Parity-doublet bands in the odd- Aisotones237U and239Pu by a particle-number-conserving method based on the cranked shell model Jun Zhang1and Xiao-Tao He2

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Parity-doublet bands in the odd-Aisotones 237U and 239Pu by a

particle-number-conserving method based on the cranked shell model

Jun Zhang1and Xiao-Tao He2, ∗

1College of Science, Nanjing University of Aeronautics and Astronautics, Nanjing 210016, China

2College of Materials Science and Technology, Nanjing University of Aeronautics and Astronautics, Nanjing 210016, China

(Dated: October 6, 2022)

Based on the reﬂection-asymmetric Nilsson potential, the parity-doublet rotational bands in odd-

Aisotones 237U and 239Pu have been investigated by using the particle-number-conserving (PNC)

method in the framework of the cranked shell model (CSM). The experimental kinematic moments

of inertia (MOIs) and angular momentum alignments are reproduced very well by the PNC-CSM

calculations. The signiﬁcant diﬀerences of rotational properties between 237U and 239Pu are ex-

plained with the contribution of nucleons occupying proton octupole-correlation pairs of π2i13/2f7/2.

The upbendings of moments of inertia of the parity-doublet bands in 237U are due to the interfer-

ence terms of alignments of protons occupying πf7/2(1/2) and the high-jintruder πi13/2(1/2,3/2)

orbitals. The splittings between the simplex partner bands of the parity-doublet bands in both 237U

and 239Pu result from the contribution of alignment of neutron occupying the νd5/2(1/2) orbital.

I. INTRODUCTION

The typical feature for octupole correlations in odd-

mass nuclei is the appearance of parity-doublet rotational

bands, which are pairs of almost degenerate states in ex-

citation energy with the same spin but opposite parities

[1–4]. The octupole correlations in nuclei are associated

with the single-particle states with orbital and total an-

gular momentum diﬀering by 3, i.e., ∆l= ∆j= 3 [5, 6].

The nuclei with Z≈88 and N≈134 in the actinide re-

gion are expected to possess the maximum octupole cor-

relations since their Fermi surface lies between the proton

π2f7/2i13/2and neutron ν2g9/2j15/2octupole correlation

pairs [7]. Experimentally, since the ﬁrst observation of

parity-doublet rotational bands in odd-mass 229Pa and

227Ac [8, 9], similar bands have been observed in many

odd-mass actinide nuclei, like 219,223,225Ac, 219,221,223Fr,

221,223Th, and so on [10–17].

The parity-doublet bands are observed experimentally

in 237U and 239Pu, in which some interesting properties

are reported [18, 19]. In Ref. [18], the angular momen-

tum alignments were compared for the bands in the Pu

isotopes with 238 6A6244, where the sharp back-

bending observed in the heavier isotopes is not present

within the same frequency range in 239Pu. The sud-

denly gained alignments in the heavier isotopes are due

to the contribution from a pair of i13/2protons. There is

at present no satisfactory explanation for the absence of

backbending phenomenon in 239Pu. As the N= 145 iso-

tones closest to the 239Pu, the behavior of the alignments

with rotational frequency for 237U is very diﬀerent. In

Ref. [19], a strong backbending in alignment occurs at

~ω≈0.25 MeV in 237U. In both work, the experimental

observations support the presence of the large octupole

correlations in the ground states and in the low-lying ex-

cited states of odd-A237U and 239Pu nuclei. The spin

∗hext@nuaa.edu.cn

and parity are assigned as Kπ= 1/2+for the ground-

state bands in both nuclei. The experimental data shows

that there exist signiﬁcant simplex splittings at low ro-

tational frequency region in the parity-doublet bands in

both nuclei. These issues need further investigations.

Many theoretical approaches were developed to in-

vestigate the octupole correlations in nuclei. These in-

clude the reﬂection-asymmetric mean-ﬁeld approach [20–

22], algebraic models [23, 24], cluster models [25–27], vi-

brational approaches [28–30], reﬂection asymmetric shell

model [31–33] and cranked shell model [34–38]. The

particle-number-conserving method in the framework of

cranked shell model (PNC-CSM) is one of the most useful

models to describe the rotational bands [39–41]. In the

PNC-CSM calculations, the cranked shell model Hamil-

tonian is diagonalized directly in a truncated Fock space.

The particle number is conserved exactly and the Pauli

blocking eﬀects are taken into account spontaneously.

The previous PNC-CSM method has been developed

to deal with reﬂection-asymmetric nuclei, and success-

fully applied to describe the alternating-parity rotational

bands in the even-even nuclei [38].

In this work, the PNC-CSM method is further applied

to describe the ground-state parity-doublet rotational

bands in odd-Anuclei 237U and 239Pu. The ground-

state spin assignments are identiﬁed for both nuclei. By

considering the octupole correlations, Pauli blocking ef-

fects and Coriolis interaction, the striking diﬀereces of

rotational properties between 237U and 239Pu and the

simplex splittings of the parity-doublet bands in both

nuclei are investigated in detail.

A brief introduction of the PNC-CSM method deal-

ing with the reﬂection-asymmetric nuclei are presented

in Sec. II. The detailed PNC-CSM analyses for the

parity-doublet rotational bands in 237U and 239Pu are

presented in Sec. III. A summary is given in Sec. IV.

arXiv:2210.02022v1 [nucl-th] 5 Oct 2022

II. THEORETICAL FRAMEWORK

The detailed understanding of PNC-CSM method

dealing with the reﬂection-asymmetric nucleus can be

found in Ref. [38]. Here we give a brief description of

the related formalism. The cranked shell model Hamil-

tonian of an axially symmetric nucleus in the rotating

frame [42–44] is

HCSM =H0+HP=HNil −ωJx+HP.(1)

HNil =PhNil(ε2, ε3, ε4) is the Nilsson Hamiltonian,

where quadrupole (ε2), octupole (ε3), and hexadecapole

(ε4) deformation parameters are included. −ωJx=

−ωPjxrepesents the Coriolis interaction with the ro-

tational frequency ωabout the xaxis (perpendicular to

the nucleus symmetry zaxis).

When ~ω= 0, for an axially symmetric and reﬂection-

asymmetric system, the single-particle Hamiltonian has

nonzero octupole matrix elements between diﬀerent shell

N. Since parity p= (−1)N, the parity is no longer a

good quantum number, but the single-particle angular

momentum projection on the symmetry axis Ω is still a

good quantum number. The single-particle orbitals can

be labelled with the quantum numbers Ω (lj), where lj

are the corresponding spherical quantum numbers.

However, when ~ω6= 0, due to the Coriolis interaction

−ωjx, the Ω is no longer a good quantum. Since the

reﬂection through plane yoz,Sxinvariant holds [45]. The

single-particle orbitals can be labelled with the simplex

quantum numbers s(s=±i), which are the eigenvalues

of Sxoperator.

The pairing HPincludes the monopole and quadrupole

pairing correlations HP(0) and HP(2),

HP(0) = −G0X

ξη

a†

ξa†

ξaηaη,(2)

HP(2) = −G2X

ξη

q2(ξ)q2(η)a†

ξa†

ξaηaη,(3)

where ξ(η) labels the time-reversed state of a Nilsson

state ξ(η), q2(ξ) = p16π/5hξ|r2Y20|ξiis the diago-

nal element of the stretched quadrupole operator, and

G0and G2are the eﬀective strengths of monopole and

quadrupole pairing interactions, respectively.

By diagonalizing HCSM in a suﬃciently large cranked

many-particle conﬁguration (CMPC) space, a suﬃciently

accurate low-lying excited eigenstate is obtained as

|ψi=X

Ci|ii,(4)

where Ciis real and |ii=|µ1µ2. . . µniis a cranked

many-particle conﬁguration for an n-particle system, and

µ1µ2. . . µnare the occupied cranked Nilsson orbitals.

The conﬁguration |iiis characterized by the simplex si,

si=sµ1sµ2. . . sµn(5)

where sµis the simplex of the particle occupying in or-

bital µ.

The occupation probability of each cranked orbital µ

can be calculated as:

nµ=X

|Ci|2Piµ,(6)

here Piµ = 1 if µis occupied and Piµ = 0 otherwise.

The angular momentum alignment of eigenstate, in-

cluding the diagonal and oﬀ-diagonal parts, can be writ-

ten as

hψ|Jx|ψi=X

|Ci|2hi|Jx|ii+X

i6=j

Ci∗Cjhi|Jx|ji,(7)

and the kinematic moment of inertia is

J(1) =1

ωhψ|Jx|ψi.(8)

For reﬂection-asymmetric systems with odd number

of nucleons [45], the experimental rotational band with

simplex sis characterized by spin state Iof alternating

parity,

s= +i, Ip= 1/2+,3/2−,5/2+,7/2−,· · · ,(9)

s=−i, Ip= 1/2−,3/2+,5/2−,7/2+,· · · ,(10)

The angular momentum alignment for positive- and

negative-parity bands can be expressed as [46]

hJxip=hψ|Jx|ψi − 1

2p4Ix(ω),(11)

J(1)

p=hψ|Jx|ψi

ω−1

2p4J(1)(ω),(12)

where |ψiis the parity-independent wave function with

the rotational frequency ωcalculated by PNC-CSM

method. The 4Ix(ω) and 4J(1)(ω) are parity splitting

of the alignment and moment of inertia in the experiment

rotational bands, respectively. The corresponding values

can be obtained by the following formula,

4Ix(ω) = Ix−(ω)−Ix+(ω),(13)

4J(1)(ω) = J(1)

−(ω)−J(1)

+(ω),(14)

in which +(−) represents the positive- (negative-) parity

in the experiment rotational bands.

III. RESULTS AND DISCUSSIONS

A. Parameters

In the present calculation, the values of Nilsson param-

eters (κ, µ) are taken from Ref. [47]. The deformation

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Parity-doubletbandsintheodd-Aisotones237Uand239Pubyaparticle-number-conservingmethodbasedonthecrankedshellmodelJunZhang1andXiao-TaoHe2,1CollegeofScience,NanjingUniversityofAeronauticsandAstronautics,Nanjing210016,China2CollegeofMaterialsScienceandTechnology,NanjingUniversityofAeronauticsandAstronau...

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Parity-doublet bands in the odd- Aisotones237U and239Pu by a particle-number-conserving method based on the cranked shell model Jun Zhang1and Xiao-Tao He2

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