The degree of freedom of nuclear triaxial deformation
and K-isomeric decays

A Thesis presented
byNaoki Tajima
tothe Division of Scienceofthe Graduate Schoolforthe Degree of Doctor of Science

Institute of Physics, University of Tokyo

December, 1988

.

Contents
1 Introduction

32 Treatment of the γ-degree of freedom

9

2.1

Similarity of the γ-vibrational model and the rigid triaxial rotor model . .

2.2

Diﬀerence of the two models in energy levels . . . . . . . . . . . . . . . . . 12

2.3

Direct investigation of wave functions . . . . . . . . . . . . . . . . . . . . . 14

3 The K-selection rule and K-isomers

9

17

3.1

The K-selection rule and its violation . . . . . . . . . . . . . . . . . . . . . 17

3.2

K-isomers in Hf-W-Os region . . . . . . . . . . . . . . . . . . . . . . . . . 20

4 The model
4.1

4.2

25

The Bohr model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
4.1.1

The full hamiltonian . . . . . . . . . . . . . . . . . . . . . . . . . . 25

4.1.2

The γ-soft model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

4.1.3

The triaxial rotor model . . . . . . . . . . . . . . . . . . . . . . . . 30

4.1.4

The axially symmetric rotor model . . . . . . . . . . . . . . . . . . 30

The particle-rotor model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
4.2.1

Derivation from the shell-model hamiltonian . . . . . . . . . . . . . 31

4.2.2

Wave functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

4.2.3

The Coriolis interaction in the laboratory-frame model . . . . . . . 39

4.2.4

Advantages of the description in the laboratory frame . . . . . . . . 40

4.2.5

Electromagnetic transitions . . . . . . . . . . . . . . . . . . . . . . 41

1

5 Details of the numerical calculation

46

5.1

Parameters of the model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

5.2

Truncation of states and its convergence . . . . . . . . . . . . . . . . . . . 48

5.3

5.2.1

Full inclusion of the particle states . . . . . . . . . . . . . . . . . . 48

5.2.2

Truncation of the core states . . . . . . . . . . . . . . . . . . . . . . 48

Convergence in the Lanczos algorithm for the Diagonalization . . . . . . . 50

6 Results of numerical calculations

57

6.1

Features of axially symmetric rotor ⊗ ν i13/2 system . . . . . . . . . . . . . 57

6.2

Eﬀects of the γ-degree of freedom . . . . . . . . . . . . . . . . . . . . . . . 59

6.3

Eﬀects of the coupling with the particles on the core

6.4

Eﬀects of the potential shape . . . . . . . . . . . . . . . . . . . . . . . . . 63

6.5

Further investigation of the fast K-isomeric decays due to the γ-softness . 66

6.6

Eﬀects of the shape of the γ-potential on energy levels . . . . . . . . . . . 68

7 Summary. . . . . . . . . . . . 61

88

2

Chapter 1
Introduction
As for the description of the degree of freedom of axially asymmetric deformation in
the nucleus (the γ degree of freedom), there are two extreme models having apparently
diﬀerent features. One is the γ-vibrational model of Bohr and Mottelson [1], in which a
nucleus vibrates around an axially symmetric shape. The other is the triaxial rotor model
given by Davydov and Filippov [2], in which a nucleus has a static axially asymmetricdeformation and rotates around all three principal axes of the deformation. In the former
model γ is considered as a dynamical variable, while in the latter model it is treated asa parameter to specify a nuclear shape. There have been a wide variety of arguments
concerning the two models for thirty years. A brief review is presented in chapter 2.
The two models give roughly similar results for most of the observables (like some energy levels and collective transitions) despite the obvious distinction in their formulations[3]. It seems due to the fact that these observables are determined by some average valueof γ and scarcely dependent on the size of the fluctuation in γ : The triaxial rotor model
can be obtained by neglecting the fluctuation in γ and fixing γ at the average value.
It is interesting to find quantities which bring about the information not only on
the average value of γ but also on the fluctuation in γ. An existing method to see
the fluctuation in γ is a systematic determination of the quadrupole matrix elements in
Coulomb excitation experiments [4] [5] . In this paper we present a new example of such
quantity, i. e. the degree of violation of the K-selection rule in K-isomerism, which isfound very sensitive to the size of the fluctuation in γ. As a result, we can distinguishthe two extreme models of axial asymmetry very clearly. In this paper, we investigate

3the eﬀects of the fluctuation in γ by changing the shape of the potential energy for γdeformation and show that the potential shape can be known to some extent from thehalf-lives of K-isomers.
In electromagnetic transitions between states having definite K-quantum numbers1 ,not only the change in the total angular momentum but also the change in the K-quantum
number cannot exceed the multipolarity of the radiation (the K-selection rule). The Kquantum number is conserved fairly well in well-deformed nuclei with axial symmetry.
Hence states which have large expectation values of the K-quantum number are likely to
become long-lived isomers when they can decay only into states with small expectation
values of K. More detailed explanations are given in chapter 3.
Experimentally, very roughly speaking, the probabilities of K-forbidden transitions
are hindered by a factor of 102 for each one increment of the K-quantum number [6] [7][8]. An extreme example of K-isomer is a K = 16 state in 178 Hf, which has a half-life aslong as 31 years [9] [10].
But recently some high-K isomers in neutron deficient Os isotopes are observed decaying with much shorter half-lives than those predicted by the above-mentioned empiricalrule [11] [12] [13]. The most remarkable one is a K π = 25+ isomer in 182 Os, which decays(in part) directly into the I=24 member of the s-band [13] [14] quickly in spite of thediﬀerence of the K-quantum number of ∼ 20 (see the footnote 2 ). The configuration of
the isomer is speculated to be of six-quasi particle state [13] [15].
In this thesis we study the K π = 10+ isomers in 182 W and 184 Os which have a very
simple configuration, i. e. a two-quasi-particle (ν[624]9/2+ , ν[615]11/2+ ) state involving
only a unique-parity (intruder) orbital (νi13/2 ,ν means neutrons). (Mixing ratio of otherconfiguration (π[514]9/2− , π[505]11/2− ) might be large in the case of 184 Os from theg-factor measurement [13] but we assume only the above-mentioned configuration forboth nuclei in this paper.) Illustrating this simple configuration, we can concentrate
our attention on the study of the γ-degree of freedom, since we need not worry about
1

The K-quantum number is defined as the component of the total angular momentum along the thirdprincipal axis of nuclear shape. This axis is conventionally the symmetry axis in prolate shape.
2
The expectation value of K is not ∼ 0 in the s-band states, because K is rather widely spread forrotationally aligned orbitals.

4

ambiguities concerning the complicated microscopic structure.
The K π = 10+ isomer observed in 184 Os [13] decays into the I = 8 ( 8+gr ) and theI = 10 ( 10+gr ) members of the ground band with a partial half-life of 180 ns and 360 ns
respectively 3 . The partial half-life with these two transitions is 120 ns 4 . In an isotone
182

W of N =108 having Z smaller by 2, the isomer which is considered to have the same+
configuration decays with a half-life of 1.4 µs into the 8+
gr state and the 10gr state [16].

The half-life is about 12 times as long as the partial half-life of the isomer in 184 Os withthese two transitions, although the transition energies are very similar in both nuclei:
In 182 W (184 Os ) the energy of the γ-ray of the transition from the isomer to the 10+
grstate is 0.519 MeV (0.495 MeV) and that to the 8+
gr state is 1.087 MeV (1.092 MeV).
It should be noted that even the isomer in 182 W has a much shorter half-life than thehalf-life predicted by the above-mentioned empirical rule. The decay properties of thetwo isomers are summarized in fig. 3.2.
Chowdhury and his co-workers suggested that the diﬀerence in the half-lives can be
attributed to the diﬀerence of the softness with respect to γ-deformation (γ-softness),because γ-deformation mixes the K-quantum number and the K-selection rule breaks
down. Nuclides like 182 W and 184 Os are classified into transitional nuclei, which are locatedbetween the well-deformed rotational nuclei and the spherical vibrational nuclei on thechart of the nuclides. They are less deformed and more γ-soft than typical rotational
nuclei in the rare earth region. The degree of γ-softness can be estimated from the ratioof the excitation energy of the I = 2 state of the γ-band ( γ-bandhead or 2+γ ) to that of182
the I = 2 state of the ground band ( 2+W while it is 7.87 in
gr ). The ratio is 12.2 in
184

Os (fig. 3.1). Therefore it is quite natural to consider that the nucleus 184 Os is more

γ-soft than the nucleus 182 W.
These clear violations of the K-selection rule seem so strange that they are expected
to inform us new aspects of nuclei concerning the γ-degree of freedom. Some explanations
for these fast K-isomeric decays are presented, in which the γ-degree of freedom plays
3

The former partial half-life is explicitly given in ref. [13]. The latter one is estimated from theintensities of γ-rays given in ref. [13].
4
The total half-life of the isomer is 20 ns. It decays with a large portion into an I = 9 state whose
configuration is not identified.

5

the leading role. At first, in the papers [11] [12] reporting the observation of the K π =25+ isomer in 182 Os, Pedersen and his collaborators suggested the coexistence of axiallysymmetric shape (for the ground-band states and the isomer) and triaxial shape (forthe s-band states). They speculated that the polarizing eﬀect of the rotation-aligned
particles may triaxialize the nuclear shape if the nucleus is very γ-soft. If it is so, many
triaxial-rotation levels having various expectation values of the K-quantum number are
created just above the s-band. Passing through these states as stepping stones, the high-K
isomer can be de-excited quickly by repeating transitions with small ∆K. The observed
complicated decay pattern of the isomer is consistent with this picture: The isomer decays
by the emission of as many as 100-150 γ-rays [17] and the s-band states are populated in ascattered manner. If such three-dimension rotational levels of a triaxial nucleus are reallyobserved, a great impact will be made especially on the theories of nuclear collective
motions [18]. But no such triaxial-rotation levels are found on the decay paths in thesubsequent detailed experiment [13].
Onishi and the author studied the K π = 25+ isomer from a diﬀerent point of view[14]. We assumed that the isomer had an angular-momentum-aligned configuration of two
protons and two neutrons in the intruder orbitals 5 , with emphasis on the importance of
the interaction between a proton and a neutron. Using a model in which h11/2 protons
and i13/2 neutrons are coupled to a (γ-soft) rotor, we calculated the de-excitation pathsof the isomer and found that these paths go through many excited two-particle-rotationaligned bands before reaching the s-band states. This work pointed out the possibility ofobserving excited rotation-aligned bands by utilizing the delayed decay of isomers lying
much higher than the yrast level, although the configuration did not turn out to be theassumed one [13] [15].
Chowdhury and his coworkers presented a new picture to explain the fast decays of
the K-isomers in the next paper [13], in which the detailed decay paths of the K π = 25+
isomer in 182 Os and the finding of the K π = 10+ isomer in 184 Os are reported. If thenuclear shape is very soft toward γ-deformation, large quantum fluctuation in γ can mix

5

When we published the work, the spin of the isomer was not assigned yet and was speculated to be(20 ± 2)h̄ in refs. [11] [12].

6the K-isomeric state and the s-band (or the ground-band) state via barrier penetration
through the oblate shape in spite of the large diﬀerence in their structures: The angularmomentum is aligned with the symmetry axis in the K-isomer, while it is perpendicular
to the symmetry axis in the s-band (or the ground-band) state, though equilibrium shapesof the two states are the same ( γ=0◦ ). Calculation of barrier heights and penetration
factors (in the framework of the cranking model) are proposed for the future work in thepaper. T. Bengtsson is attempting to make such calculations [15].
It is uncertain, however, whether the above simplified picture can be applied to real
nuclei. As for the quantitative estimation of penetration factors, it is much more uncertain.
Since the size of deformation is not large (β ∼ 0.2) in transitional nuclei, the assumptionof adiabaticity for the intrinsic motions does not seem to be a good approximation. The
motions of particles may be disturbed considerably by the quantum fluctuation in therotational motion and the shape deformation. In order to settle the problems of theseambiguous points in the semiclassical picture, three kinds of motions — the rotation, theshape deformation, and the intrinsic particle motions — should be treated on an equal
footing in a quantum mechanical way. Only in that case, we can be sure that quantitative
estimations are made.
In this paper we are going to make this fully quantum mechanical calculation in order
to examine how the half-lives are altered by introducing the γ-degree of freedom, withother eﬀects taken into account quantitatively as well. It is expected that we can obtain
new information about the axial asymmetry — such as the size of the fluctuation in γ —from decays of K-isomers.
As for the framework of the calculation, we use the particle-rotor model rather thanthe cranking model. The former is a fully quantal model and conserves the angularmomentum, while the latter is a semiclassical model in which states are not eigenvectorsof angular momentum. The latter model is not suitable for our purpose because it suﬀers
from spurious mixing of states having diﬀerent angular momentum and also because it
cannot take into account the eﬀect of the fluctuation of the rotation axis. A unique pointof our approach is that we describe the particle-rotor model in the laboratory frame using
spherical bases for the particles, not in the intrinsic frame of the nuclear shape using the7

Nilsson bases. This formulation of the particle-rotor model is quite suitable for treating
the large-amplitude motion in the nuclear shape, i. e. for treating the oblate shape as wellas the prolate shape in a single model space.
Now let us mention the contents of the following parts of this paper. In chapter 2 thesimilarity and the diﬀerence between the γ-vibrational model and the triaxial rotor model
are reviewed and discussed. In chapter 3 the general empirical facts concerning K-isomers
are reviewed. In chapter 4 the framework of our calculations are given. The formulation of
the quantal liquid-drop model of Bohr [1] is reviewed. We employ the model so as to treat
the collective motions. Next, the particle-rotor model is re-formulated in the laboratory
frame in a somewhat original way. The relation of the model to the strong-coupling
particle-rotor model is also discussed.
In chapter 5 the procedures of numerical calculations are explained in detail. We haveto use accurate wave functions in order to calculate decay amplitudes of isomers, in whichsmall components of wave functions can have large eﬀects. Thus we take the utmost carein order to obtain the accurate wave functions of the eigenstates of our model hamiltonian.
For example, full (νi13/2 )n (n=0,2,4,· · ·,14) configurations are taken into account for theparticle space. Moreover, special cares are taken for the truncation of the highly-excitedγ-vibrational states. It is confirmed that this truncation has no influence on the quantities
concerning the decay of the K-isomers.
In chapter 6 the results of numerical calculations are presented and are discussed. We
summarize this paper in chapter 7.

8Chapter 2
Treatment of the γ-degree of freedom
2.1

Similarity of the γ-vibrational model and the rigid
triaxial rotor model

According to microscopic calculations, equilibrium shapes of well-deformed nuclei are
axially symmetric and prolate (γ=0◦ ). For instance, Baranger and Kumar applied thepairing-plus-quadrupole model to even-even nuclei with N =82-126, Z=50-82 and foundthat the equilibrium shapes are spherical or axially symmetric prolate except for Z=76-80(Os, Pt, Hg) [19]. In the region of Z=76-80, where the deformation disappears graduallyas Z and/or N increase, some nuclides are predicted to have axially asymmetric (in other
word, triaxial) or oblate equilibrium shapes. The nuclei treated in this paper — 182 W
and 184 Os — are predicted to have axially symmetric prolate equilibrium shapes.
A basic model was proposed by A. Bohr to treat the quadrupole surface-vibrations
and rotations (the Bohr model) [1]. In this paper we treat collective motions of thenucleus only in the framework of the Bohr model for simplicity, though there is anothersuccessful model for nuclear collective motions, i. e. the interacting boson model of Arimaand Iachello [20]. In the work [1], Bohr and Mottelson supposed that the γ-degree of
freedom, which expresses the deviation from the axially symmetric shape, manifests itself
as the vibration of nuclear shape around axially symmetric shape. It is widely acceptedthat this γ-vibration is associated with the γ-band, i. e. the rather low-lying K π = 2+band which occurs in most deformed even-even nuclei and which has large E2 matrix
elements for its excitation.
Several years later, however, Davydov and Filippov introduced an alternative model
9

which also includes a motion associated with the γ-degree of freedom [2]. It was thetriaxial rotor model, in which the nucleus is assumed to have a rigid 1 axially asymmetricshape. The triaxial nucleus can rotate around the 3-axis (the third principal axis of
the nuclear quadrupole deformation), while the axially symmetric nucleus can not rotate
around the axis because it is symmetric around the axis. The γ-band is associated withthe excitation of rotation around the 3-axis in this model. Many experimental data likeenergy levels and transition amplitudes in the ground and the γ-band of even-even nuclei
are explained rather well in the framework of this model. The model was extended inorder to take into account the β-vibration (the nucleus is still stiﬀ in the direction of γ )and the β-band was reproduced successfully [21].
Considering the shallowness of microscopically calculated energy surfaces for γ-deformation,
it seems that the nucleus can not have such stable rigid triaxial shape. According to Bohr’s
model, the amplitude of zero-point motion in the variable γ is as large as the size of static
γ-deformation assumed in Davydov’s calculation. Bohr expressed sharp criticism to thetriaxial rotor model: The rigid triaxial deformation does not hold in the real nucleus since
the concept of deformation can be applied only if the average value is greater than thefluctuation [22]. Moreover it was pointed out by many authors that many features reproduced by the triaxial rotor model are also accounted for equivalently by the conventional
γ-vibrational model [23] [24] [25].
At first sight, it is rather strange that the predictions of the two models are so similarin spite of the diﬀerence in their pictures. In the γ-vibrational model, the variable γ
fluctuates around γ=0◦ , while in the triaxial rotor model, γ is fixed at a constant value
γ0 where γ0 > 0◦ .
We can demonstrate the origin of this similarity through the examination of the matrix
elements of functions of γ in the subspace of low excitations as for the γ-degree of freedom.
This point of view was mentioned by Preston in the Kingston Conference [22], and an
explicit explanation was given by Yamazaki [26]. When |γ| ≪ 1 in the γ-vibrational modelwith a harmonic potential, we can restrict the motion in γ to the ground state with K=0

1

The word “rigid” means that the shape of the surface is stiﬀ against deformations, not that thenucleus is regarded as a rigid body.

10

(|K=01 ⟩ 2 ) and the first excited state with K=2 (|K=21 ⟩) to a good approximation.
For |γ| ≪ 1, γ can be regarded as the radial variable for a two-dimensional motion of aparticle, where the azimuthal angle is expressed by 2θ3 , where θ3 is one of the Euler angles
representing the rotation around the 3-axis [1]. When we consider terms up to O(γ 2 ),there are only three non-vanishing matrix elements on account of this interplay between
γ and the rotation around the 3-axis,⟨K = 01 | γ |K = 21 ⟩ = γ00 ,

(2.1)

2
⟨K = 01 |γ 2 |K = 01 ⟩ = γ00
,

(2.2)

2
,
⟨K = 21 |γ 2 |K = 21 ⟩ = 2γ00

(2.3)

where γ00 is the amplitude of zero-point motion in the variable γ in the ground state.
Therefore we can replace the dynamical variable γ by a static parameter γ00 to an accuracyof O(γ). Through this replacement, the γ-vibrational model is reduced into the triaxial
rotor model with γ0 =γ00 . Although the excitation energy of γ-vibration is twice as large
as the energy of rotation around the 3-axis in rigid triaxial rotor model with γ0 =γ00 , it
can be said that gross features of the similarity are accounted for by this argument. Hence
we do not have to conclude the rigidity of triaxial shape even if the triaxial rotor model
can reproduce many features of the states with zero or one γ-vibrational phonon.
The rigid triaxiality is revived by Meyer-ter-Vehn [27]. He applied the triaxial-rotorplus-particle model to some transitional odd-A nuclei (A=135,190). As a result, he foundthat the experimental energy spectrum of the odd-A nuclei are reproduced fairy well by
this model. To his surprise, the value of static γ-deformation (γ0 ) can be determinedwithin ±2◦ to reproduce the energy spectra of an odd-A nucleus and the adjacent evennuclei. Considering the strong γ-driving eﬀects of the last odd particle, the core seems
to have much deeper potential well around γ0 than the soft potentials predicted by microscopic calculations [19]. He suggested the existence of rigid triaxial shape for thesetransitional nuclei. Toki and Faessler drew the same conclusion using an improved modelwith a variable moment of inertia [28].
Against Meyer-ter-Vehn’s suggestion, Yamazaki and his collaborators argued that the2

The subscript 1 means that the state is the lowest one having the quantum number K=0.

11

coupling of the particle motion to the γ-degree of freedom is also equivalent between
the two models as far as |γ| is small [29]. The coupling is expressed by a quadrupolequadrupole interaction and the quadrupole matrix elements are already shown to beequivalent. Now it is commonly believed that the odd particle is not a probe sensitive
enough to identify triaxial deformation [30] [31], at least for weakly deformed nuclei.
As for the microscopic calculation of the potential energy surface for deformation,
Hayashi and his collaborators discussed the eﬀects of three-dimensional angular momentum projection on the Hartree-Fock-Bogoliubov ground states [32]. Angular momentum
projection has large eﬀects in the case of weak symmetry violation like in the case ofγ-deformation of transitional nuclei. They calculated the microscopic potentials and obtained the wave functions in the β-γ plane for such potentials. They found for 188 Os that,while the unprojected wave function is γ-unstable, the projected wave function rather
localizes in the region 20◦ < γ < 40◦ . But the value of the wave function at γ = 0◦ is still
half as large as the value at the peak (γ ∼ 30◦ ). Hence we cannot say that the triaxial
deformation is realized in 188 Os.
In the real nucleus, which is composed of finite number of nucleons, the concept of the
shape of the surface is more or less ambiguous. Recently Otsuka and Sugita pointed out
in terms of the interacting boson model (IBM) that the extreme limits of the two models— the γ-unstable model (i. e. the O(6) limit of IBM) and the rigid triaxial model with
γ=30◦ (a coherent state of γ=30◦ deformed boson with angular momentum projection) —have equivalent low-lying states when the number of bosons is small enough [33]. This
result is interpreted that when the number of degrees of freedom is small enough, they
are exhausted in order to define the principal-axis frame of the shape and more specificdetermination of the shape — like determination of the value of γ — is impossible on
account of the lack of available degrees of freedom.

2.2

Diﬀerence of the two models in energy levels

As we have just seen in the last section, the similarity of the two models originates in thefact that we look at only γ00 ( the root mean square value of γ in the ground-band states).
Therefore if we have a quantity that can distinguish the two models, we can obtain more
12

information on the motion in γ than γ00 .
Although the two descriptions of the triaxiality resemble each other at low-lying states,
higher excited states can be utilized to distinguish the two models. For example, theenergy splitting of the doublet of the double-γ-vibrational states ( K=02 and K=41 )can be used to investigate the shape of the potential for γ-deformation [34] [35]. If thepotential is harmonic around γ=0◦ , the two double-γ-vibrational states are degenerate.
As the potential minimum is shifted from zero to positive values of γ, the K=41 level
goes downward while the K=02 level goes upward. In the limit of rigid triaxial rotor, theK=02 level disappears completely. The K=02 double-γ-vibrational state has not beenfound yet.
From a diﬀerent viewpoint, the situation can be expressed that the energy ratio of
the K=41 state to the K=21 state is 2 in the case of harmonic potential, but it is 4 inthe case of the rigid triaxial rotor. For example, Yamazaki and his coworkers pointed outthat the energy of the K=j+4 band in Ir isotopes, in which the odd particle with spinj (=Ω=11/2) is confirmed to be coupled to the K=41 double γ-vibrational state, is too
low to be consistent with the prediction of the triaxial-rotor-plus-particle model [29].
The spectrum of the (K=21 ) γ-band can be also utilized in order to distinguish thetwo models, though the physical meaning of the phenomena is less transparent than in
the case of the double-γ-vibrational states. In the γ-vibrational model, the spectrum
show a (3+ ,4+ ), (5+ ,6+ ), · · · bunching pattern, which is opposite to the (4+ ,5+ ), (6+ ,7+ ),· · · clustering pattern predicted in the triaxial rotor model [36] [37]. In the transitional
nuclei, the spectrum clearly shows the former bunching pattern and these nuclei seemrather γ-vibrational than rigid triaxial.
It is, however, worth while noting that energy levels of these highly excited states
can be shifted easily because the states tend to couple with quasi-particle excitations.
Hence more direct investigation of the wave functions are desirable in order to distinguishbetween the γ-vibration and the rotation of rigid triaxial nucleus.

13

2.3

Direct investigation of wave functions

Now let us consider how we can investigate the shape of the wave function for γ in more
direct manners. Though there are many detailed investigations of the transition matrix
elements, we do not describe them here. In stead, we give more intuitive considerations.
In the part (a) of figure 2.1 we show the schematic shape of wave functions for γ-vibration
around γ=0◦ and for stiﬀ triaxial deformation. Two regions of γ are missing in the latterwave function, i. e. the region A (γ < γ0 ) and the region B (γ > γ0 ). The crucial point
to decide if the deformation is triaxial or not is whether the wave function is vanishing or
not in the region A.
The information on the wave function for γ is not included in M1 transition amplitudes, because M1 operator is only the total angular momentum operator multiplied bya constant factor and does not concern to the intrinsic shape.
By E2 transitions, we can only obtain some mean deviation in γ from γ=0◦ and can notinvestigate the more detailed shape of wave functions, because the E2 operator depends
on γ only through cos γ and sin γ,
3ZeR02M(E2, µ) =
α2µ
4π
{
}
2
2
+ Dµ−2
Dµ2
3ZeR022√=
β cos γ Dµ0 + sin γ
.
4π
2

(2.4)
(2.5)

Details of the above equations are explained in chapter 4 . As is illustrated in the part(b) of figure 2.1, the E2 operator changes much slowly compared to the wave functionsfor γ.
Therefore, to learn the shape of wave functions in detail, we must use operators whose
variation in the interval 0 ≤ γ ≤ γ0 is very large. Cline and his collaborators developed
a computer code to analyze complicated data of Coulomb excitation experiment [4] [5].
Using this code, they can derive the matrix elements of not only cos γ and sin γ but also
more complicated functions of γ like cos 3γ and cos2 3γ from experiments . These functions
are sketched in the part (c) of figure 2.1. Thus the shape of wave functions for γ canbe known to some extent by this method. It is, however, worth while noting that when
we make inner products of these wave functions, we have to include the volume element14

| sin 3γ| (sketched in the part (d) of figure 2.1 ), which diminishes the contribution fromthe regions near γ=0◦ and γ=60◦ . Hence it is rather diﬃcult to examine the shape of the
wave function in the region A.
For another example, some particle transfer reactions may be employed to investigate
the shape of wave functions [38]. Cross sections of particle transfer are sensitive to single
particle levels near the Fermi level, while these single particle levels are changed largelyby the nuclear shape. Reactions which depend on γ very sensitively can teach us the
distribution in γ if deformation β is so large that particles are very adiabatically coupled
to the nuclear shape.
In this paper we show that K-isomeric decays can be used as very sensitive probes
in order to obtain information about the wave functions in the region B (γ > γ0 ), andwe investigate the mechanism associating the fluctuation in γ into γ > γ0 with the Kisomeric decays. In the next chapter, we discuss about the K-isomerism and the recent
experiments in which flagrant violations of the K-selection rule are observed.

15

Figure 2.1: Schematic pictures of the wave functions for γ in the γ-vibrational model andthe triaxial rotor model. Some functions of γ are also shown.

16

Chapter 3
The K-selection rule and K-isomers
3.1

The K-selection rule and its violation

In axially symmetrically well-deformed nuclei the component of the total angular momentum along the symmetry axis (K) is conserved fairly well. Therefore, in such nuclei, astate can be separated to a good approximation into the rotational part for the orientation
angles (Ω) and other intrinsic-motion part
|IM Kα⟩ = |IM K(Ω)⟩ ⊗ |Kα⟩ ,(3.1)

where I, M , and K are the quantum numbers for rotation, while other necessary quantum
numbers are expressed by α. If the nuclear shape is invariant with respect to π-rotation
around an axis perpendicular to the symmetry axis, the state should be a linear combination of two states having K and −K in order to fulfill the single-valuedness of the wave
function with respect to nuclear deformation seen in the laboratory frame. Even if this isapplied to eq. (3.1), the following discussion about the K-selection rule is not altered.
The moments associated with electromagnetic multipole quanta can be expressed using
the moments referring to the intrinsic frame of the nucleus,M(λµ) =

λ∑λ
Mintr (λν)Dµν
(Ω) ,

(3.2)

ν=−λ
λ
(Ω) is the rotation-matrix in the notation of ref. [8]. Matrix elements of M(λµ)
where Dµν

are given as follows.
⟨I2 M2 K2 α2 |M(λµ)|I1 M1 K1 α1 ⟩
17

=

λ
∑

λ
⟨K2 α2 |Mintr (λν)|K1 α1 ⟩⟨I2 M2 K2 (Ω)|Dµν(Ω)|I1 M1 K1 (Ω)⟩

ν=−λ

=

λ
∑

√

⟨K2 α2 |Mintr(λν)|K1 α1 ⟩ν=−λ2I1 + 1⟨I1 M1 λµ|I2 M2 ⟩⟨I1 K1 λν|I2 K2 ⟩ .
2I2 + 1

(3.3)

In the last line of the above equations, the first Clebsch-Gordan coeﬃcient demands |∆I|
≡ |I2 − I1 | ≤ λ and |∆M | ≡ |M2 − M1 | ≤ λ, while the second one requires further|∆K| ≡ |K2 − K1 | ≤ λ.

(3.4)

This restriction (eq. 3.4) is called the K-selection rule: The change in the K-quantum
number can not exceed the multipolarity of the radiation in transitions between states
having definite K-quantum numbers.
The K-selection rule is violated to some extent on account of the mixing of states
having diﬀerent K-quantum numbers (K-mixing) in real nuclei. The hindrance factor
FW for K-forbidden transitions is defined usually on the basis of the Weisskopf unit 1 fortransition probabilities,
FW =BWeisskopf unit (E or M, λ).
Bexperimental (E or M, λ)(3.5)

Empirically, FW is known to obey an exponent-rule,
FW = (fW )n ,n = |∆K| − λ ,(3.6)

where n is the K-forbiddenness and fW is the hindrance factor per n. Rusinov [6] proposedthat
fW ∼ 100 ,(3.7)

which implies that the amplitude for band-mixing with |∆K|=1 is ∼ 10−1 [8]. Löbnercompiled many experimental data and confirmed that the reduced transition probabilities
decrease approximately by a factor of 100 per n. He found, however, that the usage of the
Weisskopf unit for basis is not always meaningfull since absolute values of FW are likely
to be shifted to higher values in comparison with eqs. (3.5 - 3.7) [7]. This is not a serious
problem for largely K-forbidden cases because of the exponent-dependence of FW on n.
1

The Weisskopf unit [39] [8] is calculated using a nuclear radius parameter r0 =1.2 fm in this paper.

18

It is worth while noting that eq. (3.7) gives only a rough estimation and the values of fW
for various nuclei scatter in the interval<<

1 ∼ log10 fW ∼ 2,

(3.8)

which is found e. g. in ref. [16].
The main origin for the mixing of the K-quantum number is the Coriolis interaction, i.
e. excitation of intrinsic motions due to collective rotation. The expression for the Coriolis
interaction is obtained naturally in the strong-coupling-scheme particle-rotor model [8],which includes the collective rotational motion of the entire nucleus in addition to singleparticle motions in deformed nuclear potential,
strong
HPR= Hintr +2∑h̄2
κ=1 2J(Iκ − Jκ )2,(I3 = J3 )h̄2h̄2 ⃗2h̄2 ⃗22(I − I3 ) −(J+ I− + J− I+ ) +(J − J32 ) , (3.9)= Hintr +2J
2J
2J
where I⃗ and J⃗ are the total and the non-collective angular momenta respectively. The
four terms in the last line of eqs. (3.9) are the intrinsic, rotation, Coriolis, and the recoil
terms, respectively, in which only the Coriolis term can mix states having diﬀerent I3(=K). The Coriolis interaction has no eﬀects only when I=0 (and hence K=0). Whatshould be noted is that K-mixing occurs even in bandhead states — like K-isomers —when the band has non-zero K.
It is not certain whether we can apply the empirical rule (eqs. (3.5 - 3.8)), which
is obtained mainly for well-deformed nuclei, to the K π =10+ isomers in 182 W and 184 Os
treated in this paper. The first reason is that these nuclei are transitional nuclei having
relatively small deformation (β ∼ 0.2). Since the strength of the Coriolis interaction
is proportional to the reciprocal of the moment of inertia 1/J (eq. (3.9)), its eﬀect isenhanced in these less deformed nuclei. The second reason is that these isomers have a
configuration involving only an intruder orbital νi13/2 . The intruder orbital is aﬀected
strongly by the Coriolis interaction on account of the largeness and the purity of its
angular momentum. Hence the Coriolis interaction should be treated carefully in orderto study the structure of these K-isomers.

19

Another possible origin for K-mixing is the triaxial deformation of the nucleus, whichmixes K even in single-particle orbitals. The γ-vibration can also mix K as well as thestatic triaxiality, as is described in the last chapter. It is probable that K-isomeric decays
are sensibly aﬀected by the triaxiality besides the Coriolis interaction.

3.2

K-isomers in Hf-W-Os region

Many high-K isomers are found at or near the yrast levels in Hf isotopes and in theneighboring nuclei. This situation is caused by a strong shell eﬀect: There are many
Nilsson orbitals having large Ω 2 near the Fermi level for Hf, W and Os nuclei. By
exciting quasi-particles in these orbitals, high-spin states can be formed at excitationenergies as low as the energy of collective rotation [40].
The K-isomers observed in Hf isotopes have the typical characters of K-isomer likelong half-lives and K-selective decay patterns. The neighboring even-even nuclides — W
and Os — are also expected to have similar isomers judging from the location of the Fermi
level. They are, however, transitional nuclei, which have smaller deformation and larger
γ-softness than Hf isotopes. We show the energy levels of the low-spin members of theground and the γ-band for some N = 108 isotones in fig. 3.1 , where energies are divided
182
W and 184 Os are still deformed rotational nuclei
by those of the 2+
gr states. The nuclei

since their ground-band spectra show the pattern of rotational motion. The energy of theγ-bandhead is, however, lowered rapidly as Z increases. It is interesting to see what isbrought about to the typical K-isomers in Hf isotopes by this increase in γ-softness.
Some transitions largely violating the K-selection rule in these nuclei are compiled intable 3.1. The K π =8− isomers having N ∼ 106 are thought to have a two-quasi-particle
configuration ( ν[514]7/2− , ν[624]9/2+ ) whose excitation energy is very low. In 178 Hf 106 ,180

W106 , and 182 Os106 , the values of fW are 74.8, 54.6, and 48.7 respectively. It is suggested

that the degree of K-mixing becomes large as Z increases in this region.
The K π =10+ isomers in 182 W and 184 Os have much smaller fW (∼ 5) than the K π =8−
isomers (fW ∼ 50). As is already described in chapter 1, these isomers are observed to2

The quantum number Ω is the third component (in the intrinsic frame) of the angular momentum ofa single-particle orbital.

20

decay very rapidly in spite of the K-selection rule. The experimental data for the decays ofthese isomers are summarized in fig. 3.2. In 1977 Jeltema and his collaborators observed
the K π =10+ isomer in 182 W which decays directly into the ground-band states with arelatively short half-life of 1.4 µs despite the change in K as large as 10 [16]. They
considered that the small values of fW for this decay is due to the sensitiveness of thei13/2 orbital to the Coriolis interaction. In 1988 Chowdhury and his collaborators foundthe K π =10+ isomer in 184 Os, which has still smaller fW than the isomer in 182 W [13].
They suggested that the γ-softness of these nuclei plays the important role for these large
violations of the K-selection rule, while they depreciated the role of the νi13/2 orbital.
The large change in fW between the K π =8− isomer and the K π =10+ isomer would be
due to the diﬀerence in their configurations. The Z-dependence in fW for each isomer is,
however, likely to be attributed to the change in the deformation and in the γ-softness.
Although the K π =8− isomers are found in more nuclei than the K π =10+ ones, the former
are hard to deal with on account of the complicated structure of the normal-parity orbitals.
The latter are easier to treat because of the simple configuration involving solely the νi13/2
orbital 3 . Therefore we take up the K π =10+ isomers in this paper. We try to obtain
information on the γ-degree of freedom from the decays of these K-isomers, through theestimation of the two important factors of K-mixing — the Coriolis interaction and thetriaxiality — equally in a quantitative way.

3

The two K π =10+ isomers in 182 W and 184 Os are the only K-isomers observed so far which involvethe νi13/2 orbital exclusively in prolately deformed nuclei (not oblate yrast traps [41]) [16]. This isomerhas not been found in 180 Hf yet.

21

Table 3.1: Electromagnetic transitions largely violating the K-selection rule in even-even
Yb - Pt nuclei. Only largely K-forbidden transitions whose partial half-lives are known
are included in this table, although more isomers are found in these nuclei. Compilationof the data of other isomers are found e. g. in refs. [10] [41] [45] [46].

Nucleus

176

Yb
Hf

176

(I, K)πof
initialstate(8, 8)−(6, 6)+(14, 14)−Hf(8, 8)−(16, 16)+

Hf
W
182
W

(8, 8)−(8, 8)−(10, 10)+182(8, 8)−(10, 10)+

178

180

180

184

Os
Os

184

Pt

(8, 8)−(I, K)π
of
final
state
(8, 0)+(6, 0)+(4, 0)+(13, 8)−(12, 8)−(12, 8′ )−(8, 0)+(13, 8)−(12, 8)−(8, 0)+(8, 0)+(10, 0)+(8, 0)+(8, 0)+(10, 0)+(8, 0)+(8, 0)+∆E[ keV ]
96.1
736.2
1043.0
38.7
302.2
227.9
88.9
12.7
309.5
57.5
390.3
518.5
1086.5
554.
495.
1092.
610.1

Half-lifeof theinitialstate
11.7 [s]9.5 [µs]
401. [µs]4.0 [s]31 ± 1 [y]5.5 [h]5.2 [ms]
1.4 [µs]
0.8 [ms]
20 [ns]1.0 [ms]

Feeding ratio of
the final state,internal conversion
coeﬃcients
100 % , α = 0.38
Iγ =78Iγ =45Iγ =5.0 , α = 12
Iγ =44
Iγ =9.8
100 % , α = 0.49
99.8 % ,α = 1.5 × 107(K) 0.13 % , αK = 5.2—
100 %Iγ =63
Iγ =73100 %Iγ =6/100 (delayed)
Iγ =12/100 (delayed)
Iγ =64/120 (delayed)

22

λ

n

fW

Refs.

E1
M1
E2
M1
E2
E2
E1
E3
M4
E1
E1
M1
E2
E1
M1
E2
E1

7
5
4
5
4
4
7
5
4
7
7
9
8
7
9
8
7

94
49
43
30
23
23
79
66
64
233
55
6.4
5.1
49
5.0
3.6
58

[10] [42]
[43]
[43]
[43]
[43]
[43]
[10] [42] [44]
[10]
[10]
[10]
[10] [42][16][10] [16][10] [13] [42]
[13]
[13]
[10] [42]Figure 3.1: The γ-softness of N =108 isotones. The ratios of excitation energies of 4+
gr ,
+
+
+
+
6gr , and 2γ states to that of the 2gr state are shown. The energy level of the 2gr state isgiven in the parentheses.
23

Figure 3.2: Experimental data about the K π =10+ isomers.
24

Chapter 4
The model
We use a kind of particle-rotor model to simulate the decay of these short-lived K-isomers
into the states of the ground band. In our model non-collective motions, such as particlehole excitation and rotational alignment, occur only in particles in the unique-parity
orbital (intruder orbital), i. e. the νi13/2 orbital in this case. Other particles in normalparity orbitals are called the core and are replaced by some collective model. In thischapter the collective part of the model is described at first. Then the scheme of coupling
between the collective modes and the particles in the unique-parity orbital is presented.

4.1

The Bohr model

4.1.1

The full hamiltonian

As we aim at the study of the axially asymmetric deformation of well-deformed nuclei,
we employ the Bohr model of quadrupole surface vibrations and rotations [1] as the coreof our model : It has explicitly a variable γ which represents the degree of freedom of axially asymmetric deformation. The Bohr model may be relevant to the giant quadrupole
resonances, rather than to the low-lying quadrupole vibrations and rotations. The irrotational flow is rather a good approximation for the giant resonances but vortex flow
should be introduced for the low-lying quadrupole motions [47] [48]. Nevertheless weutilize the Bohr model because of its nice feature of symmetry of a quantum liquid dropwith quadrupole shape motions, e. g. the moment of inertia about the symmetry axis iszero. We do not use the value of the mass parameter B given by Bohr [1], but we reserve
it for a energy-scaling parameter to simulate the low-lying quadrupole motions.
25

Quadrupole surface deformations are uniquely determined by the normal coordinates
α2µ as{2∑

R(θ, φ) = R0 1 +}∗,(−1)µ α2µ Y2−µ (θ, φ) , α2µ = (−1)µ α2−µ

(4.1)

µ=−2

where R(θ, φ) is the distance of the surface from the center of the nucleus. In the Bohr
model, the intrinsic variables ( β and γ ) and Euler angles ( Ωc =(φc ,θc , ψc )) are used instead of α2µ . These variables are defined using the deformation in the intrinsic frame (aν )2and the rotation matrix Dµν
in the notation 1 of ref. [8] .

aν =

2
∑

2
α2µ Dµν(Ωc ) ,

(4.2)

µ=−2

with
a2 = a−2 = 2−1/2 βsinγ ,(4.3)

a1 = a−1 = 0 ,(4.4)

a0 = βcosγ .

(4.5)

In order to fulfill the single-valuedness of wave functions with respect to α2µ , Bohr’s
symmetry is required: The wave functions must be invariant with respect to transformations relabeling the principal axes of deformation. These transformations can be generated
by repeated application of the three basic transformations, i. e. R1 , R2 , and R3 [1] .
The hamiltonian of the full Bohr model is
HBohr

2
h̄2 ∑
∂2
(−1)µ
+ V (β, γ) ,= −2B µ=−2∂α2µ ∂α2−µ(4.6)= Tβ + Tγ + Trot + V (β, γ) ,

(4.7)

where
(4.8)

Tγ

(4.9)

Trot
1

h̄2 1 ∂ 4 ∂β,
2B β 4 ∂β ∂β
h̄2
1
∂
∂
= −
sin 3γ,22B β sin 3γ ∂γ
∂γ
3
∑
2
h̄2 2
=
Rκ , Jκ = 4Bβ 2 sin2 (γ − πκ) .
3
κ=1 2JκTβ = −∗IDM
K (Ω)=⟨IM |R(Ω)|IK⟩ .

26(4.10)

In the above equations Tβ , Tγ , and Trot represent the kinetic energies of β-vibration, γvibration, and rotation, respectively. V (β, γ) is the potential energy for deformations of
the intrinsic shape. In eqs. (4.10), Rκ and Jκ denote the angular momentum operator
and the moment of inertia for the rotation around the κth principal axis, respectively.
The sum of these kinetic energy terms is proportional to the five-dimensional laplacianwith respect to α2µ (eq. (4.6)).

4.1.2

The γ-soft model

In this paper, we fix the variable β at a constant value β0 and treat it as a static parameter,
because we aim at studying intensively the eﬀects of the γ-degree of freedom on Kisomerism. We call this β-fixed Bohr model as the γ-soft model [49] [50] from now on.
This model was introduced by Wilets and Jean [51] for the γ-unstable potential (V (β0 , γ)
= constant). Qualitative properties of the model at high spin are discussed by Turnerand Kishimoto [52]. Its hamiltonian isHγ−soft = Tγ

β=β0

+ Trotβ=β0+ V (γ) ,(4.11)

where
V (γ) = V (β0 , γ) .

(4.12)The energy is scaled only through a quantity (2h̄2 /Bβ0 2 ) except for V (γ). In later sections
we designate this value by ϵc , which is the energy of the first excited state for V (γ) = 0.
Eigenstates of Hγ−soft are expressed as|RM αc ⟩ =R∑

c|RM Kc (Ωc )⟩ ⊗ |ΦαRK(γ)⟩ ,
c(4.13)

Kc =−R

where√

2R + 1 RDM Kc (Ω0c ) ,8π 2cc⟨δ(γ − γ0 )|ΦαRK(γ)⟩ = ΦαRK(γ0 ) .
cc

⟨δ(Ωc − Ω0c )|RM Kc (Ωc )⟩ =(4.14)(4.15)In the above expression, R is the quantum number of angular momentum (R=0,2,3,4,· · ·).
M and Kc are its components along the z-axis (of the laboratory frame) and the thirdprincipal axis, respectively. Other quantum numbers to specify a state are represented by27

αc . Because of Bohr’s symmetry, the quantum number Kc takes on only even integers.
Amplitudes for the component having Kc and that having −Kc are related to each otherasccΦαR,−K(γ) = (−1)R ΦαRK
(γ) ,
c
c

(4.16)

on account of the symmetry.
We consider wave functions only in the interval 0 ≤ γ ≤ π/3, utilizing Bohr’s symmetry. The orthonormality of eigenstates,⟨R′ M ′ αc ′ |RM αc ⟩ = δR′ R δM ′ M δαc ′ αc ,(4.17)

is satisfied by the orthonormality of D-functions,⟨R′ M ′ Kc′ (Ωc )|RM Kc (Ωc )⟩√=(2R′ + 1)(2R + 1)
8π 2 ∫∫ π0= δdθc sin θc

R′ R

δ

M ′M

δ

×
∫ 2π

2π

dφc

0

Kc ′ Kc

0

′

∗ R
R
dψc {DM′ K ′ (φc , θc , ψc )} DM K (φc , θc , ψc )cc,

(4.18)

and by a requirement for intrinsic wave functions,
R
∑

α′c
c
⟨ΦRK
(γ)|ΦαRK(γ)⟩ =cc

Kc =−RR∑
Kc =−R

∫ π/3

3
0

α′

c
c
ΦRK
(γ)ΦαRK(γ) sin 3γdγ = δα′c αc .
cc(4.19)The potential energy for γ-deformation ( γ-potential ) should be a function of cos 3γ
as a consequence of Bohr’s symmetry. It can be expanded by Legendre polynomials ofcos 3γ. We take only the first two terms at most in this paper,22h̄2 ∑
V (γ) =Vl Pl (cos 3γ) .
Bβ0 2 l=1

(4.20)

We vary V1 to control the γ-softness and vary V2 to shift the minimum point of thepotential.
When V (γ)=0 (the γ-unstable model [51] [53] [54]), eigenstates are labeled by thequantum numbers µ and nγ in addition to R and M . For each R, these quantum numbers
take on the values ofµ = µmin , µmin + 1, µmin + 2, · · · , R
nγ = 0, 1, 2, 3, 4, 5, · · ·
28

,

,

(4.21)(4.22)

where{µmin = µmin (R) =R/2(R + 3)/2

for even R ,for odd R .

(4.23)The seniority quantum number λ [53] is related to µ and nγ as
λ = µ + 3 nγ .

(4.24)The energies of states are determined only through λ,Eλ =2h̄2 λ(λ + 3)
.
4
Bβ0 2

(4.25)

Number of states for each R and each λ is given in table 4.1.
λnWave functions of eigenstates of the γ-unstable model, GRKγc (γ), are given in themanner of ref. [50] as follows:λn
GRKγc (γ) =R∑µ′ =µmin(R) nγ ′ =0

where

j≡

Rλn

′

′

µ
Bµ′ nγγ′ (cos 3γ)nγ gRK
(γ) ,
c

j
∑

(4.26)

j+k(cos γ)j−k
jk (sin γ)
√
fRK
,
c
(j + k)!(j − k)!
k=−j(4.27)µ′, k = −j, −j + 1, −j + 2, · · · , j .
2(4.28)

µ′
gRK
(γ) =
c

with

nγ
∑

Rλn

jk
can be obtained by diagonalizing PI , which is the proThe coeﬃcients Bµ′ nγγ′ and fRKcjection operator into the invariant subspace with respect to transformations of Bohr’s
symmetry group. Among the eigenstates of PI , those having eigenvalue of 1 and being
regular at γ=0◦ and γ=60◦ should be chosen.
When V (γ) ̸= constant, we diagonalize Hγ−soft using the bases ofλn|RM Kc (Ωc )⟩ ⊗ |GRKγc (γ)⟩ ,(4.29)λ ≤ λmax = 24 ,(4.30)

which are truncated byor equivalently,
Eλ ≤ E24 =2h̄2× 162 ∼ 30MeV ,
Bβ0 2
29

(4.31)

in this paper. Wave functions of γ for the eigenstates are expressed as
c
ΦαRK
(γ) =
c

∞
∑

R
∑

3n +µ,nγcaαR,3nGRKγcγ +µ,nγ(γ) .

(4.32)

nγ =0 µ=µmin (R)

|

{z

}

3nγ +µ≤λmax

The method to solve the eigenvalue equation for Hγ−soft is described in detail in ref. [50].

4.1.3

The triaxial rotor model

The triaxial rotor model of Davydov and Filippov in which γ is fixed at a value γ0 is
considered as a special case of the γ-soft model. It is obtained in the limit of V1 → −∞while keeping V1 /3V2 = − cos 3γ0 in eq. (4.20). In the model, the wave-function for thevariable γ is assumed to be trapped in a deep potential well which has a minimum at γ0 .
The hamiltonian is
Htriax = Trot

β=β0 ,γ=γ0.

(4.33)

Considering the requirement of Bohr’s symmetry, eigenstates of Htriax are labeled by R
and |Kc |, which take on the values of
R = 0, 2, 3, 4, 5, · · ·{|Kc | =

0, 2, 4, · · · , R
2, 4, 6, · · · , R − 1,(4.34)

for even R ,for odd R .

(4.35)The number of states for each R is given in table 4.2.

4.1.4

The axially symmetric rotor model

For comparison, we also use the axially symmetric rotor model which does not have theγ-degree of freedom. It is obtained by setting γ=0◦ in the hamiltonian of eq. (4.33). The
moments of inertia are changed in such a way that
J1 = J2 = J ≡ 3Bβ0 2 ,J3 = 0 .

(4.36)The hamiltonian is written as
Hax.sym. =

h̄2
h̄2 2
(R1 + R22 ) =R(R + 1) ,2J
2J

R3 = 0 .

(4.37)The eigenstates are labeled only by R which takes on non-negative even integers,
R = 0, 2, 4, 6, 8, · · ·The number of states for each R is given in table 4.3.
30.

(4.38)

4.2

The particle-rotor model

4.2.1

Derivation from the shell-model hamiltonian

4.2.1.1

The shell-model hamiltonian

We use a kind of particle-rotor model in which the particle states are described notin the intrinsic frame of the core but in the laboratory frame and are expanded withspherical bases. This model was originally used for spherical nuclei [55]. As for the earlier
applications to deformed nuclei, see refs. [56]-[59]. Later works utilizing this model arefound in refs. [60] - [64]. At first we demonstrate how our particle-rotor model is derived
from a (spherical) shell-model hamiltonian and elucidate the bases and the scope of ourmodel. The shell-model hamiltonian is written as [65]
HSM =

∑

Hj0 +

j

∑ ∑

Vj1 j2 ;j3 j4 ,(4.39)

j1 ≤j2 j3 ≤j4

where
Hj0 =

∑

ϵj a†jm ajm ,(4.40)gJ (j1 j2 ; j3 j4 )A†JM (j1 j2 )AJM (j3 j4 ) .

(4.41)m

Vj1 j2 ;j3 j4 =∑JM

In the above expressions, the subscript j denotes the angular momentum, and also indicates symbolically all the quantum numbers to designate a single-particle state (having
an energy level ϵj ) except for the magnetic quantum number m. The creation operator
A†JM generates a normalized state of a nucleon-pair coupled to an angular momentum J
and its z-component M :A†JM (j1 j2 ) = (1 + δj1 j2 )−1/2∑⟨j1 m1 j2 m2 |JM ⟩a†j1 m1 a†j2 m2 .

(4.42)

m1 m2

The coeﬃcient gJ is a matrix element of a residual two-body interaction V between
normalized states of two nucleons with a coupled angular momentum J :gJ (j1 j2 ; j3 j4 ) = ⟨0|AJM (j1 j2 )V A†JM (j3 j4 )|0⟩ .

(4.43)The K π = 10+ isomers in 182 W and 184 Os have a configuration of a unique-parity(intruder) orbital νi13/2 , which is not mixed with energetically nearby orbitals by the31single-particle potential with quadrupole deformation on account of parity conservation.
Therefore we may treat only the unique-parity orbital as the particle degrees of freedom.
Thus, using notations of
≡

j0
∑′

ν i13/2 ,∑≡j(4.44),(4.45)j (j̸=j0 )we divide HSM into three parts
p
int
c
,
+ HSM
+ HSM
HSM = HSM

(4.46)

where
c
HSM
=
p
HSM
=
int
HSM
=

∑′

Hj01 +∑ ′ ∑′

Vj1 j2 ;j3 j4 ,∑′Vj0 j1 ;j0 j2 +∑′(4.48)(Vj0 j0 ;j1 j2 + Vj1 j2 ;j0 j0 )j1 j2

j1 ≤j2+(Vj0 j0 ;j0 j1 + Vj0 j1 ;j0 j0 ) +

∑′

∑′ ∑′

(Vj1 j2 ;j0 j3 + Vj0 j3 ;j1 j2 ) .

(4.49)

j1 ≤j2 j3

j1

4.2.1.2(4.47)

j1
j1 ≤j2 j3 ≤j40
Hj0 + Vj0 j0 ;j0 j0 ,The core partcby a certain
In order to derive the particle-rotor model hamiltonian, we replace HSM
ccollective model hamiltonian, Hγ−soft in this paper. Among the dependence of HSM
onthe number of particles in the core (nc ), the dependence of the ground state energy is
included to the linear term of nc . The dependence of other quantities, excitation energies
and quadrupole matrix elements, is thought to be small and is ignored. (If required, we
can change the parameters of the core model for each nc without largely changing ourcformalism.) Thus we replace HSM
by Hc :

Hc = λc n̂c + Hγ−soft (, Htriax , Hax. sym ) .

(4.50)The physical meaning of λc is the Fermi level of the core and appears only in combination
with ϵj0 as ϵj0 − λc . Eigenstates of Hc are expressed as|RM αc , nc ⟩ = |RM αc ⟩ ⊗ |nc ⟩ ,(4.51)

where |RM αc ⟩ is given in eq. (4.13) and is operated on only by the second term in Hc .
The state |nc ⟩ is concerned only with the first term in Hc ; n̂c |nc ⟩ = nc |nc ⟩ .
32

4.2.1.3

The particle partp
We treat HSM
in a fully microscopic way. As the residual interaction V in the j0 orbital,

we use the delta interaction [65] between a particle at r⃗1 and another one at r⃗2 :V (⃗r1 , r⃗2 ) = −4πV0 δ (3) (⃗r1 − r⃗2 ) .
We define a quantity G as
G = V0∫ ∞
0(4.52){R0 (r)}4 r2 dr ,(4.53)

where R0 (r) is the radial part of the wave function of the j0 orbital. The matrix elementgJ of this interaction is given by(2j0 + 1)2 ⟨j0 − 12 j0 21 |J0⟩2
gJ (j0 j0 ; j0 j0 ) = −G.
2
2J + 1(4.54)For J=0, it reads as
g0 (j0 j0 ; j0 j0 ) = −G

2j0 + 1.
2

(4.55)The value of G is considered to be the same as that of the strength GBCS of the ordinary
pairing interaction used in the BCS model of nuclei (found e. g. in ref. [66]),
j1
j2
∑ ∑
∑ ∑
1
BCS
Hpair
= − GBCSa†j1 m1 (−1)j1 −m1 a†j1 −m1 (−1)j2 −m2 aj2 −m2 aj2 m2 .
4
j1 m1 =−j1 j2 m2 =−j2
(4.56)

Corresponding values of g0 to this interaction is√

g0BCS (j1 j1 ; j2 j2 ) = −GBCS(2j1 + 1)(2j2 + 1)
2,

(4.57)

which has the same form as eq. (4.55) for j1 =j2 =j0 . The value of GBCS recommended by
Baranger and Kumar [66] is
G = GBCS ≃ 22MeV/A

for neutrons ,(4.58)

where A is the mass number. The values of gJ for the νi13/2 orbital of A=184 nuclei (G=22 MeV/184=0.120 MeV) are shown in table 4.4. The eﬀective interaction of Onishiand Negele [67] for the shell-model calculations are also given for comparison.
Basis-states of n particles in the j0 orbital are constructed using the coeﬃcients offractional parentage (c.f.p.)[68]. The c.f.p.’s are obtained by diagonalizing the antisymmetrization operator (the projection operator into the fully antisymmetric subspace) on
33

the bases composed of the direct product of the antisymmetric states of n − 1 particles
and the states of another particle. The states thus obtained are labeled by the numberof particles n, the total angular momentum J, its z-component M , seniority v, and thenecessary additional quantum numbers αp as|j0 n ( v αp ) J M ⟩ .

(4.59)

The following values are taken on by n and v :n = 0, 2, 4, · · · , 2j0 + 1 ,(4.60)

v = 0, 2, 4, · · · , min(n, 2j0 + 1 − n) .

(4.61)

Only even values of n have to be considered because we intend to treat positive parity
states. The number of states for each set of values of n, v, and J are tabulated in tables
4.5 and 4.6 . Matrix elements of operators between these bases (eq. 4.59) are calculatedusing formulae given in ref. [68].
4.2.1.4

The QQ interaction

int
by Hint which is suitable for interactions between a particle
Now we approximate HSM

and the core described with the Bohr model. The first term in eq. (4.49) does not include
transfer of particles and corresponds to the interaction between the surface deformation
α2µ and the particles in the particle-rotor model. Potential energy for a particle at ⃗rexerted by the deformed core is considered to be expressed as(V (⃗r) = V0

)

R0
r ,
R(θ, φ)(4.62)

where R(θ, φ) is given in eq. (4.1). Expanding R(θ, φ) with respect to α2 ,(V (⃗r) = V0 r − r

2
∑

)

µ

(−1)

α2µ Y2−µ (θ, φ) + O(α2 2 )µ=−2= V0 (r) − r

2
dV0 (r) ∑(−1)µ α2µ Y2−µ (θ, φ) + O(α2 2 ) .
dr µ=−2(4.63)

The first term in the last line of above equations is the spherical mean potential and haveno relation with surface deformation α2 . The second term is a quadrupole-quadrupole34

interaction between the core and the particle. Thus the first term in eq. (4.49) may bereplaced by HQQ ,
HQQ = −κ

∑

µ α2µ

(−1)

µ

β0

j0
∑

j0
∑

m′ =−j0 m=−j0

where⟨j0 m′ (θ, φ)|Y2−µ (θ, φ)|j0 m(θ, φ)⟩a†j0 m′ aj0 m , (4.64)⟩⟨

dV0 (r)
R0 (r) .
κ = β0 R0 (r) rdr(4.65)

If we assume a harmonic oscillator single-particle potential,
1
V0 (r) = mω 2 r2 ,2h̄ω = 41 MeV × A−1/3 ,(4.66)

the matrix element of r2 is related to Nj0 (the total oscillator quantum number for thej0 -orbital, Nj0 =6 for j0 =i13/2 ) as
⟨R0 (r)|r2 |R0 (r)⟩ =3h̄(Nj0 + ) .
mω
2(4.67)

Thus we obtain3κ = h̄ωβ0 (Nj0 + ) .
2
4.2.1.5(4.68)The Cooper-pair exchange interaction

The second term in eq. (4.49) expresses an exchange of a pair of particles between theunique-parity orbital and the core. Because the range of the nucleon-nucleon interaction
is short, the dominantly exchanged pair is the J = 0 one. Thus we approximate this termby a Cooper pair exchange interaction,G
Hpair = − (Pp† Pc + Pc† Pp ) ,4(4.69)

where the strength G is the same as that in eq. (4.58). Pp† (Pc† ) is a creation operator ofa Cooper pair in the j0 orbital (in the core):
Pp† =

√

2(2j0 + 1)A†00 (j0 j0 ) =j0∑(−1)j0 −m a†j0 m a†j0 −m ,(4.70)

m=−j0

Pc†

=

∑′

√

Sj1

2(2j1 + 1)A†00 (j1 j1 )

=

j1

j1
∑′ ∑
j1 m=−j135(−1)j1 −m Sj1 a†j1 m a†j1 −m , (4.71)

whereSj1 =

2gJ (j0 , j0 ; j1 , j1 )√.

(4.72)

G (2j0 + 1)(2j1 + 1)

For the matrix elements of Pc† and Pc , we assume the following form [62]:G∆′⟨R′ M ′ αc′ , n′c |(− Pc† )|RM αc , nc ⟩ = −δR′ R δM ′ M δα′c αc δn′c ,nc +2 ,
4
2
′
G
∆
.
⟨R′ M ′ αc′ , n′c |(− Pc )|RM αc , nc ⟩ = −δR′ R δM ′ M δα′c αc δn′c ,nc −242(4.73)(4.74)

When the ordinary pairing interaction of eq. (4.56) is adopted, Sj1 turns out to be unity.
Employing the BCS ground state as the state of the core, we find that ∆′ is the pairing
gap energy which does not include the contribution from the j0 orbital,⟨∆′ = 2 BCS

j1⟩G ∑′ ∑(−1)j1 −m a†j1 m a†j1 −m BCS
4 j1 m=−j1

j1
G ∑′ ∑=
uj m vj m
2 j1 m=−j1 1 1

= ∆−

j0
G ∑
uj m vj m .
2 m=−j0 0 0(4.75)(4.76)(4.77)

where ∆ is the (full) pairing gap energy of the nucleus.
We can rewrite Hpair as
Hpair = −∆′ ∑(−1)j0 −m (T a†j0 m a†j0 −m + T † aj0 −m aj0 m ) ,2 m(4.78)

where T † and T operate only on |nc ⟩ : T † |nc ⟩ = |nc + 2⟩, T |nc ⟩ = |nc − 2⟩. In this paper
the label nc is practically dummy. If we identify core states diﬀering only in nc , T † and T
are replaced by 1 and Hpair turns out to be a pairing field in the j0 orbital, which is used
e. g. in ref. [69].
4.2.1.6

The resulting particle-rotor model hamiltonian

The third term and the fourth term in eq. (4.49) vanish due to parity conservation because
we neglect contributions from much higher-lying orbitals having the same parity as thej0 orbital. Thus Hint is finally expressed as
Hint = HQQ + Hpair .

(4.79)The resulting hamiltonian of the particle-rotor model is given byp
HPR = Hc + HSM+ Hint .

36(4.80)

4.2.2

Wave functions

Eigenstates of the total system are expanded in terms of so-called weak coupling bases as|IM α⟩ =

∑

n
IM
AIα
.
Rαc nvαp J [ |Rαc , A − n⟩ ⊗ |j0 (vαp )J⟩ ](4.81)

Rαc nvαp J

On the left hand side of above equation, I is the total angular momentum and M is its
z-component, while α denotes the necessary additional quantum numbers. On the righthand side, amplitudes are denoted by AIαRαc nvαp J . The bases of the core are designated by
its angular momentum R, number of particles in the core A − n (A is the mass number),and the additional quantum number αc . The bases of particles are labeled by the numberof particles in the j0 orbital n (n = 0, 2, 4, · · · , 2j0 + 1) , seniority v, angular momentum
J, and the additional quantum number αp . It should be noted that states of particles aredescribed in the laboratory frame, not in the intrinsic frame of the core. Consequently
spherical bases are used for the particles.
Since wave functions described in the intrinsic frame are convenient to understandthe structure of states, we now express the bases on the right hand side of eq. (4.81) interms of so-called strong coupling bases in which particles are described in the frame ofthe principal axis of the deformation of the core. Quantum numbers having nothing to
do with rotations are included in αc and αp from now on in this paper:|j0 n (vαp )JM ⟩→|JM αp ⟩ ,(4.82)|RM αc , A − n⟩ → |RM αc ⟩ .

(4.83)

Eq. (4.81) is expressed briefly as|IM α⟩ =

∑

IM
.
AIα
Rαc Jαp [ |Rαc ⟩ ⊗ |Jαp ⟩ ](4.84)

Rαc Jαp

Due to the fact that the particle bases have definite angular momentum, we can rotate
the particle bases together with the intrinsic frame of the core by a relatively simple
transformation,|JM αp ⟩ =J∑J|JKp αp ⟩Ωc DM
Kp (Ωc ) .

Kp =−J37(4.85)

The subscript Ωc attached to the ket-vector indicates that the state is expressed in the
intrinsic frame. Using eqs. (4.13)-(4.15) and (4.85), we obtain| [Rαc ⊗ Jαp ]IM ⟩ =

J
∑

R
∑

RJI
c
BK
|ΦαRK
(γ)⟩|JKp αp ⟩Ωc |IM, Kc + Kp (Ωc )⟩ , (4.86)cc Kp

Kc =−R Kp =−J

where

√
RJI
BK
=
c Kp

2R + 1⟨RKc JKp |I, Kc + Kp ⟩ .
2I + 1(4.87)The state of the core as for γ can be expanded by employing bases independent of R and
Kc ,
c
|ΦαRK
(γ)⟩ = 3
c

∫ π/3
0

c
dγ0 sin 3γ0 ΦαRK(γ0 ) |δ(γ − γ0 )⟩ .
c

(4.88)

Expanding the bases on the right hand side of eq. (4.84) using eq. (4.86) and eq. (4.88),
we obtain|IM α⟩ = 3

∫ π/3
0

dγ0 sin 3γ0

∑ ∑

CγIα
|δ(γ − γ0 )⟩ |JKp αp ⟩Ωc |IM Kc + Kp (Ωc )⟩ ,0 Jαp Kc Kp

Jαp Kc Kp(4.89)
whereCγIα=
0 Jαp Kc Kp

∑

αc
RJI
AIα
Rαc Jαp BKc Kp ΦRKc (γ0 ) .

(4.90)Rαc

As an application of the new expansion (4.89), we show the expression for the probability distribution of the K-quantum number. The K-quantum number is the componentof the total angular momentum I along the third principal axis of the core 2 . The third
components of R and J in the intrinsic frame of the core are denoted by Kc and Kp
respectively (K = Kc + Kp ) . The quantum number Kc takes on only even integers dueto the symmetry as for π-rotation around the 3-axis, while Kp and K take on both evenand odd integers.
We can define the probability of K-quantum numbers as follows,∫ π/3ρIα (Kc , Kp ) = 3=0dγ0 sin 3γ0∑
Jαp R′ α′c Rαc

∑

|CγIα
|2
0 Jαp Kc Kp

Jαp
′

α′

αc
Iα
RJI
∗ R JIc.
(AIαR′ α′c Jαp ) BKc Kp ⟨ΦR′ Kc (γ)|ΦRKc (γ)⟩ARαc Jαp BKc Kp (4.91)

2

It should be noted that the third principal axis is always the longest axis even in a γ-fluctuating
model because 0 ≤ γ ≤ π/3 in our choice. This condition is explicitly taken into account in the intervalof integration with respect to γ for inner product (eq. (4.19)).

38

Considering the R1 symmetry of the core (eq. (4.16)) and
RJI
RJI
B−K
= (−1)R+J−I BK,
c ,−Kpc ,Kp(4.92)

we can obtain∑

ρIα (−Kc , −Kp ) =

α′

′

αc
Iα
RJI
∗ R JIc(AIαR′ α′c Jαp ) B−Kc −Kp ⟨ΦR′ −Kc (γ)|ΦR−Kc (γ)⟩ARαc Jαp B−Kc −Kp

Jαp R′ α′c Rαc= ρIα (Kc , Kp ) .

(4.93)

By summing up ρ for K and ρ for −K, we define the probability for |K| as
ρIα (|K|) = 2 (1 + δ|K| 0 )−1∑ρIα (Kc , |K| − Kc ) .

(4.94)Kc

4.2.3

The Coriolis interaction in the laboratory-frame model

In the strong-coupling particle-rotor model, the Coriolis interaction is obtained from therotational energy term in the hamiltonian (eq. 3.9). In the weak-coupling model describedin the laboratory frame, the rotational energy term Trot (eq. 4.10) concerns only to thecore coordinates and it might seem that there were no Coriolis interaction.
The two models are, however, equivalent as for the treatment of rotation. When thecoordinates of the particles are transformed from those referring to the laboratory frame⃗ HQQ (eq. (4.64)) becomes(say ⃗x)to those based on the intrinsic frame of the core (say ξ),
independent of the Euler angles of the core ( Ωc ):∑α2µ −µY (x̂)β0 2µ∑
a2ν −ν ˆ= −κ (−1)ν
Y (ξ) ,β0 2
νHQQ = −κ(−1)µ(4.95)

where aν does not depend on Ωc (eq. (4.3–4.5)). Thus Ωc becomes a cyclic coordinate andits conjugate momentum becomes an integral of motion, i. e. the total angular momentum.
Other angular-momentum operators of the partial systems are expressed through thecoordinates shown below:

⃗
R
J⃗
I⃗

Intrinsic-frame modelΩc , ξ⃗ξ⃗ΩcLaboratory-frame modelΩc⃗xΩc , ⃗x39

Hence Trot turns out to contain the interaction between the core coordinates and theparticle coordinates, i. e. the Coriolis interaction. This situation is clearly shown by⃗ as I⃗ − J⃗ ( since I=⃗ R+⃗ J⃗ ),
rewriting R
Trot =

3
∑

3
∑
h̄2 2
h̄2
Rκ =
(Iκ − Jκ )2 ,
2J
2J
κ
κ
κ=1
κ=1

(4.96)⃗
where Iκ concerns only to Ωc while Jκ depends solely on ξ.

4.2.4

Advantages of the description in the laboratory frame

In order to treat the particle-rotor model, there are two approaches, the weak coupling
one and the strong coupling one. In the former (the latter), the particle-rotor model
hamiltonian is diagonalized in the weak (strong) coupling bases. Results are the same
between the two approaches if they are obtained exactly. But in practical calculations, it
is important to clarify merits and demerits of the two approaches.
The strong coupling approach has an advantage in the truncation of the particle states:
Particle bases are usually taken as Nilsson single-particle states, which are generally
thought to be good approximation. But we have to take into account more configurations than usual in treating the K π = 10+ isomers, because the Coriolis mixing eﬀect
is stronger in the high-j unique-parity orbital than in other orbitals. It is also stronger
at high spin than at low spin. Thus it is not very clear whether we may consider only afew Nilsson orbitals.
In the weak coupling approach the number of bases is likely to become enormous.
This is a shortcoming inherent in the shell model. But in treating the isomers in which
only single-j configurations are involved, the number of states is small enough, so thattruncation is not necessary using the computers in the present day. Hence it is not a
defect of the approach.
Furthermore, the weak coupling approach has an advantage that it is relatively easy toconsider various motions of the core like the γ-vibration : We can solve the core and theparticle system separately and then couple them only through the quadrupole-quadrupole
and the Cooper-pair exchange interactions which are strong but simple. Existing algorithms are available to solve the core models. In the strong coupling model, however,40

the rotations of the total system, intrinsic collective motions like the γ-vibration, andthe motions of the particles are coupled in such a complicated way that they can not be
separated eﬀectively. Thus we take the weak coupling approach.

4.2.5

Electromagnetic transitions

Operators of electromagnetic transitions are expressed as follows:√M(M1, µ) = µN
with

{

(

∑∑
3
⃗ + gν
⃗ 0 m⟩a† ′ aj0 m
gc R⟨j0 m′ |J|j
j0 m
4π
′
m m

)

,

(4.97)

gc = Z/A ,gν = −0.223 assuming gl = −0.05 and gs = 0.6 × gsfree ,(4.98)∑∑
3ZeR02α2µ +⟨j0 m′ |eeﬀ r2 Y2µ (θ, φ)|j0 m⟩a†j0 m′ aj0 m ,4π
m′ m(4.99)

andM(E2, µ) =with{

R0 = 1.2 fm × A1/3 ,eeﬀ = eZ/A .

(4.100)In the above expressions, µN (= eh̄/2mp c) is the nuclear magneton, and A and Z are themass and the atomic number of the nucleus. The value of the neutron’s g-factor is takenfrom ref. [8] (observed gν of the [624]9/2+ orbital is −0.22 for 179 Hf). In order to calculate
the matrix element of r2 , we assume the harmonic oscillator wave function (eq. (4.67)).

41

Table 4.1: Number of states of the γ-unstable core for each spin R and each seniority λ.
The multiplicity of states due to the degeneracy as for M is not counted.

R
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24

µ
0≤µ≤0
2≤µ≤1
1≤µ≤2
3≤µ≤3
2≤µ≤4
4≤µ≤5
3≤µ≤6
5≤µ≤7
4≤µ≤8
6≤µ≤9
5 ≤ µ ≤ 10
7 ≤ µ ≤ 11
6 ≤ µ ≤ 12
8 ≤ µ ≤ 13
7 ≤ µ ≤ 14
9 ≤ µ ≤ 15
8 ≤ µ ≤ 16
10≤ µ ≤ 17
9 ≤ µ ≤ 18
11≤ µ ≤ 19
10≤ µ ≤ 20
12≤ µ ≤ 21
11≤ µ ≤ 22
13≤ µ ≤ 23
12≤ µ ≤ 24

0 1
1

2

1

1

λ ( = µ + 3 nγ ; nγ = 0, 1, 2, 3, · · · )3 4 5 6 7 8 9 10 11 12 13
1
1
1
1
1

1

1

1
1
1 1 1 1 1
1 1
1 1 1 2
1 1
1 1 1
1
1 1

1
1
1
1
2
1
1
1
1 1
1

1

1

1
1 1
1
1 2
1 1
2 1
1 2
2 2
1 1
1 2
1 1
1 1
1
1 1
1

1
1
1
1
2
1
2
2
2
1
2
1
1
1
1
1

1
1
1
1
1
2
1
2
2
2
2
2
1
2
1
1
1
1
1

1
1
1
2
1
1
2
2
1
3
2
2
2
2
1
2
1
1
1
1
1

42

1
1
1
1
2
1
2
2
2
2
3
2
3
2
2
1
2
1
1
1
1

14

15
1

1
1
1
1
1
2
1
2
2
2
2
3
2
3
2
2
2
2
1
2
1
1

1
1
2
1
1
2
2
1
3
2
2
3
3
2
3
2
2
2
2
1
2

Table 4.2: Number of states of the triaxial rotor model for each R. The multiplicity ofstates due to the degeneracy as for M is not counted.

R
min Kcmax Kc# states

0
0
0
1

1
2
0
0

2
0
2
2

3
2
2
1

4
0
4
3

5
2
4
2

6
0
6
4

7
2
6
3

8
0
8
5

9
2
8
4

10
0
10
6

11
2
10
5

12
0
12
7

13
2
12
6

14
0
14
8

15
2
14
7

16
0
16
9

17
2
16
8

18
0
18
10

19
2
18
9

20
0
20
11

Table 4.3: Number of states of the axially symmetric rotor model for each R. The
multiplicity of states due to the degeneracy as for M is not counted.

R# states

0
1

1
0

2
1

3
0

4
1

5
0

6
1

7
0

8
1

9
0

10
1

11
0

12
1

13
0

14
1

15
0

16
1

17
0

18
1

19
0

Table 4.4: The two-body interaction for the neutrons in the i13/2 orbital.

delta int.
J
0
2
4
6
8
10
12

gJ /G
−7.0000−1.7231−0.9329−0.6062−0.4168−0.2818−0.1642delta int.
G=0.120 MeV
gJ [MeV]−0.840−0.207−0.112−0.073−0.050−0.034−0.02043

Onishi-Negele
A=176,Z=76
gJ [MeV]−0.907−0.400−0.142−0.078−0.046−0.027+0.01020
1

Table 4.5: Number of states of n fermions in a single j orbital (1/2 ≤ j ≤ 7/2). The
multiplicity of states due to the degeneracy as for M is not counted.

j=1/2
2 × J 0 Total
v=0 1
1
2 × J 1 Total
v=1 1
1

2×J
v=0
v=2
2×J
v=1

2×J
v=0
v=2
2×J
v=1
v=3

2×J
v=0
v=2
v=4
2×J
v=1
v=3

0
1

1

0
1
1

j=3/2
0 2 4
1
1
1 3 5
1

Total
1
1
Total
1

j=5/2
2 4 6

8

1
5
1

3

7

1

2

4

j=7/2
6 8 10

3

1
1
5

1
1
9

1
11

1

1

1

1

7
1

1

Total
1
2
Total
1
2

12

14

16

15

1
17

1
9

1

44

13

1

Total
1
3
4
Total
1
5

Table 4.6: Number of states of n fermions in a single j orbital (9/2 ≤ j ≤ 13/2). The
multiplicity of states due to the degeneracy as for M is not counted.

2×J
v=0
v=2
v=4
2×J
v=1
v=3
v=5

0
1
1
1

2

4

6

3

1
1
5

1
7

1
1

1
1

1
1

2×J
v=0
v=2
v=4
v=6
2×J
v=1
v=3
v=5

0
1

2×J
v=0
v=2
v=4
v=6
2×J
v=0
v=2
v=4
v=6
2×J
v=1
v=3
v=5
v=7
2×J
v=1
v=3
v=5
v=7

0
1

8
1
2
9
1
1
1

10

j=9/2
12 14

16

18

20

22

24

1
11

1
2
13

1
15

1
1
17

1
19

1
21

23

1
25

1
1

1
1

1
1

1
1

1

1

Total
1
4
13
Total
1
9
10

22

24

26

28

30

1

12

14

16

j=11/2
18 20

1
3
3
13

2
2
15

1
3
2
17

2
2
19

1
2
2
21

1
1
23

2
2
25

1
1
27

1
1
29

1
31

1
3

2
3

1
3

1
3

1
2

1
2

2

1
1

1

1

8

10

12

14

16

j=13/2
18 20

22

24

26

28

30

32

34

36

1
4
44

1
4
6
46

3
4
48

1
4
8

3
6

1
5
7

3
7

1
4
7

3
5

1
3
7

2
5

3
5

1
4

2
4

1
2

1
3

1
5

1
7

9

1
11

1
2
1
41

1
4
2
43

1
4
4
45

2
5
3
47

2
5
4
49

1
1

1

2

4

6

8

10

1
1
1

3

1
2
1
5

1
2
7

1
3
2
9

1

1
1

1
2

1
3

2
2

2
1
11
1
1
3

2

4

6

1
2
38

1
40

1
3
3
42

2
1

1
1
3

1
2
39

1
1

32

34

36

35

1
37

1
33

1

Total
1
5
27
25
Total
1
14
34

Total
1
6
48
96
13
1
1
6
5

15

17

19

21

23

25

27

29

31

33

35

37

2
6
4

2
6
5

1
6
5

2
6
4

1
5
4

1
5
5

1
4
3

1
4
3

3
3

1
2
2

2
2

2
2
Total
1
20
81
67

1
1

45

Chapter 5
Details of the numerical calculation
5.1

Parameters of the model

The following values are used for the parameters of the model:
parameter
ϵc
β0
κ
G
∆′
ϵj0 − λc184(182 W)Os
given in table 5.1
0.24(0.22)
0.20=h̄ωβ0 (Nj0 + 3/2)
0.120 MeV
0.527 MeV
(0.564 MeV)
0.639 MeV
−0.629 MeV (−0.552 MeV) −0.480 MeV
182

W

As for 182 W, besides the best fitted parameter set (β0 = 0.24, · · ·), we used another set(β0 = 0.22, · · ·) for the sake of comparison. These values are determined in the manners
described below:
1. ϵc : The energy-scaling parameter of the core (=2h̄2 /Bβ0 2 ).
It is adjusted so as to fit the excitation energy of the 2+
gr state to the experimental
value ( 0.100 MeV for 182 W and 0.120 MeV for 184 Os ) including the eﬀects ofcoupling with the particles.
2. β0 : The deformation parameter.
The value of β0 is determined in order to reproduce the experimental transition+
amplitudes between the 0+
gr state and the 2gr state [70] including the eﬀects ofcoupling with the particles. Calculated reduced matrix elements for this transition46

are given in table 5.2. It can be seen from the table that the experimental values
are reproduced within ±2% by using β0 =0.24 for 182 W and within the interval from
−3% to +1% by setting β0 =0.20 for 184 Os. Thus we determine not to change thevalue of β0 from core to core and to fix it at the single value for each nucleus. As
for 182 W, we also use β0 =0.22 for comparison.
3. κ : The strength of the quadrupole-quadrupole interaction.
Its value is obtained using eq. (4.68), where a harmonic oscillator single-particle
potential is assumed. We calculated the value of κ for a Woods-Saxon potential
with parameters given in ref. [8] (for A=182 and Z=76) and obtained κ = 1.07 ×h̄ωβ0 (Nj0 + 3/2) for the νi13/2 orbital. But we do not employ this value because
the parameters for the spherical potentials do not seem very reliable especially in
deformed nuclei.
4. G : The strength of the two body interaction in the νi13/2 orbital.
We used eq. (4.58) to determine its value.
5. ∆′ : The strength of the Cooper-pair exchange interaction.
Its value is determined so that the experimental excitation energy of the K π = 10+
isomer ( 2.231 MeV for 182 W and 2.365 MeV for 184 Os ) is reproduced when theaxially symmetric rotor is employed as the core. The obtained values of ∆′ are used
for all the cores because the diﬀerence of core models does not aﬀect significantly
the energy level of the isomer.
By using eq. (4.76), we can estimate the value of ∆′ microscopically. By employing
the single-particle space of two oscillator shells (N =5,6), where the parameters of
the Nilsson potential are taken from ref. [8], we solved the BCS gap equation with
G=0.120 MeV under a constraint ⟨#neutron⟩ = 108 and obtained∆′micro = 0.633 MeV(∆micro = 0.821 MeV) for 182 W, β0 = 0.24 ,∆′micro = 0.714 MeV(∆micro = 0.943 MeV) for 184 Os, β0 = 0.20 .

These values are only a little larger (by 11-20%) than the adopted values.

47

6. (ϵj0 − λc ) : The energy necessary to transfer a neutron from the Fermi level to thei13/2 orbital.
It is determined so that the expectation value of n (the number of neutrons in thei13/2 orbital) becomes 10 in the ground state when the axially symmetric rotor isused as the core. The value of n is expected to be 10 from the Nilsson diagram ofenergy levels for N =108 isotones. These values are used for all the cores because
the expectation value of n is found not sensitive to the shape of the γ-potential.

5.2

Truncation of states and its convergence

5.2.1

Full inclusion of the particle states

We solve the eigenvalue equation for HPR by numerical diagonalization. The number ofparticle states |j0n ⟩ (n = 0, 2, 4, · · · , 14) is 428 (see table 4.6, the multiplicity due to thedegeneracy with respect to m is not counted) and all of them are taken into account.
Therefore we do not have to assume the weakness of the coupling between the core andthe particles, although we use so-called weak coupling bases.
In the early works using the laboratory-frame particle-rotor model, only low-seniority(v ≤ 2) states are included in order to keep the matrix dimensions in manageable size[56]-[58]. The seniority truncation was, however, shown to be a poor approximation atleast when the j0 -orbital is half occupied [59]. This result is plausible considering thedeformation of the nucleus. Therefore we should not truncate states according to thevalue of seniority, which is a suitable quantum number for labeling states in spherical
nuclei.
We take into account all the allowed values also for n. The probability distributionof n are shown in table 5.3 for states of the particles ⊗ core system. It seems that stateswith n ≤ 2 may be excluded on account of their small probabilities. But reduction of
matrix size due to such truncation is only very small. Thus such truncation is useless.

5.2.2

Truncation of the core states

The angular momentum of the core is restricted by R ≤ 20. This truncation is confirmed
to have no eﬀects on the decay amplitudes of the K π = 10+ isomers. The number of48

states of the γ-soft core is, however, infinite at each spin R and we must truncate themin numerical diagonalization. We use two schemes of truncation.
1. The λ′ -truncation scheme
In this scheme, core states with λ ≤ λ′max are included in the case of the γ-unstablemodel, where λ′max is an arbitrarily chosen number. When V (γ) is not constant,the same number of states from the yrast level for each R are taken into accountas are included in the γ-unstable case using the same λ′max . An illustration of thistruncation scheme is given in figure 5.1 for a rather γ-soft core. The energy spectrumof this core is hardly disturbed compared to the unperturbed (V (γ) = 0) spectrum(eq. (4.25)) except for in the neighborhood of the yrast levels.
It should be noted that the core states are obtained by diagonalizing Hγ−soft in thespace of λ ≤ 24 when V (γ) ̸= 0. Thus for more γ-stiﬀ potentials, less number ofcore eigenstates are necessary since the coupling with the particles has smaller eﬀects
compared to V (γ). The energy spectrum for a rather γ-stiﬀ core is shown in figure5.2. By using the same λ′max , higher-lying states can be included in comparison withthe case of the γ-soft core (figure 5.1).
2. The Nγ -truncation scheme
In the harmonic γ-vibrational model (neglecting the rotation-vibration coupling),
states are labeled by R, Nγ , and |Kc |, where Nγ is the number of γ-vibrational
phonons. The following values are taken on by |Kc | for each Nγ :{|Kc | = 2Nγ , 2Nγ − 4, 2Nγ − 8, · · · ,02for even Nγ ,for odd Nγ .

(5.1)

In the Nγ -truncation scheme, we include core states having Nγ not greater thanNγmax , where Nγmax is an arbitrarily chosen number. For anhamonic cores (like coresof the γ-soft model and the triaxial rotor model), the same number of states fromthe yrast level for each R are taken into account. The number of states for eachNγmax and each R is given in table 5.4. Figure 5.3 illustrates this truncation scheme.
We show the eﬀects of truncation of core states on isomeric decay probabilities andsome energy levels for 182 W in table 5.5 and for 184 Os in table 5.6. The employed cores are49those of the γ-soft model whose γ-softness V1 is determined to reproduce the experimental
energy of the γ-bandhead while V2 is assumed to be zero.
In the Nγ -truncation scheme, some isomeric decay probabilities do not converge withrespect to Nγmax even if 4-phonon states are included. In order to calculate small quantities
like transition matrix elements of isomeric decays, small components of wave functions
must be treated very carefully. This truncation scheme is not suitable for treating isomeric
decays in γ-soft nuclei.
On the other hand, in the λ′ -truncation scheme, it seems suﬃcient to take into accountthe core states with λmax =11 in order to obtain enough accuracy in the calculation of the
decay probabilities. Hence in the calculations where the γ-soft model is employed, we
include core states with R ≤ 20 and λmax = 11. In other words, in terms of energy levels,
we include the core states with energies about less than 8 MeV. (ϵc ≃ 0.2 MeV.)As for the triaxial rotor model, number of states at each spin R are finite and relatively
small, and we can include all the states (satisfying Bohr’s symmetry) with R ≤ 20. Theenergy spectrum for γ0 =15◦ rotor is shown in figure 5.4.

5.3

Convergence in the Lanczos algorithm for the Diagonalization

The dimension of the matrix to be diagonalized at I = 10 is 3158 if only the ground bandstates of the core are taken into account. It is 40362 when core states with λmax = 11
and R ≤ 20 are employed. (See table 5.6.) Thus some eﬃcient methods are necessary innumerical diagonalization.
We use the Lanczos algorithm [71]. In this method an initial vector is multiplied by the
hamiltonian to generate the second vector with intervening orthogonalization. And thedescendent vectors are generated by iterating this procedure. The sequence of vectors thus
obtained makes new bases which almost exhaust the eigenstates whose eigenvalues arenear the both ends of the spectrum, even when the procedure is stopped in the half-way.
The convergence of isomeric decay amplitudes are confirmed as for the number of
iterations, i. e. the number of new bases thus obtained. It is found suﬃcient to iterate
about 110 times. Energy levels converge more quickly. We obtained the results presented
50

in this paper by iterating 120 ∼ 150 times.

51

Table 5.1: The energy-scaling parameter ofthe core ϵc (=2h̄2 /Bβ0 2 ) used in the calculation. They are determined so as to fit
the experimental values of E(2+gr ) ( and also+E(2γ ) in parts (C)∼(E)) including the effects of coupling with the particles.

Core model

Table 5.2: Calculated and experimental reduced E2 matrix elements between the 0+
grstate and the 2+
gr state.

Core model

ϵc [MeV]

182
184
W
W
Os
β0 =0.24 β0 =0.22
β0 =0.20(A) γ-vibrational model(0, 0)——
173(−2, 0)200
184173(−4, 0)200
184174(−6, 0)201
185174(−8, 0)202
185
175(−10, 0)
202
186
175
(−12, 0)
202
186
175
(B) Triaxial rotor model
γ0 =30◦——179
γ0 =25◦201
186
175
γ0 =20◦200184174
γ0 =15◦202
186176
γ0 =12◦
204
188
177
γ0 =10◦205
189178
γ0 =0◦
208
191
181
184
+
Os, β0 =0.20(C) Those reproducing E(2γ ) of(−2.99, 0)——
173(−5.71, 3)——
173(−10.05, 7)——
173(−20.12, 15)——173(−41.59, 30)——
173(−72.96, 50)——
173
γ0 =14.89◦
—
—
176
182
(D) Those reproducing E(2+
)
of
W,
β
0 =0.22γ(−6.96, 0)—185—
γ0 =12.18◦
—
188
—
+
182
(E) Those reproducing E(2γ ) ofW, β0 =0.24(−6.05, 0)201——
γ0 =12.32◦
204——
Experiment
204
204
179

182

182
184
W
W
Os
β0 =0.24 β0 =0.22
β0 =0.20(A) γ-vibrational model(0, 0)—
0.16300.1780(−2, 0)0.18310.17540.1973(−4, 0)0.19140.18440.2115(−6, 0)0.19740.19090.2215(−8, 0)0.20190.19570.2286(−10, 0)0.20540.19940.2337(−12, 0)0.20810.20230.2377(B) Triaxial rotor model
γ0 =30◦——0.1758γ0 =25◦0.16550.16170.1916
γ0 =20◦0.18810.18350.2173
γ0 =15◦0.21000.20480.2426
γ0 =12◦0.22130.21580.2556
γ0 =10◦0.22770.22200.2630γ0 =0◦
0.24290.23670.2805+
184(C) Those reproducing E(2γ ) ofOs, β0 =0.20(−2.99, 0)——0.2050(−5.71, 3)——
0.2071(−10.05, 7)——
0.2097(−20.12, 15)——
0.2129(−41.59, 30)——
0.2165(−72.96, 50)——
0.2194
γ0 =14.89◦——
0.2431182)
ofW,
β0 =0.22(D) Those reproducing E(2+γ(−6.96, 0)—0.1935—γ0 =12.18◦—
0.2152—182(E) Those reproducing E(2+
)
of
W,
β
=0.24
0
γ
(−6.05, 0)0.1975——γ0 =12.32◦
0.2202——(V1 , V2 ) or γ+
2⟨0+
gr ||M(E2)||2gr ⟩ [e fm ](V1 , V2 ) or γ

52

182

Table 5.3: Probability distribution of the number of particles in the νi13/2 orbital.

184

n

0
2
4
6
8
10
12
14
Total

Os, (V1 , V2 ) = (−2.99, 0), β0 =0.20

Probability Probability The number ofat 10+
at 10+
particle states
K=10
gr
9.05×10−8
1.49×10−5
8.40×10−4
1.99×10−2
1.94×10−1
6.05×10−1
1.73×10−1
6.83×10−3
1

1.×10−12
6.98×10−7
1.14×10−4
6.09×10−3
1.25×10−1
8.30×10−1
3.89×10−2
1.38×10−7
1

1
7
55
151
151
55
7
1
428

The Number of statesof the total system
at I=10
11
591
5247
14332
14332
5247
591
11
40362

Table 5.4: The number of states of the γ-vibrational model.
Truncation
schemeNγmax =0Nγmax =1
Nγmax =2
Nγmax =3
Nγmax =4
Nγmax =5

R
0
1
1
2
2
3
3

1

2
1
2
3
4
5
6

3
1
1
2
2
3

4
1
2
4
5
7
8

5
1
2
3
4
5

6
1
2
4
6
8
10

7
1
2
4
5
7

8
1
2
4
6
9
11

53

9
1
2
4
6
8

10
1
2
4
6
9
12

11
1
2
4
6
9

12
1
2
4
6
9
12

13
1
2
4
6
9

14
1
2
4
6
9
12

15
1
2
4
6
9

16 · · ·
1
2
4
6
9
12

Table 5.5: Eﬀects of the truncation of the core states on the isomeric decays and some+energy levels for 182 W. B(M1) and B(E2)a belong to the transition from the 10+
2 to 10gr .
B(E2)b belongs to the transition to 8+
gr . The last column gives the number of states of thecore without counting the degeneracy with respect to the magnetic quantum number.
182

Truncation
scheme′λmax = 11
λ′max = 10
λ′max = 9λ′max = 8
λ′max = 7
Nγmax = 4
Nγmax = 3
Nγmax = 2
Nγmax = 1
Nγmax = 0B(M1)[ µ2N ]
5.87×10−8
5.87×10−8
5.97×10−8
6.08×10−8
5.87×10−8
5.76×10−8
7.68×10−8
1.90×10−9
5.90×10−7
6.63×10−6

W, (V1 , V2 ) = (−6.05, 0), β0 =0.24, R ≤ 20
B(E2)a[ e2 fm4 ]1.77×10−4
1.77×10−4
1.80×10−4
1.82×10−4
2.14×10−4
1.67×10−4
2.86×10−4
7.17×10−5
3.53×10−3
1.43×10−2

B(E2)b[ e2 fm4 ]6.10×10−5
6.10×10−5
6.21×10−5
6.70×10−5
7.66×10−5
6.10×10−5
6.63×10−5
2.84×10−5
2.26×10−4
7.81×10−3E(2+γ)[ MeV ]
1.222
1.222
1.222
1.224
1.230
1.224
1.237
1.304
1.452
—

E(10+
gr )
[ MeV ]
1.626
1.626
1.627
1.636
1.679
1.625
1.625
1.623
1.613
1.723

E(10+
Number of2)[ MeV ] core states
2.199
123
2.199
98
2.199
76
2.199
57
2.200
42
2.199
133
2.198
92
2.197
58
2.191
30
2.222
11

Table 5.6: Eﬀects of the truncation of the core states on the isomeric decays and some+energy levels for 184 Os. B(M1) and B(E2)a belong to the transition from the 10+
2 to 10gr .
B(E2)b belongs to the transition to 8+
gr . The last column gives the number of states of
the coupled system consisting of the core and the i13/2 neutrons at I=10 and at a certain
value of M .
184

Truncation
scheme′λmax = 11
λ′max = 10
λ′max = 9λ′max = 8
λ′max = 7
Nγmax = 4
Nγmax = 3
Nγmax = 2
Nγmax = 1
Nγmax = 0B(M1)[ µ2N ]
2.64×10−5
2.64×10−5
2.64×10−5
2.68×10−5
3.04×10−5
2.63×10−5
2.77×10−5
2.21×10−5
1.25×10−4
9.45×10−4

Os, (V1 , V2 ) = (−2.99, 0), β0 =0.20, R ≤ 20
B(E2)a[ e2 fm4 ]4.51×10−2
4.50×10−2
4.50×10−2
4.51×10−2
5.15×10−2
4.50×10−2
4.70×10−2
2.97×10−2
2.10×10−1
3.95×10−1

B(E2)b[ e2 fm4 ]1.45×10−2
1.45×10−2
1.46×10−2
1.56×10−2
2.16×10−2
1.45×10−2
1.51×10−2
1.25×10−2
5.25×10−2
7.02×10−1

54

E(2+
γ)
[ MeV ]
0.943
0.943
0.943
0.944
0.946
0.944
0.949
0.991
1.189
—

E(10+
gr )
[ MeV ]
1.791
1.791
1.792
1.799
1.836
1.791
1.791
1.788
1.780
1.916

E(10+
Dimension
2)[ MeV ]
at I=10
2.354
40362
2.354
31848
2.354
24256
2.354
17786
2.354
12456
2.354
42274
2.354
29068
2.354
17966
2.357
9160
2.423
3158

Figure 5.2: The scheme of the λ′ -truncation
for the states of the core. A rather γ-stiﬀ
core is used.

Figure 5.1: The scheme of the λ′ -truncation
for the states of the core. A rather γ-softcore is used.

55

Figure 5.3: The scheme of the Nγ truncation for the states of the core.

Figure 5.4: Energy spectrum of the triaxial
rotor model with γ0 =14.89◦ .

56

Chapter 6
Results of numerical calculations
6.1

Features of axially symmetric rotor ⊗ ν i13/2 system

At first let us learn the decay of high-K isomers in a model in which the axially symmetricrotor is coupled with the i13/2 neutrons in order to see the eﬀects of Coriolis mixing
separately from those concerning the γ-degree of freedom.
We calculate the partial half-lives of the K π = 10+ isomers, considering only the M1+
and the E2 transitions into the members of the ground band ( the 8+
gr and the 10gr states). Transition energies used in calculating transition probabilities are taken from experiments to concentrate our attention on the changes in wave functions. The contributionof internal electron conversion process is negligible (αK < 6 × 10−2 ) for the enegies andthe multipolarities of these transitions [13] [72].
Using parameters described in chapter 5, we obtain the follwoing partial half-lives.
182

W
β0 =0.24
Experiment
1.4µs↓ × 2300
Calculation 3.20 ms

184

×12

←

×1600

←

Os
β0 =0.20
120 ns↓ × 172.05µs

The calculated half-lives are much longer than the experimental ones.
Now we show the eﬀects of changing parameters. Among the parameters of the model,
ϵc and κ (or β0 ) have large eﬀects on the decay amplitudes of the isomer, while G, ∆′ ,57

and ϵj0 − λc do not aﬀect them much (see figure 6.1).
Eﬀects of changing the energy scaling parameter of the core (ϵc ) are shown in table
6.1. Other parameters are those adjusted for 184 Os described in chapter 5. The energy
level of the 2+
gr state (including the eﬀects of coupling with the particles) is shown in thesecond column (E(2+gr )). The half-life of the isomer decrease rapidly with the increase inϵc : The B(M1) and the B(E2) values of the decay from the isomer increase by a factor of
about 10 when E(2+gr ) increases by 10 keV from 120 keV. But the large ϵc reproducing theexperimental half-life can not consistently reproduce the experimental energy spectrumof the ground band.
This strong dependence of isomeric decay probabilities on ϵc is due to the fact that ϵc
is the factor of the Coriolis mixing perturbation (see eqs. (3.9) and (??)). The probability
distribution of the K-quantum number is shown in figure 6.2 for the 10+
gr state and theK π = 10+ isomer. We can see that the degree of K-mixing becomes large when ϵcincreases. The anomalous enhancement of the probability around K=10 (K=0) for theπ= 10+ isomer) means the mixing of the two states caused by the10+
gr state (the K

crossing of their energy levels.
In table 6.2, we show the eﬀects of changing the strength κ of the quadrupolequadrupole interaction for 184 Os. The quantity κ/h̄ω(Nj0 + 3/2) given in the first column
is equal to the deformation parameter β0 when a harmonic oscillator single-particle potential is assumed, although we did not change the value of β0 in the definition (4.99) of theE2 transition operator M(E2) (α2µ ∝ β = β0 ). Since the change in κ largely aﬀects thenumber of neutrons (n) in the i13/2 orbital, we adjust the value of ϵj0 − λc for each valueof κ in order to maintain ⟨n⟩ = 10 in the ground state. From the table, it can be said thatκ has also large eﬀects on the decay amplitudes. In order to reproduce the experimental
half-life of the isomer, κ/h̄ω(Nj0 + 3/2) must be 0.17, which means, however, too small
deformation for 184 Os.
These large eﬀects of the change in κ on the half-lives of the isomer can be explained
as follows: The quadrupole-quadrupole interaction HQQ separates the energy levels oforbitals having diﬀerent Ω. When its strength κ becomes large, the energy denominatorof the Coriolis perturbation increases and the mixing of K is depressed. The changes in58

the K-mixing are shown in figure 6.3.
Considering this strong dependence of the Coriolis mixing on parameters, we can saythat the diﬀerence in the half-lives of the isomer between the two nuclei are not simply
ascribed to the diﬀecrence in the γ-softness. In fact, from table 6.1, when the energy
184
level of the 2+Os) to 100 keV (like in 182 W),
gr state is decreased from 120 keV (like inthe half-life becomes longer by a factor of 40. From table 6.2, when the deformation
parameter β0 (=κ/h̄ω(Nj0 + 3/2) ) is increased from 0.20 (like in 184 Os ) to 0.24 (like in
182W), the half-life becomes longer by a factor of 30. Totally, the calculated half-life for182

W is 1600 times as large as that for 184 Os, which exceeds the experimental diﬀerence(12 times) without the γ-degree of freedom.
In conclusion, using the axially symmetric rotor, the absolute values of the experimental half-lives are overestimated with reasonable values of κ (or deformation β0 ) and ϵc (ormoment of inertia). The analyses in this section invoke the introduction of the γ-degreeof freedom.

6.2

Eﬀects of the γ-degree of freedom

Next, we take into account the γ-degree of freedom and calculate the half-lives of theisomers. In this section we employ the two extreme models treating the γ-degree of
freedom. One is the γ-vibrational model, i. e. the γ-soft model of eq. (4.11) which has aγ-potential of eq. (4.20) with V2 =0:
V (γ) = ϵc V1 cos 3γ .

(6.1)

In the model, the variable γ fluctuates around γ=0◦ . The other is the triaxial rotor modelof eq. (4.33), in which γ is fixed at some constant value γ0 .
The parameters of the model are fixed at the values described in chapter 5. In thecalculation of half-lives, we take into account only the M1 and the E2 transitions into the++ground band states (8+gr and 10gr ). Although the isomer can decay into the 8γ state in

our model, its contribution to the total half-life is very small according to our calculation:+
The transition from the isomer to the 8+
γ state and the transition to the 8gr state haveamplitudes of the same order, but the energy of the γ-ray in the former transition is about59one-tenth as large as that in the latter transition. The I=9 state to which the isomer
decays with a large fraction in experiment for 184 Os does not exist in our model space.
The transition energies are taken from the experiments as in the last section.
In figures 6.4-6.6 we show the relationship between the half-life of the K π = 10+ isomer
and the energy level of the γ-bandhead for 184 Os using β0 =0.20, 182 W using β0 =0.22, and182

W using β0 =0.24, respectively. Figure 6.5 (β0 =0.22) is given for reference, although+
the experimental value of B(E2;2+
gr → 0gr ) is reproduced well when β0 =0.24 (as in figure6.6). The two models with some values of V1 (for the γ-vibrational model) and γ (forthe triaxial rotor model) are used in each fugure. We can see a general tendency, as we
expected, that the half-lives of the isomers become shorter as the core becomes more γ-soft
or more γ-deformed. When the triaxial rotor model is used in figures 6.5–6.6, however,the partial half-life behaves irregularly with respect to γ0 . This anomaly would be due tothe interference between the Coriolis interaction and the triaxial potential exerted by thecore.
The most remarkable point in figures 6.4–6.6 is that the half-lives calculated with theγ-vibrational model are shorter than those calculated with the triaxial rotor model bymore than two orders of magnitude. This characteristic feature is discussed in detail in
the following sections in comparison with quantities of collective character which are notchanged much between the two models.
When the γ-softness (V1 ) or the size of the γ-deformation (γ0 ) is determined by fitting
the energy of the γ-bandhead to the experimental value (including the eﬀects of couplingwith the particles) for each core model, the experimental half-lives are located between(or in the neighborhoods of) those calculated with the two extreme models. Assuming
the present set of parameters are reliable, we can say that the nucleus 182 W is much likethe γ-vibrational model while the nucleus 184 Os is rather like the triaxial rotor modelfrom the viewpoint of K-isomeric decays. It should be noted that this result does notnecessarily mean the rigid triaxial nature of some transitional nuclei (184 Os), as the successof Meyer-ter-Vehn’s model did not (see chapter 2). In later sections, we show that the
crucial point in these K-isomeric decays is how large the quantum fluctuation in the sizeof triaxial deformation is, not whether the equilibrium shape is triaxial or not. It can be
60

said that we can extend our knowledge about the shape of the γ-potential from the degreeof violation of the K-selection rule.
We show the probability distribution of the K-quantum number, defined by eq. (4.94),
184
for the 10+
Os in figs. 6.7 and 6.8. The γ-vibrational model
gr state and the isomer inis employed in figure 6.7, while the triaxial rotor model is utilized in figure 6.8. If weutilize the axially symmetric rotor model (γ=0◦ ), the probability distribution is linear in
the logarithmic scale: The Coriolis interaction is just perturbative. If we introduce theγ-degree of freedom, the probabilities of large-K (small-K) components increase largely in
the ground band state ( in the isomer ) for the both models. Since the probabilities of themixing of the K-quantum number have strong correlations with the calculated half-lives
shown in figure 6.4, we can say, as we expected, that the reduction in the half-lives follows
the mixing of K-quantum number.

6.3

Eﬀects of the coupling with the particles on the
core

Before discussing the K-isomeric decays more intensively, we just mention how the properties of the core are changed through the coupling with the ∼10 neutrons in the i13/2
orbital.
1. The moment of inertia increase by ∼20% (judging from the energy of the 2+
gr statewhen the axially symmetric rotor model is used).
2. The E2 transition amplitudes increase by ∼7% (in the transition from the 2+
gr stateto the 0+
gr state).
3. The M1 transitions occur through the coupling with the particles, while they do not⃗ where R⃗ is a conserved
exist between the states of bare cores (since M(M1)core ∝ R,
quantity).
4. The γ-softness is decreased when the γ-soft model is used. The probability distribution function of γ for the state |IM α⟩ (eq. (??)) of the system consisting of the61core and the particles are defined as
ρIα (γ0 ) =

∑ ∑
2
1
αc
⟨IM α|δ(γ − γ0 )|IM α⟩ =AIα, (6.2)Rαc Jαp ΦRKc (γ0 )3 sin 3γ0
Jαp RKc αcwhich satisfies the normalization condition of
∫ π/3

3
0

ρIα (γ0 ) sin 3γ0 dγ0 = 1 .

(6.3)

In figure 6.10, these probability distributions are shown for the γ-potentials given in
the upper right hand portion of figure 6.9. Comparing the distributions with thosefor the bare cores (also given in the figure), we can see the γ-stiﬀening eﬀects of theparticles.
The γ-driving eﬀects of the particles are likely to be simulated by an eﬀective γpotential, which can be estimated using the single-j Nilsson model: We put 10
neutrons into the i13/2 orbital with a quadratically deformed single-particle potential
specified by β0 and γ0 . The energy of the ground state configuration is regarded as
the potential energy for deformation V p (β0 , γ0 ), where the superscript p attached
to V indicates that the potential is of the particles. We determine the value of
V1p so that 2ϵc V1p = V p (β0 , 0◦ ) − V p (β0 , 60◦ ). We obtain ϵc V1p = −1.27 MeV (i.
e. V1p ≃ −6) for 182 W (β0 =0.24) and ϵc V1p = −1.05 MeV (i. e. V1p ≃ −5) for184

Os. These eﬀective γ-potentials of the particles and the γ-potentials of the barecore are of the same order of magnitude. We must include this contribution fromthe particles together with the γ-potential of the bare core in order to obtain theeﬀective γ-potential for the entire nucleus.
5. The eﬀective size of γ-deformation of the coupled system (γ eﬀ ) is hardly changed
from the value of γ0 of the bare core when the triaxial rotor model is used. We
can define γ eﬀ for each state in such a way that the quadrupole moment 1 of a baretriaxial rotor having γ0 = γ eﬀ agrees with that of the coupled system. In the caseof γ0 = 15◦ for 184 Os, γ eﬀ turns out nearly equal to γ0 at low spin and slightly less(by ∼ 1◦ ) than γ0 at I ∼ 10 in the states of the ground band.
Quadrupole moments (∝ ⟨I||M(E2)||I⟩) are sensitive to the γ-deformation, while E2 transitions(⟨I − 2||M(E2)||I⟩) are not.
1

62

6.4

Eﬀects of the potential shape

We investigate the eﬀects of the shape of the γ-potential to explain the diﬀerence inthe half-life of the K-isomer between the two extreme models. The γ-soft model having
positive values of V2 mediates between the two models studied in the last section. We
consider the isomer only in 184 Os in this section.
We use the γ-potentials of the form of eq. (4.20). We vary V2 to change the shapeof the γ-potential, while V1 is determined for each V2 so that the experimental energy
level of the γ-bandhead is reproduced including the eﬀects of coupling with the particles.
Other parameters are fixed at the values adjusted for 184 Os as described in chapter 5.
The shapes of the γ-potentials used in the calculation are illustrated in the upper righthand portion of figure 6.9. The γ-potential changes in the shape from a potential having
a relatively shallow minimum at γ=0◦ (the γ-vibrational model) to a deep potential wellhaving a minimum at a finite value of γ (close to the limit of the triaxial rotor model). In
the upper left hand portion of figure 6.9, the shapes of γ-potentials for the γ-vibrationalmodel (V2 =0) are also shown for comparison.
On the left hand side of figure 6.11, we show the eﬀects of changing the shape of the
γ-potential on the decay probabilities of the K π = 10+ isomer. The probabilities of the
decay of the isomer decrease rapidly as the shape of the γ-potential approaches the limitof the triaxial rotor model.
Collective transitions are, however, not aﬀected much. Some of them are shown on
the right hand side of figure 6.11. Among the transitions and moments in the groundband, the γ-band, and the K π = 10+ band (I ≤ 12), the most largely changing ones ofcollective character are the quadrupole moments of the high spin states of the γ-band.
They are presented as B(E2) values in the figure, which change by a factor of order one
with the replacement of the γ-vibrational core by the triaxial rotor one. But their changes
are much less than the changes in the isomeric transitions which vary by a factor of order
two. Calculated transition probabilities from the isomer are summarized in table 6.3.
Other reduced E2 transition probabilities at low spin are shown in table 6.4 for reference.
The lower right hand portion of figure 6.9 shows the probability distribution of the63variable γ in the ground state. (The particles are not coupled with 2 .) It is defined as
ρ′Rαc (γ0 ) = ⟨RM αc |δ(γ − γ0 )|RM αc ⟩ = 3 sin 3γ0

R
∑

c
|ΦαRK
(γ0 )|2 .
c(6.4)

Kc =−R

The prime (′ ) is attached to ρRαc in order to designate that the volume element (3 sin 3γ)
is inlcuded. The follwoing normalization condition is fulfilled by ρ′Rαc ,∫ π/30ρ′Rαc (γ0 )dγ0 = 1 .

(6.5)

We can see that the peak of the distribution of γ stays at almost the same location,
but the tail of the distribution decreases drastically with the change in the shape of the
γ-potential. Therefore the origin of the diﬀerence between collective transitions and theisomeric ones seems that the former are determined by the average value of γ, while thelatter are determined by the fluctuation in γ, i. e. the tail of the wave function at large
γ: The components of large γ-deformation are necessary to mix the K-quantum number
and to violate the K-selection rule. Further investigations are given in the next section.
As for this enhancement of the K-isomeric transitions due to the γ-softness, otherexplanations are conceivable, which turn out unsuccessful in this case.
1. The γ-soft core can change its shape at high spin (so as to increase the momentof inertia), while the triaxial rotor can not. But in our calculation such a change
in shape is found very small up to I=10, which is seen from figure 6.10. Morespecifically, the value of γ eﬀ (defined in section 6.3) varies from γ eﬀ ≃ 17◦ (at I=2)
to γ eﬀ ≃ 15.5◦ (at I=10) in the ground band states of the system consisting of
the particles and the γ-soft core with (V1 , V2 ) = (−2.99, 0). This change in γ eﬀ is
opposite [50] [52] to explain the diﬀerence in the half-life between the two models.
2. The isomeric state is expected to be more γ-soft than no-quais-particle states on
account of the change in the configuration, when γ-soft cores are used. The γdriving forces of the particles in the isomer are estimated to be weaker than thosein the ground band states by a factor of 0.7 judging from the eﬀective γ-potential
calculated for the lowest one-particle one-hole state in the framework of the single-j
2

Probability distribution of γ for the coupled system are found in figure 6.10, where the volume elementof (3 sin 3γ) is not included.

64

Nilsson model described in section 6.2 . But such eﬀects are also small, as is shownin figure 6.10.
Now let us check the details of the isomeric decay for 184 Os. The multipolarity ofthe transition from the isomer to the 10+
gr state is dominantly M1 in experiment. In our
calculation, the contribution of M1 is ≥ 0.97 for the cases given in the part (C) of table
6.3, in good agreement with the experiment.
+
The experimental branching ratio of the 8+
gr state to the 10gr state is 2.0 in the decayof the isomer. The calculated ratios are 0.47 to 0.67 for the cases given in the part (C) of
table 6.3, which are not in quantitative agreements with the experiment.
Before ending this section, we discuss the shape of the γ-potential preferred by the
K-isomeric decay rates for 184 Os. In order to reproduce the experimental half-life of theisomer in addition to E(2+γ ), we have to employ a γ-potential having (V1 , V2 ) ≃ (−62, 43),
which is obtained by interpolating the calculated half-lives given in table 6.3. The shapeof this γ-potential is shown in figure 6.12. It is worth while noting that the form of the
γ-potential is restricted to linear combinations of P1 (cos 3γ) and P2 (cos 3γ) (eq. 4.20) in
our calculations. Inclusion of higher order terms (P3 (cos 3γ), P4 (cos 3γ),· · ·) will bringabout diﬀerent γ-potentials which also reproduce the half-life and E(2+γ ). In particular,
the shape of V (γ) near γ = 0◦ may be easily changed through such inclusion of the higher
order terms, since neither the K-isomeric decay rates nor collective quantities like E(2+γ)
have much information on the wave function at small γ. The potential energy at γ = 60◦
also has large ambiguities: it is very high (29 MeV from the ground state energy) and isunlikely to have significant meaning. The determination of the coeﬃcints of the higher
order terms (V3 , V4 , · · ·) is a problem for future works.

65

6.5

Further investigation of the fast K-isomeric decays due to the γ-softness

In this section, we investigate the mechanism through which the K-isomer decays quickly.
At first we calculate the probability distribution of the K-quantum number for each valueof γ. It is defined for the state |IM α⟩ (eq. (??)) as
ρIα (γ0 , Kc , Kp ) =

∑

|CγIα
|2
0 Jαp Kc Kp

Jαp

=

∑

(6.6)
′

Jαp R′ α′c Rαc

α′

∗ Iα
αc
R JI
RJI
c
{AIα
R′ α′c Jαp BKc Kp ΦR′ Kc (γ0 )} ARαc Jαp BKc Kp ΦRKc (γ0 ) .

It fulfills the normalization condition of
∑
Kc Kp

∫ π/3

3
0

ρIα (γ0 , Kc , Kp )dγ0 sin 3γ0 = 1 .

(6.7)

The expression for the total K-quantum number, ρIα (γ0 , |K|), is given in the same manneras eq. (4.94).
π+
In figures 6.13 and 6.14, ρIα (γ, |K|) is shown for the 10+
gr state and the K =10 isomer,

respectively, for 184 Os. The γ-potentials are those reproducing the experimental E(2+γ)
including the eﬀects of coupling with the particles. The distribution of the K-quantum
number,∫ π/3ρIα (|K|) = 30ρIα (γ0 , |K|) sin 3γ0 dγ0 ,(6.8)

is also shown. (The definition (6.8) is equivalent to eq. (4.94).) From figure 6.14, we cansee that the large deviation from the Coriolis-like K-mixing (i. e. probability distributions
linear in logarithmic scale) is mainly due to the components of the wave function withlarge γ-deformations. When we alter the core from the γ-vibrational one (part A ofthe figure) to the γ-stiﬀ one (part C), the components with large γ-deformations are
depressed on account of the decrease in the amplitude of the quantum fluctuation in γ.
Consequently, the anomalous K-mixing diminishes.
Another point to be seen in figure 6.14 is the existence of the second peak around K=0
when γ ∼ 0◦ : The two states mix with each other in spite of the large diﬀerence in the
structure and in the energy level (∼ 0.5MeV). The height of the second peak is lowered
when the core becomes γ-stiﬀ (A → C). Therefore this mixing is likely to be caused by66the γ-softness. It may be regarded as the result of the barrier penetration process throughthe γ=60◦ shape presented by Chowdhury and his coworkers [13] described in chapter 1
3.
Next, we study the distribution of the isomeric transition amplitudes with respect toγ. The E2 transition amplitude can be expressed as an integral with respect to the sizeof γ-deformation,⟨final state||M(E2)||initial state⟩ = 3∫ π/3t(γ0 ) sin 3γ0 dγ0 ,

(6.9)

1
⟨final state||M(E2)δ(γ − γ0 )||initial state⟩ .
3 sin 3γ0(6.10)0

wheret(γ0 ) =The values of t(γ0 ) are shown in figure 6.15 for the transitions from the isomer, where theeﬀective charge of the i13/2 neutrons (eeﬀ ) is set to 0 (i. e. only the core part of M(E2) is
considered): The transition amplitudes due to the particles are smaller than those by thecore by an order of magnitude. It is seen from the figure that the main contribution to
the isomeric transitions comes from small γ-deformation components, in spite of the factthat the large-K mixing is principally due to large γ-deformation componets.
In conclusion, the anomalously large K-mixing in the γ-vibrational model is attributed
to the quantum fluctuation in shape into large γ-deformations, as is speculated in section
4. The K-isomeric transitions occur, however, mainly at small γ-deformations. Therefore
the isomeric transitions are likely to occur principally from the small-amplitude groundband-like components (with small γ) to the main component of the ground-band state(and from the main component of the isomer to the small-amplitude K=10 component
in the ground-band state), not directly between the components having large γ. The
fluctuation in γ plays the crucial role in mixing the ground-band state and the isomer. A
probable picture for such mixing is the the barrier penetration mechanism via the largely
γ-deformed shape. But treatments based on such picture can not be quantitative, since
the transition amplitudes are rather widely spread as for γ.
3

It should be noted, however, that we can not see the second peak in the probability distribution ofK, on the contrary to the conjecture in the figure 10 of ref. [13], if we integrate the probability withrespect to γ: The peak is covered by the larger-amplitude components having large γ-deformations, i. e.
the components at the potential barrier, at least for the K π =10+ isomer. This situation is attributed tothe mismatching in energy. It is also worth while noting that the γ-potential of 184 Os is like that in (C),not like that in (A).

67

6.6

Eﬀects of the shape of the γ-potential on energy
levelsIn this section we compare the calculated energy levels (including the eﬀects of couplingwith the particles) with the experimental ones for the nucleus 184 Os. When we limit
the shape of the γ-potential to those with V2 = 0, we have to use V1 ≃ −3 so as toreproduce the experimental energy level of the γ-bandhead. But if we fit the spectrumof the ground band to the experimental one, the best value for V1 is about −5. Thus
γ-potentials including higher multipolarity terms (eq. (4.20)) are required.
In figure 6.16 we show the energy spectrum of the ground, the γ-, and the K π = 10+
bands for the γ-potentials shown in the upper right hand portion of figure 6.9, in which
the energy level of the γ-bandhead is fitted to the experimental value. The agreement of
the spectrum of the ground band with the experimental one is improved by using V2 > 0.
The odd-even staggering of the spectrum of the γ-band is also reproduced rather well byusing a core which lies between the γ-vibrational model and the triaxial rotor model. The
γ-potentials required by these energy levels are consistent with those preferred by the halflife of the K-isomer in a qualitative way. They are also consistent with microscopicallycalculated potentials for nearby nuclei [19] in a qualitative way. The energy spectra of
the same bands for the γ-vibrational potentials (V2 =0) are also shown in figure 6.17 forreference.
The energy spectrum of the yrast band at the first backbending region is shown infigure 6.18. Experimental and calculated spectra are expressed in terms of the angularvelocity ωrot and the kinematical moment of inertia J (1) . Assuming that the energy levels√of a rotational band are expressed by a function of L2 where L = h̄ I(I + 1), these newquantities are defined as follows.
dE
EI+1 − EI−1
≃
dL
2h̄
L
h̄2 (2I + 1)=≃
ωrot
EI+1 − EI−1

ωrot =

at I ,(6.11)J (1)at I .

(6.12)We used three kinds of cores, i. e. , the axially symmetric rotor model, the triaxial rotor
model with γ0 = 14.89◦ , and the γ-soft model with (V1 , V2 )=(−2.99, 0). The latter two68models reproduce the experimental E(2+γ ) including the eﬀects of coupling with the particles. In the γ-soft model, we truncate the core states according to the Nγ -truncation
scheme with Nγmax =4 so as to include enough number of states at high spin. Among thethree models, the triaxial rotor model reproduces the experimental backbending behavior
most excellently. This result is consistent with the prediction of the K-isomeric decayrate. But other factors like the β-vibration would also aﬀect the yrast spectrum as strongas the γ-degree of freedom. Therefore more elaborate studies seem necessary to draw any
conclusion about the γ-degree of freedom from the backbending behavior.
Now we point out the defects of our model in the energy levels. The moment of inertiaof the calculated γ-band is smaller than that of the experimental γ-band. It is also smaller
than the moment of inertia of the calculated ground band: This is a characteristic of theγ-soft model, at least when the γ-potential of the form of eq. (4.20) is assumed. On
the contrary, the calculated moment of inertia of the K π = 10+ band is larger than theexperimental one.
To summarize, the γ-potential which fits the partial half-life of the K-isomer also
improves the agreement of the some calculated energy levels with the experimental ones.
But for some of other levels, we do not obtain very good agreements between the calculated
energy levels and the experimental ones. This is not a very surprising result because
we concentrated on the γ-degree of freedom and suppressed many others, e. g. the βvibrational degree of freedom.

69

Table 6.1: Eﬀects of changing the energy scaling parameter of the core (ϵc ) on the partial
half-life of the K π = 10+ isomer in 184 Os. When ϵc =ϵ0c (=0.2805 MeV), the experimental
E(2+gr ) is reproduced including the eﬀects of coupling with the particles. The last column+
gives the branching ratio of the 8+
gr state to the 10gr state in the depopulation of theπ+
K =10 isomer.
ϵc /ϵ0c
0.4
0.5
0.6
0.7
0.8
0.9
1.0
1.1
1.2
exp.

E(2+
gr )
[ MeV ]
0.052
0.064
0.076
0.087
0.098
0.109
0.120
0.130
0.141
0.120

τ1/2
[ sec ]
1.43×100
4.75×10−2
4.10×10−3
5.26×10−4
8.12×10−5
1.34×10−5
2.05×10−6
2.28×10−7
5.11×10−9
1.20×10−7+Br(8+gr /10gr )2 ×10−5
0.030
0.095
0.201
0.353
0.572
0.882
1.28
1.75
2.0

Table 6.2: Eﬀects of changing the strength of the quadrupole-quadrupole interaction (κ)
on the partial half-life of the K π = 10+ isomer in 184 Os. As for the last column, see thecomments of table 6.1.
κ/h̄ω(Nj0 + 3/2) τ1/2 [ sec ]
0.14
0.16
0.18
0.20
0.22
0.24
0.26
exp.

4.18×10−9
4.64×10−8
3.38×10−7
2.05×10−6
1.08×10−5
5.82×10−5
3.32×10−4
1.20×10−7

70

+
Br(8+
gr /10gr )

1.22
1.12
0.995
0.882
0.780
0.685
0.620
2.0

Table 6.3: Summary of decay properties of the K π = 10+ isomer.
Summary of the transitions from the K π =10+ isomers
Core model E(2+
B(M1)
B(E2)
γ)
+
(V1 , V2 )to 10grto 10+gror
γ0[ MeV ]
[ µ2N ]
[ e2 fm4 ](A) 184 Os, γ-vibrational model(0, 0)
0.631
4.06×10−4 1.99×10+2(−2, 0)0.838
4.42×10−5 8.30×10−2(−4, 0)1.048
1.57×10−5 2.68×10−2(−6, 0)
1.247
6.15×10−6 1.13×10−2(−8, 0)1.429
2.85×10−6 5.51×10−3(−10, 0)1.596
1.55×10−6 3.12×10−3(−12, 0)1.751
9.77×10−7 2.00×10−3(B) 184 Os, triaxial rotor model
γ0 = 30◦
0.243
8.40×10−6 1.34×10−1◦
γ0 = 25
0.311
3.59×10−6 4.15×10−6
γ0 = 20◦
0.497
1.84×10−6 1.09×10−3
γ0 = 15◦
0.929
5.85×10−7 8.81×10−4◦
γ0 = 121.492
2.64×10−7 5.45×10−4
γ0 = 10◦—1.65×10−7 3.70×10−4◦
γ0 = 0—8.08×10−8 1.96×10−4(C) 184 Os, Those reproducing E(2+γ)
−5(−2.99, 0)
0.943
2.64×10
4.51×10−2−5(−5.71, 3)
0.943
1.32×10
1.83×10−2(−10.05, 7)
0.944
6.65×10−6 8.08×10−3(−20.12, 15)
0.943
3.34×10−6 3.71×10−3(−41.59, 30)
0.943
2.11×10−6 2.26×10−3(−72.96, 50)
0.943
1.62×10−6 1.76×10−3
γ0 = 14.89◦
0.944
5.70×10−7 8.68×10−4
Experiment
0.943
9.0×10−7—–
182(D)W, β0 =0.22(−6.96, 0)
1.222
1.96×10−7 4.65×10−4
γ0 = 12.18◦
1.222
2.70×10−9 1.83×10−6
◦
γ0 =0
—
6.27×10−10 1.04×10−6(E) 182 W, β0 =0.24(−6.05, 0)
1.222
5.87×10−8 1.77×10−4◦
γ0 = 12.32
1.222
8.93×10−15 2.85×10−8
γ0 =0◦—
4.85×10−11 2.28×10−7
Experiment1.222
9.3×10−8
—–

71

B(E2)
to 8+
gr

τ1/2 to+8gr and 10+ gr+
Br(8+gr /10 gr)[ e fm ](Using experimental
energies)

6.41×10−2
2.14×10−2
9.56×10−3
4.34×10−3
2.18×10−3
1.25×10−3
8.17×10−4

84.6 ps
5.04 ns13.2 ns
32.0 ns
66.9 ns
120. ns
188. ns

0.015
0.413
0.521
0.603
0.653
0.685
0.711

2.08×10−3
1.31×10−3
9.47×10−4
4.56×10−4
2.41×10−4
1.59×10−4
8.44×10−5

25.9 ns
68.5 ns121. ns324. ns
670. ns
1.05 µs
2.05 µs

0.171
0.321
0.447
0.667
0.774
0.815
0.882

1.45×10−2
7.43×10−3
3.87×10−3
2.00×10−3
1.30×10−3
1.03×10−3
4.46×10−4
2.0×10−3

8.13 ns
16.2 ns
31.9 ns62.9 ns
98.9 ns
127. ns
332. ns
120 ns

0.470
0.485
0.501
0.517
0.532
0.548
0.671
2.0

1.21×10−4
6.41×10−6
4.17×10−7

957 ns67.3 µs
294 µs

0.442
0.485
0.483

6.10×10−5
2.35×10−7
4.72×10−8
1.4×10−4

2.62 µs
11.0 ms
3.20 ms
1.4 µs

0.738
4.85
0.671
1.2

2

4

Table 6.4: Reduced E2 transition probabilities at low spin. Parameters are those adjusted
for 184 Os which reproduce the experimental E(2+γ ). Experimental values are taken fromthe figures in ref. [73].
B(E2) values in units of [ e2 fm4 ] at low spin for 184 Os

21 → 01
22 → 01
21 → 2122 → 2122 → 2231 → 2131 → 2231 → 3141 → 2141 → 2242 → 2142 → 2241 → 3142 → 3141 → 4142 → 4142 → 4251 → 3151 → 41
51 → 4251 → 51
61 → 4161 → 4262 → 4162 → 4261 → 51
62 → 51
61 → 61
62 → 61
62 → 62(−2.99, 0)
5.98E + 3
4.57E + 2
7.84E + 3
1.40E + 3
7.52E + 3
8.10E + 2
8.78E + 3
2.36E − 4
8.77E + 3
5.42E + 1
8.28E + 1
4.09E + 3
7.59E + 2
5.67E + 3
6.24E + 3
1.49E + 3
3.64E + 2
5.81E + 3
4.46E + 2
3.89E + 3
1.37E + 3
1.00E + 4
1.34E + 2
2.04E + 1
7.58E + 3
1.03E + 3
2.69E + 3
5.20E + 3
1.44E + 3
1.16E + 3(−5.71, 3)
5.98E + 3
4.84E + 2
7.84E + 3
1.39E + 3
7.45E + 3
8.53E + 2
9.43E + 3
2.25E − 4
8.77E + 3
6.66E + 1
8.52E + 1
3.68E + 3
8.10E + 2
6.24E + 3
6.13E + 3
1.52E + 3
8.82E + 2
5.77E + 3
4.62E + 2
4.44E + 3
1.60E + 3
1.00E + 4
1.64E + 2
1.92E + 1
7.10E + 3
1.12E + 3
2.80E + 3
5.04E + 3
1.46E + 3
2.30E + 3

Core models, (V1 , V2 ) or γ0(−10.05, 7) (−20.12, 15) (−41.59, 30)
5.98E + 3
5.98E + 3
5.98E + 3
4.96E + 2
4.98E + 2
4.88E + 2
7.92E + 3
7.92E + 3
8.00E + 3
1.36E + 3
1.32E + 3
1.26E + 3
7.52E + 3
7.60E + 3
7.76E + 3
8.78E + 2
8.84E + 2
8.71E + 2
9.80E + 3
1.00E + 4
1.02E + 4
2.24E − 4
2.26E − 4
2.34E − 4
8.77E + 3
8.77E + 3
8.77E + 3
7.39E + 1
7.80E + 1
7.86E + 1
8.71E + 1
8.89E + 1
9.02E + 1
3.48E + 3
3.40E + 3
3.40E + 3
8.35E + 2
8.39E + 2
8.21E + 2
6.50E + 3
6.72E + 3
6.88E + 3
6.08E + 3
6.13E + 3
6.18E + 3
1.52E + 3
1.49E + 3
1.46E + 3
1.22E + 3
1.46E + 3
1.62E + 3
5.72E + 3
5.72E + 3
5.72E + 3
4.71E + 2
4.76E + 2
4.72E + 2
4.85E + 3
5.23E + 3
5.50E + 3
1.83E + 3
2.07E + 3
2.24E + 3
1.00E + 4
1.00E + 4
1.00E + 4
1.87E + 2
2.08E + 2
2.20E + 2
1.77E + 1
1.59E + 1
1.44E + 1
6.83E + 3
6.64E + 3
6.51E + 3
1.18E + 3
1.22E + 3
1.24E + 3
2.86E + 3
2.92E + 3
2.98E + 3
4.92E + 3
4.84E + 3
4.88E + 3
1.44E + 3
1.38E + 3
1.32E + 3
3.20E + 3
4.06E + 3
4.69E + 3(−71.96, 50)
5.98E + 3
4.76E + 2
8.00E + 3
1.22E + 3
7.84E + 3
8.47E + 2
1.04E + 4
2.42E − 4
8.77E + 3
7.68E + 1
8.96E + 1
3.40E + 3
7.97E + 2
7.05E + 3
6.24E + 3
1.41E + 3
1.68E + 3
5.72E + 3
4.63E + 2
5.68E + 3
2.35E + 3
1.00E + 4
2.23E + 2
1.36E + 1
6.51E + 3
1.22E + 3
3.07E + 3
4.96E + 3
1.28E + 3
5.00E + 3

14.89 deg.
6.19E + 3
3.24E + 2
8.48E + 3
8.03E + 2
8.48E + 3
5.79E + 2
1.10E + 4
4.41E − 4
8.96E + 3
5.23E + 1
6.61E + 1
3.56E + 3
5.33E + 2
7.80E + 3
7.00E + 3
9.52E + 2
1.65E + 3
5.91E + 3
3.24E + 2
6.05E + 3
2.72E + 3
1.00E + 4
1.72E + 2
8.80E + 0
6.83E + 3
8.64E + 2
3.65E + 3
5.98E + 3
8.80E + 2
5.24E + 3

Ratios of B(E2) values at low spin for 184 Os

B(E2;31 → 21 )/B(E2;31 → 41 )B(E2;22 → 01 )/B(E2;22 → 21 )B(E2;42 → 31 )/B(E2;42 → 22 )B(E2;22 → 21 )/B(E2;21 → 01 )B(E2;42 → 41 )/B(E2;42 → 22 )B(E2;31 → 21 )/B(E2;31 → 22 )(−2.99, 0)
1.89
0.325
1.385
0.234
0.365
0.092

Core models, (V1 , V2 ) or γ0(−10.05, 7) (−41.59, 30) (−71.96, 50)
1.87
1.88
1.89
0.363
0.385
0.390
1.87
2.02
2.07
0.228
0.212
0.204
0.437
0.432
0.417
0.090
0.085
0.081

72

14.89 deg.
1.93
0.404
2.19
0.130
0.267
0.053

Exp.
1.4
0.5
—
—
0.2
—

Figure 6.1: The eﬀects of changing parameters of the model on the reduced transition
probabilities of the decay of the K π = 10+ isomer. The axially symmetric rotor is employed as the core. Other parameters as those adjusted for 184 Os. See the comments intables 6.1 and 6.2.
73

Figure 6.2: Eﬀects of changing ϵc on the probability distribution of the K-quantum number. The axially symmetric rotor is employed as the core. Other parameters as thoseadjusted for 184 Os. When ϵc =ϵ0c (=0.2805 MeV), the experimental E(2+gr ) is reproduced
including the eﬀects of coupling with the particles.

Figure 6.3: Eﬀects of changing κ on the probability distribution of the K-quantum number. The axially symmetric rotor is employed as the core. Other parameters as thoseadjusted for 184 Os.
74

Figure 6.4: The excitation energy of the γ-bandhead versus the partial half-life of theK π = 10+ isomer with M1 and E2 transitions into the ground band states in 184 Os.
Transition energies are taken from experiments in calculating the partial half-life.

75

Figure 6.5: The same as figure 6.4, but for 182 W using β0 =0.22.

Figure 6.6: The same as figure 6.4, but for 182 W using β0 =0.24.
76Figure 6.7: The probability distribution of the K-quantum number for the 10+
gr state andthe K π = 10+ isomer in 184 Os using the γ-vibrational model (V2 =0).

Figure 6.8: The same as figure 6.7, but using the triaxial rotor model.

77Figure 6.9: The γ-potential V (γ) in units of ϵc (∼ 0.2 MeV) and the probability distribution of γ of the bare (i. e. the particles are not coupled with) core in the ground state. The
γ-potential is measured from the ground state energy. The values of V1 of the potentials
given in the upper right-hand portion are given in the part (C) of table 5.1.

78

Figure 6.10: The probability distribution of γ in the coupled system of the core andthe particles. Parameters are those adjusted for 184 Os which reproduce the experimental
E(2+
γ ).
79

Figure 6.11: Reduced probabilities of isomeric and collective transitions using various
shapes of the γ-potential. In the right-hand side, we give collective E2 transitions in theK π =10+ band (in the top) and in the ground and the γ-band (for high spin in the middle
and for low spin in the bottom).

80

Figure 6.12: The γ-potential reproducing the half-life of the K-isomer and E(2+γ ) for184
Os.
81

Figure 6.13: Probabilty distribution of γ and |K| in the 10+
gr state. Parameters are thoseadjusted for 184 Os which reproduce the experimental E(2+
).
γ

82

Figure 6.14: Probabilty distribution of γ and |K| in the K π = 10+ isomer. Parameters
are those adjusted for 184 Os which reproduce the experimental E(2+
γ ).

83

Figure 6.15: Distribution of the isomeric E2 transition amplitudes with respect to the sizeof γ-deformation.
84

Figure 6.16: Energy levels of the ground band, the γ-band, and the K π = 10+ band for
Os. V2 is varied from 0 to 50, while V1 is adjusted so that the experimental E(2+γ)
is reproduced. (See γ-potentials given in the upper right hand portion of figure 6.9.)
The limiting case of the triaxial rotor model (γ = 14.89◦ ) is included (denoted by TR).
Experimental spectra are also shown (denoted by EXP).

184

85

Figure 6.17: Energy levels of the ground band, the γ-band, and the K π = 10+ band for
Os using the γ-vibrational model (V2 =0). Parameters are those adjusted for 184 Os.
The limiting case of the axially symmetric rotor model is denoted by AS.
184

86

Figure 6.18: Kinematical moment of inertia versus the angular velocity for the yrast bandof 184 Os.

87

Chapter 7
Summary
We examined the eﬀects of the γ-degree of freedom on the decay properties of K-isomers.
Aiming at the dynamical treatment of γ, we employed a γ-soft version of Bohr’s quantum
liquid drop model as the rotor of the particle-rotor model and treated the coupling between
the large-amplitude surface motions and the motions of particles in an intruder orbital
in a fully quantum mechanical way. The feature of our model is the description in thelaboratory frame using the so-called weak coupling bases. Special eﬀorts are made sothat the results of calculations converge with respect to the truncation of bases of highly
excitated γ-vibrational character.
We studied the strong violation of the K-selection rule observed in the decays of
the K π =10+ isomers in 182 W and 184 Os. Before introducing the γ-degree of freedom,
we investigated the eﬀects of the Coriolis interaction on the mixing of the K-quantum
number. It was found that the short half-lives of the isomers can not be reproduced
assuming axially symmetric nuclear shape, although the Coriolis interaction is rather
strong in these less-deformed transitional nuclei.
Next, we took into account the γ-degree of freedom. We changed the shape of thepotential for γ-deformation and mediated two extreme models: One is the γ-vibrationalmodel of Bohr and Mottelson, in which γ fluctuates around γ=0◦ in a relatively shallow
potential. The other is the triaxial rotor model of Davydov and Filippov, which treats γ
as a static parameter and fix it at some constant value γ0 .
The result is that the relatively short half-lives of the isomers can be reproduced when
we introduce the γ-degree of freedom. The half-lives calculated with the γ-vibrational

88

model are shorter than those calculated with the triaxial rotor model by a factor of morethan order two. The experimental half-lives are located between (or in the vicinities of)the values calculated with the two models. It is rather surprising that the two models canbe distinguished so clearly: They have no distinct diﬀerence in the quantities of collective
character like intraband E2 transitions.
This diﬀerence originates in the quantum fluctuation in γ. The γ-vibrational model
has a very shallow potential for γ-deformation and the wave function has a long tail. The
amplitudes of isomeric decays are aﬀected remarkably by this large-γ tail of the wave
function, as is illustrated by the fact that the components at large γ-deformations play
the crucial role to mix the K-quantum number.
It is worth while noting, however, that the transitions occur mainly from the small-γ
components, not from these large-γ ones. Hence we had better suppose indirect processes
through which the quantum fluctuation in shape hastens the isomeric decays. The most
likely process is the barrier penetration mechanism given by Chowdhury and his coworkers,
where the ground-band state and the isomer mix with each other — in spite of the large
diﬀerence in their structures — via largely γ-deformed states. But any quantitative
approximations do not seem possible on the basis of such semiclassical picture at leat for
the K π =10+ isomers because of the rather wide spread of the strength distribution of theisomeric transition.
On the contrary, quantities of collective character are determined by the bulk properties of the wave function. When we use the triaxial rotor model, we can simulate thebulk properties by adjusting the size of the static γ-deformation γ0 . But there is no tail,because γ is fixed at a constant value, and the mixing of the K-quantum number is muchsmaller than in the γ-vibrational model. Hence we can estimate the size of the fluctuationin γ from the degree of violation of the K-selection rule by utilizing the average value ofγ, which has already been obtained from e. g. the excitaion energy of the γ-bandhead.
From the present calculations, it seems that the nucleus 182 W is much like the γvibrational model while the nucleus 184 Os is rather like the triaxial rotor model, in agreements with other microscopic calculations. But it is worth while noting that there aremany degrees of freedom excluded in our model. At first, our γ-potential has only two89degrees of freedom. More detailed study of the potential shape is a future problem. Another important thing to be considered is the β-vibration, which enables us to take into
account the change in the size of deformation due to high spin in a dynamical way.

90

.

Acknowledgements
I wish to express my sincere gratitude to Professor Naoki Onishi for all the guidanceand discussions concerning this thesis. I am also grateful to the members of of the nuclear
theory group of the Institute of Physics, College of General Education, University ofTokyo for their stimulating comments.
I also express my gratitude to Drs. J. Pedersen, J. Borggreen, and G. Sletten forsending us experimental data before publication. I thank also to Dr. Tord Bengtsson forsuggesting our studying the K π =10+ isomers.
The computer calculation for this work was financially supported in part by the Grantin-Aid for Scientific Research of the Japan Ministry of Education, Science and Culture,and in part by Research Center for Nuclear Physics, Osaka University.

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