-----[ 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.