International Journal of Scientific & Engineering Research Volume 3, Issue 12, December -2012 1

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NUCLEAR STRUCTURE OF EVEN-EVEN

104-110Mo ISOTOPES

Jundi Khalat Yousif

AbstractFrom the E-Gamma Over Spin (E-GOS) curve and the energy level ratios, we have deduced that the 104-110Mo isotopes lies between the vibrational U(5) and axial deformed rotor SU(3) limits and most of them display the X(5) symmetry features. W e have compared the results obtained by interacting boson model IBM-2 for the above isotopes with those of the X(5) limit and then given a clear description about the validity of the Hamiltonian parameters used in this study.

Index Terms— Critical point symmetry, interacting boson model (IBM-2), even-even Mo isotopes.

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tomic nuclei are known to exhibit changes of their
energy levels and electromagnetic transition rates among them when the number of protons and/or neutrons is modified, resulting in phase shape transition from one kind of collective behavior to another. These transitions are not phase transition of the usual thermodynamic type. They are quantum phase transitions [1] (initially called ground state phase transition) occurring in the Hamiltonians.
In the framework of the Interacting Boson Model [2], which describes the nuclear structure of even-even nuclei within the U(6) symmetry, possessing the U(5), SU(3), and O(6) limiting dynamical symmetries, appropriate for vibrational, axial deformed, and -unstable nuclei respectively.
Shape phase transitions have been studied 25 years ago [3] using the classical limit of the model [4,5], pointing out that there is a second order shape phase transition between U(5) and O(6), a first order shape phase transition between U(5) and SU(3). It is instructive to place these shape phase transitions on the symmetry triangle of the IBM [6], at the three corners of which the three limiting symmetries of the IBM appear. The X(5) critical point symmetry, introduced by Iachello [7,8] has an analytical solution to describe the transition from a spherical harmonic vibrator U(5) to an axially deformed rotor SU(3), which has stimulated

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Author is currently one of the staff members of Department of Physics, Faculty of Science, University of Zakho, Iraq

E-mail: jundite@yahoo.com
considerable efforts both experimentally and theoretically.
Initial work concentrated on the nuclei in the rare –earth region with N=90. Extensive studies have shown that 152Sm [9] and 150Nd [10] are close manifestation of the X(5) critical point symmetry. Additional examples of X(5) behavior have been suggested in 154Gd [ 11] and 104Mo [12 ]. Later experiments provided that the yrast band energies as well as the B(E2) values are in good agreement with the X(5) predictions for

162Yb[13], 166Hf [14] and 176Os [15]. Lifetimes of the first 4+ and

6+ states in 104Mo and 106Mo have been measured using the recoil distance following spontaneous fission of 252Cf [12 ]. Excitation energies and electromagnetic transition strength in even- even 96-108Mo have been described systematically by using the proton-neutron boson model IBM-2 [16]. The dynamical symmetry X(5) arises when the potential in the Bohr Hamiltonian [17] is decoupled into two components-an infinite square well potential for the quadrupole deformation parameter  and a harmonic potential well for the triaxiality deformation parameter  [18,19]. The experimental signature for X(5) behavior are the following [19]. (a) the energies of the yrast states E(J1+), should show characteristic ratios between those of a vibrator and a rotor; (b) the strength of transitions between yrast states as reflected in the B(E2;JJ-2) values should increase with angular momentum at a rate between the values for a vibrator and rotor; (c) the position of the first excited collective 02+ is 5.67 times the energy of 21+; (d) the
B(E2;JJ-2) values for intrasequence transitions should be

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lower for the nonyrast sequence relative to those of the yrast sequence.
The aim of this paper is to know how the shape changes in

104-110Mo isotopes by using the E-Gamma Over Spin (E-GOS) curve [20], and the energy level ratios. Then determine the energy levels and electromagnetic transition strength for the above isotopes using the proton-neutron IBM-2.

2 THEORETICAL FRAMEWORK

IBM Hamiltonian takes different forms depending on the applicable regions U(5), SU(3), and O(6) of the traditional IBM triangle. The Hamiltonian that is considered is in the form [21]

( ( (1)

Where is the standard Hamiltonian of the IBM
(2)
In the Hamiltonian, and terms the characteristics of U(5) and O(6) structure respectively.
In the IBM-2 model, the degrees of freedom of neutrons and protons are explicitly taken into a count. Thus the Hamiltonian [22] can be written as:
(3)
Where ( is the neutron (proton) d-boson number
operator
, ( (4)
Where and represent the s- and d-boson
creation and annihilation operators. The rest of the operators in equation (3) are defined as:
( ) (
(( ( ( ( ( ; (5)
And
( ( ( ( ( ( (

( (6)

In this case, affects only the position of non-fully
symmetric states relative to symmetric state. For this reason is referred to as the Majorana force [22].
The electric quadrupole (E2) transitions are one of the most important factors within the collective nuclear structure. In IBM-2 model, the general linear E2 operator is expressed as [2]:
( ( ( (7)
Where
( is an absolute transition probability of the electric
quadrupole (E2) transition.
and are the proton and neutron effective charges respectively .

3 RESULTS AND DISCUSSION

Energy Gamma Over Spin (E-Gos) is a new modern and active method used to determine the symmetry limit of nuclei. This method is employed in this study through plotting a curve between E/J or R and angular momentum J for the three limits U(5), SU(3) and O(6) and also with the critical point symmetry X(5) as shown in Fig. 1. It is shown that the considered isotopes lie in the transitional region U(5) to SU(3) and may be analogues to the X(5) limit feature.
The energy level ratios R4/2= (E41+) / (E21+) are characteristics
of different collective motions of the nucleus [18]. Fig. 2 shows R4/2 ratios as a function of neutron number changing from 62 to 68. A harmonic vibrator should have R4/2 =2.00, an axial symmetric rotor has R4/2 =3.33, -unstable has R4/2 =2.5, while X(5) behavior should have R4/2=2.91. The searched nuclei in the present study have 2.5 R4/2 3.33. From these ratios, we deduced that the considered isotopes have an X(5) critical point symmetry features.
The 104-110Mo isotopes have Nπ=4 and Nvaries from 6 to 9. Whereas the parameters K, Kπ and CLwith L=0, 2 were treated as free parameter and their values were estimated by fitting with the measured energies. This procedure was made by traditional values of parameters and then allowing one parameter to vary keeping the others constant until the best fit was obtained. Table1 shows the most appropriate Hamiltonian parameters of calculations for examining

104-110Mo nuclei.

Fig. 3, shows that how the strength of the quadrupole parameter K decreases with increasing neutron number.
In Table 2, we present the calculated data and available
experimental ones for 104-110Mo nuclei. In addition, Fig.4 shows that the energy ratio of yrast J+ states and yrast 2+ state as a function of angular momentum J for 104-110Mo. 104Mo and 110Mo not only shows a behavior very close to X(5) predictions, but there also seems to be a trend from vibrational to rotational behavior, suggesting the possibility of a phase transition; furthermore, the 106Mo and 108Mo nuclei are analogous the X(5)
prediction.

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The reduced transition strength ( which is normalized to their respective ( values are
calculated and compared to the experimental data, and with
X(5), and are presented in Table 3.
The proton and neutron effective charges and where determined using the experimental reduced transition
probability ( for each isotope except the 110Mo
nucleus because it has no available experimental B(E2) data.
In Fig.5 it is seen that the relative calculated B(E2) values for 104Mo and 106Mo are very close to SU(3) symmetry, while
the relative experimental B(E2) values are far from it.

Table 2.The calculated and experimental energy values of

104-110Mo nuclei with R4/2 ratios. The experiments are taken from Refs. [23, 24, 25, 26]

Table.1. The most appropriate IBM-2 Hamiltonian

parameters (all in MeV), except 𝑒𝜈 and 𝑒𝜋 in e.b unit for

104-110Mo isotopes

Table 3. The calculated and experimental B (E2; JJ-2)

normalized to their respective ( values for

104-110Mo are compared with X(5) limit [27]. The

experiments are taken from Refs. [18,25,26]

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Fig.1 E-Gos energy plot for three limiting cases, U(5),


SU(3) and O(6) with X(5) behavior and 104-110Mo isotopes
Fig.2. The R4/2 (E41+/ E21+) ratios as a function of neutron
number changing from 62 to 68

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Fig.3. The quadrupole strength K versus neutron numbers

Fig.4. The energies of the yrast sequences (normalized to

the energy of their respectively 21+ levels) in 104-110Mo nuclei

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Fig.5. The reduced transition strength (

which is normalized to their respective (
values versus angular momentum J

4 CONCLUSITION

It is shown that from E(J+)/E(21+) and E-Gos curve values in the yrast bands of 104Mo,106Mo,108Mo and 110Mo they follow a transitional region between U(5) and SU(3), and close to the critical point symmetry X(5) prediction. The energy levels and reduced transition probability values of the ground state band of even-even 104-110Mo are calculated by interacting boson model IBM-2 and compared them with the available experimental data and with X(5) limit.
The relative calculated B(E2) values for 104Mo and 106Mo are
very close to the SU(3) symmetry and follow X(5) at low angular momentum J, while the relative experimental B(E2) values are far away from a rotational interpretation. Our investigations suggest that future study should focus on more detailed measurements of the excited states in Mo isotopes to get detailed information on states Beta and gamma bands.

REFERENCES

[1] F.Iachello, Int.,J. Mod. Phys. B20, 2687-2694 (2006)
[2] F.Iachello and A. Arima, The Interacting Boson Model, Combridge University Press, Combridge, (1987).
[3] D.Bonatsos, Shape/Phase transitions and the critical point symmetries in atomic nuclei arXiv: 0807.4492v1, 28 Jul (2008) [4] J.N.Ginocchio and M.W.Kirson, Phys.Rev.Lett. 44,1744-
1747 (1980)
[5] J.N.Ginocchio and M.W.Kirson, Nucl.Phys.A350,31-60 (1980)
[6] R.F.Casten, Nuclear structure from a simple perspective,
Oxford University Press, Oxford, (1990)
[7] F.Iachello, Phys.Rev.Lett.,87, 052503, (2001) [8] F.Iachello, Phys.Rev.Lett.,85, 3580, (2000)
[9] R.F.Casten and N.V.Zamfier, Phys.Rev.Lett.,87, 052502, (2001)
[10] R.Kruken, et al Phys.Rev.Lett.,88, 232501, (2002)
[11] D.Tonev et al, Phys.Rev.C69,034334, (2004) [12] C.Hutter et al, Phys.Rev.C67,054315, (2003)
[13] E.A.Mccutchan et al, Phys. Rev. C69, 024308, (2004) [14] E.A.Mccutchan et al, Phys. Rev. C71, 024309, (2005) [15]A.Dewald et al, J.Phys.G31,S1427-S1432, (2005)
[16] Zhang Jin et al, mixed symmetry states in even-even 96-
108Mo, comment theory (Beijing, China), 37, 335-340 (2002)
[17] A.Bohr, quadrupole degree of freedom for the nuclear shape, Mat, Fys.Medd.K.Dan.Vidensk.Selsk. 26,1-5, 1952 [18] R.M.Clark, M.Cromas, et al, Searching for X(5) behavior in nuclei, Phys.Rev. C68,037301 (2003)
[19] Sait Inan, Nurettin Turkan, IIyas Inci, devut Olgun, Comparison of the IBM-2 with X(5) critical point summetry for low lying states in 144-154Nd, mathematical and Computational application, 13, 101-112, 2008
[20] P.H.Regan et al, Signature for vibrational to rotational evolution along the Yrast line, Phys.Rev.Lett. 90, 152502, (2003)
[21] K.Hayde,P.van Isckar, M.Waroquier, J.Moreau, Triaxial
shapes in interacting boson model, Phys.Rev. C29,1420-1427, (1984)
[22] O.Scholten, Program package PHINT,KVI Internal
Report,63,(1990)
[23] J.Tuli, Nuclear Data Sheets, 108, 2035 (2007)
[24] D.DE. Frenne, and A.Negret, Nuclear Data Sheets, 109,
943, (2008)

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[25] Jean Blachot, Nuclear Data Sheets, 91, 135, (2000)
[26] D.DE.Frenne, and E.Jacobs, Nuclear Data Sheets, 89, 481, (2000)
[27] D.Bonatsos et al, E(5) and X(5) critical point symmetries
obtained from Davidson potentials through a variational procedure, arXive, Nucl Th.0402088v1, (2004)
.

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