symmetry energy was found, while the volume symmetry energy turned out to be less sensitive to the temperature. As a result, the symmetry energy coefficient of finite nuclei falls down as the temperature rises De2012a (), an observation that has been confirmed in Ref. Antonov2017 (). A study of the decomposition of NSE into spin and isospin components is carried out in Ref.(Guo2018 ()) within the Brueckner-Hartree-Fock approximation with two- and three-body forces.

In our work Antonov2016 () the volume and surface contributions to the NSE and their ratio were calculated within the CDFM using two EDF’s, namely the Brueckner Brueckner68 (); Brueckner69 () and Skyrme (see Ref. Wang2015 ()) ones. The CDFM weight function was obtained by means of the proton and neutron densities obtained from the self-consistent deformed HF+BCS method with density-dependent Skyrme interactions. The obtained results in the cases of Ni, Sn, and Pb isotopic chains were compared with results of other theoretical methods and with those from other approaches which used experimental data on binding energies, excitation energies to isobaric analog states (IAS), neutron-skin thicknesses and with results of other theoretical methods. We note that in Antonov2016 () the obtained values of the volume and surface components of NSE and their ratio concern the case at MeV.

The aim of the present work is to evaluate the above mentioned quantities for temperatures different from zero. The -dependent local density distributions and computed by the HFBTHO code are used to calculate the -dependent CDFM weight function. Such an investigation of the thermal evolution of the NSE components and their ratio for isotopes belonging to the Ni, Sn, and Pb chains around the double-magic nuclei, will extend our previous analysis of these nuclei considering them as cold systems Antonov2016 (). At the same time, the obtained results within the CDFM provide additional information on the thermal mapping of the volume and surface symmetry energies that has been poorly investigated till now (e.g., Ref. De2012a ()). In addition, we study the sensitivity of the calculated -dependent quantities on different available forms of the density dependence of the symmetry energy.

The structure of this paper is the following. In Sec. II we present the main relationships for the NSE and its volume and surface components depending on the temperature that we use in our study, as well as the CDFM formalism that provides a way to calculate the mentioned quantities. Section III contains the numerical results and discussions. The main conclusions of the study are given in Sec. IV.

## Ii Theoretical formalism

We start this section with the expression for the nuclear energy given in the droplet model that is an extension of the Bethe-Weizsäcker liquid drop model to incorporate the surface asymmetry. It can be written as Steiner2005 (); Myers1969 ():

(1) | |||||

In Eq. (1) MeV is the binding energy per particle of bulk symmetric matter at saturation. , , , and are coefficients that correspond to the surface energy of symmetric matter, the Coulomb energy of a uniformly charged sphere, the diffuseness correction and the exchange correction to the Coulomb energy, while the last term gives the pairing corrections ( is a constant and for odd-odd nuclei, 0 for odd-even and -1 for even-even nuclei). is the volume symmetry energy parameter and is the modified surface symmetry energy one in the liquid model (see Ref. Steiner2005 (), where it is defined by ).

In our previous work Antonov2017 () we studied the temperature dependence of the NSE . For the aims of the present work we will rewrite the symmetry energy (the third term in the right-hand side of Eq. (1) in the form

(2) |

where

(3) |

with

(4) |

In the case of nuclear matter, where and , we have . Also at large Eq. (3) can be written in the known form (see Ref. Bethe71 ()):

(5) |

where and . From Eq. (3) follow the relations of and with :

(6) |

(7) |

In what follows we use essentially the CDFM scheme to calculate the NSE and its components (see Refs. Ant80 (); AHP (); Antonov2016 ()) in which the one-body density matrix is a coherent superposition of the one-body density matrices for spherical ”pieces” of nuclear matter (”fluctons”) with densities and . It has the form:

(8) |

with

(9) | |||||

where is the first-order spherical Bessel function and

(10) |

with

(11) |

is the Fermi momentum of the nucleons in the flucton with a radius . The density distribution in the CDFM has the form:

(12) |

It follows from (12) that in the case of monotonically decreasing local density () the weight function can be obtained from a known density (theoretically or experimentally obtained):

(13) |

We have shown in our previous works Gaidarov2011 (); Gaidarov2012 (); Antonov2016 () that the NSE in the CDFM for temperature can be obtained in the form:

(14) |

where the symmetry energy for the asymmetric nuclear matter that depends on the density has to be determined using a chosen EDF (in Antonov2016 () Brueckner and Skyrme EDF’s have been used).

In this work the -dependent NSE is calculated by the expressions similar to Eq. (14) but containing -dependent quantities:

(15) |

In Eq. (15) the weight function depends on the temperature through the temperature-dependent total density distribution :

(16) |

where

(17) |

and being the proton and neutron -dependent densities that in our work Antonov2017 () were calculated using the HFB method with transformed harmonic-oscillator basis and the HFBTHO code Stoitsov2013 ().

As mentioned, in the present work we consider the -dependence of the NSE , but also of its volume and surface components and their ratio [Eq. (4)].

Following Refs. Dan2003 (); Dan2006 (); Diep2007 (); Antonov2016 () an approximate expression for the ratio can be written within the CDFM:

(18) |

where is determined by Eq. (16), Diep2007 () and is the NSE at equilibrium nuclear matter density and MeV. For instance, the values of for different Skyrme forces in the Skyrme EDF are given in Table II of Ref. Antonov2016 (). In what follows we use the commonly employed power parametrization (firstly, in subsection III.1) for the density dependence of the symmetry energy (e.g., Diep2007 (); Dan2004 (); Dan2006 ())

(19) |

There exist various estimations for the value of the parameter . For instance, in Ref. Diep2007 () and in Ref. Dan2006 () . The estimations in Ref. samm17 () (given in Table 2 there) of the NSE based on different cases within the chiral effective field theory and from other predictions are (NLO), (NLO), (DBHF) and 0.79 (APR akmal98 ()). Another estimation of is also given in Ref. russ16 ().

## Iii Results of calculations and discussion

### iii.1 Results with the density dependence of the symmetry energy given by Eq. (19)

The calculations of the -dependent nuclear symmetry energy , its volume and surface components, as well as their ratio were performed using the relationships (15), (18)-(21) with the weight function from Eqs. (16) and (17). As mentioned in Sec. II, the -dependent proton and neutron density distributions, as well as the total density [Eq. (17)] were calculated using the HFBTHO code from Ref. Stoitsov2013 () with the Skyrme EDF for SkM* and SLy4 forces. We note that in the calculations of [Eq. (20)] and [Eq. (21)] we use the weight function from Eq. (16), where the density distributions for finite nuclei are used. The quantity in Eqs. (15) and (18) is the symmetry energy for asymmetric nuclear matter chosen in the parametrized form of Eq. (19). Our calculations are performed for the Ni, Sn, and Pb isotopic chains.

Studying the -dependence of the mentioned quantities we observed (as can be seen below) a certain sensitivity of the results to the value of the parameter in Eq. (19). In order to make a choice of its value we imposed the following physical conditions: i) the obtained results for the considered quantities at MeV to be equal or close to those obtained for the same quantities in our previous works for the NSE, its components and their ratio (Ref. Antonov2017 (); Antonov2016 ()), and ii) their values for MeV to be compatible with the available experimental data (see, e.g., the corresponding references in Antonov2016 ()).

In Figs. 1-6 the results for , , , and are given as functions of the mass number for the isotopic chains of Ni, Sn, and Pb nuclei for temperatures –3 MeV calculated using the SkM* and SLy4 Skyrme forces. The results are presented for two values of the parameter and . The reason for this choice is related to the physical criterion mentioned above. It can be seen that at MeV and the value of is around . This result is in agreement with our previous result obtained in the case of the Brueckner EDF in Ref. Antonov2016 (), namely . The latter is compatible with the published values of extracted from nuclear properties presented in Ref. Diep2007 () from the IAS and skins Dan2004 () () and from masses and skins Dan2003 () (). In the case of our result for MeV is that is in agreement with the analyses of data in Ref. Diep2007 () (), as well as with the results of our work Antonov2016 () in the case of Skyrme EDF, namely, for the Ni isotopic chain , for the Sn isotopic chain , and for the Pb isotopic chain , all obtained with SLy4 and SGII forces.

Before making the comparison of our results for at MeV with our previous results from Ref. Antonov2017 () (there the NSE is denoted by ) we mention that though the latter are in good agreement with theoretical predictions for some specific nuclei reported by other authors, we showed that they depend on the suggested definitions of this quantity. The comparison of the results in the present work for at MeV with those from our work Ref. Antonov2017 () (the latter illustrated there in Fig. 12) shows that they agree with our present values of within the range of –0.4, except in the case of Pb with SLy4 force, where the present results are somewhat lower.

It can be seen from Figs. 1–6 that the quantities , , and decrease with increasing temperatures (–3 MeV), while slowly increases when increases. This is true for both Skyrme forces (SkM* and SLy4) and for the three isotopic chains of the Ni, Sn, and Pb nuclei. Here we would like to note that the values of between 0.3 and 0.4 that give an agreement of the studied quantities with data, as well as with our previous results for MeV, are in the lower part of the estimated limits of the values of (e.g., in the case of Diep2007 ()).

It can be seen also from Figs. 1–6 that there are “kinks” in the curves of , , , and for MeV in the cases of double closed-shell nuclei Ni and Sn and no “kinks” in the Pb chain. This had been observed also in our previous work for Antonov2017 (), as well as for its volume and surface components and their ratio at MeV in Ref. Antonov2016 ().

In Figs. 7–9 are given the results for the -dependence of , , , and for the double-magic Ni, Sn, and Pb nuclei obtained using both SkM* and SLy4 Skyrme forces. The results are presented by grey areas between the curves for the values of the parameter and . It can be seen that , , and decrease, while slowly increases with the increase of the temperature for both Skyrme forces.

### iii.2 Comparison with results from alternative parametrizations of the density dependence of the symmetry energy

In this part of our work we study the sensitivity of the obtained results towards different density dependences of the symmetry energy. First, we note the Eq. (6) from Ref. Dong2012 ():

(22) |

This relationship, which coincides with the shape from the density-dependent M3Y interaction Muk2007 (), is similar to that of Eq. (9) of Ref. Agrawal2013 (), where the last term of Eq. (22) is exchanged by . Second, we use also the dependence of the symmetry energy Tsang2009 (); Dong2012 ()

(23) |

It was noted in Tsang2009 () that the analysis of both isospin diffusion and double ratio data involving neutron and proton spectra by an improved quantum molecular dynamics transport model suggests values of .

Concerning the values of the coefficients and in Eq. (22), we determined them by fitting the three curves of (for different values of ) presented in Fig. 4 of Ref. Dong2012 () and note them as when MeV, when MeV, and when MeV.

We mention also the remark in Ref. Dong2012 () that the expressions for in Eqs. (19) and (23) ”do not reproduce the density dependence of the symmetry energy as predicted by the mean-field approach around nuclear saturation density”. Nevertheless, in what follows we present for a comparison our results from subsection III.1 with those calculated not only using Eq. (22) but also from Eq. (23) which we will note as .

Figure 10 illustrates the behavior of the density dependence of the symmetry energy by giving some of the curves for different functions, namely, , three curves using Eq. (19) that we label as with , 0.4, 0.7, as well as three curves for that correspond to , 0.3, 0.7. At this point we emphasize that the CDFM weight function which is used in Eqs. (15) and (18) has a form of a bell with a maximum around at which the value of the density is half of the value of the central density equal to []. So, namely in this region (around ), the values of the different play the main role in the calculations of [Eq. (15)] and [Eq. (18)].

In the next Fig. 11 we consider, as an example, the mass dependence of , , , and in the case of the Ni isotopic chain for temperatures –3 MeV using the SLy4 Skyrme force. The results are given when the symmetry energy has the form of at and 0.3. One can see that, e.g., the value of at MeV and is 1.90 that is close to the result obtained using Eqs. (19) and (21) for (it is ) shown in the left panel of Fig. 2. The value of when is 2.69 which is similar to that in the case of Eq. (19) and (21) for (it is ) shown in the right panel of Fig. 2. There exist in these cases similarities also of the behavior of the quantities , , and as functions of in the corresponding cases. The reason for the mentioned similarities is the closeness of the corresponding curves shown in Fig. 10 in the region around .

In Fig. 12 we give, as an example, the -dependence of , , , and in the case of Ni nucleus using the density dependence of the symmetry energy for and in the case of SLy4 force. This figure corresponds to Fig. 7 (right panel) and one can see the similarities of the presented quantities. As we mentioned before, the reason for the latter is the closeness of the corresponding curves at around .

Table 1 reflects the noted above similarities of the results for and at MeV in the cases when Eqs. (19) and (23) are used. However, the results obtained for using the density-dependence from Eq. (22) are quite different from those in Table 1 with either set of parameter values: for , for , and for (while the values of for the same symmetry energy forms are 14.62 MeV, 14.22 MeV, and 13.85 MeV, respectively). The reason for this comes from the fact that the integrand in Eq. (18) for is strongly peaked around , and the results depend on the behavior of in respect to and in this region (see Fig. 10). It is also worth mentioning that the large values of the ratio given above are comparable with those presented in Table 1 at large values of the parameter ().

0.3 | 1.66 | 21.35 | 0.2 | 1.90 | 19.00 |

0.4 | 2.64 | 19.08 | 0.3 | 2.69 | 17.48 |

0.5 | 4.09 | 17.20 | 0.4 | 3.74 | 16.23 |

0.7 | 10.67 | 14.26 | 0.7 | 9.82 | 13.58 |

## Iv Conclusions

In the present work we perform calculations of the temperature dependence of the NSE , its volume and surface components, as well as their ratio . Our method is based on the local density approximation. It uses the coherent density fluctuation model Ant80 (); AHP () with -dependent proton , neutron , and total density distributions. The latter are calculated within the self-consistent Skyrme HFB method using the cylindrical transformed harmonic-oscillator basis (HFBTHO) Stoitsov2013 (); Stoitsov2005 () and the corresponding code with SkM* and SLy4 Skyrme forces. In the calculations we used in Eqs. (15) and (18) different density dependences of the symmetry energy , namely Eq. (19) in subsection III.1 and the alternative cases, Eqs. (22) and (23) in subsection III.2. The quantities of interest are calculated for the isotopic chains of Ni, Sn, and Pb nuclei.

The main results of the present work can be summarized as follows:

(i) With increasing , the quantities , , and decrease, while slightly increases for all the isotopes in the three chains for both Skyrme forces and for all used density dependences of the symmetry energy. The same conclusion can be drawn for the thermal evolution of the mentioned quantities in the cases of the three considered double-magic Ni, Sn, and Pb nuclei.

(ii) The results for , , , and are sensitive to the choice of the density dependence of the symmetry energy in Eqs. (15) and (18). In subsection III.1 the sensitivity of the studied quantities towards the values of the parameter in Eq. (19) is shown. In subsection III.2 are considered in detail the results when other different density dependences of the symmetry energy [Eqs. (19), (22), and (23)] are used. The similarities and differences between the results from various functional forms are related to the behavior of the corresponding values of around the value of the ratio for which the CDFM weight function has a maximum.

(iii) In the cases of double-magic Ni and Sn nuclei we observe “kinks” for MeV in the curves of , , , and , but not in the case of Pb isotopes. This effect was also observed in our previous works. It is also worth mentioning how the kinks are blurred and eventually disappear as increases, demonstrating its close relationship with the shell structure.

###### Acknowledgements.

Three of the authors (M.K.G., A.N.A., and D.N.K) are grateful for support of the Bulgarian Science Fund under Contract No. DFNI-T02/19. E.M.G. and P.S. acknowledge support from MINECO (Spain) under Contract FIS2014–51971–P(E.M.G. and P.S.) and FPA2015-65035-P(E.M.G.).## References

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