Journal of Nuclear and Radiochemical Sciences,Vol. 3, No. 1, pp. 169–172, 2002

169

Collective Properties and Structure of Heavy and Superheavy Nuclei
I. Muntian,a,b Z. Patyk,a and A. Sobiczewski∗,aaSoltan Institute for Nuclear Studies, Hoża 69, PL-00-681 Warsaw, Polandb
Institute for Nuclear Research, Kiev, Ukraine
Received: November 11, 2001; In Final Form: November 11, 2001
Ground-state collective properties of heaviest nuclei, such as deformations and rotational energies, are studied. In
particular, the role of deformations of various multipolarities in the rotational energies is discussed. A large regionof even-even nuclei with proton, Z = 82–126, and neutron, N = 126–190, numbers is considered.

1. Introduction
The activity of various groups in studies of heaviest nuclei isfast increasing. Number of institutes performing experimentalstudies is growing. To laboratories, in which experimental studies are being done for already a long time1–4 join new ones.5–7
Besides synthesis and investigations of decay modes of thesenuclei, also spectroscopic studies of them are performed.8–10
Also theoretical studies intensify. Besides traditional macroscopic-microscopic approaches (e.g. References 11–14), fully
microscopic approximations (e.g. References 15–22) are used.
One of the objectives of theoretical research are shell effects.
These effects are especially well seen in heaviest nuclei, as thesenuclei just exist due to these effects (e.g. Reference 23). Also
shapes of these nuclei is a quite interesting question, as theseshapes are expected to be quite complex. In particular, deformations of higher multipolarities are expected to play a more
important role in these nuclei than in lighter ones. For example, deformations of as high multipolarity as 6 may decrease theenergy (i.e. increase the binding energy) of a nucleus by up toabout 1.5 MeV.24
The objective of this paper is a short discussion of collective
properties of heaviest deformed nuclei. These are such properties as deformations and rotational properties. In particular, therole of deformations of various multipolarities played in rotational properties of these nuclei is discussed.

Figure 1. Contour map of the ground-state shell correction energy Esh .

Figure 2 illustrates neutron single-particle spectra for thethree double magic nuclei, for which the minima of the energy
Esh have been obtained in Figure 1. One can see that a large
energy gap, obtained in a rather uniform spectrum of doubly de-

2. Method of the Calculations
The ground-state energy of a nucleus is calculated in a macroscopic-microscopic approach. The Yukawa-plus-exponential
model25 is taken for the macroscopic part of the energy and theStrutinski shell correction is used for the microscopic part. TheWoods-Saxon single-particle potential, with the universal variant of its parameters found in Reference 26 and also specified
explicitly in Reference 11, is taken as a basis for the shell correction.
Moment of inertia of a nucleus is calculated in the cranking approximation (cf. e.g. Reference 27 and references given
therein).
3. Shell Structure
Figure 1 shows a contour map of the ground-state shell correction, Esh , calculated for a large region of nuclei with proton
number Z = 82–120 and neutron number N = 126–190.28 One
can see that, besides reproducing minimum of energy for the experimentally well-known doubly magic spherical nucleus 208 Pb,
the figure predicts two minima in the superheavy region: onearound the nucleus 270 Hs, predicted to be deformed, and theother around the nucleus 298 114, predicted to be spherical. It
is interesting to note that the two minima, corresponding to deformed and spherical nuclei, are of comparable depths.
∗Corresponding author. E-mail: sobicz@fuw.edu.pl.

Figure 2. Neutron single-particle energy levels calculated for the doubly magic nuclei: 208 Pb, 270 Hs, and 298 114.

c 2002 The Japan Society of Nuclear and Radiochemical Sciences
Published on Web 6/30/2002

170 J.Nucl.Radiochem.Sci.,Vol. 3, No. 1, 2002

MuntianFigure 3. Contour maps of the equlibrium deformations β0λ , λ = 2, 4, 6, 8, plotted as functions of proton Z and neutron N numbers.

generate neutron levels, at the neutron number N = 162 (as well
as a smaller gap at N = 152), is a quite spectacular effect.
4. Role of Deformations of Various Multipolarities
The equilibrium deformation parameters β0λ of a nucleus arefound by minimization of the potential energy E in the deformation space {βλ }. As also higher multipolarity components are found to be important for heaviest nuclei,29 a large,
7-dimensional deformation space {βλ }, λ = 2, 3, ..., 8, is takenin the analysis.
Figure 3 shows values of β0λ , λ = 2, 4, 6, 8, obtained in such ananalysis.30 Here, only even-multipolarity components β0λ of deformation are shown, as the odd ones β0λ , λ = 3, 5, 7, are obtained
different from zero only for a very few nuclei in the consideredregion.
One can see in Figure 3 that most of the investigated nuclei
are deformed. Only two, relatively small, regions of spherical
nuclei appear: one (smaller) region of nuclei near to those withclosed neutron shell at N = 126, and the other (larger) near tonuclei with closed neutron shell at N = 184. Relatively small
effects of weaker proton spherical closed shells at Z = 82 andZ = 114, on shapes of nuclei, are also visible.
One can also see that the main, quadrupole, component of
the deformation, β02 , is the biggest. It is large (β02 ≈ 0.24)
and about constant in a large part of the studied region andrapidly decreases as one moves to the boundaries of this region.
The higher-multipolarity components change sign as one moves
across the region. Although smaller than β02 , these components
play a significant role, more significant than for lighter nuclei,
as mentioned in the Introduction.
Here, we will illustrate this role in the calculated value ofenergy of the lowest 2+ state of an even-even nucleus. Thisstate is expected to be of rotational nature. Measurement of thisenergy, for a nucleus in the region around the nucleus 270 Hs, may
solve an important problem, if these nuclei are really deformed,
as expected on the theoretical ground. The problem has been
extensively discussed in Reference 27.
Figure 4 shows the energy E2+ calculated for a large regionof even-even heaviest nuclei. One can see that the lowest valuesof E2+ are in the range of 40–55 keV. They are obtained fornuclei with closed neutron shells at N = 152 and 162. When one
moves, however, with N off the closed shells, values of E2+ fast
increase.
It is interesting to learn the dependence of E2+ on the dimension λmax of the deformation space, with the use of which the energy E2+ is calculated. Figure 5 shows this dependence for four
heavy nuclei: 252 No, 254 No, 266 Sg, and 270 Hs. One can see that,for the nucleus 254 No, the inclusion of the hexadecapole component β4 of deformation to the analysis has almost no effect on
E2+ , while the inclusion of deformation of so high multipolarity
as 6, β6 , is very important. It decreases E2+ by about 20%.
For 270 Hs, however, the hexadecapole deformation β4 is crucial. Its inclusion decreases E2+ by more than 30%. Exampleof 266 Sg shows that also β8 may have a significant influence onE2+ . Figure 5 tells us how important is the role of deformationsof various multipolarities for E2+ , and how fast it changes when
one moves from one nucleus to another. Thus, although indirectly, measurement of E2+ may tell us quite much about the(complex) shape of a very heavy nucleus.

Figure 4. Dependence of the energy E2+ on neutron number N, calculated for elements with proton number Z = 102–112. For each element,values of considered N are specified below the value of Z.

Collective Properties and Structure

J.Nucl.Radiochem.Sci.,Vol. 3, No. 1, 2002

171Figure 5. Dependence of energy E2+ on the dimension λmax of deformation space used in the calculation of this energy.

Acknowledgements. The authors would like to acknowledge
a support by the Polish State Committee for Scientific Research(KBN), Grant No. 2 P03B 117 15.
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