Evolution of single-particle states beyond $^{132}$Sn

0.15 MB·English

by  L. Coraggio

Evolution of single-particle states beyond 132Sn
L. Coraggio1, A. Covello1,2, A. Gargano1, and N. Itaco1,2
1Istituto Nazionale di Fisica Nucleare, Complesso Universitario di Monte S. Angelo, I-80126 Napoli, Italy
2Dipartimento di Fisica, Universita` di Napoli Federico II,
Complesso Universitario di Monte S. Angelo, I-80126 Napoli, Italy
(Dated: January 23, 2013)
We have performed shell-model calculations for the two one valence-neutron isotones 135Te and
137Xe and the two one valence-proton isotopes 135,137Sb. The main aim of our study has been
to investigate the evolution of single-particle states with increasing nucleon number. To this end,
we have focused attention on the spectroscopic factors and the effective single-particle energies.
In our calculations, we have employed a realistic low-momentum two-body effective interaction
derived from the CD-Bonn nucleon-nucleon potential that has already proved quite successful in
3 describing the spectroscopic properties of nuclei in the 132Sn region. Comparison shows that our
1 results reproduce very well the available experimental data. This gives confidence in the evolution
0 of thesingle-particle states predicted by thepresent study.
2
PACSnumbers: 21.60.Cs,21.30.Fe,21.10.Jx,27.60.+j
n
a
J
I. INTRODUCTION stable beams in inverse kinematics to test new detectors
2
specifically designedfor use with heavy beams. A recent
2
notable example of this kind of study is the 136Xe (d,p)
The evolution of single-particle states when moving
] away from double shell closures has long been a subject reaction performed at the ATLAS facility at Argonne
h National Laboratory [6], which has provided a stringent
ofprimaryinterestinnuclearstructurephysics. Itwasin
t
- the early 1960s [1], in fact, that quantitative experimen- test of the HELIOS spectrometer used to analyze the
cl tal information on this subject started to become avail- outgoing protons.
u The experimentalstudies ofnucleontransfer reactions
ablethankstothestudyofone-nucleontransferreactions
n currently goingon andthe brightperspectives for future
suchas(d,p)and(d,t). Duringthe1960sandearly1970s
[
research with RIBs are of course a great stimulus for
thefielddevelopedrapidlyandspectroscopicfactorswere

1 extracted from pickup and stripping reactions for many theoreticalstudies aimed atunderstanding the evolution
v of single-particle states. This is indeed the case of our
nuclei. In particular, the stable Sn isotopes and N =82
1 very recent shell-model study [7] of the single-neutron
isotoneswerethesubjectofextensivestudies[2,3]which
9 properties of 137Xe, which received motivation from the
provided relevant information on the shell-model struc-
1
experiment of Ref. [6].
5 ture of these nuclei. By the end of the 1970s, however,
. mostofthefeasibleexperimentshadbeenperformedand Based on the results obtained in [7], we have found it
1 the study of transfer reactions began loosing its role as very interesting to perform a similar study for other nu-
0 a most powerful tool for the understanding of nuclear cleibeyond132Snwhicharewithinreachof(d,p)transfer
3
1 structure. reactionswithRIBs. Moreprecisely,weconsiderthetwo
: As it was to be expected, the advent of radioactive one valence-neutron isotones 135Te and 137Xe and the
v two one valence-proton isotopes 135,137Sb. Actually, re-
Xi itornanbsefearmrsea(cRtiIoBnss),hwahsicohpiesnceudrraenntelwyreercaofgonriztehdeasstuodnyeooff sults for 137Xe have already been given in [7], where we
essentially focused on the comparison between our spec-
r the major research themes to be pursued at the second-
a troscopic factors and those reported in [6]. The present
generation RIB facilities.
paper is ofmoregeneralscope,in thatwe study how the
A main feature of transfer reactions with RIBs is that
single-particle states of 133Sb and 133Sn are modified by
they have to be performed in inverse kinematics. This
theadditionofpairsofneutronandprotons,respectively.
is for instance the case of the 132Sn(d,p) reaction re-
The article is organized as follows. In Sec. II we give
cently performed at the first-generation RIB facility at
an outline of our calculations. In Sec. III the results for
Oak Ridge National Laboratory [4, 5]. This experiment
the energies and spectroscopic factors are presented and
allowedto investigate the single-particle structure of the
one-valenceneutronnucleus133Sn,andthespectroscopic discussed in detail. In Sec. IV we present our predicted
effective single-particle energies and discuss their evolu-
factors for the groundand three excited states extracted
tion. Sec. V contains a summary of our conclusions.
from the differential cross sections turned out to be very
close to one. This is a remarkable achievement which
has setthe stagefor future studies of neutron-richnuclei
in the 132Sn region, which are also of great interest for II. OUTLINE OF CALCULATIONS
modelling the r processes of nucleosynthesis.
In this context, it is worth mentioning that it is also The results presented in this section have been ob-
of current interest the study of transfer reactions with tained with the same Hamiltonian used in our previous
2
to belong to the πg νi configuration.
7/2 13/2
TABLE I: Single-proton and -neutron energies (in MeV).
The shell-model calculations have been performed by
using the OXBASH code [20].
π(n,l,j) ǫ ν(n,l,j) ǫ
0g 0.00 1f 0.00
7/2 7/2
1d 0.962 2p 0.854
5/2 3/2 III. ENERGIES AND SPECTROSCOPIC
2d3/2 2.440 2p1/2 1.363 FACTORS: RESULTS AND DISCUSSION
0h 2.792 0h 1.561
11/2 9/2
2s1/2 2.800 1f5/2 2.005 In this section, we present and discuss our results for
0i13/2 2.690 135Teand137Xewithtwoandfourprotonsinadditionto
133Sn, as well as those for 135Sb and 137Sb with two and
four neutrons in addition to 133Sb. We focus attention
shell-model studies of neutron-rich nuclei beyond 132Sn on the energies and spectroscopic factors of states with
[8–10]. In this Hamiltonian, the single-particle energies angularmomentumandparitycorrespondingto thoseof
are taken from experiment while the two-body effective thesingle-neutronand-protonorbitsforN =83isotones
interaction is derived within the framework of pertur- andSb isotopes, respectively. In fact, it is a main aimof
bation theory [11, 12] starting from the CD-Bonn NN our study to find out whether states with single-particle
potential [13] renormalized by way of the Vlow−k ap- charactersurvivewhenaddingnucleonpairsandtowhich
proach [14]. Some details on the calculation of this two- extent this depends on the nature of the added pairs.
body interaction can be found in [7]. We discuss here
only the choice of the single-particle energies that may
be relevant to the discussion of our results. A. 135Te and 137Xe
We assume that the valence protons and neutrons oc-
cupy the first major shells outside doubly magic 132Sn. Let us start with the N = 83 isotones. Excitation
Namely, protons are allowed to move in the five orbits energies and spectroscopic factors for 135Te and 137Xe
0g7/2, 1d5/2, 1d3/2, 0h11/2, and 2s1/2 of the 50-82 shell are shown in Tables II and III, respectively, where only
and neutrons in the six orbits 1f7/2, 2p3/2, 0h9/2, 2p1/2, states with C2S ≥ 0.07 are included. In Table III, we
1f5/2, and 0i13/2 of the 82-126 shell. alsoshowtheenergiesandspectroscopicfactorsobtained
The adopted single-proton and -neutron energies are in [6] for levels which can be safely identified with the
shown in Table I. For protons, the first four are taken calculated ones. For a detailed comparison between ex-
to be the experimental energies of the levels with cor- periment and theory, we refer to [7] while here some
responding spin and parity in 133Sb [15]. The energy points relevant to the present discussion will be consid-
of the s orbit, which is still missing, is taken from ered. For 135Te only experimental energies are avail-
1/2
[16], where it was determined by reproducing the exper- able [15], although a (d,p) reaction on 134Te was per-
imental energy of the 1/2+ state at 2.150 MeV in 137Cs. formed at HRIBF [21]. In fact, preliminary tentative re-
Thisis,infact,populatedwithsignificantstrengthinthe sults from this experiment do not allow to extract spec-
136Xe(3He, d)137Cs transfer reaction [3], while no 1/2+ troscopic factors but only to identify the 7/2− ground
− −
statehasbeenyetidentifiedinthelighterN =82isotone, state and the 1/2 and 3/2 yrast states as resulting
135I. It is worth noting that no spectroscopic factors are from the l=3 and 1 transfers, respectively.
available for the states of 133Sb, which has been mainly From Tables II and III we see that the agreementbe-
studied by β-decay experiments in the ’90s [17]. tweenexperimentalandcalculatedenergiesisverygood.
As already mentioned in the Introduction, the single- Thelargestdiscrepancyinboth135Teand137Xeisfound
neutron nature of the observed states in 133Sn has been for the 13/2+ state, which is overestimated by 150 and
recently studied in a (d,p) transfer reactionwith a 132Sn 330keV,respectively. Inthisconnection,itisworthmen-
RIB [4, 5]. In this experiment, the spectroscopic factors tioning that its position is directly related to the energy
− − −
of the previously observed [18] 7/2 , 3/2 , and 5/2 of the 0i level, which, as mentioned in the previous
13/2
stateswereextractedevidencingalittlefragmentationof section, is still missing.
the single-particle strength. Furthermore, a strong can- Our calculations for 135Te confirm the preliminary re-
didate for the 2p single-particle orbit was identified sults of [21], the 7/2−, 1/2− and 3/2− yraststates hav-
1/2
at 1.363 MeV excitation energy, which is about 300 keV ing, in fact, the largest spectroscopic factors. Actually,
lower than the previous proposed value [18]. A state at it turns out that this 7/2− state, as well as the yrast
1.561 MeV was not significantly populated in this (d,p) 13/2+ state, carries the largest fraction of the single-
reaction. Thisstate,associatedwithJπ =9/2− inprevi- particlestrength,whileanon-negligiblepercentageofthe
ousβdecay[18]andspontaneousfission[19]experiments, p and p strengths is distributed over higher-energy
1/2 3/2
is expected to correspond to the 0h9/2 orbit. The 13/2+ states. As for the Jπ = 9/2− states, we find that it is
state has not yet been observed in 133Sn and the energy theyrarestateat1.346MeVto havethelargestspectro-
reported in Table I has been estimated from that (2.434 scopic factor, 0.51, to be compared to 0.18 for the yrast
MeV) of the experimental 10+ state in 134Sb, assumed one. These predicted two 9/2− states are quite close
3
TABLE II: Calculated energies and spectroscopic factors for
3.0
statesin135Te. Theavailableexperimentaldataarereported
i
for comparison (see text for details). 13/2
Calc. Expt. 0.72
0.41 0.75
Jπ E(MeV) C2S Jπ E(MeV) 2.0 f5/2 0.20
(1/2−)1 1.110 0.45 (1/2−) 1.083
− h
((11//22−−))23 12..944070 00..3126 − MeV) p19//22 00..4551 00..7423
((33//22−))21 10..772216 00..2673 (3/2 ) 0.659 E( 1.0 p3/2 0.63 0.57
(5/2−)1 1.119 0.12 (5/2−) 1.127
−
(5/2 )6 2.238 0.41
(7/2−)1 0.000 0.88 (7/2−) 0.000 0.0 f7/2 0.88 0.86
− −
(9/2−)1 1.302 0.18 (9−/2 )− 1.246 133Sn 135Te 137Xe
(9/2 )2 1.346 0.51 (7/2 ,9/2 ) 1.380
(9/2−)5 2.214 0.07
−
(9/2 )6 2.308 0.07 FIG.1: Calculated energylevels withcorresponding spectro-
(13/2+)1 2.268 0.72 (13/2+) 2.109 scopicfactorsintheN =83isotones135Teand137Xe(seetext
(13/2+)2 3.356 0.19 fordetails). Single-neutronlevels for133Snarealso reported.
TABLEIII:Calculated energiesandspectroscopicfactorsfor tationisfoundin137Xewithrespectto135Te,thehighest
statesin137Xe. Theavailableexperimentaldataarereported spectroscopic factor being 0.20 for the 5th excited state.
for comparison (see text for details). Ourfindingsfor137Xeareconsistentwiththeexperimen-
talresults,althoughthe spectroscopicfactorofthe yrast
Calc. Expt. −
9/2 state is overestimated by the theory. This point is
Jπ E(MeV) C2S Jπ E(MeV) C2S discussed in Ref. [7].
(1/2−)1 1.127 0.43 1/2−,3/2− 0.986 0.35 Based on the above considerations, we have found it
(1/2−)2 1.926 0.13 interesting to compare the adopted single-neutron ener-
(1/2−)4 2.305 0.18 gies for 133Sn with the calculated spectra of 135Te and
− 137Xe, including, for each Jπ, only the level with the
(1/2 )5 2.407 0.07
− − largest spectroscopic factor. This is done in Fig. ??,
((33//22−))21 10..770288 00..0577 3/2 0.601 0.052 where we see that when adding proton pairs to 133Sn
the single-neutron spectrum preserves its original struc-
−
(3/2 )3 1.783 0.15 ture, becoming on the whole slightly more compressed.
(5/2−)1 1.349 0.17 5/2− 1.303 0.22 Actually, only the 5/2− state in both nuclei moves up
−
(5/2 )5 2.039 0.20 in energy. In this context, it may be worth mentioning
(7/2−)1 0.000 0.86 7/2− 0.000 0.94 that the lower 5/2− states, which are not of a single-
(9/2−)1 1.327 0.72 9/2− 1.218 0.43 particlenature,arepredictedbyourcalculationstoarise
(13/2+)1 2.082 0.75 (13/2+) 1.751 0.84 fromthecouplingofthesingle-neutronf7/2 statetotwo-
proton excitations. The 2+, 4+, and 6+ in 134Te lie, in
fact, at an energy ranging from 1.3 to 1.7 MeV, which is
lower than that, 2.0 MeV, of the f level in 133Sn.
in energy, the difference being only a few tens of keV. 5/2
Note that a state at 1.380 MeV has been observed with
− −
no firm spin assignment (7/2 , 9/2 ), which could be
associated either with our second 9/2− state or our sec- B. 135Sb and 137Sb
ond 7/2− state at 1.336 MeV with C2S =0.06. Finally,
we see that the largest strength of the f5/2 orbit is pre- The energies and spectroscopic factors for 135Sb and
dictedto be carriedbythe 6th5/2− state at2.238MeV, 137Sb are reported in Tables IV and V, respectively, in-
C2S =0.41,while thespectroscopicfactorsofthelowest cluding only states with C2S ≥0.07 selected among the
five states are much smaller. lowest twenty states for each angular momentum. For
When one adds two more protons going from 135Te 137Sbonlythegroundstateisknown,whilefor135Sbsev-
to 137Xe, a state of a single-particle nature can be still eral states have been observed, most of them, however,
identifiedforeachJπ,whichistheyraststate,exceptfor without firm spin assignment. Furthermore, no spectro-
Jπ =5/2−. Asforthef strength,astrongerfragmen- scopic factors have been obtained. In Table IV the ex-
5/2
4
TABLEIV:Calculated energies andspectroscopic factorsfor 135Sb
statesin135Sb. Theavailableexperimentaldataarereported gth 1.0
for comparison (see text for details). n
e
str
Calc. Expt. on 0.8 g7/2
Jπ E(MeV) C2S Jπ E(MeV) prot h
(1/2+)1 0.659 0.07 (1/2+) 0.523 gle– 0.6 11/2
n
((11//22++))1121 32..181890 00..3027 med si 0.4 d5/2
m
(1/2+)15 3.476 0.17 Su
(3/2+)1 0.497 0.07 (3/2+) 0.440 0.2 d3/2
(3/2+)3 1.320 0.07 s1/2
0.0
(3/2+)12 2.600 0.32 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
(3/2+)13 2.697 0.10
(3/2+)14 2.768 0.09 137Sb
(5/2+)1 0.387 0.42 (5/2+) 0.282 gth 1.0
n
((55//22++))52 10..695278 00..0293 on stre 0.8 g7/2
(5/2+)8 1.950 0.14 prot
–
(7/2+)1 0.000 0.74 (7/2+) 0.000 gle 0.6 d5/2
((171//22+−))21 20..695424 00..5128 med sin 0.4 h11/2
− m
((1111//22−))52 33..512322 00..2111 Su 0.2 d3/2
s
1/2
0.0
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
TABLE V: Calculated energies and spectroscopic factors for
states in 137Sb. FIG.2: Summedspectroscopicfactorsfor135Sband137Sbas
a function of theincluded states (see text for details).
Calc.
Jπ E(MeV) C2S
(1/2+)1 0.403 0.11 tive sum of the spectroscopic factors. This is shown in
(3/2+)1 0.333 0.12 Fig. ?? for each single-proton orbit as a function of the
(5/2+)1 0.186 0.53 number of states included in the sum. We see that the
(5/2+)10 1.538 0.09 fragmentationisparticularlystrongforthes1/2 andd3/2
(7/2+)1 0.000 0.71 strengths, which in 137Sb do not reach significant values
(7/2+)2 0.913 0.09 even when including the 15 lowest-lying states.
− As a direct consequence of the large fragmentationin-
(11/2 )1 2.587 0.38
ducedbyadditionofpairsofneutrons,thesingle-particle
−
(11/2 )2 2.848 0.15 spectrum of 133Sb is substantially modified, as shown in
Fig. ?? which is the counterpart of Fig. ??. This occurs
already in 135Sb with only two more neutrons, the most
perimental energies of 135Sb [15] for states which can be noticeable change being the lowering of the 5/2+ state
safely associatedwith the calculatedonesare shownand by about 600 keV.
we see that they are very well reproduced by the theory. The low position of the 5/2+ state in 135Sb has been
As regards the spectroscopic factors, the calculated indeed in focus of great attention [22, 23]. In some
values for both nuclei evidence a strong fragmentation studies, this was traced to a decrease of the proton
of the single-particle strength with respect to 135Te and d − g spacing produced by the two neutrons be-
5/2 7/2
137Xe. We find, in fact, that in 135Sb only for three Jπ, yond the N = 82 shell closure. On the other hand, our
which reduce to two in 137Sb, there is one state with predicted spectroscopic factor, 0.42, evidences that this
a spectroscopic factor larger than 0.4. The remaining state has no strong single-particle nature, namely non-
strength in both nuclei is shared between many states. negligible components with seniority larger than one are
We have seen in Sec. III.A that for the N =83 isotones containedinits wavefunction. This wasdiscussedinde-
this occurs only for Jπ =5/2− in 137Xe. tail in Ref. [22], where we showed that the seniority-one
The spreading of the single-proton strength in the state |πd (νf )2 >, with an unperturbed energy
5/2 7/2 J=0
Z =51 isotopes can be better seen through the cumula- of 0.962 MeV, is pushed down by the neutron-protonin-
5
TABLE VI: Neutron-proton monopole components
0.31 VM(j πg ) (in MeV).
ν 7/2
3.0 h
11/2
s1/2 0.52 0.38 νp1/2 νp3/2 νf5/2 νf7/2 νh9/2 νi13/2
d 0.32 -0.09 -0.11 -0.11 -0.18 -0.20 -0.33
3/2
V) 2.0
e
M
TABLE VII: Neutron-proton monopole components
(
E VM(νf7/2jπ) (in MeV).
1.0 d
5/2 πs πd πd πg πh
1/2 3/2 5/2 7/2 11/2
0.42 0.11 -0.27 -0.35 -0.28 -0.18 -0.13
0.12
0.53
g 0.74 0.71
0.0 7/2
133Sb 135Sb 137Sb ′
where ρ and ρ stand for neutron and proton index, re-
spectively, or viceversa. For a given j , ǫ denotes the
ρ jρ
corresponding energy in the one-valence system and N
FIG.3: Calculated energy levelswith correspondingspectro- jρ
scopicfactorsintheZ =51isotopes135Sband137Sb(seetext the occupation number in the ground state of the even-
for details). Single-proton levels for 133Sb are also reported. even system, while
teraction getting admixed with the seniority-three state VM(jρjρ′)= PJ(2J +1)<jρjρ′;J|V|jρjρ′;J > (2)
|πg7/2(νf7/2)2J=2 >, which lies at 0.368 MeV. PJ(2J +1)
This mechanism explains indeed the fragmentation
isthe monopolecomponentofthe neutron-protoninter-
of the single-particle strength we predict for almost all
statesinboth135Sband137Sb. Bythesametoken,wesee action.
ItisworthnotingthattheESPEscoincide[25]withthe
whynostrongfragmentationisfoundfortheN =83iso-
energy centroidsdefined in [27] more than 40 years ago.
tones,mostofthe seniority-onestates lying,inthis case,
The practical use of the latter definition, however, re-
sufficiently lower in energy with respect to the seniority-
quiresknowledgeofallenergiesandspectroscopicfactors
threestates. Thisdifferenceintheevolutionofthesingle-
corresponding to a given angular momentum, whereas
proton versus single-neutron states may be then traced
expression (1) makes the computation of the ESPEs
to the different pairing force acting betweenprotons and
straightforward.
neutrons.
WehaveemployedEq.(1)tocalculatetheneutronand
Itisindeedafeatureofoureffectiveinteractiontohave
proton ESPEs whose values are shown in Fig. ??. By
different pairing properties for protons above Z = 50
examiningthevarioustermsof(1),weseethatonlysome
andneutronsaboveN =82,whichresults ina“normal”
ofthemgiveanimportantcontributiontotheseenergies.
pairing for protons anda significantlyweakerpairing for
In particular, for each effective single-neutron energy in
neutrons. In our derivation of the effective interaction,
the N = 83 isotones, the dominant contribution is that
thisarisesfromthesecond-ordercorepolarizationeffects,
corresponding to πg due to its large occupancy. We
the main role being played by one-particle-one hole ex- 7/2
predict,infact,that1.6outof2protonsareintheπg
citations [24]. Experimentally, the reduction of neutron 7/2
pairing is clearly manifested by the large difference be- orbit for 134Te, and 3.6 out of 4 for 136Xe. Note that
tween the proton and neutron gap in 134Te and 134Sn, our predicted occupancies for the ground state of 136Xe
the latter being about 0.5 MeV smaller than the former. areingoodagreementwiththeresultsobtainedfromthe
experiment of Ref. [3]. Similar results are found for the
Sbisotopesasregardsthe occupancyofthe neutronf
7/2
IV. EFFECTIVE SINGLE-PARTICLE orbit. The monopolematrixelementsenteringthe domi-
ENERGIES nantterms for the neutron and protonESPEsare there-
fore VM(j πg ) and VM(νf j ), respectively, which
ν 7/2 7/2 π
are reported in Tables VI and VII.
In this section we discuss the single-proton and -
neutronenergieswhichcanbe associatedtothe underly- From Fig. ??, we see that all the neutron and proton
ESPEsgodownwhen pairsof nucleons areadded. How-
ing mean field when neutron and proton pairs are added
to 133Sb or 133Sn, respectively. These quantities are the ever,thedecreaseinenergyisnotthesameforthevarious
levels leading to single-particle spectra substantially dif-
effective single-particle energies (ESPEs) [25, 26]
ferent from those of the one valence-particle nuclei. For
instance, the five neutron ESPEs above the lowest one
ǫ¯jρ =ǫjρ +XVM(jρjρ′)Njρ′, (1) tend to lie in a smaller energy interval, the i13/2−f5/2
jρ′ spacingreducingfromthe initialvalue of700to 100keV
6
3.0 (a) 3.0 (b)
)
V ) h
e V 11/2
M2.0 e2.0
( M
PE i13/2 E ( s1/2
ES f5/2 SP
Neutron 1.0 ph19//22 Proton E1.0 d3/2
p
3/2
0.0 0.0
d
5/2
g
f 7/2
7/2
–1.0 –1.0
133Sn 135Te 137Xe 133Sb 135Sb 137Sb
FIG. 4: Evolution of (a) neutron and (b) proton ESPEs.
in 137Xe. On the other hand, the spacing between the matrix element corresponding to the orbit with opposite
lowest and the highest proton levels remains practically spin orientation relative to proton g /neutron f is
7/2 7/2
unchanged while the three other levels move toward the the most attractive. This accounts for the variation in
g withanoveralldownshiftofabout500keVin137Sb. the f −f and p −p neutron spacings, as well
7/2 5/2 7/2 1/2 3/2
Inthis context,itshouldbe mentionedthatthe evolu- as in the d3/2 −d5/2 proton one, and may be traced to
tion of shell structure has received great attention in re- the effect of the tensor component of our effective inter-
cent years owing to new available data on exotic nuclei, action [29, 30]. We also predict a significant decrease
which have evidenced the appearance or disappearance in the neutron i13/2−h9/2 spacing, which comes essen-
of magic numbers. Several theoretical studies have then tially from VM(νi πg ) being more attractive than
13/2 7/2
beencarriedouttotrytoidentifythemechanismbehind VM(νh πg ). The less attraction of the latter may
9/2 7/2
the variation of single-particle energies. In particular, be again attributed to the tensor force, the neutron h
9/2
the role of the monopole components of the spin-isospin orbit,unlikethei one,beingspin-downastheproton
13/2
and tensor interaction has been pointed out, the latter g orbit [29]. In this connection, it is worth mention-
7/2
producing an attractive/repulsive effect for two unlike ing that this force has been shown to be responsible of
nucleonswithopposite/alignedspinorientation [28,29]. the reductionin the separationofthese two orbitsin the
By using the spin-tensor decomposition of the interac- N =83 isotones [6, 29].
tion, the interplay between the tensor force and other
components of the neutron-proton interaction has been
investigated[30]. This has providedevidence for the im-
V. CONCLUDING REMARKS
portanceofboththecentralandtensorforceindetermin-
ingtheevolutionofESPEs,thelatterplayingadominant
role for the energy difference of spin-orbit partners. In this work, we have performed shell-model calcula-
It would be clearly desirable to perform such a de- tions for the four neutron-rich nuclei 135Sb, 137Sb, 135Te
composition also in the present study to clarify how the and 137Xe. The main aim of our study was to follow the
behaviorofthe ESPEsshowninFig. ??isrelatedtothe evolution of the single-proton and -neutron states out-
various terms of our effective interaction. This is, un- side doubly magic 132Sn when adding neutron and pro-
fortunately, not feasible since our shell-model space, on ton pairs, respectively. It is worth emphasizing that in
whichtheinteractionisdefined,doesnotcontainforeach our calculationswe haveconsistently employedthe same
orbital angular momentum the two spin-orbit partners. Hamiltonian which has given a very good description of
This is, in fact, a prerequisite to performthe spin-tensor other nuclei beyond 132Sn [8–10] without use of any ad-
decomposition. We cannot therefore disentangle the ef- justable parameters.
fects of the central and tensor force, so as to unambigu- We havecalculatedenergies,spectroscopicfactorsand
ously assess the role of the various terms of the effective effective single-particle energies for states of all the four
neutron-proton interaction in determining the evolution nuclei considered and compared our results with the
of the ESPEs. available experimental data which are, however, rather
FromTablesVIandVII,however,weseethatforeach scanty. In particular, no spectroscopic factors are avail-
pair ofneutron/protonspin-orbitpartnersthe monopole able for 135Te and 135Sb while no spectroscopic informa-
7
tion at all exists for 137Sb. This amounts to say that very good agreementwith the measuredexcitation ener-
this study is largelypredictive in nature and is meant to gies in 135Te, 135Sb and 137Xe as well as with the spec-
stimulate, and be of guide to, experimental efforts with troscopic factors extracted from the data for the latter.
RIBs in the beyond 132Sn region. This makes us confident in the predictive power of our
Of noteworthy interest is the stronger fragmentation calculations.
of the single-particle strengths predicted for 135Sb and
137Sb as compared to that for 135Te and 137Xe. This
stems essentially from the fact that the neutron pairing
Acknowledgments
beyond N = 82 is significantly smaller than the proton
pairing beyond Z =50. We have shown in [24] that this
difference inthe protonandneutroneffective interaction This work has been supported in part by the Italian
finds its origin in core-polarizationeffects. Ministero dell’Istruzione e della Ricerca (MIUR) under
As shown in Tables II, III and IV, our results are in PRIN 2009.
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