Orbits, shapes and currentssfrauend/conferences/mesoscopic/... · 2011. 9. 22. · density ρ(rE). Finding the energies i and wavefunctions ψ i of the nucleons in the potential V[ρ(rE)]

Orbits shapes and currents

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INSTITUTE OF PHYSICS PUBLISHING PHYSICA SCRIPTA

Phys Scr T125 (2006) 1ndash7 doi1010880031-89492006T125001

Orbits shapes and currentsStefan Frauendorf

Department of Physics University of Notre Dame Notre Dame IN 46556 USAandIKH FZ-Rossendorf PF 510119 01314 Dresden Germany

E-mail sfrauendndedu

Received 14 July 2005Accepted for publication 1 September 2005Published 28 June 2006Online at stacksioporgPhysScrT1251

AbstractShapes and current distributions of nuclei and alkali clusters are discussed in terms ofthe single particle motion of fermions in the average potential For small particle numberan interpretation in terms of the lowest spherical harmonics is presented For largeparticle number an interpretation in terms of classical periodic orbits is presented

PACS numbers 2110Re 2110Dr 2320Lv 2410Pa 3640Cg 7322minusf 7323Ra

1 Introduction

Sven Goumlsta Nilssonrsquos work [1] that is commemorated at thismeeting treated for the first time the motion of nucleons in arealistic spheroidal potential generated by all other nucleonsThe famous Nilsson diagrams which show the single particleenergies as functions of the deformation parameter gave us afirst quantitative estimate of the nuclear shape This seminalpaper comprises in an ingeniously simple way the concept ofthe mean field approach

inte[ρ(Er)] dτ rarr t + V [ρ(Er)] minusωlzψi = εiψi rarr ρ(Er)

The average potential V [ρ(Er)] is generated by variationof some energy density functional with respect to the nucleondensity ρ(Er) Finding the energies εi and wavefunctions ψi ofthe nucleons in the potential V [ρ(Er)] permits us to calculatethe density ρ(Er) Making the potential self-consistent withdensity determines the shape of the density distribution andthe potential which is called the nuclear shape Inclusion ofthe cranking term minusωlz permits us to calculate the currentdistribution Ej(Er) of a rotating nucleus that carries a finiteamount of angular momentum

Since Nilssonrsquos paper nuclear shapes have been studiedin great detail using a variety of mean field approaches Theyall gave very similar shapes Moreover the shapes of alkaliatom clusters were calculated in a mean field approximationsimilar to that of nuclei Here one calculates the energiesand wavefunctions of one electron in the average potentialgenerated by all other delocalized conduction electrons andthe positive ions which are treated as a lsquojelliumrsquo of positivecharge As discussed later the cluster shapes are very similar

to nuclear shapes These examples illustrate the generalobservation that the shapes of all finite fermion systems thathave near constant density in the interior which goes tozero within a thin surface layer are essentially the sameThese shapes reflect the quantized motion of the fermionsin a leptodermic potential The relation between this quantalmotion and the shape that the system takes will be discussedThere are two simple situations For small particle number theshape of the system reflects the shape of the lowest quantalorbitals Since the shape of the molecules reflects the shape ofthe s- and p-valence orbitals it is called the chemical regimeFor large particle number the combined geometry of manysingle particle orbitals determines the shape of the system Ifone is interested only in the gross structure the shapes can bediscussed in terms of periodic classical orbits of the fermionsin the average potential There is a certain analogy to theconstruction of a concert hall In order to avoid echoes onestudies how sound impulses bounce from the walls and travelthrough the interior (periodic orbits) It is called the acousticregime

Clusters are simpler than nuclei because only thedelocalized conduction electrons determine the shape Innuclei both the protons and the neutrons do this If bothprefer the same shape the resulting shape is similar to thoseof clusters If the protons and neutrons prefer different shapesthe result will be a compromise A comparisons of systematiccalculations for clusters as shown in figure 1 with those ineg [3] clearly reveals this simple interplay of protons andneutrons Hence only one kind of fermion as in the case ofclusters may be considered

The current distribution of rotating nuclei and alkaliclusters exposed to a magnetic field is also largely determinedby the quantization of the fermionic motion The two

0031-894906125001+07$3000 copy 2006 The Royal Swedish Academy of Sciences Printed in the UK 1

Stefan Frauendorf

Figure 1 The deformation parameters of heavy alkali clusters ascalculated in [2] using the shell correction method The deformationparameters αl approximately correspond to the standard deformationparameters βl introduced by a multipole expansion of the surface

phenomena are related as follows The external magneticfield induces electric currents in a cluster that are carried bythe conduction electrons These currents generate a magneticmoment A nucleus does not rotate like a rigid body Thecurrents that generate the total angular momentum consist oftwo parts

Ej(Er)= Ej rig(Er)+ Ej sh(Er) Ej rig(Er)= ρ(Er) Eωtimes Er (2)

The term Ej rig(Er) is the current of a rigid body that rotateswith the angular velocity Eω The deviation Ej sh(Er) is caused byquantized motion of the fermions It represents the residualcurrent in the rotating frame of reference that is fixed to thebody This residual currents correspond to the electric currentsin a cluster where the magnetic field should be expressed bythe Larmor frequency

2 The chemical regime

In the case of molecules the direction of the chemical bonds isdetermined by the density distribution of the valence s-orbital(isotropic spherical harmonic Y00) and the valence p-orbitals(the anisotropic spherical harmonics Y1m) The interactionwith the partners in the bonds may cause a strong mixing ofthe valence s- and p-orbitals which is called hybridizationThe hybrid orbitals have new shapes like the three bonds ofthe sp2 hybrid which lie in a plane with a mutual angle of120 In the lightest fermion systems only the s- p- and d-orbitals are occupied The shape of the system reflects theshape of these orbitals Figure 2 shows the shapes of thelightest alkali clusters and nuclei In most cases they closelyresemble the shapes of the density distributions obtainedby filling the s- p- and d-orbitals in a spherical potentialThe nodal structure of the first eigenmodes in a compactpotential has to be similar to these orbitals which means asimilar density distribution The admixture of higher shellsmodifies the radial profile of the orbitals such that the densitychanges from the interior value to zero within a thin surfacelayer Such a profile minimizes the total energy of the energydensity functional which has a minimum at the saturationdensity As a consequence the shapes (equidensity surfaces)resemble the surfaces R =

sumlm nlm |Ylm |

2 which are alsoshown Moreover the energy density functionals favour asmall surface area because of the positive surface energySince the surface tension of nuclei is larger than that ofclusters the nuclear surface follows less well the shapesof the valence orbitals because a more rounded shape hasa lower surface energy If the state lm 6= 0 is only halffilled the hybrid orbital (Ylm + Ylminusm)

radic2 is formed which

has a lower surface energy The hybridization is also drivenby the tendency to clusterization The N = 10 system is anexample As seen in figure 3 the calculated shape is somewhatreflection asymmetric The asymmetry can be generated bysome admixture of the harmonics Y2plusmn1 to the Y1plusmn1 and ofY30 to Y20 (see figure caption) The mixing coefficients of thishybridization are determined by the tendency to form the twoparticularly stable subsystems N = 8 and N = 2 The energygain in forming two closed shell clusters has to compete withthe loss due to the larger surface of the two fragments Theshape is a compromise showing partial clusterization As seenin figure 2 the shapes of the N = 12 16 and 18 systemsalso differ somewhat from the reflection-symmetric shapesgenerated by filling the spherical orbitals The asymmetry iscaused by a similar hybridization mechanism as discussed forN = 10 (see [4])

The rotational and magnetic response reflects the currentdistribution generated by combining the lowest orbitalssuch that the cranking term minusωlz is taken into accountFigure 4 shows the velocity field in the rotating frame ofreference Evsh(Er)= Ej(Er)ρ(Er)minus Eωtimes Er for the N = 4 systemThe presence of strong vortices demonstrates the dramaticdeviation of the current from rigid rotation The pattern isdominated by the transition current between the l = 1m = 0-like and l = 1m = 1-like orbitals without radial nodes It iscombined with some irrotational current The example is aharmonic oscillator potential at equilibrium deformation asdiscussed in [5] In this case the moment of inertia takes

282000

228 ))12(21(2000 minus+

2220 ))12(21(2000 minusminus minus+

1120 ))12(21(2000 minusminus minus+2000

283000

228 212000

182000

181000 minus

281000 minus

121000

221000

122 ))11(11(1000 minus+

22 00)00( === =nml

Figure 2 The shapes of the lightest alkali clusters and N = Z nuclei as calculated in [4] using the ultimate jellium energy densityfunctional Shown are the surfaces of half density In addition the surfaces R =

sumlm nlm |Ylm |

2 are shown where the occupation number nlm

of each orbital is given as (lm)n

Figure 3 The shape of the N = 10 cluster as in figure 2 The upperpanel shows the surfaces generated from spherical harmonicswhere the configuration 0021021121(minus1)2202 is left andthe hybridized configuration 002102(11 + 05 times 21)2

(1(minus1) +05 times 2(minus1))2(20 + 023 times 30)2 is rightThe lower panel shows the shape calculated in [4]

the value for rigid rotation Jrig ie the contributions of thevortices to the total angular momentum cancel In the caseof a more realistic leptodermic potential the contributions

Figure 4 The velocity field in the rotating frame of four fermionsin a spheroidal harmonic oscillator potential the axis ratio of whichis chosen to satisfy the condition for equilibrium deformation(see [5])

from the currents in the body fixed frame do not cancel ingeneral The deviations of the moment of inertia from therigid body value will be discussed for large systems They arestronger than for small systems because the potential deviates

Stefan Frauendorf

more from the oscillator one However it is stressed that theflow pattern deviates strongly from rigid rotation even if themoment of inertia is close to the rigid body value

3 The acoustic regime

Now the case of large particle number is discussed whichcorresponds to a mesoscopic scenario The level densitybinding energy and other quantities may be considered asa sum of a smooth part that represents the properties of asmall but macroscopic droplet and a shell part that describesthe consequences of the quantal motion of the fermions nearthe Fermi surface The shapes are the result of a compromisebetween the liquid drop energy which prefers a sphericalshape to reduce the surface energy and the shell energywhich seeks a shape for which the level density near theFermi surface is low The gross dependence of shapes onthe particle number is discussed which corresponds to someaverage of adjacent clusters or nuclei The gross shell structureis described by means of the periodic orbit theory (POT) adetailed presentation of which was given in [6] The energy isdecomposed into a smooth part E which is the energy of thedroplet and an oscillating part Esh which describes the shellenergy

E = E + Esh Esh =

Eβ (3)

where β labels the periodic orbits that generate the shellenergy The gross shell structure is given by the few shortestorbits For simplicity the case of a cavity ie an infinitepotential step at the surface is discussed Then the classicalorbits are composed of straight lines and specular reflectionon the surface The energy e is conveniently expressed bythe wavenumber k =

radic2meh For the spherical cavity the

periodic orbits are equilateral polygons The upper panel offigure 5 shows the triangle and the square which are thesimplest Each polygon belongs to a threefold degeneratefamily which corresponds to the possible orientations ofthe polygon in space that are generated by the three Eulerangles All orbits of a family have the same length L thecircumference of the polygon As seen in figure 5 there aretwo types of orbits in the case of a cavity with moderatespheroidal deformation The equator orbits are the regularpolygons in the equatorial plane They belong to a onefolddegenerate family generated by the possible rotations of thepolygon in the plane The meridian orbits are the polygonsin a plane that goes through the symmetry axis They belongto a twofold degenerate family which is generated by therotation of the meridian plane about the symmetry axis and ashift of the reflection points on the surface within the meridianplane

The contribution from each family to the shell energy isgiven by

Aβ(kF) sin(Lβk + νβ)D

(kFLβγ Ro

where Ro = ro N 13 is the radius of a sphere with the samevolume as the spheroidal cavity The period of revolution ofa particle moving with the Fermi momentum hkF =

radic2meF

on the orbit is denoted by τβ Each term in the sum oscillates

Figure 5 Classical periodic orbits in a spheroidal cavity atmoderate deformation

Figure 6 Shell energy for a cavity-like spheroidal potential Forα = 0 the shape is spherical for α = 05 the shape is prolate withan axis ratio of

radic3 and for α = minus05 the shape is oblate with

an axis ratio of 1radic

3 The dashed and full lines correspond tosin(L4kF + ν4)= 1minus1 respectively The figure is relevant for alkaliclusters Taken from [7]

with the frequency kF as a function of the length Lβ of theorbital The Maslov index νβ is a constant phase (see [6]) TheamplitudeAβ depends on the degeneracy of the periodic orbitthe more symmetries a system has the greater the degeneracyand the more pronounced are the oscillations of the shellenergy The damping factor D(kLβγ Ro) is a decreasingfunction of its argument which filters out the shortest orbitsthat are responsible for the gross shell structure

31 Shapes

For a qualitative discussion of the shapes it is sufficientto consider only the family of tetragons Figure 6 showsthe shell energy of N fermions in a spheroidal cavity withdeformation α The volume of the cavity is constant V =43πr3

o N The figure includes the lines of constant length ofthe tetragons L4kF + ν4 = π(2n + 12) (dashed) and L4kF +ν4 = 2π(2n minus 12) (full) which correspond to the maximaand minima of the sin-function in equation (4) respectivelyAt spherical shape the shell energy oscillates as functionof the particle number N because L4 prop N 13 The minimaat N = 58 92 136 are the spherical shell closures Whenthe cavity takes a prolate deformation (α gt 0) the meridianorbits become longer In order to keep the length constantthe size of the cavity must decrease which corresponds toa smaller particle number N Hence the lines of constant

length of the meridian orbits are down sloping The equatororbits become shorter at prolate deformation In order tokeep the length constant N must increase and the lines ofconstant length are up sloping Figure 6 shows the interferencepattern between the meridian and equator orbits There isa system of down sloping valleys and ridges emanetingfrom the minima and maxima at spherical shape whichfollow the lines of constant length of the meridian orbitsSuperimposed is a system of up sloping valleys and ridgesthat follows the lines of constant length of the equator orbitsThe contribution from the meridian orbits is stronger becausethey are twofold degenerate whereas the equator orbits areonly onefold degenerate On the oblate side the meridianorbits become shorter and the equator orbits longer and theparticle number N must increase and decrease respectivelyin order to keep the length constant The landscape for theshell energy of the realistic nuclear potential is similar tothe cavity case except that the spinndashorbit term changes themagic numbers for spherical shell closure to N = 50 82 126Hence the following interpretation applies to nuclei as well ifthe appropriate degree of shell filling is considered

The shape is spherical (α = 0) for closed shells (N =

58 92 136) Taking particles away the equilibrium shape islocated in the valley on the prolate side that is generated bythe meridian orbits The deformation gradually increases inthe valley If N decreases further the equilibrium shape movesover the saddle on the ridge that is generated by the meridianorbits The deformation decreases abruptly after the saddleFigure 1 shows the smooth increase of the deformation α withdecreasing N and its sudden decrease This N -dependenceof the quadrupole deformation is also experimentally wellknown for nuclei Reference [8] first pointed out that it reflectsthe down sloping meridian valleys

The meridian ridges and valleys are less down sloping onthe oblate side than on the prolate side The different slopecan be explained in terms of geometry In the lower panel offigure 5 consider the meridian square orbit that has two of itssides perpendicular and two parallel to the symmetry axis ofthe cavity If a is the length of the symmetry semi-axis and bthe length of the other two axes then the length of the orbitL4 prop a + b The volume of the cavity is V = 4πab23 It hasto be the same as for spherical shape V = 4πR3

o3 Thismeans L4 prop bRo + 1(bRo)

2 This function has a largernegative slope for bRo lt 1 (prolate) than for bRo gt 1(oblate) In fact the slope becomes positive for bRo gt

213 The system tries to avoid the mid-shell mountain atspherical shape by taking a deformed shape It is energeticallyfavourable to go to the prolate side because the argument ofthe sin-function changes more rapidly This is the explanationof [9] for the preponderance of prolate over oblate shapeswhich can be seen in figure 1 and which is well known fornuclei

The weaker valley-ridge structure generated by the equa-tor orbits is less important for the quadrupole deformationHowever it generates the hexadecapole deformation As seenin figure 6 the meridian and equator valleys cross in the mid-dle of the shell at N asymp 76 α asymp 03 and N asymp 114 α asymp 025The constructive interference generates the local minimumwhich corresponds to the favoured shape Figure 1 showsthat these are the quadrupole deformation parameters α of

Figure 7 Shapes of heavy alkali clusters as calculated in [2] usingthe shell correction method The arrows have the same length

mid-shell clusters and that the hexadecapole deformationα4 asymp 0 for them Moving along the meridian valley to largeror smaller N brings the system out of the equator valleywhich makes the shell energy less negative It is energeticallyfavourable to change the shape such that the equator diame-ter remains the same as in the middle of the shell Then thelength of the equator orbits does not change and their contri-bution to the shell energy remains at the most negative valueThe hexadecapole deformations α4 in figure 1 are positivebelow and negative above the middle of the shell Figure 7shows that the equator diameter of the equilibrium shapes isapproximately constant indeed The measured nuclear shapesfollow the same sequence through a shell lemon-like in thelower part of the shell spheroidal in the middle and barrel-like in the upper part of the shell

32 Currents and moments of inertia

Here the case of slow rotation is considered which meansthat the periodic orbits are close to the discussed polygons (fordetails concerning this simplification see [7]) The importantquantity is the rotational flux through the orbit It is defined inthe same way as the magnetic flux which takes its role in thecase of clusters in a magnetic field

h8(θ)= 2m Aβω cos θ (5)

Here Aβ is the area enclosed by the orbit and θ is the anglebetween the normal of its plane and the axis of rotation Therotation manifests itself in the appearance of an additionalmodulation factor M in the expression for the shell energysum

Eβ rarr

MβEβ (6)

The modulation factor is an oscillating function of 8β whichis the flux through the orbit perpendicular to the rotationalaxis These oscillations determine the shell energy at highspin which is discussed in [7] Only the first term (quadratic)of the expansion into powers of8β is considered which givesan expression for the moment of inertia

J = Jrig +Jsh Jsh =

Jβ (7)

Stefan Frauendorf

Figure 8 The experimental ground state shell energy (upper panel)and the shell moment of inertia of unpaired nuclei (lower panel) asfunctions of the neutron number N The different symbols give theproton number Z Taken from [7]

where Jrig is the rigid body value For the discussion onlyone term in the sum over the periodic orbits is kept Then themoments of inertia are given by

Jsh =h2

(k2F A)

2 Esh Jshperp =h2

(k2F Aperp)

2 Eshperp

(8)where E(kF) is the contribution of the equator orbits to theground-state shell energy (4) and E(kF)perp the correspondingcontribution of the meridian orbits The areas of the respectiveorbits are A and Aperp

Figure 8 shows the experimental ground state shellenergies of heavy nuclei as function of the neutron numberwhich are obtained in the usual manner by subtractingthe energy of a spherical droplet from the experimentalground state energies In the lower panel the experimentalshell contribution to the moment of inertia (lsquoshell m oirsquo) of unpaired nuclei is shown It is obtained by fittingthe expression I 22Jexp to the experimental yrast energiesabove spin I = 20h where the pair correlations are essentiallyquenched The experimental shell moment is the differenceJsh = Jexp minusJrigid (for details see [7]) The deviations of themoments of inertia from the rigid body value are strong Their

N -dependence is similar to the one of the ground state shellenergy which is expected from the relation (8) Equation(8) gives a relative scale of Jsh asymp (h21000 MeV2)A43 Eshwhich is correct as the comparison of the two panelsshows However concerning POT there is one importantdifference between ground state energies and moments ofinertia Whereas all orbits contribute to the ground state shellenergy only the orbits that carry rotational flux contributeto the shell moment of inertia That is only the meridianorbits contribute if the rotational axis is perpendicular tothe rotational axis and only the equator orbits contributeif the rotational axis is parallel to the rotational axis Thisis indicated by the subscripts and perp for the parallel andperpendicular orientation of the rotational axis On the otherhand both the meridian and equator orbits contribute to theground state shell energy ie Esh = Esh + Eshperp The twoorientations of the rotational axis can be easily distinguishedIf it is perpendicular to the symmetry axis one observesa regularly spaced rotational band If it is parallel to thesymmetry axis the yrast line becomes an irregular sequenceof states that contains many high-K isomers

The strongly negative value of Jsh at the magic numberN = 126 reflects the shell closure The POT interpretation isthe same as discussed above for Esh The dip at N = 82 is lessdeep because Z is mid shell As for Esh the positive value ofJsh around N = 90 is caused by the meridian ridge Nuclei inthis region rotate about an axis perpendicular to the symmetryaxis and show rapid alignment of nucleon angular momentumwith this axis (back bending) The bump in Jsh demonstratesthat the meridian orbits are responsible because only they areperpendicular to the axis of rotation and thus carry rotationalflux The bump of Jsh around N = 110 is caused by theequator ridge Nuclei in this region rotate about the symmetryaxis which is reflected by the appearance of many high-K isomers This demonstrates that the equator orbits mustbe responsible because only they are perpendicular to therotational axis and carry rotational flux1 The equator orbitsgenerate only a shoulder in Esh because the meridian orbitsmake a negative contribution that increases with N

Figure 9 shows the velocity field of a nucleus within thegroup around N = 90 which have a positive shell momentof inertia As expected there is a substantial current in thebody fixed frame which has the same direction as the rotationIt causes the positive shell moment of inertia The velocityfield resembles the one generated by nucleons moving on themeridian orbits shown in the insert As discussed in [10]nuclei outside this region have a similar current distributionin the rotating frame However the velocities are opposite tothe rotation which is consistent with a negative shell momentof inertia A more quantitative relation between the currentsand the motion of particles on classical orbits remains to beestablished2

The negative shell energy that causes super deformationof rotating nuclei is substantial At a first glance one mightexpect that this would correspond to a negative shell momentof inertia However superdeformed nuclei are known to

1 For most of the yrast states the rotational axis is not completely parallel tothe symmetry axis That is why Jsh does not become positive It does becomepositive in microscopic calculations that assume parallel rotation [7]2 Rotation about the symmetry axis was not studied in [10]

Figure 9 The velocity field of 162Yb in the laboratory system (left)and the body fixed system (right) at angular momentum of about15h Taken from [10] The inset shows the tetragonal orbits that areresponsible for the current in the rotating frame The arrows indicatethat more particles run anti-clockwise than clockwise

Figure 10 Classical orbits in a superdeformed cavity

have moments of inertia that are close to the rigid bodyvalue The reason is that super deformation is caused byother types of orbits [11] examples of which are shown infigure 10 Superdeformed nuclei rotate about an axisperpendicular to the symmetry axis Therefore the equatororbits do not contribute to the shell moment of inertia Themeridian orbits are of the butterfly type They do not carry fluxbecause the contributions of the two wings compensate eachother (The motion is clockwise in one wing and anticlockwisein the other) Since none of the orbits carries rotational fluxthe moment of inertia takes the rigid body value

4 Conclusion

The shapes of nuclei and alkali clusters reflect the quantizedmotion of the fermions at the Fermi surface in the averagepotential They are not sensitive to the interaction thatgenerates the potential ie their properties are universal toall leptodermic potentials The relation between the quantalsingle particle motion and the shapes and currents is relativelytransparent for small and large particle number For smallparticle number the pattern originates from the densityand current distributions of the first (hybridized) sphericalharmonics For large particle number the pattern is generatedby the shortest classical orbits in the potential The orbit lengthcontrols the deformation Constant length of the orbits in themeridian plane determines the quadrupole deformation andconstant length of the orbits in the equator plane determinesthe hexadecapole deformation The current pattern is universalas well In the absence of pair correlations the currents ofrotating nuclei strongly deviate from rigid flow In heavynuclei this is reflected by the moments of inertia at high spindiffering from the rigid body value The difference can berelated to the classical periodic orbits It suggests a strongmagnetic response of alkali clusters

Acknowledgments

This work was supported by the US Department of Energyunder contract DE-FG02-95ER40934

References

[1] Nilsson S G 1955 Mat-Fys Medd K Dan Vidensk Selsk29 1

[2] Frauendorf S and Pashkevich V V 1996 Ann Phys Lpz 5 34[3] Moumlller P Nix J R Myers W D and Swiatecki W J 1995

At Data Nucl Data Tables 59 185[4] Koskinen M Lipas P O and Manninien M 1995 Z Phys D

35 285[5] Bor A and Mottelson B 1975 Nuclear Structure II

(New York Benjamin)[6] Brack M and Bhaduri R K 1997 Semiclassical Physics

(Reading MA Addison-Wesley)[7] Deleplanque M A Frauendorf S Pashkevich V V and Chu S Y

Z Phys A 283 269[8] Strutinsky V M Magner A G Ofengenden S R and Dossing T

1977 Z Phys A 283 269[9] Frisk H 1990 Nucl Phys A 511 309

[10] Fleckner J Kunz J Mosel U and Wuumlst E 1980 Nucl Phys A339 227

[11] Yamagami M and Matsuyanagi K 2000 Nucl Phys A 672 123

1 Introduction

31 Shapes


4 Conclusion

Acknowledgments

References