# Investigation of Nuclear Phase Transition by Solvababe supersymmetric
algebraic model and its application in Ru-Rh and Zn-Cu Isotopes

M. A. Jafarizadeh<EMAIL_ADDRESS>Department of Theoretical Physics
and Astrophysics, University of Tabriz, Tabriz 51664, Iran.  Research
Institute for Fundamental Sciences, Tabriz 51664, Iran  M.Ghapanvari
<EMAIL_ADDRESS>Department of Nuclear Physics, University of
Tabriz, Tabriz 51664, Iran.  The Plasma Physics and Fusion Research School,
Tehran, Iran.  N.Fouladi<EMAIL_ADDRESS>Department of Nuclear Physics,
University of Tabriz, Tabriz 51664, Iran.  Z.Ranjbar<EMAIL_ADDRESS>Department of Nuclear Physics, University of Tabriz, Tabriz 51664, Iran.
A.Sadighzadeh  The Plasma Physics and Fusion Research School, Tehran, Iran.
<EMAIL_ADDRESS>
###### Abstract

Solvable supersymmetric algebraic model for descriptions of the spherical to
$\gamma-unstable$ shape-phase transition in even and odd mass nuclei is
proposed. This model is based on dual algebraic structure and Richardson -
Gaudin method, where the duality relations between the unitary and quasispin
algebraic structures for the boson and fermion systems are extended to mixed
boson-fermion system. The structure of two type of nuclear supersymmetry
schemes, based on the U(6/2) and U(6/4) supergroups, is discussed. We
investigate the change in level structure induced by the phase transition by
doing a quantal analysis. By using the generalized quasispin algebra, it is
shown that the nuclear supersymmetry concept can be also used for transitional
regions in addition to dynamical symmetry limits. Experimental evidence for
the U(5)-O(6) transition in Ru-Rh and Zn-Cu supermultiplets is presented. The
low-states energy spectra and B(E2)values for these nuclei have been
calculated and compared with the experimental data.

††preprint: APS/123-QED

##  I Introduction

Symmetry is one of the fundamental concepts in physics. In the nuclei,
symmetry is in the form of dynamical symmetry. The dynamical symmetries were
introduced in nuclear physics with the development of the interacting boson
model and its extensions1 ; 2 . The IBM describes the collective excitations
of even-even nuclei in terms of correlated pairs of nucleons with L = 0, 2
treated as bosons (s, d bosons)1 . The IBM represents a simple description of
nuclear collective excitations based of an algebraic theory where symmetry
transformations play a significant role. The N-boson system space is spanned
by the irrep $\mathrm{[N]}$ of $\mathrm{U^{B}(6)}$ 1 . The interacting boson -
fermion model also explains odd-A nuclei in terms of correlated pairs, s and d
bosons , and unpaired particles of angular momentum j (j fermions)2 . The
states of the boson - fermion system can be classified according to the
irreducible representation $\mathrm{[N]\times[1]}$ of $\mathrm{U^{B}(6)\times
U^{F}(M)}$ where M is the dimension of the single particle space. The IBM and
IBFM can be unified into a supersymmetry (SUSY) approach that was discovered
into nuclear structure physics in the early 1980 3 ; 4 .The experiments
performed at various laboratories have confirmed the predictions made using a
SUSY scheme 5  . Originally, nuclear supersymmetry was considered as symmetry
among pairs of nuclei consisting of an even-even and an odd-even nucleus3 ; 4
. The supersymmetric representations $[\mathcal{N}\\}$,
$\mathcal{N}=\mathrm{N_{B}+N_{F}}$, of $\mathrm{U(6/M)}$ spanned a space that
explains the lowest states of an even-even nucleus with $\mathcal{N}$ bosons
and an odd-A nucleus with $\mathcal{N}-1$ bosons and a unpaired fermion2 . In
supersymmetry approach, the core-particle interaction entirely are determined
by the even-even Hamiltonian and most of the parameters in the core-particle
interaction can be taken from the neighboring even-even nucleus thus the
nuclear properties of neighboring even-even and odd-A nuclei can be described
with a single Hamiltonian and a single set of transition and transfer
operators2 ; 6 . The nuclear supersymmetry have been used successfully in
description dynamical symmetry limits of the even-even and odd-A nuclei2 ; 3 ;
4 ; 7 . The IBFM could be an ideal ground for testing supersymmetry7 .
Virtually simultaneously, with the introduction of the nuclear supersymmetry,
the idea of spherical-deformed phase transitions at low energy in finite
nuclei germinated8 ; 9 . Studies of QPTs in odd-even nuclei with
supersymmetric scheme had implicitly been initiated years before by A. Frank
et al.6 . They have studied successfully a combination of $\mathrm{U^{BF}(5)}$
and $\mathrm{SO^{BF}(6)}$ symmetry by using $\mathrm{U(6/12)}$ supersymmetry
for the Ru and Rh isotopes. Iachello 10  extended the concept of critical
symmetry to critical supersymmetry and provided a benchmark for the study of
shape phase transition in odd-even nuclei and also J.Jolie et al.11  studied
QPTs in odd-even nuclei using a supersymmetric approach in interacting Boson-
Fermion model.

In this paper, concept of supersymmetry and phase transitions are brought
together by using the generalized quasi-spin algebra and Richardson - Gaudin
method. Exactly solvable solution for the spherical to gamma - unstable
transition in transitional nuclei based on dual algebraic structure and
nuclear supersymmetry concept is proposed. Two separate cases are considered
with the fermions lying in a j=1/2 shell or in a j=3/2 shell. The existence of
duality symmetries has proven to be a powerful tool in relating the
Hamiltonians with the number-conserving unitary and number-nonconserving
quasispin algebras for system with pairing interactions that first introduced
in nuclear physics in Ref.12  and further developed in Refs.13 ; 14 . These
relations were obtained for both bosonic and fermionic systems13 ; 14 . We
have established the duality relations for mixed boson-fermion system. We
evaluate exact solutions for eigenstate and energy eigenvalues for
transitional region and dynamical-symmetry limits for even-even and odd-A
nuclei in Supersymmetry scheme by using Richardson - Gaudin method and
changing the control parameters that based on affine $\mathrm{\widehat{GQA}}$
generalized quasi-spin algebra. In order to investigation of phase
transition,we calculate observables such as level crossing, expectation values
of the d-boson number operator and expectation values of the fermion number
operator. The low-lying states of even-mass ruthenium and the odd-mass rhodium
isotopes and Zn-Cu supermultiplets have been studied within suggested model.
The results of calculations for these nuclei will present for energy levels
and transitions probabilities, two neutron separation energies and will
compare with the corresponding the experimental data.

This paper is organized as follows: Section 2 briefly summarizes theoretical
aspects of transitional Hamiltonian and generalized quasi-spin algebraic
technique. Sections 3 and 4 include the quantal analysis and experimental
evidence and sect. 5 is devoted to the summary and some conclusions.

##  II Theoretical framework

###  II.1 The Model

The quasispin algebras have been explained in detail in Refs 12 ; 13 ; 14 .
The SU(1,1) algebra is produced by $\mathrm{S^{\nu}}$, $\mathrm{{\nu=0}}$ and
$\mathrm{{\pm}}$ , which satisfies the following commutation relations12 ; 13
; 14

$[S_{B}^{0},S_{B}^{\pm}]={\pm}S_{B}^{\pm}\quad,\,\,\,[S_{B}^{+},S_{B}^{-}]=-2S_{B}^{0}$
(1)

In IBM , the generators of $\mathrm{SU^{d}(1,1)}$ generated by the d-boson
pairing algebra12

$S^{+}(d)=\frac{1}{2}(d^{+}.d^{+})\quad\quad,S^{-}(d)=\frac{1}{2}(\widetilde{d}.\widetilde{d}),\quad\quad
S^{0}(d)=\frac{1}{4}{\sum_{\nu}({d_{\nu}^{+}d_{\nu}+d_{\nu}d_{\nu}^{+}}})=\frac{1}{4}(2\hat{n}_{d}+5)$
(2)

Similarly, s- boson pairing algebra forms another $\mathrm{SU^{s}(1,1)}$
algebra generated by 12

$S^{+}(s)=\frac{1}{2}s^{+^{2}}\quad,\quad S^{-}(s)=\frac{1}{2}s^{2},\quad
S^{0}(s)=\frac{1}{4}(s^{+}s+ss^{+})=\frac{1}{4}(2\hat{n}_{s}+1)$ (3)

$\mathrm{SU^{sd}(1,1)}$is the s and d boson pairing algebras generated by12

$S^{+}(sd)=\frac{1}{2}(d^{+}.d^{+}\pm s^{+^{2}})\quad,\quad
S^{-}(sd)=\frac{1}{2}(\widetilde{d}.\widetilde{d}\pm
s^{2})\quad,S^{0}(sd)=\frac{1}{4}{\sum_{\nu}({d_{\nu}^{+}d_{\nu}+d_{\nu}d_{\nu}^{+}})}+\frac{1}{4}(s^{+}s+ss^{+})$
(4)

The fermionic quasispin operators are13 ; 14

$S_{F}^{+}=\frac{1}{2}\sum\limits_{m^{\prime}}(-1)^{j\mp
m^{\prime}}a_{jm^{\prime}}^{+}a_{j-m^{\prime}}^{+}=\frac{1}{2}(a_{j}^{+}\cdot
a_{j}^{+})$ (5) $S_{F}^{-}=\frac{1}{2}\sum\limits_{m^{\prime}}(-1)^{j\mp
m^{\prime}}\widetilde{a}_{jm^{\prime}}\widetilde{a}_{j-m^{\prime}}=\frac{1}{2}(\widetilde{a}_{j}\cdot\widetilde{a}_{j})$
(6) $S_{F}^{0}=\frac{n_{f}}{2}-\frac{2j+1}{4}$ (7)

satisfy the commutation relations of the fermion quasispin SU(2) algebra13 ;
14 .

$[S_{F}^{0},S_{F}^{\pm}]={\pm}S_{F}^{\pm}\quad,\,\,\,[S_{F}^{+},S_{F}^{-}]=2S_{F}^{0}$
(8)

The generalized quasispin algebra contain both bosonic (B) and fermionic (F)
operators defined as 15

$S_{BF}^{0}=S_{B}^{0}+S_{F}^{0},\quad S_{BF}^{+}=S_{B}^{+}+S_{F}^{+},\quad
S_{BF}^{-}=S_{B}^{-}+S_{F}^{-}$ (9)

Thus

$S_{BF}^{0}=\frac{1}{4}(N-M)+\frac{1}{2}(n_{b}+n_{f})$ (10) $\displaystyle
S_{BF}^{+}=\frac{1}{2}\sum\limits_{m}(-1)^{2\mp
m}b_{m}^{+}b_{-m}^{+}\mp\frac{1}{2}\sum\limits_{m^{\prime}}(-1)^{j\mp
m^{\prime}}a_{jm^{\prime}}^{+}a_{j-m^{\prime}}^{+}=\frac{1}{2}(b^{+}\cdot
b^{+})\mp(a_{j}^{+}\cdot a_{j}^{+})$ (11)
$S_{BF}^{-}=\frac{1}{2}(\widetilde{b}.\widetilde{b})\pm\frac{1}{2}(\widetilde{a}_{j}.\widetilde{a}_{j})$
(12)

Where $\mathrm{n_{b}}$ and $\mathrm{n_{f}}$ are the boson and fermion number
operator, respectively. Thus the operators
$\mathrm{(S_{BF}^{\pm}}$,$\mathrm{S_{BF}^{0})}$ form a generalization of the
usual fermionic and bosonic quasispin algebras with commutation relations
given as15

$[S_{BF}^{0},S_{BF}^{\pm}]=\pm
S_{BF}^{\pm}\quad,\quad[S_{BF}^{+},S_{BF}^{-}]=-2S_{BF}^{0}$ (13)

The generalized quasispin algebra(GQA) may be labeled in terms of the
eigenvalues of the second-order Casimir invariant and those of the quasispin
operator $\mathrm{S_{BF}^{0}}$ 15 . The second-order Casimir invariant for the
GQA is defined as

$C_{2}(GQA)=S_{BF}^{0}(S_{BF}^{0}-1)-S_{BF}^{+}S_{BF}^{-}$ (14)

The basis states of an irreducible representation (irrep)GQA,
$\mathrm{|k,\mu\rangle}$ ,are determined by a single number $k$ , where can be
any positive number and $\mathrm{\mu=k,k+1,....}$ Therefore,

$S_{BF}^{0}|k,\mu\rangle=(\frac{N-M}{4}+\frac{(N_{B}+N_{F})}{2})|k,\mu\rangle$
(15)
$C_{2}(su(1,1))|k,\mu\rangle=k(k-1)|k,\mu\rangle=(\frac{N-M}{4}+\frac{(\nu_{B}+\nu_{F})}{2})(\frac{N-M}{4}+\frac{(\nu_{B}+\nu_{F})}{2}-1)|k,\mu\rangle$
(16)
$|k,\mu\rangle=|\frac{N-M}{4}+\frac{\nu_{B}+\nu_{F}}{2},\frac{N-M}{4}+\frac{N_{B}+N_{F}}{2}\rangle$
(17)

The basis states are determined considering the fact that fermionic part of
the action of $\mathrm{S_{BF}^{+}}$is restricted by the Pauli Exclusion
Principle and the action of $\mathrm{S_{BF}^{-}}$ terminates when a state of
$\mathrm{\nu}$ unpaired particles is reached i.e
$\mathrm{S_{BF}^{-}|\nu_{B},\nu_{F}\rangle=0}$ 15 .

The infinite dimensional generalized quasispin algebra that is generated by
use of 12

$S_{BF,n}^{\pm}=c_{s}^{2n+1}S_{B}^{\pm}(s)+c_{d}^{2n+1}S_{B}^{\pm}(d)+c_{f}^{2n+1}S_{F}^{\pm}(f)$
(18)
$S_{BF,n}^{0}=c_{s}^{2n}S_{B}^{0}(s)+c_{d}^{2n}S_{B}^{0}(d)+c_{f}^{2n}S_{F}^{0}(f)$
(19)

Where $\mathrm{c_{s}}$ , $\mathrm{c_{d}}$ and $\mathrm{c_{f}}$ are real
parameters and n can be $\mathrm{0,\pm 1,\pm 2,....}$. These generators
satisfy the commutation relations

$[S_{BF,m}^{0},S_{BF,n}^{\pm}]=\pm
S_{BF,m+n}^{\pm}\quad\quad,\quad\quad[S_{BF,m}^{+},S_{BF,n}^{-}]=-2S_{BF,m+n+1}^{0}$
(20)

Then, $\mathrm{{S_{BF,m}^{\mu},\mu=0,+,-;m=\pm 1,\pm 2,...}}$generate an
affine generalized quasispin algebra $\mathrm{\widehat{GQA}}$ without central
extension. For evaluating the eigenvalues, the eigenstates are considered as

$|k;\nu_{s}\nu
n_{\Delta}LM\rangle=\textit{N}S_{BF}^{+}(x_{1})S_{BF}^{+}(x_{2})S_{BF}^{+}(x_{3})...S_{BF}^{+}(x_{k})|lw\rangle^{BF}$
(21)
$S_{x_{i}}^{+}=\frac{c_{s}}{1-c_{s}^{2}x_{i}}S_{B}^{+}(s)+\frac{c_{d}}{1-c_{d}^{2}x_{i}}S_{B}^{+}(d)+\frac{c_{f}}{1-c_{f}^{2}x_{i}}S_{F}^{+}(f)$
(22)

The lowest weight state, $\mathrm{|lw\rangle^{BF}}$, is defined as

$\displaystyle|lw\rangle^{BF}$
$\displaystyle=|N=N_{B}+N_{F},k_{d}=\frac{1}{2}(\nu_{d}+\frac{5}{2}),\mu_{d}=\frac{1}{2}(n_{d}+\frac{5}{2}),k_{s}=\frac{1}{2}(\nu_{s}+\frac{1}{2}),\mu_{s}=\frac{1}{2}(n_{s}+\frac{1}{2}),k_{f}=\frac{1}{2}$
$\displaystyle(\nu_{f}-\frac{2j+1}{2}),\mu_{f}=\frac{1}{2}(n_{f}-\frac{2j+1}{2}),J,M\rangle$
(23)
$S_{n}^{0}|lw\rangle^{BF}=\Lambda_{n}^{0}|lw\rangle^{BF}\quad,\quad\Lambda_{n}^{0}=c_{s}^{2n}(\nu_{s}+\frac{1}{2})\frac{1}{2}+c_{d}^{2n}(\nu_{d}+\frac{5}{2})\frac{1}{2}+c_{f}^{2n}(\nu_{f}-\frac{2j+1}{2})\frac{1}{2}$
(24)

In order to obtaining an algebraic solution for transitional region, we have
used of dual algebraic structures. Such relations have often been used to
effect simplifications of the calculations for two-level and multi-level
systems. These relations were obtained for both bosonic and fermionic
systems13 ; 14 . We have established the duality relations for mixed boson-
fermion system. Because of duality relationships 12 ; 13 ; 14 , It is known
that in even-even nuclei the base of $\mathrm{U(5)\supset SO(5)}$ and
$\mathrm{SO(6)\supset SO(5)}$ are simultaneously the basis of
$\mathrm{SU^{d}(1,1)\supset U(1)}$ and $\mathrm{SU^{sd}(1,1)\supset U(1)}$,
respectively. By use of duality relations 12 ; 13 ; 14 , the Casimir operators
of SO(5) and SO(6) can also be expressed in terms of the Casimir operators of
$\mathrm{SU^{d}(1,1)}$ and $\mathrm{SU^{sd}(1,1)}$, respectively

$\hat{C}_{2}(SU^{d}(1,1))=\frac{5}{16}+\frac{1}{4}\hat{C}_{2}(SO^{B}(5))$ (25)
$\hat{C}_{2}(SU^{sd}(1,1))=\frac{3}{4}+\frac{1}{4}\hat{C}_{2}(SO^{B}(6))$ (26)

For a mixed boson-fermion system, the chain of subalgebras of unitary
superalgebras U(6/M ) for j=1/2 and j=3/2 is shown in Fig.1 and Fig.2,
respectively. The two-level pairing system has two dynamical symmetries
defined with respect to the generalized quasispin algebras, corresponding to
either the upper or lower subalgebra chains in Eq.(27).

$GQA_{1}^{sf}\otimes
GQA_{2}^{df}\supset\left\\{\begin{array}[]{cl}GQA_{1,2}^{sdf}\\\
U_{1}^{sf}(1)\otimes U_{2}^{df}(1)\\\ \end{array}\right\\}\supset
U_{1,2}^{sdf}$ (27)

The upper subalgebra chain is corresponding to strong-coupling dynamical
symmetry limit while lower chain is weak-coupling limit. For odd-A nuclei with
j=1/2, we have obtained the relation between the Casimir operators SO(5) and
$\mathrm{GQA^{df}}$ (generalized quasispin algebra of d bosons and single
fermion with j=1/2) and the Casimir operators SO(6) and $\mathrm{GQA^{sdf}}$
(generalized quasispin algebra of s and d bosons and single fermion with
j=1/2) according with Eqs.[25] and [26], respectively. The correspondence
between the basis vectors in this case was shown in Ref.12 .For odd-A nuclei
with j=3/2, we have obtained the relation between the Casimir operators
$\mathrm{Spin^{BF}(5)}$ and $\mathrm{GQA^{df}}$ (generalized quasispin algebra
of d bosons and single fermion with j=3/2) as

If $\mathrm{\tau_{1}=v_{d}-\frac{1}{2}}$ and $\mathrm{\tau_{2}=\frac{1}{2}}$

$\hat{C}_{2}(GQA^{df})=\frac{1}{4}\hat{C}_{2}(Spin^{BF}(5))-\frac{1}{4}(\tau_{1}+\frac{3}{4})$
(28)

If $\mathrm{\tau_{1}=v_{d}+\frac{1}{2}}$ and $\mathrm{\tau_{2}=\frac{1}{2}}$

$\hat{C}_{2}(GQA^{df})=\frac{1}{4}\hat{C}_{2}(Spin^{BF}(5))-\frac{1}{4}(3\tau_{1}+\frac{7}{4})$
(29)

By use of duality relations, the correspondence between the basis vectors
$\mathrm{Spin^{BF}(5)}$ and $\mathrm{GQA^{df}}$ is

$|\mathcal{N};[N_{B}=N],{N_{F}=1},\nu_{d},(\tau_{1}=v_{d}-\frac{1}{2},\tau_{2}),n_{\Delta}JM\rangle=|\mathcal{N};k^{d}=\frac{1}{2}(\nu_{d}+\frac{5}{2}),k^{df}=\frac{\tau_{1}+2}{2},\mu^{df}=\frac{1}{4}+\frac{1}{2}(n_{d}+n_{f}),n_{\Delta}JM\rangle$
(30)
$|\mathcal{N};[N_{B}=N],{N_{F}=1},\nu_{d},(\tau_{1}=v_{d}+\frac{1}{2},\tau_{2}),n_{\Delta}JM\rangle=|\mathcal{N};k^{d}=\frac{1}{2}(\nu_{d}+\frac{5}{2}),k^{df}=\frac{1+\tau_{1}}{2},\mu^{df}=\frac{1}{4}+\frac{1}{2}(n_{d}+n_{f}),n_{\Delta}JM\rangle$
(31)

The Casimir operator of $\mathrm{Spin^{BF}(6)}$ and $\mathrm{GQA^{sdf}}$
(generalized quasispin algebra of d and s bosons with fermion j=3/2 ) has the
following correspondence If $\mathrm{\sigma_{1}=\sigma-\frac{1}{2}}$ and
$\mathrm{\sigma_{2}=|\sigma_{3}|=\frac{1}{2}}$

$\hat{C}_{2}(GQA^{sdf})=\frac{1}{4}\hat{C}_{2}(Spin^{BF}(6))-\frac{3}{4}(\sigma_{1}+\frac{3}{4})$
(32)

If $\mathrm{\sigma_{1}=\sigma+\frac{1}{2}}$ and
$\mathrm{\sigma_{2}=|\sigma_{3}|=\frac{1}{2}}$

$\hat{C}_{2}(GQA^{sdf})=\frac{1}{4}\hat{C}_{2}(Spin^{BF}(6))-\frac{1}{4}(\sigma_{1}+\frac{3}{2})$
(33)

By use of duality relations, the correspondence between the basis vectors
$\mathrm{Spin^{BF}(6)}$ and $\mathrm{GQA^{sdf}}$ is

$|\mathcal{N};[N_{B}=N],{N_{F}=1},\sigma,(\sigma_{1},\sigma_{2},\sigma_{3}),(\tau_{1},\tau_{2}),n_{\Delta}JM\rangle=|\mathcal{N};k^{sdf}=\frac{1}{2}(\sigma_{1}+\frac{3}{2}),\mu^{sdf}=\frac{1}{4}+\frac{1}{2}(n_{s}+n_{d}+n_{f}),n_{\Delta}JM\rangle$
(34)
$|\mathcal{N};[N_{B}=N],{N_{F}=1},\sigma,(\sigma_{1},\sigma_{2},\sigma_{3}),(\tau_{1},\tau_{2}),n_{\Delta}JM\rangle=|\mathcal{N};k^{sdf}=\frac{1}{2}(\sigma_{1}+\frac{5}{2}),\mu^{sdf}=\frac{1}{4}+\frac{1}{2}(n_{s}+n_{d}+n_{f}),n_{\Delta}JM\rangle$
(35)

In this article, we investigate U(6/2) and U(6/4) supersymmetry. The detailed
description of the U(6/2) and U(6/4) supersymmetry in U(5) and O(6) limits can
be found in Refs. 2 ; 3 ; 4 . Here we restrict ourselves only to a brief
discussion of the main ideas.

###  II.2 U(6/2) Supersymmetry Model

First we considered case that fermions occupy a single particle state with
angular momentum j =1/2. Thus, the underlying graded algebra is U(6/2)2 . In
order to study the occurrence of supersymmetries, one must first identify the
set of nuclei belonging to the representation $\mathcal{N}$. Supermultiplets
here contain 3 nuclei, if $\mathcal{N}\geq 2$. The three nuclei are
characterized by2

$\mathrm{N_{B}}=\mathcal{N},\mathrm{N_{F}}=0$
$\mathrm{N_{B}}=\mathcal{N}-1,\mathrm{N_{F}}=1$
$\mathrm{N_{B}}=\mathcal{N}-2,\mathrm{N_{F}}=2$

These nuclei are alternately even-even and odd-even nuclei and have an
increasing number of unpaired fermions 2 . States with $\mathrm{N_{F}}=0$ and
$\mathrm{N_{F}}=1$ are the lowest states of the corresponding nuclei, while
states with $\mathrm{N_{F}>2}$ are at higher energies 2 . The lattice of
algebras in this model is shown in Fig.(1).

In the following we will explain how derive the even-even and odd-A
Hamiltonians within the U(6/2) supersymmetry scheme.By employing the
generators of $\mathrm{\widehat{GQA}}$ and casimir operators of subalgebras,
the following Hamiltonian for transitional region between U(5)-O(6) limits is
prepared

$\hat{H}=S_{BF,0}^{+}S_{BF,0}^{-}+\alpha S_{BF,1}^{0}+\beta\hat{C}_{2}
̂(O^{B}(5))+\delta\hat{C}_{2} ̂(O^{B}(3))+\gamma\hat{C}_{2}(spin^{BF}(3))$
(36)

$\mathrm{\alpha}$ , $\mathrm{\beta}$ ,$\mathrm{\delta}$ ,$\mathrm{\gamma}$ are
real parameters. The eigenvalues of Hamiltonian Eq.(36) can then be expressed;

$E^{(k)}=h^{(k)}+\alpha\Lambda_{1}^{0}+\beta\nu_{d}(\nu_{d}+3)+\delta
L_{B}(L_{B}+1)+\gamma J(J+1)\quad,\quad
h^{(k)}=\sum_{i=1}^{k}{\frac{\alpha}{x_{i}}}$ (37)
$\frac{\alpha}{x_{i}}=\frac{c_{s}^{2}(\nu_{s}+\frac{1}{2})}{1-c_{s}^{2}y_{i}}+\frac{c_{d}^{2}(\nu_{d}+\frac{5}{2})}{1-c_{d}^{2}x_{i}}+\frac{c_{f}^{2}(\nu_{f}-1)}{1-c_{f}^{2}x_{i}}-{\sum_{j\neq
i}{\frac{2}{x_{i}-x_{j}}}}$ (38)

###  II.3 U(6/4) Supersymmetry Model

The case that fermions occupy a single particle state with angular momentum j
=3/2 constitute the superalgebra U(6/4). Supermultiplets here contain 5
nuclei, if $\mathcal{N}\geq 4$.The five nuclei are characterized by2 ; 3 ; 4

$\mathrm{N_{B}}=\mathcal{N},\mathrm{N_{F}}=0$
$\mathrm{N_{B}}=\mathcal{N}-1,\mathrm{N_{F}}=1$
$\mathrm{N_{B}}=\mathcal{N}-2,\mathrm{N_{F}}=2$
$\mathrm{N_{B}}=\mathcal{N}-3,\mathrm{N_{F}}=3$
$\mathrm{N_{B}}=\mathcal{N}-4,\mathrm{N_{F}}=4$

In order to obtain Hamiltonian within the U(6/4) supersymmetry scheme, We have
considered the lattice of algebras is shown in Fig.2.

So,following Hamiltonian for transitional region between U(5)-O(6) limits is
prepared

$\hat{H}=S_{BF,0}^{+}S_{BF,0}^{-}+\alpha
S_{BF,1}^{0}+\beta^{\prime}\hat{C}_{2}(Spin^{BF}(5))+\gamma\hat{C}_{2}(spin^{BF}(3))$
(39)

The eigenvalues of Hamiltonian Eq.(39) can then be expressed;

$E^{(k)}=h^{(k)}+\alpha\Lambda_{1}^{0}+\beta^{\prime}(\tau_{1}(\tau_{1}+3)+\tau_{2}(\tau_{2}+1))+\gamma
J(J+1)\quad,\quad h^{(k)}=\sum_{i=1}^{k}{\frac{\alpha}{x_{i}}}$ (40)
$\frac{\alpha}{x_{i}}=\frac{c_{s}^{2}(\nu_{s}+\frac{1}{2})}{1-c_{s}^{2}x_{i}}+\frac{c_{d}^{2}(\nu_{d}+\frac{5}{2})}{1-c_{d}^{2}x_{i}}+\frac{c_{f}^{2}(\nu_{f}-2)}{1-c_{f}^{2}x_{i}}-{\sum_{j\neq
i}{\frac{2}{x_{i}-x_{j}}}}$ (41)

It can be shown that Hamiltonians Eq.(36) and Eq.(39) are equivalent to a
boson Hamiltonian for the even-even nuclei if acting on the
$\mathrm{[N]\times[0]}$ representation of $\mathrm{U^{B}(6)\times U^{F}(4)}$
and with boson-fermion Hamiltonian for odd-A nuclei if acting on the
$\mathrm{[N]\times[1]}$ representation of $\mathrm{U^{B}(6)\times U^{F}(4)}$.
By considering the neutron and proton degree of freedom in this model, we can
describe all nuclei belonging to the representation $[\mathcal{N}\\}$. In
odd-A nuclei, Hamiltonians Eq.(36) and Eq.(39) are equivalent to
$\mathrm{O^{BF}(6)}$ Hamiltonian when $\mathrm{c_{s}=c_{d}=c_{f}}$ and with
$\mathrm{U^{BF}(5)}$ Hamiltonian if $\mathrm{c_{s}=0}$ and $\mathrm{c_{d}\neq
c_{f}\neq 0}$. So, the $\mathrm{c_{s}\neq c_{d}\neq c_{f}\neq 0}$ situation
just corresponds to $\mathrm{U^{BF}(5)\leftrightarrow O^{BF}(6)}$ transitional
region. Hamiltonians Eq.(36) and Eq.(39) in even-even nuclei is equivalent to
$\mathrm{O(6)}$ Hamiltonian when $\mathrm{c_{s}=c_{d}}$ and with
$\mathrm{U(5)}$ Hamiltonian if $\mathrm{c_{s}=0}$ and $\mathrm{c_{d}\neq 0}$
and Hamiltonian in transitional region with $\mathrm{c_{s}\neq c_{d}\neq 0}$.
In our calculation, we take $\mathrm{c_{d}(=1)}$ constant value and
$\mathrm{c_{s}}$ and $\mathrm{c_{f}}$ change between 0 and $\mathrm{c_{d}}$

In order to obtain the numerical results for energy spectra
$\mathrm{(E^{(k)})}$ of considered nuclei in two kinds supersymmetry approach,
a set of non-linear Bethe-Ansatz equations (BAE) with k- unknowns for k-pair
excitations must be solved also constants of Hamiltonian with least square
fitting processes to experimental data is obtained 16 . To achieve this aim,
we have changed variables as

$\mathrm{C_{s}=\frac{c_{s}}{c_{d}}\leq 1,C_{f}=\frac{c_{f}}{c_{d}}\leq
1,y_{i}=c_{d}^{2}x_{i}}$
$\frac{\alpha}{y_{i}}=\frac{C_{s}^{2}(\nu_{s}+\frac{1}{2})}{1-C_{s}^{2}y_{i}}+\frac{(\nu_{d}+\frac{5}{2})}{1-y_{i}}+\frac{C_{f}^{2}(\nu_{f}-\frac{2j+1}{2})}{1-C_{f}^{2}y_{i}}-{\sum_{j\neq
i}{\frac{2}{y_{i}-y_{j}}}}$ (42)

The quantum number (k) is related to $\mathcal{N}$ by
$\mathcal{N}=\mathrm{2k+\nu_{s}+\nu_{d}+\nu_{f}}$. The quality of the fits is
quantified by the values of
$\mathrm{\sigma=(\frac{1}{N_{tot}}\sum_{i,tot}{|E_{exp}(i)-E_{Cal}(i)|^{2}})^{\frac{1}{2}}}$
(keV) and
$\mathrm{\phi=\frac{\sum\limits_{i}|E_{i}^{theor}-E_{i}^{exp}|}{\sum\limits_{i}E_{i}^{exp}}}$
(%) ($N_{tot}$ the number of energy levels where included in the fitting
processes) 2 ; 3 ; 4 . A measure of the breaking of the supersymmetry scheme,
the average absolute deviation divided by the average excitation energy, give
with $\phi$ parameter 2 ; 4 . The method for optimizing the set of parameters
in the Hamiltonian $\mathrm{(\beta,\gamma,\delta)}$ includes carrying out a
least-square fit (LSF) of the excitation energies of selected states 16 .

###  II.4 E2 transition probabilities

The observables such as electric quadrupole transition probabilities, $B(E2)$,
as well as quadrupole moment ratios within the low-lying state provide
important information about the nuclear structure and $QPTs$. In this section
we discuss the calculation of E2 transition strengths for j=1/2 and j=3/2.
Supersymmetry implies that all transitions in nuclei belonging to the same
supermultiplet be explained by the same operator2 . The electric quadrupole
transition operator $\hat{T}^{(E2)}$ in odd-A nuclei consists of a bosonic and
a fermionic part2 ; 17 ; 18 . In the $j=1/2$ case, the E2 transitions for
even-even and odd-A nuclei are completely determined by the bosonic part of
the E2 operator2  . The bosonic part have the specific selection rules, where
for former term $\Delta\nu_{d}=\pm 1$, $|\Delta L|\leq 2$ and for latter
$\Delta\nu_{d}=0,\pm 2$, $|\Delta L|\leq 0,4$. In the $j=3/2$ case, we also
consider the portion of fermionic term2 ; 17 ; 18 .

$\hat{T}^{(E2)}=\hat{T}_{B}^{(E2)}+\hat{T}_{F}^{(E2)}$ (43)

With

${T}_{B,\mu}^{(E2)}=q_{2}[s^{+}\times\tilde{d}+d^{+}\times\tilde{s}]_{\mu}^{(2)}+q^{\prime}_{2}[d^{+}\times\tilde{d}]_{\mu}^{(2)}=q_{B}Q_{B,\mu}$
(44)
$Q_{B,\mu}=[s^{+}\times\tilde{d}+d^{+}\times\tilde{s}]_{\mu}^{(2)}+\chi[d^{+}\times\tilde{d}]_{\mu}^{(2)}$
(45)
${T}_{F}^{(E2)}=q_{f}\sum_{jj^{\prime}}Q_{jj^{\prime}}[a_{j}^{+}\times\tilde{a}_{j^{\prime}}]^{(2)}$
(46)

Where $\mathrm{Q_{B}}$ and $\mathrm{Q_{jj^{\prime}}}$ are the boson and
fermion quadrupole operator and $\mathrm{q_{B}}$ and $\mathrm{q_{f}}$ are the
effective boson and fermion charges2 ; 17 ; 18 .

The reduced electric quadrupole transition rate between the $J_{i}\rightarrow
J_{f}$ states is given by 1 ; 2

$B(E2;\alpha_{i}J_{i}\rightarrow\alpha_{f}J_{f})=\frac{|\langle\alpha_{f}J_{f}||T^{(E2)}||\alpha_{i}J_{i}\rangle|^{2}}{2J_{i}+1}$
(47)

For evaluating B(E2), we have calculated the matrix elements of T(E2)
operators between the eigenstates of Eq.(21) that the normalization factor
obtain as:

$N=\sqrt{{\frac{1}{\prod_{p=1}^{k}\sum_{i=p}^{k}(\frac{2C_{s}^{2}(k-p+\frac{1}{2}(\nu_{s}+\frac{1}{2}))}{(1-C_{s}^{2}y_{k+1-p})(1-C_{s}^{2}y_{i})}+\frac{2(k-p+\frac{1}{2}(\nu_{d}+\frac{5}{2}))}{(1-y_{k+1-p})(1-y_{i})}-\frac{2C_{f}^{2}(k-p+\frac{1}{2}(\nu_{f}-\frac{2j+1}{2}))}{(1-C_{f}^{2}y_{k+1-p})(1-C_{f}^{2}y_{i})})}}}$
(48)

We calculate fermion part for j=3/2 by using Eq.(49)17 ; 18

$\displaystyle\langle\nu_{B},\nu_{F},(\tau_{1},\tau_{2}),L_{BF},J||T_{F}^{(\lambda,\mu)}||\nu^{\prime}_{B},\nu^{\prime}_{F},(\tau^{\prime}_{1},\tau^{\prime}_{2}),L^{\prime}_{BF},J^{\prime}\rangle=\sum_{\nu,L}\sum_{\nu^{\prime},L^{\prime}}\xi_{N+1/2,\tau_{1},J}^{N,\nu,L}\xi_{N+1/2,\tau^{\prime}_{1},J^{\prime}}^{N,\nu^{\prime},L^{\prime}}(-1)^{L+J^{\prime}+3/2}(-1)^{J-J^{\prime}}$
$\displaystyle\left\\{\begin{array}[]{ll}3/2\quad J\quad L\\\ J^{\prime}\quad
3/2\quad 2\\\ \end{array}\right\\}\langle 3/2||T_{F}^{(E2)}||3/2\rangle$ (51)

Where $\xi$ is isoscalar factor and $\langle 3/2||T_{F}^{(E2)}||3/2\rangle$ is
a single-particle matrix element that is obtained as2 ; 17 ; 18

$\langle
j||(a_{j_{1}}^{+}\times\widetilde{a}_{j_{2}})^{(\lambda)}||j^{\prime}\rangle=-\sqrt{2\lambda+1}\delta_{j_{2},j^{\prime}}\delta_{j_{1},j}$
(52)

A test of supersymmetry is to see the extent to which the electric quadrupole
transition probabilities of even-even and odd-A nuclei can be describe with
the same coefficients of Eq.(47). To determine boson effective charge, we have
extracted these quantities from the empirical B(E2) values via least square
technique. In the fitting process, empirical transition rates of both of even-
even and odd-A nuclei were used.

##  III Quantal analysis

This section presented the calculated phase transition observables such as
level crossing,expectation values of the d-boson and the fermion number
operators.

###  III.1 energy spectrum and level crossing

In order to display how the energy levels change within the whole range of the
$C_{s}$ and $C_{f}$ control parameters, the energy surfaces $E(C_{s},C_{f})$
with the other parameters fixed can be defined and calculated using procedure
that explained in sec.($\prod$). Fig.3 and Fig.4 show the energy surfaces of
Hamiltonian of Eq.(36) and Eq.(39) for the neighboring even-even (left panel)
and odd-A nuclei (right panel) with $N=10$, respectively. The calculation are
performed by considering the same fit parameters for neighboring even-even and
odd-A nuclei, where in Fig.3 the fixed parameters are $\alpha=1000$keV,
$\beta=3.14$keV, $\delta=-5.26$keV, $\gamma=0.0439$keV and Fig.4 obtained with
$\alpha=1000$keV, $\beta^{\prime}=-1.29$keV, $\gamma=6.05$keV. Figs show how
the energy levels evolve from one phase to the other as a function of
$C_{s}$and $C_{f}$ control parameters. These Figures clearly show how the
transition from one dynamical symmetry to another occurs. It can be seen from
Figs that numerous level crossings occur. The crossings are due to the fact
that$\nu_{d}$, O(5) quantum number called seniority, is preserved along the
whole path between O(6) and U(5)19 ; 20 .

###  III.2 expectation values of the d-boson number operator

An appropriate quantal order parameter is:

$\langle\hat{n_{d}}\rangle=\frac{\langle\psi|\hat{n_{d}}|\psi\rangle}{N}$

In order to obtain $\langle\hat{n_{d}}\rangle$, we act $s_{m}^{0}$ on the
eigenstate, $|k;\nu_{s}\nu n_{\Delta}LM\rangle$

$\langle\hat{n_{d}}\rangle=\frac{2C_{s}^{2}C_{f}^{2}(\Lambda_{0}^{0}+k)-2(C_{s}^{2}+C_{f}^{2})(\Lambda_{1}^{0}+ky_{1}^{-1})+2(\Lambda_{2}^{0}+ky_{2}^{-2})}{N(1-C_{s}^{2})(1-C_{f}^{2})}-\frac{5}{2N}$
(53)

Fig.5 and Fig.6 show the expectation values of the d-boson number operator for
the lowest states even-even (left panel) and odd-A nuclei (right panel) as a
function of control parameters for j=1/2 and j=3/2 with N=10 bosons,
respectively. Fig.5 and Fig.6 (left panel) display that the expectation values
of the number of d bosons for each L, $n_{d}$, remain approximately constant
for $C_{s}<0.45$ and only begin to change rapidly for $C_{s}>0.45$. The near
constancy of $n_{d}$ for $C_{s}<0.45$, is a obvious indication that $U(5)$
dynamical symmetry preserves in this region to a high degree and also the
$n_{d}$ values change rapidly with $C_{s}$ over the range $0.65\leq C_{s}\leq
1$.

###  III.3 expectation values of the fermion number operator

The expectation values of the fermion number operator are obtained as

$\langle\hat{n_{f}}\rangle=\frac{\langle\psi|\hat{n_{f}}|\psi\rangle}{N}=\frac{2(1+C_{s}^{2})(\Lambda_{0}^{0}-\Lambda_{1}^{0}+k(1-y_{1}^{-1}))-2(\Lambda_{0}^{0}-\Lambda_{2}^{0}+k(1-y_{2}^{-2}))}{N(1-C_{f}^{2})(C_{s}^{2}-C_{f}^{2})}+\frac{2j+1}{2N}$
(54)

Fig.7 and Fig.8 show also $\langle n_{f}\rangle$ as a function of the $C_{s}$
and $C_{f}$ control parameters.

It can bee seen from figs that combination of multi-symmetry in the point of
phase transition cause dissimilar behavior with neighboring points that
representing the transition of u(5) limit to O(6) limit.

##  IV Experimental evidence

This section presented the calculated results of low-lying states of Rh-Ru
Supermultiplets and Zn - Cu Supermultiplets. The results include energy levels
and the B(E2)values and Key observables of E(5) and E(5/2j+1) symmetry.

###  IV.1 The Rh-Ru Supermultiplets

Nuclei in the mass region around $\mathrm{A\sim 100}$ have transitional
characteristics intermediate between spherical and gamma-unstable shapes 21 ;
22 . Stachel et al.23  have been studied the Ru isotopes and are found that
these isotopes have U(5)-O(6) transition features. The odd-A Rh isotopes were
studied in the framework IBFM by Vervier and Janssens 24  and Refs.16 ; 21 ;
22 ; 25 . A. Frank and collaborators have studied the Rh-Ru Supermultiplets by
using supersymmetry approach 6 . They have studied successfully a combination
of $\mathrm{U^{BF}(5)}$ and $\mathrm{SO^{BF}(6)}$ symmetry by using
$\mathrm{U(6/12)}$ supersymmetry for the Ru and Rh isotopes. We have also
analyzed the negative parity states of the odd-proton
nuclei,$\mathrm{{}_{45}^{101-109}Rh}$ and positive parity states of the even-
even nuclei,$\mathrm{{}_{44}^{100-108}Ru}$.The negative parity states in the
odd-even nuclei Rh are built mainly on the $\mathrm{2p_{\frac{1}{2}}}$ shell
model orbit21 . In order to obtain energy spectrum and realistic calculation
for these nuclei, we need to specify Hamiltonian parameters Eq.(36). According
to the supersymmetry between bosons and fermions, the fermion is transformed
into a boson, the system is described by the same set of parameters thus to
achieve a better fit, states of both of even-even and odd-A nuclei were used.
Eigenvalues of these systems are obtained by solving Bethe-Ansatz equations
with least square fitting processes to experimental data to obtain constants
of Hamiltonian. The best fits for Hamiltonian’s parameters, namely
$\mathrm{\alpha}$, $\mathrm{\beta}$, $\mathrm{\delta}$ and $\mathrm{\gamma}$,
used in the present work are shown in table 1. Fig.9 and Fig.10 show a
comparison between the available experimental levels and the predictions of
our results for the $\mathrm{{}_{45}^{101-109}Rh}$ and
$\mathrm{{}_{44}^{100-108}Ru}$ isotopes in the low-lying region of spectra.
The quantum numbers of different states of each isotope and a comparison
between theoretical prediction and experimental values for considered isotopes
are presented in Tables.2(a,b…e). An acceptable degree of agreement is obvious
between them.

The most important successful nuclear model characteristic is a good
description of electromagnetic properties of the nucleus in addition to its
energy spectrum. So, we have calculated B(E2) transition rates for Ru-Rh
Supermultiplets. In the fitting process, because there is no empirical data
for Rh isotopes except ${}^{103}Rh$ we have used only of the even-mass Ru
experimental data for extracting of effective charge parameters. In
${}^{102}Ru-^{103}Rh$ supermultiplet, we have used of experimental transition
rates of both nuclei. Extracted values for effective charge parameters have
presented in Table 3. Tables.4(a,b,c,d) show experimental and calculated
values of B(E2) for negative parity states of Rh and positive parity states of
Ru.

###  IV.2 The Zn-Cu Supermultiplets

In order to test the predictions of the U(6/4) scheme in dynamical symmetry
limits and transition region, we choose the negative parity states of the odd-
proton nuclei,${}^{61-69}Cu$ and positive parity states of the even-even
nuclei,${}^{62-70}Zn$. These nuclei have not considered in other theoretical
studies completely but for some of them which we have found similar
counterparts. The Cu isotopes across the N=40 shell closure and have a single
proton outside the Z=28 closed shell. The ground state spin/parity of the
odd-A copper isotopes( Between ${}^{57}Cu$ and ${}^{69}Cu$) is
$(\frac{3}{2})^{-}$. The low-lying levels of $\mathrm{{}^{63,65}Cu}$ were
studied by Bijker and Kota in 1984 37 . They showed that the Spin(5) symmetry
could occur in the odd-proton nuclei in the Cu region with the odd-proton
occupying the $\mathrm{2p_{3/2}}$ orbit. Since the low-lying negative parity
states in $\mathrm{{}^{63,65}Cu}$ are built mainly on the $\mathrm{2p_{3/2}}$
shell model orbit and the adjacent even-even nuclei $\mathrm{{}^{64,66}Zn}$
show a vibrational type of spectrum, it has been suggested 37 ; 38 , that the
Cu-Zn mass-region provides experimental evidence for the existence of the
Spin(5) spinor symmetry and the U(6/4) supersymmetry 37 . The odd-even nuclei
$\mathrm{{}^{63,65}Cu}$ alone provide a test of the Spin(5) symmetry, while
the pairs of nuclei Zn-Cu provide a test of the U(6/4) supersymmetry 37 .
Since vibrational nuclei are known to exhibit other excitation modes such as
quasi particle modes or intruder states due to the excitation of two protons
into the next major shell in Zn 39 , great care must be taken in comparing
with experiment. We have used a similar procedure as has done for Ru-Rh
supermultiplets calculations to extract the parameters of Hamiltonian Eq.(39)
for Zn-Cu supermultiplets. In the fitting process, in addition to the states
in Cu, the states of Zn isotopes were considered. The best fits for
Hamiltonian’s parameters, namely $\alpha$, $\beta^{\prime}$ and $\gamma$, used
in the present work are shown in table 5. Figs.11 and 12 show the comparison
of the experimental and the theoretical level scheme. Experimental data of
${}^{61-69}Cu$ and ${}^{62-70}Zn$ isotopes was taken from Refs.[40-49]. The
quantum numbers of different states of each isotope and a comparison between
theoretical prediction and experimental values for considered isotopes are
presented in Tables.6(a,b,..,e).

we have also calculated B(E2) transition rates for Zn-Cu Supermultiplets. To
determine boson effective charge, we have extracted these quantities from the
empirical B(E2) values via least square technique. In the fitting process, we
have used of experimental transition rates of both nuclei in a Supermultiplet.
Extracted values for effective charge parameters have presented in Tables 7.
Tables.8(a,b,.,e) show experimental and calculated values of B(E2) for
negative parity states of Cu and positive parity states of Zn. In Zn-Cu
Supermultiplet, improved results my be obtained by considering neutron and
proton degree of freedom separately.

In the following, we concentrated upon those distinguishing observables which
vary along the U(5)- SO(6) transition. The observables such as the energy
ratios, the B(E2) values that are most sensitive to the U(5)-O(6) structural
transition. M.A.Caprio and F.Iachello 18  obtained analytic descriptions for
transitional nuclei near the critical point. The solutions provided baselines
for experimental studies of even- even [E(5)] and odd-mass E(5/4) nuclei near
the critical point of the spherical to gamma-unstable phase transition. Their
results have provided benchmarks for nuclei near the critical point of the
U(5)-O(6) phase transition and can be used as a basis for comparison with
experiment. So, we have also calculated these quantities for the Ru-Rh and Zn-
Cu supermultiplets by using their method and performed an analysis for these
isotopes. One of the most basic structural predictions of $\mathrm{U(5)-O(6)}$
transition is a $R_{\frac{4}{2}}=\frac{E(4_{1}^{+})}{E(2_{1}^{+})}$ value. The
ratio equal to 2.2-2.3 indicates the spectrum of transitional nuclei 18 ; 50 ;
51 . Thus, we calculated this quantity for even-even Ru and Zn Isotopes.
Fig.13(a) and Fig.14(a) show $R_{\frac{4}{2}}$ prediction along with the
experimental values for Ru and Zn isotopes, respectively. We have also
calculated observables such as $\frac{E(0_{2}^{+})}{E(2_{1}^{+})}$ ,the
vibrational excitation measure, values for control parameter, namely the ratio
of shape phase transition, and ratios $B(\frac{E2;4_{1}^{+}\longrightarrow
2_{1}^{+}}{E2;2_{1}^{+}\longrightarrow 0_{1}^{+}})$ and
$B(\frac{E2;0_{2}^{+}\longrightarrow 2_{1}^{+}}{E2;2_{1}^{+}\longrightarrow
0_{1}^{+}})$. The comparison with experiment for these quantities in Ru and Zn
isotopes is summarized in Table.9 and Table.10, respectively.

Because of the occurrence of single particle orbitals that can considerably
perturb the spectrum and electromagnetic transition strengths, comparison of
the critical point description with experimental data in odd-mass nuclei is
more difficult than even-even nuclei18 . For Cu and Rh isotopes, we calculated
only the energy ratio quantity. Fig.13(b) and Fig.14(b) show
$\mathrm{\frac{E(\nu_{d}=2)}{E(\nu_{d}=1)})}$ prediction along with the
experimental values for these isotopes.

From these figures and tables, one can conclude, the calculated energy spectra
in this approach are approximately in good agreements with the experimental
data. It means that our suggestion to use this transitional Hamiltonian for
the description of the Rh-Ru and Zn-Cu supermultiplets would not have any
contradiction with other theoretical studies done with special hypotheses
about mixing of intruder and normal configurations. On the other hand,
predictions of our model for the control parameters of considered
supermultiplet, $C_{s}$ and $C_{f}$ , describe the vibrational, i.e.$C_{s}=0$,
or rotational, namely $C_{s}=1$, and also $C_{f}\neq 0$ for odd-A nuclei and
$C_{f}=0$ for even-even nuclei, confirm this mixing of both vibrating and
rotating structures in these nuclei when $C_{s}\sim 0.5\rightarrow 0.65$. The
values of the control parameter, $C_{s}$, suggest structural changes in
nuclear deformation and shape-phase transitions in Rh-Ru and Zn-Cu
supermultiplets. we proposed $C_{s}\sim 0.5\rightarrow 0.65$ as critical
point. We conclude from the values of control parameter which has been
obtained, observables such as the energy ratios and the B(E2) values that
${}^{104}Ru-^{105}Rh$ and ${}^{64}Zn-^{63}Cu$ supermultiplets are as the best
candidates for $U(5)-O(6)$ transition. Also, theoretical B(E2)transition
probabilities of the Rh-Ru and Zn-Cu supermultiplets, which have obtained by
using the model perspectives, exhibit nice agreement with experimental ones.

###  IV.3 two-neutron separation energies

Shape phase transitions in nuclei can be studied experimentally by considering
the behavior of the ground state energies of a series of isotopes, or, more
conveniently, the behavior of the two-neutron separation energies, $S_{2n}$ 1
; 52 . On the other hand, the ground-state two-neutron separation
energies,$S_{2n}$, are observables very sensitive to the details of the
nuclear structure and indirect test of the supersymmetry scheme. The
occurrence of continuities in the behavior of two-neutron separation energies
describe a second-order shape-phase transition between spherical and
$\mathrm{\gamma-unstable}$ rotor limits 1 ; 52 .In due to, we have
investigated the evolution of two-neutron separation energies along the
Ru,Rh,Zn and Cu isotopic chains by both experimental and theoretical values,
which have been presented in Fig.15 and Fig.16.

Two-neutron separation energies for even-even and odd-even nuclei defined by2
; 4

$S_{2}(\mathcal{N},N_{f}=0)=E_{B}(\mathcal{N}+1,N_{f}=0)-E_{B}(\mathcal{N},N_{f}=0)$
(55)
$S_{2}(\mathcal{N},N_{f}=1)=E_{B}(\mathcal{N}+1,N_{f}=1)-E_{B}(\mathcal{N},N_{f}=1)$
(56)

Where $E_{B}$ denotes the binding energy. It can be shown that if
supersymmetry applies, the separation energies in even-even and odd-even
nuclei should be linear functions of $\mathcal{N}$ as2 ; 4

$S_{2}(\mathcal{N},N_{f}=0)=S_{2}(\mathcal{N},N_{f}=1)=D_{1}+C_{1}\mathcal{N}$
(57)

Using the $\mathrm{S_{2n}}$ empirical values for these isotopic chains 36  we
have extracted $\mathrm{D_{1}=2.28\times 10^{{4}},3.0814\times 10^{{4}}}$ MeV
and $\mathrm{C_{1}=-898.5,-2.5\times 10^{{3}}}$ MeV for Ru-Rh and Zn-Cu
supermultiplets, respectively. Then, we obtained the two-neutron separation
energies, which are shown in Fig.15 and Fig.16, together with the experimental
values.

It can be seen from Fig.15 and Fig.16 that exist continuities (linear
variation) in the behavior of two-neutron separation energies thus the phase
transition for Ru, Rh, Zn and Cu isotopic chains is of second order and also
approximately equal slopes for Ru(Zn) even-even and Rh(Cu) odd-mass nuclei is
a result of supersymmetry scheme. Our result Confirmed the predictions of done
in refs. [1,2,4,52].

##  V Conclusions

In this paper, we have proposed exactly-solvable supersymmetry Richardson-
Gaudin (R-G) model for transitional region. We apply supersymmetry ideas to
the description of even and odd nuclei and display that several states in many
nuclei can be explained by schemes based on the U(6/2) and U(6/4) supergroups.
We have employed the nuclear supersymmetry approach for description of the
transitional region between spherical and gamma -unstable phase shape in
addition to dynamical symmetry limits in one chain isotopic. Key observables
of phase transition such as level crossing, expectation values of the d-boson
and fermion number operator were calculated. New experimental data on the
even-mass ruthenium and the odd-mass rhodium isotopes and Zn-Cu
Supermultiplets were used to test the predictions of the new Supersymmetry
scheme in dynamical symmetry limits and transition region and performed an
analysis for these isotopes via a GQA-based Hamiltonian. The results indicate
that the energy spectra of the Rh-Ru and Zn-Cu Supermultiplets can be
reproduced approximately well. The observables such as the energy ratios, the
B(E2) values that are most sensitive to the U(5)-O(6) structural transition
were calculated and compared with the available experimental data. Our results
show that the Rh-Ru and Zn-Cu Supermultiplets have gamma-unstable rotor
features but the vibrational character is dominant and also
${}^{104}Ru-^{105}Rh$ and ${}^{64}Zn-^{63}Cu$ Supermultiplets are as the best
candidates for $U(5)-O(6)$ transition.

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Table 1.Parameters of Hamiltonian Eq.(36) used in the calculation of the Ru-Rh
Supermultiplets. All parameters are given in keV.
---

Nucleus | $\mathcal{N}$ | $C_{s}$ | $C_{f}$ | $\alpha$ | $\beta$ | $\delta$ | $\gamma$ | $\sigma$ | $\phi$
---|---|---|---|---|---|---|---|---|---
${}^{100}Ru-^{101}Rh$ | 6 | 0.38 | 0.99 | 279.2 | 3.974 | 7.89 | -2.217 | 164.53 | 11.23%
${}^{102}Ru-^{103}Rh$ | 7 | 0.23 | 0.8 | 170.32 | 0.8854 | -1.207 | 6.874 | 104.74 | 6.38 %
${}^{104}Ru-^{105}Rh$ | 8 | 0.54 | 0.99 | 70.61 | 0.1915 | 46.69 | -15.123 | 137.8 | 13.2%
${}^{106}Ru-^{107}Rh$ | 9 | 0.34 | 0.7 | 197.52 | 0.329 | 51.14 | -35.72 | 128.73 | 11.79 %
${}^{108}Ru-^{109}Rh$ | 10 | 0.58 | 0.99 | 248.2 | 4.464 | 1.21 | -7.863 | 128.54 | 12.8 %

Table 2a.Energy spectra for ${}_{44}^{100}Ru$ \- ${}_{45}^{101}Rh$ isotope.The
experimental data for ${}_{44}^{100}Ru$ \- ${}_{45}^{101}Rh$ supermultiplet
are taken from 26 ; 27 ; 36 .
---

Nuclei | $J^{\pi}$ |  $K$ |  $\nu_{d}$ |  $E_{exp}$(keV) |  $E_{cal}$(keV)
---|---|---|---|---|---
${}_{44}^{100}Ru$ | $0_{1}^{+}$ |  3 |  0 |  0 |  0
| $2_{1}^{+}$ |  2 |  1 |  539.5 |  503
| $0_{2}^{+}$ |  3 |  2 |  1130.3 |  1284.9
| $2_{2}^{+}$ |  2 |  2 |  1362.16 |  1219.1
| $4_{1}^{+}$ |  2 |  2 |  1226.48 |  1126.72
| $0_{3}^{+}$ |  1 |  3 |  1740.98 |  1938.6
| $3_{1}^{+}$ |  1 |  3 |  1881.04 |  2018.3
| $2_{3}^{+}$ |  2 |  1 |  1865.1 |  2018.3
| $4_{2}^{+}$ |  1 |  3 |  2062.51 |  2086.5
| $6_{1}^{+}$ |  1 |  3 |  2076.1 |  2256.9
| $0_{4}^{+}$ |  3 |  0 |  2051.65 |  2290.8
| $2_{4}^{+}$ |  2 |  2 |  2099.1 |  2207.7
${}_{45}^{101}Rh$ | $(1/2)_{1}^{-}$ |  2 |  0 |  0 |  0
| $(3/2)_{1}^{-}$ |  2 |  1 |  305.5 |  315.4
| $(5/2)_{1}^{-}$ |  2 |  1 |  305.5 |  292.3
| $(3/2)_{2}^{-}$ |  1 |  2 |  355.3 |  536.9
| $(5/2)_{2}^{-}$ |  1 |  2 |  355.3 |  425
| $(7/2)_{1}^{-}$ |  1 |  2 |  851.4 |  860.4
| $(9/2)_{1}^{-}$ |  1 |  2 |  851.4 |  826.7
| $(9/2)_{2}^{-}$ |  1 |  2 |  899.3 |  1104.2
| $(5/2)_{3}^{-}$ |  2 |  1 |  996.4 |  914.4
| $(3/2)_{4}^{-}$ |  1 |  2 |  1058 |  895.7

Table 2b.Energy spectra for${}_{44}^{102}Ru$ \- ${}_{45}^{103}Rh$ isotope.The
experimental data for ${}_{44}^{102}Ru$ \- ${}_{45}^{103}Rh$ supermultiplet
are taken from 28 ; 29 ; 36 .
---

Nuclei | $J^{\pi}$ |  $K$ |  $\nu_{d}$ |  $E_{exp}$(keV) |  $E_{cal}$(keV)
---|---|---|---|---|---
${}_{44}^{102}Ru$ | $0_{1}^{+}$ |  3 |  0 |  0 |  0
| $2_{1}^{+}$ |  3 |  1 |  475.08 |  510.8
| $0_{2}^{+}$ |  3 |  0 |  943.69 |  828.9
| $2_{2}^{+}$ |  2 |  2 |  1103.15 |  1267.4
| $4_{1}^{+}$ |  2 |  2 |  1106.36 |  1215
| $0_{3}^{+}$ |  2 |  3 |  1837.10 |  1702.6
| $3_{1}^{+}$ |  2 |  3 |  1521.67 |  1621.4
| $2_{3}^{+}$ |  3 |  1 |  1580.56 |  1577.1
| $4_{2}^{+}$ |  2 |  3 |  1602.9 |  1671.4
| $6_{1}^{+}$ |  2 |  3 |  1873.23 |  1809
| $0_{4}^{+}$ |  3 |  0 |  1968.66 |  2131
| $2_{4}^{+}$ |  2 |  2 |  2036.9 |  2360.3
${}_{45}^{103}Rh$ | $(1/2)_{1}^{-}$ |  3 |  0 |  0 |  0
| $(3/2)_{1}^{-}$ |  2 |  1 |  294.984 |  276.8
| $(5/2)_{1}^{-}$ |  2 |  1 |  357.408 |  325.5
| $(1/2)_{2}^{-}$ |  3 |  0 |  803.07 |  966.5
| $(3/2)_{2}^{-}$ |  2 |  2 |  803.07 |  647.1
| $(7/2)_{1}^{-}$ |  2 |  2 |  847.58 |  717.6
| $(5/2)_{2}^{-}$ |  2 |  2 |  880.47 |  689.9
| $(9/2)_{1}^{-}$ |  2 |  2 |  920.1 |  794.5
| $(3/2)_{3}^{-}$ |  2 |  2 |  1277.04 |  1247.9
| $(13/2)_{1}^{-}$ |  2 |  3 |  1637.64 |  1551.1
| $(15/2)_{1}^{-}$ |  1 |  4 |  2221.2 |  2255.5
| $(17/2)_{1}^{-}$ |  1 |  4 |  2345.35 |  2400.9

Table 2c.Energy spectra for${}_{44}^{104}Ru$ \- ${}_{45}^{105}Rh$ isotope.The
experimental data for ${}_{44}^{104}Ru$ \- ${}_{45}^{105}Rh$ supermultiplet
are taken from 30 ; 31 ; 36 .
---

Nuclei | $J^{\pi}$ |  $K$ |  $\nu_{d}$ |  $E_{exp}$(keV) |  $E_{cal}$(keV)
---|---|---|---|---|---
${}_{44}^{104}Ru$ | $0_{1}^{+}$ |  4 |  0 |  0 |  0
| $2_{1}^{+}$ |  3 |  1 |  358.02 |  406.3
| $0_{2}^{+}$ |  4 |  0 |  988.3 |  1294.3
| $2_{2}^{+}$ |  3 |  2 |  893.1 |  1139.4
| $4_{1}^{+}$ |  3 |  2 |  888.5 |  944.9
| $3_{1}^{+}$ |  2 |  3 |  1242.4 |  1233.5
| $2_{3}^{+}$ |  3 |  1 |  2285 |  2103.8
${}_{45}^{105}Rh$ | $(1/2)_{1}^{-}$ |  3 |  0 |  129.781 |  175.9
| $(3/2)_{1}^{-}$ |  3 |  1 |  392.65 |  447.4
| $(5/2)_{1}^{-}$ |  3 |  1 |  455.61 |  371.8
| $(3/2)_{2}^{-}$ |  2 |  2 |  762.11 |  642.9
| $(3/2)_{3}^{-}$ |  2 |  1 |  783 |  642.9
| $(5/2)_{2}^{-}$ |  2 |  2 |  817 |  1006.7
| $(7/2)_{1}^{-}$ |  2 |  2 |  817 |  900.8
| $(5/2)_{3}^{-}$ |  2 |  3 |  866 |  1112.6
| $(7/2)_{2}^{-}$ |  2 |  3 |  898 |  1115.1
| $(7/2)_{3}^{-}$ |  2 |  2 |  976 |  1009.2
| $(9/2)_{1}^{-}$ |  2 |  2 |  976 |  802.5
| $(3/2)_{4}^{-}$ |  2 |  3 |  1147 |  1174.2
| $(5/2)_{4}^{-}$ |  3 |  1 |  1147 |  987.7

Table 2d.Energy spectra for${}_{44}^{106}Ru$ \- ${}_{45}^{107}Rh$ isotope.The
experimental data for ${}_{44}^{106}Ru$ \- ${}_{45}^{107}Rh$ supermultiplet
are taken from 32 ; 33 ; 36 .
---

Nuclei | $J^{\pi}$ |  $K$ |  $\nu_{d}$ |  $E_{exp}$(keV) |  $E_{cal}$(keV)
---|---|---|---|---|---
${}_{44}^{106}Ru$ | $0_{1}^{+}$ |  4 |  0 |  0 |  0
| $2_{1}^{+}$ |  4 |  1 |  270.07 |  321.1
| $0_{2}^{+}$ |  4 |  0 |  990.62 |  1086.5
| $2_{2}^{+}$ |  3 |  2 |  792.31 |  613.1
| $4_{1}^{+}$ |  3 |  2 |  714.69 |  811.3
| $3_{1}^{+}$ |  3 |  3 |  1091.55 |  974
| $2_{3}^{+}$ |  4 |  1 |  1392.21 |  1173.6
| $4_{2}^{+}$ |  3 |  3 |  2367 |  2321
| $6_{1}^{+}$ |  3 |  3 |  1295.8 |  1232.6
${}_{45}^{107}Rh$ | $(1/2)_{1}^{-}$ |  4 |  0 |  268.36 |  320
| $(3/2)_{1}^{-}$ |  3 |  1 |  485.66 |  613.4
| $(5/2)_{1}^{-}$ |  3 |  1 |  543.84 |  451.3
| $(9/2)_{1}^{-}$ |  3 |  2 |  559.97 |  662.7
| $(3/2)_{2}^{-}$ |  3 |  2 |  752.55 |  711.7
| $(5/2)_{2}^{-}$ |  3 |  2 |  877.75 |  1139.8
| $(3/2)_{3}^{-}$ |  3 |  2 |  974.44 |  1146.5
| $(5/2)_{3}^{-}$ |  2 |  3 |  974.44 |  967.9
| $(7/2)_{1}^{-}$ |  3 |  2 |  974.44 |  994.3
| $(3/2)_{4}^{-}$ |  2 |  3 |  1009.76 |  1146.5

Table 2e.Energy spectra for${}_{44}^{108}Ru$ \- ${}_{45}^{109}Rh$ isotope.The
experimental data for ${}_{44}^{108}Ru$ \- ${}_{45}^{109}Rh$ supermultiplet
are taken from 34 ; 35 ; 36 .
---

Nuclei | $J^{\pi}$ |  $K$ |  $\nu_{d}$ |  $E_{exp}$(keV) |  $E_{cal}$(keV)
---|---|---|---|---|---
${}_{44}^{108}Ru$ | $0_{1}^{+}$ |  5 |  0 |  0 |  0
| $2_{1}^{+}$ |  4 |  1 |  242.24 |  308.9
| $0_{2}^{+}$ |  5 |  0 |  975.96 |  1051.8
| $2_{2}^{+}$ |  4 |  2 |  707.82 |  825.3
| $4_{1}^{+}$ |  4 |  2 |  665.2 |  752.3
| $3_{1}^{+}$ |  3 |  3 |  974.8 |  1110.2
| $2_{3}^{+}$ |  4 |  1 |  1249.19 |  1126.9
| $4_{2}^{+}$ |  3 |  3 |  1183.03 |  1254.2
| $6_{1}^{+}$ |  3 |  3 |  1240 |  1139.3
| $0_{3}^{+}$ |  3 |  3 |  1218.8 |  1358.5
${}_{45}^{109}Rh$ | $(1/2)_{1}^{-}$ |  4 |  0 |  374.1 |  557.7
| $(3/2)_{1}^{-}$ |  4 |  1 |  568.2 |  634.7
| $(5/2)_{1}^{-}$ |  4 |  1 |  623.2 |  806.1
| $(3/2)_{2}^{-}$ |  3 |  2 |  704.9 |  914.3
| $(5/2)_{2}^{-}$ |  3 |  2 |  856.1 |  914.5
| $(5/2)_{3}^{-}$ |  4 |  1 |  926.9 |  1154.7
| $(3/2)_{3}^{-}$ |  4 |  1 |  1162.3 |  1040.4
| $(3/2)_{4}^{-}$ |  3 |  2 |  1214.3 |  1332.7
| $(5/2)_{4}^{-}$ |  3 |  2 |  1283.9 |  1332.9
| $(1/2)_{2}^{-}$ |  4 |  0 |  1631 |  1503.8

Table 3.The coefficients of $T(E_{2})$used in the present work for the Ru-Rh
Supermultiplets.Experimental values are taken form Refs. [26-36] .
---

Nucleus | $q_{B}$ | $q_{f}$
---|---|---
${}^{100}Ru-^{101}Rh$ | 4.27 | 0
${}^{102}Ru-^{103}Rh$ | 1.5221 | 0
${}^{104}Ru-^{105}Rh$ | 2.2782 | 0
${}^{108}Ru-^{109}Rh$ | 2.2602 | 0

Table 4a.B(E2)values for ${}_{44}^{100}Ru$ and ${}_{45}^{101}Rh$
isotopes.Experimental values are taken form Refs.[26,27] and are presented in
Weisskopf units(W.u.).
---

Nucleus | $J_{i}^{\pi}\longrightarrow J_{j}^{\pi}$ | $\frac{B(E_{2}(W.u.)}{exp.\quad\quad\quad calc.}$
---|---|---
${}_{44}^{100}Rh$ | $2_{1}^{+}\longrightarrow 0_{1}^{+}$ | 35.6 35.6
| $0_{2}^{+}\longrightarrow 2_{1}^{+}$ | 35 33.39
| $4_{1}^{+}\longrightarrow 2_{1}^{+}$ | 51 55.26
| $2_{2}^{+}\longrightarrow 2_{1}^{+}$ | 30.9 15.71
| $2_{2}^{+}\longrightarrow 0_{1}^{+}$ | 1.9 1.9
${}_{45}^{101}Rh$ | $(3/2)_{1}^{-}\longrightarrow(1/2)_{1}^{-}$ |  2.2323
| $(5/2)_{1}^{-}\longrightarrow(1/2)_{1}^{-}$ |  0
| $(5/2)_{2}^{-}\longrightarrow(1/2)_{1}^{-}$ |  0
| $(5/2)_{2}^{-}\longrightarrow(3/2)_{1}^{-}$ |  0.9945
| $(5/2)_{2}^{-}\longrightarrow(5/2)_{1}^{-}$ |  0.0043
| $(7/2)_{1}^{-}\longrightarrow(3/2)_{1}^{-}$ |  5.3894
| $(9/2)_{1}^{-}\longrightarrow(5/2)_{1}^{-}$ |  0.9934

Table 4b.B(E2)values for ${}_{44}^{102}Ru$ and ${}_{45}^{103}Rh$
isotopes.Experimental values are taken form Refs.[28,29] and are presented in
Weisskopf units(W.u.).
---

Nucleus | $J_{i}^{\pi}\longrightarrow J_{j}^{\pi}$ | $\frac{B(E_{2}(W.u.)}{exp.\quad\quad\quad calc.}$
---|---|---
${}_{44}^{102}Ru$ | $2_{1}^{+}\longrightarrow 0_{1}^{+}$ | 45.1 32.1487
| $0_{2}^{+}\longrightarrow 2_{1}^{+}$ | 35 39.46
| $4_{1}^{+}\longrightarrow 2_{1}^{+}$ | 66 56.34
| $2_{2}^{+}\longrightarrow 2_{1}^{+}$ | 32 8.47
| $2_{2}^{+}\longrightarrow 0_{1}^{+}$ | 1.14 0.11
${}_{45}^{103}Rh$ | $(3/2)_{1}^{-}\longrightarrow(1/2)_{1}^{-}$ | 36 55.98
| $(5/2)_{1}^{-}\longrightarrow(1/2)_{1}^{-}$ | 44 29.3311
| $(5/2)_{2}^{-}\longrightarrow(1/2)_{1}^{-}$ | 1.4 1.4002
| $(5/2)_{2}^{-}\longrightarrow(3/2)_{1}^{-}$ | 2.7 0
| $(5/2)_{2}^{-}\longrightarrow(5/2)_{1}^{-}$ | 4 1.34
| $(7/2)_{1}^{-}\longrightarrow(3/2)_{1}^{-}$ | 34 37.03
| $(9/2)_{1}^{-}\longrightarrow(5/2)_{1}^{-}$ | 46 29.86

Table 4c.B(E2)values for ${}_{44}^{104}Ru$ and ${}_{45}^{105}Rh$
isotopes.Experimental values are taken form Refs.[30,31] and are presented in
Weisskopf units(W.u.).
---

Nucleus | $J_{i}^{\pi}\longrightarrow J_{j}^{\pi}$ | $\frac{B(E_{2}(W.u.)}{exp.\quad\quad\quad calc.}$
---|---|---
${}_{44}^{104}Ru$ | $2_{1}^{+}\longrightarrow 0_{1}^{+}$ | 57.9 35.8
| $0_{2}^{+}\longrightarrow 2_{1}^{+}$ | 25 15.98
| $4_{1}^{+}\longrightarrow 2_{1}^{+}$ | 83 53.36
| $2_{2}^{+}\longrightarrow 2_{1}^{+}$ | 0.6 4.94
| $2_{2}^{+}\longrightarrow 0_{1}^{+}$ | 2.8 2.8
${}_{45}^{105}Rh$ | $(3/2)_{1}^{-}\longrightarrow(1/2)_{1}^{-}$ |  1.37
| $(5/2)_{1}^{-}\longrightarrow(1/2)_{1}^{-}$ |  0.16
| $(5/2)_{2}^{-}\longrightarrow(1/2)_{1}^{-}$ |  0.006
| $(5/2)_{2}^{-}\longrightarrow(3/2)_{1}^{-}$ |  0.048
| $(5/2)_{2}^{-}\longrightarrow(5/2)_{1}^{-}$ |  1.32
| $(7/2)_{1}^{-}\longrightarrow(3/2)_{1}^{-}$ |  2.6
| $(9/2)_{1}^{-}\longrightarrow(5/2)_{1}^{-}$ |  0.82

Table 4d.B(E2)values for ${}_{44}^{108}Ru$ and ${}_{45}^{109}Rh$
isotopes.Experimental values are taken form Refs.[34,35] and are presented in
Weisskopf units(W.u.).
---

Nucleus | $J_{i}^{\pi}\longrightarrow J_{j}^{\pi}$ | $\frac{B(E_{2}(W.u.)}{exp.\quad\quad\quad calc.}$
---|---|---
${}_{44}^{108}Ru$ | $2_{1}^{+}\longrightarrow 0_{1}^{+}$ | 58 58.03
| $4_{1}^{+}\longrightarrow 2_{1}^{+}$ | 102 95.3138
| $2_{2}^{+}\longrightarrow 0_{1}^{+}$ | 0.5 0.5
| $6_{1}^{+}\longrightarrow 2_{2}^{+}$ | 0.08 0
| $6_{1}^{+}\longrightarrow 0_{1}^{+}$ | 0.004 0
${}_{45}^{109}Rh$ | $(3/2)_{1}^{-}\longrightarrow(1/2)_{1}^{-}$ |  2.8083
| $(5/2)_{1}^{-}\longrightarrow(1/2)_{1}^{-}$ |  0.3349
| $(5/2)_{2}^{-}\longrightarrow(1/2)_{1}^{-}$ |  2.1831
| $(5/2)_{2}^{-}\longrightarrow(3/2)_{1}^{-}$ |  0.0026
| $(5/2)_{2}^{-}\longrightarrow(5/2)_{1}^{-}$ |  0.0203
| $(7/2)_{1}^{-}\longrightarrow(3/2)_{1}^{-}$ |  0.5573
| $(9/2)_{1}^{-}\longrightarrow(5/2)_{1}^{-}$ |  1.0983

Table 5. Parameters of Hamiltonian (30) used in the calculation of the Zn-Cu
Supermultiplets. All parameters are given in keV.
---

Nucleus | $\mathcal{N}$ | $C_{s}$ | $C_{f}$ | $\alpha$ | $\beta^{\prime}$ | $\gamma$ | $\sigma$ | $\phi$
---|---|---|---|---|---|---|---|---
${}^{62}Zn-^{61}Cu$ | 3 | 0.4 | 0.87 | 889.63 | 0.077 | 15.067 | 217.2 | 10.86%
${}^{64}Zn-^{63}Cu$ | 4 | 0.51 | 1 | 946.2 | 0.2498 | 8.3402 | 189 | 8.92 %
${}^{66}Zn-^{65}Cu$ | 5 | 0.65 | 1 | 739.6 | -0.136 | 5.947 | 181.92 | 7.05%
${}^{68}Zn-^{67}Cu$ | 6 | 0.03 | 0.9 | 988.7 | 2.367 | -43.11 | 202.43 | 6.9 %
${}^{70}Zn-^{69}Cu$ | 6 | 0.01 | 0.4 | 914.99 | 1.537 | -0.0468 | 228.18 | 10.23 %

Table 6a.Energy spectra for${}_{30}^{62}Zn$\- ${}_{29}^{61}Cu$ isotope.
Experimental values are taken form Refs.[40,41]
---

Nuclei | $J^{\pi}$ |  $K$ |  $\nu_{d}$ |  $E_{exp}$(keV) |  $E_{cal}$(keV)
---|---|---|---|---|---
${}_{30}^{62}Zn$ | $0_{1}^{+}$ |  1 |  0 |  0 |  0
| $2_{1}^{+}$ |  1 |  1 |  954 |  785.4
| $2_{2}^{+}$ |  0 |  2 |  1804.7 |  2266.4
| $4_{1}^{+}$ |  0 |  2 |  2186.1 |  1773.3
| $0_{2}^{+}$ |  1 |  0 |  2330 |  2425.7
| $3_{1}^{+}$ |  0 |  3 |  2384.5 |  2686.5
| $2_{3}^{+}$ |  1 |  1 |  2810 |  2826.6
| $4_{2}^{+}$ |  0 |  3 |  2743.5 |  2807.1
| $2_{4}^{+}$ |  0 |  2 |  2890 |  3156
| $2_{5}^{+}$ |  1 |  1 |  3470 |  3387.5
| $2_{6}^{+}$ |  0 |  2 |  3640 |  3484.7
${}_{29}^{61}Cu$ | $(3/2)_{1}^{-}$ |  1 |  0 |  0 |  0
| $(1/2)_{1}^{-}$ |  0 |  1 |  475.1 |  524.3
| $(5/2)_{1}^{-}$ |  0 |  1 |  970.06 |  726.1
| $(7/2)_{1}^{-}$ |  0 |  1 |  1310.55 |  1431.6
| $(5/2)_{2}^{-}$ |  0 |  2 |  1394.2 |  1855.7
| $(7/2)_{2}^{-}$ |  0 |  2 |  1732.61 |  1961.3
| $(5/2)_{3}^{-}$ |  0 |  2 |  1904.18 |  1855.7
| $(7/2)_{3}^{-}$ |  0 |  2 |  1942.49 |  1961.3
| $(1/2)_{2}^{-}$ |  0 |  2 |  2088.86 |  1734.7
| $(5/2)_{4}^{-}$ |  0 |  1 |  2203.4 |  2415.81
| $(9/2)_{1}^{-}$ |  0 |  2 |  2295.09 |  2096.9

Table 6b.Energy spectra for${}_{30}^{64}Zn$\- ${}_{29}^{63}Cu$
isotope.Experimental values are taken form Refs.[42,43]
---

Nuclei | $J^{\pi}$ |  $K$ |  $\nu_{d}$ |  $E_{exp}$(keV) |  $E_{cal}$(keV)
---|---|---|---|---|---
${}_{30}^{64}Zn$ | $0_{1}^{+}$ |  2 |  0 |  0 |  0
| $2_{1}^{+}$ |  1 |  1 |  991.54 |  1040.878
| $2_{2}^{+}$ |  1 |  2 |  1799.41 |  1406.2
| $4_{1}^{+}$ |  1 |  2 |  2306.68 |  2487.7
| $3_{1}^{+}$ |  1 |  1 |  2979.81 |  2820.4
| $2_{3}^{+}$ |  1 |  1 |  2793.7 |  2770.4
| $4_{2}^{+}$ |  0 |  3 |  2736.6 |  2887.2
| $2_{4}^{+}$ |  1 |  2 |  3005.72 |  3170.3
| $2_{5}^{+}$ |  0 |  4 |  3094.6 |  2820.4
| $0_{2}^{+}$ |  2 |  0 |  1910.3 |  1725.6
| $0_{3}^{+}$ |  2 |  0 |  2609.45 |  2324.5
| $4_{3}^{+}$ |  0 |  3 |  3078.43 |  2887.2
| $3_{2}^{+}$ |  0 |  3 |  3094.6 |  2820.4
${}_{29}^{63}Cu$ | $(3/2)_{1}^{-}$ |  1 |  0 |  0 |  0
| $(1/2)_{1}^{-}$ |  1 |  1 |  669.67 |  724.4
| $(5/2)_{1}^{-}$ |  1 |  1 |  962.1 |  1295.6
| $(7/2)_{1}^{-}$ |  1 |  1 |  1327.01 |  1105.4
| $(5/2)_{2}^{-}$ |  0 |  2 |  1412.05 |  1844.4
| $(3/2)_{1}^{-}$ |  1 |  1 |  1547.05 |  1404.1
| $(7/2)_{2}^{-}$ |  0 |  2 |  1904.5 |  1861.61
| $(7/2)_{3}^{-}$ |  0 |  2 |  2092.6 |  2204.5
| $(3/2)_{2}^{-}$ |  1 |  0 |  2011.25 |  1802
| $(1/2)_{2}^{-}$ |  0 |  2 |  2062.23 |  1777.7
| $(9/2)_{1}^{-}$ |  0 |  2 |  2207.88 |  2179.6
| $(5/2)_{3}^{-}$ |  0 |  2 |  2336.58 |  2595.9

Table 6c.Energy spectra for${}_{30}^{66}Zn$\- ${}_{29}^{65}Cu$
isotope.Experimental values are taken form Refs.[44,45]
---

Nuclei | $J^{\pi}$ |  $K$ |  $\nu_{d}$ |  $E_{exp}$(keV) |  $E_{cal}$(keV)
---|---|---|---|---|---
${}_{30}^{66}Zn$ | $0_{1}^{+}$ |  2 |  0 |  0 |  0
| $2_{1}^{+}$ |  2 |  1 |  1039.39 |  1069.4
| $2_{2}^{+}$ |  1 |  2 |  1872.94 |  2245.9
| $4_{1}^{+}$ |  1 |  2 |  2451.13 |  2660.1
| $2_{3}^{+}$ |  2 |  1 |  2780.55 |  2772.5
| $4_{2}^{+}$ |  1 |  3 |  2765.69 |  2913.6
| $2_{4}^{+}$ |  1 |  2 |  2938.45 |  2985.5
| $0_{2}^{+}$ |  2 |  0 |  2372.53 |  2524.7
| $0_{3}^{+}$ |  2 |  0 |  3030 |  3262.3
| $4_{3}^{+}$ |  1 |  3 |  3077.93 |  3280.8
${}_{29}^{65}Cu$ | $(3/2)_{1}^{-}$ |  2 |  0 |  0 |  0
| $(1/2)_{1}^{-}$ |  1 |  1 |  770.64 |  794.4
| $(5/2)_{1}^{-}$ |  1 |  1 |  1115.556 |  1523
| $(7/2)_{1}^{-}$ |  1 |  1 |  1481.83 |  1564.6
| $(5/2)_{2}^{-}$ |  1 |  2 |  1623.43 |  1736.1
| $(3/2)_{2}^{-}$ |  1 |  1 |  1725 |  1922.9
| $(7/2)_{2}^{-}$ |  2 |  1 |  2094.34 |  2208.1
| $(7/2)_{3}^{-}$ |  0 |  2 |  2092.6 |  2204.5
| $(3/2)_{3}^{-}$ |  2 |  0 |  2329.05 |  2449.2
| $(1/2)_{2}^{-}$ |  1 |  2 |  2212.84 |  2118.9
| $(5/2)_{3}^{-}$ |  1 |  2 |  2107.44 |  2365.5
| $(5/2)_{4}^{-}$ |  1 |  1 |  2532.04 |  2691.6

Table 6d.Energy spectra for${}_{30}^{68}Zn$\- ${}_{29}^{67}Cu$
isotope.Experimental values are taken form Refs.[46,47]
---

Nuclei | $J^{\pi}$ |  $K$ |  $\nu_{d}$ |  $E_{exp}$(keV) |  $E_{cal}$(keV)
---|---|---|---|---|---
${}_{30}^{68}Zn$ | $0_{1}^{+}$ |  3 |  0 |  0 |  0
| $2_{1}^{+}$ |  2 |  1 |  1077.37 |  1183.3
| $2_{2}^{+}$ |  2 |  2 |  1883.14 |  1989.8
| $4_{1}^{+}$ |  2 |  2 |  2417.44 |  2379.2
| $2_{3}^{+}$ |  2 |  1 |  2338.29 |  2470
| $4_{2}^{+}$ |  1 |  3 |  2955.9 |  2893.5
| $2_{4}^{+}$ |  2 |  2 |  2821.58 |  2978.5
| $0_{2}^{+}$ |  3 |  0 |  1655.94 |  1627.5
| $0_{3}^{+}$ |  3 |  0 |  3102.45 |  3213.6
| $3_{1}^{+}$ |  1 |  3 |  3009.2 |  3238.4
${}_{29}^{67}Cu$ | $(3/2)_{1}^{-}$ |  2 |  0 |  0 |  0
| $(1/2)_{1}^{-}$ |  2 |  1 |  2272 |  2212.6
| $(3/2)_{2}^{-}$ |  2 |  1 |  1937.1 |  2070
| $(3/2)_{2}^{-}$ |  1 |  1 |  1725 |  1922.9
| $(3/2)_{3}^{-}$ |  2 |  0 |  2272 |  2564.8
| $(1/2)_{2}^{-}$ |  1 |  2 |  2623.1 |  2707.9
| $(3/2)_{4}^{-}$ |  1 |  2 |  2623.1 |  2595.1
| $(3/2)_{5}^{-}$ |  2 |  1 |  2680.1 |  3058.7
| $(1/2)_{3}^{-}$ |  2 |  1 |  2841.1 |  3201.3
| $(3/2)_{6}^{-}$ |  1 |  3 |  2841.1 |  3090.4

Table 6e.Energy spectra for${}_{30}^{70}Zn$\- ${}_{29}^{69}Cu$
isotope.Experimental values are taken form Refs.[48,49]
---

Nuclei | $J^{\pi}$ |  $K$ |  $\nu_{d}$ |  $E_{exp}$(keV) |  $E_{cal}$(keV)
---|---|---|---|---|---
${}_{30}^{70}Zn$ | $0_{1}^{+}$ |  3 |  0 |  0 |  0
| $2_{1}^{+}$ |  2 |  1 |  884.8 |  952.3
| $2_{2}^{+}$ |  2 |  2 |  1759.1 |  2074
| $4_{1}^{+}$ |  2 |  2 |  1786.5 |  1923.1
| $2_{3}^{+}$ |  2 |  1 |  1957.7 |  2389.6
| $4_{2}^{+}$ |  1 |  3 |  2693.6 |  2458.6
| $2_{4}^{+}$ |  2 |  2 |  2538 |  2464.3
| $0_{2}^{+}$ |  3 |  0 |  1068.3 |  1163.7
| $0_{3}^{+}$ |  3 |  0 |  2140.4 |  2016.9
| $3_{1}^{+}$ |  1 |  3 |  2949.6 |  2838.1
| $2_{6}^{+}$ |  2 |  1 |  3635 |  3438.1
| $0_{4}^{+}$ |  1 |  3 |  3680 |  3465.2
| $2_{7}^{+}$ |  2 |  2 |  3710.6 |  3904.6
${}_{29}^{69}Cu$ | $(3/2)_{1}^{-}$ |  2 |  0 |  0 |  0
| $(1/2)_{1}^{-}$ |  2 |  1 |  1096 |  1540.4
| $(7/2)_{1}^{-}$ |  2 |  1 |  1710.8 |  1539.7
| $(7/2)_{2}^{-}$ |  1 |  2 |  1870 |  2007.7

Table 7. The coefficients of $T(E_{2})$used in the present work for the Ru-Rh
Supermultiplets..Experimental values are taken form Refs. [40-49] .
---

Nucleus | $q_{B}$ | $q_{f}$
---|---|---
${}^{61}Cu-^{62}Zn$ | 2.4197 | -0.9731
${}^{63}Cu-^{64}Zn$ | 2.1278 | -0.28
${}^{65}Cu-^{66}Zn$ | 8.775 | 0
${}^{67}Cu-^{68}Zn$ | 4.9527 | 0
${}^{69}Cu-^{70}Zn$ | 7.7603 | 0

Table 8a. B(E2)values for ${}^{61}Cu$ and ${}^{62}Zn$ isotopes. Experimental
values are taken form Refs.[40,41]and are presented in Weisskopf units(W.u.).
---

Nucleus | $J_{i}^{\pi}\longrightarrow J_{j}^{\pi}$ | $\frac{B(E_{2}(W.u.)}{exp.\quad\quad\quad calc.}$
---|---|---
${}^{61}Cu$ | $2_{1}^{+}\longrightarrow 0_{1}^{+}$ | 16.8 15.3989
| $2_{2}^{+}\longrightarrow 2_{1}^{+}$ | 18 17.2585
| $4_{1}^{+}\longrightarrow 2_{1}^{+}$ | 26 25.7274
| $2_{2}^{+}\longrightarrow 0_{1}^{+}$ | 0.32 0.14
| $3_{1}^{+}\longrightarrow 2_{2}^{+}$ | 110 45.21
| $3_{1}^{+}\longrightarrow 2_{1}^{+}$ | 0.5 0.5466
${}^{62}Zn$ | $(5/2)_{1}^{-}\longrightarrow(1/2)_{1}^{-}$ | 17 9.006
| $(5/2)_{1}^{-}\longrightarrow(3/2)_{1}^{-}$ | 7.2 1.4164
| $(7/2)_{2}^{-}\longrightarrow(5/2)_{1}^{-}$ | 0.03 0
| $(7/2)_{2}^{-}\longrightarrow(3/2)_{1}^{-}$ | 18 20.4437
| $(3/2)_{2}^{-}\longrightarrow(3/2)_{1}^{-}$ | 1.5 1.1757

Table 8b. B(E2)values for ${}^{64}Zn$ and ${}^{63}Cu$ isotopes. Experimental
values are taken form Refs.[42,43]and are presented in Weisskopf units(W.u.).
---

Nucleus | $J_{i}^{\pi}\longrightarrow J_{j}^{\pi}$ | $\frac{B(E_{2}(W.u.)}{exp.\quad\quad\quad calc.}$
---|---|---
${}^{64}Zn$ | $2_{1}^{+}\longrightarrow 0_{1}^{+}$ | 20 28.9867
| $0_{2}^{+}\longrightarrow 2_{1}^{+}$ | 0.057 3.2783
| $4_{1}^{+}\longrightarrow 2_{1}^{+}$ | 12.2 9.7
| $2_{2}^{+}\longrightarrow 2_{1}^{+}$ | 39 35.186
| $2_{2}^{+}\longrightarrow 0_{1}^{+}$ | 0.24 0.92
| $0_{3}^{+}\longrightarrow 2_{2}^{+}$ | 1.3 0.00
${}^{63}Cu$ | $(1/2)_{1}^{-}\longrightarrow(3/2)_{1}^{-}$ | 15.2 15.7863
| $(5/2)_{1}^{-}\longrightarrow(3/2)_{1}^{-}$ | 15.7 12.7599
| $(7/2)_{1}^{-}\longrightarrow(3/2)_{1}^{-}$ | 12.7 11.05
| $(5/2)_{2}^{-}\longrightarrow(1/2)_{1}^{-}$ | 6 5.3356
| $(5/2)_{2}^{-}\longrightarrow(3/2)_{1}^{-}$ | 1 0.002
| $(3/2)_{2}^{-}\longrightarrow(3/2)_{1}^{-}$ | 3.7 0.00
| $(7/2)_{2}^{-}\longrightarrow(3/2)_{1}^{-}$ | 1.4 0.94

Table 8c. B(E2)values for ${}^{66}Zn$ and ${}^{65}Cu$ isotopes. Experimental
values are taken form Refs.[44,45]and are presented in Weisskopf units(W.u.).
---

Nucleus | $J_{i}^{\pi}\longrightarrow J_{j}^{\pi}$ | $\frac{B(E_{2}(W.u.)}{exp.\quad\quad\quad calc.}$
---|---|---
${}^{66}Zn$ | $2_{1}^{+}\longrightarrow 0_{1}^{+}$ | 17.5 8.0597
| $2_{2}^{+}\longrightarrow 2_{1}^{+}$ | 330 330.3182
| $2_{2}^{+}\longrightarrow 0_{1}^{+}$ | 0.032 0.032
| $4_{1}^{+}\longrightarrow 2_{1}^{+}$ | 18 14.7453
| $2_{3}^{+}\longrightarrow 2_{1}^{+}$ | 0.13 0
| $2_{3}^{+}\longrightarrow 0_{1}^{+}$ | 0.54 4.6768
${}^{65}Cu$ | $(1/2)_{1}^{-}\longrightarrow(3/2)_{1}^{-}$ |  0.003
| $(5/2)_{1}^{-}\longrightarrow(3/2)_{1}^{-}$ |  31.5248
| $(7/2)_{1}^{-}\longrightarrow(5/2)_{1}^{-}$ |  0.001
| $(5/2)_{2}^{-}\longrightarrow(1/2)_{1}^{-}$ |  0.3517
| $(5/2)_{2}^{-}\longrightarrow(3/2)_{1}^{-}$ |  5.5851

Table 8e. B(E2)values for ${}^{68}Zn$ and ${}^{67}Cu$ isotopes. Experimental
values are taken form Refs.[46,47]and are presented in Weisskopf units(W.u.).
---

Nucleus | $J_{i}^{\pi}\longrightarrow J_{j}^{\pi}$ | $\frac{B(E_{2}(W.u.)}{exp.\quad\quad\quad calc.}$
---|---|---
${}^{68}Zn$ | $2_{1}^{+}\longrightarrow 0_{1}^{+}$ | 14.69 9.61
| $4_{1}^{+}\longrightarrow 2_{1}^{+}$ | 10.8 7.7922
| $2_{2}^{+}\longrightarrow 0_{1}^{+}$ | 16 8.425
| $0_{2}^{+}\longrightarrow 2_{1}^{+}$ | 5.5 3.366
| $2_{2}^{+}\longrightarrow 0_{2}^{+}$ | 16 8.425
| $2_{2}^{+}\longrightarrow 2_{1}^{+}$ | 28.6 29.683
${}^{67}Cu$ | $(1/2)_{1}^{-}\longrightarrow(3/2)_{1}^{-}$ |  0.058
| $(5/2)_{1}^{-}\longrightarrow(3/2)_{1}^{-}$ |  0.3646
| $(7/2)_{1}^{-}\longrightarrow(5/2)_{1}^{-}$ |  0.00
| $(5/2)_{2}^{-}\longrightarrow(1/2)_{1}^{-}$ |  0.6664
| $(5/2)_{2}^{-}\longrightarrow(3/2)_{1}^{-}$ |  3.2829

Table 8d. B(E2)values for ${}^{70}Zn$ and ${}^{69}Cu$ isotopes. Experimental
values are taken form Refs.[48,49]and are presented in Weisskopf units(W.u.).
---

Nucleus | $J_{i}^{\pi}\longrightarrow J_{j}^{\pi}$ | $\frac{B(E_{2}(W.u.)}{exp.\quad\quad\quad calc.}$
---|---|---
${}^{70}Zn$ | $2_{1}^{+}\longrightarrow 0_{1}^{+}$ | 16.5 12.3661
| $0_{2}^{+}\longrightarrow 2_{1}^{+}$ | 37.3 37.1785
| $2_{2}^{+}\longrightarrow 2_{1}^{+}$ | 69 42.23
| $2_{2}^{+}\longrightarrow 0_{1}^{+}$ | 2.4 1.275
| $2_{4}^{+}\longrightarrow 0_{1}^{+}$ | 0.15 0.8133
${}^{69}Cu$ | $(1/2)_{1}^{-}\longrightarrow(3/2)_{1}^{-}$ |  0.145
| $(5/2)_{1}^{-}\longrightarrow(3/2)_{1}^{-}$ |  1.002
| $(7/2)_{1}^{-}\longrightarrow(5/2)_{1}^{-}$ |  0.002
| $(5/2)_{2}^{-}\longrightarrow(1/2)_{1}^{-}$ |  0.328
| $(5/2)_{2}^{-}\longrightarrow(3/2)_{1}^{-}$ |  0.418

Table 9.Comparison of key observable in Ru isotopes with the E(5) symmetry.
---

Nucleus | $R_{\frac{4}{2}}$ | $C_{s}$ | $\frac{E(0_{2}^{+})}{E(2_{1}^{+})}$ | $B(\frac{E2;4_{1}^{+}\longrightarrow 2_{1}^{+}}{E2;2_{1}^{+}\longrightarrow 0_{1}^{+}})$ | $B(\frac{E2;0_{2}^{+}\longrightarrow 2_{1}^{+}}{E2;2_{1}^{+}\longrightarrow 0_{1}^{+}})$
---|---|---|---|---|---
E(5) | 2.19 | - | 3.03 | 1.68 | 0.68
${}^{100}Ru$ | 2.24 | 0.38 | 2.5544 | 1.55 | 0.9379
${}^{102}Ru$ | 2.37 | 0.23 | 1.6227 | 1.7524 | 1.2274
${}^{104}Ru$ | 2.32 | 0.54 | 3.18 | 1.49 | 0.446
${}^{106}Ru$ | 2.526 | 0.34 | 3.38 | - | -
${}^{108}Ru$ | 2.435 | 0.65 | 3.4 | 1.64 | -

Table 10.Comparison of key observable in Zn isotopes with the E(5) symmetry.
---

Nucleus | $R_{\frac{4}{2}}$ | $C_{s}$ | $\frac{E(0_{2}^{+})}{E(2_{1}^{+})}$ | $B(\frac{E2;4_{1}^{+}\longrightarrow 2_{1}^{+}}{E2;2_{1}^{+}\longrightarrow 0_{1}^{+}})$ | $B(\frac{E2;0_{2}^{+}\longrightarrow 2_{1}^{+}}{E2;2_{1}^{+}\longrightarrow 0_{1}^{+}})$
---|---|---|---|---|---
E(5) | 2.19 | - | 3.03 | 1.68 | 0.68
${}^{62}Zn$ | 2.257 | 0.51 | 3.088 | 1.55 | 0.9379
${}^{64}Zn$ | 2.391 | 0.4 | 1.657 | 1.7524 | 1.2274
${}^{66}Zn$ | 2.48 | 0.65 | 2.36 | 1.49 | 0.446
${}^{68}Zn$ | 2.014 | 0.03 | 1.375 | - | -
${}^{70}Zn$ | 2.0194 | 0.01 | 1.22 | 1.64 | -

Figure 1: The lattice of algebras in the U(6/2) supersymmetry scheme. Figure
2: The Lattice of algebras in the U(6/4) supersymmetry scheme.

Figure 3: Energy levels as a function of $C_{s}$ control parameter for a even-
even nuclei (left panel) and for odd-A nuclei as a function of the $C_{s}$ and
$C_{f}$ control parameters(right panel) in the Hamiltonian (27) for $N=10$
bosons with$\alpha=1000$, $\beta=3.14$, $\delta=-5.26$, $\gamma=0.0439$.

Figure 4: Energy levels as a function of $C_{s}$ control parameter for a even-
even nuclei (left panel) and for odd-A nuclei as a function of the $C_{s}$ and
$C_{f}$ control parameters(right panel) in the Hamiltonian (30) for $N=10$
bosons with $\alpha=1000,\beta^{\prime}=-1.29,\gamma=6.05$.

Figure 5: The expectation values of the $d$-boson number operator for the
lowest states as a function of $C_{s}$ control parameter for an even-even
nuclei (left panel) and for odd-A nuclei as a function of the $C_{s}$ and
$C_{f}$ control parameters(right panel) for j=1/2.

Figure 6: The expectation values of the $d$-boson number operator for the
lowest states as a function of $C_{s}$ control parameter for an even-even
nuclei (left panel) and for odd-A nuclei as a function of the $C_{s}$ and
$C_{f}$ control parameters(right panel) for j=3/2. Figure 7: The expectation
values of the fermion number operator for odd-A nuclei for the lowest states
as a function of $C_{s}$ and $C_{f}$ control parameters. Figure 8: The
expectation values of the fermion number operator for odd-A nuclei for the
lowest states as a function of $C_{s}$ and $C_{f}$ control parameters.

Figure 9: Comparison between calculated and experimental spectra of positive
parity states in Ru Isotopes. The parameters of the calculation are given in
Tables 2. In the experimental spectra, taken from 26 ; 28 ; 30 ; 32 ; 34 ; 36
.

Figure 10: Comparison between calculated and experimental spectra of negative
parity states in Rh Isotopes. The parameters of the calculation are given in
Tables 1. In the experimental spectra, taken from 27 ; 29 ; 31 ; 33 ; 35 ; 36
.

Figure 11: Comparison between calculated and experimental spectra of positive
parity states in Zn Isotopes. The parameters of the calculation are given in
Tables 2. In the experimental spectra, taken from 41 ; 43 ; 45 ; 47 ; 49 ; 36
.

Figure 12: Comparison between calculated and experimental spectra of negative
parity states in Cu Isotopes. The parameters of the calculation are given in
Tables 1. In the experimental spectra, taken from 40 ; 42 ; 44 ; 46 ; 48 ; 36
.

Figure 13: $(E(\nu_{d}=2))/(E(\nu_{d}=1))$ prediction values for Ru and Rh
Isotopes. In the experimental spectra, taken from 26 ; 28 ; 30 ; 32 ; 34 ; 36

Figure 14: $(E(\nu_{d}=2))/(E(\nu_{d}=1))$ prediction values for Zn and Cu
Isotopes. In the experimental spectra, taken from 27 ; 29 ; 31 ; 33 ; 35 ; 36

Figure 15: A comparison between theoretical and experimental two neutron
separation energies,$S_{2n}$ (in keV) for Ru isotopes (1eft panel) and Rh
isotopes (right panel). Experimental data from 36 .

Figure 16: A comparison between theoretical and experimental two neutron
separation energies,$S_{2n}$ (in keV) for Zn isotopes (1eft panel) and Cu
isotopes (right panel). Experimental data from 36 .


# CLaC at SemEval-2023 Task 2: Comparing Span-Prediction and Sequence-Labeling
approaches for NER

Harsh Verma, Sabine Bergler
CLaC Labs, Concordia University
{h_ver, bergler} @cse.concordia.ca

###### Abstract

This paper summarizes the CLaC submission for the MultiCoNER 2 task which
concerns the recognition of complex, fine-grained named entities. We compare
two popular approaches for NER, namely Sequence Labeling and Span Prediction.
We find that our best Span Prediction system performs slightly better than our
best Sequence Labeling system on test data. Moreover, we find that using the
larger version of XLM RoBERTa significantly improves performance. Post-
competition experiments show that Span Prediction and Sequence Labeling
approaches improve when they use special input tokens (`<s>` and `</s>`) of
XLM-RoBERTa. The code for training all models, preprocessing, and post-
processing is available at this Github repo.

##  Introduction

MultiCoNER 2 Fetahu et al. (2023b, a) is the second version of the MultiCoNER
Malmasi et al. (2022) shared task. It focuses on identifying complex named
entities in short sentences that lack context and addresses the challenge of
identifying and distinguishing 30 fine-grained entity types and of handling
simulated errors, such as typos.

Competition data is tokenzied and tagged with the BIO scheme, i.e. the label B
for _begin_ , I for _inside_ , and O for _outside_ target entities. For this
challenge, the 30 entity types were distinguished using suffixes to the B and
I labels, e.g. B-Corp and I-Corp, as illustrated in Example 2.1.

NER is frequently formulated as a sequence labeling problem Chiu and Nichols
(2016); Ma and Hovy (2016); Wang et al. (2022), in which a model learns to
label each token individually using the BIO labeling scheme. In contrast, span
prediction approaches Jiang et al. (2020); Li et al. (2020); Fu et al. (2021);
Zhang et al. (2023) label entire text spans with entity types. Recently, Fu et
al. (2021) compared the two approaches and found that sequence labeling
performed better for long entities with low label consistency but that span
prediction performed better for out of vocabulary words and entities of short
to medium length. Because MultiCoNER 2 tests both, the ability to identify out
of vocabulary entities and entities of varying length111Creative work titles
can be long, e.g. _To Kill A Mocking Bird_ , it is interesting to compare the
two approaches on this task.

Our submission to the English Track does not use any external knowledge and
holds rank 21 of 41 on
Codalab222https://codalab.lisn.upsaclay.fr/competitions/10025#results with a
macro-F1 of 59.

##  Submitted Systems

###  Tokenization

The competition input consisted of tokenized data. The gold labels are given
in form of BIO tags, where each token is labeled with a corresponding BIO tag
as shown in Example 2.1. Sample input is given to XLM RoBERTa Conneau et al.
(2020) as the raw input string, without RoBERTa input tags such as `<s>` and
`</s>` in the submission systems.

Token pharma company eli lilly and company announced // BIO-Tag O O B-Corp
I-Corp I-Corp I-Corp O //

###  Sequence Labeling Model

Our sequence labeling model uses XLM RoBERTa large Conneau et al. (2020). Like
all BERT-derived models, RoBERTa splits the input tokens further into word
piece subtokens, for which the pre-trained embedding vectors are included, and
for which the subtoken-level output is generated. These subtokens are
classified into the BIO sequence. For submission and scoring, this subtoken
sequence has to be decoded into word-level tokens.

#### Token Representation Step

Given a sentence $\mathbf{x}=[w_{1},w_{2},...,w_{n}]$ with $n$ word piece
tokens, we generate for each token $w_{i}$ a pretrained embedding
$\mathbf{u}_{i}$ using the last hidden layer of XLM RoBERTa large Conneau et
al. (2020)

$\displaystyle\text{Embed}(\textbf{x})$
$\displaystyle=\text{Embed}([w_{1},w_{2},...,w_{n}])$
$\displaystyle=[\mathbf{u_{1}},\mathbf{u_{2}},...,\mathbf{u_{n}}]$

#### Token Classification Step

In this layer, we classify every token representation into a set of named
entity types corresponding to the BIO(beginning, inside, outside) tagging
scheme. Assuming $\mathbf{\Theta}$ is the set of all named entity types , the
set of all BIO tags $\mathbf{B}$ is of size $(2\times|\mathbf{\Theta}|)+1$. We
use a linear layer to map each subtoken $\mathbf{u}_{i}\in\mathbb{R}^{d}$ to a
prediction $\mathbf{p}_{i}\in\mathbb{R}^{|\bf{B}|}$, where $d$ is the length
of the token embedding. The predictions are used to calculate loss of sample
$\mathbf{x}$ with $n$ tokens as follows:

$\text{Loss}(\mathbf{x})=\frac{-1}{n}\sum_{i=1}^{n}\text{log}(\text{Softmax}(\mathbf{p}_{i})_{y_{i}})$
(1)

Here $y_{i}$ represents the index of the gold BIO label of the $i^{th}$ token.

#### Decoding Step

For this task, the boundaries of each predicted span must align with actual
word boundaries, which poses an issue due to the word piece tokenization. We
align every predicted span with the nearest enveloping words. Concretely, let
$(b,e)$ represent the beginning and end offsets of a predicted span $s$. If
$b$ and $e$ are contained in words $w_{b}$ and $w_{e}$ respectively, they are
remapped to the beginning and end offsets of the containing words.

###  Span Prediction Model

#### Token Representation Layer

The token representation layer is identical to that of the Sequence Labeling
model.

#### Span Representation Layer

Let a span $\mathbf{s}$ be a tuple $\mathbf{s}=(b,e)$ where $b$ and $e$ are
the begin and end word piece token indices, and $\mathbf{s}$ represents the
text segment $[w_{b},w_{b+1},...,w_{e}]$ where $w_{i}$ is the $i^{th}$ word
piece token. In this layer, we enumerate all possible spans, represented by
the tuple $(b,e)$. Because $b\leq e$, there are $\frac{n^{2}}{2}$ possible
spans. We follow Fu et al. (2021) and encode each span $s_{i}$ as the
concatenation of their begin and end word piece token embeddings
$\mathbf{v}_{i}=[\mathbf{u}_{b_{i}};\mathbf{u}_{e_{i}}]$. The output of the
decoding layer is $\textbf{V}\in\mathbb{R}^{k\times(2\times d)}$ where
$k=\frac{n^{2}}{2}$ and $d$ is length of the token embedding vector.

#### Span Classification Layer

In this layer, we classify each span representation with a named entity type .
We introduce an additional label `Neg_Span` which represents the absence of a
named entity . In particular, a linear layer maps each span representation
$\mathbf{v}_{i}\in\mathbb{R}^{(2\times d)}$ to a prediction
$\mathbf{p}_{i}\in\mathbb{R}^{|\Omega|}$, where $\Omega$ is the set of all
named entity types (including `Neg_Span`) and $d$ is the size of the token
embedding. The predictions are used to calculate the loss for sentence
$\mathbf{x}$ with $l$ possible spans as follows:

$\text{Loss}(\mathbf{x})=\frac{-1}{l}\sum_{i=1}^{l}\text{log}(\text{Softmax}(\mathbf{p}_{i})_{y_{i}})$
(2)

Here $y_{i}$ represents the index of the gold label of the $i^{th}$ span.

#### Decoding

Similarly to the Sequence Labeling model, we align all spans with word
boundaries. Because the Span Prediction model predicts overlapping spans, we
remove overlaps with the following procedure:

  1. 1.

For each span $s$, if $s$ is completely contained within another span $S$, we
remove $s$. We keep removing fully contained spans until none are left.

  2. 2.

For each span $s_{1}$ which partially overlaps another span $s_{2}$, we
randomly select one of the two spans and remove it. We keep removing until no
overlapping spans are left.

The remaining spans are then mapped to BIO tags .

###  Training

XLM RoBERTa large is fine-tuned on the training data using the Adam optimizer
Kingma and Ba (2015) with a learning rate of `1e-5` and a batch size of 1. The
best model is selected using early stopping. Training for 10 epochs takes
around 6 hours on one Nvidia RTX 3090 gpu.

##  Results and Discussion

We evaluate all of our systems on the Codalab Competition
Website333https://codalab.lisn.upsaclay.fr/competitions/10025, where
submissions are evaluated using the entity level macro F1 metric. All models
use the XLM-RoBERTa large LM unless otherwise indicated.

Table 1 shows the performance of the two submitted systems S(submitted) in
bold.

Post-competition experimentation showed that increasing the batch size from 1
to 4 improved results by at least $3.2$, this improvement is shown in Table 1
for systems B.

We also found that adding the embeddings for RoBERTa special tokens `<s>` and
`</s>` further improved performance, especially raising performance of the
Sequence model Seq by $2.1$, nearly erasing the performance advantage of the
span-based model. The results are shown in Table 1 for systems B+E.

System Type | Seq | Span
---|---|---
| F1 | P | R | F1 | P | R
S(submitted) | 53.2 | 52.8 | 54.5 | 55.0 | 56.5 | 55.5
B | 56.9 | 57.3 | 57.5 | 59.0 | 61.2 | 58.4
B+E | 59.0 | 60.7 | 58.9 | 59.7 | 61.3 | 59.5

Table 1:  Performance on the test set in bold. The submitted systems improve
when the batch size is increased from 1 to 4 (systems B), and they improve
further when special RoBERTa token embeddings are added (systems B+E).

Table 2 shows the performance of our systems on the validation set.

The Span Prediction models showed significantly higher performance on the
validation set than the test set. The post-competition improvements affect the
sequence and span-based models differently and nearly erase the performance
difference.

Model | $\text{Span}_{\text{S}}$ | $\text{Seq}_{\text{S}}$ | $\text{Span}_{\text{B}}$ | $\text{Seq}_{\text{B}}$
---|---|---|---|---
Macro F1 | 58.5 | 59.3 | 63.6 | 60.2

Table 2: Performance on validation set for systems S and B

Table 3 compares the performance of large and base versions of XLM RoBERTa.
$\text{Span}_{\text{base}}$ and $\text{Seq}_{\text{base}}$ are identical to
$\text{Span}_{\text{B}}$ and $\text{Seq}_{\text{B}}$ except that the base
models use XLM RoBERTa base instead of large. The larger models seem to be
performing significantly better.

Model | $\text{Span}_{\text{B}}$ | $\text{Seq}_{\text{B}}$ | $\text{Span}_{\text{base}}$ | $\text{Seq}_{\text{base}}$
---|---|---|---|---
Macro F1 | 59.0 | 56.9 | 52.5 | 51.1

Table 3: Performance of large and base pretrained models on test set for
systems B

##  Conclusion

We submitted two systems to MultiCoNER 2, one inspired by a Sequence Labeling
approach and another inspired by a Span Prediction approach. We find that our
best Span Prediction system performs slightly better than our best Sequence
Labeling system on test data. We showed significant increases in our systems’
performance when using a larger pretrained language model, batch size 4, and
special tokens `<s>` and `</s>`.

## References

  * Chiu and Nichols (2016) Jason P.C. Chiu and Eric Nichols. 2016.  Named entity recognition with bidirectional LSTM-CNNs.  _Transactions of the Association for Computational Linguistics_ , 4:357–370.
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