Physics Letters B 526 (2002) 322–328

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Cluster interpretation of parity splitting in alternating parity bands

T.M. Shneidmana,b, G.G. Adamiana,b,c, N.V. Antonenkoa,b, R.V. Jolosa,b, W. Scheida

a Institut für Theoretische Physik der Justus-Liebig-Universität, D-35392 Giessen, Germanyb Joint Institute for Nuclear Research, 141980 Dubna, Russiac Institute of Nuclear Physics, Tashkent 702132, Uzbekistan

Received 31 July 2001; received in revised form 16 November 2001; accepted 27 December 2001

Editor: J.-P. Blaizot

Abstract

The parity splitting in actinides is described with a cluster model of oscillations in mass asymmetry coordinate. The spindependence of the calculated parity splitting is in a good agreement with the experimental data. 2002 Elsevier Science B.V.All rights reserved.

PACS: 21.60.Ev; 21.60.Gx

Keywords: Cluster states in nuclei; Dinuclear system; Parity splitting; Actinides

The observation of low-lying negative parity statesnear the ground state have shown that many ac-tinides isotopes have reflection-asymmetric shapes[1,2]. However, in these nuclei the positive and neg-ative parity states being considered together do notform undisturbed rotational bands as in the case of theasymmetric molecules. At small spinsI the negativeparity states are shifted up with respect to the posi-tive parity states. This shift (parity splitting) decreaseswith increasingI and disappears at some value ofspin which varies from nucleus to nucleus, although inseveral nuclei small oscillations of the parity splittingaround zero are observed for largeI [3]. Thus, in con-trast to molecules, in nuclei the potential barrier be-tween a shape with reflection asymmetric deformationand its mirror image, if existing, is not large enough to

E-mail address: werner.scheid@theo.physik.uni-giessen.de(W. Scheid).

prevent a barrier penetration. The penetration throughthis barrier lowers the energies of levels with evenIwith respect to the energies of levels with oddI . How-ever, with increasing spin the barrier becomes higherand the penetration probability goes to zero. Then wefind almost ideal alternating parity bands.

Thus, there are two experimental characteristicsof the alternating parity bands which should be ex-plained: the parity splitting at the beginning of the ro-tational band and the critical value of spin at which theparity splitting disappears.

It was recognized that low-lying negative-paritystates can be described by an octupole or dipole de-gree of freedom, and many studies have explored thispossibility. The parity splitting related to octupole de-formation was treated in Refs. [4,5] within the self-consistent microscopic model with parity projection.Phenomenological interpretation of the spin depen-dence of the parity splitting was realized in [6,7] where

0370-2693/02/$ – see front matter 2002 Elsevier Science B.V. All rights reserved.PII: S0370-2693(01)01512-X

T.M. Shneidman et al. / Physics Letters B 526 (2002) 322–328 323

the suggested simple exponential formula with two pa-rameters describes the experimental data quite well.However, the parameters vary significantly from iso-tope to isotope and we need a nuclear structure modelto understand these variations.

The aim of the present Letter is a formulation of aquite simple model which provides a quantitative ex-planation of the variations from nucleus to nucleus ofthe observed values of the parity splitting at the be-ginning of the rotational bands and the critical angu-lar momentum at which the parity splitting disappears.It is shown below that collective oscillations of a nu-clear shape, which lead to the formation of cluster-typeshapes, explain the observed properties of the paritysplitting in actinides.

The idea that alpha clustering [8–10] explains anappearance of low lying negative parity states in ac-tinides was explored in Ref. [11] within a phenomeno-logical model based on group theoretical methods. In[11,12] nuclear ground states correspond to dipole vi-brations rather than to rigid molecular-like dipole de-formations. Corresponding wave functions consist ofα-cluster and mononucleus components. In contrast tothis approach a cluster configuration with fixed massasymmetry is proposed in [13] to describe the proper-ties of the low-lying positive and negative parity statesin actinides. The cluster heavier thanα-cluster is takenin [13] as a light cluster.

The nuclear systems consisting of a light clusterA2 plus a heavy clusterA1 belong to the class ofdinuclear-type shapes. They were first introduced toexplain data on deep inelastic and fusion reactionswith heavy ions [14]. Instead of the parameterizationof the nuclear shape in terms of quadrupole (β2), oc-tupole (β3) and higher multipole deformations, themass asymmetryη= (A1 −A2)/(A1 +A2), (η= 1 ifA2 = 0 andA1 =A) and the distanceR between thecenters of clusters are used as relevant collective vari-ables [15]. Sinceη is a dynamical variable, the groundstate wave function is a superposition of differentcluster-type configurations including the mononucleusconfiguration with|η| = 1. In the present Letter weconsider a motion inη which leads to an appearance ofdifferent cluster states and a mononucleus configura-tion with certain probability. The relative contributionsof each cluster component in the total wave functionare determined by the potential energy of the collectiveHamiltonian described below. Our calculations have

shown that in the considered cases the dinuclear con-figuration (η = ηα) AZ → (A−4)(Z− 2) + 4He withan alpha cluster has a potential energy which is closeor even smaller than the energy of the mononucleus at|η| = 1 [16]. Since the energies of the configurationswith a light cluster heavier than anα-particle increaserapidly with decreasing|η|, we can restrict our inves-tigations to configurations with light clusters not heav-ier than Li (η= ηLi ), i.e., to cluster configurations near|η| = 1 and not too high spins. For large angular mo-menta, the other cluster configurations can be also im-portant for treating the oscillations inη. The symmet-ric cluster configurations withη ≈ 0 have a relativelylow potential energy as well. However, these config-urations are characterized by very large quadrupoledeformations and rather correspond to hyperdeformedstates [16]. They are separated from the mononucleusconfiguration by a large barrier.

The potential energy of the DNS is expressed as

(1)U(R,η, I)= B1(η)+B2(η)+ V (R,η, I),whereB1 andB2 are the experimental binding ener-gies of the DNS nuclei at a given mass asymmetryη.Shell effects and pairing correlations are included inthese binding energies. It is known [17,18] that thereis a strong correlation between the setting in of anoctupole deformation and the quenching of the pair-ing correlations. However, the pairing interaction be-tween the nucleons of different clusters is not takeninto consideration because the pairing interaction ma-trix elements are rather small for touching clusters.The quantityV (R,η, I) in (1) is the nucleus–nucleuspotential. It is given asV (R,η, I) = Vcoul(R,η) +VN(R,η)+Vrot(R,η, I) with the CoulombVcoul, cen-trifugal Vrot = h̄2I (I + 1)/(2�(η,R)) and nuclear in-teractionVN potentials. The nuclear potentialVN canbe obtained by averaging the nucleon–nucleon interac-tion over the generator coordinate (GC) wave functionof a system of two clusters. The distanceR plays therole of a generator coordinate and the potential is givenby the diagonal part of a corresponding integral kernelof the GC method. The 0+ ground state wave func-tions are taken as the cluster wave functions. In fact,only the direct part of the integral GCM kernel whichneglects antisymmetrization effects is taken into ac-count and as a consequence a double folding form ofthe potentialVN with the ground state nuclear densi-ties is obtained. Antisymmetrization between nucleons

324 T.M. Shneidman et al. / Physics Letters B 526 (2002) 322–328

belonging to different clusters is imitated by a den-sity dependence of the nucleon–nucleon forces whichis responsible for the repulsive core in the cluster–cluster interaction potential. This density dependencedescribes a change of the nucleon–nucleon interactionfrom an attraction in the region of low nuclear densityto a repulsion when the nuclear density is sufficientlylarge, thus taking into account the effect of the Pauliexclusion principle. Details of calculations ofVN aregiven in [19]. The parameters of the nucleon–nucleoninteraction are fixed in the nuclear structure calcula-tions [20]. The nucleon density distribution is approx-imated by the Fermi distribution with the radius para-meterr0 = 1.15 fm. While the diffuseness parametera for 4He and7Li is taken as 0.48 fm (in Ref. [16] (seeFig. 5 of [16]) we used 0.55 fm for allη that is toolarge for |η| � 0.85 and produces too small potential

energy at|η| = |ηα|), we seta = 0.56√B(0)n /Bn fm for

heavy nuclei, whereBn andB(0)n are the neutron bind-ing energies of the studied nucleus and of the heaviestisotope considered for the same element, respectively.For example, in the case of Ra, Th and U isotopes,B(0)n corresponds to226Ra, 232Th and 238U, respec-

tively.The potential taken as a function ofR has a pocket

and the DNS is localized in the minimum of thispocket atR = Rm corresponding to the touchingconfiguration with a possible deformation of the heavycluster whose characteristics are taken from [21]. Therelative orientation of the deformed nuclei in the DNSfollows the minimum of the potential energy whichcorresponds to the pole-to-pole orientation.

The nucleus–nucleus potentialV (R,η, I) and po-tential U (“driving potential”) were successfully ap-plied to the analysis of the experimental data on fusionand deep inelastic reactions with heavy ions [22,23].

In Ref. [13] the cluster configuration with a lightercluster heavier than4He is determined as the mostimportant one by the “maximum stability condition”which selects the configurations with the largest devi-ation ofB1 + B2 from the corresponding liquid dropvalue. Since in our treatment the overlap of clustersis much smaller than in the model given in [13], thechoice of relevant cluster configuration follows theminimum of the driving potentialU , i.e., the inter-actionV (R,η, I) is taken in (1) into account. As aresult we describe the same nuclear properties as in

[13] with cluster configurations having larger|η| butsmaller overlap of the clusters.

To calculate the potential energy atI = 0, themoment of inertia�(η,Rm) for the cluster systemshas to be defined. It is known that the moments ofinertia of superdeformed states are about 85% of therigid-body limit [24]. As was shown in [16], the highlydeformed states are well described as cluster systems.Therefore, we assume that the moment of inertia ofcluster configurations withα and Li as light clusters isdescribed by the expression

(2)�(η)= c1(

�r1 + �r2 +m0A1A2

AR2m

).

Here, �i (i = 1,2) are the rigid body moments ofinertia for the nuclei constituting the DNS,c1 = 0.85for all considered nuclei andm0 is the nucleon mass.

For |η| = 1, the value of the moment of inertia isnot known from the data because the experimentalmoment of inertia takes intermediate value betweenthose for the mononucleus (|η| = 1) and for clusterconfigurations arising due to the oscillations inη. Weparameterize�(|η| = 1) as

(3)�(|η| = 1) = c2�r ,

where�r is the rigid body moment of inertia of themononucleus calculated with deformation parameters[21] and c2 is a scaling parameter which is fixed todescribe the energy of the first 2+ state (it can bedone also for any other even parity state). The cho-sen values ofc2 vary in the small interval 0.1< c2<0.3. So, in our calculations there is a single free pa-rameter which is used to fit the energies of one ofthe known rotational states. It should be noted thatthe rotational states of the alternating parity bandsare characterized by the energies of the positive par-ity states and the angular momentum dependence ofthe parity splitting. Our aim is a description of theparity splitting characteristics only. However, the pa-rameterc2 is used to describe the gross behaviorof the rotational band. In220,222,224,226Ra isotopes�(|η| = 1)= 12, 17, 22 and 32̄h2/MeV, respectively.In 222,224,226,228,230,232Th isotopes�(|η| = 1)= 12,20, 30, 48, 52 and 55̄h2/MeV, respectively. In theconsidered U isotopes�(|η| = 1)≈ 56 h̄2/MeV. Usu-ally, the pairing interaction is important for a correctdescription of the moment of inertia of the mononu-cleus. However, in our calculations this moment of

T.M. Shneidman et al. / Physics Letters B 526 (2002) 322–328 325

inertia is a single parameter fixed to describe the en-ergy of one of the experimental rotational even paritystates. Note that the angular momentum is treated inthis Letter very roughly. We assume that it consists oftwo parts, of the angular momentum of the collectiverotation of a heavy cluster and of the orbital momen-tum of the relative motion of two clusters. Single parti-cle effects are neglected. In principle, alignment of thesingle-particle angular momentum can produce a mix-ing of Kπ = 0− state considered in this Letter withKπ = 1−,2−,3− states.

Determining the potential energy (1), we substi-tute the experimental masses of the clusters and cal-culate their nuclear and Coulomb interactions. A spe-cific point is |η| = 1. Since the ground state wavefunction is distributed inη, the potential energy at|η| = 1 is fixed so as to reproduce the experimen-tal binding energy of theZA nucleus with respectto U(ηα). We vary the value ofU at |η| = 1 andsolve the Schrödinger equation forη-motion up to themoment when the experimental value of the groundstate energy is obtained. If the potential energy oftheα-cluster configuration is smaller than the exper-imental binding energy in the ground state like in220,224,226Th and220,222,224,226Ra, then the potentialenergy has a minimum atη = ηα and a maximum at|η| = 1. In the other cases the potential has a minimumat |η| = 1.

In order to solve the Schrödinger equation inη, asmooth parameterization of the potential

U(x, I)=U(xα, I)+4∑k=1

a2k(I )(x2k − x2k

α

),

x = −η+ 1 if η > 0,

(4)x = −η− 1 if η� 0,

is used which goes through the calculated values of thepotential energy at|η| = 1, η = ηα andη = ηLi . Thecalculations with other parameterizations show almostno difference in the description of parity splitting inthe considered nuclei. If the minimum of the potentialis located atη = ηα , three parametersa2, a4 anda6 aredetermined so as to fix the position of the minimum atthe right place, to get correct values ofU for a clusterconfiguration with Li as the light cluster and to fix theground state energy, which we obtain after solving theSchrödinger equation, at the right position with respectto U(ηα). The fourth parametera8 is necessary to

Fig. 1. Comparison of experimental (points) and theoretical (lines)rotational spectra for238,236,234,232U. The experimental data aretaken from [26].

avoid a fall down of the potentialU when |η| � ηLiwhich appears because of the negative value ofa6needed to describe correctlyU(ηLi ). Since this falldown ofU can influence our calculations, we take theminimal necessary positive value ofa8 to guarantee anincrease ofU for |η| � ηLi . If the minimum is locatedat |η| = 1, only two parameters are necessary.

The Hamiltonian describing a motion inη has thefollowing form:

(5)H = − h̄2

2Bη

d2

dx2 +U(x, I),

326 T.M. Shneidman et al. / Physics Letters B 526 (2002) 322–328

Fig. 2. The same as in Fig. 1, but for232,230,228,226,224,222Th.

where Bη is a constant effective mass parameter.The method of calculation of the mass parametersfor the DNS is given in [25]. Calculations show thatBη is a smooth function of the mass number. Sincewe study nuclei in the same mass region, we takeBη = 20× 104m0 fm2 for all considered nuclei. Usingthis value ofBη and the connection between massasymmetry and multipole expansion coordinates ofRef. [16], one can find that the mass parameter forβ3vibrations is close to the value 200̄h2/MeV knownin literature [17]. Solving the eigenvalue problemof the Hamiltonian (5), we obtain the spectrum ofη-vibrations and the parity splitting as a function ofI .

Fig. 3. The same as in Fig. 1, but for226,224,222,220Ra.

The eigenfunctions of the Hamiltonian (5) have a welldefined parity with respect tox→ −x reflection. Thechange of the sign ofx is equivalent to the change ofthe sign ofη and therefore is equivalent to the spatialreflection because, as it is seen from the definitionof η, the change of the sign ofη is equivalent to thepermutation of clusters.

Results of the calculations are shown in Figs. 1–3.They demonstrate a good agreement with the exper-imental data [26] for all nuclei considered excludingthe lightest isotopes of Ra. A satisfactory descriptionof the experimental data, especially of the variation of

T.M. Shneidman et al. / Physics Letters B 526 (2002) 322–328 327

Fig. 4. Potential energy (solid curve) and the wave functionsof the lowest positive (long-dashed curve) and negative parity(short-dashed curve) states for224Ra.

the parity splitting at lowI and of the value of thecritical spin at which the parity splitting disappearswith A means that the variation of the potential en-ergy as a function ofη andI for the isotopes of Ra, Thand U is correctly described by our cluster model. Theuse of the correct value of the inertia coefficientBη isalso important. For the same potential energy with aminimum atη = ηα , we can obtain a maximum of theground state wave function at about|η| = 1 for smallBη or a concentration of the ground state wave func-tion near|η| = ηα for very largeBη.

The calculated ground state wave function has itsmaximum in the vicinity of|η| = 1 even when thepotential energy has a minimum atη = ηα becausethis minimum is rather shallow (it does not exceed0.8 MeV), and the inertia coefficientBη used in thecalculations is not large (see Fig. 4). Thus, the groundstate energy level lies near the top of the barrier andthe estimated weight of theα-cluster configuration isabout 2×10−2 for 226Ra, that is close to the calculatedspectroscopic factor [27]. This means that our modelis in agreement with the knownα-decay widths for thenuclei considered.

Using the wave functions obtained, we calculatedthe multipole momentsQ1, Q2, Q3 andQ4. The ob-tained values are in good agreement with the knownexperimental data forQexp

λ [2]. For example, for226Ra we findQ1 = 0.10 e fm (Qexp

1 = 0.1 e fm)Q2 = 740e fm2 (Qexp

2 = 750e fm2),Q3 = 3200e fm3

(Qexp3 = 3100 e fm3) andQ4 = 19 000e fm4. These

first results allow us to hope for a good descriptionof transitions probabilitiesB(Eλ) and multipole mo-mentsQλ for different isotopes of Ra, Th and U in aforthcoming publication.

In conclusion, we suggested a cluster interpretation(oscillations in the mass asymmetry coordinate) of theparity splitting in different Ra, Th and U isotopes. Theexisting experimental data on the spin dependence ofparity splitting were quite well described. The char-acteristics of the Hamiltonian used were determinedby investigations of completely different phenomena,namely, heavy ion reactions at low energies.

Acknowledgements

G.G.A. and T.M.S. are grateful to the Alexandervon Humboldt-Stiftung and BMBF, respectively. Thiswork was supported in part by DFG and RFBR, byVolkswagen-Stiftung, STCU, SCST and UFBR.

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