Abstract
In this paper, the nuclear structure of ground-state bands of even-even 104-122Cd isotopes has been reviewed under the framework of interacting boson model-1 (IBM-1). The theoretical energy levels for Z = 48, even N = 56–74 up to 8+ state have been obtained by using the PHINT computer program. The values of the parameters in the IBM-1 Hamiltonian yield the best fit to the experimental energy spectrum. The results are compared with the most recent experimental values where an acceptable degree of agreement is achieved. In this paper, the properties of even 104-122Cd isotopes have been considered to the U(5) ~ SO(6) transitional region of the IBM-1.
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Review
Introduction
Atomic nuclei are known to exhibit changes of their energy levels and electromagnetic transition rates among them when the numbers of protons (or neutrons) are modified, resulting in the shape transition from one kind of collective behavior to another [1–3]. The even-mass cadmium isotopes have been extensively investigated both theoretically and experimentally in recent years with special emphasis on interpreting experimental data via collective models [4]. The collective character in medium-mass nuclei has been successfully described by Arima and Lachello using the interacting boson model-1 (IBM-1) [5]. There is no distinction of IBM-1 for proton and neutron degree of freedom. Recently, many researches on the structure of electromagnetic transition properties and energy levels of doubly even Cd isotopes have been investigated [6–10]. Cadmium isotopes are good examples of quadrupole vibration nuclei [7, 11–13]. However, during the last few years, new experimental data and calculations have led to a modified picture on these nuclei. These descriptions with particular emphasis on describing the experimental data via collective models suggest these nuclei to be soft with regard to the γ deformation with an almost maximum effectivity trivially of γ = 30° [14]. The even-even cadmium isotopes are part of an interesting region near the closed proton shell at Z = 50, while the number of neutrons in the open shell is larger, as such these nuclei have been commonly considered to exhibit vibration-like properties [4].
Longi et al. described the low-lying levels and high-spin states in 116,118,120Cd the framework of interacting boson model [14]. Morrison and Smith described four separate constant boson interactions to explain the structure of 108-116Cd for states with L up to six [15, 16]. Using in-beam and off-beam γ-ray and conversion electron spectroscopy, the low-lying (low-spin) collective states in even-even 106-112Cd and 116Cd were investigated [17]. Octupole excitations in vibration nuclei with mass between 98 and 150 were studied by Pignanelli et al. and Kathryn et al. [18, 19] using the interacting boson model. Recently, we have investigated the energy level of even-even 104-122Cd isotopes under the framework of IBM-1 [20, 21]. Electromagnetic reduced transition properties of the ground-state band of even 102-106Pd and 104-112Cd isotopes were studied by means of IBM-1 [22, 23]. At present, the review on interacting boson model is used to predict the ground-state energy bands up to 8+ in doubly even isotopes 104-122Cd and to compare them with experimental results. Furthermore, even 104-122Cd isotopes in U(5)-O(6) transitional symmetry would be investigated and studied systematically, as well as the ratios of those nuclei in the ground-state band.
Energy levels
The IBM-1 provides theoretical level energies while including anharmonicities from residual interactions. The vibrational model uses a geometric approach, the IBM employs a severely truncated model space, and as such, calculations are possible for nuclei with N nucleons, providing a quantitative mechanism to compare experimental results and calculated values [21]. In the first approximation of IBM-1, only pairs with angular momentum L = 0 (called S bosons) and L = 2 (called d bosons) are considered [24]. The energy levels are calculated using the computer program PHINT [25].
The Hamiltonian of the IBM-1 [26] is given as
where ϵ is the intrinsic boson energy and V ij is the interaction between bosons i and j. The multipole form of the Hamiltonian [6] is given as
The operator gives the number of d boson, p is the pairing operator for the S and d bosons, J is the angular momentum operator, Q is the quadrupole operator, and T3 and T4 are the octupole and hexadecapole operators, respectively. And the interaction parameters in the PHINT program are given below:
The Hamiltonian as given in Equation 2 tends to reduce to three limits, the vibration U(5), γ-soft O(6), and the rotational SU(3) nuclei, starting with the unitary group U(6) and finishing with group O(2) [27]. In the U(5) limit, the effective parameter is ϵ; in the γ-soft limit O(6), the effective parameter is the pairing a0; and in the SU(3) limit, the effective parameter is the quadrupole a2.
The eigenvalues for the three limits are given as follows [28]:
K 1 , K 2 , K 3 , K 4 , and K 5 are other forms of strength parameters. Many nuclei have a transition property between two or three of the above limits, and their eigenvalues for the yrast line are given [28] by the following:
Results and discussion
Boson numbers
A simple correlation exists between the nuclei showing identical spectra and their valence neutron proton (Np) and neutron number (Nn). The identification of such a correlation scheme provided the clue to understand the identical band phenomena. It is natural to assume that the nuclei with equal total boson number Nb = Np + Nn should have the same deformation and identical spectra. The number of valance proton Np and neutron Nn has a total N = (Np + Nn)/2 = n π + n v bosons. The boson number and calculated parameters (in keV) for 104-122Cd even-even isotopes are presented in Table 1. Experimental and IBM-1 calculations of ground-state energy band (in keV) for 104-122Cd are presented in Table 2.
The R4/2 classification
Collective dynamics of energies in even-even nuclei are grouped into classes; within each class, the ratio of excitation energies of the first 4+ and the first 2+ excited states is . A harmonic vibrator has , an axially symmetric rotor should have , while X(5) behavior should have . The variation of values as a function of even neutron numbers of cadmium isotopes for experimental values, IBM-I, U(5), O(6), and SU(3) limits are presented in Figure 1. We identify U(5)-O(6) transitional symmetry in even-even nuclei with atomic number Z = 48 and even neutron number N = 56–74, and most of them are near the U(5) symmetry.
In Figure 2, we present energies of the yrast sequences of ground-state band using IBM-1 (normalized to the energy of their respective levels) in these nuclei and have compared them with previous experimental values [29–39]. We have compared ratios in the ground-state band (a usually adopted measure of nuclear collectivity) as a function of even neutron number N = 56–74. From Figures 1 and 2, we can see that the IBM-1 calculation fits the U(5)-O(6) predictions generally. However, we find that the R L values are consistently smaller in the IBM calculations than in the experimental values.
In order to achieve the low-lying energy levels of Cd isotopes in the mass region A = 104–122 by IBM-1, it is important to specify the shape symmetry of a nucleus, which can be predicted from the energy ratio , where is the energy level at and is the energy level at . R has a limit value of 2 for the vibration nuclei U(5), 2.5 for the γ-unstable nuclei O(6), and finally 3.33 for rotational nuclei SU(3).
IBM-1 calculated the energy of different states (i.e., 0+, 2+, 4+, 6+, 8+) for doubly even 104-122Cd isotopes. Considering one parameter to vary and keeping constant the others until an overall best fit was archived for the ground-state bands of doubly even 104-122Cd isotopes. A suitable free parameter was determined to find the close excitation energy of all positive parity levels (2+, 4+, 6+, 8+) for which a good indication of the spin value exists [20, 21]. Each nucleus at the evolving states is determined using Equation3. Table 1 shows the values of these parameters that were used to calculate the energy of the yrast states for the isotopes Z = 48 and N = 66–76 under this study. The energy level fits with IBM-1 are presented in Table 2. During the calculation, PHINT code [25] was used and we made a comparison with experimental levels [29–39]. The agreement between the calculated theory and experiment is excellent and reproduced well. The values of the first excited state and the ratio show that 104-122Cd isotopes are vibration nuclei.
Conclusions
We have reviewed the ground-state energy band for even-even 104-122Cd isotopes within the framework of interacting boson model approximations. The energy levels 0+, 2+, 4+, 6+, and 8+ of even 104-124Cd nuclei were obtained by the best fitted values of the parameters in the Hamiltonian of the IBM-1. The validity of the presented parameters in IBM-1 formulations has been investigated, and it is seen that there is an existence of a satisfactory agreement between IBM-1 results and previous experimental data [29–39]. We have concluded that the general characteristics of the Cd isotopes are well accounted in this review, and the idea of shape coexistence in this region is supported. It is established that the framework of interacting boson approximations showed that the Cd isotopes under study are considered as vibration nuclei and close to the U(5) ~ SO(6) transitional region of IBM-1.
Authors’ information
IH is working as a professor at the Department of Physics, Rabigh College of Sciences and Arts, King Abdulaziz University. He received his D.Sc. in Energy Sciences at the Tokyo Institute of Technology in 1999. His research interests are theoretical and experimental nuclear structure and applied radiation physics. HYA is an assistant professor of the Department of Physics, College of Education, Salahaddin University. His research interest is theoretical nuclear structure. IMA is a professor at the Department of Physics, College of Education, Mosul University. He is an expert in the field of nuclear structure.
References
Iahello F: Analytic description of critical point nuclei in a spherical-axially deformed shape phase transition. Phys. Rev. Lett. 2001, 87: 052502-(1–5).
Iahello F: Dynamic symmetries at the critical point. Phys. Rev. Lett. 2000, 85: 3580-3583.
Cejnar P, Jolie J, Casten RF: Quantum phase transitions in the shapes of atomic nuclei. Rev. Mod. Phys. 2010, 82: 2155-2212.
Ahmad ST, Kakil DO: The prediction of the electromagnetic properties and δ(E2/M1) of 110-116Cd isotopes in IBM model. Baghdad Sci. J. 2011,8(3):772-780.
Aritma A, Iachello F: The Interacting Boson Model. Cambridge: Cambridge University Press; 1987.
Kumar R, Sharma S, Gupta JB: Character of quasi-band in 150Sm using IBM. Armenian J. Phys. 2010,3(3):150-154.
Aprahamian A, Brenner DS, Casten RF, Gill RL, Piotrowski A, Heyde K: Observation of 0+ states in 118Cd and systematic of intruder states. Phys. Lett. B 1984, 140: 22-28.
Deleze M, Drissi S, Kem S, Vorlet JP: Systematic study of the mixed ground state and intruder bands in 110-114Cd. Nucl. Phys. A 1993, 551: 269-294.
Piiparinen M, Julin R, Juutinen S, Virtanen A, Ahonen P, Fahlander C, Hattula J, Lampinen A, Lonnroth T, Maj A, Mitarai S, Muller D, Nyberg J, Pakkanen A, Sugawara M, Thorslund I, Tormanen S: Lifetimes of yrast states in 110Cd. Nucl. Phys. 1993, A565: 671-686.
Abdulla HY, Hossain I, Ahmed IM, Ahmed ST, Karwan WQ, Kasimin MK, Chong JC, Viswanathan KK, Ibrahim N: Calculation of 8+ isomers of even-even nuclei 76Ni to 94Pd for N = 48 neutrons. Int J. Phys. Sci. 2011, 6: 901-906.
Meyer M, Daniere J, Latessier J, Quentin P: A description of odd neutron-deficient cadmium isotopes within the rotor-plus-particle model using self-consistent core single-particle states. Nucl. Phys. A 1979, 316: 93-115.
Bentley I, Frauendorf S: Microscopic calculation of interacting boson model parameters by potential-energy surface mapping. Phys. Rev. C 2011, 83: 064322-(1–10).
Giannatiempo A, Nannini A, Perego A, Sona P: 0 2 + and 0 3 + states in 110Cd. Phys. Rev. C 1990, 41: 1167-1171.
Long GL, Zhut SJ, Sun HZ: Description of 116,118,120Cd in the interacting boson model. J. Phys. G; Nucl. Part. Phys. 1995, 21: 331-338.
Boelaert N, Smirnova N, Heyde K: Shell model description of the low-lying states of the neutron deficient Cd isotopes. Phys. Rev. C 2007, 75: 014316-(1–6).
Morrison I, Smith R: The interacting boson approximation and spectroscopy of even cadmium and tin isotopes. Nucl. Phys. A 1980, 350: 89-108.
Kumpulainen J, Julin R, Kantele J, Passoja A, Trzaska WH, Verho E, Vaaramaki J: Systematic study of low-spin states in even Cd nuclei. Phys. Rev. C 1992, 45: 640-661.
Pignanelli M, Blasi N, Micheletti S, Hofstee MA, Schippers JM: Octupole excitations in vibration nuclei and the sdf interacting boson model. Nuclear Physics A 1990,519(3):567-601.
Green KL: Nuclear structure of 112Cd through studies of β decay. M.Sc. thesis. The University of Green; 2009.
Abdullah HY, Hossain I, Ahmed IM, Ahmad ST, Saeed MA, Ibrahim N: Investigation of energy level of even-even 104-112Cd isotopes under the framework of IBM-1. Armenian J. Phys. 2012,5(2):58-61.
Hossain I, Abdullah HY, Ahmed IM, Saeed MA, Ahmad ST: Study on ground state energy band of even 114-124Cd isomers under the framework of interacting boson model (IBM-1). IJMPE 2012, 21: 1250072-(1–6).
Hossain I, Saeed MA, Ghani NNAMB, Sa'adeh H, Hussein M, Abdullah HY: Electromagnetic reduced transition properties of the ground state band of even-even 102-106Pd isotopes by means of interacting boson model. Indian J Phys 2013., 87: doi:10.1007/s12648-013-0374-5
Abdullah HY, Hossain I, Ahmed IM, Ahmad ST, Kasmin MK, Saeed MA, Ibrahim N: Electromagnetic reduced transition properties of the ground state band of even-even 102-106Pd isotopes by means of interacting boson model-I. Indian J. Phys. 2013, 87: 571-575.
Eisenberg JM, Greiner W: Nuclear Theory vol. 1, Nuclear Models, 3rd end. North Holland Physics Pub1. Amsterdan: Elsevier; 1985.
Scholten O: The program package PHINT. Kernfysischversneller Institute Groniger KVI-63. 1979. https://www.kvi.nl/~scholten
Scholten O, Lachello F, Arima A: Interacting boson model of collective nuclear states. The transition from SU(5) to SU(3). Annl. Phys. 1978, 115: 325-366.
Lachello F: Nuclear Structure. Edited by: Abraham K, Allaart K, Dieperink AEL. New York: Plenum; 1981.
Burrows TW, McLaughlin PK, Nichols AL: Nuclear Structure and Decay Data (NSDD) Workshop. Italy: ICTP Triste; 2005.
Blachot J: Nuclear data sheets for A = 104. Nucl. Data Sheets 2007, 108: 2035-2172.
De Frenne D, Negret A: Nuclear data sheets for A = 106. Nucl. Data Sheets 2008, 109: 943-1102.
Blachot J: Nuclear data sheets for A = 108. Nucl. Data Sheets 2000, 91: 135-296.
Gurdal G, Kondev FG: Nuclear data sheets for A = 110. Nucl. Data Sheets 2012, 113: 1315-1561.
De Frenne D, Jacobs E: Nuclear data sheets for A = 112. Nucl. Data Sheets 1996, 79: 639-758.
Blachot J: Nuclear data sheets for A = 114. Nucl. Data Sheets 2002, 97: 593-763.
Blachot J: Nuclear data sheets for A = 116. Nucl. Data Sheets 2010, 111: 717-895.
Kitao K: Nuclear data sheets update for A =118. Nucl. Data Sheets 1995, 75: 99-198.
Kitao K: Nuclear data sheets for A =120. Nucl. Data Sheets 2002, 96: 241-390.
Tamura T: Nuclear data sheets for A = 122. Nucl. Data Sheets 2002, 108: 455-632.
Turkan N, Maras I: Search on results of IBM for region between 120 ≤ A ≤ 150: 120-128Te and 122-134Xe nuclei. Math. and Com. App. 2011, 16: 467-476.
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The authors thank King Abdulaziz University for providing the fund to complete the projects.
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IH carried out the confirmation of the calculation and writing of the paper. HYA collected all papers included in the sequence alignment and drafted the manuscript. IMA participated in the sequence alignment. MAS participated in the IBM-1 calculation. All authors read and approved the final manuscript.
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Hossain, I., Abdullah, H.Y., Ahmed, I.M. et al. Ground-state energy band of even 104-122Cd isotopes under the framework of interacting boson model-1: a review. J Theor Appl Phys 7, 46 (2013). https://doi.org/10.1186/2251-7235-7-46
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DOI: https://doi.org/10.1186/2251-7235-7-46