403
Progress of Theoretical Physics, Vol. 119, No. 3, March 2008
Cluster Structures in Oxygen Isotopes
Naoya Furutachi,1 Masaaki Kimura,2 Akinobu Doté,3
Yoshiko Kanada-En’yo4 and Shinsho Oryu1
1 Department
of Physics, Faculty of Science and Technology,
Tokyo University of Science, Noda 278-8510, Japan
2 Institute of Physics, University of Tsukuba, Tsukuba 305-8571, Japan
3 High Energy Accelerator Research Organization (KEK), Tsukuba 305-0801, Japan
4 Yukawa Institute for Theoretical Physics,
Kyoto University, Kyoto 606-8502, Japan
(Received August 15, 2007)
Cluster structures of 16 O,18 O and 20 O are investigated using the antisymmetrized molecular dynamics (AMD) plus generator coordinate method (GCM). We have found the K π =0+
2
and 0− rotational bands of 18 O that have prominent 14 C+α cluster structure. The clustering
+
20
π
systematics are richer in O. We suggest the presence of a K =02 band that is a mixture of
−
the 12 C+α+4n and 14 C+6 He cluster structures. K π =0+
3 and 0 bands that have prominent
16
C+α cluster structure are also found.
§1.
Introduction
Investigation of the cluster structure of atomic nuclei has been mainly carried
out for light N =Z nuclei, and much less is known about it for N=Z nuclei. A
famous example of clustering in N=Z nuclei is the 2α clustering of neutron-rich Be
isotopes.1)–10) Studies employing AMD have successfully described many properties
of neutron-rich Be isotopes and have shown that most of them have the 2α cluster
core.4), 6), 8), 10) The degree of the 2α clustering changes dynamically, depending on
the motion of the valence neutrons. This feature is understood well in terms of the
molecular-orbital.1)–3), 5), 9) The study in Ref. 11) with AMD predicts the existence
of cluster structures also in 22 Ne, which is a system expected to have the 16 O+α
cluster core as an analogue of neutron-rich Be isotopes. In Ref. 11), it is shown that
the 16 O+α cluster core is formed and dissolved depending on configurations of two
valence neutrons. Thus, the cluster structures of these nuclei suggest a rich variety
of cluster structure in N=Z nuclei.
To investigate clustering in N=Z nuclei, a series of oxygen isotopes is another
good candidate, because the 12 C+α cluster structure in 16 O has been investigated in
detail for a long time.2), 12) Our interest in this study is to determine what happens
to the 12 C+α cluster structure when we add neutrons to 16 O.
Many of the low-lying excited states of 18 O have been described well within the
shell model space of two neutrons in sd-orbitals above the N=Z=8 shell closure.
However, it is also known that the core excited 4p-2h states coexist in the same
energy region.13), 14) They were associated with the 14 C+α cluster structure, and
many works have been carried out to investigate the cluster structure of 18 O.14)–22)
404
N. Furutachi, M. Kimura, A. Doté, Y. Kanada-En’yo and S. Oryu
Consequently, the K π =0+ band that consists of the 0+ (3.63 MeV), 2+ (5.26 MeV),
4+ (7.11 MeV) and 6+ (11.69 MeV) states has been theoretically and experimentally
established well as the 14 C+α molecular band. The cluster model calculations17), 18)
that show the 14 C+α cluster structure of the K π =0+ band also predict two K π =0−
bands that have the 14 C+α cluster structure. More recently, the α-cluster structure
was investigated by elastic α scattering on 14 C24) and α breakup reactions.25)–27) In
Ref. 26), the assignment of the K π =0− band built on the 1− state at 8.04 MeV
was proposed as the parity doublet partner of the K π =0+ molecular band.25) This
assignment is consistent with the prediction of the cluster model.17), 18) In the case
of 20 O, nothing is known about the clustering. On the basis of two neutron transfer
reactions of 18 O(t, p)20 O, it has been suggested that the core excited 6p-2h states
coexist with the normal 4p-0h states.28) We believe that the core excited states
(0+ (4.46 MeV), 2+ (5.30 MeV), and 4+ (7.75 MeV)) could be associated with the
cluster state as in the case of 18 O.
The purpose of the present study is to investigate the cluster structures of 18 O
and 20 O. We apply AMD+GCM (antisymmetrized molecular dynamics plus the
generator coordinate method) framework for this purpose. AMD has proved to be
a useful method for describing cluster structures in neutron-rich nuclei. This is
due to the flexibility of the AMD wave function, which can describe various types of
cluster structure. Therefore, the framework of AMD+GCM is useful for investigating
the existence of the cluster structures in 18 O and 20 O. We also calculate 16 O to
investigate how the 12 C+α cluster states are described in this framework. We suggest
various kinds of cluster states in 18 O and 20 O. We find the K π =0± bands that have
prominent 14 C+α cluster structure, and some negative parity states that have a
non-negligible amount of 14 C+α cluster state components. In 20 O, the motion of the
valence neutrons around the 12 C+α cluster core enriches the variety of the clustering.
Depending on the motion of the valence neutrons, 12 C+α+4n, 14 C+6 He and 16 C+α
cluster structures appear.
The contents of this article are as follows. In the next section, the AMD+GCM
framework is briefly outlined. In §3, the cluster structures of 16 O, 18 O and 20 O are
discussed. In the last section, we summarize this work.
§2.
Theoretical framework
2.1. AMD wave function and calculational procedure
In this subsection, the AMD+GCM framework is briefly outlined. For more
detail, readers are directed to Refs. 29) and 30). The AMD intrinsic wave function
of a A-nucleon system is described by a Slater determinant:
1
Φint = √ det[ϕ1 , ϕ2 , · · · , ϕA ],
A!
ϕi (r) = φi (r)χi τi .
(2.1)
(2.2)
Here, ϕi is a single particle wave packet, which is composed of a spatial part, φi (r),
a spin part, χi , and an isospin part, τi . The spatial part is given by the Gaussian
Cluster Structures in Oxygen Isotopes
function,
φi (r) =
2ν
π
3/4
Zi
exp −ν r − √
ν
2
Z 2i
,
+
2
405
(2.3)
where Z i is a complex three-dimensional vector. The width parameter ν is the same
for all nucleons and fixed to 0.17 fm−2 . The spin part is parameterized by a complex
parameter ξi :
1
1
+ ξ i χ↑ +
− ξ i χ↓ .
(2.4)
χi =
2
2
The isospin part is fixed to up (proton) or down (neutron). Z i and ξi are variational
parameters and optimized by the frictional cooling method. The parity-projected
wave function, which is generated from Φint , is used as a variational wave function,
Φ± = P̂ ± Φint =
(1 ± P̂x )
Φint ,
2
(2.5)
where P̂x is the space-inversion operator.
The Hamiltonian used in this study is given by
Ĥ = T̂ + V̂n + V̂c − T̂g .
(2.6)
Here, T̂ is the total kinetic energy, T̂g is the energy of the center-of-mass motion,
and V̂c is the Coulomb force. As the effective nuclear force V̂n , the modified Volkov
force (MV1)31) with a spin-orbital part of the G3RS32) force are used. The details
and parameter sets for these forces are given in the next subsection.
The energy variation is performed under a constraint on the matter quadrupole
deformation parameter β. The constraint potential
Vcnst = υcnst (β − β0 )2
(2.7)
is added to the total energy of the system. Here, υcnst takes an appropriate positive
value, and β0 is a given number. We note that we do not put any constraint on
the quadrupole deformation parameter γ. Therefore γ takes the optimal value for a
given value of β. The definitions of β and γ are given in Ref. 6).
After the variation, the optimized wave function Φ± (β) is projected to an eigenstate of the total angular momentum J:
J
±
J
J∗
(β)
=
P̂
Φ
(β),
P̂
=
dΩDM
(2.8)
ΦJ±
MK
MK
K (Ω)R̂(Ω).
MK
The integrals over three Euler angles are calculated numerically.
Finally, we perform GCM calculations by employing β as the generator coordinate. The same choice of the generator coordinate has been made in HartreeFock+GCM calculations, and also in the study of the clustering properties of 20 Ne33)
and 44 Ti.34) The final wave function is given by a superposition of basis wave functions ΦJ±
M Ki (βi ), with the generator coordinate βi and K quantum number, where
406
N. Furutachi, M. Kimura, A. Doté, Y. Kanada-En’yo and S. Oryu
|K| ≤ 4 and |K| ≤ 3 are taken for positive and negative parity states, respectively.
The wave function that describes a certain state is given by
cni ΦJ±
(2.9)
ΨnJ± =
M Ki (βi ),
i
where ci is determined by solving the Hill-Wheeler equation,
δ(ΨnJ± |Ĥ|ΨnJ± − n ΨnJ± |ΨnJ± ) = 0.
(2.10)
After the GCM calculation, we obtain the ground state and many excited states.
We investigate the dominant component of each state by calculating the squared
overlap between the GCM wave function ΨnJ± and the basis wave function ΦJ±
M K (βi )
J±
J±
J±
J±
2
as |ΦM K (βi )|Ψn | /ΦM K (βi )|ΦM K (βi ).
In the present study, most of the obtained GCM wave functions are mixtures of
basis wave functions having different values of the K quantum number. However, it
is possible to assign several bands. The ground band is assigned on the basis of the
dominant component of the GCM wave function, while the excited rotational bands
are assigned on the basis of the enhancement of the E2 transition probabilities. Each
band is dominated by the basis wave functions with a certain K quantum number,
and for this reason, we call it the ‘K band’.
2.2. Interactions
We use the MV1 case3 force31) for the central force, and the G3RS force32) for
the spin-orbit force. The MV1 force consists of finite-range two-body and zero-range
three-body terms:
V̂M V 1 = V̂ (2) + V̂ (3) ,
(1 − m − mP̂σ P̂τ )
V̂ (2) =
(2.11)
i<j
×{V̂A exp[− (r̂ ij /rA )2 ] + V̂R exp[− (r̂ ij /rR )2 ]},
r̂ ij = r̂ i − r̂ j ,
υ (3) δ(r̂i − r̂ j )δ(r̂i − r̂ k ).
V̂ (3) =
(2.12)
(2.13)
(2.14)
i<j<k
The spin-orbit part of the G3RS force is given by
2
2
u{e−κI r̂ij − e−κII r̂ij }P̂ (3 O)l̂ij · (ŝi + ŝj ),
V̂LS =
(2.15)
i<j
where P̂ (3 O) is the projection operator onto the triplet odd state. The adopted force
parameters are summarized in Table I. The force parameters are modified from the
original values so that the binding energies of 16,18,20 O, the threshold energies of
+
12 C are reasonably
α+12,14 C and the excitation energies of the 2+
1 and 41 states of
reproduced by the present model (Table II).
Cluster Structures in Oxygen Isotopes
407
Table I. The force parameters for the MV1 case3 force and the spin-orbit part of the G3RS force.
m
0.61
VA [MeV]
−83.34
u [MeV]
3000
VR [MeV] rA [fm]
104.86
1.60
κI [fm−2 ]
5.0
Table II. The binding energies of 16,18,20 O,
12
and 4+
C are also shown.
1 states of
4
He
6
He
14
C
16
C
16
O
18
O
20
O
12
C
4,6
B.E [MeV] (Cal.)
28.9
31.3
102.8
108.4
127.3
137.5
153.2
B.E [MeV](0+ )
Cal.
89.4
Exp.
92.16
He and
rR [fm] υ (3) [MeV]
0.82
4000
κII [fm−2 ]
2.778
12,14,16
C. The excitation energies of the 2+
1
B.E [MeV] (Exp.)
28.29
29.26
105.28
110.75
127.62
139.81
151.36
Ex.(2+
)
[MeV]
Ex.(4+
1
1 ) [MeV]
5.5
13.3
4.44
14.1
2.3. Analysis of single-particle orbitals
We also investigate the single-particle structure of the obtained wave functions
Φ± (β) by diagonalizing the single-particle Hamiltonian.6) First, we transform the
single-particle wave packets ϕi into the orthonormal bases ϕ̃α :
A
1 ciα ϕi .
ϕ̃α = √
µα
(2.16)
i=1
Here, µα and ciα are the set of eigenvalues and eigenvectors of the overlap matrix
Bij ≡ ϕi |ϕj :
A
Bij cjα = µα ciα .
(2.17)
j=1
Using the ϕ̃α , we construct the single-particle Hamiltonian matrix h:
hαβ = ϕ̃α |t̂|ϕ̃β +
A
ϕ̃α ϕ̃i |υ̂|ϕ̃β ϕ̃i − ϕ̃i ϕ̃β i=1
+
1
2
A
ϕ̃α ϕ̃i ϕ̃j |υ̂3 |ϕ̃β ϕ̃i ϕ̃j + ϕ̃j ϕ̃β ϕ̃i
i,j=1
+ϕ̃i ϕ̃j ϕ̃β − ϕ̃β ϕ̃j ϕ̃i − ϕ̃i ϕ̃β ϕ̃j − ϕ̃j ϕ̃i ϕ̃β .
(2.18)
408
N. Furutachi, M. Kimura, A. Doté, Y. Kanada-En’yo and S. Oryu
Then we obtain the single-particle energy p and the single-particle wave function
φs by diagonalization of the matrix h:
hαβ gβp = p gαp ,
(2.19)
β
φs =
A
gαp ϕ̃α .
(2.20)
α=1
In this study, we calculate the density distribution of the single-particle wave function
φs to investigate the motion of valence neutrons.
§3.
3.1.
Results
16 O
16 O
is well known to have a prominent 12 C+α cluster structure in its excited
First, we investigate 16 O to determine how the cluster structure is described
in the present framework.
The energy curves before and after the angular momentum projection for the
(a) positive- and (b) negative-parity states are shown in Fig. 1. Before the angular
momentum projection, the positive-parity curve [dotted line in Fig. 1 (a)] has a
minimum at the spherical point. As the deformation becomes larger, the energy
rapidly increases. The angular momentum projection drastically changes the energy
curve. Here, we discuss the energy curves after the angular momentum projection
with K=0 for the sake of simplicity. The 0+ curve has a minimum at β=0.20 and
a shallow local minimum at β=0.66. The minimum state is dominated by the 0ω
configuration and corresponds to the ground state, while the local minimum state
has the 4ω configuration (proton 2ω and neutron 2ω) and dominates the 0+
2
state. Here, an ω excitation means a particle-hole excitation from the p-shell to the
sd-shell, and the particle-hole configuration of each state is evaluated from analysis
of single-particle orbitals. The density distributions of the intrinsic wave functions at
these minima are shown in Figs. 2 (a) and (b). Figure 2 (b) shows the formation of a
two-center structure. There are twelve wave packets located on the left side, and four
on the right side, which corresponds to the developed 12 C+α cluster structure. In the
case of the 2+ , 4+ and 6+ curves, they have two energy minima around β=0.30 and
0.65. The energy curves exhibit different behavior in the β > 0.53 region, because
the structure drastically changes to the 4ω configuration from the 0ω configuration
around β=0.53. The energy curve of the negative-parity state is also steep before
the angular momentum projection. After the angular momentum projection, the
energy spectra are different in the moderately deformed region (β < 0.53) and the
strongly deformed region (β > 0.53). In the former region, the lowest state is the 3−
state, and the 1− state is approximately 5 MeV above the 3− state. In the strongly
deformed region, the spectrum has a rotational nature. This is due to the structure
change of the intrinsic wave function. In the moderately deformed region, the wave
function has the 1ω configuration, while in the strongly deformed region, it has the
12 C+α cluster structure, as shown in Figs. 2 (c) and (d).
states.2)
Cluster Structures in Oxygen Isotopes
409
Fig. 1. Energy curves of 16 O as functions of the matter quadrupole deformation parameter β for
the (a) positive-parity and (b) negative-parity states. The solid line represents the energy of
each parity and angular momentum (K=0) state, and the dashed lines represent the energy
before the angular momentum projection.
Fig. 2. Matter density distributions of the intrinsic wave functions of 16 O. The contour lines are
plotted from 0.01 to 0.16 fm−3 , with an interval of 0.02 fm−3 . The centroids of the single-particle
wave packets are plotted with white squares. Here, Φint(+) and Φint(−) denote the intrinsic wave
function on the positive and negative parity curves, respectively.
After the angular momentum projection, we performed the GCM calculation.
The states with non-zero K quantum number are also included in the GCM calculation. Figure 3 displays the calculated and observed level scheme. The E2 transition
+
probabilities are listed in Table III. We have obtained the excited K π =0+
2 , 2 and
0− rotational bands, together with the ground state and many excited states. The
ground state consists dominantly of the wave functions around the minimum at
−
β=0.20. The excited K π =0+
2 and 0 rotational bands mainly consist of the wave
functions in the largely deformed region (β=0.5−0.7 and β=0.6−0.8, respectively)
that have the prominent 12 C+α cluster structure. These bands are dominated by
the K=0 components. The squared overlaps between the GCM wave functions of
−
the K π =0+
2 and 0 band members and the K=0 wave functions in the largely deformed region are 60−80%. It is noted that the K π =0− band exhibits K-mixing.
−
−
The overlaps between the GCM wave functions of the 3−
2 , 51 and 71 states and the
K=1 wave function at β=0.52 are 15%, 35% and 30%, respectively. Almost all of the
other excited states, including the K π =2+ band, also consist of the wave functions
−
−
that have the 12 C+α structure, except for the 3−
1 , 11 and 21 states, which have the
π
+
1ω configuration. The K =2 band is dominated by the K=2 component, which
amounts to 60−80%.
410
N. Furutachi, M. Kimura, A. Doté, Y. Kanada-En’yo and S. Oryu
Fig. 3. Excitation energies of the low-lying states of 16 O. The dotted lines represent the theoretical
and experimental threshold energies. The theoretical threshold energy was calculated using the
binding energies listed in Table II.
Table III. Calculated E2 transition probabilities for
Ref. 35).
16
O. The experimental data are taken from
Jiπ
+
2 (6.92)
2+ (6.92)
2+ (9.84)
2+ (9.84)
4+ (10.36)
4+ (11.10)
2+ (11.52)
2+ (11.52)
Exp.
Jfπ
+
0 (0)
0+ (6.05)
0+ (0)
+
0 (6.05)
2+ (6.92)
2+ (6.92)
0+ (0)
+
0 (6.05)
e2 fm4
7.4±0.2
65±7
0.074±0.007
2.9±0.7
156±14
2.4±0.7
3.6±1.2
7.4±1.2
1− (7.12)
2− (8.87)
2− (8.87)
3− (6.13)
3− (6.13)
1− (7.12)
50±12
20±2
25±2
Jiπ
2+
1
2+
1
2+
3
2+
3
4+
3
4+
1
2+
2
2+
2
4+
2
4+
3
1−
1
2−
1
2−
1
Cal.
Jfπ
0+
1
0+
2
0+
1
0+
2
2+
1
2+
1
0+
1
0+
2
2+
1
2+
2
3−
1
3−
1
1−
1
e2 fm4
4.2
40
0.65
1.6
29
9.1
1.0
21
21
49
20
4.2
3.1
−
−
The excitation energies of the 3−
1 , 11 and 21 states and the moment of inertia
of the excited K π = 0− band are qualitatively reproduced. The present result is
in almost the same situation with other microscopic calculations2), 36), 37) that also
−
12 C+α
overestimate the excitation energies of the K π =0+
2 and 0 bands having the
cluster structure. In the case of microscopic cluster models, one reason for the overestimation has been conjectured to be the problem in reproducing the 12 C+α threshold
energy.2), 37) Though the 12 C+α threshold energy is approximately reproduced in
the present calculation, the situation is not improved drastically.
The other excited states that have 12 C+α cluster structure (around 17−23 MeV)
are also considered to overestimate the excitation energy. We have assigned some
of them to the observed states by comparison of the E2 transition probabilities, as
Cluster Structures in Oxygen Isotopes
411
shown in Table III. The underestimation of the in-band E2 transition probabilities
+
+
of the K π =0+
2 band, especially for 43 → 21 , is mainly due to fragmentation. The
K=0 component of the 12 C+α cluster structure fragments into several states such
+
+
+
+
+
as the 4+
2 and 22 states, as indicated by large B(E2) values for 42 → 21 , 43 → 22
+
+
+
+
and 22 → 02 in the present calculation. The 42 and 22 states have considerable
contributions from the wave functions around β=0.2−0.3 that do not have the 12 C+α
structure, and therefore the mixing of these states with the K π =0+
2 -band states
results in the decrease of the in-band transition strengths. Because the mixing is
sensitive to the relative energies, we conjecture that the underestimation of the
strengths results from the fact that the excitation energies of the 12 C+α cluster
states are not sufficiently described in the present calculations.
In the present results, the parity doublet bands that have prominent 12 C+α
cluster structure are obtained, and the single-particle excitations in the low-lying
negative parity states are also described. Although the quantitative reproduction
of these states is not good compared to an earlier work using 12 C+α OCM,12) the
present model is believed to be effective for investigating the cluster structure in
18 O and 20 O. The present framework enables us to describe the mixing between the
cluster structure and the shell-like structure, which is considered to be important in
18 O and 20 O. Moreover, we can investigate the clustering property of 20 O, for which
it is difficult to make an a priori assumption of a certain cluster core, because 16 C
and 6 He are weakly-bound nuclei.
3.2.
18 O
In this subsection, we investigate how the α cluster structure changes by adding
two neutrons to 16 O. The same calculational procedure as in the case of 16 O is applied to 18 O. The obtained energy curve and the density distributions of the core
and valence neutrons are shown in Figs. 4 and 5, respectively. Here, we have defined
the valence neutrons as two neutrons in the most weakly bound neutron orbitals,
and the core as the nucleons in the lowest 16 orbitals. In the states shown in Fig. 5,
two valence neutrons occupy the orbitals that have the same spatial density distributions. The 0+ energy curve has a minimum at β=0.20, and the 2+ and 4+ energy
curves have minima in the β ∼ 0 region. These minimum states have a shell-like 0ω
configuration, although the density distribution of the 0+ minimum state [Fig. 5
(a)] exhibits small deformation without space-inversion symmetry. Around β=0.45,
the 6+ energy curve has a minimum, and the 0+ , 2+ and 4+ curves have shoulders.
In this region, the wave functions approximately correspond to the proton 2ω configuration. The density distribution given in Fig. 5 (b) shows that the system is
separated into two clusters. There are 14 wave packets on the left side and 4 on
the right side. This indicates the formation of 14 C+α cluster structure. Indeed, the
density distribution of the two valence neutrons shows that the valence neutrons stay
only around the 12 C cluster. The formation of the 14 C+α cluster structure leads to
the rotational nature of the 0+ , 2+ , 4+ and 6+ energies. The wave functions around
β=0.51 become the dominant component of the K π =0+
2 rotational band after the
GCM calculation.
In the case of negative parity states [Fig. 4(b)], the energy minimum of the 3−
412
N. Furutachi, M. Kimura, A. Doté, Y. Kanada-En’yo and S. Oryu
Fig. 4. Energy curves of 18 O as functions of the matter quadrupole deformation parameter β for
the (a) positive-parity and (b) negative-parity states. The notation is the same as in Fig. 1.
Fig. 5. Density distributions of the core (black contour lines) and the valence neutrons (color plots)
of 18 O. The contour lines are plotted from 0.01 to 0.16 fm−3 , with an interval of 0.02 fm−3 . The
centers of the single-particle wave packets are shown by the white squares. The intrinsic wave
−
function of (a) and (c) gives the minimum energy for the 0+
1 and 11 states, and (b) and (d)
+
π
π
−
become the dominant components of the K =02 and K =0 rotational bands.
curve is at β=0.23, where the wave function has the proton 1ω configuration. The
energy minimum of the 1− curve is at β=0.34. The density distribution of this state
[Fig. 5 (c)] shows a slight development of the cluster structure. As the deformation
becomes larger, this cluster structure develops. Figure 5 (d) shows the pronounced
14 C+α cluster structure of the largely deformed negative-parity state. Although the
energy curves have no local minimum, they become the dominant component of the
K π =0− rotational band after the GCM calculation. Again, the development of the
14 C+α cluster structure is confirmed from the distributions of the wave packets and
the localization of the valence neutrons around 12 C [Fig. 5 (d)].
The energy levels obtained from the GCM calculation are shown in Fig. 6, together with the experimental results. The energy levels calculated using the weaker
spin-orbit interaction (u=2000 MeV) are also shown. These are compared with the
present results below. The E2 transition probabilities are listed in Table IV. We
have obtained the ground band and many excited states, including the K π =0+
2 and
+
−
14
0 bands, which have the C+α cluster structure. The ground band (the 0+
1 , 21
+
and 41 states) consists dominantly of the wave functions around β=0.20 that have
shell-like 0ω configurations. Compared to the experimental energy spectrum, the
level spacing in the ground band is considerably underestimated. The E2 transition
Cluster Structures in Oxygen Isotopes
413
Fig. 6. Low-lying level scheme of 18 O. In the middle panel, we also show the level scheme calculated
with a different spin-orbit strength, u=2000 MeV, which is weaker than the default strength
−
(u=3000 MeV) in the present work. The experimental candidates of the K π =0+
bands
2 and 0
are quoted from Refs. 23) and 25).
probabilities involving the ground band also considerably underestimate the experimental values. The underestimation of the B(E2) values for the ground band is due
to the strong proton closed shell nature of our shell-like 0ω configuration. We conclude that the proton excitation in the ground band is not satisfactorily described
in this calculation. The calculation also fails to reproduce the large B(E2) value
for the transition from 2+ (5.26 MeV) to 4+ (3.56 MeV). This strong E2 transition
found in the experiment implies that the 4+ (3.56 MeV) state has a rather different
structure from the ground state. However, such a description is insufficient in this
−
−
−
calculation. The 5−
1 , 31 , 11 and 21 states consist dominantly of the wave functions
in the moderately deformed region (β=0.2−0.4), which have the proton 1ω config−
−
uration. We have assigned the 5−
1 and 31 states to the experimental 3 (5.10 MeV)
−
and 5 (8.10 MeV) states, because the enhancement of the E2 transition probability
qualitatively agrees with experiments. However, the theoretical result that the 5−
1
state is the lowest negative parity state is inconsistent with the experimental level
scheme, in which the 1− state is the lowest. The experimental B(E2) value for the
transition from 3− (6.40 MeV) to 1− (4.46 MeV) is also large. However, we cannot
find enhancement of the E2 transition probability between the calculated low-lying
−
negative-parity states, except for the transition from 3−
1 to 11 . The level scheme
of the shell-like 0ω and 1ω states is not satisfactorily described in the present
calculations. This problem is mainly due to the difficulty in reproducing the cluster
threshold energy and shell-like states simultaneously with an effective interaction.
We discuss this point below.
+
+
+
We have found that the 0+
2 , 23 , 42 and 61 states consist dominantly of the wave
functions around β=0.51. The E2 transition probabilities between these states are
extremely enhanced, and hence we have classified them as the K π = 0+
2 band. As
14
mentioned above, the intrinsic wave functions around β=0.51 have the C+α cluster
414
N. Furutachi, M. Kimura, A. Doté, Y. Kanada-En’yo and S. Oryu
Table IV. Calculated E2 transition probabilities for
Ref. 38).
18
O. The experimental data are taken from
Jiπ
(1.98)
(3.56)
(3.63)
(3.92)
(5.26)
(5.26)
(5.26)
(5.34)
(7.12)
(7.12)
(7.12)
Exp.
Jfπ
+
0 (0)
2+ (1.98)
2+ (1.98)
0+ (0)
0+ (0)
+
4 (3.56)
0+ (3.63)
2+ (1.98)
2+ (1.98)
2+ (3.92)
2+ (5.26)
e2 fm4
9.3±0.3
3.3±0.2
48±6
3.9±0.6
5.97±0.34
62±34
70±42
5.3±1.1
8.7±2.0
7.3±1.7
(15.7±4.5)
3− (6.40)
5− (8.13)
1− (4.46)
3− (5.10)
25±17
14±14
+
2
4+
0+
2+
2+
2+
2+
0+
4+
4+
4+
Jiπ
2+
1
4+
1
0+
2
2+
2
2+
3
2+
3
2+
3
0+
3
4+
2
4+
2
4+
2
4+
3
6+
1
Cal.
Jfπ
0+
1
2+
1
2+
1
0+
1
0+
1
4+
1
0+
2
2+
1
2+
1
2+
2
2+
3
2+
3
4+
2
5−
1
3−
1
3−
1
1−
1
e2 fm4
0.20
0.17
2.1
1.1
0.69
0.09
69
0.70
0.89
9.1
50
47
27
8
15
structure. Therefore this band is regarded as having the 14 C+α cluster structure.
The overlaps between the GCM wave functions of the K π =0+
2 band members and
+
the K=0 wave functions around β=0.51 are 70−80%. We assigned the 0+
2 and 23
+
+
states to the experimental 0 (3.63 MeV) and 2 (5.24 MeV) states, because the
calculated B(E2) value is close to the experimental data. The experimental 4+ (7.11
MeV) and 6+ (11.69 MeV) states are regarded as possibly corresponding to the other
K π =0+
2 members, because they have been proposed to be the band, together with
the 0+ (3.63 MeV) and 2+ (5.26 MeV) states, in many theoretical works13), 16)–18)
and the α-transfer reaction.23) The level spacing between the K π =0+
2 band members
+
and
4
is in good agreement with these experimental states. The 2+
2
3 states exhibit
14
mixing between the C+α cluster state and the shell-like state, namely, they have
a considerable amount of the 14 C+α component added to the shell-like 0ω state
+
component. The overlaps between the GCM wave functions of the 2+
2 and 43 states
and the wave function at β=0.20, which has a shell-like 0ω configuration, are 40%
and 15%, while the overlaps with the wave function at β=0.51, which has a 14 C+α
+
structure, are 20% and 30%. This leads to a large B(E2) value for 4+
3 → 23 and the
+
+
suppression of the 42 → 23 transition.
−
−
The K π =0− rotational band (the 1−
3 , 32 and 52 states) consists dominantly of
the wave functions around β=0.72, which have a prominent 14 C+α cluster structure.
Therefore, the K π =0− band can be regarded as the parity doublet partner of the
π
−
K π =0+
2 band. The overlaps between the GCM wave functions of the K =0 band
members and the K=0 wave functions around β=0.72 are 40−50%. Experimentally,
the assignment of the K π =0− band, which consists of the 1− (8.04 MeV), 3− (9.67
MeV), and 5− (11.62 MeV) states, has been proposed on the basis of results obtained
from the α breakup reaction.25) Therefore, we consider that they correspond to the
Cluster Structures in Oxygen Isotopes
415
calculated K π =0− band. The spacing between the band head states of the K π = 0+
2
−
+
−
and 0− bands (0+
2 -13 , 4.6 MeV) is close to that of the experiment [0 (3.63 MeV)-1
(8.04 MeV), 4.41 MeV]. As in the case of the positive parity states, there is mixing
between the 14 C+α structure and the proton 1ω configuration. This leads to the
fragmentation of the 14 C+α cluster structure into many states. In particular, the
−
−
−
−
14 C+α cluster state components,
1−
1 , 12 , 33 , 52 and 14 states have considerable
though they are much smaller than those in the K π =0− band members.
In the cluster model calculation,17) adding to two K π =0+ bands, two K π =0−
bands and K π =1+ and 1− bands have been proposed. In our calculation, the K π =1+
band that was mainly described by the 14 C(2+ )+α configuration in Ref. 17) has not
been obtained, because the 14 C+α cluster structure obtained in the β-constrained
energy curve has a structure that is almost axially symmetric and has a very small
amount of the K=0 component. The lower K π =0− band suggested in Ref. 17) has
been assigned to the experimental 1− (4.46 MeV), 3− (5.10 MeV) and 5− (8.10 MeV)
states. We have found the enhancement of the E2 transition probability between
−
−
π
−
the 1−
1 , 31 and 51 states. However, these states are mixed with the K =1 states,
and are out of the rotational row. Therefore, we have not considered them as the
K π =0− band. There are no K π =1− states that exhibit the enhancement of the E2
transition probability. Therefore we have not classified the K π =1− band.
As mentioned above, the reproduction of the level scheme of the low-lying states
is insufficient in the present calculations. The effective interaction used in this calculation is effective for reproducing the 14 C+α threshold energy. However, the strength
of the spin-orbit force is not appropriate for describing the shell-like 0ω and 1ω
states in 18 O. Therefore, we have investigated the dependence of the low-lying states
on the spin-orbit force by changing its strength. We have performed calculations
using smaller spin-orbit strength parameters, u=2000 and 1500 MeV, while fixing
the other interaction parameters. The level scheme calculated with u=2000 MeV
is shown in the middle panel of Fig. 6. Compared with the case of u=3000 MeV,
the energy levels of the low-lying 0ω and proton 1ω states are improved by the
use of the weaker spin-orbit strength. Specifically, the level spacing in the K π =0+
1
ground band exhibits better agreement with the experiment, and the order of the
−
−
low-lying negative-parity states (3−
1 , 11 and 51 ) is also improved. In contrast to
the case of these shell-like states, the cluster states are not as sensitive to the spin−
orbit strength as the shell-like states. We obtain the K π =0+
3 and 0 bands that
14
have C+α cluster structure even with the weaker spin-orbit strength, although
their excitation energies are much larger than in the case u=3000 MeV. The energy
splitting between the K π =0± bands is almost unchanged, and fragmentation of the
cluster states also takes place. Use of the strength u=1500 MeV does not change
+
the situation drastically. The excitation energies of the 2+
1 and 41 states increase to
14 C+α
1.2 and 2.4 MeV, and the band head state of the K π =0+
3 band, which has
cluster structure, increases to 12.4 MeV. It is important that the properties of the
cluster states are almost unchanged with the weaker spin-orbit interaction, which
gives a better description of the shell-like states.
416
3.3.
N. Furutachi, M. Kimura, A. Doté, Y. Kanada-En’yo and S. Oryu
20 O
If we add four neutrons to 16 O, a variety of cluster states appears. The obtained
energy curves and the density distributions of the core and four valence neutrons
are shown in Figs. 7 and 8, respectively. We define the valence neutrons as the four
neutrons in the most weakly bound neutron orbitals and the core as the nucleons in
the lowest 16 orbitals. In the states shown in Fig. 8, there are always two orbitals that
have different density distributions, and two valence neutrons occupy each orbital.
The energy minimum of the 0+ curve is at β=0.20 [Fig. 8 (a)], and the energy
minima of the 2+ and 4+ curves are around β=0.10 and β=0, respectively. In
this region, the wave functions have the 0ω configuration. Around β=0.41, the
structure changes from the 0ω to the proton 2ω configuration, and the 0+ and 2+
(4+ and 6+ ) curves have shoulders (local minimum). In this region, two different
cluster structures appear. Let us compare the wave functions at β=0.41 and β=0.53
[Figs. 8 (b) and (c)]. They have similar core density distributions, which show the
development of the 12 C+α cluster core. The difference between them is clearly seen
in the density distributions of the valence neutrons. The density distribution Fig.
8 (b) shows that four valence neutrons orbit around the entire 12 C+α core. By
contrast, Fig. 8 (c) shows that two of the four valence neutrons are localized around
the 12 C cluster, and the others are localized around the α cluster. Therefore, we
conclude that the wave function at β=0.41 has 12 C+α+4n structure, and the wave
function at β=0.53 has 14 C+6 He structure. In the former, the valence neutrons move
in the mean field of the whole system of the 12 C+α core, while in the latter, the
spatial correlations of two neutrons with the 12 C and α core are enhanced. From
β=0.58, where the 0+ , 2+ , 4+ and 6+ curves have shoulders, another cluster structure
appears. The density distribution of the wave function at β=0.62 [Fig. 8 (d)] shows
the formation of developed 16 C+α structure, in which all valence neutrons orbit only
around the 12 C cluster.
Various structures also appear on the negative parity curve. The 1− , 3− and 5−
curves have energy minimum around β=0.17, where the intrinsic wave functions have
the proton 1ω configuration. Around β=0.50, where the negative parity curves exhibit a rotational nature, the 14 C+6 He structure appears. The density distributions
of the core and valence neutrons in this state [Fig. 8 (e)] are quite similar to those
of the 14 C+6 He structure that appears on the positive parity curve [Fig. 8 (c)]. In
the strongly deformed region, the 16 C+α cluster structure appears around β=0.61
[Fig. 8 (f)], which is quite similar to that found on the positive parity curve [Fig. 8
(d)].
The level scheme of 20 O obtained from the GCM is shown in Fig. 9, together
with the experimental results. The ground band (K π =0+
1 ), many excited rotational
+
+
π
+
−
bands (K =02 , 03 , 1 and 0 ), and the other excited states have been obtained.
+
−
The in-band E2 transition probabilities of the K π =0+
2 , 03 and 0 bands are listed in
+
+
Table V. It is seen that the level spacing in the ground band (01 , 2+
1 and 41 states)
is considerably underestimated compared to the experiments. This is believed to
be due to the strong spin-orbit interaction, as in the case of 18 O. We have found
−
that the 1−
1 and 31 states, which have proton 1ω configurations, are close to the
Cluster Structures in Oxygen Isotopes
417
Fig. 7. Energy curves for 20 O as functions of the matter quadrupole deformation parameter β for
the (a) positive-parity and (b) negative-parity states. The notation is the same as in Fig. 1.
Fig. 8. Density distributions of the core (black contour lines) and the valence neutrons (color plots)
of 20 O. The contour lines are plotted from 0.01 to 0.16 fm−3 with an interval of 0.02 fm−3 . Two
orbitals for four valence neutrons around the 16 O-core are shown for each intrinsic state. The
orbital shown on the right (left) side is the most (second most) weakly bound orbital.
experimental 1− at 5.35 MeV39) and the tentative (3− ) state at 5.61 MeV. More
information concerning excited states in 20 O is given in Ref. 40), though the spin
+
+
+
+
and parity have not been determined. The K π =0+
2 band (02 , 23 , 43 and 62 states)
consists dominantly of the wave functions around β=0.41−0.58. In this region,
12 C+α+4n and 14 C+6 He cluster structures appear, as mentioned above. Therefore,
the K π =0+
2 band is a mixture of these structures. The overlaps between the GCM
wave functions of the K π =0+
2 band members and the K=0 wave functions around
418
N. Furutachi, M. Kimura, A. Doté, Y. Kanada-En’yo and S. Oryu
Fig. 9. Level scheme of
20
O.
Table V. In-band E2 transition probabilities for
band
K π = 0+
1
Jiπ
2+
1
4+
1
Cal.
Jfπ e2 fm4
0+
0.44
1
2+
0.17
1
K π = 0+
3
2+
7
4+
6
6+
5
0+
3
2+
7
4+
6
96
112
87
band
K π = 0+
2
K π = 0−
Jiπ
2+
3
4+
3
6+
2
3−
6
5−
5
20
O.
Cal.
Jfπ e2 fm4
0+
42
2
2+
54
3
4+
60
3
1−
22
5
3−
31
6
β=0.41−0.58 are 70−80%. Experimentally,28) these 0+ (4.46 MeV), 2+ (5.30 MeV)
and 4+ (7.75 MeV) states have been assigned to the proton 2ω states predicted
by analysis with shell model calculations. These experimental states are good candidates of our K π =0+
2 band, because the particle-hole configuration of the wave
functions that have 12 C+α+4n and 14 C+6 He structures are proton 2ω, as mentioned above. Because of the fragmentation of the 14 C+6 He cluster state, we can
not classify a clear K π =0− band, which should be the parity doublet partner of the
14 C+6 He cluster state components,
K π =0+
2 band. Many negative parity states have
−
−
−
−
−
−
such as the 13 , 35 , 54 , 14 , 34 and 53 states. The K π =0+
3 band consists dominantly
16
of the wave functions around β=0.62, which have the C+α cluster structure. The
K π =0− band also consists of the wave functions that have the 16 C+α cluster structure. As mentioned above, the 16 C+α structures in the positive and negative parity
states around β=0.62 are quite similar. Therefore, these bands are regarded to be
parity doublet bands. As in the case of the 14 C+6 He cluster state, the fragmentation
of the 16 C+α cluster state also takes place in the negative parity states, and it leads
to the suppression of the in-band E2 transition probabilities of the K π =0− band.
Let us discuss characteristics of the cluster features of 20 O in a series of O iso-
Cluster Structures in Oxygen Isotopes
419
topes. In the present results, we have found α-cluster structures in all the 16 O,
18 O and 20 O nuclei. The K π =0+ band of 16,18 O and the K π =0+ band of 20 O that
2
3
have α-cluster structures have almost the same relative energies to the theoretical α+12,14,16 C threshold energy, as shown in Figs. 3, 6 and 9. Contrastingly, the
20 O appears at a much smaller excitation energy than the theoK π =0+
2 band of
14
6
retical C+ He threshold energy, although this band has a large component of the
14 C+6 He cluster structure. We note that the 14 C+6 He wave function [Fig. 8 (c)] has
a large overlap with the 12 C+α+4n wave function [Fig. 8 (b)], and hence the system
has a molecular-orbital-like nature. Therefore, we believe that the valence neutrons
play an important role to lower the energy of the K π =0+
2 band. The presence of the
20
molecular-orbital-like band in O is believed to be related to the weakly bound nature of 16 C and 6 He. Because the last two neutrons in both nuclei are weakly bound,
16 C+α and 14 C+6 He do not appear at small excitation energies. However, when two
neutrons are covalently bound and shared by 14 C and α clusters, the energy of the
system is lowered.
§4.
Summary
We have investigated the cluster structures of 16 O, 18 O and 20 O using the
− bands of
AMD+GCM framework. First, we confirmed that the K π =0+
2 and 0
16 O have the 12 C+α cluster structure. In the case of 18 O, the K π =0+ and 0− bands,
2
which have the 14 C+α cluster structure and can be regarded as parity doublet bands,
were obtained. It is also noted that the 14 C+α cluster structure is fragmented into
many states. Although the description of the low-lying states is insufficient, we have
shown that they can be improved by weakening the spin-orbit force, and the properties of the cluster states are insensitive to the spin-orbit force. We found that the
valence neutrons give richer structure for 20 O. The analysis of the valence neutron
orbitals revealed the presence of cluster structures that exhibit different types of motion of the valence neutrons. The first is the 12 C+α+4n cluster structure, in which
four valence neutrons orbit around entire 12 C+α core. The second is the 14 C+6 He
cluster structure, in which the valence neutrons are localized around the 12 C or the
α cluster. These structures are mixed and form the K π =0+
2 band. This band has
14
6
a much smaller excitation energy than the C+ He threshold energy. The third is
−
the 16 C+α cluster structure, which forms the K π =0+
3 and 0 parity doublet bands.
20
The appearance of a variety of cluster states in O is believed to be related to the
weakly bound nature of the subsystems, 16 C and 6 He. They are not rigid cluster
subunits, because the last two neutrons are weakly bound in both nuclei. As a consequence, the four valence neutrons orbit around the 12 C+α cluster core in different
ways. This leads to the variety of cluster structures.
Acknowledgements
We would like to thank Professor H. Horiuchi for valuable discussion. One of
the authors (N.F.) also thanks Professor K. Saito, Dr. T. Watanabe and members of
the nuclear theory group at the Tokyo University of Science for the encouragement
420
N. Furutachi, M. Kimura, A. Doté, Y. Kanada-En’yo and S. Oryu
and discussions. The numerical calculations were carried out on Altix3700 BX2 at
YITP in Kyoto University.
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