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Geometric Structure Features in Se Isotopes -.:: Natural Sciences J. Rad. Nucl. Appl. 5, No. 2, 147-152 (2020) 147 Journal of Radiation and Nuclear Applications An International Journal http://dx.doi.org/10.18576/jrna/050209 Geometric structure features in 72-80Se isotopes Heiyam Najy Hady1* and Mohsin Kadhim Muttalb2 1Department of Physics, Education College for girls, University of Kufa, Kufa, Iraq. 2Department of Physics, Science College, University of Babylon, Iraq. Received: 21 Feb. 2020, Revised: 22 Mar. 2020, Accepted: 24 Mar. 2020. Published online: 1 May. 2020. Abstract: The geometric properties of the interacting boson model, interacting boson model-1and interacting boson model-2 have been used to perform studying of 72-80Se isotopes. The low –lying positive party states, energy ratios and the potential energy surface for 72-80Se have been investigated by using software package 퐼퐵푀푃 푎푛푑 퐼퐵푀 computer code for 퐼퐵푀 − 1 and Neutron Proton Boson 푁푃퐵푂푆 software package for 퐼퐵푀 − 2. 푈(5) are the dominant in 72−80푆푒 with addition of a small effect of rotational parameters started from 74Se to 80Se isotopes, energy ratios in which 72푆푒 isotopes as the nearest isotopes to typical vibrational limit while 74−80푆푒 isotopes tend towards the rotational region lied on 푈(5)- 푆푈(3), leg of “Casten's triangle”. 72푆푒 isotope potential energy curve has one minimum point at 훽 = 0, this is the same as spherical equilibrium shapes (a vibrational-like spectrum). The counter diagram represents as regular concentric, 74−80푆푒 isotopes that have evolved the deformation. Keywords: 퐼퐵푀푃, 퐼퐵푀 − 1, and 퐼퐵푀 − 2. 1 Introduction (푠, 푑푚)operators where the index 푚 = 0, ±1, ±2.The most general Hamiltonian ,which includes on-boson terms in The interacting boson model has associated with it an boson – boson interaction is [1]:- intrinsic geometric structure. The natural space for the 1 퐻 = 휀 (푠†푠) + 휀 ∑ 푑† 푑 + ∑ (2퐿 + geometric properties of system with group structure is the 푠 푑 푚 푚 푚 퐿=0,2,4 2 1 so called cost space.[1] Imagining a liquid drop vibrational ( ) (0) 1 ( ) ( ) 1)2 퐶 [(푑†푑†) 퐿 . (푑푑)퐿] + υ [(푑†푑†) 2 . (푑푠) 2 + at high frequency have been gotten a good idea of the 퐿 √2 2 (0) 1 physics of nuclear vibration. Although the average shape is (푑†푠†)(2). (푑푑)(2)] + υ [(푑†푑†)(0). (푠푠)(0) + 2 0 spherical. Rotational motion can be observed only in nuclei ( ) ( ) † † (0) (0) 0 † † (2) 2 0 with no spherical equilibrium shape. These nuclei can have (푠 푠 ) . (푑푑) ] + u2[(푑 푠 ) . (푑푠) ] + 1 (0) substantial distortions from spherical shape and are called 푢 [(푠†푠†)(0). (푠푠)(2)] (1) deformed nuclei[2]. There are many studies and research 2 0 that attempted to understand and explain the behavior of 72- Where 휀푑, 휀푠 are 푠 and 푑 bosons energy and 푉 represent the 80Se nuclei by using different models[3-8] in present study boson-boson interacting energy. The most commonly used the relation between geometric structure and nuclear form of 퐼퐵푀1 Hamiltonian is [9,10]:- † structure has been investigated. The 72-80Se isotopes have 퐻 = 휀푛푑 + 푎0푃 푃 + 푎1퐿. 퐿 + 푎2푄. 푄 + 푎3푇3푇3 Z=34 then, 3 particle bosons the number of protons and +푎4푇4푇4 (2) neutrons are lying between 28 and 50 magic shells, 72Se has Where 휀 = 휀푑 − 휀푠 is the boson energy,for simplicity 휀푠 38 neutrons that mean 5 particle neutron bosons. While 74- was set equal to zero only 휀 = 휀푑 appears, 푎0, 푎1, 푎2, 푎3, 푎4 80Se have 40-46 neutrons indicating 5-2 hole neutron designate the strengths of the quadrupole, angular momentum, pairing, octupole, and hexadecapole interacting bosons respectively. between bosons respectively. The five component of 푑 2 Theoretical Parts boson and the single component of the 푠 boson extended across a six dimensional space. For a fixed number of In the 퐼퐵푀 − 1, it is assumed that the Hamiltonian operator boson 푁 the group structure of the problem is 푈(6). contains only one body and two body terms thus, Considering the different reductions of 푈(6), three † † dynamical symmetries emerge, namely 푈(5), 푆푈(3) and introducing creation (푠 , 푑푚) and annihilation * Corresponding author e-mail: [email protected] © 2020 NSP Natural Sciences Publishing Cor. 148 H. N. Hady and M. K. Muttalb.: Geometric structure features … 푂(6); these symmetries are related to the geometrical idea interacting boson model-1and interacting boson model-2 of the spherical vibrator, deformed rotor and a symmetric( have been used to perform studying of 72-80Se series of 훾 −soft) deformed rotor, respectively [8-12]. The general isotopes, which that medium mass with (Z=34). The formula for the potential energy surface as a function of software package IBM, IBMP computer code for IBM − 1 geometrical variables β and γ is given by [1, 9,11]:- and Neutron Proton Boson NPBOS software package have N(ε +ε β2) N(N+1) V(β, γ) = s d + (α β4 + α β3Cos3γ + been used by estimating set of parameters described in the 1+β2 (1+β2)2 1 2 Hamiltonian operator as they are shown in equations (2) 2 α3β + α4) (3) and(6). The parameters estimated for the calculations of excited energy levels for 72-80Se isotopes and geometric 퐶0 퐶2 9퐶4 8 interacting boson model are given in table (1). where훼 = + + , 훼 = −√ 휐 , 훼 = 1 10 7 35 2 35 2 3 (휐0+푢2) Table 1: The parameters have been used in the 푰푩푴 − ퟏ, , 훼4 = 푢0 (4) √5 IBMP and 푰푩푴 − ퟐ Hamiltonian for even-even 72-80Se isotopes (in MeV ) except χ , χν and χπ were unit less. where N is the total boson number β is the quadruple deformation parameter operator from β = 0 − 2.4 . γ is the 퐼퐵푀 − 1parameters in MeV unless χ distortion parameter operator or (asymmetry angle) Isotopes 72Se 74Se 76Se 78Se 80Se ° ° for 0 ≤ γ ≤ 60 .The variables α1, α2, α3, α4 are related to N 8 8 7 6 5 the parameters CL, υL, uL which are given in equation (1). ε 0.6002 0.6 0.54 0.6 0.6 ′ The relationships between the variables (α s) and these a0 0.0 0.0 0.0 0.0 0.0 parameters have been expressed by F. Iachello [1] as one a1 0.02 0.02 0.02 0.02 0.02 must take into account the asymmetry angle occurs only in a2 0.0 -0.007 -0.009 -0.038 -0.036 the term cos3γ. Thus, the energy surfaces has minima only a3 0.001 0.0 0.0 0.0 0.0 at γ = 0°and 60°. These expressions give at large N, a4 0.001 0.0 0.0 0.0 0.0 βmin = 0, √2, 1 for U(5), SU(3),and O(6) respectively. The χ 0.0 -1 -0.86 -1 -1 Hamiltonian operator in 퐼퐵푀 − 2 has been given 퐼퐵푀 − 2 Parameters in MeV unless χ , χπ= -0. 74 , Nπ =3 by[1,9,13]:- Nν 5 5 4 3 2 퐻 = 퐻 + 퐻 + 푉 (5) 휋 휈 휋휈 εd 0.99 0.88 0.78 0.78 0.8 A simple schematic Hamiltonian guided by microscopic 휅 -0.076 -0.077 -0.086 -0.076 -0.16 consideration is given by[14] 휒휈 0.88 0.82 0.84 0.84 -0.6 퐻 = 휀(푛푑휋 + 푛푑휈) + 휅푄휋. 푄휈 + 푉휋휋 + 푉휈휈 + 푀휋휈 (6) 휁2 0.002 0.012 0.01 0.03 0.001 where † † 2 † 2 휁1,3 -0.01 -0.02 -0.02 -0.016 -0.01 푄휌 = (푑휌푠휌 + 푠휌 푑휌)휌 + 휒휌(푑휌푑휌)휌 휌 = 휋, 휈 (7) 1 -0.99, -0.9, -0.41, 0.2, 1 휌 † † (0) -0.48, 푉 = ∑ (2퐿 + 1)2퐶 [(푑 푑 )(퐿). (푑 푑 )(퐿)] (8) 퐶 퐿 0.76,0. 0.76,0. 0.86,0. 0.86,0. 휌휌 퐿=0,2,4 2 퐿 휌 휌 휌 휌 휈 44,0.69 12 1 32 5 휀휋, 휀휈 are proton and neutron energy respectively, they assumed to be equal 휀 = 휀 = 휀 the last term in Eq. (5) 0.2, 0.2, 0.72, 휋 휈 퐿 0.2,0.86, 0.22, contains the Majorana operator 푀 and it is usually added 퐶휋 0.76 0.76,0. 21,0.69 휋휈 0.22 0.9,0.5 in order to remove states of mixed proton neutron ,0.12 09 symmetry ,this term can be written as [15] The geometric interacting boson model parameters in MeV εs 0 -0.035 -0.045 -0.19 -0.18 † † † † 푀 = 휁 (푠 푑 − 푑 푠 )(2). (푠 푑 − 푑 푠 )(2) + 휋휈 2 휈 휋 휈 휋 휈 휋 휈 휋 εd 0.723 0.706 0.644 0.682 0.684 † † (푘) (푘) ∑푘=1,3 휁푘(푑 푑휋) − (푑휈푑휋) (9) 휈 α1 0.001 -0.002 -0.002 0 0 0 -0.015 -0.017 0.008 -0.008 it is possible to obtain spectra which are similar to those of α2 the 퐼퐵푀 − 1with only one kind of boson[15]. The 푈(5) α3 0 -0.028 -0.036 -0.152 -0.144 limit when ≫ 휅 , 푆푈(3) limit when 휀 ≪ 휅 and 휒 = 휒 = 휋 휈 α4 0 0 0 0 0 −√7⁄2, and 푂(6) limit when 휀 ≪ 휅 and 휒휈 = −휒휋 . Most + + + + nuclei do not strictly belong to any of these three limiting Calculated energy ratios of (E41 /E21 ), (E61 /E21 ) and + + cases, but are somewhere between two of them. In the 퐼퐵푀, (퐄ퟖퟏ /퐄ퟐퟏ ) for 72-80Se isotopes have been indicated in it is possible to make a smooth transition between the figure (1) as a function of mass numbers.
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