Table Of ContentExotic neutron-rich medium-mass nuclei with realistic nuclear forces
Naofumi Tsunoda,1 Takaharu Otsuka,1,2,3,4 Noritaka Shimizu,1
Morten Hjorth-Jensen,5,6 Kazuo Takayanagi,7 and Toshio Suzuki8
1Center for Nuclear study, the University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo, Japan
2Department of physics and Center for Nuclear Study,
the University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo, Japan
3National Superconducting Cyclotron Laboratory, Michigan State University, East Lansing, MI, 48824, USA
4Instituut voor Kern- en Stralingsfysica, Katholieke Universiteit Leuven, B-3001 Leuven, Belgium
5National Superconducting Cyclotron Laboratory and Department of Physics and Astronomy,
Michigan State University, East Lansing, MI, 48824, USA
6Department of Physics, University of Oslo, N-0316 Oslo, Norway
7Department of Physics, Sophia University, 7-1 Kioi-cho, Chiyoda-ku, Tokyo 102, Japan
6
8Department of Physics, College of Humanities and Sciences,
1
0 Nihon University, Sakurajosui 3, Setagaya-ku, Tokyo 156-8550, Japan
2 (Dated: November1, 2016)
t We present the first application of the newly developed EKK theory of the effective nucleon-
c
nucleon interaction to shell-model studies of exotic nuclei, including those where conventional ap-
O
proaches with fitted interactions encounter difficulties. This EKK theory enables us to derive the
1 interactionsuitableforseveralmajorshells(sd+pf inthiswork). Byusingsuchaneffectiveinterac-
3 tion obtained from the Entem-Machleidt QCD-based χN3LO interaction and the Fujita-Miyazawa
three-bodyforce,theenergies,E2propertiesandspectroscopicfactorsoflow-lyingstatesofneutron-
] rich Ne, Mg and Si isotopes are nicely described, as the first shell-model description of the “island
h ofinversion”withoutfitoftheinteraction. Thelong-standingquestionastohowparticle-holeexci-
-t tations occuracross thesd-pf magic gap is clarified with distinct differencesfrom theconventional
l approaches. The shell evolution is shown to appear similarly to earlier studies.
c
u
n PACSnumbers: 21.30.Fe,21.60.Cs,27.30.+t
Keywords: Effectiveinteraction,shellmodel,islandofinversion,shellevolution,exoticnuclei
[
2
v Introduction. – The nuclearshellmodel[1, 2]provides Hilbert space represented by the degrees of freedom of
2 a unified and successful description of both stable and one major (oscillator) shell. As we move towards ex-
4
exotic nuclei, as a many-body framework which can be otic nuclei, some new features and phenomena arise. A
4
related directly to nuclear forces. Exotic nuclei are lo- notable example can be the shell evolution due to nu-
6
0 cated far from the β-stability line on the Segr`e chart, clear forces between a proton and a neutron in different
. exhibiting very short life times, mainly due to an unbal- shells [5, 11, 12]. This leads to significant particle-hole
1
0 ancedratioofproton(Z)andneutron(N)numbers. Ex- excitations between two shells, for example, in Z=8-14
6 otic nuclei differ remarkably in some other aspects from neutron-rich exotic nuclei [3–5, 9, 11–17]. A microscopic
1 their stable counterparts,providing us with new insights understanding of many of these phenomena requires the
:
v in understanding atomic nuclei and nuclear forces [3– degrees of freedom of at least two major shells.
i 5]. Asexperimentaldataonexoticnucleiare,ingeneral,
X To derive SM effective Hamiltonians is a challenge to
less abundant compared to stable nuclei, theoretical cal-
nuclear theory. Several attempts have been made re-
r
culations, interpretations and predictions play an ever
a cently in this direction [18–24], while the issue of two
increasing role.
major shells is still unsettled.
Shell-model (SM) calculations handle the nuclear The aim of this work is first to derive SM interac-
forces in terms of two-body matrix elements (TBMEs). tion for the model space consisting of the sd and pf
In the early days, TBMEs were empirically determined shells based on the so-called Extended Kuo-Krenciglowa
in order to reproduce certain observables. A well-known (EKK) method [25–27]. Secondly, we apply this inter-
example is the effective interaction for p-shell nuclei by action to our studies of exotic neutron-rich Ne, Mg and
Cohen and Kurath [6]. A breakthrough towards more Si isotopes. These are nuclei in and aroundthe so-called
microscopically-derived TBMEs was achieved by Kuo “island of inversion” [14], where the degrees of freedom
andBrownforsd-shellnuclei[7]. Althoughbasicfeatures of sd and pf shells are essential. We thus present, in
ofthenucleon-nucleon(NN)forcefortheSMcalculation this Letter, the first application of the EKK method to
are included in these effective interactions, empirical ad- actual cases. Three-nucleon forces (3NFs) are also in-
justments of TBMEs were needed in order to reproduce cluded since they play an important role in reproducing
various observables [8–10]. basic nuclear properties [28–32].
These effective interactions were all derived for a Hamiltonian and model spaces. – Many-body pertur-
2
bationtheory(MBPT)hasbeenthemethodofchoicefor forZ=N=20willturnoutclosetoSPEsknownfor40Ca.
derivingeffectiveinteractionsforthenuclearshellmodel, The SPEs are isospin invariant, because of the 16O core.
seeforexampleRefs.[33–37]. TheconventionalMBPTs, In this work we study Ne (Z=10), Mg (Z=12) and Si
for instance, Kuo-Krenciglowa (KK) method [33–35], (Z=14) isotopes with a focus on N ∼ 20 nuclei. Eigen-
are constructed for degenerate single-particle states in valuesandeigenstatesareobtainedviastandardLanczos
the model space, which usually refers to one major shell diagonalizationmethod,includingupto8-particle-8-hole
[33–37]. The present model space, however, includes all (8p8h) excitations from the sd-shell to the pf-shell. For
single-particle states of the sd and the pf shells, labeled the cases where the Lanczos diagonalizationis infeasible
sdpf hereafter. This leads to possible divergences when (e.g., N >22)because of too largedimension, we switch
constructing an effective interaction with conventional to a Monte Carlo Shell Model calculation [45, 46].
MBPTsduetothenon-degeneraciesofthesingle-particle TheTBMEsvarygraduallyasafunctionofAbecause
states. A practical, but not satisfactory way to circum- ~ωchanges. AlthoughTBMEscanbecalculatedforeach
vent such divergences has been to enforce in an ad hoc nucleus explicitly, we introduce an overall scaling factor
way degenerate single-particle energies for all involved (A/A )−0.2 with A =28. The value of the power is de-
0 0
single-particle states spanning a model space, see for ex- terminedsothatTBMEscalculatedexplicitlyforseveral
ample Refs. [29, 38, 39]. Employing the EKK method nucleicanbebestreproducedbythissimpleparametriza-
as done in this work, removes the above-mentioned di- tion.
vergence problems and allows for a correct treatment of Results and discussions. – Figure 1 shows the ex-
the different single-particle energies [27]. Starting from perimental ground-state energies (Coulomb-force effects
a Hamiltonian which contains kinetic energy and a two- subtracted) of Ne, Mg and Si isotopes compared with
body NN interaction and following the formalism de- ourtheoreticalcalculationswithandwithoutthree-body
scribed in Ref. [27], we can define an effective Hamilto- forces.
nian through
∞ 1 dkQˆ(ξ) -50 Ne
Heff =HBH(ξ)+Xk! dξk {Heff −ξ}k, (1) -100
k=1 V)
Mg
e
where H and Qˆ are the Bloch-Horowitz Hamiltonian M-150
BH (
and the so-calledQˆ-box [36, 37], respectively. The latter y
g-200 Si
contains only linked and unfolded Feynman-Goldstone er
n
diagrams in MBPT. The quantity ξ is a parameter (the E
-250
originoftheQˆ-boxexpansion),andthepolesofQˆ(ξ)can EXP.
CALC. w 3NF
be avoided with appropriate value. -300 CALC. w/o 3NF
WerenormalizetheNN interactionusingtheso-called
8 14 16 20 28 40
V approach with a cutoff 2.0 fm−1 [40–42]. We solve N
lowk
Eq. (1) with a harmonic oscillator basis with the oscilla-
torparameter~ω =45A−1/3−25A−2/3MeV=12.10MeV FIG. 1. (Color online) Ground-state energies of Ne, Mg and
where A = 28 (A = Z +N). The Qˆ-box is calculated Si isotopes. Circles represent the experimental values. Shell-
upto the thirdorderinEq. (1) andintermediateexcita- modelresultswith(without)3NFareshownbysolid(dashed)
tions up to 17 ~ω are included to guarantee convergence lines. Arrowsindicatethepredicteddriplines. Experimental
data are taken from Ref. [47]
inthesumoverintermediatestates. Three-nucleonforces
are included through the Fujita-Miyazawa term with its
strength given by a standard π-N-∆ coupling [43]. This Figure 1 indicates that the ground-state energies are
term is in turn transformed into a medium-dependent wellreproducedwhen3NFeffects areincluded. Itisalso
two-body interaction, see Refs. [28, 39, 44] for details. worth noticing that the repulsion due to the 3NF grows
The single-particle energies (SPEs) in the sdpf space as N increases in all the isotopic chains. This is consis-
areinput too. Becauseofon-goingefforts towardsa bet- tent with earlier studies [28, 39]. We point out that this
ter description of density profiles [32], which is directly repulsion becomes stronger also as Z increases, suggest-
relatedtothesizeofshellgap,weemployhere16Oasthe ingthatthepresent3NFcontributionisrepulsivealsoin
closed-shell core and fit the values of SPEs to selected the proton-neutronchannel. Figure 1 shows also present
observables such as ground-state energies of N < 20, predictions for the drip lines [48, 49].
34Si 2+ level, etc. The energies of the 2p and the Figure 2 (a) shows the 2+ and 4+ energy levels. Ex-
1 1/2
1f orbits are constrained by the GXPF1 Hamiltonian perimental levels are well reproducedby the present cal-
5/2
ofRef.[10],duetolackofappropriatedata. TheseSPEs, culation, including both gradual and steep changes as a
i.e., one-bodypartoftheHamiltonian,willbeconfirmed function of N. For instance, the 2+ level drops down
1
tobereasonable,becauseeffectiveSPEstobecalculated steeplyby∼1MeVfromN =16to18inNechain,while
3
+
EXP. (2 ) EXP. (B(E2))
(a) + (b)
6.0 EXP. (4 ) 600 CALC. (B(E2))
+
Ne CALC. (2 ) Ne
+
4.0 CALC. (4 ) 400
2.0 200
0.0 0
)
) 6.0 Mg 4m 600 Mg
V
f
e 2
M 4.0 e 400
(
( )
x 2.0 2 200
E
E
(
B
0.0 0
6.0 Si 600 Si
4.0 400
2.0 200
0.0 0
4.0 1.0
c. (c) y (d) 0p0h
x 3.0 t
e Mg li Mg
h 2.0 bi 0.5
p a
b 2p2h
# of 1.0 Pro 4p4h 6p6h
0.0 0.0
8 12 16 20 24 8 12 16 20 24
N N
FIG. 2. (Color online) (a) Excitation energies of first 2+ and 4+ states and (b) B(E2 : 0+ → 2+) values of Ne, Mg and Si
1 1
isotopes. Experimental data (symbols) [47, 50] and present calculations (lines) are compared. (c) Expectation values of the
number of the particle-hole excitations in the ground state of Mg isotopes. The plain histograms are present results, while
the hatched ones imply thevalue by [14]. The dashed line shows the trend. The two-way arrow indicates the additional 2p2h
excitation (see the text). (d) Decomposition of ph-excitation probabilities for theground state of Mg isotopes.
a similar change occurs from N =18 to 20 in Mg chain. fective charges,(e ,e )=(1.25,0.25)e,calculatedB(E2)
p n
The 4+ levels show a different behavior as functions of values exhibit systematic behaviors in agreement with
1
N. For instance, in Mg isotopes, from N = 16 to 18, experiment. This agreement is, however, not as good as
it starts to come down whereas the 2+ level stays con- that obtained for the energy levels. This probably in-
1
stant, implying that not only the degree of deformation dicates the need to derive E2 operator using the same
but also the shape is changing. On the other hand, in Si microscopic theory. While the N =20 shell closure in Si
isotopes,the2+ leveljumpsupatN =20. Thesebehav- isotopes is evident also in B(E2) systematics, the B(E2)
1
iors of excited levels are reproduced well by the present values of Ne and Mg isotopes are larger at N =20 than
calculation. The low excitation energy of the 2+ levels at N = 18, a feature which is consistent with growing
1
in30Neand32Mgindicatesthatthesenucleiarestrongly deformation seen in the 2+ level.
1
deformedandthatsd-to-pf particle-hole(ph)excitations We now turn to properties of the wave functions of
over the N =20 gap occur significantly. In contrast, the Mg isotopes. Figure 2 (c) depicts the expectation value
higher-lying 2+ level of 34Si suggests weaker degrees of of the number of ph excitations over the Z=N=20 shell
1
ph excitations. gapforthegroundstate. Onenoticestheabruptincrease
Figure 2 (b) shows B(E2 : 0+ → 2+) values. With ef- atN=18and furthermore atN=20. These increasesare
1 1
4
associated with the onset of large deformation. More mental levels rather well, for example, the ordering of
particles (predominantly neutrons) in the pf-shell and lowest four levels. The configurations of the positive-
many holes in the sd-shell enhance collective motion to- party states are of 0p0h (normal) by 1-3%, 2p2h by 66-
−
wards more deformed nuclear shapes. The averagevalue 68%, and 4p4h by 29-30%. In contrast, the lowest 3/2
ofthephexcitationsisabout3.5for32Mg. Thisisalarge and7/2−statesconsistof33-35%1p1h(normal),55-57%
number, compared to the conventional picture for 32Mg 3p3h, and 11% 5p5h configurations. Thus, the ph prop-
being basically a 2p2h state [9, 14–17]. The value as- erties differ significantly between positive- and negative-
sumed in Warburton-Brown-Becker’s island-of-inversion paritystates,whereassimilarphexcitationscanbefound
model [14] is shown in Fig. 2 (c) as a reference. The within the same parity. It is clear that the coupling be-
basic trend remains unchanged in other shell-model cal- tweendifferentphconfigurationshastobeevaluatedpre-
culations, although the actual values can be somewhat cisely. Figure3includesthe resultsofothercalculations.
larger [9, 15–17]. Among them, only the shell-model calculationof [17] re-
In the present calculation, this ph value starts low produces the near-degeneracy of the three lowest levels.
(<0.5) at N=12, and increases almost monotonically to Further fits of TBMEs were not attempted in Ref. [17].
N=16. This increaseis a “modest” effectof the effective Thisisnaturalbecauseoftoomanyparameterswithtwo
interaction shifting nucleons between the two shells, e.g. major shells. The present work however, generates all
pairing effects. Considering that the 2+ level remains TBMEs fromfirst principles thanks to the EKK method
1
high (∼ 2 MeV) up to N=18, this ph excitation mode and no further fine-tuning of the effective interaction is
doesnotproduceastrongdeformation. Weextrapolateit done.
linearlyuptoN=20inFig.2(c). Thedifferencebetween We briefly discuss a test of the present wave functions
the extrapolatedand actualvalues is about 2,which can by transfer reactions. The spectroscopic factors are cal-
be interpreted as an additional 2p2h excitation essential culated as 0.77 and 1.41 for one-neutron removal from
for promoting the strong deformation. Although this in- 32Mggroundstateto3/2− and7/2− states,respectively.
1 1
terpretation is intuitive, the 2p2h excitation on top of The experimental values are 0.59(11) and 1.19(36) [52],
the “modest” correlationappears to be analogousto the respectively, which are in good agreement after scaling
2p2h picture seen in the conventionalapproaches. thetheoreticalvaluesbytheusualquenchingfactor∼0.7
Figure 2 (d) shows more details of the ph excitations [53,54]. Thus,thepresentwavefunctions,whichlookdif-
involved in the Mg ground states. The probability of ferent from those of conventional calculations, turn out
the 0p0h configuration comes down slowly for N ≤ 16, to be quite consistent with these experimental values.
anddropsdownsharplyafterthat. The2p2hprobability Thelastprimarysubjectisthe effectivesingle-particle
increases gradually until N=16. The 4p4h is negligible energy (ESPE). The ESPE represents the combined ef-
up to N=16, but increases abruptly for N ≥ 18, espe- fectonthesingle-particleenergiesfromtheinertcoreand
ciallyforN=20. Notethatthe2p2hprobabilityevende- the monopole effects from other valence nucleons, pro-
creasesatN=20. Suchchangesdrivethenucleustowards ducing important effects on the shell structure of exotic
stronger deformation by including higher ph configura- nuclei [5, 11, 12, 28]. Figure 4 shows the neutron ESPEs
tions. The present work thus resolves, for the first time, of the N = 20 isotones, calculated in the normal filling
the long-standing puzzle as to how sd-to-pf excitations scheme, with and without 3NF and tensor force compo-
occur,showing notable differences fromconventionalap- nent. Here the tensor force component is extracted from
proaches. a spin-tensor decomposition of the present NN effective
In constructing effective NN interactions for the sd- interactiongeneratedby the EKKmethod[27]. Figure 4
shell,e.g. USD[8],certaineffectsofsd-pf phexcitations (a)indicatesthattheprimaryeffectof3NFistoraiseall
are renormalized into TBMEs within the sd shell. If an orbits by similar amounts [28]. Without 3NF effect, the
sd-pf interactionisconstructedontopofthissd-shellin- neutron orbits are bound too deeply, yielding excessive
teraction (with minor changes), a renormalized fraction binding energies, as seen in Fig. 1. On the other hand,
ofthoseexcitationsshouldnotappearexplicitly. Infact, Fig. 4 (b) demonstrates that the tensor force lowers or
this is the case with the sdpf-m interaction (probably raises the ESPEs depending on the cases, and thereby
aswellasothers),withthe above-mentionedexpectation change rather rapidly the spacing between certain orbits
value being as small as 0.05, 0.08, 0.20, 0.85, 2.25 and [11]. The ESPEs obtained at N = 20, including other
1.96 for 24−34Mg, respectively. In some cases, however, sd−pf orbits, appear to be reasonable for 40Ca, sug-
the explicit treatment of those excitations becomes cru- gesting the validity of the input SPEs being used.
cial. Figure 4 (a,b) indicates that the neutron sd-pf gap
Fortransitionalnucleiaround32Mg,the mixingofdif- grows rapidly as Z increases from Z =8 to Z =14, and
ferentph configurationsis indeedcrucialsince the defor- stays relatively constant after Z = 14 up to Z = 20 as
mations contribute less to its binding energy. We show being the N = 20 magic gap. This gap, however, disap-
low-lying positive- and negative-parity levels of 31Mg in pears around Z = 8, and the N = 16 magic gap arises
Fig. 3. The present calculation reproduces the experi- instead [55–57]. This is consistentwith the trendshown
5
1.5 (a) (b) (c) (d) 31
Mg
3/2+
55//22--
(e)
V)1.0 (5/2+) 5/2+
e 3/2- 3/2+
M (3/2+) 5/2-
( 7/2- 7/2-
0.5 (7/2-) 1/2+ 3/2-
x
3/2-
E
(3/2-) 3/2+
1/2+
(3/2+) 3/2- 3/2+
0.0 3/2+ 7/2- 3/2+
1/2+ 1/2+ 3/2- 1/2+
FIG. 3. Energy levels of 31Mg. (a) experimental values, (b) present work, (c) sdpf-m [16], (d) sdpf-U-mix [17] and (e)
AMD+GCM calculation [51], respectively.
(a) neutron ESPE (b) neutron ESPE agreement with experiment. Our TBMEs are not fitted
0
and the quality of the agreement is similar to or better
V) -5 p3/2 p3/2 than SM approacheswith fitted TBMEs. Definite differ-
Me f7/2 f7/2 encesfromconventionalapproachesareseen,particularly
y (-10 inthe patternofparticle-holeexcitationsbetweenthe sd
nerg-15 d3/2 s1/2 d3/2 atunrdepofftshheelilssl.anIndtohfisincvoenrtseioxnt,stehrevecsoansveanntiionntuailt2ivpe2hinpteicr--
E Full s1/2 Full pretation. Wepointoutthattheabove-mentioneddiffer-
-20 w/o 3NF w/o tensor
ences can be smaller, if ph excitations over the relevant
8 10 12 14 16 18 20 8 10 12 14 16 18 20 magic gap are included explicitly in the fit of TBMEs
Z Z
like the case for pf+1g orbit, e.g., [58, 59]. The shell
9/2
evolutionisderivedfromthechiralEFTforceandFujita-
FIG. 4. (Color online) ESPEs of N=20 isotones for neutrons Miyazawa 3-body force for the first time. All these fea-
obtained in the normal filling scheme. Solid (dotted) lines in
tures contribute to further studies of exotic nuclei where
(a) show the case with (without) three-nucleon forces, while
microscopic theories play crucial roles. Further progress
the solid (dot-dashed) lines in (b) represent the case with
innuclearforcesandmany-bodytreatmentswillimprove
(without) thetensor force.
the agreement to experiment and provide us with more
predictive power.
bythe simplifiedinteraction,calledVMU[12],wherethe
We thank Dr. Y. Utsuno for useful discussions.
central force gives a steady enlargement of the gap from
The Lanczos shell-model calculation is performed with
Z =8toZ =20,whilethetensorforcemakesthechange
the code “KSHELL” [60]. This work was sup-
morerapidforZ <14butcancelsthechangeforZ >14.
ported in part by Grants-in-Aid for Scientific Research
The neutron ESPEs of f and p at Z = 20 are also
7/2 3/2
(23244049,15K05090). ItwassupportedinpartbyHPCI
consistent with those from empirically determined inter-
Strategic Program (hp140210,hp150224,hp160221), in
actions [16]. Two other upper orbitals, p and f
1/2 5/2
part by MEXT and JICFuS as a priority issue (Elu-
appear atthe energies similar to those ofanother empir-
cidation of the fundamental laws and evolution of the
ical interaction [10], although not shown in Fig. 4. We
universe) to be tackled by using Post “K” Computer,
thus demonstrate, for the first time, what type of shell
evolution a chiral NN interaction such as the χN3LO and also by CNS-RIKEN joint project for large-scale
nuclear structure calculations. MHJ acknowledges U.S.
interaction produces in medium-mass exotic nuclei. It is
NSF Grant No. PHY-1404159 (Michigan State Univer-
seenalsothattheloweringasafunctionofZ isdampened
sity)andtheResearchCouncilofNorwayundercontract
by three-nucleonforces. The resulting trendis similar to
ISP-Fysikk/216699.
that obtained with the VMU interaction of Ref. [12] as
wellasotherinteractionsfitted toexperiment[9, 16,17].
Summary. –We have presented a consistent micro-
scopic description of neutron-rich nuclei Z=10-14 and
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