CHINESE JOURNAL OF PHYSICS
VOL. 46 , NO. 3
JUNE 2008
Nature of Excited States of Gadolinium Isotopes
Harun Reşit Yazar1 and Ümit Erdem2
1
Faculty of Art and Science, Nevşehir University, Nevşehir, Turkey
2
Erdemir Çelbor A. Kayseri Yolu 7 .Km Kırıkkale, Turkey
(Received August 23, 2007)
Recent observations of a large number of 0+ states in the heavy deformed regions opened
a new window and emphasized the importance of microscopic approach to atomic nucleus.
A large number of results of a (p,t) experiment revealed 13 excited 0+ states below ∼ 3.2
MeV in 158 Gd [4]. Such data provide a challenge to theory. In this work, excited K= 0+
states in the 158 Gd isotope are studied by using the interacting boson models sd-IBM and
df-IBM. The Sd-IBA and df-IBA models can be related to each other, and the states of the
IBA-1 model can be identified with the fully symmetric states in the sdf-IBA model. It is
the purpose of this work to study this relation and apply it to Gadolinium isotopes. We
show that the interacting boson approximation may account for many of these 0+ states. It
was found that the calculated energy spectra of the Gadolinium isotopes agrees quite well
with the experimental data. The observed B(E2) values were also calculated and compared
with the experimental data.
PACS numbers: 21.10.-k, 21.60.-n
I. INTRODUCTION
The nature of the 0+ states is the most controversial subject in even-even deformed
nuclei. Especially, the structure of the lowest one has become a research field by itself.
Observation of a large number 0+ states in 158 Gd in a recent (p,t) experiment stimulated
new studies in this field. For that reason, it is important to understand the origin of such a
large number of 0+ modes. So far, although many theoretical studies on the 0+ excitations
in deformed nuclei have been made in the last decades, there has not been any study
considering the situation up to ∼ 3 MeV because of the lack of any such data.
Recently the 0+ states in 158 Gd have been studied by many workers. Sun et al. [1]
have studied the 0+ excitations in 158 Gd using the Projected Shell Model in the framework
of the Tamm-Dancoff approximation (TDA). Iudice et al. [2] used the Quasi ParticlePhonon (QPM) model including monopole and quadrupole pairing with a quadrupolequadrupole force term. Making a detailed analysis, they presented the calculations on
the microscopic properties including the energies, the E2 and E0 transitions, and the twonucleon spectroscopic factors with the shell and multiphonon structure of the 0+ states.
Another study has been made in [3] by using the pairing-plus-quadrupole model (PPQ)
including only monopole pairing, and a good description has been given for the distribution
and the nature of the 0+ states.
Recently, a remarkable (p,t) experiment [4] using the Q3D spectrometer at the Uni-
http://PSROC.phys.ntu.edu.tw/cjp
270
c 2008 THE PHYSICAL SOCIETY
OF THE REPUBLIC OF CHINA
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HARUN REŞIT YAZAR AND ÜMIT ERDEM
271
versity of Munich MP tandem accelerator laboratory established the existence of 13 excited
0+ states below ∼ 3.2 MeV in 158 Gd. A few of these were previously known [5, 6] but many
are new. One or two are somewhat tentative. Nevertheless, it is clear that there are a large
number of 0+ excitations at relatively low energies in this deformed nucleus.
The 158 Gd nucleus is at the beginning of the deformation region 150 < A < 190.
The nucleus is a rotor which shows a developed γ-vibrational band. To explain the form
of a nucleus, the binding energy of the nucleus, the transition probabilities between different energy levels, the electric and magnetic multipole moments, the quadrupole moments,
and the rest of the observable quantities must be known properties. The pairing and the
quadrupole forces are important in deformed nuclei. These forces especially influence the
particles in the unfilled states. The pairing force keeps the nucleus in spherical symmetry.
The quadrupole charge distribution causes what is known as the quadrupole force. This
force takes the nucleus to the deformed state. The relation between the pairing and the
quadrupole forces determines the form of the nucleus.
As a result, all the developments mentioned above imply that new microscopic models
and interactions that can give new contributions are necessary. Certainly, various collective
0+ modes can exist and it is clearly of interest to determine how many such excitations
appear below ∼3 MeV. In this paper, in order to study the nature of the 0+ excitations in
the 158 Gd isotope and investigate the general behaviour of the quantities that characterize
them, we use the model Hamiltonian including sd- and df- boson terms in the framework
of the Interacting Boson Approximation (IBA).
In this paper, we considered the 158 Gd isotope in order to study the nature of the 0+
excitations and analyze the general behaviour of the quantities and characterize it. And we
also calculate the energy levels and the electric quadrupole transition probabilities B(E2;Ii
→If) of 154 Gd and 158 Gd isotopes in terms of the sd-df boson Hamiltonian.
II. THE INTERACTING BOSON MODEL AND ENERGY LEVELS
The empirical 0+ states and theoretical results of 158 Gd are shown in Figure 1 and
the empirical states and theoretical results of the energy states of 154 Gd and 158 Gd are
given in Figure 2(a),(b). It is known that the sd-boson IBM can account only for about five
or six excited 0+ states below ∼ 3.2 MeV but that the inclusion of the octupole degree of
freedom allows one to predict, perhaps unexpectedly, nearly eleven excited 0+ states below
3.2 MeV and about 14 below ∼ 4 MeV.
The calculations were done in a simple and straightforward way. No attempt was
made to fit individual 0+ states, nor is any claim made that the specific predicted 0+
states have a correspondence to specific empirical states. The point of the calculations was
rather to see the number of 0+ excitations in the energy range up to ∼ 3–4 MeV. The IBM
NATURE OF EXCITED STATES OF GADOLINIUM . . .
272
VOL. 46
FIG. 1: Distribution of the calculated and experimental (0+ states) energies of
158
Gd.
calculations were similar. First of all we express the sd-boson Hamiltonian as follows [7].
ε0 n
1
2 η (L.L)
1
2 κ (Q.Q)
Hsd =
+
d+
(4) (0)
(4) ,
+15ξ d+ d˜
x d+ d˜
(3) (3) (0)
√
+
+
˜
˜
x d d
− 5 7ω d d
0
(1)
0
where
√ + (1) + (13) (0)
˜
˜
L.L = −10 3 d d
x d d
(2)
0
and
Q.Q =
√
5
+˜
˜+
s d + ds
(2)
(2) (0)
(2)
(2) +
+˜
+˜
˜
˜
. (3)
x
+χ d d
s d + ds
+χ d d
+
0
The parameters were fit by reproducing the deformed nature of this nucleus (e.g.,
the ground state rotational band) and the properties of the γ-vibration. The following
parameters resulted: ε0 = 0.308 MeV, ω = 0.001 MeV, η = 0.0155 MeV, κ = −0.02 MeV,
χ = −0.910, and ξ = 0.0001 MeV. These parameters are similar to those found for other
nuclei in this region. Again, the model is capable only of accounting for relatively few of
the now known 0+ states. The simple df-boson Hamiltonian is [7]
(4)
Hdf = η 0 (Ld .Lf ) + κ0 (Qd .Qf ) − χ0 O(3) .O(3) ,
VOL. 46
HARUN REŞIT YAZAR AND ÜMIT ERDEM
273
FIG. 2: Comparison of the calculated and experimental energy spectra of (a)
154
Gd, (b)
158
Gd.
where
(1) (1) (0)
√
+˜
+˜
Ld .Lf = −2 210 d d
x f f
,
(5)
0
√
Qd .Qf = −2 35
(2)
(2) (2) (0)
+
+˜
+˜
+˜
s d+d s
−χ d d
x f f
,
(6)
0
and
(3)
(3)
+ χ d+ f˜ + f + d˜
O(3) = s+ f˜ + f + s
(7)
was used, where η 0 = 0.0155 MeV, κ0 = −0.02 MeV, and χ0 = −0.97 are the strengths of
the L.L-force, the Q.Q-force, and the octupole force, respectively. Therefore, we carried
out further IBM calculations with the help of the sd-boson Hamiltonian and df-boson
Hamiltonian. The simple Hamiltonian (in terms of the sd-df bosons) is,
H = Hsd + Hdf ,
(8)
274
NATURE OF EXCITED STATES OF GADOLINIUM . . .
VOL. 46
and the quadrupole operator is
Qsdf = Qsd + Qdf .
(9)
The parameters chosen, ε0 , ω, η, and ξ, were the same as in the sd-IBM calculation.
Calculations with the boson energy ε0 = 0.308 MeV reproduce rather well the experimental
level states, as seen in Fig. 1, which shows the predicted 0+ states below ∼ 3.2 MeV.
We made no attempt to fine tune the calculations to the empirical 0+ states (there is
insufficient data on the detailed structure of these states to accurately fix the parameter of
the full Hamiltonian), there is no point in invoking a precise energy cut-off. Therefore, it
is also appropriate to look slightly above 3.2 MeV where there is a continuing spectrum of
0+ states, amounting to 14 excited 0+ states below 4.0 MeV. Given that 158 Gd is typical of
many rare earth and actinide deformed nuclei, one can expect that similar numbers of 0+
excitations appear throughout the deformed regions of nuclei. It is therefore an important
issue to understand the origin of such a large number of Kπ = 0+ modes
III. ELECTROMAGNETIC TRANSITION PROBABILITIES
A successful nuclear model must yield a good description not only of the energy
spectrum of the nucleus but also of its electromagnetic properties. The most important
electromagnetic features are the E2 transitions. The B(E2) values were calculated by using
the E2 operator. The E2 transition operator must be a hermitian tensor of rank two and
therefore the number of bosons must be conserved. Since, with these constraints the general
E2 operator can be written as [7]
Tm (E2) = eπ Qπ + eν Qν ,
(10)
ρ
+ ˜ (2)
˜ (2)
+ χρ [d+
Qρ = [d+
ρ dρ ] ,
ρ s + sρ dρ ]
(11)
where ρ corresponds to the π(proton) or ν(neutron) bosons, and χρ determines the structure
of the quadrupole operator, and is determined empirically. Here Qρ are the Qπ and Qν
boson quadrupole operators and eπ and eν are the “effective charges” for the proton bosons
and the neutron bosons. For simplicity the “effective charges”, eπ and eν , were taken as
equal. The B(E2) strength for the E2 transitions is given by
B(E2; Li → Lf ) = 1/ (2L1 + 1)1/2 |hLf kT (E2)k Li i |2 .
(12)
Some calculated B(E2) values from the ground state band and B(E2) ratios are given in
Table 1.
Since the gadolinium nucleus has a rather vibrational character, we used the multiple
expansion form of the Hamiltonian for our approximation, taking into account the dynamic
symmetry location of the even-even gadolinium nuclei at the IBM phase triangle where
their parameter sets are at the O(6)-U(5) transition region and closer to U(5) vibrational
VOL. 46
HARUN REŞIT YAZAR AND ÜMIT ERDEM
TABLE I: B(E2;I → I − 2) values for the ground state bands of the
N
90
94
B(E2) ( in e2 b2 )
41 → 21
21 → 01
[8]
Theory Exp.
Theory Exp.[8]
1.17 1.062
0.77
0.78
1.55
1.01 1.095
154
275
Gd and
158
Gd isotopes.
B(E2) ratios
(41 → 21 ) / (21 → 01 )
Theory
Exp.[8]
1.519
1.361
1.534
character and possessing good vibrational states. In order to find the value of the effective
charge we have fitted the calculated absolute strengths B(E2) of the transitions within the
ground state band to the experimental ones. The best agreement is obtained with the value
eπ = eν = e = 0.14 eb [10], as shown in Table 1. The B(E2) values depend quite sensitively
on the wave functions, which suggest that the wave functions obtained in this work are
reliable.
In Table 1, the B(E2: Kπ = 0+ → 2+
gsb ) transition probabilities have been presented
and compared with experiments. The theoretical E2 decays of the 0+ states to the 2+
ground state are all weak except for the first two. The present model gives the best values
for the first state and also the second state. While the first state is in good agreement with
the data, a relatively large B(E2) value can be obtained for the second state.
IV. RESULTS AND DISCUSSIONS
The even-even Gadolinium isotopes have been described by the sd-df IBA Hamiltonian. The single d-boson energies were determined from the experimental data — the
0–2 spacing in the appropriate semi-magic nuclei, where this data is known. It was found
that the Hamiltonian yields a good description of the energy levels of the 154 GD and 158 Gd
isotopes.
The present work demonstrates that the sd-df IBA Hamiltonian parameters based on
the IBA-1 model gave good results for the excitation energies and the electric quadrupole
transition probability B(E2; Ii →If ) of the 154 Gd and 158 Gd isotopes. For the non-fullysymmetric states, we renormalized the parameters (ε and κ) and obtained good results. In
the present calculations, we have shown the ability of the projection in correlating different
properties in Gadolinium isotopes in terms of a few parameters.
Recent observations of a large number of 0+ states in the heavy deformed regions
opened a new window and emphasized the importance of the microscopic approach to the
atomic nucleus. In fact studies in this field imply the need for new approximations based
on the microscopic approach.
In Fig. 1, we have marked with a circle those 0+ states having a double octupole
character, i.e., with the expectation number of f bosons, nf ∼ 2. Clearly there are large
NATURE OF EXCITED STATES OF GADOLINIUM . . .
276
VOL. 46
numbers of 0+ states at relatively low energy with dominant predicted two-phonon octupole
character. As stressed, there is no assurance that these correspond to specific empirical 0+
states. Nevertheless, the results highlight the importance of including the octupole degree
of freedom if one is studying deformed nuclei above the pairing gap. We interpret these
results as being an existence proof that models including the octupole degree of freedom
are able to predict a large number of relatively low-lying 0+ states in accord with recent
experiments and theoretical work [3, 4].
The observed γ-decay 0+ states [5, 6] support the concept of widespread octupole
character. Every 0+ state in 158 Gd below 2 MeV, whose γ-decay is known, deexcites by E1
transitions to lower lying negative parity states. This is consistent with a strong two-phonon
octupole character in a number of these 0+ states. Recent measurements [2] of the lifetimes
+
of almost all of the 0+ states show that the B(E2;0+
i → 21 ) values are at most a few W.u.
Combined with the decay to octupole modes, this is consistent with a double octupole
phonon character of some of them. These results, along with further measurements of the
γ ray branches from higher lying 0+ states, will allow one to define clearly the parameters
of the Hamiltonian and the electromagnetic transition operators.
Experimental B(E2) values for transitions between positive parity states are compared
in Table 1 with our IBM results, which were obtained from the program PHINT [9] by using
the wave functions to fit the energy levels as described in Section 3.
The B(E2) values depend quite sensitively on the wave functions, which suggests that
the wave functions obtained in this work are reliable. This method of approximation may
be applied to many other even-even nuclei and many other nuclear properties [10].
In conclusion, we carried our further IBM calculations with the help of the sd-boson
Hamiltonian and df-boson Hamiltonian, and we try to show that our specific (sd)(df) IBM
model is more efficient than the (sdpf) IBM model [11] and the calculated energies of 0+
states are closer to the experiment data.
Acknowledgements
This work was supported by the Research Foundation of Kırıkkale University and the
Scientific and Technological Research Council of Turkey under the grant numbers BAP2007-36 and TUBITAK-107T557.
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