$(h_{11/2})^2$ alignments in neutron-rich $^{132}$Ba with negative-parity pairs

0.19 MB·English

by  Yang Lei

(h )2 alignments in neutron-rich 132Ba with negative-parity pairs
11/2
Y. Lei (雷杨)1, and Z. Y. Xu (徐正宇)2
∗
1Key Laboratory of Neutron Physics, Institute of Nuclear Physics and Chemistry,
China Academy of Engineering Physics, Mianyang 621900, China
2INPAC, Physics Department, Shianghai Jiao Tong University, Shianghai 200240, China
(Dated: July 27, 2015)
The shell-model collective-pair truncation with negative-parity pairs is adopted to study the
(h )2 alignment in 132Ba. The proton (h )2-alignment state is predicted as an E ∼4.6 MeV
11/2 11/2
and τ ∼ 0.5 µs isomer with relatively strong E3 decay channels. The oblately deformed neutron
(h )−2 alignment intheyrastbandandfournegative-paritybandsareconfirmed,evenalthough
11/2
two of these negative-parity bands favor the prolate deformation, which directly manifests the γ
unstability of 132Ba.
5
1
PACSnumbers: 21.10.Re,21.60.Cs,23.35.+g,27.60.+j
0
2
l 132Baisatypicalγ-unstablenucleusinthetransitional
u region of Z >50 and N 6 82 with a large variety of co- TABLE I. Hamiltonian parameters from Ref. [16] in MeV.
J
4 existing nuclear shapes. Its negative-parity bands with s1/2 d3/2 d5/2 g7/2 h11/2
2 Iπ =5−and6− bandheads(bandheadenergies: 2.12and επ 2.990 2.708 0.962 0.000 2.793
2.358MeV,respectively)maintainanoblateshape,while εν 0.332 0.000 1.655 2.434 0.242
] prolately deformed negative-parity bands with Iπ = 7− G0π G2π G0ν G2ν κπ κν κπν
h and 8 bandheads (bandhead energies: 2.902 and 3.105 0.130 0.030 0.130 0.026 0.045 0.065 0.070
−
t
- MeV, respectively) were also reported in the 132Ba level
l
c scheme [1, 2]. There four negative-parity bands here are
u denoted by “5 , 6 , 7 and 8 bands” in sequence. In Model has be proved to be an efficient approach for the
− − − −
n theyrastband,theτ =8.94(14)ns10+isomerisassigned (h11/2)2-alignment description [15–17]. In such trunca-
[ astheneutron(h11/2)−2 alignmentwithanoblateshape, tion, collective pairs with spin 0~ and 2~ are normally
3 which has two h11/2 neutron holes rotate alignedly. On employed to represent the low-lying collectivity [18–24],
v the other hand, the proton (h )2 alignment with the which naturally provide ∆I = 2 band structures, as
11/2
6 prolate deformation is also expected in 132Ba (unfortu- 5−, 6−, 7− and 8− bands of 132Ba behave. Recently,
4 negative-paritypairs were introduced into the pair trun-
natelyunobservedyet),becausethecompetitionbetween
3 protonandneutron(h )2 alignmentsaregenerallyex- cation [25–27], which enables a shell-model description
3 11/2
ofnegative-paritystatesinaheavyeven-nucleonsystem,
0 hibited in this nuclear region [3–12].
e.g.,negative-paritystatesof132Badiscussedhere. Thus,
. There may exist strong electromagnetic transitions
1 the collective-pair truncation with negative-parity pairs
from the (πh )2-alignment state to states in 7 and
0 11/2 −
is adopted for our negative-parity-staterelated study on
5 8− bands, because initial and final states share a simi- the (h )2 alignment of 132Ba.
1 lar prolate shape. It’s desirable to theoretically estimate 11/2
Ourcalculationadoptsaphenomenologicalshell-model
: these transition rates before an experimental search for
v Hamiltonian as in Ref. [16]:
i the (πh11/2)2 alignment in 132Ba.
X
r 6 T,h7e (aνnhd118/2)−b2anadlsigancmcoenrdtinwgastoatlshoeisrubgagnedsteirdreignul5a−r-, H =− ( εjσnˆjσ + GsσPσs†·P˜σs
a − − − σ=Xπ, ν Xj sX=0,2
ity around 5 6 MeV [1, 2]. However, the experimental
evidence is n∼ot as solid as the (νh11/2)−2 alignment in +κσQσ·Qσ)+κπνQπ·Qν, (1)
the yrast 10+ isomer [13, 14]. It’s essential to confirm with
the (νh ) 2 alignment in these negative-parity bands
11/2 −
from a shell-model perspective. P0† = √22a+1(Ca†×Ca†)0, (2)
The present work aims at studying the (h )2 align- Pa
11/2
mentinbothpositiveandnegative-paritystatesof132Ba P2† = q(ab)(Ca†×Cb†)2,Q= q(ab)(Ca†×C˜b)2.
within the shell-model framework. The model-space Pab Pab
truncation is required to reduce the gigantic dimen- In Eq. (2), q(ab) = a r2Y2 b /r2, where r is the os-
sion of the shell-model description for 132Ba. It’s note- cillator parameter, h~|/|(mω)|.| Hiam0iltonian p0arameters,
worthy that the collective-pair truncation of the Shell i.e., εjσ, GSσ, κσ, anpdκπν in Eq. (1), are alsotaken from
Ref. [16] as listed in Table I.
Ourpair-truncatedshell-modelspacefor132Baisgiven
by the coupling of three proton pairs and three (hole-
∗ [email protected] like)neutronpairs in the 50-82shell. These pairs canbe
2
by two s and/or d neutrons along with two h
TAB−LEII.S−tructurecoefficients,i.e., βaIbπ definedinEq. (3), neutrons.1/T2hesetwos31//22 and/ord3/2 neutronsonlyp11r/o2-
of 5 and 6 pairs.
vide little angular-momentum contribution, so that the
Iπ =5− Iπ =6− I > 10~ total angular momentum of the 5 6 cou-
− −
ab=h11/2×s1/2 +0.551 +0.616 pling almost comes from the rest two h11/2 ne⊗utrons. In
ab=h11/2×d3/2 −0.832 +0.780 otherwords,these twoh11/2 neutronshaveto contribute
ab=h11/2×d5/2 +0.038 +0.097 I 10~ angular momentum, which forces them to ro-
ab=h11/2×g7/2 −0.050 +0.053 tat∼e alignedly. Thus, the (νh11/2)−2 alignment emerges
within the 5 6 coupling.
− −
⊗
For 132Ba with 3 valence proton pairs and 3 valence
formally defined by
neutron-hole pairs in 50-82 major shell, our proton or
whAerIeπ†C=iXas6tbhβeaIbπcrAeIaπt†io(anb)o,peArIaπt†o(raob)f =the(Ca√a†o1×r+bCitδb†.a)bITπh,(u3s), neutro|τnσJ=pπa,irν-tir=un(cid:16)c(cid:0)aAterd1†b×asAisri2s†(cid:1)g(iJv2e)n×bAyr3†(cid:17)(J)|0i, (4)
AIπ (ab)a†is a non-collective pair with two nucleons at a where Ar1†, Ar2† and Ar3† can be any type of pairs we
†
and b orbits; AIπ represents a collective pair with op- described above. Proton and neutron basis are coupled
†
tniumcilzeeadr lsotwru-clytiunrge ccoolelefficctiivenittys., βEaIsbπp,ectioalblye,stβIdπe=sc0+ribceortrhee- mtoagtertihxeerleamse|nτtJs=uJnπ×deJrνi|τ=Ji|bτaπJsπeis×can|τνbJνeic.alHcualmatieldtowniiatnh
ab the formalism of Ref. [32]. These elements are denoted
spondstotheCooper-pairstructureduetothestrongnu-
by Hτ, where i and j are indexes to identify bases in
clearpairingcollectivity,andis determinedfollowingthe ij
the diagonalization algorithm. The Hamiltonian should
principle of projected-particle-number BCS theory [28].
Other βIπ with Iπ = 0+ is obtained by the diagonaliza- be diagonalizedunder a setof linearly independent, nor-
ab 6 malized and orthogonalbases, which can be constructed
tion in the ν = 2 broken-pair model space [29]. More
details about the determination of βIπ are described in by the linear combination of |τJi bases as described in
ab Ref. [33]. In detail, we calculate the overlap matrix of
Ref. [30].
τJ bases,anddiagonalizeit with a serialofeigenvalues
In our calculation, four types of pairs are adopted: | i
(e )andcorrespondingeigenvectormatrixelements(v ).
i ij
1 Following previous pair-truncation calculations The ith normalized and orthogonalbasis is given by
[18–24], collective pairs with Iπ = 0+ and 2+ are
1
introduced. φJ = v τJ . (5)
2 Bandbeads of 5 and 6 bands correspond to the | ii √ei Xj ij| j i
− −
mixture of neutron h s and h d
configurations[1,2]. I1n1/t2he×501-/822major1s1h/2el×l,su3c/h2 To handle the overcompleteness, we neglect |φJiis with
e = 0, and reserved φJ bases construct the pair-
mixture can only emerge in collective Iπ =5− and triuncatedspacefor132Ba|. Tihus,theHamiltonianmatrix
6 neutron pairs. Thus, these two pairs are intro-
− under φJ bases is produced by
ducedtodevelop5 and6 bands,anddenotedby | i
− −
“5− and 6−” pairs. Hφ = 1 v v Hτ, (6)
3 7− and8− bands arebuilt onthe coupling ofh11/2 ij √eiej Xkl ik jl kl
andg protons[1,2]. Consideringtheshort-range
7/2
and inputted into Hamiltonian diagonalization. Resul-
property of nuclear force, the coupling with total
tant wave-functions are used for further calculations on
spin I = 9 should give the lowest binding energy
[31]. Thus,thenon-collective(πh πg )Iπ=9− electromagneticproperties andexpectation values of the
11/2× 7/2 pair numbers.
pair is introduced.
The electromagnetic-transition operators adopted in
4 We take the (πh πh )Iπ=10+ pair to de- our calculation are
11/2 11/2
×
scribe the (πh11/2)2 alignment. T(E2)= e r2Y2, T(E3)= e r3Y3,(7)
σ σ σ σ σ σ
In previous pair-truncation calculations [15–17], the σ=Xπ, ν σ=Xπ, ν
(νh11/2)−2 alignment is represented by the (νh11/2 3
νh )Iπ=10+ pair. However, we don’t introduce suc×h T(M1)=r4π glσ−→l σ+gsσ−→sσ.
11/2 σ=Xπ, ν
pair, because the coupling of two 5 and/or 6 pairs
− −
with total angular momentum I > 10~ (denoted by the Here, e , g and g (in units of e and µ /~) are the
σ lσ sσ N
“5 6 coupling”) can also describe the (νh ) 2 effective charge, orbital and spin gyromagnetic ratios of
− − 11/2 −
⊗
alignment. To demonstrate this point, we list structure valence nucleons, respectively. They are taken from Ref.
coefficients of 5 and 6 pairs in Table II, which sug- [16] as e =2, e = 1, g =5.58 0.7, g = 3.82
− − π ν πs νs
− × − ×
gests that the 5 6 coupling are mainly constructed 0.7, g =1.05 and g =0.05.
− − πl νl
⊗
3

21- 21- TABLEIII.E3 transition rates (in W.u.) from theIπ =10+
8 16+ isomer with the (πh )2 alignment to states in 7− and 8−
19- 19- 11/2 ∗
bands. The superscript “ ” labels themain decay branch.
V)6 17- 16- 16+ 1157-- 16- 1157-- 1164-- 16+ 1157-- 1146-- 1124++ J7−fπ B0.(0E083) J8−fπ B0.(1E453)
E (Me24 579111---135--- 68111--024--- 68111++024+++ 7911--13-- (811131-024-----) 175913-1---- 6811--02-- 68111++024+++ 7911--13-- 811-02-- 11340---+ Ex91p−1− 00..1000(91h∗11/2)-2 al11ig02n−−m10e0.n0−2t43
Cal
4+ 4- 4+ V) 1.2 B(E
FcuIlGat.e10d.s(pCeocltorrumon(lib02n)++e,)(ianEc)xlEupdexripnimgetnhtealy(raa)st[022b++,a3n4d],a5n(−db,)t6Ch−e,ac7la−l-, E-E(MeII-2 0.8 (a) ( b) 235050 2, ->) (WII-2
l8i−ghbteadndbsy,raenddctohloer.(πh11/2)2-alignment band, which is high- 0.4 5 . u.)
) 0.4 (c) (d) 2.0 pa
/N 0.2 1.5 ir n
Fig. 1 presents the calculated 132Ba spectrum com- (g 1.0 um
pared with the experimental one, including the yrast 0.0 0.5 ber
band, 5 , 6 , 7 , 8 bands andthe potential(πh )2-
− − − − 11/2 -0.2
0.0
alignment band. A rough spectral consistency between 0 4 8 12 16 0 4 8 12 16
our calculation and the experiment is achieved for these
I ( )
bands we are concerned about. One may also note that
sthtaetrees,iswshtiicllhianrceonasllisoteuntcoyffoourrloscwie-lnytiinfigc 1m−o,ti3v−atiaonnda4n−d aFsIGw.e2ll.aEsIt−heEsIu−m2(ao)f,5B−(Ea2n,dI6→− pIa−ir2)nu(bm)baenrdsg(df)a,ctoofrysr(acs)t,
states in 132Ba from experiments (Exp) [34] and our calcula-
our model space constructed with four types of pairs
tion (Cal). The yrast (νh )2 alignment is highlighted.
we itemize above [2, 35, 36]. Thus, we have no inten- 11/2
tion to fix the inconsistency by introducing redundant
pairs, which obviously will complicate our study, and in
decay to observed levels. Resultant E2 and M1 decay
what follows we limit ourselves to current model space.
ratestoyrast8+,10+ and12+ statesareallsmallerthan
Furthermore, calculated 7 and 8 bands based on the
(πh πg )Iπ=9− pai−r are sti−ll observed to be sys- 10−13 W.u.. Such weak decays can be attributed to the
11/2 × 9/2 parity conservation, which forbids E2 and M1 transition
tematicallyhigherthanthosefromexperimentsby 0.3
MeV. This is because the single-particle energy o∼f the operators from scattering the negative-parity h11/2 nu-
cleonto otherpositive-parityorbitsof50-82majorshell.
132Ba h11/2 proton, i.e., the [550]21− level, is depressed In other words, any state related to the (πh11/2)2 align-
by 0.3MeV relatively to the Fermi surface [37], due
mentbystrongE2orM1transitionsmustbeconstructed
∼
to the β 0.12 prolate deformation of 7 and 8
2 − − withtwoh valenceprotons. However,yraststates,as
∼ 11/2
bands [38]. This effect shall also have direct impact
well as most of low-lying positive-parity states in 132Ba,
on the prolately deformed (πh )2 alignment, which
11/2 have few valence h protons due to the large πh
the shell-model framework can not intrinsically consider 11/2 11/2
single-particle energy. Thus, strong E2 and M1 decays
yet. Therefore, our calculated excitation energy of the
from the (πh )2 alignment to these low-lying positive
(πh )2 alignment should be reduced by 0.6 MeV as 11/2
11/2 ∼ levels are absent, which may partially explain the inac-
anamendment. Bysearchingthe(πh11/2×πh11/2)Iπ=10+ cessibility of the (πh11/2)2 alignment in observed level
pair in output wave-functions, the experimentally unob- scheme of 132Ba. On the other hand, we obtain rela-
servedbandbeyondthe(πh11/2)2 alignmenthasbeenlo- tively strong E3 decays from the Iπ = 10+ state with
catedincalculatedspectrum,asdemonstratedbyredlev- the (πh )2 alignment to states of 7 and 8 bands as
11/2 − −
els in Fig. 1(b). The excitation energy of corresponding shown in Table III. Such strong E3 decays are expected,
10+ bandhead (i.e., the (πh11/2)2 alignment) is E ∼ 5.2 becausethe(πh11/2)2 alignmentsharestheprolatedefor-
MeV.Thus,the132Ba(πh11/2)2-alignmentenergyispre- mationwith7−and8−bands. Basedonabovecalculated
dicted to be around 4.6 = 5.2 0.6 MeV, which agrees decay rates,the Iπ =10+ level with the (πh )2 align-
− 11/2
with the systematics of the alignment energy in light Ba ment is suggested to be a 0.511µs isomer with the main
isotopes [2, 11, 39]. decay branch of E 1.3MeV to the Iπ = 9 level in
γ −
∼
To probe electromagneticpropertiesofthe 4.6MeV the 7 band.
−
(πh )2-alignment state, we calculate its E∼2, M1, E3 Now, let’s turn to the (νh ) 2 alignment. The
11/2 11/2 −
4

in both experimental and calculated level schemes. We
) 1.2
V present the sum of 5 and 6 pair numbers in Fig. 3(b)
− −
Me 0.8 (a) correspondingly. Below I =14~, only one 5− or 6− pair
( exists in 5 and 6 bands as expected by Ref. [1, 2];
− −
EI 5- band from exp beyondthispoint,twomore5 and/or6 pairs,i.e.,the
- 0.4 5- band from cal − −
EI+2 6- band from exp 5−⊗6− coupling,areexcited. T he(νh11/2)−2 alignment
6- band from cal
0.0
er 2.8 ers 2.0
b b
m 2.4 (b) m
u u
air n 12..60 Sum of 5- and 6- pair numbers air n 1.5
p p
1.2 6-
d 1.0
6 8 10 12 14 16 18 20 n
a
I ( ) -
5
FIG. 3. (Color online) EI+2 −EI (a) and the sum of 5− of 0.5
and6− pairnumbers(b)in5− and6− bands. Abbreviations m
“exp”and“cal”correspondtotheexperimentaldata[34]and u
S
calculated results, respectively. The grey zone highlights the
(νh )−2 alignment. 0.0
11/2 8 10 12 14 16
I ( )
− − − −
FIG.4. Sumof5 and6 pairnumbersin7 and8 bands.
yrast Iπ = 10+ isomer is the most typical (h11/2)−2- The grey zone highlight the potential (νh )−2 alignment.
11/2
alignment observationin 132Ba. We would like to revisit
the yrast alignment before discussing (νh ) 2 align-
11/2 −
ments in negative-parity bands. According to Fig. 1, in 5 and 6 bands is confirmed.
− −
our calculation spectrally reproduces the excitation en- The (νh ) 2 alignment is also expected in 7 and
11/2 − −
ergy of the yrast alignment. To present this alignment 8 bands around I = 16~ [2], even though the spectral
−
more explicitly, we additionally compare calculated and
evidence was insufficient. We plot the sum of 5 and 6
− −
experimentalassociatedenergies[E E ],reducedE2
I I 2 pair numbers in 7 and 8 bands in Fig. 4, where the
transition rates [B(E2, I I 2)] a−nd m−agnetic g fac- 5 6 coupling−is obser−ved beyond I = 16~. Thus,
→ − − −
tors of the yrast band in Fig. 2. The backbending of we ⊗confirm the (νh ) 2 alignment with oblate shape
11/2 −
E E ,thesharpdroppingdownofB(E2,I I 2)
anId−g fIa−ct2oratthe Iπ =10+ isomerbothexperim→ent−ally in 7− and 8− bands, although these two bands favor the
prolate deformation.
andtheoretically evidentthe intruding ofthe (νh ) 2
11/2 − To summarize, we adopt the pair-truncation of the
alignment in the yrast band. To observe this alignment
Shell Model with negative-parity pairs to describe the
atthe wave-functionlevel,wealsocalculatethe expecta-
(h )2 alignment in 132Ba. Our calculation is spec-
tion value of 5 and 6 pair numbers, and plot the sum 11/2
− − trally consistent with experiments. The Iπ = 10+
of 5 and 6 pair numbers in Fig. 2(d). Correspond-
− − state with the (πh )2 alignment is predicted to be an
ingly to the (νh ) 2 alignment at I = 10~, two 5 11/2
11/2 − − E 4.6MeVandτ 0.5µsisomerwithrelativelystrong
and/or6 pairsaresuddenly excitedas arepresentation ∼ ∼
− E3transitionsto7 and8 bands,whichrequiresfurther
of the 5 6 coupling. In other words, the (νh ) 2 − −
−⊗ − 11/2 − experimental verifications. (νh11/2)−2-alignment obser-
alignment indeed is induced by the 5 6 coupling.
− − vationsinbothyrastbandandnegative-paritybandsare
⊗
Therefore, we take the sudden increasing of 5 and 6
− − also well reproduced by the 5 6 coupling, which is
pair numbers by two as a signal of the (νh11/2)−2 align- suggested to be another repres−en⊗tat−ion of the (νh )2-
11/2
ment.
alignment configuration. This oblately deformed align-
The (νh ) 2 alignment in 5 and 6 bands was ment in prolate-favor 7 and 8 bands typically evi-
11/2 − − − − −
proposed according to the band irregularity around I = dences the γ unstability of 132Ba.
14~ in References [1, 2]. In Fig. 3(a), level spacings This workwassupportedbythe NationalNaturalSci-
(E E )alsooccurasuddendecreasearoundI =14~, ence Foundation of China under Grant No. 11305151.
I+2 I
−
and more implicitly demonstrate this band irregularity DiscussionwithProf. Y.M.Zhaois greatlyappreciated.
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