SCIENCE CHINA
Physics, Mechanics & Astronomy
• Article •
January 2015 Vol. 58 No. 1: 012001
doi: 10.1007/s11433-014-5576-0
Structure of high spin state in proton-rich 74,76,78Kr isotopes: A
projected shell model description
LIU YanXin*, YU ShaoYing* & SHEN CaiWan
School of Science, Huzhou Teachers College, Huzhou 313000, China
Received February 17, 2014; accepted June 13, 2014; published online August 28, 2014
The N≈Z nuclei in the mass A~80 region has been researched because of an abundance of nuclear structure phenomena. The
projected shell model (PSM) was adopted to investigate the structure of high spin state in proton-rich 74,76,78Kr isotopes including yrast spectra, moment of inertia, electric quadrupole transitions and the behavior of single particle. The calculated results
are in good agreement with available data and the shape coexistence in low-spin is also discussed.
projected shell model, high spin state, moment of inertia, electric quadrupole transition
PACS number(s): 21.60.Cs, 21.10.Re, 27.50.+e
Citation:
Liu Y X, Yu S Y, Shen C W. Structure of high spin state in proton-rich 74,76,78Kr isotopes: A projected shell model description. Sci China-Phys Mech
Astron, 2015, 58: 012001, doi: 10.1007/s11433-014-5576-0
1 Introduction
In recent years, the structure of collective bands in proton-rich krypton has attracted considerable research interest
in experimental and theoretical studies [1–7] because the
nuclei in the A~80 mass region exhibit definite patterns of
deformation, shape coexistence and rich nuclear structure.
Proton-rich krypton isotopes are particularly noteworthy
since both the proton and neutron shells are half-filled in
these nuclei. As a result, these nuclei show large ground
state deformation. Therefore, both the protons and neutrons
can align the associated angular momentum at similar frequencies. It is necessary to investigate the nature of alignments in the ground state especially at high spin.
Most of nuclei in the A~80 mass region are strongly deformed. At the deformed potential minimum, the high j g9/2
orbital of the proton and neutron intrudes into the pf-shell
near the Fermi levels. Therefore, the g9/2 orbital dominate
*Corresponding authors (LIU YanXin, email: [email protected]; YU ShaoYing, email:
[email protected])
© Science China Press and Springer-Verlag Berlin Heidelberg 2014
the low-lying structure of these nuclei and these orbitals
need to be included in the model space. However, it is technically difficult for a conventional shell model to directly
consider the g9/2 orbital. This case suggests a proper construction of a shell model basis that is capable of describing
the physics within a manageable space. The projected shell
mode (PSM) [8] is different from the conventional shell
model because the deformed basis is adopted to build the
model space. In this way, a large model space can be easily
included and many nuclear correlations can be constructed
before a configuration mixing is executed. Current PSM
have been successfully applied to treat the nuclear structure
away from the stable valley [9–13].
In the framework of PSM, the residual neutron-proton
interaction in the N=Z krypton has been well researched
[14,15] and the properties of yrast state in 74,76Kr at low-spin
is presented by Palit et al. [16]. However, the nature of high
spin state was not obtained at that time because of the lack
of sufficient data. In the present paper, by adopting the PSM,
we will focus on the properties along the yrast line of proton-rich 74,76,78Kr isotopes in the mass region A~80, and attempt to explain the physics associated at high spin region.
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Liu Y X, et al.
Sci China-Phys Mech Astron
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January (2015) Vol. 58 No. 1
tion with inclusion of both the monopole and quadrupole-pairing terms:
2 Outline of the model
The PSM closely follows the shell-model philosophy which
is a shell model truncated in a deformed basis. It proceeds
as follows: first, the basis truncation is done in the multi-qusiparticle (qp) basis by selecting low-lying states based
on the Nilsson+BCS representation; then, the rotational
symmetry is restored for these deformed multi-qp states by
the projection method to form a spherical (many-body) basis in the laboratory frame. Finally, the Hamiltonian is diagonalized in this projected basis. The truncation achieved
in this way is efficient. Quite satisfactory results can be obtained by choosing only a few orbitals near the Fermi surface since the deformed qp basis already contains most of
important (pairing and quadrupole) correlations. Deformation parameters in the Nilsson model [17] are well studied quantities, so that researchers know precisely where the
optimal basis is. Herein we construct deformed basis with
quadrupole deformation 2 and hexadecapole deformation 4
which are taken from the data tables [18] and adjusted the
data in order to obtain a good basis. The deformation parameters are listed in Table 1.
The set of multi-qp states {|>}, which needs to take
into account in the shell-model configuration space by projecting onto a good angular momentum I, is selected as
thus:
0 , a†1a† 2 0 , a†1a†2 0 , a†1a† 2 a†1a†2 0 ,
(1)
where a† is the qp creation operators and () denotes proton (neutron) Nilsson quantum numbers which run over
properly selected (low-lying) orbitals. Five or more nucleon-like qps are omitted because they have higher excitation
energies and thus have little effect to the results in the energy (and the spin) range that is detailed herein.
The PSM wave function can be written as:
I
IM   f IK  PˆMK
 ,
(2)
K
I
is the
where |> is the qp basis state in eq. (1) and PˆMK
H QP  
 QQ
2
 Qˆ  Qˆ   G
†
2
2
M
Pˆ † Pˆ  GQ  Pˆ2† Pˆ2  .
In eq. (4), the QQ-force strength is determined in such a
way that it has a self-consistent relation [8] with the quadrupole deformation 2. The monopole-pairing strength GM is
of the form GM=[G1G2(NZ)/A]/A with G1=20.25, G2=
16.20 for neutron and GM=G1/A for proton. The quadrupole-pairing strength GQ is assumed to be proportional to
GM, with the proportionality constant 0.24.
3
Results and discussion
The properties of yrast bands at high spins are studied by
the PSM calculation. The results are given in the following
five parts.
3.1
Deformed single particle states
The deformed single particle (SP) states are calculated by
using the standard Nilsson parameters [19]. These parameters were obtained to fit abundant experimental data and
thus can produce reliable SP states. Three major shells are
include, that is, N=2, 3, 4 for both protons and neutrons in
the PSM calculations. The Nilsson diagram for neutrons is
plotted in Figure 1 which is the basis for our subsequent
discussions. The diagram for protons is not shown here
since the neutrons and protons have similar Nilsson energy
levels for N=2, 3, 4 major shells. In Figure 1, the solid lines
and dashed lines denote positive parity states and negative
parity states, respectively. The dotted rectangles indicate the
deformation range of ε2=0.32–0.35 and the main single particle states near the Fermi surface. In this deformation range,
the positive parity state of K=3/2, K=5/2 and K=7/2 from
the g9/2 orbital, the negative parity state of K=3/2 from f5/2
and p3/2 orbital fall in the rectangle. Therefore, these orbitals
angular-momentum projection operator.
In the present work, Hamiltonian consists of the following separable forces:
Hˆ  Hˆ 0  Hˆ QP ,
(3)
where H0 is the spherical single particle Hamiltonian and
HQP is the pairing plus uadrupole-quadrupole (QQ) interacTable 1 Quadrupole and hexadecapole deformation parameters employed
for generating the deformed basis
Kr
2
4
N=38
0.350
0.053
N=40
0.328
0.060
N=42
0.345
0.060
(4)

Figure 1
(Color online) The Nilsson diagram for neutrons.
Liu Y X, et al.
Sci China-Phys Mech Astron
can construct low-lying 2-qp states.
3.2
Energy spectrum
The calculated energy levels of yrast bands and the comparison with experimental data for 74,76,78Kr isotopes are shown
in Figure 2. The available experimental data are reproduced
consistently by the PSM calculations up to I ≈30+. Valiente-Dobón et al. [1] discuss the phenomenon of nontermination of rotational bands in 74Kr experimentally. The
band termination is defined by a continuous transition with
the same configuration from collective rotation at low spin
to a non-collective SP (terminating) state at the maximum
spin which can be built in the pure configuration. The calculated energy levels in Figure 2(a) is in good agreement
with available data [1–3] which indicates that the yrast band
does not terminate at Imax and keeps large prolate deformation. Though a band termination effect in the side bands
Figure 2
Table 2
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is suggested by experimental measurement and cranked
Nilsson-Strutinsky calculations [2], the result in Figure 2(b)
shows that the yrast band favors a collective rotation. As for
78
Kr, the present data and the PSM calculation manifest a
mainly prolate configuration for yrast band.
3.3
Band diagram
In order to understand the structure of yrast bands, we plot
band diagram in Figure 3 which can be the basis for the
following discussions. To facilitate the discussion, we term
the relevant two-qp configurations 2n1, 2n2, 2p for 74,76Kr
isotopes and 2n3, 2n4, 2p for 78Kr, with the details listed in
Table 2. The single-particle compositions of these configurations are consistent with the Nilsson diagram shown in
Figure 1.
In Figure 3, we distinguish different qp bands by lines, so
that it can be easily followed with the increasing of neutron
(Color online) Calculated energy levels of ground state band and comparison with experimental data.
Two-quasiparticle configurations for neutrons and protons
Label
2n1 (K=1+)
2n2 (K=4+)
2n3
2n4
Figure 3
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Configuration
3/2+[431]×5/2+[422]
3/2+[431]×5/2+[422]
5/2+[422]×7/2+[413]
3/2-[301]×5/2-[303]
Label
2p
4qp (1)
4qp (2)

(Color online) Band diagram. Dots denote yrast band and the different lines denote 2-qp and 4-qp bands.
Configuration
1/2+[440]×3/2+[431]
2n1×2p
2n3×2p

Liu Y X, et al.
Sci China-Phys Mech Astron
number. The dots labeled yrast are the lowest state at each
spin obtained after diagonalization. These are the theoretical
results compared with the data (see Figure 2). At low spin
region, the 0-qp band dominate the yrast band and two qp
states of protons and neutrons are flat and close to each other, then they cross with 0-qp band at I≈10 almost simultaneously. Because both the proton and neutron shells are
half-filled in proton-rich 74,76,78Kr, the proton and neutron
align the associated angular momentum at similar frequencies. After band crossing, the proton 2-qp bands (labeled 2p)
are lower in energy than the neutron 2-qp (labeled 2n1 and
2n3) with bands then becoming essential to the yrast band.
For two light 74,76Kr isotopes (Figures 3(a) and (b)), the
configuration of 2n1 is 3/2+[431]×5/2+[422] from neutron
g9/2 orbital. With increasing neutron number, the neutron
7/2+[413] orbital is near the Fermi surface and then has a
role to construct 2-qp bands, namely 5/2+[422]×7/2+[413]
(labeled 2n3) for 78Kr (see Figure 3(c)). Another noteworthy
phenomenon in Figure 3 is that the 4-qp bands approach the
0-qp band and cross with ground band at I≈12 or 14 then
dominate the yrast bands. The positive parity K=4 bands are
predicted in Figure 3 which are expected to be obtained in
the future experimental studies. The configuration of K=4
bands is neutron 2-qp 3/2+[431]×5/2+[422] for 74,76Kr isotopes and 3/2[301]×5/2[303] (labeled 2n4) for 78Kr. We
predict that the bandhead energies of K=4 bands in 74,76,78Kr
isotopes are 3.298, 3.240 and 2.551 MeV, respectively. The
change of configuration in K=4 bands is attributed to the
rise of neutron Fermi surface which may lead to the falling
of neutron 3/2[301] and 5/2[303] orbitals from pf shell
into the rectangle of Nilsson diagram (see Figure 1).
3.4
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012001-4
(defined as J(I)=(2I−1)/[E(I)−E(I−2)]) and compared with
the available data (filled circles). The PSM results are consistent with experimental data except at I=2. This can be
explained by the shape coexistence at low spin [20,21].
Though the theoretical results are inconsistent with experimental data for 78Kr isotope at high spins, the tendency of
variation is in good agreement with the data (Figure 4(c)). A
possible reason is the change of shape at high spins of 78Kr
isotope [22].
It can be observed that for these isotopes, J increases
nearly linearly with spin I for the low spin states. However,
the irregularities are seen at I≈10, with the actual pattern
differing in different isotopes. Subsequently, the MoI remains increasing and then shows down-bending at I=14 or
16. To understand the variation, we recall the band diagram
in Figure 3. Because of the first band crossing at I≈10, the
MoI has an abrupt increase. When the spins extend to 14 or
16, the down-bending of MoI occurs where the 4-qp bands
cross with the ground band.
In order to understand the shape of 78Kr isotope, in Figure 5, we show two sets of calculated MoI by assuming a
prolate shape with ε2=+0.345 and an oblate shape with
ε2=−0.265, respectively. It can be seen that at low spin I<6,
the oblate result is in good agreement with the data because
of shape coexistence [21], however, in I=8–16 spin region,
the PSM calculation suggests a prolate shape. Mixing of
different shapes can drive the system away from an ideal
rotor behavior. The departure behavior of the theoretical
MoI for spins I>20 for the positive-parity yrast band is indicative of band termination [22]. However, the PSM theory
assumes a fixed deformation in the model basis and is difficult to describe the loss of the collectivity.
Moment of inertia
The moment of inertia (MoI) is a characteristic quantity for
the description of rotational behavior. In Figure 4, the calculated results (filled squares) for the yrast bands in 74,76Kr
(Figures 4(a) and (b)) are presented with regard to the MoI
3.5
Electric quadrupole transitions
Electric quadrupole transition probabilities are critical
quantities to test nuclear wavefunctions. The B(E2) value
that measures the electric quadrupole transition rate from an
Figure 4 (Color online) Comparison of the calculated moments of inertia (filled squares) for yrast bands in 74,76,78Kr isotopes with the known experimental
data (filled circles).
Liu Y X, et al.
Sci China-Phys Mech Astron
Figure 5 (Color online) Comparison of the MoI for yrast band in 78Kr
isotope by assuming a prolate shape with ε2 =+0.345 and an oblate shape
with ε2 =−0.265, respectively.
initial state I to a final state I − 2 is given by
B ( E 2, I  I  2) 
1
 I  2 Qˆ 2  I
2I  1
2
,
(5)
where the wavefunctions |I> are defined in eq. (2). The
January (2015) Vol. 58 No. 1
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effective charges used in our calculations are the standard:
e = 1.5e and eν= 0.5e, which are fixed for all nuclei herein.
Thus, any variations in calculated B(E2) represent a structural change in the wavefunction.
In Figure 6, near-constant B(E2) values seen in most
parts of the curve reflect a stable rotor character of the system with large collectivity. However, a clear dip can be seen
at I=14 in Figure 6(a) and I=12 in Figures 6(b) and (c)
where the experimental data are seemingly lost. The drop in
the calculated B(E2) can be understood by the band crossing
between the 4-qp band and 0-qp band shown in Figure 3,
where the yrast structure changes. After the dip, the theoretical B(E2) values prefer to another constant in 74,76Kr but
decrease at I=20 in 78Kr which indicate the yrast deformation change. This variation can be confirmed by the
down-bending in moment of inertia as seen in Figure 4.
From Figure 6(a), it can be seen that the calculated B(E2)
values is smaller than the experimental data. To understand
the discrepancy, effects of the basis deformation are shown
in Figure 7. In Figure 7 we perform calculations by using
four different sets of input deformation parameters. In Figure 7(a), with the deformation parameters increasing, the
curves of the calculated B(E2) values are enhanced. For the
calculation with quadrupole deformation ε2=0.410, the
Figure 6 (Color online) Comparison of calculated B(E2, I→I-2) (in e2b2) (filled triangle) for yrast bands in 74,76,78Kr isotopes with the known experimental
data (filled squares).
Figure 7
(Color online) Variation of B(E2) and MoI for the yrast band of 74Kr calculated with four different sets of deformation parameters.
Liu Y X, et al.
Sci China-Phys Mech Astron
results agree with the data points. Conversely, for the calculation for moments of inertia (Figure 7(b)), small differences are found for the different basis parameters particularly at low spins. The result indicates that large deformation can improve the calculated B(E2) values in 74Kr
isotope [6]. Furthermore, the small differences of MoI for
different basis appears to remain valid.
4
5
6
7
8
4 Conclusions
The PSM calculations are carried out to explore the structure of high spin state in the proton-rich krypton. The obtained energy of yrast bands are consistent well with the
experimental data and another positive parity K=4 bands are
predicted for 74,76,78Kr isotopes which are expected to be
detected by the experiment. The good agreement with the
experimental data suggests that the collectivity of the yrast
bands for 74,76Kr isotopes retains at high spins, however, the
structure of yrast band of 78Kr is influenced by the deformation change at high spin.
Properties along the yrast lines in 74,76,78Kr isotopes were
also investigated. The variation of moments of inertia in
these nuclei is described by the crossings of 2-qp and 4-qp
bands with the ground band. The variation of structure in
yrast bands can be attributed to the interaction between g9/2
orbital and pf shell. We discuss the changes in B(E2) along
the yrast lines and suggest the dominant composition in
wave functions at different spin ranges. The physics in B(E2)
along the yrast lines at high spins needs to be further verified by experimental measurement in future work.
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This work was supported by the National Natural Science Foundation of
China (Grant Nos. 11305059, 11275067, 11275068 and 11135005).
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