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REF"EFiEN CG ic/89/27












O. Dumitrescu


M. Horoi



Internationa] Atomic Energy Agency and

United Nations Educational Scientific and Cultural Organization INTERNATIONAL CENTRE FOR THEORETICAL PHYSICS


O. Duraitrescu ** and M. Horoi **

International Centre for Theoretical Physics, Trieste, Italy, Istituto Nazionale di Fisica Nucleare, Sezione di Trieste, Trieste, Italy


Istituto Nazionale di Fisica Nucleare, Sezione Sanita Roma, Rome, Italy.


A superfluid isomer by analogy with the fission (shape) isomers is theoretically predicted and found among the 0+ excited states of 1MSm.


Submitted for publication.

Permanent address: Department of Fundamental Physics, Institute of Physics and Nuclear Engineering, Central Institute of Physics, P.O.Box MG-6, Magurele, Bucharest, R-76900, Romania.

The name "superfiuid isomer" we have chosen in analogy to the fission isomers [l] which are actually shape iflomers corresponding to second minima in the potential energy along the elongation degree of shape deformation. The analogy can be done in the Bohr-Mottelson spirit of symmetry breaking [2]. The appearance of the pairing rotations [3], for instance, as a mode of elementary excitations, is related to the symmetry breaking of the nucleon number conservation. This type of symmetry breaking generates superfiuid phases [4-6] of atomic nuclei characterized by non-zero static deformations (Art(p)} of the pairing

fields by analogy with the rotational symmetry breaking that generates the non-zero static deformations (/3ao and 0io) of the selfconsistent field and, hence, the rotational levels in

the excitation spectrum of the atomic nuclei.

Pairing interactions are typical effective residual interactions in medium and heavy deformed nuclei together with the long range ones. They lead to the pairing and long range correlations in the mentioned nuclei. By including an adequate four-nucteon effective interaction in addition to the pairing and long range (e.g. multipole-multipole) ones, global correlations between proton and neutron fluids other than pairing and long range correlations are induced [6j and the phase structure is enriched by a new superfiuid phase - the so-called alpha type superfiuid phase - dominated by the alpha type correlations (ATC) [6|.The superfiuid isomers correspond to second (local) minima of the correlation energy versus static deformations Ap(nj of the pairing and four-nucleon fields.

Our problem is to minimize the energy functional [6|

= 2 (1)


4 I

with respect to the variational parameters vti subject to the constraints

where Nn(p) is the number of protons (neutrons) contained in their respective A - shells

that defines the cutoff and E(. are the renormaiized single particle level energies. This

problem is analogous to minimizing the functional


with respect to vtp(,t and Xp^ from which we obtain the equations


= « + A?)*

getting,thus, A; = A,(A?) and the gap equations

(6) (7)


or to minimize the functional (1)

E = E(u,j,(A2),ui n(A^), AP(A2), A^(A^)) (9)

with respect to A ^ , ) subject to the constraints (7). The functions v,p{n](A^, -.} should

have the parametrization (5).

The role of the potential energy for the fission isomers, in our case is played by

E as function of Ap and A j - static deformations of the pairing and four-nucleon fields.

Let us analyse the g22^mfto - case. This nucleus presents well defined rotational

bands [7] corresponding to the following intrinsic K" - states: 0+ - ground state, 0+

-/?-vibrational state (E = 0.685 MeV), 2+ - f-vibrational and l~-octupole - vibrational

states.There are three additional intrinsic 0+ - states at E2 = 1.082 MeV, Es = 1.656 MeV

and E4 = 1.736 MeV. On the second Oj - state (E2 = 1.082 MeV) is built a rotational

band with a smaller moment of inertia than for the ground and ^-vibrational bands mo-ments of inertia. This is an indication for the existence of the superfluid isomer in this nucleus, because the second minimum corresponds to larger gap parameters (see Table 1 and Figs.l and 2) implying thus a smaller moment of inertia. A possible interpretation of the second 0+ - state as a two - quasiparticle state or a neutron pairing vibrational state

does contradict the experimental evidence concerning the moment of inertia (see Fig.3). The calculations [8,9] for the theoretical data in Table 1 and Figs.l and 2 have been done using the Woods-Saxon single particle levels with the deformation (02 = °-3, /?4 = 0.05) and shape parameters taken from Ref,4 page 21 and the ESM [6,8,9] parameters Cp = 22.4

MeV ,Cn — 16.2 MeV and C4 = 28 MeV. In Table 1 we show the experimental and

theo-retical energies corresponding to the ground state superfluid band (APl = 0.597 MeV,Ari

= -6.938 UeV;Ani =0,586 MeV,An, = -6.372 MeV) and the isomeric state superfiuid band

(A,,, = 4.567 MeV,AP3 = -7.117 MeV;Anj = 3.920 MeV, An, = -6.518 MeV) i.e including

the /9-vibrational states [9] elementary excitations. In this table we did not include the corresponding rotational levels (see Fig.3). The existence of the first positive RPA solution smaller than the energy of the first pole corresponding to the /J-vibrational state of the superfluid isomer confirms the stability of our model in describing this superfluid isomer and the new superfluid phase of the atomic nucleus - the alpha type auperfluid phase.Fig.l shows the positions of the global and local minima verus the proton and neutron gap pa-rameters, while in Fig.2 we reproduced a section of Fig.l along the straight line connecting

the two minima. We see a barrier height of 987 MeV with respect to the global minimum (i.e. the ground state) telling us that the gamma or E0 deexcitation of states belonging to the isomerie superfluid band to the states belonging to the ground state superfluid band (pairing superfluid band) should be inhibited [10].

The situation described here could be compared to the case of the superdeformed fission isomers. Traditionally, most of us would think of a superdeformed nucleus as a pro-late shape object with the length-to-width (axis ratio) very close to 2:1. The latter picture evolved from studies of fission isomers in actinide region. The quadrupole deformation of these isomers is about /?2o = 0.65 i.e. three times larger than the corresponding

deforma-tion of the ground states.This indeed corresponds well with the 2:1 axis ratio geometry which, moreover, is supported in qualitative terms by the characteristic shell pattern of the harmonic oscillator spectrum. The gap parameters for the superfluid isomer of l s sSm

are approximately in the same ratio with the gap parameters of the 1MSin-ground state

as for the above mentioned fission isomers. The analogy perhaps could once more confirm the existence of the superlluid isomers prdicted by the ESM.

To conclude, in this paper we predict theoretically the so-called superfluid isomer which may be an evidence for a new superfluid phase of atomic nuclei - the so-called alpa type superfluid phase - mentioned for the first time in the Refs.6.This isomer in the gauge space associated to the particle number conservation plays the same role as rotational symmetry breaking fission (shape) isomers do with respect to the ordinary geometrical space. New bands of rotational and vibrational excitations corresponding to these super, fluid isomers are also predicted.


The authors would like to thank Professor Abdus Salam, the International Atomic Energy Agency and UNESCO for hospitality at the International Centre for Theoretical Physics, Trieste. They would also like to thank Profs. Claudio Ciofi degii Atti and Luciano Fonda for many useful discussions and suggestions and also Profs. L.Fonda, C.Ciofi degli Atti, N.Paver and E.Castelli for their warm hospitality at ICTP, Trieste and INFN, Roma and Trieste.



[lj S.Polikanov, V.A.Druin, V.A.Karnauchov, V.L.Micheev, A.A.Pleve, V.K.Skobelev, V.G.Subbotin, G.M.Ter-Akopyan and V.A.Fomincev Journ.Exper.i teor.Fiz. 42 (1962) 1464; Sov.Phys.JETP 15(1962) 1016.

[2] A.Bohr and B.Mottelson,Nuclear Structure,vol.II (Benjamin, N.Y., 1974). |3] D.R.Besa and R.A.Broglia, Proc.International School "Enrico Fermi" Course


\i\ V.G.Soloviev, Theory of Complex Nuclei(Nauka, Moscow, 1971); (Pergamon Press,

New York, 1976).

[5] D.R.Bess and R.A.Sorensen,Advance Nucl.Phys.2(1969)129.

[6] M.Apostol, I.Bulboaca, F.Carstoki, O.Dumitrescu and M.Horoi, Europhys. Lett.4(2),

(1987) 197; Nucl.Phys. A470 (1987) 64. [7j C.M.Baglin, Nucl.Data Sheets 30(1980)1.

[8] O.Dumitrescu, M.Horoi and I.Bulboaca, ICTP, Trieste preprint IC/89/14 (1989). [9] O.Dumitrescu and M.Horoi, ICTP, Trieste preprint IC/89/19 (1989).

[10] I.Btilboaca, O.Dumitrescu and M.Horoi Rev.Roum.Phys. 34, (1989) in press.

Table 1

The experimental [7] and the calculated, within the ESM, energies of the first low-lying 0+-states separated in two superfluid bands.

Superfluid band ground state isometric state I'K 0+0 0+0 0+0 0+0 EclI,p(Ue\ 0. 0.685 1.082 1.737? r) Eth 0. 0.670 0.837 4.500


Figure captions

Fig.l The correlation energy (9) versus proton and neutron gap energies.The two min-ima have the co-ordinates described in the text.

Fig.2 The profile of the correlation energy versus proton gap along the path between the two minima (see Fig. 1).

Fig.3 The experimental rotational bands built on the ground and first two excited (Te-states of 1MSm nucleus.


5 h

4 h


1 r

A -O.91835E.-O3 B -0.918O7E+03 C -0.91780E+03 D -0.91752E+O3 E -0.91724E+03 I' -0.91696E+03 G -0.91669E+03 H ^0.91641Ei-03 I -0.91613E+03 J _0.9158bEt03


Z 3 4 5 6 A



Fig. 1


E'(Mev) 5 6 AD (Mev) F i g . 2 2.O 1.5 1.0 O.5 Excitation Energy 2.149 12+O t6O9 1OO 1.730 4*0 1.736 O*0 1.656 O*O 1.31O 6*0 1.293 2*0 1.125 8*O 1.O82 O O 1O22 4*O Superftuid isomer a81O 2*O -band O.7O7 - band O.366 4*0 Q.121 2 ' 0 O O*O M e V 152 Sm 6 2 90 I'ig-10


Stampato In proprlo nella tlpografia

del Centro Intemazlonale dl Fislca Teortca


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