# Identical Bands Around the Isobaric Rare Earth Even-Even Nuclei with the
Mass Number A = 164

M. A. Abdelsalam  Physics Department, Faculty of Science, Al-Azhar University,
Cairo, Egypt  H. A. Ghanim  Physics Department, Faculty of Science, Al-Azhar
University, Cairo, Egypt  M. Kotb  Physics Department, Faculty of Science, Al-
Azhar University, Cairo, Egypt  A. M. Khalaf  Physics Department, Faculty of
Science, Al-Azhar University, Cairo, Egypt

###### Abstract

Eight pairs of rare-earth normally - deformed (ND) nuclei around the isobaric
nuclei with A = 164 and have identical values of F-spin, $\pm$ $F_{0}$ and
$N_{p}$ $N_{n}$ ($N_{p}$ and $N_{n}$ are the number of valence protons and
valence neutrons respectively ) have been studied. These pairs of identical
bands (IB’s) cover 16 mass units and are classified as (i) 3 pairs of nuclei
separated by (2p,2n) :(${}^{162}Yb-^{166}Hf$), (${}^{162}Er-^{166}Yb$),
(${}^{162}Dy-^{166}Er$) (ii) 2 pairs of nuclei separated by (4p,4n):
(${}^{160}Dy-^{168}Yb$), (${}^{160}Er-^{168}Hf$) (iii) 2 pairs of nuclei
separated by (6p,6n): (${}^{158}Er-^{170}W$) (${}^{158}Dy-^{170}Hf$) and (iv)
one pair of nuclei separated by (8p,8n): (${}^{156}Dy-^{172}W$).

We suggested a theoretical collective rotational formula containing three
parameters (CRF3) as an extended version of Bohr-Mottelson model to calculate
the ground state positive parity excitation energies. Also, the sd-version of
the interacting boson model (IBM) has been used to describe the nuclear shapes
by using the intrinsic coherent-state. The optimized models parameters for
each nucleus are adjusted by using a simulation search program to minimize the
root mean square deviation between the theoretical calculation and
experimental excitation energies. The best adopted model parameters of the
CRF3 are used to calculate the rotational frequencies $\hbar\omega$, the
kinematic $J^{(1)}$ and dynamic $J^{(2)}$ moments of inertia and the evolution
of $J^{(1)}$ and $J^{(2)}$ with increasing $\hbar\omega$ are systematically
analyzed. A smooth gradual increase in both moments of inertia was seen.

The calculated results agree excellently with the experimental ones which give
strong support to the suggested CRF3.

The adopted IBM parameters are used to calculate the potential energy surfaces
(PES’s) which describe the nuclear deformation. The PES’s for our nuclei shows
two wells corresponding to prolate and oblate sides which indicate that these
nuclei are deformed and have rotational behaviors.

The correlation quantities which identify the IB’s are extracted. It is found
that the nuclei having $N_{p}N_{n}/\bigtriangleup$ where $\bigtriangleup$ is
the average pairing gap, exhibit identical excitation energies and energy
ratios in their ground state rotational bands.

Keywords : Interacting Boson model (IBM) - Identical Bands - Potential Energy
Surface

##  1 Introduction

The discovery of rotational bands in adjacent even-even and odd-mass
superdeformed (SD) nuclei in which the $\gamma$-ray transition energies are
nearly identical to within a few KeV was an exotic and unexpected phenomenon
in nuclear structure physics [1, 2, 3, 4, 5]. Since the identical bands (IB’s)
have essentially identical transition energies, then the associated dynamical
moment of inertia are thus identical. Several explanations were put forward
[4, 5, 6, 7, 8, 9, 10, 11, 12] to understand the origin of IB’s phenomenon
assuming the occurrence of such IB’s to be a specific property of the SD
states in nuclei. The explanations of these IB’s includes: the Coriolis force,
the particle alignment and pairing [13], the roles of special high-N orbitals
of intruder configuration and band crossing[14, 15, 16, 17], the pseudo-spin
in supersymmetry [7, 18, 19] and the supersymmetry with many-body interactions
[20].

Soon the phenomenon of low-spin identical bands was found in pairs of even-
even normal deformed (ND) nuclei [21], and in neighboring even-even and odd-
mass nuclei in rare-earth region where they have similar moments of inertia
[22, 23]. If was noted that low spin IB’s are not limited to nearby nuclei but
are widespread and found in pairs of even-even nucleoside as separated by 24
mass unit (like ${}^{156}Dy,^{180}Os$) [24]. Attempts were made to understand
the low-spin IB’s in terms of some simple systematics of the moments of
inertia in the rare-earth region [25, 26, 27, 28, 29, 30] or from several
types of consideration [31].

For the description of normally deformed (ND) bands, some useful models were
proposed. Bohr and Mottelson [32] pointed out that, under the adiabatic
approximation, the rotational energy of an axially symmetric nucleus may be
expanded for $K=0$ band as a power series in the I(I+1) term. The expansion
for the $K\neq 0$ band takes the same form, but includes a band head energy
and the I(I+1) is replaced by $\left[I(I+1)-K^{2}\right]$. Another useful
models for nuclear rotational spectra are the particle-rotor model (PRM) [33],
the variable moment of inertia (VMI) model [34, 35], the soft rotor model [36]
and the interacting boson model [37].

In the concept of F-spin and its projection [38] any pairs of conjugate nuclei
with the same F-spin and $F_{0}$ values in any F-multiplet will have the same
$N_{p}N_{n}$ [24, 39, 40] where $N_{p}$ and $N_{n}$ are respectively the
number of valence protons and valence neutrons. The product $N_{p}N_{n}$ was
used in the classification of the changes that occur in nuclear structure [41,
42]. It was assumed that [25, 43] the moment and the P-factor depends also on
the product $N_{p}N_{n}$.

The purpose of the present paper is (i) to analyse the excitation energies for
even-even normally deformed nuclei in rare earth region in framework of
suggested new collective rotational formula (CRF3). (ii) to exhibit the
occurrence of IB’s in eight pairs of nuclei in rare earth region. (iii) to
present the parameters which characterize the appearance of IB’s. (iv) use the
sd version of interacting boson model (sdIBM) to calculate the potential
energy surfaces (PES’s).

##  2 Outline of the Suggested Collective Rotational Formula with Three
Parameters (CRF3)

Rotational states in normal deformed (ND) nuclei can be characterized by their
excitation energies E(I) as a function of spin I, which generally lie low as
compared to the single-particle excitation. In the strong coupling limit, the
rotational ground state energy for an axially symmetric even-even nucleus
obeys the I(I+1) rule, i.e form bands of levels that fulfill the relation

$\displaystyle E(I)$
$\displaystyle=\dfrac{\hbar^{2}}{2J}I(I+1)=\alpha\,\textit{\text{\^{I}}}^{2}$
(1)

where $\alpha$ = $\hbar^{2}/2J$ and Î = I(I+1)

The relation (1) defines in addition the nuclear moment of inertia J as a
constant for an ideal rotor. This simple rotational formula gives deviations
from experimental data, So Bohr and Mottelson pointed out that agreement was
improved by adding to it a second team to yield

$\displaystyle E(I)$ $\displaystyle=\alpha I(I+1)+\beta[I(I+1)]^{2}$
$\displaystyle=\alpha\,\text{\^{I}}^{2}+\beta\,\text{\^{I}}^{4}$
$\displaystyle E(I)$
$\displaystyle=\alpha\,\text{\^{I}}^{2}(1+\gamma\,\text{\^{I}}^{2})$ (2)

where $\gamma=\beta/\alpha$

Since the moment of inertia J increases on rotation of the nucleus, the
observed deviations from the experiment were still more evident.

According to the variable moment of inertia(VMI) model[34, 35], there is a
gradual increase in moment of inertia J with increasing the spin I, so we
suggest that the moment inertia J can be written as

$J=J(I)=J\,(1\,+\,\sigma\,\text{\^{I}}^{2})$ (3)

Substituting in equation (2), yield

$E(I)=\alpha\,\text{\^{I}}^{2}\left(\dfrac{1+\gamma\,\text{\^{I}}^{2}}{1+\sigma\,\text{\^{I}}^{2}}\right)$
(4)

Therefore, the two-term Bohr-Mottelson formula becomes an extended new formula
with three parameters. We denote formula (4) as the collective rotational
formula with three parameters (CRF3). The parameters are
$\alpha,\beta,\gamma$.

The suggested CRF3 is more general because it leads to the following three
predictions:

a) when $\sigma=\gamma$ it gives pure rigid rotor equation(1)

b) when $\sigma=0$ it gives the two parameters Bohr-Mottelson equation (2)

c) when $\gamma=0$ it gives soft rotor model [36]

$E(I)=\dfrac{\hbar^{2}}{2J}\dfrac{I(I+1)}{1+\sigma(I+I^{2})}$ (5)

Two types of moments of inertia were suggested by Bohr-Mottelson which reflect
two different aspects of nuclear dynamics. The first moment of inertia is the
kinematic $J^{(1)}$, it is equal to the inverse of the slope of the curve of
energy E versus $\text{\^{I}}^{2}$ (or I(I+1)) times $\hbar^{2}/2$, while the
second moment of inertia is the dynamic $J^{(2)}$, it is related to the
curvature in the curve of E versus Î (or $\sqrt{I(I+1)}$ ).

The kinematic $J^{(1)}$) and dynamic $J^{(2)}$ moments of inertia are defined
as:

$\displaystyle J^{(1)}$
$\displaystyle=\dfrac{\hbar^{2}}{2}\left[\dfrac{dE}{dI(I+1)}\right]^{-1}=\hbar\dfrac{\sqrt{I(I+1)}}{\omega}$
$\displaystyle=\dfrac{\hbar^{2}}{2}\left(\dfrac{dE}{d\text{\^{I}}^{2}}\right)^{-1}=\hbar\dfrac{\text{\^{I}}}{\omega}$
(6) $\displaystyle J^{(2)}$
$\displaystyle=\hbar^{2}\left[\dfrac{d^{2}E}{d(\sqrt{I(I+1)})^{2}}\right]^{-1}=\hbar\dfrac{d\sqrt{I(I+1)}}{d\omega}$
$\displaystyle=\hbar^{2}\left(\dfrac{d^{2}E}{d\text{\^{I}}^{2}}\right)^{-1}=\hbar\dfrac{d\text{\^{I}}}{d\omega}$
(7)

In the case of our CRF3, the two moments of inertia becomes

$J^{(1)}(I)=\dfrac{\hbar^{2}}{2\alpha}\dfrac{(1+\sigma\text{\^{I}}^{2})^{2}}{[1+\gamma\text{\^{I}}^{2}(2+\sigma\text{\^{I}}^{2})]}$
(8)
$J^{(2)}(I)=\dfrac{\hbar^{2}}{2\alpha}\dfrac{(1+\sigma\text{\^{I}}^{2})^{3}}{[(1+6\gamma\text{\^{I}}^{2})+\sigma\text{\^{I}}^{2}(3\gamma\text{\^{I}}^{2}+\alpha\gamma\text{\^{I}}^{4}-3)]}$
(9)

Experimentally $\hbar\omega$, $J^{(1)}$and $J^{(2)}$ are extracted in terms of
the transition energy $E_{\gamma}(I)=E(I)-E(I-2)$ as:

$\hbar\omega(I)=\frac{1}{4}[E_{\gamma}(I+2)+E_{\gamma}(I)]\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;(MeV)$
(10)
$J^{(1)}(I)=\dfrac{2I-1}{E_{\gamma}(I)}\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;(\hbar^{2}MeV^{-1})$
(11)
$J^{(2)}(I)=\dfrac{4}{E_{\gamma}(I+2)-E_{\gamma}(I)}\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;(\hbar^{2}MeV^{-1})$
(12)

As a special case, the lowest dynamical moment of inertia reads

$J^{(2)}_{lowest}=\dfrac{4}{E_{\gamma}(4^{+}_{1}\rightarrow
2_{1}^{+})-E_{\gamma}(2^{+}_{1}\rightarrow 0_{1}^{+})}$ (13)

##  3  Determination of Ground State Band Properties of Even-Even Nuclei and
the Physical Identical Parameters

In order to understand the behavior of low lying states of an axially
symmetric normally deformed nuclei, it is insightful to examine some physical
observables which exist in a pair of IB’s, the observables include:

1\. The P- Factor, Structure Factor (SF), and Saturation Parameter (SP)

Casten [43] introduced the P-Factor

$P=\dfrac{N_{p}N_{n}}{N_{p}+N_{n}}$ (14)

where $N_{p}$ and $N_{n}$ are the numbers of valence protons and valence
neutrons respectively which are counted as particles or holes from the nearest
closed shell

$\displaystyle N_{p}$ $\displaystyle=min[(Z-50),(82-Z)]$ (15) $\displaystyle
N_{n}$ $\displaystyle=min[(N-82),(126-N)]$ (16)

The P- Factor represents the average number of interactions of each valence
nucleon with those of the other type. It can be viewed as the ratio of the
number of valences p-n residual interactions to the number of valence like-
nucleon pairing interactions, or if the p-n and pairing interactions are orbit
independent, then P is proportional to the ratio of the integrated p-n
interaction strength to the integrated pairing interaction strength. The
nuclear collectivity and deformation depend sensitively on the P- Factor.

The structure factor (SF) and the saturation parameter (SP) are given by

$\displaystyle SF$ $\displaystyle=N_{p}N_{n}(N_{p}+N_{n})$ (17) $\displaystyle
SP$ $\displaystyle=\left(1+\dfrac{SF}{SF_{max}}\right)^{-1}$ (18)

It is found that the lowest dynamical moment of inertia $J^{(2)}_{lowest}$ is
proportional to $\sqrt{SF}$.

2\. The Concept of F-Spin

A nucleus with $N_{p}$ valence protons and $N_{n}$ valence neutrons has a
total boson number

$N_{B}=\dfrac{N_{p}+N_{n}}{2}=N_{\pi}+N_{\nu}$ (19)

The $N_{\pi}$ proton bosons and neutron bosons are assigned F-Spin,
$F=\frac{1}{2}$ with projection $F_{0}=+\frac{1}{2}$ for proton bosons and
$F_{0}=-\frac{1}{2}$ for neutron bosons. A given nucleus is characterized by
two quantum numbers [38]:

$F=\dfrac{N_{\pi}+N_{\nu}}{2}$ and its projection
$F_{0}=\dfrac{N_{\pi}-N_{\nu}}{2}$

Squaring and subtracting, yield

$4(F^{2}-F^{2}_{0})=4N_{\pi}N_{\nu}=N_{p}N_{n}$ (20)

That is any pair of conjugate nuclei with the same F-spin and $F_{0}$ values
in any F-spin multiplet have identical $N_{p}N_{n}$ values.

In our chosen nuclei, the F-spin multiplet is given by: (A+4, Z+2), (A+8,
Z+4), (A+12, Z+6) and (A+16, Z+8) for Dy, Er, Yb, Hf, and W isotopes.

Any pair of nuclei which show identical excitation energies have nearly equal
value of the product of their valence nucleon numbers $N_{p}$ and $N_{n}$
[41]. However, the analysis of experimental data shows that the converse is
not true. The simple quantity $N_{p}N_{n}$ helps also in the evolution of
nuclear deformation and collectivity in nuclei [40]. On the other hand, the
product $N_{p}N_{n}$ or the P- Factor plays an important role in studying the
orbit dependence, shell gaps, and intruder orbitals.

3\. Pairing Interaction Energy

The pairing interaction energy $\bigtriangleup$ in an even-even nucleus is the
average pairing gap ($(\bigtriangleup_{p}+\bigtriangleup_{n})/2$ where
$\bigtriangleup_{p}$ and $\bigtriangleup_{n}$ are respectively the proton and
neutron pairing gaps which are determined from the difference in binding
energies of the neighboring odd and even nuclei

$\displaystyle\bigtriangleup_{p}$
$\displaystyle=\frac{1}{4}[B(N,Z-2)-3B(N,Z-1)+3B(N,Z)-B(N,Z+1)]$ (21)
$\displaystyle\bigtriangleup_{n}$
$\displaystyle=\frac{1}{4}[B(N-2,Z)-3B(N-1,Z)+3B(N,Z)-B(N+1,Z)]$ (22)

The pairing gaps $\bigtriangleup_{p}$ and $\bigtriangleup_{n}$ are determined
empirically from the relation

$\displaystyle\bigtriangleup_{p}\simeq\bigtriangleup_{n}=\dfrac{12}{\sqrt{A}}\texttt{
}\;\;\;\;\;\;\;\;\;\;\;(MeV)$ (23)

The average pairing gap of the nucleus is then

$\bigtriangleup=\dfrac{\bigtriangleup_{p}+\bigtriangleup_{n}}{2}=\dfrac{12}{\sqrt{A}}\texttt{
}MeV$ (24)

It is observed that [39, 43] the even-even nuclei belong to different mass
number having identical $(N_{p}N_{n}/\bigtriangleup)$ values exhibit identical
excitation energies and identical energy ratios.

4\. Quadrupole Transition Probabilities and Deformation Parameters

The quadrupole transition probability per unit time for the transition
$I_{i}\rightarrow I_{f}$ is given by

$T(E_{2})=\dfrac{4\pi}{75}\left(\dfrac{5}{\hbar}\right)\left(\dfrac{E_{2^{+}_{1}}}{\hbar
c}\right)^{5}B(E_{2};I_{i}\rightarrow I_{f})$ (25)

where $B(E_{2})$ is the reduced transition probability and $E_{2^{+}_{1}}$ is
the energy of the $2_{1}^{+}$ state.

Experimentally $T(E_{2})$ for transition $2_{1}^{+}\rightarrow 0_{1}^{+}$ is
obtained by

$T(E_{2},2_{1}^{+}\rightarrow
0_{1}^{+})=\dfrac{ln2}{(1+\alpha)T_{1/2}}=\dfrac{0.693}{(1+\alpha)T_{1/2}}$
(26)

where $\alpha$ is the total conversion coefficient taken from the tabulated
values given by Rose [44] and $T_{1/2}$ is the lifetime of the rotational
level.

The $B(E_{2},2_{1}^{+}\rightarrow 0_{1}^{+})$ values carry important
information about the collectivity of nuclear rotation and can be extracted
from the equations (25,26).

The relation between the intrinsic nuclear quadrupole moment $Q_{0}$ and
$B(E_{2})$ is given by

$Q_{0}^{2}=\dfrac{16\pi}{e}B(E_{2},2_{1}^{+}\rightarrow 0_{1}^{+})$ (27)

Practically the most reliable method of determining the quadrupole deformation
parameter $\beta_{2}$ in framework of geometric collective model (GCM) is to
extract $\beta_{2}$ from $Q_{0}$ according to the formula

$\beta_{2}(exp)=\dfrac{\sqrt{5\pi}}{3ZR_{0}^{2}}Q_{0}$ (28)

assuming a uniformly charged nucleus of spheroidal shape, where the nuclear
radius has the value $R_{0}=1.2A^{1/3}$(fm) and Z is the nuclear charge
number.

The expression (28) for $\beta_{2}$ is widely used to compare the quadrupole
deformation of different nuclei. It is noticed that the
$B(E_{2},2_{1}^{+}\rightarrow 0_{1}^{+})$ values increase when going from the
closed shell at N=82 toward midshell where maximum values are occur, while
from midshell toward the shell closure at N= 126 its values are decreases.

In a second way , specially where the $B(E_{2},2_{1}^{+}\rightarrow
0_{1}^{+})$ value is not known, we estimate $\beta$ by using the approximate
empirical Grodzins relation [45]:

$E_{2^{+}_{1}}B(E_{2},2_{1}^{+}\rightarrow 0_{1}^{+})=2.5\times
10^{-3}\texttt{ }\dfrac{Z^{2}}{A}$ (29)

where

$\displaystyle B(E_{2},2_{1}^{+}\rightarrow
0_{1}^{+})=\dfrac{1}{16\pi}e^{2}Q^{2}_{0}=\dfrac{9}{80\pi^{2}}e^{2}Z^{2}R^{4}_{0}\beta^{2}\;\;\;\;\;\;\;(\texttt{in
units of }e^{2}b^{2})$ (30)

We can relate $\beta$ and $E_{2^{+}_{1}}$ as:

$\beta^{2}_{G}=\dfrac{1224}{E_{2^{+}_{1}}A^{7/3}}$ (31)

where $E_{2^{+}_{1}}$ is in MeV.

Also $\beta_{2}$ can be determined by using the SU(3) rotational limit of
interacting boson model(IBM)[37], the square of the deformation parameter
$\beta^{2}$ in a state of angular momentum I is given by [46]:

$\langle\beta^{2}\rangle_{I}=\dfrac{\alpha^{2}}{6(2N-1)}[I(I+1)+8N^{2}_{B}+22N_{B}-15]$
(32)

where $N_{B}$ is the total number of valence bosons and $\alpha$ is a
normalization constant ($\alpha=0.101$ for rare-earth nuclei). The expectation
value of $\beta^{2}$ in the ground state becomes

$\langle\beta^{2}\rangle_{0}=\alpha^{2}\dfrac{8N^{2}_{B}+22N_{B}-15}{6(2N-1)}$
(33)

which is an almost linearly increasing function of the boson number $N_{B}$
and has the same value for nuclei having the same number of valence nucleons

$N=[N_{p}+N_{n}],N=[(N_{p}-1)+(N_{n}-1)]$ (34)

It is evident that $\beta_{IBM}$ extracted from IBM is much larger than
$\beta_{GCM}$ extracted from GCM because $\beta_{GCM}$ refer to the
deformation of all A nucleons while $\beta_{IBM}$ describe only 2N valence
bosons, the approximate relation between them is given by:

$\beta_{GCM}=1.18\left(\dfrac{2N}{A}\right)\beta_{IBM}$ (35)

The deformation parameter $\beta$ reflects the equilibrium shape and structure
of the nucleus such as the energy ratio $R_{4/2}=E(4_{1}^{+})/E(2_{1}^{+})$
and the reduced transition probability $B(E_{2},2_{1}^{+}\rightarrow
0_{1}^{+})$ which are the best indicators to exhibit the collective properties
of the even-even nuclei.

5\. Energy Ratios and Percentage Difference in Transition Energies

The energy ratios and the percentage difference in transition energies give
the characteristic of the evolution of the collectivity in the even-even
nuclei. Only deformed nuclei show rotational levels and particularly the even-
even nuclei display a simple structure energies proportional to I(I+1) with
only even values of the spin I considering that the moment of inertia is
constant (rigid rotator), therefore the energy ratio $R_{4/2}=3.333$. The
observed moment of inertia extracted from the experiment is only one-quarter
to one-half of what one would expect from a rigid rotator which means that not
the whole nucleons are participating in the collective motion.

On the other hand for an ideal harmonic quadrupole spectrum for spherical
nuclei a system of equidistant states is formed by the composition of
vibrational quanta. The first excited state is $2_{1}^{+}$ followed by the
degenerate $0_{2}^{+},2_{2}^{+},4_{1}^{+},$ and so forth. Therefore energy
ratio$R_{4/2}=2$.

To compare level spacing in two nuclei with masses $A_{1}$, and $A_{2}$ where
$A_{2}>A_{1}$, we define the percentage differences ratios in transition
energies as :

$\delta=\dfrac{\bigtriangleup E_{\gamma}(I)}{E_{\gamma_{2}}(I)}$ (36)

where

$\displaystyle E_{\gamma}=E(I)-E(I-2)$ (37) $\displaystyle\bigtriangleup
E_{\gamma}(I)=E_{\gamma_{1}}(I)-E_{\gamma_{2}}(I)$ (38)

So that

$\displaystyle E_{\gamma_{1}}=(1+\delta)E_{\gamma_{2}}$ (39)

For rigid rotor the ratio

$\displaystyle\delta_{R}=\left(\dfrac{A_{2}}{A_{1}}\right)^{5/3}-1$ (40)

define the fractional change in $A^{5/3}$.

The fractional change in transition energies $\delta$ divided by the rigid
rotor ratio $\delta_{R}$ is denoted by $\delta_{\gamma}$. If the spacings are
identical, then $\delta=0,\delta_{\gamma}=0$ and if they scale as $A^{5/3}$
then $\delta_{\gamma=1}$.

Similarly, the percentage difference in kinematic moment of inertia $J^{(1)}$
is given by

$\displaystyle K=-\dfrac{\bigtriangleup J^{(1)}(I)}{J^{(1)}_{2}(I)}$ (41)

where

$\displaystyle J^{(1)}(I)$ $\displaystyle=\dfrac{2I-1}{E_{\gamma}(I)}$ (42)
$\displaystyle\bigtriangleup J^{(1)}(I)$
$\displaystyle=J^{(1)}_{1}(I)-J^{(1)}_{2}(I)$ (43)

So that

$J^{(2)}_{2}=(1+K)J^{(1)}_{1}$ (44)

Substituting for $J^{(1)}$, yield $K=\delta$.

##  4 The Interacting Boson Model to Calculate the Potential Energy Surfaces
and Electric Quadrupole Transition Probability

We consider the Hamiltonian of the first order U(5)- SU(3) quantum shape phase
transition in the form

$H=\epsilon_{d}\hat{n}_{d}+a_{2}\hat{Q}^{(x)}\hat{Q}^{(x)}$ (45)

where $\hat{n}_{d}$ and $\hat{Q}^{(x)}$ are respectively the d-boson number
operator and quadrupole operator defined as

$\displaystyle\hat{n}_{d}$
$\displaystyle=\sum_{\mu}d_{\mu}^{\dagger}\stackrel{{\scriptstyle\sim}}{{d}}_{\mu}$
(46) $\displaystyle\hat{Q}^{(x)}$
$\displaystyle=\left[d^{\dagger}s+s^{\dagger}\stackrel{{\scriptstyle\sim}}{{d}}\right]^{(2)}+x\left[d^{\dagger}\times\stackrel{{\scriptstyle\sim}}{{d}}\right]^{(2)}$
(47)

where $\left(s^{\dagger},d^{\dagger}\right)$ and
$\left(s,\stackrel{{\scriptstyle\sim}}{{d}}\right)$ are the boson creation and
annihilation operators respectively, and $x$ is the structure parameter of the
quadrupole operator of IBM ($x$ for pure rotational SU(3) limit is equal to
$-\sqrt{7}/2$). Here $d_{\mu}=(-1)^{\mu}d_{-\mu}$ and standard notation of
angular momentum coupling is used.

To get the potential energy surface (PES) of the Hamiltonian, we introduce the
intrinsic coherent frame in which the ground state of a nucleus with N bosons
can be expressed as a boson condensate. The bosonic intrinsic coherent state
for the ground state band of a given even-even nucleus can be written in the
form[47, 48, 49]

$\lvert
N\beta\gamma\rangle=\dfrac{1}{\sqrt{N!}}[b^{\dagger}(\beta,\gamma)]^{N}\lvert
0\rangle$ (48)

where $\lvert 0\rangle$ is the boson vacuum and $b^{\dagger}$ is the boson
creation operator which acts in the intrinsic system and is given by:

$\displaystyle b^{\dagger}=\dfrac{1}{\sqrt{1+\beta^{2}}}[s^{\dagger}+\beta
cos\gamma(d_{0}^{\dagger})+\dfrac{1}{\sqrt{2}}\beta
sin\gamma(d_{2}^{\dagger}+d_{-2}^{\dagger})]$ (49)

where $\beta$ is the quadrupole deformation parameter which measures the axial
deviation from spherical symmetry and the parameter $\gamma$ controls the
departure from axial symmetries.

The ground state PES is the expectation value of the Hamiltonian in the
intrinsic coherent state

$PES=\langle N\beta\gamma\rvert H\rvert N\beta\gamma\rangle$ (50)

The associated PES of the Hamiltonian (45) for $x=-\sqrt{7}/2$ reads

$\displaystyle E(N,\beta,\gamma)$
$\displaystyle=\epsilon_{d}\dfrac{N\beta^{2}}{1+\beta^{2}}+a_{2}\left[\dfrac{N}{1+\beta^{2}}(5+\dfrac{11}{4}\beta^{2})+\dfrac{N(N-1)}{(1+\beta^{2})^{2}}(4\beta^{2}-2\sqrt{2}\beta^{3}cos3\gamma+\dfrac{1}{2}\beta^{4})\right]$
(51)

Equation (51) can be written in another form as

$\displaystyle
E(N,\beta,\gamma)=g_{1}\dfrac{N\beta^{2}}{1+\beta^{2}}+\dfrac{N(N-1)}{(1+\beta^{2})^{2}}[g_{2}\beta^{2}+g_{3}\beta^{3}cos3\gamma+g_{4}\beta^{4}]+c$
(52)

where the coefficients involve linear combination of the Hamiltonian
parameters

$\displaystyle g_{1}$
$\displaystyle=\epsilon_{d}-\dfrac{9}{4}a_{2},\;\;\;\;\;\;\;\;\;\;g_{2}=4a_{2}$
$\displaystyle g_{3}$
$\displaystyle=2\sqrt{2}a_{2},\;\;\;\;\;\;\;\;\;\;\;\;\;\;g_{4}=\dfrac{1}{2}a_{2},\;\;\;\;\;\;\;c=5Na_{2}$

Also, equation (51) can be rewritten in general form as

$E(N,\beta,\gamma)=\dfrac{A_{2}\beta^{2}+A_{3}\beta^{3}cos3\gamma+A_{4}\beta^{4}}{(1+\beta^{2})^{2}}+A_{0}$
(53)

where the coefficients read

$\displaystyle A_{2}$
$\displaystyle=\left[\epsilon+\left(4N-\dfrac{25}{4}\right)a_{2}\right]N,\;\;\;\;\;\;\;\;\;\;\;A_{3}=2\sqrt{2}a_{2}(N-1)N$
$\displaystyle A_{4}$
$\displaystyle=\left[\epsilon+\left(\dfrac{2N+5}{4}-4\right)a_{2}\right]N,\;\;\;\;\;\;A_{0}=5a_{2}N$

For $a_{2}=0$, we get the pure spherical vibrator U(5) limit and for
$\epsilon_{d}=0$, we get the pure deformed rotational Su(3) limit.

Another important quantity that tests the nature of the shape phase transition
of low lying collective states the reduced electric quadrupole transition
probabilities $B(E_{2})$.

In IBM, the general form of the electric quadrupole operator is written in the
form [50]

$T(E_{2})=eQ(sdIBM)$ (54)

The coefficient e is the boson’s effective charge.

The reduced electric quadrupole transition probabilities are given by

$B[E_{2},I_{i}\rightarrow I_{f}]=\dfrac{1}{2I_{i}+1}\rvert\langle
I_{f}\rvert\rvert T(E_{2})\rvert\rvert I_{i}\rangle\rvert^{2}$ (55)

For rotational SU(3), yield

$\displaystyle B(E_{2},I+2\rightarrow I)$
$\displaystyle=e^{2}\,\dfrac{3}{4}\dfrac{(I+2)(I+1)}{(2I+3)(2I+5)}(2N-1)(2N+I+3)$
(56) $\displaystyle Q(I)$
$\displaystyle=-e\sqrt{\dfrac{16\pi}{40}}\dfrac{I}{2I+3}(4N+3)$ (57)

For the special case for I=0, we have

$\displaystyle B(E_{2},2_{1}^{+}\rightarrow
0_{1}^{+})=e^{2}\dfrac{1}{5}N(2N+3)$ (58)

##  5 Numerical Calculations and Discussion

In this section, we applied our formalism to eight pairs of nuclei having
identical bands (IB’s) in rare-earth region namely:
$(^{162}Yb-^{166}Hf),(^{162}Er-^{166}Yb),(^{162}Dy-^{166}Er),(^{160}Dy-^{168}Yb),(^{160}Er-^{168}Hf),\\\
(^{158}Er-^{170}W),(^{158}Dy-^{170}Hf)$ and $(^{156}Dy-^{172}W)$.

To calculate the ground state positive parity excitation energy E(I) for each
nucleus, we suggested the CRF3.

The parameters $\alpha,\gamma,\sigma$ of CRF3 have been determined by a
fitting procedure using a computer-simulated search program to minimize the
root mean square deviation of the calculated excitation energies from the
experimental ones. The quality of the fitting is indicated by the standard
common definition of $x$

$\displaystyle
x=\sqrt{\dfrac{1}{N}\Sigma_{i}\left(\dfrac{E_{exp}(I_{i})-E_{cal}(I_{i})}{\delta
E_{exp}(I_{i})}\right)^{2}}$

where N is the number of experimental data points entering the fitting
procedure and $\delta E_{exp}(I_{i})$ is the experimental error in the
excitation energies - The experimental excitation energies are taken from
[51]. The optimized best adopted values of parameters for each nucleus of our
studied nuclei are listed in Table (LABEL:tab:1).

Figure 1: Systematic of the calculated (solid curves) ground state energies for our selected even-even rare earth Dy, Er, YB, Hf, W isotopes versus neutron number N and comparison with the experimental ones (dashed curves). The spin-parity are labeled by $I^{\pi}$.  Table 1: Values of optimized best parameters $\alpha,\gamma,\sigma$ of the collective rotational formula(CRF3) for ground state bands in our selected even-even rare-earth nuclei. $N_{p}$ and $N_{n}$ are the number of valance protons and the number of valance neutrons respectively. Nuclide |  $\alpha$ (KeV) |  $\gamma$ ($10^{-3}$) |  $\sigma$ ($10^{-3}$) |  $N_{p}$ |  $N_{n}$
---|---|---|---|---|---
Dy 156 | 22.96 | 6.964 | 14.54 | 16 | 8
158 | 16.48 | 2.163 | 4.339 | 16 | 10
160 | 14.49 | 0.8683 | 2.021 | 16 | 12
162 | 13.49 | 1.398 | 2.233 | 16 | 14
Er 158 | 32.76 | 9.699 | 23.52 | 14 | 8
160 | 20.73 | 3.017 | 6.641 | 14 | 10
162 | 17.01 | 1.440 | 3.212 | 14 | 12
166 | 13.49 | 0.2573 | 1.188 | 14 | 16
Yb 162 | 27.87 | 6.334 | 14.27 | 12 | 10
166 | 17.08 | 2.053 | 3.95 | 12 | 14
168 | 14.72 | 1.039 | 2.425 | 12 | 16
Hf 166 | 26.60 | 5.565 | 12.67 | 10 | 12
168 | 20.58 | 3.116 | 6.849 | 10 | 14
170 | 15.92 | -0.00749 | 1.391 | 10 | 16
W 170 | 26.44 | 5.714 | 13.55 | 8 | 14
172 | 20.68 | 3.944 | 9.279 | 8 | 16
Figure 2: The calculated energy ratio $R_{4/2}=E(4^{+}_{1})/E(2^{+}_{1})$
versus neutron number N characterizes the low lying spectrum in Dy, Er, Yb,
Hf, and W isotopes. The symbols $o,\ast,\Square,\triangle,$ and x denote
${}_{66}Dy,_{68}Er,_{70}Yb,_{72}Hf,$ and ${}_{74}W$ respectively.

The systematic of the excitation energies of the low spin states as a function
of neutron number N in the considered even-even Dy, Er, Yb, Hf, W isotopes in
the mass region A= 156 - 172 in the normally deformed nuclear are shown in
Figure(1) and compared with the experimental ones. Only the ground state of
positive parity and spin $I^{\pi}=2^{+},4^{+},6^{+},8^{+},10^{+},12^{+}$ has
been indicated. We can see that the excitation energies decrease with
increasing the neutron number. Also, Figure(2) illustrate the calculated
energy ratio $R_{4/2}$ as a function of neutron number N for our studied
nuclei. We observe that for each isotopic chain the value of $R_{4/2}$
increases with increasing N (that is the deformation increased), and the
difference in $R_{4/2}$ for all pairs of IB’s is ranging from 0.4 % to 2.5 %
except the two pairs including the two isotopes ${}^{170,172}W$ (the
difference is about 5%).

Figure 3: The calculated results of kinematic $J^{(1)}$ (dashed curves) and
dynamic $J^{(2)}$ (solid curves) moments of inertia plotted as a function of
rotational frequency $\hbar\omega$ for the studied eight pairs of identical
bands in the rare-earth region. The $\ast$ and o correspond to the lighter and
heavier nucleus respectively.

For the eight pairs of IB’S, the kinematic $J^{(1)}$ and the dynamic $J^{(2)}$
moments of inertia derived from the transition energies are plotted versus the
rotational frequency $\hbar\omega$ as shown in Figure(3). It can be seen that
for all bands $J^{(1)}$ is smaller than $J^{(2)}$ and a smooth gradual
increase in both $J^{(1)}$ and $J^{(2)}$ with increasing $\hbar\omega$ are
seen and the similarities between each pair of IB’S are observed.

The IB’s correlation quantities exist between the considered pairs of nuclei
which exhibit the same identical excitation energies in their ground state
bands are listed in Table (LABEL:tab:21). These quantities include the P.
Factor, structure Factor SF, Saturation parameter SP, the F-Spin and its
projection $F_{0}$, pairing gaps $\bigtriangleup$, and the deformation
parameter $\beta$. The maximum structure factor for our region of nuclei is
SF= 6720. It is seen that the ratio $N_{p}N_{n}/\bigtriangleup$ rather than
the product $N_{p}N_{n}$ may be a better parameter for studying the IB’s. Note
that nuclei with symmetric $\pm F_{0}$ values have identical $N_{p}N_{n}$
values. For example the pair (${}^{160}Er$ and ${}^{168}Hf$) have
$(N_{p},N_{n})=(14,10)$ and $(10,14)$ respectively, so that $N_{p}N_{n}=140$
and $F_{0}=\pm 1$. Therefore if any F-spin multiplet has $F_{0}=\rvert
N_{p}-N_{n}\rvert/4$, those indicate that the pair of nuclei are similar in
structure if they have identical $(\rvert F_{0}\rvert,N_{p}N_{n})$.

Table 2: The identical band quantities of our eight pairs of nuclei. | $N_{p}N_{n}$ | P | SF | SP | $\lvert\delta\rvert\%$ | $\lvert k\rvert\%$
---|---|---|---|---|---|---
(${}^{158}Er\;-\;^{170}W\;$) | 112 | 5.090 | 2464 | 0.7317 | 1.28 | 1.27
(${}^{162}Yb\;-\;^{166}Hf$) | 120 | 5.4545 | 2640 | 0.7179 | 2.94 | 2.45
(${}^{156}Dy\;-\;^{172}W\;$) | 128 | 5.333 | 3072 | 0.6862 | 6.73 | 6.28
(${}^{160}Er\;-\;^{168}Hf$) | 140 | 5.833 | 3360 | 0.6666 | 1.35 | 1.22
(${}^{158}Dy\;-\;^{170}Hf$) | 160 | 6.1538 | 4160 | 0.6176 | 1.28 | 1.27
(${}^{162}Er\;-\;^{166}Yb$) | 168 | 6.6461 | 4368 | 0.6060 | 0.22 | 0.20
(${}^{160}Dy\;-\;^{168}Yb$) | 192 | 6.6857 | 5376 | 0.5555 | 0.10 | 0.30
(${}^{162}Dy\;-\;^{166}Er$) | 224 | 7.466 | 6720 | 0.5 | 1.29 | 1.26
| $(N_{\pi},N_{\nu})$ | N | $\dfrac{N_{\nu}}{N_{\pi}}$ | $(F,F_{0})$ |  $\bigtriangleup$ (MeV) |  $\dfrac{N_{p}N_{n}}{\bigtriangleup}$(MeV-1) | $\beta_{G}$
---|---|---|---|---|---|---|---
$\;\;{}^{158}Er\;\;\;\;$ | (7,4) | 11 | 0.571 | (5.5,1.5) | 0.954 | 117.4 | 0.2173
$\;\;{}^{170}W\;\;\;\;$ | (4,7) | 11 | 1.750 | (5.5,-1.5) | 0.920 | 121.739 | 0.2206
$\;\;{}^{162}Yb\;\;\;\;$ | (6,5) | 11 | 0.833 | (5.5,0.5) | 0.942 | 127.388 | 0.2270
$\;\;{}^{166}Hf\;\;\;\;$ | (5,6) | 11 | 1.2 | (5.5,-0.5) | 0.931 | 128.893 | 0.2254
$\;\;{}^{156}Dy\;\;\;\;$ | (8,4) | 12 | 0.5 | (6,2) | 0.960 | 133.333 | 0.2601
$\;\;\;\;{}^{172}W\;\;\;\;$ | (4,8) | 12 | 2.0 | (6,-2) | 0.914 | 140.043 | 0.2459
$\;\;{}^{160}Er\;\;\;\;$ | (7,5) | 12 | 0.714 | (6,1) | 0.948 | 147.679 | 0.2643
$\;\;{}^{168}Hf\;\;\;\;$ | (5,7) | 12 | 1.4 | (6,-1) | 0.925 | 151.351 | 0.2517
$\;\;{}^{158}Dy\;\;\;\;$ | (8,5) | 13 | 0.625 | (6.5,1.5) | 0.954 | 167.714 | 0.3026
$\;\;{}^{170}Hf\;\;\;\;$ | (5,8) | 13 | 1.6 | (6.5,-1.5) | 0.920 | 173.913 | 0.2754
$\;\;{}^{162}Er\;\;\;\;$ | (7,6) | 13 | 0.857 | (6.5,0.5) | 0.942 | 178.343 | 0.2896
$\;\;{}^{166}Yb\;\;\;\;$ | (6,7) | 13 | 1.166 | (6.5,-0.5) | 0.931 | 180.451 | 0.2814
$\;\;{}^{160}Dy\;\;\;\;$ | (8,6) | 14 | 0.75 | (7,1) | 0.948 | 202.531 | 0.3181
$\;\;{}^{168}Yb\;\;\;\;$ | (6,8) | 14 | 1.333 | (7,-1) | 0.925 | 207.567 | 0.2993
$\;\;{}^{162}Dy\;\;\;\;$ | (8,7) | 15 | 0.875 | (7.5,0.5) | 0.942 | 237.791 | 0.3256
$\;\;{}^{166}Er\;\;\;\;$ | (7,8) | 15 | 1.142 | (7.5,-0.5) | 0.931 | 240.601 | 0.3167

The percentage differences ratios in transition energy $\delta$ and the rigid
rotor ratio $\delta_{R}$ between pairs of levels in two nuclei are calculated
and listed in Table(4) for our eight pairs of IB’s. In spite of the parameters
$N_{p}N_{n}$, P, SF and SP are the same for the pairs $(^{156}Dy,^{172}W)$,
this pair is not really identical according to their high average percentage
differences in transition energies (approximately 6.7%).

For each nucleus in isotopic chains of ${}_{66}Dy,_{68}Er,_{70}Yb,_{72}Hf$ and
${}_{74}W$, the values of lowest dynamical moments of inertia
$J^{(2)}_{lowest}$ were calculated and displayed against the neutron number N
in Figure(4) - It can be seen that $J^{(2)}_{lowest}$ increases with
increasing the neutron number N and the difference in$J^{(2)}_{lowest}$ for
each pair of IB’s is very small ( approximately a horizontal line). As an
example of two nuclei that exhibit good IB’s, the pair
${}^{162}_{68}Er(J^{(2)}_{lowest}=31.525\hbar^{2}MeV^{-1})$ and
${}^{166}_{70}Yb(J^{(2)}_{lowest}=31.519\hbar^{2}MeV^{-1})$, that is nearly
the same $J^{(2)}_{lowest}$.

Table 4: The percentage differences ratios in transition energies $\delta$, the fractional change in transition energies divided by the rigid rotor ratio $\delta R$ and the ratio $R\;=\delta/\delta R$ for the eight pairs of identical bands. Identical pairs | $\lvert\delta\rvert=\dfrac{\bigtriangleup E_{\gamma}}{E_{\gamma_{2}}}\;\;\%$ | $\delta R$ | $\langle R_{\delta}\rangle$
---|---|---|---
(${}^{162}Yb\;-\;^{166}Hf$) | 2.964 | 4.149 | 0.714
(${}^{162}Er\;-\;^{166}Yb$) | 0.415 | 4.149 | 0.100
(${}^{162}Dy\;-\;^{166}Er$) | 1.297 | 4.149 | 0.312
(${}^{160}Er\;-\;^{168}Hf$) | 1.352 | 8.471 | 0.159
(${}^{160}Dy\;-\;^{168}Yb$) | 1.131 | 8.471 | 0.133
(${}^{158}Er\;-\;^{170}W\;$) | 10.826 | 12.976 | 0.834
(${}^{158}Dy\;-\;^{170}Hf$) | 1.765 | 12.976 | 0.136
(${}^{156}Dy\;-\;^{172}W\;$) | 7.410 | 17.671 | 0.419
Figure 4: The lowest dynamical moment of inertia $J^{(2)}_{lowest}$ against
the neutron number N for the eight pairs of identical bands. The solid line
connects each pair and symbols $o,\ast,\triangle,\Square,$ and $\diamondsuit$
denotes ${}_{66}Dy,_{68}Er,_{70}Yb,_{72}Hf,$ and ${}_{74}W$ respectively.

We classified our selected pairs of IB’s into four multiplets = (A+4), Z+2),
(A+B,Z+4), (A+12,Z+6), and (A+16,Z+8) and the percentage differences in
transition energies $\delta=\bigtriangleup E_{\gamma}/E_{\gamma_{2}}$ as a
function of spin I (up to I=10) have been calculated and illustrated Figure
(5). It is seen that the pairs of IB’s have approximately similar $\delta$ (
less than 2.5 %) except the two pairs which include the tungsten isotopes
${}^{170,172}W$ where the value of $\delta$ reaches $\sim 6-10$% in spite of
they have the same $N_{p}N_{n}$ value ($N_{p}N_{n}=112$ for
${}^{158}Er,^{170}W$ and $N_{p}N_{n}=128$ for ${}^{156}Dy,^{172}W$).

To further investigation for IB’s we used the SU(3) rotational limit of the
IBM to extract the quadrupole deformation $\beta_{IBM}$ for each nucleus. The
calculated $\beta_{IBM}$ is plotted against the ratio $N_{\nu}/N_{\pi}$ (where
$N_{\nu}$ and $N_{\pi}$ are the number of valence neutron and valence proton
bosons respectively) in Figure(6). It is seen that $\beta_{IBM}$ is the same
for each pair of IB’s (horizontal line).

Figure 5: Percentage difference in transition energies $\delta=\bigtriangleup
E_{\gamma}/E_{\gamma_{2}}$ for the eight pairs of multiplet (A+4,Z+2),
(A+8,Z+4), (A+12,Z+6), and (A+16,Z+8) for Dy, Er, Yb, Hf, and W isotopes. The
dashed curve represents the ratio of the rigid rotor. Figure 6: The quadrupole
deformation parameter $\beta_{IBM}$ was calculated from SU(3) limit of IBM as
a function of $N_{\nu}/N_{\pi}$ for our eight pairs of identical bands. Figure
7: Sketch of the potential energy surface PES calculated from the U(5)-SU(3)
shape phase transitions of IBM with intrinsic coherent state versus the
deformation parameters $\beta$ for the eight pairs of even-even nuclei having
identical bands.

For each nucleus, by using the IBM Hamiltonian equation (45) and its
eigenvalues equation (53), the PES’s have been calculated as a function of
deformation parameter $\beta$ along the axial trajectory $\gamma$ = 0°, 60°.
The results are illustrated in Figure(7) and the corresponding calculated
parameter of the PES’s $A_{2},A_{3},A_{4}$ and $A_{o}$ which are linear
combinations of the original parameters $\epsilon_{0}$ and $a_{2}$ are listed
in Table(4). From the graphs presented in Figure(7), we observe the similarity
in PES’s for each pair of IB’s. All studied nuclei are deformed and have
rotational characters, the prolate deformation is deeper than the oblate
deformation.

Table 5: Values of the adopted best (PES) parameters $A_{2},A_{3},A_{4},A_{0}$ ( in KeV ) for the studied eight pairs of identical bands. $N_{B}$ is the total number of bosons. |  $N_{B}$ |  $A_{2}$ |  $A_{3}$ |  $A_{4}$ |  $A_{0}$
---|---|---|---|---|---
$\;\;{}^{162}Dy\;\;\;\;$ | 15 | -2.4667 | -0.5863 | 1.6665 | -0.3265
$\;\;{}^{166}Er\;\;\;\;$ | 15 | -1.6586 | -2.0341 | 4.4739 | -0.7875
$\;\;{}^{162}Er\;\;\;\;$ | 13 | -5.0526 | -2.5496 | 3.7667 | -0.9375
$\;\;{}^{166}Yb\;\;\;\;$ | 13 | -5.3088 | -3.1366 | 4.0554 | -0.925
$\;\;{}^{162}Yb\;\;\;\;$ | 11 | -4.84 | -1.6163 | 3.6775 | -0.9
$\;\;{}^{166}Hf\;\;\;\;$ | 11 | -2.8484 | -1.9547 | 3.9131 | -0.8625
$\;\;{}^{160}Dy\;\;\;\;$ | 14 | -1.9568 | -0.8838 | 1.1005 | -0.3
$\;\;{}^{168}Yb\;\;\;\;$ | 14 | -5.3088 | -3.1366 | 4.0554 | -0.925
$\;\;{}^{160}Er\;\;\;\;$ | 12 | -3.0403 | -2.3636 | 4.1401 | -0.8625
$\;\;{}^{168}Hf\;\;\;\;$ | 12 | -3.463 | -2.4694 | 4.039 | -0.875
$\;\;{}^{158}Dy\;\;\;\;$ | 13 | -1.6288 | -1.1822 | 1.0095 | -0.288
$\;\;{}^{170}Hf\;\;\;\;$ | 13 | -3.1845 | -3.395 | 4.497 | -0.8375
$\;\;{}^{158}Er\;\;\;\;$ | 11 | -1.6586 | -2.0541 | 4.4739 | -0.7875
$\;\;{}^{170}W\;\;\;\;$ | 11 | -0.9761 | -2.4841 | 4.7606 | -0.7546
$\;\;{}^{156}Dy\;\;\;\;$ | 12 | -1.5043 | -1.2135 | 0.9961 | -0.3
$\;\;{}^{172}W\;\;\;\;$ | 12 | -0.8852 | -1.4675 | 1.0599 | -0.313

##  6 Conclusion

By using a novel three parameters collective rotational formula (CRF3), the
positive parity ground state excitation energies are calculated for sixteen
nuclei in rare-earth region. The optimized three parameters are deduced by
using a computer simulated search program in order to obtain a minimum root
mean square deviation of the calculated excitation energies from the measured
ones. The potential energy surfaces are calculated by using the sd-version of
the interacting boson model.

The problem of low-spin identical bands in normal deformed nuclei in rare-
earth region is treated. We have exhibited identical bands in eight pairs of
conjugate even-even nuclei of widely dispersed spanning as much as sixteen
mass unit. Each pair with the same F-spin and projections $\pm F_{0}$ values
have identical product of valence proton and neutron numbers $N_{p}N_{n}$
values. Also, the values of dynamical moments of inertia for each identical
band pair are approximately the same. We extracted all the identical band
symmetry parameters like P-factor, saturation parameter, and structure factor
which all depend on $N_{p}$ and $N_{n}$. The pairing interaction energy, the
quadrupole transition probabilities, and the energy ratios are also treated.

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