Pairing interactions and the vanishing pairing correlations in hot nuclei

a r X i v :n u c l -t h /0612074v 2 1 M a r 2007

Pairing interactions and vanishing pairing correlations in hot nuclei

E.Khan a ),Nguyen Van Giai a ),N.Sandulescu a ),b )

a)Institut de Physique Nucl′e aire,Universit′e Paris-Sud,

IN 2P 3-CNRS,91406Orsay Cedex,France b)Institute for Physics and Nuclear Engineering,P.O.Box MG-6,76900Bucharest,Romania

Finite temperature Hartree-Fock-Bogoliubov calculations are performed in Sn isotopes using Skyrme and zero-range,density-dependent pairing interactions.For both stable and very neutron-rich nuclei the critical temperature at which pairing correlations vanish is independent of the vol-ume/surface nature of the pairing interaction.The value of the critical temperature follows approx-imatively the empirical rule T c ?0.5?T =0for all the calculated isotopes,showing that the critical temperature could be deduced from the pairing gap at zero temperature.On the other hand,the pairing gap at temperatures just below T c is strongly sensitive to the volume/surface nature of the pairing interaction.

I.INTRODUCTION

The competition between the temperature and pairing correlations in hot nuclei has been studied for more than four decades.The ?rst studies were based on the BCS approximation [1]but later on more involved calculations based on the Bogoliubov approach have been performed [2,3].More recently the Bogoliubov approach has been employed together with self-consistent Hartree-Fock mean ?elds in order to study the pairing properties of hot nuclei.One of the ?rst ?nite-temperature HFB (FT-HFB)calculations was based on a ?nite-range force of Gogny type,which is used for describing both the mean ?eld and the pairing properties of hot nuclei [4].FT-HFB calculations using zero-range forces have been done for hot nuclei [5]and for the inner crust matter of neutron stars [6].In the latter case the mean ?eld is obtained by using a Skyrme force while the pairing correlations are calculated with a density-dependent delta interaction.Also,shell-model approaches [7,8]have been used in order to probe the impact of the temperature on both pairing and deformation degrees of freedom.

The interplay between temperature and pairing correlations was also studied intensively in nuclear and neutron matter [9,10].Typically,the pairing gaps are calculated in the BCS approximation and using single-particle states de-termined by self-consistent Brueckner-Hartree-Fock or Green’s function methods (see [11,12]and references therein).In more fundamental approaches,which go beyond BCS approximation,is still unclear how much the pairing correla-tions are a?ected by the medium dependence of the nucleon-nucleon interaction (see [13,14]and references therein).One open issue in current HFB calculations is how much the form of the pairing interaction a?ects the properties of nuclei,especially when one approaches the drip lines.It is also not clear yet if one really needs to introduce an explicit density dependence in the pairing interaction in order to enforce a pairing ?eld evenly distributed in the nucleus (”volume type pairing”)or strongly localized in the surface region (”surface type pairing”).Since a realistic pairing force derived from ?rst principles is missing,one hopes to disentangle between various types of pairing forces by analyzing their consequences on measurable quantities.However,up to now these studies are not conclusive.For instance,in Ref.[15]a mixed surface-volume pairing interaction is considered to better explain the odd-even mass di?erences of some isotopic chains,whereas in Ref.[16]the surface or the volume type of the pairing interaction is found to be not so relevant for the neutron separation energies.It is also worth stressing that the pair density,which gives indications upon the localization of pair correlations in ?nite nuclei,is not strongly correlated to the surface or volume character of the pairing force but rather to the localization of the single-particle states close to the chemical potential [17].

Apart from the e?ects mentioned above,the type of the pairing force could also a?ect the vanishing of pairing correlations in hot nuclei.Besides constant G studies there have not been such systematic studies with e?ective density-dependent pairing interactions.It is known that,in a simple BCS approach with a constant pairing G,the vanishing of pairing correlations is expected to occur at T c ?0.5?T =0[2].The aim of the present work is to analyze if the volume or surface character of the pairing force could signi?cantly in?uence vanishing pairing correlations using density-dependent pairing interactions.It should be noted that experimentally,the critical temperature could be extracted from the change of the speci?c heat in the vanishing pairing correlations region,using level densities measurements,as shown in Refs.[18,19].

II.FINITE-TEMPERATURE HARTREE-FOCK-BOGOLIUBOV WITH SKYRME INTERACTIONS In this work we employ the FT-HFB approach with zero-range forces.Details can be found elsewhere[5,6],and we recall only the main equations.The FT-HFB equations,in coordinate representation,have the following form:

h T(r)?λ?T(r)?T(r)?h T(r)+λ U i(r)V i(r) =E i U i(r)V i(r) ,(1)

where E i is the quasiparticle energy,U i,V i are the components of the radial FT-HFB wave function andλis the chemical potential.The quantity h T(r)is the thermal averaged mean?eld Hamiltonian and?T(r)is the thermal averaged pairing?eld.The latter is calculated with a density-dependent contact force of the following form[20]:

V(r?r′)=V0[1?η(

ρ(r)

2V eff(ρ(r))

1

2

V eff(ρ(r))κT(r),(3)

whereκT(r)is the thermal averaged pairing tensor.Due to the density dependence of the pairing force,the thermal averaged mean?eld Hamiltonian h T(r)depends also onκT.In addition,the averaged mean?eld Hamiltonian depends on thermal averaged particle density,spin density and kinetic energy density.The thermal averaged particle density is given by:

ρT(r)=

1

d rκT(r)(5)

Figure2shows the thermal evolution of the mean neutron gap in124Sn,in the case of a surface pairing interaction. The critical temperature above which pairing correlations vanish is T c=0.7MeV.It should be noted that the T c?0.5?T=0rule is still qualitatively veri?ed.

An alternative de?nition of the pairing gap is to use the particle densityρT instead of the pairing densityκT:

= d rρT(r)?T,n(r)

?n

ρ

respect to pairing localization properties,also in very neutron-rich nuclei.There is,however,a di?erence in the thermal evolution of the two mean gaps in170Sn,as noticed in previous Sn isotopes.For instance,at T=0.7MeV the gap is two times higher for a surface pairing interaction than for a volume one.Hence pairing correlations at temperatures below T c in neutron-rich as well as in stable nuclei may provide information about the surface or volume type of the pairing interaction.

Calculations have also been performed in Ni isotopes,and the results are illustrated here for the case of the84Ni nucleus.The T c?0.5?T=0rule is still veri?ed,both with surface and volume type interactions.The surface and the volume pairing interactions lead to similar critical temperatures,with100keV variation between the two cases.The situation is analogous in104Sn(Fig.3)where the surface and volume interactions give the largest variation in T c, namely100keV.This upper limit shows that the absolute value of the critical temperature itself is rather independent of the nature of the pairing interaction.However,if measurements can reach such a resolution it would be of high interest to measure pairing properties at temperatures located just below T c:for instance,in104Sn at T=0.8MeV, the pairing gap is close to zero in the volume case whereas the gap remains?=1MeV in the surface case.This is also the case for84Ni.

Fig.7displays the speci?c heat for84Ni,obtained in the FT-HFB calculations.Due to the?nite step in temperature used in the HFB calculations,the speci?c heat displays a kink at the critical temperature instead of the usual singularity.Actually,due to the?nite size of the nucleus,the speci?c heat should have a smooth s shape behavior around the critical temperature[19].In order to get this behavior in the calculations,one needs to go beyond the HFB approach,e.g.,projecting out the number of particles and taking into account the thermal?uctuations[23].

IV.CONCLUSION

In conclusion,the critical temperature for vanishing of pairing correlations in hot nuclei appears to be rather insensitive to the surface or volume localization of the pairing force used in FT-HFB calculations.For all the neutron-rich tin isotopes studied,we have found that the critical temperature is given approximatively by T c?0.5?T=0for both type of pairing forces and for stable and unstable nuclei.Hence,the critical temperature could be deduced from the gap value at zero temperature.On the other hand,the pairing gap is strongly sensitive to the nature of the pairing interaction for temperatures just below the vanishing of pairing correlations,for the large majority of nuclei studied.This result should open an experimental investigation for the pairing interaction in nuclei.

[1]M.Sano,S.Yamasaki,Prog.Theor.Phys.29(1963)397

[2]A.L.Goodman,Nucl.Phys.A352(1981)30.

[3]J.L.Egido and P.Ring,J.Phys.G:Nucl.Part.Phys.19(1993)1.

[4]J.L.Egido and L.M.Robledo,Phys.Rev.Lett.85(2000)26.

[5]E.Khan,Nguyen Van Giai,M.Grasso,Nucl.Phys.A731(2004)311.

[6]N.Sandulescu,Phys.Rev.C70(2004)025801.

[7]K.Kaneko,M.Hasegawa,Nucl.Phys.A740(2004)95.

[8]https://www.sodocs.net/doc/031668574.html,nganke,D.J.Dean,W.Nazarewicz,Nucl.Phys.A757(2005)360.

[9]D.J.Dean and M.Hjorth-Jensen,Rev.Mod.Phys.75(2005)1.

[10]T.Frick and H.Muther,Phys.Rev.C68(2003)034310.

[11]M.Baldo,Nuclear Methods and the Nuclear Equation of State,Int.Rev.of Nucl.Physics Vol9(World-Scienti?c,

Singapore,1999).

[12]H.Muther and W.H.Dickho?,Phys.Rev.C72(2005)054313.

[13]L.G.Cao,U.Lombardo,P.Schuck,Phys.Rev.C74(2006)064301.

[14]G.Gori,F.Ramponi,F.Barranco,P.F.Bortignon,R.A.Broglia,G.Colo’,E.Vigezzi,Phys.Rev.C72(2005)011302.

[15]J.Dobaczewski,W.Nazarewicz,P.-G.Reinhard,Nucl.Phys.A693(2001)361.

[16]Y.Yu and A.Bulgac,Phys.Rev.Lett.90(2003)222501.

[17]N.Sandulescu,P.Schuck,X.Vinas,Phys.Rev.C71(2005)0254303.

[18]M.Guttormsen,M.Hjorth-Jensen,E.Melby,J.Rekstad,A.Schiller,S.Siem,Phys.Rev.C64(2001)034319.

[19]A.Schiller,A.Bjerve,M.Guttormsen,M.Hjorth-Jensen,F.Ingebretsen,E.Melby,S.Messelt,J.Rekstad,S.Siem,S.W.

Odegard,Phys.Rev.C63(2001)0213306(R).

[20]G.F.Bertsch and H.Esbensen,Ann.Phys.209(1991)327.

[21]E.Chabanat,P.Bonche,P.Haensel,J.Meyer,R.Schae?er,Nucl.Phys.A635(1998)231.

[22]Nguyen Van Giai and H.Sagawa,Nucl.Phys A371(1981)1.

[23]S.Liu,Y.Alhassid,Phys.Rev.C87(2001)022501.

116Sn128Sn170Sn

V0V ol(MeV.fm3)-220-197-220-240

-520-570-510

FIG.3:Mean value of the neutron pairing gap in104,116,124,128Sn,calculated with a volume pairing interaction(solid line)and a surface one(dashed line).

FIG.4:Mean value of the neutron pairing gap in130Sn(with SGII interaction)and170Sn(with SLy4interaction),calculated with a volume pairing interaction(solid line)and a surface one(dashed line)

FIG.5:Neutron pairing gaps in104Sn corresponding to the quasiparticle states of the N=50-82valence shell.

FIG.6:Neutron single quasiparticle energies in104Sn corresponding to the states of the N=50-82valence shell.

FIG.7:Speci?c heat in84Ni calculated with a surface pairing interaction.

-2

-1.5

-1-0.5

0012345678910

Vol.

Surf.

r (fm)

V p a i r (M e V )

0.5

1

1.5

T (MeV)

?n (M e V )

0.5

11.52

0.51

1.5T (MeV)

?(M e V )

0.51

1.5

T (MeV)

?(M e V )

0.511.5

2

T (MeV)

?n (M e V )

2

4

6

T (MeV)

E q p (M e V )

0102030

4050

T (MeV)

C V