Rapid structural change in low-lying states of neutron-rich Sr and Zr isotopes

1.7 MB·English

by  H. Mei

Rapid structural change in low-lying states of neutron-rich Sr and Zr isotopes
H. Mei1,2, J. Xiang1, J. M. Yao2,1, Z. P. Li1, J. Meng3,4,5
1School of Physical Science and Technology, Southwest University, Chongqing 400715, China
2Physique Nucl´eaire Th´eorique, Universit´e Libre de Bruxelles, C.P. 229, B-1050 Bruxelles, Belgium
3School of Physics, Peking University, Beijing 100871, China
4School of Physics and Nuclear Energy Engineering, Beihang University, Beijing 100191, China and
5Department of Physics, University of Stellenbosch, Stellenbosch, South Africa
(Dated: January 18, 2012)
Therapidstructuralchangeinlow-lyingcollectiveexcitationstatesofneutron-richSrandZriso-
topesisstudiedbysolvingafive-dimensionalcollectiveHamiltonianwithparametersdeterminedby
bothrelativisticmean-fieldandnon-relativisticSkyrme-Hartree-FockcalculationsusingthePC-PK1
and SLy4 forces respectively. Pair correlations are treated in BCS method with either a separable
2 pairing force or a density-dependent zero-range force. The isotope shifts, excitation energies, elec-
1 tricmonopoleandquadrupoletransitionstrengthsarecalculatedandcomparedwithcorresponding
0 experimentaldata. Thecalculated resultswithboththePC-PK1andSLy4forcesexhibitapicture
2 of spherical-oblate-prolate shape transition in neutron-rich Sr and Zr isotopes. Compared with the
experimental data, the PC-PK1 (or SLy4) force predicts a more moderate (or dramatic) change in
n
mostofthecollectivepropertiesaroundN =60. Theunderlyingmicroscopicmechanismresponsible
a
J for therapid transition is discussed.

7
PACSnumbers: 21.10.-k,21.10.Re,21.60.Jz, 27.60.+j
1

]
h I. INTRODUCTION large deformation properties for 100Zr [29]. The mecha-
t nismresponsible for the rapidtransitionin the structure
-
l of low-lying states in Sr and Zr isotopes around N =60
c The spectroscopy of nuclear low-lying states provides
requires further investigation.
u rich information about the interplay of nuclear collec-
n Nuclear energy density functional theory (DFT) is
tivity and single-particle structure. For neutron-rich Sr
[ nowadays one of the most important microscopic ap-
and Zr isotopes, lots of spectroscopic data have been
proaches for large-scale nuclear structure calculations in
1 measured [1–11]. The abrupt changing in the lifetimes
medium and heavy nuclei. The main ingredient of DFT
v and excitation energies of 2+ states, two-neutron sepa-
1 is the energy density functional (EDF) that depends on
6 ration energies and rms charge radii indicates a sudden
5 densitiesandcurrentsrepresentingdistributionsofnucle-
onset of large quadrupole deformation at neutron num-
5 onic matter, spins, momentum, and kinetic energy and
ber N = 60. In the past decades, various models have
3 their derivatives. In the past decades, a lot of efforts
1. beenemployedtostudysuchadramatictransitioninthe have been devoted into finding out an EDF with reli-
low-energystructureofthesenuclei [12–25],thedescrip-
0 able and good predictions by optimizing about 10 uni-
tion of which has been a challenge in theoretical nuclear
2 versal parameters to basic properties of nuclear matter
physics.
1 and some selected nuclei. At present, there are three
v: Two main mechanisms based on shell models and types of most successful EDFs, i.e., the non-relativistic
i mean-fieldapproacheswereproposedto explainthe sud- Skyrme and Gogny forces and the effective relativistic
X
denonsetoflargenuclearcollectivityatN =60,namely, Lagrangian, being employed extensively in the descrip-
r a strong isoscalar proton-neutron interaction between tion of nuclear structure properties [30].
a
particlesoccupyingtheg9/2-g7/2 spin-orbitpartners[12], The nuclear DFT of single-reference state (SR-DFT)
and the occupation of low-K components of the h11/2 with constraint on quadrupole moments, in the context
neutron orbit [26, 27] respectively. Besides these two ofHartree-Fock(HF)orHartree-Fock-Bogoliubov(HFB)
factors,thesimultaneouspolarizationof2p1/2,2p3/2,and methods, has already been adopted to study the evo-
1f5/2protonorbitsasonegoesfrom96Srto98Srandfrom lution of deformation energy surfaces in β-γ plane (see
98Zr to 100Zr was pointed out to be the the major factor the recent work [31]) for the neutron-rich Sr and Zr iso-
inthemorerecentprojectedshellmodelstudy[28]. How- topes [27, 32, 33]. Even though, all of these studies have
ever,inmostofthepreviousstudiesfornuclearlow-lying indeedshowntheincreasingofdeformationsbasedonthe
states with, for instance the (projected) shell model, ei- shift of global minimum. However, most of these stud-
ther a phenomenological effective interaction or effective ies are focused on the properties of nuclear mean-field
chargesforneutronsandprotonsdefinedwithinaspecific ground states. In particular, a competing weakly oblate
valence space was introduced. In particular, it has been deformedminima coexistingwitha largeprolateone has
shown recently that the large-scale shell model calcula- been found in nuclei around N = 60, in which case the
tions with a more extended model space and carefully dynamic correlation effects from quadrupole fluctuation
adjusted effective interaction are able to reproduce the are expected to be significant. Moreover, the evolution
low-spin spectroscopic data of 90−98Zr, but fail to give behavioroflow-lyingexcitedstatesissensitivetothebal-
2
ance between the oblate and prolate minima. Therefore, quadrupolevibrations,rotations,andtheircouplingscan
it is necessary to extend these studies for the low-lying be written in the form [46, 47, 49]
excited states in Sr and Zr isotopes.
Inthecontextofgenerator-coordinatemethod(GCM), Hˆ =Tˆvib+Tˆrot+Vcoll , (1)
the framework of SR-DFT has been extended for study-
ing the nuclear low-lying states. In Ref. [34], the collec- where Vcoll is the collective potential. The vibrational
tive quadrupole and octupole excitations in Zr isotopes kinetic energy reads,
were studied using a basis generated by the HF+BCS
~2 1 ∂ r ∂
calculations with the SkM* effective interaction. How- Tˆ = β4B
ever, the low-lying states have not been described quite vib −2√wr (cid:26)β4 (cid:20)∂βrw γγ∂β
well, probably due to the missing of effects from tri- ∂ r ∂ 1 ∂
β3B +
axiality and angular momentum projections. In recent −∂βrw βγ∂γ(cid:21) βsin3γ (cid:20)−∂γ
years, the GCM has been extended much further by im-
r ∂ 1 ∂ r ∂
plementing the exact one-dimensional [35–37] or three- sin3γB + sin3γB (2),
dimensional [38–41] angular momentum projection or rw βγ∂β β∂γrw ββ∂γ(cid:21)(cid:27)
together with particle number projection before or af-
and rotational kinetic energy,
ter variation for the mean-field states in the modern
EDF calculations. The dynamic correlation effects re-
1 3 Jˆ2
lated to the symmetry restoration and quadrupole fluc- Tˆ = k, (3)
rot
tuation(alongbothβandγ directions)aroundthemean- 2kX=1 Ik
field minimum are included naturally without introduc-
ing any additionalparameters. However,the application withJˆ denotingthecomponentsoftheangularmomen-
k
of these methods with triaxiality for systematic study tum in the body-fixed frame of a nucleus. It is noted
is still much time-consuming. Up to now, such kind of that the mass parameters B , B , B , as well as the
ββ βγ γγ
studyismostlyrestrictedtolightnuclei[42,43]andsome moments of inertia , depend on the quadrupole defor-
k
I
specific medium heavy nuclei [44, 45]. mation variables β and γ,
AsaGaussianoverlapapproximationofGCM,thecol-
lective Hamiltonian with parameters determined by self- k =4Bkβ2sin2(γ 2kπ/3), k =1,2,3. (4)
I −
consistent mean-field calculations is much simple in nu-
Two additional quantities that appear in the expres-
mericalcalculations,andhasachievedgreatsuccessinde-
sion for the vibrational energy: r = B B B , and
scriptionofnuclearlow-lyingstates[46–50]andimpurity 1 2 3
w =B B B2 ,determinethevolumeelementinthe
effect of Λ hyperon in nuclear collective excitation [51]. ββ γγ− βγ
collective space. The corresponding eigenvalue problem
In particular, a systematic study of low-lying states for
is solved using an expansion of eigenfunctions in terms
a large set of even-even nuclei has been carried out with
of a complete set of basis functions that depend on the
theGognyD1SforcemappedcollectiveHamiltonianand
deformationvariablesβ andγ,andthe Eulerangles φ, θ
good overall agreement with the low-lying spectroscopic
and ψ [53].
data has been achieved [52]. However, some fine struc-
The dynamics of the 5DCH is governed by the seven
tures along the isotonic or isotopic chains are still not
functions of the intrinsic deformations β and γ: the col-
reproduced satisfactorily. For Sr and Zr isotopes, the
lective potential V , the three mass parameters: B ,
evolution of isotope shifts and the change in the proper- coll ββ
B , B , and the three moments of inertia . These
ties oflow-lyingstatesarefoundto be moremoderatein βγ γγ k
I
functions are determined by the relativistic mean-field
comparison with the data. Therefore, in this work, we
(RMF)+BCS calculations using the PC-PK1 force [54]
will examine the evolution of low-lying states obtained
for the particle-hole (ph) and the separablepairing force
from the calculations with two other popular EDFs, i.e.,
(adjusted to reproduce the pairing properties of the
the non-relativistic Skyrme force and the effective rela-
Gogny force D1S in nuclear matter) [55, 56] for the
tivisticLagrangian. Thedifferenceinthe resultsofthese
particle-particle(pp)channelsorbytheSkyrme-Hartree-
two calculations will be emphasized.
Fock (SHF)+BCS calculations using the SLy4 force [57]
The paper is organized as follows. In Sec. II we will
forthephchannelandadensity-dependentδ-forceinthe
introducebriefly ourmethodusedto study the low-lying
pp channel,
states in neutron-rich Sr and Zr isotopes. The results
and discussions will be presented in Sec.III. A summary ρ(r)
is made in Sec. IV. V(r ,r )=Vpp 1 δ(r r ), (5)
1 2 0 (cid:20) − ρ (cid:21) 1− 2
0
with a strength of Vpp = 1000MeV fm3 and ρ =0.16
II. THE METHOD fm−3forbothneutro0nsan−dprotonsandwithaso0ftcutoff
at 5 MeV above and below the Fermi energy as defined
The quantized five-dimensional collective Hamilto- inRef.[58]. Aconstraintonthe deformationparameters
nian (5DCH) that describes the nuclear excitations of of both β, ranging from 0 to 0.8 (∆β = 0.05) and γ,
3
-1 MeV) 3400 -1 MeV) 3400
2 20 2 20
I (110 I (110

-1 MeV)4000 SLy4 PC-PK1 -1 MeV)4000 SLy4 PC-PK1
2 2
(200 (200
B B

-1 eV) 2000 -1 eV) 0
M M200
2 2
(100 (
100
B B

0 0
96 96 98 98
24 Sr Sr 24 Zr Zr
22 (fm)rp2202 91800SSrr 91800SSrr 22 r (fm)p2202 110002ZZrr 110002ZZrr
18
18
-0.8 -0.4 0.0 0.4 0.8 -0.4 0.0 0.4 0.8 -0.8 -0.4 0.0 0.4 0.8 -0.4 0.0 0.4 0.8
FIG.1: (Coloronline)Momentofinertiaalongx-directionI1,massparametersBββ,Bγγ andsquaredprotonradiirp2 in5DCH
for (left panel) 96,98,100Sr and (right panel) 98,100,102Zr as functions of axial deformation parameter β determined from the
mean-field calculations with both theSLy4and PC-PK1 forces.
◦ ◦ ◦
ranging from 0 to 60 (∆γ = 6 ) is imposed in both The potential V in Eq.(1) is given by,
coll
calculations.
The moments of inertia Iκ are calculated using the Vcoll(β,γ)=Etot(β,γ)−∆Vvib(β,γ)−∆Vrot(β,γ), (10)
Inglis-Belyaev formula [59, 60]
where E is the total energy from constrained mean-
tot
k = (uivj −viuj)2 iJˆk j 2, (6) fipeolidnt-ceanlceurglaytioofnvs.ibrTathieon∆aVnvdibroatnadtio∆nVrreostpaecretivtehley,zero-
I E +E h | | i|
Xi,j i j
1
where k = 1,2,3 denotes the axis of rotation, and the ∆Vvib(β,γ) = 4Tr[M−(31)M(2)], (11)
summationofi,jrunsovertheprotonandneutronquasi-
particlestates. ThemassparametersBµν(β,γ)aregiven ∆Vrot(β,γ) = ∆Vµµ(β,γ), (12)
by [61] µ=−X2,−1,1
~2 where the matrix is determined by Eq.(8) with in-
B (β,γ)= −1 −1 , (7) M(n)
µν 2 hM(1)M(3)M(1)iµν dquicaedsrµu,pνolreuonpneirnagtoorvsedre0finaendda2saQˆnd cor2rze2spoxn2dingy2maansds
20
with Qˆ x2 y2. Moreover,∆V (β,γ)≡is cal−culat−ed by
22 µµ
≡ −
i Qˆ j j Qˆ i
M(n),µν(β,γ)=Xi,j h | (2Eµi|+ihE|j)n2ν| i(uivj +viuj)2. ∆Vµν(β,γ)= 41M((23)),,µµνν((ββ,,γγ)), (13)
M
(8)
The mass parameters B in Eq.(7) can be converted where (β,γ) is also determined by Eq.(8), but
µν M(n),µν
into the forms of B ,B ,B by using the following withtheintrinsiccomponentsofquadrupoleoperatorde-
ββ βγ γγ
relationships [53], fined as,
cos2γ sin2γ sin2γ 2iyz, µ=1
B B a
Bβββγ=−21sin2γ cos2γ 21sin2γB0020a0020, Qˆ2µ ≡ −−2i2xxyz,, µµ==−21 (14)
Bγγ  sin2γ −sin2γ cos2γ B22a22  −
(9) The details about the solution of constrained
where the coefficients a02 = a00/√a22 = a00/2, with RMF+BCS and SHF+BCS equations have been given
a =9r4A10/3/16π2, and r =1.2. in Refs. [33] and [62] respectively.
00 0 0
4
III. RESULTS AND DISCUSSION N = 58 to N = 60. However, the subtle balance be-
tween these two minima is quite different in these two
A. Collective parameters in Hamiltonian calculations, as shown in Fig. 5, where the total en-
ergy in 96,98,100Sr and 98,100,102Zr as a function of axial
deformation parameter β is plotted. The evolution of
Figure 1 displays the moment of inertia along x-
deformation energy curves, i.e., from weakly oblate de-
direction, mass parameters and squared proton radii in
formed96Sr,98Zrtolargeprolatedeformed98Sr,100Zrin
5DCH for 96,98,100Sr and 98,100,102Zr isotopes as func-
SLy4 calculations, is much more rapid than that in PC-
tions of axial deformation parameter β determined from
PK1calculations. Moreover,the oblateandprolatemin-
the mean-fieldcalculationswithboth SLy4andPC-PK1
ima in the deformation energy curves of 96,98,100Sr and
forces. We note that the collective parameters in 5DCH
98,100,102Zrarequitecloseinenergy. Averylargemixing
do not change much in 96,98,100Sr and 98,100,102Zr iso-
of oblate and prolate configurations is expected in their
topes at all the β-γ deformation regions, except for the
◦ ground states. In contrary to the PC-PK1 calculations,
B with β 0.4 and γ = 60 as well as the B with
ββ γγ
≃ ◦ theSLy4forcegivesabroaderordeeperoblateminimum
β 0.5 and γ =0 . Moreover,the parameters in 5DCH
≃ in 96Sr and 98Zr respectively and deeper or broader pro-
determinedfrombothSLy4andPC-PK1mean-fieldcal-
late minimum in 98,100Sr and 100,102Zr respectively. In
culationsarequite similar. Forthe meansquaredproton
other words, the 96Sr and 98Zr are more oblate, while
radii r2, the SLy4 predicts slightly larger value than the
p 98,100Sr and 100,102Zr are more prolate in the SLy4 cal-
PC-PK1 by 0.5% systematically.
∼ culations.
The structuralchange in low-lying states is illustrated
clearly in Fig. 6, where the distribution of the squared
B. Spectroscopy of nuclear low-lying states
collective wave functions ρ in the β-γ plane for the
Iα
0+ and 2+ states from the 5DCH calculations with both
Figure 2 displays the electric quadrupole transition 1 1
the SLy4 and the PC-PK1 forces is plotted. The ρ is
strength B(E2 : 2+ 0+), excitation energy E (2+) of Iα
2+ state and the r1at→io R1 ( E (4+)/E (2+))xas 1func- defined as
1 4/2 ≡ x 1 x 1
tions of neutron number in Sr and Zr isotopes from the ρ (β,γ)= ΨI (β,γ)2β3 sin3γ , (15)
Iα | α,K | | |
5DCH calculationswith both the SLy4 and the PC-PK1 XK
forces. Systematically,theB(E2:2+ 0+)(orE (2+))
1 → 1 x 1 which follows the normalization condition,
values from the 5DCH calculations with both forces are
increasing (or decreasing) with the neutron number up ∞ 2π
to N = 62 in both Sr and Zr isotopes, which indicates βdβ dγρIα(β,γ)=1. (16)
Z Z
the existence of transition from spherical to prolate de- 0 0
formed shapes. Compared with the experimental data, Here,ΨI (β,γ)isthecollectivewavefunctionthatcor-
α,K
the evolutionof B(E2:2+ 0+) andR with respect responds to the solution of 5DCH. The distributions of
1 → 1 4/2
to the neutron number around N = 60 is much more the squaredcollective wave functions in these two calcu-
dramatic (moderate) in the results of SLy4 (PC-PK1) lations are quite different. The SLy4 predicts a sudden
calculations. Similar evolution behavior as that by the transitionfromoblate 96Sr to verygoodprolate98,100Sr,
PC-PK1 force is also observed in the 5DCH calculations while, the PC-PK1 gives a coexistence picture for the
with the Gogny force D1S [52]. However, this dramatic ground states of 96,98,100Sr, with the dominant compo-
decreasing in E (2+) from N = 58 to N = 60 in Sr nent changing from oblate to prolate and back to oblate
x 1
and Zr isotopes, together with the sudden increasing of again moderately. For the 2+ state in the calculations
1
E (2+) in 96Zr is not reproduced in both calculations. with the SLy4 force, the dominant component in 96Sr
x 1
is oblate and it becomes well prolate in 98Sr. While
in the calculations with the PC-PK1 force, the domi-
C. Deformation energy surfaces and collective nant component is already prolate in 96Sr. This pic-
wave functions ture provides a interpretation for the evolution char-
acter of B(E2 : 2+ 0+), E (2+) and R shown
1 → 1 x 1 4/2
Tounderstandtheevolutioncharacterofspectroscopic in Fig. 2. Compared with the experimental data for
quantities around N = 60 shown in Fig. 2, we plot the B(E2:2+ 0+)andR ,the SLy4(orPC-PK1)force
deformationenergysurfacesof96,98,100Srand98,100,102Zr overestim1at→es (1or undere4s/t2imates) somewhat the slop of
isotopesinβ-γplanefromtheconstrainedmean-fieldcal- shape transition around N =60.
culations with both SLy4 and PC-PK1 forces in Figs. 3
and 4 respectively. In both calculations, there is always
a weakly oblate deformed minimum coexisting with a D. Isotope shifts and monopole transition
prolate minimum in the deformation energy surfaces of strengths
96,98,100Sr and 98,100,102Zr. Furthermore, the absolute
minimum in both calculations is shifted from the oblate Nuclear chargeradiior isotopic shifts are goodindica-
to the prolate side asthe neutronnumber increasesfrom tors of shape changes along isotopic chains. In Refs. [32]
5

SLy4 PC-PK1
2 b)0.4 2 b)0.4
2 e (a) 2 e (a)
+ 0) (10.3 Exp. + 0) (10.3 Exp.
+ 210.2 5DCH + 210.2 5DCH
2: 2:
E E
B(0.1 B(0.1
Sr Zr Sr Zr
0.0 0.0
V) 2.5 (b) V) 2.5 (b)
e e
+ E (2) (Mx1112...050 + E (2) (Mx1112...050
0.5 0.5
0.0 0.0
(c) (c)
R4/2 3 R4/2 3

2 2
1 1
50 52 54 56 58 60 62 50 52 54 56 58 60 62 50 52 54 56 58 60 62 50 52 54 56 58 60 62
neutron number neutron number neutron number neutron number
FIG. 2: (Color online) The electric quadrupole transition strength B(E2:2+ → 0+), excitation energy E (2+) and the ratio
1 1 x 1
R (≡E (4+)/E (2+))asfunctionsofneutronnumberinSrandZrisotopesfromthe5DCHcalculationswith(leftpanel)the
4/2 x 1 x 1
SLy4force and (right panel) the PC-PK1 force, in comparison with theexperimental data [63]. The R values for vibration
4/2
(2.00) and rotation (3.33) limits are indicated with horizontal dotted lines.
60 60 60 60 60 60
96 (deg) 98 (deg) 100 (deg) 96 (deg) 98 (deg) 100 (deg)
Sr 40 Sr 40 Sr 40 Sr 40 Sr 40 Sr 40
20 20 20 20 20 20
0 0 0 0 0 0
0.00.2 0.4 0.6 0.8 0.00.2 0.4 0.6 0.8 0.000..22 00..44 00..66 00..88 0.00.2 0.4 0.6 0.8 0.00.2 0.4 0.6 0.8 0.000..22 00..44 00..66 00..88
60 60 60 60 60 60
98 (deg) 100 (deg) 102 (deg) 98 (deg) 100 (deg) 102 (deg)
Zr 40 Zr 40 Zr 40 Zr 40 Zr 40 Zr 40
20 20 20 20 20 20
0 0 0 0 0 0
0.00.2 0.4 0.6 0.8 0.00.2 0.4 0.6 0.8 0.000..22 00..44 00..66 00..88 0.00.2 0.4 0.6 0.8 0.00.2 0.4 0.6 0.8 0.000..22 00..44 00..66 00..88
FIG. 3: (Color online)The deformation energy surfaces of FIG. 4: (Color online) Same as Fig. 3, but from the con-
96,98,100Srand98,100,102Zrisotopesintheβ-γ plane,from the strained RMF+BCS calculations with the PC-PK1 force for
constrained SHF+BCS calculations with the SLy4 force for thephchannel and a separable force for the ppchannel.
thephchannelanddensity-dependentδforcefortheppchan-
nel. All energies are normalized to the absolute minimum.
Each contour line is separated by 0.5 MeV.
quadrupolecorrelationeffectonchargeradiiofalargeset
ofeven-evennuclei has been studied in the frameworkof
configurationmixingofprojectedaxiallydeformedstates
and [33], the evolution of charge radii with the neutron
based on a topological Gaussian overlap approximation.
number in Sr and Zr isotopes around N = 60 has been
It has been shown that the dynamical correlation leads
studied with the self-consistent constrained mean-field
to an overall increase of radii, and it might also reduce
calculations. The charge radii corresponding to differ-
the charge radii for some specific nuclei. Therefore, it is
ent local minima in the deformation energy surface are
interestingtoshowtheevolutionofchargeradiiwiththe
comparedwith the data. It has been shownthat a rapid
correlationeffect from quadrupole fluctuation.
change in nuclear shape is essential to reproduce the ex-
perimental charge radii. However, in these two studies, Figure 7 displays the isotope shifts corresponding to
thebeyondmean-fieldcorrelationeffectonnuclearcharge different configurations in Sr and Zr isotopes from the
radii has not been examined. In Ref. [64], the dynamic mean-field calculations with both SLy4 and PC-PK1
6
2
V) 1
e
M
0
y(
rg-1
e
En-2 96 98
-3 98 Sr SLy4 10 0Zr
Sr Zr
-4 10 0Sr 102Zr
V) 1
e
M
0
y(
rg-1
e
En-2
-3 PC-PK1
-4
-0.4 0.0 0.4 0.8 -0.4 0.0 0.4 0.8
FIG. 5: (Color online) The total energy in 96,98,100Sr and
98,100,102Zr as a function of axial deformation parameter β
from the constrained mean-field calculations with both the
SLy4 and the PC-PK1 forces. All energies are normalized to
thevalue at β=0. FIG. 7: (Color online) The isotope shift (normalized to
N =50)inSrandZrisotopesasafunctionofneutronnumber
from the mean-field calculations of both SHF+BCS with the
SLy4forceandRMF+BCSwiththePC-PK1force. Thecor-
responding 5DCH calculated results and experimental data
from Refs. [65, 66] are given as well. The “obl.”, “pro.”,
96 98 100 “sph.” are used to label the character of mean-field configu-
Sr Sr Sr
SLy4 60 60 60 rations.
(deg)

40 40 40
+
01 forces. In addition, we present the 5DCH predicted iso-
20 20 20
tope shifts, in which, the quadrupole fluctuation effect
with triaxiality is included. It is shown clearly that the
0 0 0
0.0 0.2 0.4 0.6 0.8 0.0 0.2 0.4 0.6 0.8 0.0 0.2 0.4 0.6 0.8 overall of isotope shifts could be reproduced much bet-
60 60 60
(deg) ter by the 5DCH calculations with dynamic correlation

40 40 40 effect. The shape evolution picture consistent with that
+ exhibited in the B(E2 : 2+ 0+), R as well as the
21 20 20 20 1 → 1 4/2
distribution of squared wave functions is shown again in
the isotope shifts, a sudden rising of which from N =58
0 0 0
0.0 0.2 0.4 0.6 0.8 0.0 0.2 0.4 0.6 0.8 0.0 0.2 0.4 0.6 0.8 to N = 60 due to the transition from weakly oblate de-
60 60 60
PC-PK1 (deg) formed to large prolate deformed shapes is slightly over-

40 40 40 estimated (or underestimated) in the 5DCH calculations
0+1 20 20 20 with the SLy4 (or PC-PK1) force.
Electric monopole transition strengths are a model-
independent signature of the mixing of configurations
0 0 0
0.0 0.2 0.4 0.6 0.8 0.0 0.2 0.4 0.6 0.8 0.0 0.2 0.4 0.6 0.8 with different mean-square charge radii [69] and there-
60 60 60
(deg) fore provide a good way to illustrate the occurrence of

40 40 40 shape-coexistence. Figure 8 shows the evolution of elec-
+
21 tric monopole transition strength ρ2(E0 : 0+ 0+),
20 20 20 2 → 1
given by the off-diagonal element of charge radius,
0.0 0.2 0.4 0.6 0.80 0.0 0.2 0.4 0.6 0.80 0.0 0.2 0.4 0.6 0.80 0+ e r2 0+ 2
ρ2(E0;0+ 0+)= h 2| k k k| 1i , (17)
2 → 1 (cid:12) PeR2 (cid:12)
(cid:12) 0 (cid:12)
FIG. 6: (Color online) The distribution of squared collective (cid:12) (cid:12)
9w6a,9v8e,10fu0Snrctfiroonms inthteh5eDβC-γHpclaanlceulfaotriotnhsew0+1ithanbdot2h+1 tshteatSeLsyin4 wnuitmhbeRr0in=ne1u.2trAo1n/-3ricfmh,S(cid:12)rwiatnhdrZesrpeiscottotpoes(cid:12)t.heTnheeutsryosn-
and PC-PK1 forces. tematic of ρ2(E0 : 0+ 0+) is similar in the calcula-
2 → 1
tions with the SLy4 and PC-PK1 forces. Both calcula-
7
ergy levels, both calculations predict a large shell gaps
around the Fermi energy level in the prolate side, which
0.16 SLy4 PC-PK1 is mainly formed by two levels split from the degener-
+ 0) 10.12 ate π1g9/2 state due to the deformation (or Jahn-Teller)
effect. The size of this energy gap does not change too
+ 02 Sr S r much from 96Sr to 98Sr in both calculations. It means
E0:0.08 that the proton plays a minor role in the rapid nuclear
(
shape transition from the mean-field point of view. For
0.04
the neutron single-particle energy levels, both calcula-

tions predict two evident shell gaps around the Fermi
0.00
energylevelin oblate andprolatesides, however,the de-
0.16 Zr Zr tails of the single-particle structure are quite different.
+ 0) 1
0.12 ComparedwiththePC-PK1force,theSLy4forcepre-
+ 0:02 dicts a stronger spin-orbit splitting for neutrons (by a
E0.08 factor of 1.1) for all the states, as shown in Fig. 10,
( ∼
where the splitting of neutron spin-orbit doublet states,

0.04
0.00 ǫ ǫ
50 5n2eu5t4ron56 nu5m8b6e0r62 50 5n2eu5t4ron56 nu5m8b6e0r62 ∆Eso = nlj2<ℓ−+1nlj>, j≷ =ℓ±1/2, (18)
FIG. 8: (Color online) The electric monopole transition
strength ρ2(E0:0+2 →0+1) from the 5DCH calculations with in the spherical states of 96Sr and 98Sr as a function
both the SLy4 and PC-PK1 forces, in comparison with the
of the orbital angular momentum ℓ is plotted. ǫ is
experimental data [67, 68]. nlj<
the energy of single-particle state with quantum num-
bers(n,ℓ,j ). Consequently,thepositionofν1g state
< 7/2
is pushed up and the ν1h state is pulled down com-
tions predict the same peak position, i.e., at 96Sr and 11/2
pared with the PC-PK1 results as shown in Fig. 9. As
100Zr,whicharethe nucleibeforeandafterthe dramatic
a result, the shell gaps around the Fermi energy at the
transition respectively. Quantitatively, however, as ex-
spherical point and the minima of the deformation en-
pected from the distribution of squared wave functions
ergy curves are quite different. For the spherical point
for 96,98,100Sr in Fig. 6, the SLy4 (or PC-PK1) force
(β = 0), a relatively large shell gap at N = 56, which
predicts a much weaker (or stronger) mixing of oblate
providesa mechanismresponsiblefor the observedmuch
and prolate configurations in their ground states, and higher E (2+) in 96Zr, is shown in the SLy4 calculation,
therefore smaller (larger) ρ2(E0 : 0+ 0+) values in x 1
2 → 1 but not in the PC-PK1 calculations. Moreover, com-
96,98,100Sr. Comparedwiththeexperimentdatafor98Sr,
pared with the PC-PK1 results for 96Sr, the position of
the SLy4 (or PC-PK1) force underestimates (or overes-
ν2d state is almost the same, but the ν1h state is
timates) the ρ2(E0 : 0+ 0+) value by a factor of two 5/2 11/2
2 → 1 muchlowerintheSLy4results. Asaresult,theshellgap
approximately. Similar evolution behavior is also shown
aroundtheFermilevelintheprolateside,mainlyformed
in Zr isotopes. However, except for 98Zr, the difference
by the K =3/2component of ν2d orbit, the intruded
in magnitude between the results of SLy4 and PC-PK1 5/2
K = 1/2 component of ν2f orbit, and other two lev-
calculations is smaller than that in Sr isotopes. 7/2
els with K = 3/2,5/2 split from ν1h orbit, is much
11/2
smaller than that in the PC-PK1 calculations. In 98Sr,
the energyofν1h state is shifted up, whichbroadens
11/2
E. Single-particle energy levels and spin-orbit the shell gap significantly in the prolate side. This big
splitting
changeinthe energygaparoundFermienergyisrespon-
sible for the sudden onset of large prolate deformation
The nuclear low-lying states reflect information about at N = 60 given by the SLy4 calculations. In the PC-
the underlying single-particle structure and vice versa. PK1calculations,however,thischangeintheshellgapof
Therefore, to understand the evolution character of col- prolate side is more moderate. We note that, similar as
lectivity in Sr and Zr isotopes around N = 60 in the the PC-PK1 results, the shift of ν1h state in 96,98Sr
11/2
calculations with both the SLy4 and PC-PK1 forces, in fromtheGognyD1Scalculationsissmallandthechange
Fig.9,weplottheneutronandprotonsingle-particleen- in the shell gap of prolate side is not significant [70].
ergylevelsin96,98Srasfunctionsoftheaxialdeformation On the other hand, compared with the SLy4 force, the
parameter β. In general, the minima in the deforma- PC-PK1 predicts a larger shell gaps in the oblate side
tion energy curve are associated with a shell effect due of 96,98Sr, which provides the mechanism responsible for
to the low level density around the Fermi energy. It is the strong mixing of prolate and oblate shapes in their
shown in Fig. 9 that, for the proton single-particle en- ground states.
8

PC-PK1 SLy4 PC-PK1 SLy4
-2 70 -2 1h11/2 -2 17h011/2 -2 12hd131/2/2
-4 2d3/2 -4 1g7/2 -4 2d3/2 -4 1g7/2
) )
eV -6 1g7/2 -6 56 eV -6 1g7/2 -6 56
(Mn -8 2d5/2 58 -8 2d5/2 58 (Mn -8 2d5/2 58 -8 2d5/2 58
50 50 50 50
-10 1g9/2 -10 1g9/2 -10 1g9/2 -10 1g9/2
-12 96-12 -12 98-12
-8 S-8r -8 S-8r
50
-10 1g9/2 -10 -10 -10 50
)-12 2p1/2 -12 21pg19//22 )-12 12gp91//22 -12 1g9/2
MeV-14 2p3/2 38 -14 1f5/2 38 MeV-14 2p3/2 38 -14 12f5p/21/2 38
(p-16 1f5/2 -16 2p3/2 (p-16 1f5/2 -16 2p3/2

-18 28 -18 28 -18 28 -18 28
-20 -20 -20 -20
-0.4 -0.2 0.0 0.2 0.4 0.6 0.8 -0.4 -0.2 0.0 0.2 0.4 0.6 0.8 -0.4 -0.2 0.0 0.2 0.4 0.6 0.8 -0.4 -0.2 0.0 0.2 0.4 0.6 0.8
FIG. 9: (Color online) Neutron and proton single-particle energy levels in (left panel) 96Sr and (right panel) 98Sr from the
constrainedmean-fieldcalculationswithbothSLy4andPC-PK1forces. ThedotsdenotethecorrespondingFermienergylevels.
corresponding experimental data. The calculated re-

sults with both the PC-PK1 and the SLy4 forces ex-
1.0
hibit a picture of spherical-oblate-prolate shape transi-
=0
tion in neutron-rich Sr and Zr isotopes. However, com-
n=1
0.8 paredwiththeexperimentaldata,thePC-PK1(orSLy4)
)
V
e force predicts a more moderate (or dramatic) change in
M
most of the collective properties around N = 60 and a
(so0.6 much stronger (or weaker) mixing between oblate and
E 96
n=2 Sr: SLy4 prolate configurations in their ground states. The dif-
96
0.4 98Sr: PC-PK1 ference betweenthese twocalculations is mainly because
Sr: SLy4 of the quite different structure in neutron single-particle
98
Sr: PC-PK1
states, mostly caused by the different spin-orbit inter-
0.2 action strengths. Moreover, the sudden broadening of
1 2 3 4
neutron shell gap in the prolate side, mainly formed
orbital angular momentum ( )
by the K = 3/2 component of ν2d , the intruded
5/2
K = 1/2 component of ν2f , and other two compo-
FIG.10: (Coloronline)Splittingofneutronspin-orbitdoublet 7/2
states [c.f.(18)]in thespherical states of96Sr(open symbols) nents of ν1h11/2 state, has been shown to be responsible
and 98Sr (filled symbols) as a function of the orbital angular fortherapidshapetransitionatN =60. However,ithas
momentum ℓ from the mean-field calculations with both the tobe pointedoutthatthe rapidchangeinthe excitation
SLy4(triangle) and thePC-PK1 (circle) forces. energyofthefirst2+statehasnotbeenreproducedinthe
calculations with both the PC-PK1 and SLy4 forces. In
particular, even though the SHF+BCS calculation with
IV. SUMMARY the SLy4 force indeed predicts a sizable spherical shell
gap at N = 56, the corresponding 5DCH calculation is
not able to reproduce the suddenly increased excitation
In summary, the rapid structural change in low-lying
energy for the first 2+ state at 96Zr. A further beyond
collective excitation states of neutron-richSr and Zr iso-
mean-field investigation is required.
topes has been studied by solving a 5DCH with pa-
rameters determined from both the RMF and SHF cal-
culations. Pair correlations are treated in the BCS
Acknowledgments
methodwitheitheraseparablepairingforceoradensity-
dependent zero-range force. The isotope shifts, excita-
tion energies, electric monopole and quadrupole transi- JMY gratefully acknowledges a postdoctoral fellow-
tion strengths have been calculated and compared with ship from the F.R.S.-FNRS (Belgium) and fruitful dis-
9
cussions with Paul-Henri Heenen. This work was partly sities (XDJK2010B007 and XDJK2011B002), the Re-
supportedbytheMajorStateBasicResearchDeveloping search Fund for the Doctoral Programof Higher Educa-
Program 2007 CB815000, the National Science Founda- tionunderGrantNo. 20110001110087andtheSouthwest
tionofChinaunderGrantsNo. 11105111,No. 11105110, University Initial Research Foundation Grant to Doctor
No. 11175002, No. 10975008 and No. 10947013, the (SWU109011 and SWU110039).
Fundamental Research Funds for the Central Univer-
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