Superdeformed structures and low $\Omega$ parity doublet in Ne$-$S nuclei near neutron drip-line

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by  Shailesh K. Singh

SuperdeformedstructuresandlowΩparitydoubletinNe−Snucleinearneutrondrip-line
Shailesh K. Singh, S.K. Patra and C.R. Praharaj
Institute of Physics, Bhubaneswar-751 005, India
(Dated:January27,2014)
ThestructureofNe,Na,Mg,Al,Si,PandSnucleineartheneutrondrip-lineregionisinvestigatedinthe
frame-workofrelativisticmeanfieldtheoryandnon-relativisticSkyrmeHartree-Fockformalism. Therecently
discoverednuclei40Mgand42Al,whicharebeyondthedrip-linepredictedbyvariousmassformulaearelocated
withinthesemodels.Wefindmanylargelydeformedneutron-richnuclei,whosestructuresareanalyzed.From
thestructureanatomy, wefindthatatlargedeformation, lowΩorbitsofoppositeparities(e.g. 1+ and 1−)
2 2
occurclosetoeachotherinenergy.
PACSnumbers:21.10.-k,21.10.dr,21.10.Ft,21.30.-x,24.10.-1,24.10.Jv
4 I. INTRODUCTION predict the experimental observables, the recent forces are
1 good enough to reproduce the bulk properties not only near
0 the β−stability line but also far away from it. Here, we use
The structure of light nuclei near the neutron drip-line is
2 thesetwosuccessfulmodels[11–27]tolearnabouttheprop-
oneoftheinterestingtopicforagoodnumberofexoticphe-

n nomena.Nucleiinthisregionarequitedifferentincollectivity ertiesofdrip-linenucleiNe−S.
a
and clustering features than the stable counterpart in the nu-
J
clear chart. For example, the neutron magic property is lost
4 forN=8in12Be[1]andN=20in32Mg[2]. Theunexpect- A. TheSkyrmeHartree-Fock(SHF)Method
2
edly large reaction cross-section for 22C gives the indication
] ofneutronhalostructure[3]. Thediscoveryoflargecollectiv- The general form of the Skyrme effective interaction used
h ity of 34Mg by Iwasaki et al. [4] is another example of such in the mean-field model can be expressed as a Hamiltonian
-t exotic properties. The deformed structures, core excitation densityH[13–19]. ThisHiswrittenexpressedasafunction
cl and the location of drip-line for Mg and neighboring nuclei ofsomeempiricalparametersgivenas:
u are few of the interesting properties of investigation. In this
n context,thediscoveryof40Mgand42Al,oncepredictedtobe H=K+H0+H3+Heff +··· , (1)
[
nuclei beyond the drip-line by various mass formulae [5, 6]
1 showtoneedthemodificationofthemassmodels. where K is the kinetic energy term, H0 the zero range, H3
the density dependent and H the effective-mass depen-
v Ontheotherhand,theappearanceofN=16asmagicnum- eff
dent terms, which are relevant for calculating the properties
9 ber in 24O and the existence of neutron halo in 11Li are es-
7 tablishedobservations[7]. However,theproposedproton[8] ofnuclearmatter. Thesearefunctionsof9parametersti, xi
2 (i=0,1,2,3)andη,andaregivenas
(8B) and neutron [9, 10] halo (14Be, 17B, 31Ne) in the ex-
6
1. otic nuclei are currently under investigations. In addition to H = 1t (cid:2)(2+x )ρ2−(2x +1)(ρ2+ρ2)(cid:3), (2)
these,theclusterstructureofentirethelightmassnucleiand 0 4 0 0 0 p n
0
14 tshtuedsykionffloigrmhtatmioanssindnriepu-tlrinoen-nduricpleiis.otIonptehsismpoatpiveart,eouusrfaoirmthies H3 = 214t3ρη(cid:2)(2+x3)ρ2−(2x3+1)(ρ2p+ρ2n)(cid:3), (3)
: tostudytheneutrondrip-lineforNe−Sisotopicchaininthe 1
iv frame-work of a relativistic mean field (RMF) and nonrela- Heff = 8[t1(2+x1)+t2(2+x2)]τρ
X tivisticSkyrmeHartree-Fock(SHF)formalismsandanalyzed
1
thefeaturesoflargequadrupoledeformationoftheseisotopes. + [t (2x +1)−t (2x +1)](τ ρ +τ ρ ()4.)
r 8 2 2 1 1 p p n n
a Thepaperisorganizedasfollows: TheRMFandSHFfor-
malisms are described briefly in Section II. The results ob- The kinetic energy K = (cid:126)2 τ, a form used in the Fermi gas
tained from our calculations are discussed in Section III. Fi- 2M
model for non-interacting Fermions. The other terms, repre-
nally, summaryandconcludingremarksaregiveninSection
senting the surface contributions of a finite nucleus with b
4
IV. andb(cid:48) asadditionalparameter,are
4
(cid:20) (cid:21)
1 1 1
H = 3t (1+ x )−t (1+ x ) (∇(cid:126)ρ)2
II. THEFORMALISM Sρ 16 1 2 1 2 2 2
(cid:20) (cid:21)
1 1 1
− 3t (x + )+t (x + )
The mean field methods like SHF and RMF have been 16 1 1 2 2 2 2
widelyusedinthestudyofbindingenergies,rootmeansquare (cid:104) (cid:105)
× (∇(cid:126)ρ )2+(∇(cid:126)ρ )2 , (5)
radii,quadrupoledeformationandotherbulkpropertiesofnu- n p
clei [11, 12]. In general, one can say that although the older 1(cid:104) (cid:105)
parametrizations of SHF and RMF have some limitation to HSJ(cid:126) = −2 b4ρ∇(cid:126) ·J(cid:126)+b(cid:48)4(ρn∇(cid:126) ·J(cid:126)n+ρp∇(cid:126) ·J(cid:126)p) .(6)
2
(cid:80)
Here, thetotalnucleonnumberdensityρ = ρ +ρ , theki- using(cid:52) = G u v . Theoccupationnumberisdefined
n p i>0 i i
neticenergydensityτ = τ +τ , andthespin-orbitdensity as:
n p
J(cid:126) = J(cid:126) +J(cid:126) . The subscripts n and p refer to neutron and (cid:34) (cid:35)
n p 1 (cid:15) −λ
proton, respectively. The nucleon mass is represented by m. n =v2 = 1− i . (11)
The J(cid:126) = 0, q = n or p, for spin-saturated nuclei, i.e., for i i 2 (cid:112)((cid:15)i−λ)2+(cid:52)2
q
nucleiwithmajoroscillatorshellscompletelyfilledorempty. Thechemicalpotentialsλ andλ aredeterminedbythepar-
n p
The total binding energy (BE) of a nucleus is the integral of
ticlenumbersforneutronsandprotons. Thepairingenergyis
H. computedasE = −(cid:52)(cid:80) u v . Foraparticularvalue
pair i>0 i i
of (cid:52) and G, the pairing energy E diverges, if it is ex-
pair
tended to an infinite configuration space. In fact, in all real-
B. TheRelativisticMeanField(RMF)Method
istic calculations with finite range forces, the contribution of
states of large momenta above the Fermi surface (for a par-
Therelativisticmeanfieldapproachiswell-knownandthe ticular nucleus) to (cid:52) decreases with energy. We use a pair-
theoryiswelldocumented[20–25].Herewestartwiththerel- ingwindow,wheretheequationsareextendeduptothelevel
ativisticLagrangiandensityforanucleon-mesonmany-body |(cid:15) −λ| ≤ 2(41A−1/3). Thefactor2hasbeendeterminedso
i
systemas: astoreproducethepairingcorrelationenergyforneutronsin
118SnusingGognyforce[20,29,30]. Thevaluesof(cid:52) and
1 1 n
L = ψ {iγµ∂ −M}ψ + ∂µσ∂ σ− m2σ2 (cid:52) aretakenfrom[31],asinputintheBCS-eqaution.
i µ i 2 µ 2 σ p
We compare the results with various simple and sophisti-
1 1 1
− g σ3− g σ4−g ψ ψ σ− ΩµνΩ catedpairingprescriptionslikeBCS-deltaforce[32]andBCS
3 2 4 3 s i i 4 µν
density dependent delta force [33]. These calculations have
1
+ m2VµV −g ψ γµψ V been done only for 20Ne and 47Al nucleus in both SkI4 and
2 w µ w i i µ
NL3 force parameter sets. We have given these results in
1 1
− B(cid:126)µν.B(cid:126) + m2R(cid:126)µ.R(cid:126) −g ψ γµ(cid:126)τψ .R(cid:126)µ Table I along with experimental results like quadrupole de-
4 µν 2 ρ µ ρ i i formation parameter β [34], total binding energy (BE) [35]
2
−1FµνF −eψ γµ(1−τ3i)ψ A . (7) androotmeansquarechargeradius(rch)[36]. Wefindthat,
4 µν i 2 i µ for this lighter mass region of the periodic chart, pairing is
less important for majority of the cases. With pairing, the
All the quantities have their usual meanings. From the rela-
deformation becomes negligible for 20Ne and we do not get
tivistic Lagrangian, we obtain the field equations for the nu-
theexperimentaldeformationparameterinRMFcalculations.
cleonsandmesons. Theseequationsaresolvedbyexpanding
With no pairing, we reproduce substantially the deformation
the upper and lower components of the Dirac spinor and the
parameter in RMF because the density of states near Fermi
bosonfieldsinanaxiallydeformedharmonicoscillatorbasis.
surface for such light nuclei are small and not conducive to
Thesetofcoupledequationsissolvednumericallybyaself-
pairing [37]. To understand the influence of pairing on the
consistentiterationmethod. Thetotalenergyofthesystemin
openshellnuclei,wehavetakenintoaccounttheexperimental
RMFformalismisgivenby
data, wherever available. The SHF(SkI4) results are used as
E =E +E +E +E +E +E +E , (8) guidelineintheabsenceofthesedata. Werealizedaftercom-
total part σ ω ρ c pair c.m.
paringthecalculatedβ ofRMFandSHFwithexperimental
2
where Epart is the sum of the single particle energies of the data that the quadrupole deformation of SHF is closer to ex-
nucleons and Eσ, Eω, Eρ, Ec, Epair, Ecm are the contribu- perimentwithouttakingpairingcorrelationintoaccount. For
tions of the meson fields, the Coulomb field, pairing energy example, when we use the (cid:52) and (cid:52) from the experimen-
n p
andthecenter-of-massenergy,respectively. tal binding energy of odd-even values or from the empirical
formulaofRef. [31,38]tocalculateβ for20,22,24,26,28Nein
2
RMF(NL3), we find β ∼ 0.18,0.35,0.19,0.0,0.0, respec-
2
C. Pairingcorrelation tivelyfortheseisotopesagreeingwiththeresultofLalazissis
etal[39]. Theseβ stronglydisagreewiththemeasuredval-
2
Totakecareofthepairingcorrelationforopenshellnuclei, ues(β (expt.) = 0.723,0.562,0.45,0.498,0.50)[34]. Simi-
2
the constant gap BCS-approach is used in our calculations. lareffectsarealsoseeninotherconsideredisotopes. Onthe
Thepairingenergyexpressioniswrittenas: otherhand,ifweignorepairing,thenthecalculatedresultsare
often better and these β are quite close to the experimental
(cid:34) (cid:35)2 2
(cid:88) data. Theinfluenceofpairingisalsovisibleinthetotalbind-
E =−G u v , (9)
pair i i ingenergy.Insomeofthecases,evenacoupleofMeVdiffer-
i>0 enceintotalbindingenergyisfoundwithandwithouttaking
withG=pairingforceconstant,v2andu2 =1−v2aretheoc- pairingcorrelationintoaccountinRMFformalism. Contrary
i i i
to the RMF, the pairing in the SHF formalism is almost in-
cupationprobabilities[28,29]. Thevariationalapproachwith
respecttov2givestheBCSequation[28] sensitive to quadrupole deformation for the considered mass
i
region. Thus,wehaveperformedthecalculationsthroughout
2(cid:15) u v −(cid:52)(u2−v2)=0, (10) thepaperwithoutconsideringpairingintoaccount.
i i i i i
3
TABLEI:Calculationofbindingenergy(BE),quadrupoledeforma- FIG.1: ThechangeinbindingenergyBE,rootmeansquarematter
tion parameter β , root mean square of matter radius (r ) and radius(r ),quadrupoledeformationparameterβ withFermionic
2 rms rms 2
charge radius (r ) by taking various pairing methods. We have N andbosonicN harmonicoscillatorbasisforsomeselectednu-
ch F B
given these results for both SkI4 and NL3 parameter sets with ex- clei.
perimentaldata[34–36].
SkI4 48Al 49Si 56S
Nucl. TypeofPairing β r BE r
20Ne Nopairing 0.5249 2.r9m1s1 156.8 3.0c3h0 MeV) 304 324 375
BCS-BdCenSs-.ddeelpta.dfeolrtac-eforce 00..554488 22..991100 115566..88 33..003300 Energy( 229868 332106 370
47Al BCNSo-dpealtiarinfogrce 00..000076 33..997527 228878..87 33..332147 Binding 248.20 341.21 346.25
BCS-dens.dep.delta-force 0.055 3.970 288.0 3.322 4.1 4 4.1
NL3 m) 4 3.9 4
20Ne Nopairing 0.537 2.846 156.7 2.972 r(frms 33..98 3.8 3.9
BCS-delta 0.036 2.920 154.9 3.055 3.7 3.7 3.8
47Al Nopairing 0.090 3.832 294.6 3.246 0.2 0.1 0.1
BCS-delta 0.081 3.834 294.8 3.246 0.15 0.08 0.08
Exp.Results 0.1 0.06 0.06
20Ne 0.728 160.6 3.005 β2 0.04 0.04
0.05 0.02 0.02
47Al —
0 0 0
8 10 12 14 16 18 20 8 10 12 14 16 18 20 8 10 12 14 16 18 20
Number of Shells (N = N)
F B
D. PauliBlockingandHarmonicoscillatorbasis
Foreven-evennucleus±morbitsarepairwiseoccupiedand Someotherkindsofbasisliketransformedharmonicoscilla-
themeanfieldhastimereversalsymmetry. Butinthecaseof tor basis [45], Gaussian expansion method [46] and Woods-
odd nucleon the time reversal symmetry is broken. To take Saxon basis [47, 48] are also available in literature. The in-
careoftheoddnucleon,weemploytheblockingmethod[40]. clusion of sufficiently large harmonic oscillator model space
Weputthelastnucleoninoneoftheconjugatestates±mand givesreasonablyconvergentresults. Thistypeofprescription
keepingotherstateempty. Inthiswaywefollowthetimere- is already done in Ref. [49]. However, to fully include con-
versalsymmetryforodd-evenandodd-oddnuclei. Werepeat tinuum effects more work has to be done (by use of basis of
this calculation by putting the odd nucleon in all the near by finitepotentialsandinclusionofcorrelationeffectsinHartree-
stateoftheconjugateleveltodeterminethemaximumbinding Bogoliubovscheme[50]).
energyofthegroundstate[40,41].
In our present calculations the nuclei are treated as axial-
symmetrically deformed, with z-axis as the symmetric axis. E. GroundstatepropertiesfromtheSHFandRMFmodels
Sphericalsymmetryisnolongerpresentingeneralandthere-
fore j is not a good quantum number any more. Because of Certainly for light mass nuclei, the correction of centre
axialsymmetry,eachorbitisdenotedbythequantumnumber of mass motion can not be ignored and it should be done
m of Jz and is a superposition of |jm > states with various self-consistently. That means, in the evaluation of centre-of-
j values. The densities are invariant with respect to a rota- mass energy, one should evaluate E = <F|P2|F> using
tion around the symmetry axis. For numerical calculations, CM 2M
|F >= |F > wavefunction. In this case, one has to
RMF
the wavefunctions are expanded in a deformed harmonic os-
calculate the matrix elements directly. However, this pro-
cillatorpotentialbasisandsolvedself-consistentlyinaniter-
cedure is more involved and in the present calculations we
ation method. The major oscillator quanta for Fermion N
F havesubtractedthespuriouscentre-of-massmotionusingthe
and bosons N are taken as N = 12. The convergence
B max Elliott-Skyrmeapproximationandtheapproximateanalytical
of our numerical results are tested in Fig. 1 for BE, matter expressioniswrittenasE = 3.41A−1/3 MeV(harmonic
radiusr andquadrupoledeformationparameterforsome CM 4
rms oscillator approximation), where A is mass number [51–53]
selected nuclei like 48Al, 49Si and 56S. Here, the results are
and expect that the two results should not differ drastically.
estimated from N = N = 8 to N = N = 18, which
F B F B The quadrupole moment deformation parameter β is evalu-
2
are shown in Fig. 1. From this analysis, we observed that
ated from the resulting proton and neutron quadrupole mo-
theβ valuesalmostidenticalwiththevariationofoscillator
2 mentsthrough:
quanta. However, the rms radii and binding energy vary till
NF = NB = 12, beyond which the results are unchanged. Q=Q +Q =(cid:114)16π (cid:18) 3 AR2β (cid:19), (12)
Itiswellknownthatharmonicoscillatorbasisisnotsuitable n p 5 4π 2
in dripline nuclei due to the asymptotic behavior of the den-
sity distribution. To resolve this problem , efforts have been whereR=1.2A1/3.Therootmeansquareradiiofprotonsand
made for solving the equations in coordinate space [42–44]. matterdistributionaredefinedas(cid:104)r2(cid:105) = 1 (cid:82) ρ (r ,z)r2dτ,
p Z p ⊥
4
and (cid:104)r2 (cid:105) = 1 (cid:82) ρ(r ,z)r2dτ; respectively, where Z is
rms A ⊥
theprotonnumberandρ (r ,z)isthedeformedprotonand TABLEII:Thecalculatedgroundstatebindingenergyobtainedfrom
p ⊥
ρ(r ,z)isthetotalnucleondensitydistribution. Theproton SHFandRMFtheoryarecomparedwiththeexperimentallyknown
⊥
andchargermsradiusisconnectedthroughtherelationr = heaviestisotopeforNe,Na,Mg,Al,Si,PandS[35].
ch
(cid:113) Nucl. RMF SHF Expt. Nucleus RMF SHF Expt.
r2+0.64[41].
p 31Ne 216.0 213.2 211.4 32Na 234.5 233.4 230.9
We use the well known NL3 parameter set [54] for the 36Mg 263.9 260.2 260.8 38Al 283.5 281.4 280.3
RMFformalism. Thissetnot onlyreproduces theproperties 41Si 310.1 307.2 307.9 43P 331.7 329.0 330.7
of stable nuclei but also well predicts for those far from the 45S 353.4 350.4 354.7
β-stabilityvalley. Also,theisoscalarmonopoleenergyagrees
excellentlywiththeexperimentalvaluesfordifferentregions
ofthePeriodicTable.Themeasuredsuperdeformedminimum
TABLEIII:Thepredictedmassnumberofneutrondrip-lineforNe,
in194Hgis6.02MeVabovethegroundstate,whereasinRMF
Na, Mg, Al, Si, P and S nucleus in RMF (NL3) and SHF (SKI4)
calculationwithNL3set,thisnumberis5.99MeV[54].
parametersetsarecomparedwithinfinitenuclearmatter(INM)mass
ForSHFmodel,weusetheSkyrmeSkI4setwithb (cid:54)= b(cid:48)
4 4 model[56],finiterangedropletmodel(FRDM)[57]andthenuclei
[26]. This parameter set is designed for considerations of withthelargestneutronnumberssofarexperimentallydetected[35]
properspin-orbitinteractioninfinitenuclei,relatedtotheiso- alongwiththenumbershowninparenthesisaretheexperimentally
tope shifts in Pb region and is better suited for the study of extrapolatedvalues.
exotic nuclei. Several more recent Skyrme parameters such Nucl. RMF SHF INM FRDM Expt.
as SLy1-10, SkX, SkI5 and SkI6 are obtained by fitting the Ne 34 34 34 33 31(34)
Hartree-Fock (HF) results with experimental data for nuclei Na 40 37 37 36 32(37)
startingfromthevalleyofstabilitytoneutronandprotondrip- Mg 40 40 39 40 36(40)
lines[13,26,27,55]. Al 48 48 42 42 38(43)
Si 54 48 45 43 41(45)
P 54 55 49 48 43(47)
S 55 55 51 51 45(49)
III. RESULTSANDDISCUSSIONS
The binding energy BE, rms charge radius r and
ch 2,Z)≥0. Thenucleiwiththelargestneutronnumberssofar
quadrupole deformation parameter β of the isotopes of Ne,
2 experimentallydetectedinanisotopicchainalongwiththeex-
Na, Mg, Al, Si, P and S are calculated near the drip-line re-
trapolateddataarealsodisplayedinthelastcolumnofTable
gion. Forthis,boththerelativisticandnon-relativisticmodels
III.Thenumbersgivenintheparenthesisaretheexperimen-
areused.
tallyextrapolatedvalues[35].Togetaqualitativeunderstand-
ingofthepredictionofneutrondrip-line, wehavecompared
ourresultswiththeinfinitenuclearmatter(INM)[56]andfi-
A. Bindingenergyandneutrondrip-line niterangedropletmodel(FRDM)[57]massestimation. The
RMFandSHFdrip-linescoincidewitheachotherforNe,Mg,
Thegroundstatebindingenergy(BE)forNe,Na,Mg,Al, AlandS.IncaseofNaandSitheRMFdripnucleiarefound
Si,PandSisotopesareselectedbycomparingthebindingen- tobe3and6unitheavierthantheSHFprediction. TheINM
ergyobtainedfromtheprolate,oblateandsphericalsolutions prediction of drip nuclei are always in the heavier side than
for a particular nucleus. For a given nucleus, the maximum FRDM. From Table III, we find that the experimental effort
bindingenergycorrespondstothegroundstateandotherso- hasalmostreachedtotheINMandFRDMpredictionofdrip
lutionsareobtainedasvariousexcitedintrinsicstates. InTa- nucleiforlightermassregion.
bleII,thegroundstatebindingenergyfortheheaviestknown Thetheoreticalpredictionofdripnucleiareveryimportant
isotopes for the discussed nuclei are compared with the ex- after the discovery of 40Mg and 42Al [5]. These two nuclei
perimental data [35]. The binding energy for 31Ne is 216.0 are considered to be beyond the drip-line (neutron-unbound)
MeV with RMF (NL3) and these are 213.2 and 211.4 MeV in some of the mass calculations [6, 58]. The discovery of
in SHF(SkI4) and experiment, respectively. Similarly, these these two isotopes, suggests the existence of drip-line some-
results for 45S respectively are 353.4, 350.4 and 354.7 MeV whereintheheavierside. Thus,thestudyoftheseisotopesis
in RMF, SHF and experiment. Analyzing the data of Table beyondthescopeoftheexistingmassmodels[6,58]. Inthe
II, generally one finds that BE of RMF is slightly over esti- present RMF/SHF calculations, the newly discovered 40Mg
mated and SHF is underestimated than the experimental val- and 42Al are well within the drip-line. Also, a point of cau-
ues.However,theoverallagreementofthecalculatedenergies tion, it may be possible that if we allow triaxial deformation
arewithinanacceptablerangewiththeexperimentaldata. inthecalculation,thenwemaygetoneminimumasasaddle
We have listed the neutron drip-lines in Table III, which point and another one as triaxial minimum. But this calcu-
areobtainedfromthegroundstatebindingenergyforneutron lation is out of scope in our paper, we are dealing with axial
richNe, Na, Mg, Al, Si, PandSnuclei. Thedrip-lineisde- deformedcodebyusingNL3andSkI4parametersetswhere
termined by setting the condition that the minimum value of mostly we find similar results in both the formalisms. This
twoneutronseparationenergyS =BE(N,Z)−BE(N− typeofprescriptionsareusedinmanyoftheearlierpublica-
2n
5
TABLEIV:Thecalculatedvalueofchargeradius(r ),quadrupole TABLEV:SameasTableIV,forAlandSiisotopes.
ch
momentdeformationparameterβ2andbindingenergy(BE)forNe, Nucl. RMF SHF Exp.
NaandMgnucleiinRMF(NL3)andSHF(SkI4)formalisms. We r β BE r β BE r β BE
ch 2 ch 2 ch 2
compareourresultswithexperimentalβ2[34],groundstatebinding 24Al 3.097 0.388 182.3 3.174 0.413 185.0 183.6
energyBE(MeV)[35]andthechargeradiusrch(fm)[36]. 25Al 3.072 0.381 197.7 3.164 0.430 199.5 200.5
Nucl. RMF(NL3) SHF(SkI4) Exp. 26Al 3.052 -0.275 207.8 3.122 0.315 211.4 211.9
rch β2 BE rch β2 BE rch β2 BE 27Al 3.053 -0.292 221.9 3.092 0.204 222.7 3.061 225.0
20Ne 2.970 0.535 156.7 3.030 0.550 156.8 3.006 0.727 160.6
28Al 3.037 -0.208 238.6 3.105 0.202 232.5 232.7
21Ne 2.953 0.516 165.9 3.012 0.529 166.8 2.970 167.4
29Al 3.033 -0.141 245.6 3.126 0.241 241.5 242.1
22Ne 2.940 0.502 175.7 3.010 0.520 175.8 2.953 0.562 177.8
23Ne 2.913 0.386 181.8 2.975 0.382 182.2 2.910 183.0 30Al 3.070 -0.184 253.8 3.139 0.194 248.7 247.8
24Ne 2.881 -0.259 189.1 2.950 -0.250 188.5 2.901 0.45 191.8 31Al 3.101 -0.205 259.8 3.161 -0.192 256.0 255.0
25Ne 2.907 0.272 194.2 2.948 0.170 194.2 2.932 196.0 32Al 3.103 -0.111 261.2 3.162 0.020 262.6 259.2
26Ne 2.926 0.277 199.9 2.950 0.120 199.4 2.925 0.498 201.6 33Al 3.165 -0.333 269.4 3.183 0.000 269.8 264.7
27Ne 2.945 0.247 203.9 2.987 0.159 203.2 203.1 34Al 3.134 -0.108 275.1 3.198 0.090 271.7 267.3
28Ne 2.965 0.225 208.2 3.010 0.160 206.5 2.964 0.50 206.9 35Al 3.167 0.268 274.1 3.229 0.250 274.4 272.5
29Ne 2.981 0.161 211.2 3.027 0.010 210.1 207.8 36Al 3.173 -0.189 277.7 3.254 0.320 277.4 274.4
30Ne 2.998 0.100 215.0 3.050 0.000 213.7 211.3 37Al 3.208 0.355 281.5 3.278 0.371 280.1 278.6
31Ne 3.031 0.244 216.0 3.057 0.225 213.2 211.4 38Al 3.214 -0.254 283.5 3.288 0.378 281.4 280.3
32Ne 3.071 0.373 218.6 3.100 0.380 213.1 39Al 3.236 -0.299 286.7 3.383 -0.121 287.1
33Ne 3.095 0.424 219.5 3.148 0.429 213.5 40Al 3.257 -0.336 290.4 3.316 0.403 284.2
34Ne 3.119 0.473 220.9 3.180 0.490 213.5 41Al 3.278 -0.370 290.6 3.338 -0.367 285.9
24Na 2.964 0.379 192.3 3.042 0.411 194.0 2.974 193.5 42Al 3.281 -0.355 291.2 3.341 -0.339 286.2
25Na 2.937 0.273 200.6 3.024 0.314 201.4 2.977 202.5 43Al 3.282 -0.338 292.2 3.341 -0.312 286.6
26Na 2.965 0.295 207.1 3.027 0.274 208.4 2.993 208.1 44Al 3.274 -0.288 293.6 3.340 -0.282 287.0
27Na 2.993 0.323 214.2 3.043 0.282 214.9 3.014 214.8
45Al 3.271 -0.263 293.5 3.338 -0.250 287.6
28Na 2.993 0.272 219.0 3.058 0.234 219.7 3.040 218.4
29Na 3.004 0.232 224.3 3.072 0.194 224.3 3.092 222.8 46Al 3.359 0.341 294.5 3.326 -0.129 287.7
30Na 3.031 0.169 228.1 3.079 0.030 228.6 3.118 225.1 47Al 3.246 0.090 294.8 3.318 -0.004 288.7
31Na 3.047 0.108 232.7 3.103 0.000 233.5 3.170 229.3 48Al 3.319 -0.252 294.0 3.347 -0.060 287.6
32Na 3.077 0.237 234.5 3.121 0.187 233.4 230.9 28Si 3.122 -0.331 232.1 3.190 -0.350 233.6 3.122 0.407 236.5
33Na 3.113 0.356 237.9 3.172 0.352 234.9 29Si 3.035 0.001 240.7 3.176 -0.272 243.1 3.118 245.0
34Na 3.137 0.404 239.8 3.198 0.407 236.2 30Si 3.070 0.148 250.6 3.170 -0.210 252.6 3.134 0.315 255.6
35Na 3.161 0.450 242.3 3.224 0.457 237.4 31Si 3.108 -0.180 259.1 3.182 -0.199 261.7 262.2
36Na 3.175 0.481 242.5 3.235 0.501 237.5 32Si 3.137 -0.201 268.5 3.200 -0.200 270.5 0.217 271.4
37Na 3.190 0.512 243.1 3.251 0.541 237.6 33Si 3.131 -0.084 275.6 3.196 0.010 278.1 275.9
38Na 3.199 0.491 243.4 34Si 3.148 0.000 284.4 3.220 0.000 286.3 0.179 283.4
39Na 3.209 0.472 244.1 35Si 3.161 -0.083 287.4 3.226 0.010 289.5 285.9
40Na 3.228 0.477 243.4 36Si 3.186 -0.162 291.5 3.150 0.150 292.4 0.259 292.0
24Mg 3.043 0.487 194.3 3.130 0.520 195.2 3.057 0.605 198.3 37Si 3.200 0.238 295.4 3.269 0.247 295.9 294.3
25Mg 3.009 0.376 202.9 3.103 0.432 204.3 3.028 205.6 38Si 3.218 0.281 299.8 3.290 0.310 298.2 0.249 299.9
26Mg 2.978 0.273 212.5 3.080 -0.300 213.2 3.034 0.482 216.7 39Si 3.224 0.263 302.4 3.298 0.292 301.4 301.5
27Mg 3.015 0.310 220.2 3.096 0.339 221.5 223.1
40Si 3.272 -0.301 306.0 3.310 -0.280 304.0 306.5
28Mg 3.048 0.345 228.7 3.110 0.340 229.0 0.491 231.6
29Mg 3.055 0.289 234.3 3.118 0.283 235.0 235.3 41Si 3.295 -0.336 310.1 3.349 -0.329 307.2 307.9
30Mg 3.062 0.241 240.5 3.120 -0.180 240.5 0.431 241.6 42Si 3.318 -0.369 314.6 3.330 -0.350 310.0
31Mg 3.075 0.179 245.1 3.123 0.030 246.1 243.9 43Si 3.320 -0.356 315.2 3.377 -0.339 311.1
32Mg 3.090 0.119 250.5 3.150 0.000 252.0 0.473 249.7 44Si 3.322 -0.342 316.2 3.380 -0.300 311.6
33Mg 3.117 0.233 253.1 3.165 0.155 253.0 252.0 45Si 3.316 -0.308 317.5 3.374 -0.282 312.9
34Mg 3.150 0.343 257.3 3.210 0.330 255.1 256.7 46Si 3.303 -0.262 319.3 3.370 -0.240 313.5
35Mg 3.173 0.388 260.5 3.239 0.393 257.8 257.5 47Si 3.345 -0.298 319.8 3.340 0.030 314.3
36Mg 3.198 0.432 263.9 3.265 0.440 260.2 260.8 48Si 3.263 0.001 321.8 3.350 0.000 315.4
37Mg 3.212 0.462 264.9 3.279 0.469 261.0 49Si 3.290 0.045 321.1
38Mg 3.227 0.492 266.3 3.295 0.490 261.6 50Si 3.341 -0.159 321.5
39Mg 3.237 0.473 267.8 3.307 0.485 262.4 51Si 3.358 -0.135 321.2
40Mg 3.247 0.456 269.7 3.320 0.470 262.8 52Si 3.371 0.082 321.4
53Si 3.391 0.042 321.6
54Si 3.415 0.000 322.3
tions[59].
6
gradually with increase of neutron number. In case of Na,
TABLEVI: SameasTableIV,forPandSisotopes. N = 3 starts filling up at 33Na with quadrupole moment
osc
Nucl. RMF SHF Exp. deformation parameter β2 = 0.356 and −0.179 with occu-
rch β2 BE rch β2 BE rch β2 BE piedorbits[330]1− and[303]7−,respectively. Thefillingof
30P 3.138 0.130 246.3 3.189 0.026 249.9 250.6 N =3goeson2increasingfo2rNawithneutronnumberand
osc
31P 3.158 0.205 258.3 3.201 0.105 261.1 3.189 262.9 itis[330]1−, [310]1−, [321]3− and[312]5− atβ = 0.472
32P 3.174 -0.143 267.1 3.216 0.069 270.9 270.9 2 2 2 2 2
33P 3.201 -0.183 277.5 3.246 -0.167 280.5 281.0 for 39Na. Again for the oblate solution the occupation is
34P 3.201 -0.082 285.8 3.248 0.001 289.9 287.2 [301]21−,[301]32−,[303]52−and[303]72−forβ2 =−0.375for
35P 3.216 -0.001 295.4 3.265 0.000 299.2 295.6 39Na. IncaseofMgisotopes,evenfor30,32Mg,theN =3
osc
36P 3.227 0.120 299.5 3.272 0.007 303.3 299.1 shellhassomeoccupationforthelow-lyingexcitedstatesnear
37P 3.246 0.209 305.0 3.290 0.148 307.4 305.9 theFermisurface. For30Mg(atβ =0.599withBE=237.7
2
38P 3.260 0.250 310.4 3.313 0.240 311.7 309.6 MeV)theN =3orbitis[330]7− andfor32Mgis[330]1−
39P 3.275 0.288 316.1 3.334 0.301 316.1 315.9 osc 2 2
(BE=248.8MeVatβ = 0.471). Withtheincreaseofneu-
40P 3.281 0.274 320.1 3.343 0.290 319.6 319.2 2
tronnumberinMgandSiisotopicchains,theoscillatorshell
41P 3.288 0.261 324.4 3.355 0.295 322.7 324.2
withN =3getsoccupiedmoreandmore.
42P 3.306 0.301 327.3 3.371 0.320 325.6 326.3 osc
43P 3.346 -0.323 331.7 3.398 -0.320 329.0 330.7 In Tables (4−6) the results for theground state solutions
44P 3.346 -0.302 333.3 3.398 -0.293 330.6 are displayed. Thus, the prolate solutions have more bind-
45P 3.315 0.222 335.4 3.397 -0.264 332.4 ing than the oblate one for Ne, Na, Mg and S isotopes. In
46P 3.342 -0.251 337.5 3.397 -0.237 334.2 somecases,like24−30Netheprolateandoblatesolutionsare
47P 3.341 -0.232 340.0 3.399 -0.218 336.0 indegeneratestates. Forexample, 24NehasBE=188.9and
48P 3.381 -0.271 341.2 3.379 0.034 337.4 189.1MeVatβ =0.278and−0.259respectively. Contrary
2
49P 3.328 0.088 343.2 3.387 0.012 339.3 to this, the ground state solutions for Al and Si are mostly
50P 3.353 0.101 343.7 3.414 -0.061 339.2 oblate. Forexample,34AlhasBE=269.9and275.1MeVat
51P 3.397 -0.166 344.7 3.437 0.068 339.4 β = 0.159and−0.108respectively. Insuchcases,thepro-
2
52P 3.403 0.109 345.2 3.462 0.079 339.7 latesolutionsareinlow-lyingexcitedintrinsicstate.Notethat
53P 3.428 0.109 346.3 3.487 0.089 340.1
inmanycases,thereexistlowlayingsuperdeformedstates.
54P 3.447 0.074 346.6 3.502 0.016 340.5
It is important to list some of the limitations of the results
55P 3.468 0.037 347.4 3.525 0.001 341.2
due to the input parameters, mostly comes from E and
33S 3.241 0.197 275.5 3.276 0.119 278.9 280.4 pair
E energies.AsonecanseefromFig.3,inmanycasesthere
34S 3.257 -0.168 286.5 3.300 -0.160 289.3 3.285 0.252 291.8 cm
35S 3.260 -0.078 295.7 3.300 -0.006 299.6 298.8 aresolutionsofdifferentshapeslyingonlyafewMeVhigher,
36S 3.273 0.002 306.2 3.310 0.000 309.6 3.299 0.168 308.7 sometimes even degenerate with the ground states. Such a
37S 3.285 0.152 311.6 3.319 -0.008 315.1 313.0 fewMeVdifferenceiswithintheuncertaintyofthepredicted
38S 3.300 0.228 318.6 3.340 0.210 320.2 0.246 321.1 binding energies. A slight change in the pairing parameter,
39S 3.312 0.264 325.3 3.354 0.248 326.5 325.4 among others, may alter the prediction for the ground state
40S 3.325 0.299 332.4 3.370 0.300 332.1 0.284 333.2 shape. WithfewMeVuncertaintyingroundstatebindingen-
41S 3.331 0.287 337.7 3.381 0.294 336.9 337.4 ergies,byreassigningthegroundstateconfigurations,thede-
42S 3.338 0.277 343.2 3.390 0.290 341.0 0.300 344.1 formation may change completely, and make the predictions
43S 3.359 0.318 347.2 3.413 0.326 344.7 346.7 close to each other and agree with the FRDM predictions as
44S 3.381 0.367 351.0 3.440 0.370 348.3 0.254 351.8 well.
45S 3.375 0.312 353.4 3.430 0.311 350.4 354.7
46S 3.371 0.258 356.6 3.420 0.250 352.5
47S 3.385 0.257 358.5 3.428 -0.214 354.8
48S 3.400 0.259 360.8 3.430 -0.200 356.6 C. Quadrupoledeformation
49S 3.403 0.227 362.9 3.430 0.127 358.8
50S 3.403 0.189 365.5 3.440 0.120 360.8 The ground and low-lying excited state deformation sys-
51S 3.427 0.188 366.4 3.459 -0.090 361.8 tematicsforsomeoftherepresentativenucleiforNe,Na,Mg,
52S 3.451 0.183 367.6 3.490 -0.140 362.5 Al, Si, P and S are analyzed. In Fig. 2, the ground state
53S 3.463 0.158 369.1 3.508 -0.113 363.6 quadrupoledeformationparameterβ isshownasafunction
54S 3.477 0.139 371.0 3.530 0.000 364.7 2
of mass number for Ne, Na, Mg, Al, Si, P and S. The β
55S 3.494 0.105 371.4 3.541 0.030 365.4 2
value goes on increasing with mass number for Ne, Na and
Mgisotopesnearthedrip-line. Thecalculatedquadrupolede-
formationparameterβ for34Mgis0.59whichcompareswell
2
B. Neutronconfiguration withtherecentexperimentalmeasurementofIwasakietal[4]
(β = 0.58±0.06). Itfoundthatthissuperdeformedstateis
2
Analyzing the neutron configuration for these exotic nu- 3.2MeVabovethanthegroundband. Again, themagnitude
clei, we notice that, for lighter isotopes of Ne, Na, Mg, Al, ofβ forthedripnucleireduceswithneutronnumberNand
2
Si, P and S the oscillator shell N = 3 is empty in the againincreases. Aregionofmaximumdeformationisfound
osc
[N ,n ,Λ]Ωπ. However,theN = 3shellgetsoccupied foralmostallthenucleiasshowninthefigure. Itsohappens
osc 3 osc
7
0.6 8
NL3 Ne
NMag RMF (NL3) SHF (SkI4) Ne
β)20.4 Al 6 Mg
(eter SPi V) 4 SSi
m 0.2 S
a e
par M 2
mation 0 BE(o0
defor-0.2 BE-p-2
-0.4 -4
20 30 40 50 60
mass number (A)
-6
24 30 36 42 48 54 24 30 36 42 48 54 60
A
FIG.2: Thegroundstatequadrupoledeformationparameterβ ver-
2
susmassnumberAforNe,Na,Mg,Al,Si,PandSisotopesnearthe
drip-linewithNL3parameterset.
FIG.3: Thedifferenceinbindingenergybetweentheprolate-oblate
solutionsisshownforeven-evenNe,Mg,SiandSisotopesnearthe
neutrondrip-linewithNL3andSkI4parametersets.
in cases like, Ne, Na, Mg and Al that the isotopes are maxi-
mum deformed which may be comparable to superdeformed
nearthedrip-line. ForAlandSiisotopes,ingeneral,wefind actual quantitative energy difference of ground and excited
oblatesolutionsinthegroundconfigurations(seeTable5). In configurationcanbegivenbyperformingconfigurationmix-
manyofthecases,thelow-lyingsuperdeformedconfiguration ing(mixingsuchasinthegeneratorcoordinatemethod(GCM)
areclearlyvisibleandsomeofthemcanbeseeninFig. 2. [63])aftertheangularmomentumprojection[60].
D. Shapecoexistence
E. Twoneutronseparationenergy(S )
2n
Oneofthemostinterestingphenomenainnuclearstructure
The appearance of new and the disappearance of known
physicsistheshapecoexistence[59–62].Insomecasesofthe
magic number near the neutron drip-line is a well discussed
nuclei,considerednearthedrip-line,thegroundstateconfig-
topic currently in nuclear structure physics [7, 64]. Some of
urationaccompaniesalow-lyingexcitedstate. Inafewcases,
thecalculationsinrecentpastpredictedthedisappearanceof
it so happens that these two solutions are almost degenerate
the known magic number N = 28 for the drip-line isotopes
in energy. For example, in the RMF calculation, the ground
of Mg and S [65–67]. However, magic number 20 retains
statebindingenergyof24Neis189.1MeVwithβ =−0.259
2 its magic properties even for the drip-line region. In one of
and the binding energy of the excited low-lying configura-
our earlier publications, [68] we analyzed the spherical shell
tion at β = 0.278 is 188.9 MeV. The difference in BE of
2 gap at N = 28 in 44S and its neighboring 40Mg and 42Si us-
thesetwosolutionsisonly0.179MeV.Similarlythesolution
ingNL-SH[69]andTM2parametersets[53]. Thespherical
of prolate-oblate binding energy difference in SkI4 is 0.186
shellgapatN=28in44SwasfoundtobeintactfortheTM2
MeV for 30Mg with β = −0.183 and 0.202. This type of
2 and is broken for NL-SH parametrization. Here, we plot the
degeneratesolutionsareobservedinmostoftheisotopesnear
two-neutron separation energy S for Ne, Mg, Si and S for
2n
thedrip-line. Itisworthytomentionthatinthetruncationof
the even-even nuclei near the drip-line (Fig. 4). The known
the basis space, an uncertainty of ≤ 1 MeV in total binding
magic number N = 28 is noticed to be absent in 44S. On the
energymayoccur. However, thisuncertaintyinconvergence
other hand, the appearance of a sudden decrease in S en-
2n
does not effect to determine the shape co-existence, because
ergyatN=34inSHFresultisquiteprominent,whichisnot
boththesolutionsareobtainedbyusingthesamemodelspace
clearlyvisibleinRMFprediction. Thisisjusttwounitsahead
ofN =N =12.
F B thantheexperimentalshellclosureatN=32[70].
Toshowinaquantitativeway,wehaveplottedtheprolate-
oblatebindingenergydifference(BE −BE )inFig. 3. The
p o
lefthandsideofthefigureisforrelativisticandtherightside
F. SuperdeformationandLowΩparitydoublets
isthenonrelativisticresults. Fromthefigure,itisclearthatan
islandofshapecoexistenceisotopesareavailableforMgand
Si isotopes. These shape coexistence solutions are predicted Thedeformation-drivingm = 1−orbitscomedowninen-
2
takingintoaccounttheintrinsicbindingenergy. However,the ergyinsuperdeformedsolutionsfromtheshellabove,incon-
8
34Mg (1/2+ & 1/2-) 32Mg
10
0
20 RMF (NL3) SHF (SkI4)
.... .... .... ..
-10.. .. .. ..
Ne
15 Mg ..*
(MeV)2n 10 SSi ε(MeV)i--3200......**.... ......**.... ......**.... ....*......
S -40 .. ..
..
5 -50.n.eutron .. .n.eutron.. .ne.utron .. .ne.utron ..
proton proton proton proton
-60
0 β=0.588 β=0.343 β=0.471 β=0.119
2 2 2 2
20 24 28 32 36 20 24 28 32 36 40
FIG. 5: The 1+ and 1− intrinsic single-particle states for the
N 2 2
normalandsuperdeformedstatefor32Mgand34Mg. Doubletsare
noticedfortheSDintrinsicstatesonly. The±1−statesaredenoted
2
FIG.4: Thetwo-neutronseparationS2nenergyversusneutronnum- bygreenlinesandthe±1+statesaredenotedbyblack.
2
berNforneutron-richNe,Mg,SiandSisotopes.
47Al (1/2+ & 1/2-) 46Al
10
iTnAgBbLyEthVeIdI:efOocrmcuaptaiotino.nofneutronorbitsmπin47Aland46Aldriv- 0 ...... ...... ...... .....
A β2 n1+ n1− n3+ n3− n5+ n5− n7+ n7− n9+ -10 ... ... .. .. ...... ....
2 2 2 2 2 2 2 2 2
47Al 0.09 8 10 4 6 2 2 0 2 0 .. ..
47Al 0.672 10 10 4 6 2 2 0 0 0 eV) -20 .. ..* . * ....* . *
46Al 0.109 8 9 4 6 2 2 0 2 0 M * . . . . * . . . .
46Al 0.701 10 10 4 5 2 2 0 0 0 ε (i -30 .. .... ..* .. .... . .*
. . . .
-40 .. .. . . . . .. . .
. . . .
-50 .. neutron .. neutron
trasttothenormaldeformedsolutions. Theoccurrenceofap- neutron pr.oto.n neutron pr.oto.n
proton proton
proximate 1+, 1− paritydoublets(degeneracyof|m|π= 1+, -60 β=0.653 β=0.090 β=0.660 β=0.125
2 2 2
1−states)forthesuperdeformedsolutionsareclearlyseenin
F2igs. 5and6whereexcitedsuperdeformedconfigurationsfor FIG.6: SameasFig.5for46Aland47Al.
32Mg, 34Mg and for 46Al, 47Al are given (RMF solutions).
For each nucleus, we have compared the normal deformed
(β ∼ 0.1−0.3) and the superdeformed configurations and
2 1. StructureofSuperdeformedConfiguration:
analyzed the deformed orbits. The 1+ and 1− states for the
2 2
singleparticlelevelsareshowninFig.5(for32Mgand34Mg)
and Fig. 6 for 47Al and 46Al. The occupation of neutron We discus some clear and important characteristics of su-
states(denotedbymπ)in47Aland46AlisgiveninTableVII. perdeformed solutions ( β ∼ 0.5 or more) obtained in
In both 47Al and 46Al two neutrons occupying oblate driv- mean field models as compared to the normal solutions of
ing f m = 7 orbits in normal deformation are unoccupied smaller deformation. Since the lowering and occupation of
72 2 the deformation-driving Ω = 1 orbits from the shell above
inthesuperdeformed(SD)case;insteadtwoneutronsoccupy 2
theusualvalencespaceissoimportantinproducingsuperde-
the very prolate deformation driving [440]1/2 orbits (raising
formation we have emphasized their role in this discussion.
n to 10) which is a superposition of g d s orbits
12+ 92,27 52,32 12 Thereistheoccurrenceof 1+, 1−orbitsclosetogetherinen-
of N = 4 origin. In 46Al one m = 3− neutron shift 2 2
osc 2 ergy(doublets)belowandneartheFermisurfaceoftheself-
to m = 1−, enhancing the prolate deformation. It is to be consistentsuperdeformedsolutions. Thisfeaturealsooccurs
2
emphasized that the deformations of occupied orbits of self- broadly in Nilsson orbits at asymptotically large prolate de-
consistentSDsolutionsaremore(thantheirnormaldeformed formations (see the Nilsson diagrams in Bohr and Mottelson
counterparts)becauseofmixingamongtheshells. vol. II[71]).
9
2. Somefeaturesofsuperdeformedsolutions: . .
B B B
In normal deformed case, the deformed orbits of a major . . . . .
shellforma“band”-likesetoforbits,distinctlyseparatedfrom . . A+ . A+ . . A+
A- A- A-
themajorshellaboveandbelow(seeFig. 6for47Al(β=0.09) (a) (b) (c)
and 46Al (β = 0.125)). Thus physical states obtained from
such intrinsic states of low deformation will be well sepa-
FIG.7: WithparitydoubletofoccupiedorbitsA+,A−(havingm=
rated in energy from those intrinsic states where excitation
±1/2and+veand-veparities)andanunoccupiedorbitsBpossible
occursacrossamajorshell(asinglenucleonexcitationacross
occupationofneutronsareshowninconfigurations(a),(b)and(c).
amajorshellmeansachangeinparityandsignificantenergy
Thetwoexcitedconfigurations(b)and(c)havethesameK value
changeforsmalldeformation). n
andrepresenttwoexcitedbandsofdifferentparities(paritydoublet).
The above mentioned “band”-like separation of orbits of
Suchsituationcanoccurforneutronconfigurationsinsuperdeformed
major shells of unique parity is quite lost in the case of su- 47Aland32Mg,34Mg(Figs.6,5).
perdeformation (see Fig. 6, β=0.653 of 47Al and β=0.660
of 46Al). The “band”-like orbits now spread in energy (both
downwardandupward)andorbitsofsuccessivemajorshells thesuperdeformedsolutions. Wenotice(fromtheplotofthe
comeclosertoeachotherinenergy; aninter-minglingofor- orbits)thatthereisoccurrenceof 1+and 1−orbitsveryclose
bitsofdifferentparities(seeFigs. 5,6). Thisisasignificant 2 2
to each other in energy for the superdeformed (SD) shape.
structural change from the case of small deformation. This Such 1+, 1− degenerate orbits occur not only for the well-
has also been seen in the case of 84Zr in Hartree-Fock study 2 2
bound orbits but also for the unbound states. For example,
[72,73].
thedoubletofneutronorbits[220]1+and[101]1−are4MeV
Wewouldliketoemphasizethatintheself-consistentmod- 2 2
apart from each other in the normal deformed prolate solu-
els(Skyrme-HFandRMF)thedeformationofthenucleusis
tions;buttheybecomedegenerateinthesuperdeformed(SD)
theresultofthedeformationoftheself-consistentlyoccupied
solutions (shown by * in Figs. 5 and 6 for Mg, Al). More
individualorbits:
suchdoubletsareeasilyidentified(Figs. 5and6)forsuperde-
(cid:88)
Q= qi··· (13) formedsolutionsof32,34Mgand46,47Al. Infactitisnoticed
i(occupied) thattheΩ= 1 statesofuniqueparity,seenclearlywellsepa-
2
ratedinenergyfromtheusualparityorbitsinthenormalde-
The occupation of the more deformation driving orbits from
formedsolutions, occursclosertotheminenergyfortheSD
the shell above the valence space and the unoccupation of
states,showingadegenerateparitydoubletstructure. Infact,
oblate driving orbits (e.g. f , m = ±7) contribute much to
27 2 for SD solution the 1+ and 1− orbits are intermixed in the
configurationmixingandtheloweringofm= 1 orbitsandto 2 2
2 energy plot; while for the normal deformation they occur in
generationofthequadrupoledeformation.Becauseofcoming
distinctgroups. ThisistruebothintheSkyrmeHartreeFock
togetherinenergyofm= 21+and 12−orbits,itiseasytosee andtheRMFcalculations.
thatsuperdeformedintrinsicstatesoftwodifferentparitiesfor This can be seen by examining the 1+ and 1− orbits for
aparticularKquantumnumbercanbeformedwhichwillbe 2 2
smallandlargedeformationsinFig. 5. Thiscanleadtopar-
closetoeachotherinenergy. Thiswillleadtoparitydoublets
ity mixing and octupole deformed shapes for the SD struc-
inbandstructures.Fortheneutron-richnucleibeingdiscussed
tures [72]. Parity doublets and octupole deformation for su-
here,theprotonsarequitewellboundandpossiblelowenergy perdeformedsolutionshavebeendiscussedfor84Zr[72,73].
excitationswillbethoseofneutronsneartheFermisurface.
Thereismuchinterestfortheexperimentalstudyofthespec-
|φ >=|φp >|φn >··· , (14) traofneutron-richnucleiinthismassregion[74]. Thehighly
K Kp Kn deformedstructuresfortheneutron-richNe-Na-Mg-Alnuclei
whereK andK aretheKquantumnumbersforprotonand areinterestingandsignatureofsuchsuperdeformedconfigu-
p n
neutronconfigurations(K=K +K ). rations(withparitydoubletstructure)shouldbelookedfor.
p n
3. Examplesofparitydoubletconfigurations: IV. SUMMARYANDCONCLUSIONS
Weillustrateschematicallypossibleparitydoubletofcon-
Insummary,wecalculatethegroundandlow-lyingexcited
figurations for neutrons in Fig. 7, the proton configuration
state properties, like binding energy and quadrupole defor-
|φpKp >beingfixed. Weshowherethelastfewneutronoccu- mation β2 using RMF(NL3) formalism for Ne, Na, Mg, Si,
pations of superdeformed solutions and rearrangements near P and S isotopes, near the neutron drip-line region. In gen-
the Fermi surface. In Fig. 7, (b) and (c) are parity doublet eral, we find large deformed solutions for the neutron-drip
of configurations. A+ → A− transition between (b) and (c) nucleiwhichagreewellwiththeexperimentalmeasurement.
configurationsisofoddparitymultipolenature. Thecalculationisalsorepeatedintheframe-workofnonrela-
Thus, in summary, we find a systematic behaviour of the tivisticHartree-FockformalismwithSkyrmeinteractionSkI4.
low Ω (particularly 1+ and 1−) prolate deformed orbits for Both the relativistic and non-relativistic results are compara-
2 2
10
bletoeachotherfortheconsideredmassregion.Inthepresent state. Also, a spherical nucleus can have a fairly low lying
calculations a large number of low-lying intrinsic superde- collective2+state(e.g.Snnuclei)becauseofquadrupolecol-
formed excited states are predicted in many of the isotopes lectivity. Inthisstudywefindthat,fortheSDshape,thelow
andsomeofthemarereported. Frombindingenergypointof Ωorbits(particularlyΩ= 1)becomemoreboundandnearly
2
view, i.e. the sudden fall in S value, the breaking of N = degenerate with the orbits of opposite parity, i.e. they show
2n
28 magic number and the likely appearance of a new magic a parity doublet structure. Closely lying parity-doublet band
numberatN=34noticedinournon-relativisticcalculations, structures and enhanced odd parity multipole transitions are
in contrast with RMF finding. This is an indication of more possibleforthesuperdeformedshapes.
bindingthantheneighbouringisotopes. Howevertoconfirm CRP was supported during this work by Project
N=34asamagic/non-magicnumbermorecalculationsare SR/S2/HEP-37/2008ofDepartmentofScienceandTechnol-
needed. A deformed nucleus has a collective low lying 2+ ogy,Govt. ofIndia.
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