DOCSLIB.ORG

The Interacting Boson Approximation Model and Applications

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
The Interacting Boson Approximation Model and Applications -----[ RIJKSUNIVERSITEIT TE GRONINGEN THE INTERACTING BOSON APPROXIMATION MODEL AND APPLICATIONS PROEFSCHRIFT TER VERKRIJGING VAN RET DOCTORAAT IN_DE WISKUNDE EN NATUURWETENSCHAPPEN AAN DE RIJKSUNIVERSITEIT TE GRONINGEN OP GEZAG VAN DE RECTOR MAGNTFICUS DR. J. BORGMAN IN BET OPENBAAR TE VERDEDIGEN OP MAANDAG 19 MEL 1980 DES NAMIDDAGS TE 4.00 UUR DOOR OLAF SCHOLTEN GEBOREN TE MADlUN (INDONESIE) 1980 Drukkerij Dijkstra Niemeyer bv, Groningen PROMOTOR PROF, DR, F, IACHELLO CONTENTS CO-REFERENTEN: DR, A,E,L, DIEPERINK CHAPTER I INTRODUCTION PROF, DR, A, LANDE References I 3 CHAPTER II THE INTERACTING BOSON MODEL 2. I. Introduction 5 2.2. Outline of the model 6 2.2.1. The Hamiltonian 6 2.2.2. Other operators 7 2.3. Dynamical symmetries 9 2.3. I. The SU(5) limit 10 2.3.2. The SUO) limit 15 2.3.3. The 0(6) limit 21 2.4. Application to the Samarium isotopes 25 ·2.4. I. Energies 26 2.4.2. E2 and E4 transitions 31 2.4.3. EO transitions and radii 34 2.4.4. Two-nucleon transfer 37 2.5. Summary 39 References II 39 CHAPTER III THE NEUTRON-PROTON INTERACTING BOSON HODEL 3.1. Introduction 41 3.2. The microscopic formulation 42 3.2.1. The s- and d-bosans 42 3.2.2. The boson Hamiltonian 45 3.2.3. Predicted N-dependence of the parameters 50 This work was performed as part of the research program of the 3.3. Calculations on the Xe, Ba and Ce isotopes 53 IlS t ichting voar Fundamenteel Onderzoek der Materie,r (FOM), which 3.3.1. Excitation energies 54 is financially supported by the "Nederlandse Organisatie voor 3.3.2. E2 transitions 58 Zuiver Wetenschappelijk Onderzoek" (ZWO). 3.3.3. Other properties 62 3.4. Summary 65 References III 67 CHAPTER IV THE RELATION BETWEEN IBA-I AND IBA-II 4.1. Introduction 69 4.2. F-spin 69 4.3. Projection 71 CHAPTER VII Sillll1ARY AND CONCLUSIONS 119 4.4. Application to the Nd, Sm and Gd 73 References VII 122 4.5. Conclusions 77 124 References IV 78 APPENDIX A 129 CHAPTER V ODD-A NUCLEI -FORHALISH ACKNOWLEDGEMENTS 5.1. Introduction 79 SAlIENVATTING 130 5.2. The single-j shell 80 5.2.1. Definition of the basis states 80 5.2.2. The pseudo-particle operators 82 ,5.2.3. The particle-boson interaction 83 5.2.4. Projection 85 5.3. The non-degenerate major shell 86 5.3.1. Definition of the basis states 86 5.3.2. The pseudo-particle operators 89 5.3.3. The particle-boson interaction 91 5.4. Conclusions 93 References V 94 CHAPTER VI CALCULATIONS FOR ODD-A NUCLEI 6.1. Introduction 95 6.2. Special solutions 95 6.2.1. The SU(5) limit 96 6.2.2. The SU(3) limit 99 6.2.3. The 0(6) limit 105 6.2.4. Dynamical symmetries 109 6.3. The Europium isotopes 113 6.3.1. Excitation energies 114 6.3.2. Binding energies 116 6.4. Summary 117 References VI 117 ------- , CHAPTER I I NTRODUCTI ON The spectroscopy of medium mass and heavy even-even nuclei is characterized by the occurrence of low-lying collective states. The study of the nuclear collective motion is one of the most interesting topics in nuclear physics. The basis for this was laid by Rainwater}) 2 and Bohr and Mottelson ,3). In the following we will restrict ourselves to the models that describe the collective motion in terms of bosans. 5 At the level of phenomenology one can regard most boson rnodels ,6) as generalizations of the second quantized version of the Bohr Hamil­ 2 tonian ) in which the collective motion is described in terms of the five quadrupole degrees of freedom ct l-l A connnon problem connected 2 with this formulation is that the boson-boson Hamiltonian cannot be restricted to two-body boson-boson interactions. The importance of high­ er order terms in the interaction is also confirmed by Kishimoto and 7 Tamura ) using the boson expansion technique. In order to solve the problem of infinite expansions Janssen, Jolos and Do"nau 9) introduced a cut-off factor. This is analogous to IO the Holstein-Primakoff ) representation of SU(2). l In a parallel development Arima and Iachello 1) introduced in 1975 the Interacting Boson Approximation (IBA) Model in which the monopole or scalar s-boson appears in addition to the quadrupole d-boson. Their 12 formulation can be regarded as the Schwinger )type representation of SU(6). The IBA-model marks a distinct departure from other boson models models in that the boson number 1S. conserved . Furth ermore 13) , t h at the boson could be regarded as a collective pair of nucleons With the exception of the IBA and of the Janssen, Jolos, Donau models the conventional boson models were successful only for spher­ ical nuclei. The collective states of deformed nuclei were already de­ scribed successfully in terms of the rigid rotor model of Bohr and Mottelson3 ,4). Each model broke down at the other limit. }lany attempts have been made to unify the description of rotational and vibrational nuclei. However, in all boson models, with the exception of the two mentioned above9, 11), attempts to extend them into the rotational region involved introducing more and more higher order terms in the Hamiltonian. 2 3 l4 23) . h . 1e Alternatively Kumar and Baranger , 15) have investigated the Bohr Boson-Fermion Approximation (IEFA) model by coupllug t e SlUg Hamiltonian in first quantization, without recourse to bosons, and nucleon degree of freedom to the system of s- and d-bosans describing have developed a unified geometrical description of the collective the even-even core. It. is shown that the particle-boson interaction motion. Their approach, however, entails extensive calculation for can be derlved. from the neutron-proton quadrupole interaction as used each nucleus. in the description of even-even nuclei. The interaction appears to It is shown by Arima and Iachello that within the framework of depend rather strong1 y on the microscopic structure of the bosons. the IBA-model it is possible to reproduce the properties of three Results of actual calculations in the framework of the IBFA­ different classes of nuclei which, in the conventional nomenclature, model are shown in chapter 6. As an application a calculation of d 'b' 1 16 ) . 11 . d d l 7) correspon to v~ rat~ona ,ax~a y syrnmetr~c eforme and y_ excitation energies for the Europium isotopes is shown. 18) l' . unstab1 e nuc e~. At these l~m~t cases the system may be solved analytically" For intermediate cases it involves numerical diagonal­ ization of a relatively small matrix. We consider this version of REFERENCES I the lBA-model in Some detail in chapter 2 and apply it to the even 1. J. Rainwater. Phys. Rev. 79 (1950) 432. Samarium (Sm) isotopes. In the Sm isotopes a transition from Dan. 'd. Selsk. 26 (1952) no. 14. spherical to deformed has been observed I 9) . 2. A. Bohr, Mat. Fys. Me dd • Vl 21 3. A. Bohr and B.R. Hottelson, Hat. Fys. Medd. Dan. Vid. Selsk. 27 Otsuka, Arima, Iachello and Talmi ) have shown how to relate (1953) no. 16. 20 the IBA-model to the conventional ) shell model. The degrees of free­ 4. A. Bohr and B.R. Mottelson, Nuclear Structure, vol. II dom of the IBA-model are viewed as superpositions of nucleon pairs (W.A. Benjamin, Reading, Massachusetts, 1975). with boson properties, and in ref. 21 predictions are made for the 5. D.M. Brink, A.F.R. de Toledo Piza and A.K. Kerman, Phys. Lett. JJl (1965) 413. dependence of parameters of the IBA-model on neutron and proton number. 6. G. Gneuss and W. Greiner, Nucl.Phys. Al71 (1971) 449, and references In chapter 3 the IBA-model is extended by introducing explicitly therein. neutron and proton bosons. This version (lBA-2) has been applied to 7. T. Kishimoto and T. Tamura, Nucl. Phys. A270 (1977) 317, and many even-even nuclei in different regions of the periodic table, references therein. from the even Krypton up to the even Thorium ~sotopes. 22) . Calculat~ons. 8. S.T. Belyaev and V.G. Zelevinsky, Nucl. Phys. ~ (1962) 582. for the Xe, Ba and Ce region are shown. 9. D. Janssen, R ..V J 0 I as and F . Donau, Nucl. Phys. A224 (1974) 93. The parameters of the IBA-2 model have a much more direct physical 10. T. Holstein and H. Primakoff, Phys. Rev. ~ (1940) 1098. meaning than in the IBA-l model. A drawback of the IBA-2 model is 11. A. Arima and F. Iachello, Phys. Rev. Lett. ~ (1975) 1069. that the model space, and thus the matrix that has to be diagonalized, 12. J. Schwinger, in lIQuantum Theory of Angular Momentum,11 L . B~e. d en- harn and H. van Dam, ed., Academic Press, N.Y. (1965) p. 229. is much larger than in the IBA-I model. Using the method discussed in 13. A. Ar~ma,. T . Otsuka, F. Iachello and 1. Talmi, Phys. Lett. 66B chapter 4 it is possible to relate the parameters of the two versions of the IBA-model, which makes it possible to do the calculations in (1977) 205. 14. K. Kumar, Structure of Medium-Heavy Nuclei, Rhodos lust. Phys. the simple IBA-I model, while still using the same 'microscopic infor­ ConL Ser. '!1 (1979) 169. mation as in IBA-2. This becomes important in the discussion of 15.
Load more

Recommended publications
  • The Interacting Boson Model
    The interacting boson model P. Van Isacker, GANIL, France Dynamical symmetries of the IBM Neutrons, protons and F-spin (IBM-2) T=0 and T=1 bosons: IBM-3 and IBM-4 Symmetries in N~Z nuclei (III), Valencia, September 2003 The interacting boson model • Nuclear collective excitations are described in terms of N s and d bosons. • Spectrum generating algebra for the nucleus is U(6). All physical observables (hamiltonian, transition operators,…) are expressed in terms of the generators of U(6). • Formally, nuclear structure is reduced to solving the problem of N interacting s and d bosons. Symmetries in N~Z nuclei (III), Valencia, September 2003 Justifications for the IBM • Bosons are associated with fermion pairs which approximately satisfy Bose statistics: (0) ( 2) S + = a a+ ¥ a + Æ s+ , D+ = a a+ ¥ a + Æ d +  j ( j j )0 m  jj '( j j' )m m j jj ' • Microscopic justification: The IBM is a truncation and subsequent bosonization of the shell model in terms of S and D pairs. • Macroscopic justification: In the classical limit (N Æ ∞) the expectation value of the IBM hamiltonian between coherent states reduces to a liquid-drop hamiltonian. Symmetries in N~Z nuclei (III), Valencia, September 2003 Algebraic structure of the IBM • The U(6) algebra consists of the generators U(6) = s +s,s+ d ,d + s,d + d , m,m' = -2, ,+2 { m m m m' } K • The harmonic oscillator in 6 dimensions, +2 + + H = ns + nd = s s +  dm dm = C1[U(6)] ≡ N m = -2 • …has U(6) symmetry since "gi ŒU(6) : [H,gi ]= 0 • Can the U(6) symmetry be lifted while preserving the rotational
    [Show full text]
  • EVE Xe ISOTOPES by the FRAMEWORK of IBA Đsmail Maraş , Ramazan Gümüş and Nurettin Türkan Celal
    Mathematical and Computational Applications , Vol. 15, No. 1, pp. 79-88, 2010. © Association for Scientific Research THE IVESTIGATIO OF EVE-EVE 114-120 Xe ISOTOPES BY THE FRAMEWORK OF IBA Đsmail Maraş 1*, Ramazan Gümüş 2 and Nurettin Türkan 3,4 1Celal Bayar University, Faculty of Arts and Science, Manisa, Turkey [email protected] 2Celal Bayar University, Institute of Science, Manisa, Turkey [email protected] 3University of Wisconsin, Department of Physics, 53715 Madison, WI, USA 4Bozok University, Faculty of Arts and Science, 66200 Yozgat, Turkey [email protected] Abstract- In this work, the ground state, quasi beta and quasi gamma band energies of 114,116,118,120 Xe isotopes have been investigated by using the both (IBM-1 and IBM-2) versions of interacting boson model (IBM). In calculations, the theoretical energy levels have been obtained by using PHINT and NP-BOS program codes. The presented results are compared with the experimental data in respective tables and figures. At the end, it was seen that the obtained theoretical results are in good agreement with the experimental data. Key Words- Interacting Boson Model, Even-Even Xe, Band Energies (Ground State, Quasi Beta and Quasi Gamma Band). 1. ITRODUCTIO One of the most remarkable simplicities of atomic nuclei is that the thousands of 2-body nucleonic interactions in a nucleus can be reduced to and simulated by a 1-body potential [1]. This is done with the interacting boson model (IBA) [2], which is a useful model to formalize description of symmetry in nuclei. This model (IBA) has a U(6) group structure leading to sub-groups chains denoted by U(5), SU(3) and O(6), which describe vibrational, axially symmetric rotational and γ-soft rotational nuclei.
    [Show full text]
  • Generalized Interacting Boson Model and the Collective Behaviour in Nuclei
    PramS.a, Vol. 17, No. 5, November 1981, pp. 381-387. © Printed in India Generalized interacting boson model and the collective behaviour in nuclei M SUGUNA, R D RATNA RAJU and V K B KOTA* Theoretical Group, Department of Physics, Andhra University, Waltair 530 003, India *Department of Physics and Astronomy, University of Rochester, Rochester, New York, USA *Permanent Address: Physical Research Laboratory, Ahmedabad 380 009, India MS received 9 May 1981 ; revised 3 October 1981 Abstract. The effect of including the high spin bosons on the manifestation of collec- tive behaviour in nuclei is examined by plotting the B(E2; 2+ ~ 0+) rates as a function of neutron number for various values of )/, where ~ is the highest angular momentum of the bosons included in the calculation. B(E2; 2+ ~ 0 +) values of a large number of nuclei in various regions of the nuclear periodic table are calculated with a single value for the effective charge in the generalized scheme. Irreducible representations of SU(3) contained in the symmetric partition [N] of U(15) are worked out for integers N upto N = 15, to enable the explicit inclusion of the g boson into calculations. The experimentally observed odd-K bands in 284U and 184W are described as a direct conse- quence of the g boson. Keywords. Generalized interacting boson model; irreducible representations; sym- metric partitions; g boson. 1. Introduction The interacting boson model 0~M) of Arima and Iachello (1976, 1978) describes the collective properties of nuclei by considering pairs of protons and neutrons coupled to S = 0 and J -----L = 0, and 2.
    [Show full text]
  • Investigation of Shape Coexistence in Te Isotopes
    Investigation of shape coexistence in 118-128Te isotopes H. Sabria1, Z. Jahangirib, M. A. Mohammadia a Department of Physics, University of Tabriz, Tabriz 51664, Iran. b Physics Department, Payame Noor University, Tehran 19395-4697, Iran. 1 Corresponding Author E-mail: [email protected] 1 Abstract In this paper, we have considered the interplay between phase transitions and configuration mixing of intruder excitations in the 118-128Te isotopes. A transitional interacting boson model Hamiltonian in both IBM-1 and IBM-2 versions which are based on affine SU (1,1) Lie Algebra are employed to describe the evolution from spherical to deformed gamma unstable shapes along the chain of Te isotopes. The excitation energies, B(E0) and B(E2) transition rates are rather well reproduced in comparison with experimental counterparts when the weight of SO(6) limit is increased in Hamiltonian. Also our results show obvious relations between the configuration mixing ratio and quadrupole, hexadecapole and hexacontatetrapole deformation values in this isotopic chain. Keywords: intruder states; Interacting Boson Model (IBM); infinite dimensional algebra; energy levels; B(E0) and B(E2) transition probabilities. PACS: 21.60.Fw; 21.10.Re; 27.60.+j 1. Introduction Shape coexistence has been observed in many mass regions throughout the nuclear chart and has become a very useful paradigm to explain the competition between the monopole part of the nuclear effective force that tends to stabilize the nucleus into a spherical shape, in and near to shell closures, and the strong correlations (pairing, quadrupole in particular) that favors the nucleus into a deformed shapes in around mid-shell regions [1-15].
    [Show full text]
  • Interacting Boson Model of Collective Octupole States (I)
    Nuclear Physics A472 (1987) 61-84 North-Holland, Amsterdam INTERACTING BOSON MODEL OF COLLECTIVE OCTUPOLE STATES (I). The rotational limit J. ENGEL’ and F. IACHELLO A. W. Wright Nuclear Structure Laboratory, Yale University, New Haven, Connecticut 06.511, USA Received 5 March 1987 Abstract: We discuss the problem of describing low-lying collective negative-parity states within the framework of the interacting boson model. We suggest that a simultaneous description of quadrupole and octupole states in nuclei be done in terms of the group U(16), which includes f- and p-bosons in addition to the usual d- and s-bosons. We analyze the dynamical symmetries associated with the rotational limit of this model and discuss their classical (large-N) analogs. We conclude with a preliminary application of the model to the radium isotopes. 1. Introduction Low-lying collective nuclear states are dominated by the occurrence of quadrupole vibrations and deformations. Their properties can be described in terms of shape variables, aZcL (CL= 0, kl, *2), ref ‘) or alternatively in terms of interacting s- and d-bosons with Jp = O+ and Jp = 2+ respectively ‘). The role played by d-bosons is easily understood since they can be thought of as a quantization of the variables (yzcI. The introduction of s-bosons is less obviously necessary and arose from a study of the underlying microscopic structure which led to an interpretation of bosons in terms of nucleon pairs ‘). It reflects the existence in nuclei of a pairing interaction in addition to the quadrupole interaction. An important consequence of the introduc- tion of s-bosons is that it facilitates phenomenological descriptions of spectra.
    [Show full text]
  • Phase Diagram of the Proton-Neutron Interacting Boson Model
    PHYSICAL REVIEW LETTERS week ending VOLUME 93, NUMBER 21 19 NOVEMBER 2004 Phase Diagram of the Proton-Neutron Interacting Boson Model J. M. Arias,1 J. E. Garcı´a-Ramos,2 and J. Dukelsky3 1Departamento de Fı´sica Ato´mica, Molecular y Nuclear, Facultad de Fı´sica, Universidad de Sevilla, Apartado 1065, 41080 Sevilla, Spain 2Departamento de Fı´sica Aplicada, Universidad de Huelva, 21071 Huelva, Spain 3Instituto de Estructura de la Materia, CSIC, Serrano 123, 28006 Madrid, Spain (Received 2 June 2004; revised manuscript received 3 September 2004; published 15 November 2004) We study the phase diagram of the proton-neutron interacting boson model with special emphasis on the phase transitions leading to triaxial phases. The existence of a new critical point between spherical and triaxial shapes is reported. DOI: 10.1103/PhysRevLett.93.212501 PACS numbers: 21.60.Fw, 05.70.Fh, 21.10.Re, 64.60.Fr Quantum phase transitions (QPT) have become a sub- IBM-1 there are three dynamical symmetries: SU(5), ject of great interest in the study of several quantum O(6), and SU(3). These correspond to well-defined nu- many-body systems in condensed matter, quantum optics, clear shapes: spherical, deformed -unstable, and prolate ultracold quantum gases, and nuclear physics. QPT are axial deformed, respectively. The structure of the IBM-1 structural changes taking place at zero temperature as a Hamiltonian allows to study systematically the transition function of a control parameter (for a recent review, see from one shape to another. There were some pioneering [1]). Examples of control parameters are the magnetic works along these lines in the 1980s [10–12], but it has field in spin systems, quantum Hall systems, and ultra- been the recent introduction of the concept of critical cold gases close to a Feshbach resonance, or the hole- point symmetry that has recalled the attention of the doping in cuprate superconductors.
    [Show full text]
  • Description of Transitional Nuclei in the Sdg Boson Model
    UM-P-91/102 . Description of transitional nuclei in the sdg boson model V.-S. Lac and S. Kuyucak School of Physics, University of Melbottrne, Parkville, Victoria 3052, Australia Abstract We study the transitional nuclei in the framework of the sdg boson model. This extension is necessitated by recent measurements of E2 and E4 transitions in the Pt and Os isotopes which can not be explained in the sd boson models. We show how 7-unstable and triaxial shapes arise from special choices of sdg model Hamiltonians and discuss ways of limiting the number of free parameters through consistency and coherence conditions. A satisfactory description of E2 and E4 properties is obtained for the Pt and Os nuclei, which also predicts dynamic shape transitions in these nuclei. \V ' i 1. Introduction Description of transitional nuclei has been one of the most challenging tasks for collective models of nuclei. A complicating feature of these nuclei is the triaxial nature of their energy surface which is neither 7-rigid as in the Davydov-Filippov model *) nor 7-unstable as in the Wilets-Jean model 2), but rather 7-soft which necessitates introduction of more elaborate geometric models such as the Generalized Collective Model3). More recently, the interacting boson model (IBM)4) has provided a very simple description for the transitional nuclei based on the 0(6) limit and its perturbations 5-6). The 0(6) limit had been especially successful in explaining the E2 transitions among the low-lying levels of the Pt isotopes 5_6). Its main shortcomings are (i) the energy surface is 7-unstable which leads to too much staggering in the quasi-7 band 7), (ii) the quadrupole moments vanish 8), (iii) the B(E2) values fall off too rapidly due to boson cut-off 9), (iv) it fails to describe the E4 properties 10-18).
    [Show full text]
  • Extended O(6) Dynamical Symmetry of Interacting Boson Model Including Cubic D-Boson Interaction Term
    Australian Journal of Basic and Applied Sciences, 11(2) February 2017, Pages: 123-132 AUSTRALIAN JOURNAL OF BASIC AND APPLIED SCIENCES ISSN:1991-8178 EISSN: 2309-8414 Journal home page: www.ajbasweb.com Extended O(6) Dynamical Symmetry of Interacting Boson Model Including Cubic d-Boson Interaction Term Manal Sirag Physics Department, Faculty of Women for Art ,Science and Education-Ain Shams University, Cairo - Egypt. Address For Correspondence: Dr. Manal Mahmoud Sirag, Ain Shams University - Faculty of Women for Art ,Science and Education, Physics Department, AsmaaFahmy street, Heliopolis, Cairo, Egypt. E-mail: [email protected] ARTICLE INFO ABSTRACT Article history: In the interacting boson model (IBM), the low lying collective states in even-even Received 18 December 2016 nuclei are described by the interacting s-(L=0) and d-(L=2) bosons. In the original Accepted 16 February 2017 (IBM) the Hamiltonian was defined in terms of one and two body interactions. There Available online 25 February 2017 are three group chains U(5), SU(3) and O(6) of which represent vibration, rotation and γ-unstable respectively of U(6) which ends in O(3). The cubic terms (three boson interaction terms) are considered in the IBM Hamiltonian in order to obtain a stable, Keywords: triaxially shaped nucleus. The inclusion of single third order interaction between the d- Interacting Boson Model – Potential bosons for interacting boson model (IBM) in the potential of the O (6) limit and the energy surface – coherent state – even- influence on the nuclear shapes are studied. The classical limit of the potential energy even – O(6) limit.
    [Show full text]
  • Part a Fundamental Nuclear Research
    1 Part A Fundamental Nuclear Research Encyclopedia of Nuclear Physics and its Applications, First Edition. Edited by Reinhard Stock. © 2013 Wiley-VCH Verlag GmbH & Co. KGaA. Published 2013 by Wiley-VCH Verlag GmbH & Co. KGaA. 3 1 Nuclear Structure Jan Jolie 1.1 Introduction 5 1.2 General Nuclear Properties 6 1.2.1 Properties of Stable Nuclei 6 1.2.2 Properties of Radioactive Nuclei 7 1.3 Nuclear Binding Energies and the Semiempirical Mass Formula 8 1.3.1 Nuclear Binding Energies 8 1.3.2 The Semiempirical Mass Formula 10 1.4 Nuclear Charge and Mass Distributions 12 1.4.1 General Comments 12 1.4.2 Nuclear Charge Distributions from Electron Scattering 13 1.4.3 Nuclear Charge Distributions from Atomic Transitions 14 1.4.4 Nuclear Mass Distributions 15 1.5 Electromagnetic Transitions and Static Moments 16 1.5.1 General Comments 16 1.5.2 Electromagnetic Transitions and Selection Rules 17 1.5.3 Static Moments 19 1.5.3.1 Magnetic Dipole Moments 19 1.5.3.2 Electric Quadrupole Moments 21 1.6 Excited States and Level Structures 22 1.6.1 The First Excited State in Even–Even Nuclei 22 1.6.2 Regions of Different Level Structures 23 1.6.3 Shell Structures 23 1.6.4 Collective Structures 25 1.6.4.1 Vibrational Levels 25 1.6.4.2 Rotational Levels 26 1.6.5 Odd-A Nuclei 28 1.6.5.1 Single-Particle Levels 28 1.6.5.2 Vibrational Levels 28 Encyclopedia of Nuclear Physics and its Applications, First Edition.
    [Show full text]
  • BEHAVIOUR of the Ge ISOTOPES Nureddin Turkan , Ismail Maras
    Mathematical and Computational Applications , Vol. 15, No. 3, pp. 428-438, 2010. © Association for Scientific Research E(5) BEHAVIOUR OF THE Ge ISOTOPES Nureddin Turkan 1, Ismail Maras 2 1Bozok University, Department of Physics, Yozgat, 66200 Turkey, [email protected] 2Celal Bayar University, Department of Physics, Manisa, 45040 Turkey [email protected] Abstract- The sufficient aspects of model leading to the E(5) symmetry have been proved by presenting E(5) characteristic of the transitional nuclei 64-80Ge. The positive parity states of even-mass Ge nuclei within the framework of Interacting Boson Model have been calculated and compared with the Davidson potential predictions along with the experimental data. It can be said that the set of parameters used in an calculations is the best approximation that has been carried out so far. Hence, Interacting Boson Approximation (IBA) is fairly reliable for the calculation of spectra in such set of Ge isotopes. Keywords- Davidson-like potentials, nuclear phase transition, critical symmetry, even Ge, Interacting Boson Model. 1. INTRODUCTION This paper presents a computational study in the field of nuclear structure. The declared goal of the paper is to identify features of the E(5) dynamical symmetry in the critical point of the phase/shape transition for the even-even Ge isotopes with A in the range 64 to 80. The topic of dynamical symmetries (E(5) and X(5)) was theoretically predicted by Iachello [1] and found in real nuclei by Casten et al. [2], just beside experiment [3-5]. A systematic search of nuclei exhibiting this features along the nuclide chart is underway in several structure labs around the world.
    [Show full text]
  • Stdin (Ditroff)
    84Za01 Transitions to Stretched States in the Deformed Limit L. Zamick, Phys. Rev. C29, 667 (1984). Nuclear Structure: 28Si; calculated B(M6). Deformed model. 84Za02 Direct Reaction Theories and Classical Trajectory Approxiamtions V. I. Zagrebaev, Izv. Akad. Nauk SSSR, Ser. Fiz. 48, 136 (1984) Nuclear Reactions: 40Ca(d,p), E=11 MeV; 208Pb(d,p), E=15.8 MeV; calculated σ(θ); deduced parameters. T-matrix approach, direct reaction, classical trajectory approximation. 84Za03 Inelastic Electron Scattering to Negative Parity States of 24Mg H. Zarek, S. Yen, B. O. Pich, T. E. Drake, C. F. Williamson, S. Kowalski, C. P. Sargent, Phys. Rev. C29, 1664 (1984). Nuclear Reactions: 24Mg(e,e'), E=90-280 MeV; measured σ(E(e')), σ(θ). 24Mg levels deduced electromagnetic form factors, B(λ). Open-shell RPA calculations. 84Za04 Analysis of the Actinide Fission Data (E(α) ≤ 100 MeV) taking into Account the Exit-Channel Parameters and the Emission Fis- sion N. I. Zaika, Yu. V. Kibkalo, V. P. Tokarev, V. A. Shityuk, Yad. Fiz. 39, 548 (1984). Nuclear Reactions: ICPND 232Th, 235, 236U(α,F), E 20-100 MeV; measured ®ssion σ(E), σ(θ); deduced ®ssion parameter energy dependence. Optical model, parabolic barrier approximation. 84Za05 Space Parity Nonconservation Effects in Radiative Neutron Capture D. F. Zaretsky, V. K. Sirotkin, Yad. Fiz. 39, 585 (1984). Nuclear Reactions: Br, 111Cd, 117Sn, 139La(n,γ), E resonance; calculated capture γ CP, γ(θ) asymmetry coef®cient; deduced space parity nonconservation role. 84Za06 Off-Shell Effects in Nucleon-Deuteron Polarization Observables H. Zankel, W. Plessas, Z. Phys. A317, 45 (1984).
    [Show full text]
  • IBM-1 CALCULATIONS on the EVEN-EVEN 122-128Te ISOTOPES
    Mathematical and Computational Applications, Vol. 10, No. 1, pp. 9-17, 2005. © Association for Scientific Research IBM-1 CALCULATIONS ON THE EVEN-EVEN 122-128Te ISOTOPES Atalay Küçükbursa, Kaan Manisa Physics Department, Faculty of Arts & Sciences, Dumlupınar University, Kütahya,Turkey [email protected] Mehmet Çiftçi, Tahsin Babacan and İsmail Maraş Physics Department, Faculty of Arts & Sciences, Celal Bayar University, Manisa, Turkey Abstract- The 122-128Te isotopes in O(6)- SU(5) transition region were investigated. For these nuclei, the energy levels, B(E2) transition probabilities, and (E2/M1) mixing ratios were calculated within framework of the Interacting Boson Model (IBM-1). The results are compared with the experimental data and the previous calculations. It is shown that there is a good agreement between the results found and especially with the experimental ones. Keywords- energy levels, transition probabilities, mixing ratios, 1. INTRODUCTION The neutron-proton interaction is known to play a dominant role in quadrupole correlations in nuclei. As a consequence, the excitation energies of collective quadrupole excitations in nuclei near a closed shell are strongly dependent on the number of nucleons outside the closed shell. The even-even tellurium isotopes are part of an interesting region beyond the closed proton shell at Z=50, while the number of neutrons in the open shell is much larger, as such these nuclei have been commonly considered to exhibit vibrational-like properties. The even-mass tellurium isotopes have been extensively investigated both theoretically and experimentally in recent years with special emphasis on interpreting experimental data via collective models. Energy levels, electric quadrupole moments, B(E2) values of 122-128Te isotopes have been studied within the framework of the semi- microscopic model [1], the two-proton core coupling model [2], dynamic deformation model [3] and the interacting boson model-2 [4-6].
    [Show full text]