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DATE DUE DATE DUE DATE DUE MSU Is An Affinnetive Action/Equal Opportunity Institution c:\clrc\datoduo.pm3-p.1 NUCLEAR MODELS FOR BETA AND DOUBLE-BETA DECAYS By LIAN G ZHAO A DISSERTATION Submitted to Michigan State University in partial fulfillment of the requirements for the Degree of DOCTOR OF PHILOSOPHY Department of Physics and Astronomy 1992 W7» ”30 ABSTRACT NUCLEAR MODELS FOR BETA AND DOUBLE-BETA DECAYS By Liang Zhao The Zuflfl decay matrix element of 48Ca is studied with a large-basis shell-model calculation. The theoretical and experimental fl- and [3+ spectra and their relation to 2Vflfl are discussed. A new empirical effective interaction is found to give the best agreement to fl“ and fl+ spectra with the effective Gamow-Teller operator 6t=0.77(7t. Our shell-model prediction of T1/2=1.9x 1019 yr differs by a factor of two from present experimental limit of Tl/z > 3.6x1019 yr. The validity and accuracy of the anRPA as a model to study ,5“ and ,3/3 decay are examined by making a comparison of the anRPA and the full-basis shell-model calculations for the fp shell nuclei. Our comparison includes the total decay matrix elements, the relevant strength distributions and coherent one-body transition densi- ties. The coherent one-body density is introduced in order to study the single—particle state contributions to the total Gamow—Teller strength. Discrepancies between the two models are found. The anRPA overestimates the total fl+ and flfl matrix el- ements. There are large disagreements in the shape of the spectra as well as in the coherent one—body transition densities between the anRPA and shell-model results. Empirical improvements for the anRPA are discussed. The correlated BCS wave function is introduced by first-order perturbation theory, in which the four quasiparticle correlations are taken into account. The extended BCS equation is derived. The applications of the extended BCS have shown some improvements compared to the standard BCS theory. An extended anRPA equation is developed based on the extended BCS theory and applied to study fi+ and ,Bfl decay. The calculations show that the disagreements between the anRPA and the shell model in the total B(GT+) strengths and fifl decay matrix elements have been reduced, but those in the shape of the spectra have not yet been improved. Other possible improvements for the anRPA are discussed. A second anRPA equation is developed by including two and four quasiparticle excitations in the phonon creation operator. An extended second anRPA equation is obtained by combination of the second anRPA and extended anRPA equations. They should provide more accurate methods for studying the transition of the one- and two-body charge exchange modes. ACKNOWLEDGMENTS First of all, I would like to express deepest thanks to my advisor B. Alex Brown for his guidance and patience during five years of my graduate study. He has taught me what nuclear physics is and how to do good research. I will appreciate this for the rest of my life. I wish to thank my guidance committee members, Prof. George F. Bertsch, Prof. Aaron Galonsky, Prof. Wayne W. Repko and Prof. Susan M. Simkin. I have a special acknowledgement for Prof. Aurel Bulgac and Prof. Pawel Danielewicz with whom I had many helpful discussions. I thank Dr. Andy Sustich for the help and friendship I received. I am also grateful to Dr. Don Cha for his helpful suggestions for part of my thesis. I appreciate Prof. Kovacs for the help I received in my pursuit of graduate study at Michigan State University. Thanks are also due to my friends, Renyong Fan, Dr. Minzhuan Gong, Nengjiu Ju, Jianan Liu, Andre Maul, Dr. David Mikolas, Zhijun Sun, Yang Wang, Qing Yang, Weiqing Zhong, Fan Zhu and many others for their help and friendship. I would like to express my gratitude to my family members, my parents, Prof. Minlun Zhao and Prof. Shujing Shen, for their constant encouragement and for their always stressing the importance of education, and my brother Dr. Jian Zhao and his wife Dr. Xiaofeng Li for their discussions, advice, help and support, especially in this five years. Thanks are due to my mother-in-law, Dr. Pingling Liu, my aunt Dr. Shuzhao Shen and her husband Prof. Dashou Sheng. Finally, I heartily thank my wife, Dr. Rong Xu, for her love and also for her financial support in finishing my thesis. Contents LIST OF TABLES vi LIST OF FIGURES ix 1 Introduction 1 1.1 Beta and Double-Beta Decays ...................... 1 1.2 Shell Model Theory ............................ 2 1.3 anRPA Theory ............................. 8 1.4 Beyond RPA ............................... 12 1.5 Thesis Organization ............................ 13 2 Shell-model Calculation for Two-neutrino Double—beta Decay of 48Ca 16 2.1 Introduction ................................ 16 2.2 Model Spaces and Effective Interactions ................ 19 2.3 Calculations and Discussions ....................... 20 2.4 Summary ................................. 32 3 Comparison between anRPA and Shell Model I: fl+ Decay 33 3.1 Introduction ................................ 3.2 Formalism ................................. 3.2.1 anRPA equations ........................ 3.2.2 BCS equations .......................... 3.2.3 B (GT) in shell-model calculations ................ 3.2.4 The coherent one-body transition density ............ 3.3 Calculations and Discussions ....................... 3.3.1 Comparison of anRPA and shell model ............ 3.3.2 Empirical improvements of anRPA .............. 3.4 Summary and Conclusions ........................ Comparison between anRPA and Shell Model II: 2Vflfl Decay 4.1 Introduction ................................ 4.2 Formalism of 2uflfl Decay ........................ 4.3 Results and Discussions ......................... 4.4 Summary and Conclusions ........................ BCS Theory and Extension 5.1 Introduction ................................ 5.2 BCS Theory for Proton-Neutron System ................ 5.2.1 Nuclear Hamiltonian in quasiparticle space ........... 5.2.2 BCS equations ( independent quasiparticle ) .......... 5.2.3 BCS energy spectrum ....................... iii 33 35 35 39 40 41 41 49 52 52 56 69 71 71 73 73 76 80 5.3 Extended BCS Theory .......................... 5.3.1 Interactions between quasiparticles ............... 5.3.2 Correlated BCS wave function .................. 5.3.3 Construction of two-quasiproton two-quasineutron excitations in the J-scheme .......................... 5.3.4 Formalism of extended BCS theory ............... 5.4 Application ................................ 5.5 Summary ................................. 6 Extended anRPA Theory and Applications 6.1 Introduction ................................ 6.2 Extended anRPA Equations ...................... 6.3 Application and Discussions ....................... 6.4 Summary and Conclusions ........................ 7 Eirther Improvements and Considerations 7.1 Introduction ................................ 7.2 Second anRPA Equations ....................... 7.3 Extended Second anRPA Equations .................. 7.4 Summary ................................. 8 Summary and Conclusions A QRPA equations iv 81 81 83 84 87 90 93 93 94 97 102 104 104 105 109 109 111 115 B Relation between QRPA and anRPA Equations C Second QRPA Equations D Coherent One-Body Transition Density E BCS and Extended BCS in Uncoupled Representation F Spurious States in the BCS and extended BCS. LIST OF REFERENCES 117 119 122 124 127 131 List of Tables 2.1 2.2 2.3 2.4 Summary of the B (GT‘) and B(GT+) values obtained from the exper- iments and compared to the theoretical calculations with the MSOBEP and the MH interactions .......................... Comparison of the nuclear matrix elements B(cls) and Mgfi, the av- erage excited energy < Em > and half life T1 /2' The shell-model space configurations are described by f?fi"(p3/2f5/2pl/2)" with n=0 to nm” for the fp shell referring to the initial(i), intermediate(m) and final(f) states. ................................... The first ten B(GT‘), B(GT+) and M'CQ‘T values obtained with the MSOBEP interaction ........................... Comparison of M33 in different truncations. The shell-model space configurations are described by ff]; (pg /2 f5 /2p1 n)" for the fp shell and dg7z"(sl/2d3/2)" for the sd shell referring to the initial(i), intermedi- ate(m) and final(f) states with n=0 to nm”. The full-basis means nmu=8 in the fp shell or nmfl=6 in the sd shell. The MSOBEP in- teraction was used for “Ca and “Ca and the interaction [Wil 84] was used for 22O ................................. vi 24 28 29 31 3.1 3.2 3.3 4.1 5.1 5.2 46Ti: Proton and neutron: single particle energies 5,- (MeV), gap pa- rameters A,- (MeV) and occupation probabilities 12? from BCS and the shell-model Eg(SM) and 22,-2(SM) . The Fermi energies obtained from the BCS calculation are —12.968 MeV and —10.868 MeV for proton and neutron, respectively. ........................ Comparison of the coherent one-body transition density (COBTD) ob- tained in the anRPA, modified anRPA and shell-model calculations of 46Ti .................................... Comparison of the coherent transition matrix elements (CTME) ob- tained in the anRPA, modified anRPA and shell-model calculations of 46Ti. Labels (A), (B), (C), (D) and (E) are given by Table 3.2 . . . 46Ca: Neutron single particle energies 5,- (MeV), gap parameters A,- (MeV) and occupation probabilities v? and quasineutron energies E,- (MeV) from BCS, the shell-model E,(SM), v§(SM) defined in Chapter 3. The neutron Fermi energy is —8.712 MeV ............... 46Ti: The occupation probabilities < lalmajm| > /(2j + 1) for protons and neutrons obtained from the BCS, extended BCS (EBCS) and shell- model calculations, respectively ...................... The extended BCS parameters, 11?, 0?, gap parameters A.- (MeV) and quasiparticle energies E,- (MeV) for 46Ti ................ 41 45 46 65 91 6.1 Comparison of the coherent one-body transition density (COBTD) ob- tained for the anRPA, extended anRPA and shell model calcula- tions of 46Ti. The labels represent (A): COBTD in the anRPA ; (B): COBTD in the extended anRPA ; (C): COBTD in the shell model. 98 6.2 Comparison of the coherent transition matrix elements (CTME) ob- tained in the anRPA, extended anRPA and shell model calculations of 46Ti. Labels (A), (B) and (C) are given by Table 6.1 ........ 102 viii List of Figures 1.2 1.3 1.4 2.1 (a) The single-particle energies of a harmonic-oscillator potential. N is the oscillator quantum number. (b) The single-particle energies of a Woods-Saxon potential. (c) The single-particle energies of a Woods- Saxon potential plus spin—orbit coupling. Each state is labeled by (nlj). (d) The numbers 2(23' + 1) ........................ Schematic representation of the simple shell-model configurations (a) - 48Ca ground state (closed shell). (b) 49Ca ground state. (c) “Ca excitation state. (d) 49Ca excitation state. ............... Schematic representation of the TDA and RPA configurations for 48Ca (a) TDA ground state (closed shell). (b) RPA ground state. (c) TDA excitation state. (d) RPA excitation state. ............... Schematic representation of the pnTDA configurations for 48Ca (a) pnTDA ground state (close shell). (b) pnTDA excitation state. . . . . Mass spectrum for A=48 nuclei. The double-beta decay is a possible decay mode for the 48Ca ground state. See text about the single 6‘ decay of 48Ca ................................ 18 2.2 2.3 2.4 3.1 3.2 MGT(Em) as a function of Em. The first MGT(Em) is fixed at E1=2.52 MeV. In (a), the solid line is obtained from the MSOBEP interaction, and dashed line from the MH interaction. The truncation for both curves is (D: n34 for 48Ca and 48Ti, n3 5 for 4880). (b) shows the results for the MSOBEP interaction at different levels of truncation for (48Ca, 48Sc, 48Ti): A (n=0, n3 1, n=0); B (n51, n3 2, n31) and C (n32, n_<_ 3, n32). ........................... The B(GT") spectra for 48C2L-+“8Sc. The spectrum with FWHM = 100 keV obtained with the MSOBEP interaction is shown in (a). The spectra with FWHM = 400 keV obtained with the MSOBEP (solid line) and MH interaction (dashed line ) are shown in (b). The ef- fective operator defined in Eq.(2.1) is employed in our calculations. The experimental B(GT‘) from [And 85] and RD. Anderson (private communication) is presented in (c). The hatched area indicates the un- certainty resulting from subtracting the Fermi strength in the 0+(T=4) state at 6.8 MeV. ............................. The B(GT+) values for 48Ti —+4ssc. The experimental values from [Alf 91] are compared to the results obtained with the MSOBEP and MH interactions. The effective operator defined in Eq. (2.1) is employed in our calculations. ............................ Summed Gamow-Teller strength for 46Ti —>46Sc. The solid line is the shell-model result while the dashed line is the anRPA result. . . . . Summed Gamow-Teller strength for 46Ti —+46Sc with various particle- particle strength gm, where gph = 1. ................... 22 25 26 42 43 3.3 4.1 4.2 4.3 4.4 4.5 4.6 4.7 The anRPA result with the shell-model quasiparticle energies and / or occupation probabilities ( see Table 3.1 ) ................. Mass spectrum for A246 nuclei, where the double-beta decay is the only possible decay mode for the 46Ca. ground state .......... Summed Gamow-Teller strength B(GT’) of 46Ca —>4GSc. The dashed and solid lines are the anRPA and shell-model results, respectively . Summed energy dependent 211,85 decay matrix element MGT(Em) as the function of the excitation energies, the solid line and dashed line are the shell-model and anRPA results, respectively ......... Summed closure 21/66 decay matrix element BCLS(Em) as the function of the excitation energies, the solid line and dashed line are the shell- model and anRPA results, respectively ................ Summed closure matrix element BCLs(Em), the solid line is from the shell-model calculation, the dashed line is anRPA result with gm, = 0.0 and gph = 1.0 ............................. Summed closure matrix element BCLS(Em), the solid line is from the shell-model calculation, the dashed line is anRPA result with gm, = 0.5 and gph = 1.0 ............................. Summed closure matrix element BCLS(Em), the solid line is from the shell-model calculation, the dashed line is anRPA result with gm, = 1.28 and gph = 1.0. The anRPA reproduces shell-model total BCLS value. ..... ' .............................. xi 50 55 58 59 60 61 62 4.8 4.9 4.10 4.1 p—a 6.1 6.2 6.3 Summed energy dependent matrix element MGT(Em), the solid line is from the shell—model calculation, the dashed line is anRPA result with gm, = 1.2 and gph = 1.0. The anRPA reproduces shell-model total MGT value. ............................. Summed closure matrix element BCL3(E,,,), the dashed line obtained by the anRPA with E, u,v from BCS and the dotted line with E from shell model and u,v from BCS, respectively. .............. Summed closure matrix element BCL5(Em), the dashed line obtained by the anRPA with E, u,v from BCS and the dotted line with E from BCS and u,v from shell model, respectively. .............. Summed closure matrix element BCL5(Em), the dashed line obtained by the anRPA with E, u,v from BCS and the dotted line with u,v and E from shell model, respectively. .................. Summed Gamow-Teller strength for 46Ti —>46Sc. The solid line, the dashed line and the dotted line are obtained by the shell model, pn- QRPA and extended anRPA, respectively. .............. Summed energy dependent 21/66 decay matrix element MGT(Em) as the function of the excitation energies, the solid line, the dashed line and the dotted line are obtained by the shell model, anRPA and extended anRPA, respectively ..................... Summed closure 21/66 decay matrix element BCL3(E,,,) as the function of the excitation energies, the solid line, the dashed line and the dotted line are obtained by the shell model, anRPA and extended anRPA, respectively ................................ 64 66 67 68 99 100 101 Chapter 1 Introduction 1.1 Beta and Double-Beta Decays The beta (6) decay process was one of the first types of radioactivity to be observed and still provides new valuable insight into weak interaction and nuclear structure. In this process, the 6 unstable nucleus can become more stable by converting a proton ( neutron ) within the nucleus into a neutron ( proton ). So the mass number of the nucleus, A, remains the same but the nuclear charge number, Z, changes by one unit. The 6‘ decay involves emission an electron and an antineutrino, (A, Z) —+ (A, Z+1)+ e‘ + 17, and 6+ decay involves a positron and a neutrino (A, Z) —> (A, Z — 1) + 6+ + 1/. The 6 decay mechanism is well described by the standard weak interaction theory. The beta transition probability T(6) depends on two different nuclear matrix elements and can be written as T(6) oc (if/B(F) + G2AB(GT), (1.1) where CV and CA are the coupling constants associated with the vector part and the axial-vector part of charge-current, respectively. B (F) is the Fermi transition strength related to the isospin operator. B(GT) is the Gamow-Teller transition strength asso- ciated with the Pauli spin operator as well as the isospin operator. 1 Nuclear double-beta (66) decay phenomena is a rare transition between two nuclei of the same mass number having a change of two units of nuclear charge. In cases of interest, ordinary single beta decay is forbidden because of energy conservation or because of very strong suppression due to a large angular momentum mismatch between the parent and daughter states. There are two modes of double-beta de- cay [Hax 84, Doi 85, Mut 88, Ver 86], 21/mode: (A,Z)—> (A,Z+2)+2e"+217 (1.2) 0 I/mode: (A,Z)—> (A,Z+2)+2e". (1.3) The first one is called two-neutrino (21/) 66 decay, which involves the emission of two antineutrinos and two electrons (21/ mode), it occurs in second order in the standard weak interaction theory. Another is called neutrinoless (01/) 66 decay, which involves the emission of two electrons and no neutrinos. This process violates the lepton number conservation and requires the neutrino to be a Majorana particle and have a nonzero mass and/or a nonstandard right-hand coupling. It occurs in some theories beyond the standard weak interaction model. The 21/66 decay has been observed in recent experiments [Ell 87, Avi 91, Eji 91, Tur 91], but the 01/66 decay has not yet been observed. There are two important nuclear models which can be applied to the study of the 6 and 66 decays which occur in nuclei, the shell model and the anRPA model. We will briefly discuss them in the following sections. 1.2 Shell Model Theory The nuclear shell—model theory was introduced by Mayer and Jensen 40 years ago [May 55]. The basic assumption is that each nucleon (proton and neutron) moves independently in a potential that represents the average interaction with the other nucleons in the nucleus. This potential is the combination of a. central part and the spin-orbit cou- pling term. The energy levels obtained by solving the Schr6dinger equation for a nucleon in the potential are given in Figure 1.1. The energy levels in the column (a) and (b) are obtained by using harmonic-oscillator and Wood-Saxon potentials, respectively, where the spin-orbit coupling term has not been included. 3, p, d, f, etc stand for orbital momentuml = 0, 1, 2, 3, etc. The column (c) in Figure 1.1 illustrates the level splitting due to addition of spin-orbit coupling term. Each level in column (c) is called a single-particle state and is labeled as (nl j ) in this thesis, where n is the radial quantum number and j = l + s. The ground state in the simple shell model is assumed to be the configuration in which the energy levels are filled consecutively by the nucleons with the constraint of the Pauli principle. For example, the configuration of the 48Ca ground state is that the 28 neutrons have filled the single-particle states up to 1f7/2 shell, and the 20 protons have filled the single-particle states up to 1d3/2 shell. 48Ca is a closed shell nucleus. The schematic representation of 48Ca is shown in Figure 1.2 (a). For the ground state of 49Ca, the configuration in the simple shell model is that the valence neutron occupies the 2p3/2 state outside of the 48Ca closed shell as shown in Figure 1.2 (b). So the ground state spin should be J " = 3/2“. The success of this simple model is that there are many nuclei whose ground states properties agree with these assumptions. In order to simplify our discussions for the excitations in the simple shell model, we only consider two single-particle states 1f7/2 and 2113/2. Then the excitations of 48Ca can be constructed by removing one nucleon from 1f7/2 to 1p3/2 state, forming the one-particle-one-hole configurations. For 49Ca, the excitations are two-particle- ‘v- 1w: mammal—184 - 4‘ _-’.’q- -_ 4'”: (2)— 6h“) - 3d —(‘ If 237/2 -‘——— l8;- X‘fi 1,,2-(1 )- 9V." - 2‘ d‘>‘l‘— 345,2 ‘ l6)- ,. / ‘- eta—«10>— - 1 --—£ ‘\ ‘9 3 11,3,2-1141-11261—126 .u- PM; '- - 3 ‘. p —r. . _ 3"” 2f (4)- - 2 ,.- 5]: (6)- 5hr.) / —-~. 2!", (31-11001 odd ’fi "hm—— no)- ._ m —(I “— mm, (121—1321— 32 — 3: —..--— as", 121- ” "'—u3n (C)— uzu " —--~ 24,” (61-1001 m ”—“H: ‘8’- - I, —: ’ 3m - 2! ——:. .-——- if," 101-m) .. Iii—J,“ 2?” “1‘ .‘— 1hr: 18,-[2'1 -— 2. m - u —:‘ '- zi’r 12"1161 mu _ ,...-1pm 121-101 —- 8 ode 1' “- 19,, (41-!!! o — '3—o-u'n (2,-[21-2 (a) (b) (e) (d) Figure 1.1: (a) The single-particle energies of a harmonic—oscillator potential. N is the oscillator quantum number. (b) The single-particle energies of a Woods-Saxon potential. (c) The single-particle energies of a Woods-Saxon potential plus spin-orbit coupling. Each state is labeled by (nl j ) (d) The numbers 2(23' + 1). one-hole configuration. The schematic illustrations of these excitation configurations are presented in Figure 1.2 (c) and (d). In present shell-model theory, the nuclear structure properties are assumed to be determined by the valence nucleons which simultaneously occupy several different, partially filled, single-particle states within one or two given major shells. For light nuclei ( Ag 40), the major shells are the oscillator shells indicated by the N labels on the left-hand side of Figure 1.1. For heavy nuclei (AZ 40), the major shells usually include the addition of one high j -state from the (N + 1) oscillator shell. This is called the large-basis shell-model calculation. Many multinucleon configurations are taken into account in this calculation. For example, the ground state configuration of 48Ca in the full fp shell (1f7/2, 2193/2, 1 f5/2 and 2111/2) is considered as the linear combination of all possible configurations in which eight neutrons simultaneously occupy all states in the fp shell. The label |n1n2n3n4 > used below indicates that 721 neutrons occupy the 1f7/2, n; the 2113/2, n3 the 1f5/2, and 724 the 2111/2. The Pauli principle and sum n1 + n; + 713 + m = 8 must be satisfied. Then the wave function of the 48Ca ground state is expanded as l4sca,0+ > = a1|8000 > +a2|6200 > +a3|5300 > +a4|4400 > + . .. + a72IO242 > +073|0062 > , (1.4) with 2:,- a? = 1. There are 73 terms (partitions) in Eq. (1.4). Some terms are missing due to angular momentum coupling restriction [Etc 85]. Within this set of partitions, the number of independent states with J = 0 is 347, and the number of independent states with M = 0 is 12022. These numbers are referred to the J-dimension and M -dimension, respectively. In the simple shell model, all coefficients in Eq. (1.4) are zeros except a1 = 1. The corresponding J —- and M — dimensions are both equal to 1. The large-basis shell-model calculation includes three steps: (1) setting up the 93/2 I—I Ground State . i p n (a) f‘r/z -—-*— 93/2 A V Excited State l p n (c) f‘r/z f7/2 (b) (d) Figure 1.2: Schematic representation of the simple shell-model configurations (a) 48Ca ground state (closed shell). (b) 49Ca ground state. (c) 48Ca excitation state. ((1) 49Ca excitation state. single-particle basis, (2) construction of the Hamiltonian matrix, and (3) diagonal- ization of this matrix. However, carrying out the large-basis shell-model calculation requires extensive numerical computation, so better numerical methods and modern computer facilities are very important. The phrase of the full-basis shell-model calculation will refer to a large-basis shell— model calculation in which all possible configurations in a major shell are included. However, a limiting constraint on the large-basis shell-model calculations is that the dimension of the Hamiltonian matrix increases rapidly when the single—particle basis increases. For example, the dimension of J1r 2 0+ of the 48Ca ground state is one in the 1f7/2 shell, 14 in the full f shell ( lf7/2,1f5/2 ), and 347 in the full fp shell ( 1f7/2,2p3/2,1f5/2,2p1/2 ). In medium and heavy nuclei ( A250 ), the di- mensions are extremely large. For example, in (1,34Sn92, if we allow the 12 valence protons to occupy lg7/2,2d5/2,2d3/2,331/2, 1h11/2 states, and 10 valence neutrons to occupy 1h9/2,2f7/2,2f5/2,3p3/2,3111/2,12'13/2 states. the dimension of J1r = 0+ ma- trix is 41,654,193,516,917 [Iac 87]. This is of course completely beyond current com- puter capability. Thus present shell-model calculations for heavy nuclei are performed within very truncated model spaces. In the extreme limit, they may go back to the simple shell model. Because of the difficulties in carrying out the large-basis shell- model calculation, the RPA, the QRPA and the anRPA, which are discussed in the next section, are frequently employed to study the properties of medium and heavy nuclei. Another important aspect in the shell model is the eflective interaction. The first and reasonable guess is from the theoretical G-matrix calculations [Rin 80]. But more precise and reliable effective interactions are currently obtained from two methods. One of them is using an unconstrained Hamiltonian, in which all two-body matrix elements and single-particle energies are allowed to vary as free parameters in a fit to experimental data ( for example, see ref. [Wil 84]). Another is employing some well-understood nuclear potentials ( such as one pion or one boson exchange poten- tials ) and fitting the parameters in these models with the experimental data. Here the experimental data usually includes the binding energies and low-lying excitation energies ( for example, see [BAB 88, Ric 91] ). In this thesis, we will use the effective interaction from previous determination. Most experimental data for light nuclei ( A340 ) can be successfully explained and even predicted by the large-basis shell-model calculations [Bru 77, Law 80]. For example, the Gamow-Teller transition strengths of the full-basis shell-model calcula- tions in the sd shell (Ids/2,231” and 1d3/2) are in good agreement with those in the experiments [Bro 85, Bro 88]. 1.3 anRPA Theory In this section, we discuss the proton-neutron Quasiparticle Random Phase Approxi- mation (anRPA), which is widely used for 6 — and 66 —- decay calculations in heaver nuclei. The method of the anRPA is totally different from that of the shell model. In order to understand this theory, it is useful to understand the following theories and their relations: the Tamm-Dancoff Approximation ( TDA ), the Random Phase Ap- proximation ( RPA ) and the Quasiparticle Random Phase Approximation ( QRPA ). Their mathematical derivations can be found on the textbook of many-body problem such as [Row 70]. Here we only concentrate on the physical picture. First, we qualitatively describe the TDA and RPA models for “Ca. The TDA ground state is the 48Ca closed shell shown in Figure 1.3 (a), and the excited states are constructed by removing a neutron in 1f7/2 closed shell, creating another one in the empty states above 1f7/2 shell, and forming one-particle—one-hole ( lplh ) configurations, see Figure1.3 (c). Thus we find the TDA is the same as the simple shell model shown in Figure 1.2 (a) and (c). In the RPA, the ground state is not pure closed shell, but has mixtures of some types of 2p2h, 4p4h, .. -, components as shown in Figure 1.3 (b). The excitations of the RPA can be obtained by removing one neutron from the 1 f7/2 to other states above 1f7/2 like the TDA, and also can be constructed by removing a neutron above f7/2 states and creating a neutron in f7/2 state, see Figure 1.3 (d). The RPA theory cannot be simply linked to the simple shell model or large-basis shell model. For the even-even nuclei away from closed shells, the experimental results have suggested that the ground state may be dominated by pairing correlations [Row 70, Rio 80]. The BCS ( Bardeen, Cooper and Schrieffer ) theory assumes all nucleons in the ground state to be paired. ( BCS theory is discussed in more detail in Chapter 5 ). In terms of the Bogoliubov quasiparticle, which is a linear combination of the particle and hole, the BCS ground state is a vacuum with respect to the quasipar- ticle. The Quasiparticle Tamm-Dancoff Approximation ( QTDA ) suggests that the excitations are obtained by creating two quasiparticles from the BCS ground state. Similar to the relation between the TDA and RPA, the QRPA assumes the ground state configuration is the quasiparticle vacuum plus mixtures of some types of four quasiparticle, eight quasiparticle, - - o, components. Thus the excitations in the QRPA are performed by creating and destroying two quasiparticles from the QRPA ground state. In the TDA, RPA and QRPA, the excitations and ground states are with respect to the same nucleus. Now we generalize the TDA to the pnTDA in order to study the charge-exchange processes. Starting at the 48Ca closed shell, as an example, the excitations in the TDA RPA f f‘r/ :5 Ground State 7’2 2 '7/2 p n p n + 9 n 1' " (a) (b) . 91 9" in r Excited State 7’2 7’2 (c) (d) Figure 1.3: Schematic representation of the TDA and RPA configurations for 48Ca (a) TDA ground state (closed shell). (b) RPA ground state. (c) TDA excitation state. (d) RPA excitation state. ll pnTDA Ground State . f7“ (a) Excited State Figure 1.4: Schematic representation of the pnTDA configurations for 48Ca (a) pnT DA ground state (close shell). (b) pnTDA excitation state. l2 pnTDA are constructed by transferring a neutron from the 1f7/2 single-particle state to a proton which can occupy all empty proton states including the 1f7/2. So the con- figuration is the proton-particle neutron-hole shown in Figure 1.4. Then the transition matrix elements of the charge—exchange operator can be calculated in the pnTDA. One can derive the anRPA model from the pnTDA, along the line TDA —+ RPA -+ QRPA. The basic idea is that the anRPA excitations, which contribute to the charge-exchange mode, are obtained by creating and destroying one quasiproton- quasineutron pair from the even-even nucleus ground state. The even-even nucleus ground state is assumed to be a quasiparticle vacuum ( BCS ) with the addition of mixtures of some types of four quasiparticle, eight quasiparticle, -~, components, where the quasiparticle could be the quasiproton or quasineutron. The transition strengths in the anRPA describe the charge-exchange processes. The advantage of the TDA, RPA, QRPA, and anRPA is the small dimension involved. For example, the dimension of J1r = 0* matrix in the QRPA is 4 for the fp shell nuclei. The TDA, RPA, QRPA and anRPA are approximate models. The accuracies of these theories should be examined. 1 .4 Beyond RPA Improvement of the QRPA and anRPA may be an important goal in the field of nuclear structure theory as well as in the many-body problem. Since the derivation of the QRPA is similar to that of the RPA [Row 70, Rin 80], it is useful to review the development of the RPA theory. The physical picture of the RPA was discussed in section 1.3. The mathematical derivation of RPA equation can be achieved from the equation of motion method [Row 70, Rin 80], time-dependent Hartree-Fock the- ory [Row 70] and the Green’s Function method [Fet 71, Bro 71]. We will concentrate 13 here on the equation of motion method because it is the most appropriate for the extensions we will develop [Row 70, Rin 80]. In the equation of motion, the excitation states are constructed by acting a phonon creation operator Q] on the ground state. This equation corresponds exactly to the full many-body Schrédinger equation if and only if the ground state in the equation of motion is the true ground state and the phonon creatidn operator exhausts the whole Hilbert space (i.e., lplh, 2p2h, 3p3h, and so on). For the RPA theory, there are two important assumptions. Only one-particle-one—hole (lplh) creation and destruction operators are used in the phonon creation operator and the Hartree-Fock (HF) ground state is employed to calculate the matrix elements in the RPA equation. The HF ground state is obtained by the mean field. Recently three ideas for improving RPA have been suggested [Dro 90]. One of them is the second RPA [Yan 83], in which the phonon creation operator is expanded up to 2p2h creation and destruction operators and the HF ground state is retained. Several numerical calculations have been made for giant resonances in the second RPA theory. However, the second RPA is still missing some types of correlations in the ground state [Tak 88]. Another idea is the extended RPA which uses an improved HF wave function to calculate the RPA matrix elements [Ada 88]. This improved HF wave function consists of the usual HF ground state plus 2p2h correlation correc- tions. The third idea combines these previous two and is called the extended second RPA [Tak 88]. It not only includes the 2p2h expansions in the phonon creation op- erator similar to the second RPA, but also uses an improved HF wave function as a ground state. Then the extended second RPA contains all the correction terms up to secondcrder perturbations in the two-body interaction. These second-order corrections are important for the more precisely measured magnetic moments and l4 6-decay matrix elements [Ari 87, Tak 84]. Improvements for the RPA suggest the generalization to the QRPA and anRPA which we will develop in Chapter 5, 6, 7. 1.5 Thesis Organization The goals of this thesis consist of three aspects: (1) study of the 6 and 66 decay of mass A = 48 nuclei with the large-basis shell-model calculations, (2) examina- tions of the validity and accuracy of the anRPA for 6+ and 66 processes, and (3) improvements and extensions of the anRPA model. In Chapter 2, we study the 6' and 6+ Gamow-Teller transition strength with the large-basis shell-model calculations for the nuclei 48Ca and 43Ti, respectively. The agreements between the theoretical results and the experimental data are examined. The half life of two—neutrino double-beta decay of 48Ca is calculated as well. Since the anRPA is an approximate theory as we mentioned in section 1.3, we examine the validity and accuracy of the anRPA theory as a model for 6+ decay and double-beta 66 decay in Chapter 3 and Chapter 4, where the comparison of the anRPA and full-basis shell-model calculations ( exact model ) are made. We conclude that there are some correlations which are important in 6 and 66 decay but are not included in the anRPA. Some empirical improvements for the anRPA have been suggested and tested. In Chapter 5, the BCS theory for the proton-neutron system is reviewed. The correlated BCS wave function is introduced by incorporating the quasiparticle cor- relations with first-order perturbation theory. An extended BCS is derived and is applied to study the nuclear ground state properties. In Chapter 6, we develop an extended anRPA equation with the correlated BCS 15 ground state. The application of the extended anRPA is presented and compared to the anRPA and shell-model calculations. Some improvements over the anRPA are found. In Chapter 7, we present the formalism for further possible improvements for the anRPA equation. The second anRPA and the extended second anRPA equations are derived. They should provide more accurate methods for studying the transition of the charge-exchange modes. The summary and discussions are given in Chapter 8. We derive the QRPA equation in Appendix A and discuss the relation between the QRPA and anRPA in Appendix B. The second QRPA and coherent one-body transition density are given in Appendix C and D, respectively. In Appendix E, the BCS and the extended BCS theories are derived in angular momentum uncoupled space. In Appendix F, we discuss the spurious state in the BCS and the extended BCS theories. Chapter 2 Shell-model Calculation for TWO-neutrino Double-beta Decay of 48Ca 2.1 Introduction The theory and experiment of double-beta (66) decay have greatly attracted ele- mental particle and nuclear physicists for a long time. It is an important process for examining the character of the neutrino and for testing the theories beyond the standard weak interaction theory. On the other hand, the calculations for the 66 matrix elements provide a strong challenge to nuclear physicists, because 66 decays which are experimentally accessible occur in medium and heavy nuclei where we are still not clear how to precisely take into account the ground state correlations as well as calculate the excitations. In order to analyse the experimental results to determine the character of the neutrino in 66 decay, precise calculations of the nuclear matrix elements are re- quired. For example, the neutrino masses and the right-handed current coupling constants, which can be deduced from the experimental neutrinoless decay half-life, depend on the relevant nuclear matrix elements which have to be calculated theoret- 16 ically [Hax 84, Doi 85, Mut 88], However, in particular, the agreement between the experiment and theory for the standard 21/ mode is one of the prerequisites for a reliable interpretation of the more exotic 01/ mode. In this Chapter, we study the 21/66 decay of 48Ca which has the largest double-beta decay Q-value of any nucleus and where the large basis shell-model calculations are possible. The mass spectrum of A248 is given in Figure 2.1. In fact, the lowest states ( 6+,5+ and 4+ ) of 488C are located in the 48Ca’s Q- value window [Alb 85], but these single 6 decays are highly forbidden because of the angular momentum mismatch. Their half-lives are estimated to be about a factor of 10 times that of 21/66 decay [War 85]. There are serval difficulties with previous shell-model calculations for 21/66 decay of 48Ca. In cases where intermediate states in 48Sc were considered explicitly the fp shell-model space was highly truncated [Tsu 84, BAB 85, Sko 83, Ver 86], in other cases where the truncation was less severe the intermediate states were not calculated and the closure approximation was used instead [Hax 84, Ver 86, Zam 82]. Also the effective interactions used were not always well tested with regard to the nuclear spectra. In a more recent calculation [Oga 89], a new method was used to implicitly take into account the spectrum of the intermediate 1+ states exactly. However, we will emphasize below the importance of the testing the interactions with respect to the explicit intermediate spectrum. In the following, we calculate the nuclear matrix elements for 21/6 6 decay of “Ca and the related 6’ and 6+ decay in the fp shell space with a much larger basis than previously used and with a new effective interaction than previously used. —48 Mass excess (MeV) 18 1+ 0+ 48C8. 4850 Q+ 48Ti Figure 2.1: Mass spectrum for A=48 nuclei. The double-beta decay is a possible decay mode for the 48Ca ground state. See text about the single 6‘ decay of 48Ca. 19 2.2 Model Spaces and Effective Interactions Because of the present computational limitations, a truncated shell-model basis is used in our calculations. The truncated space in the fp shell is defined by the set of partitions f78/3“ (pg/2f5/2p1/2)". In this work, the partitions assumed for 48Ca(0+,T=4), 48Sc(l+,T=3) and 48Ti(0"’,T=2) are (n _<_ 4), (n S 5) and (n S 4), respectively. The n S nmax means that n = 0, - - - ,nmax are allowed. The corresponding J-scheme dimensions are 133, 5599 and 3613, respectively. This is an order of magnitude larger basis than has been used in previous calculations. Our calculations were carried out with the shell-model code OXBASH [Etc 85] on a VAX computer. The most complete fp shell calculation should be based on the full-basis space (11 S 8), but at present this is impossible because of the large dimensions involved. For example, the J-scheme dimension for the 48Ti ground state is 10872 in the full-basis space. It is at edge of our current computer capability. In later discussions, we will argue that our truncation is a good approximation with respect to the full-basis space. The model space for the intermediate nucleus (4SSc) should include all states which can be reached by a one-body operator from the initial and final nuclei. Thus for our initial (48Ca) and final (43Ti) states which have n _<_ 4, we include n S 5 configurations in the intermediate system. Then the B(GT) from the 48Ti or 48Ca ground states satisfy the sum rule, 2 B(GT’) — Z B(GT+) = 3(N — Z), (2.1) where B(GT) = (< f||0t||i >)2/(2J,-+1). The effective interactions used in this Chapter are called MH [Mut 84] and MSOBEP [Ric 91]. The MH interaction has a long history. McGrory et a1. [McG 70] started with the renormalized Kuo-Brown interaction [Kuo 68] and changed several two-body 20 matrix elements (T BME), which involved the fly; and/or 173/2 orbits. Later McGrory et a1. [McG‘81] added 50 keV to the f7/2 — f5/2 diagonal TBME and introduced new single-particle energies. Based on Ref. [McG 81], Muto and Horie shifted the T=0,J ( monopole of the inter-shell matrix elements < f7/2jIVIf5/2j > j = 193/21 P1/2 and f7/2) matrix elements by —-0.3 MeV [Mut 84]. MSOBEP is a new effective interaction based on a modified surface (MS) one- boson exchange potential (OBEP) [BAB 88]. Modified refers to the addition of monopole (infinitely long range) terms to the central part of the potential, and surface refers to an assumed density dependence which empirically is surface peaked. This MSOBEP potential has been successful in reproducing the sd-shell energy levels in terms of a few parameters associated with the strengths of the various OBEP chan- nels. Richter et a1. [Ric 91] have recently refit the parameters of this potential to 61 energy level data in the lower part of the fp shell, and this is the new interaction which we employ in the present work. 2.3 Calculations and Discussions In this section, we will discuss the results for the double-beta decay matrix element of 48Ca. At first we introduce the effective Gamow-Teller operator based on previous beta decay and (p,n) reaction studies [Bro 88] (”7 = 0.770. (2.2) This is used because experimental B (GT) strengths are uniformly 30%~50% less than the shell-model calculations. The missing strength can be explained by a combina- tion of the coupling to a A-particle-N-hole configurations [G00 81, Boh 81, Ari 87, Tow 87], and to the admixtures of 2p-2h configurations [Ber 82, Ari 87, Tow 87]. 21 For purposes of discussion, we introduce the matrix element for 21/66 decay, E E + ~ - + + ~ — + m m m <0 Hot Hlm ><1m||ot ”0, > Mama) = 2: Ma = 2: ’ E +12. ’ m=1 m=1 (2.3) which is a function of the 1+ excitation energy Em in “SC. E0 = To/2+AM, where To is the Q-value for 66 decay of 48Ca and AM is the mass difference between 483c and 48Ca, To=4.27 MeV and AM=—0.277 MeV [Wap 85]. The total matrix element for 21/66 is given by M3} = MGT(Em = 00). The Fermi transition contribution vanishes when isospin is conserved. An estimate of its contribution with isospin-mixed wave functions indicates that it is small and can be neglected [BAB 85]. The half life is given by 1 21/ 2 —T = GIMGTI , (2.4) 1/2 where G is related to fundamental constants and the phase space integral [Mut 88]. In fact, G depends somewhat on the GT strength distribution [Mut 88, Tsu 84] as well. Since the strength distribution of [Tsu 84] is close to ours, we use a value of G=1.10x10‘” yr‘1(MeV)2 deduced from the first row in Table 2.1 of Ref. [Tsu 84]. The closure approximation employed in the earlier calculations is defined by Em BCLS(E,,,) = Z < 0;“||&t-||1j,; >< 1,’;||&1-||0:r >= 2 < 0}||o},-<7”,,t,‘,‘,t,j||0f > 171,11 m=1 and BCLS M 2" l = . 2. where BCLS = BCLs(Em 2' 00). In this approximation, BCLS does not depend on the intermediate states. Estimates for the average energy < Em > of the 1+ states in 48Sc 0 22 I I l l l l l r l l' l r I I l q 0.20 l (a) _3 48 48 Ca -> T1 015 j 010 . :. D-Msosrp; As 0.05 f ' " --------------------- D-M‘H —. m : V < 33 0.00 , _< 2 0.20 ..... ___ (b) 2 "'“‘~. A-MSOBEP ; 0.15 _: 0-10 ' c-usoasP—j ‘“‘ * ...,., B-MSOBEP; 0.05 ............ 2 P1 l I 1 L 1 i 1_ I l 1 l 1 l l : MOO 5 10 15 20 En, (MeV) Figure 2.2: MGT(E,,.) as a. function of Em. The first MGT(E,,,) is fixed at E1=2.52 MeV. In (a), the solid line is obtained from the MSOBEP interaction, and dashed line from the MH interaction. The truncation for both curves is (D: n34 for “Ca and 48Ti, n5 5 for 48Sc). (b) shows the results for the MSOBEP interaction at different levels of truncation for (48Ca, 48Sc, 48Ti): A (n=0, n5 1, n=0); B (n31, n3 2, n51) and C (n52, n5 3, n_<_2). 23 were made to obtain Mé?r(cls) [Hax 84]. These previous estimates can be compared with exact results, given by the comparison between Mé'r’r and Mé‘flcls) BCLS < E... >= Mé‘i — E0. (2.7) The calculated matrix elements MGT(Em) as a function of Em for the MH and MSOBEP interactions are shown in Fig 2.2(a). There are about 300 eigenstates in each curve from 2.52 MeV ~ 15 MeV. The MGT(Em) become negligibly small after about 12 MeV even though there are still many 1+ states (over 5000) above this energy in the calculation. To understand the 66 matrix elements, we examine the 6‘ and 6‘L spectra. The theoretical B(GT‘) strengths vs Em are shown in Figure 2.3. The experimental distribution in Figure 2.3(c) represents the strength above the background line in Figure l of [And 85]. There is additional strength in the background between 4.5 and 14.5 MeV not shown in Figure 2(c) but indicated in the numerical comparisons made in Table 2.1. There may be more strength in the background above 14.5 MeV which we will comment on latter. The experimental spectrum in Figure 2.3(c) was obtained by the fitting the experimental cross section to a series of Gaussian peaks and then converting the cross section in each peak into a Gamow-Teller strength ( [And 85] and RD. Anderson, private communication). Because the experimental measurement has a finite resolution, the theoretical B(GT') spectra are smoothed by a Gaussian. The B(GT‘) spectrum with a high resolution (FWHM=100 keV) is shown in Figure 2.3(a) for the MSOBEP interaction. The low resolution spectra for the MSOBEP (solid line) and MH (dashed line) interactions shown in Figure 2.3 (b) was obtained with FWHM=400 keV. One normalized factor is introduced in Figure 2.3 to make the areas proportional to the B(GT‘) strength. The B(GT‘) values extracted from the (p,n) data are compared with the theory in Table 2.1. 24 Table 2.1: Summary of the B(GT‘) and B(GT+) values obtained from the experi- ments and compared to the theoretical calculations with the MSOBEP and the MH interactions. Em ExperimexFl MSOBEP MH (MeV) 2.52—3.5 1.30 1.32 1.24 13- 3.5—14.5 8.61+2.86b) 12.31 12.39 16.8(T24) 0.45 042(062)c 072(073)c 2.52 0.07 0.07 015 5+ 3.0—6.0 0.49 0.50 0.51 >6.0 ? 0.03 0.10 a) The experimental B(GT") and B(GT+) strengths from [And 85] and [Alf 91]. b) The B(GT) in the experimental background in the region of 4.5 SEm$145MeV [And 85]. cl The first number is the strength in the single strongest T24 state whereas the number in the bracket includes the additional strength from small states i500 keV on either side of the strongest state. For the broad peak between 4.5 ~ 14.5 MeV, the minimum experimental value of 8.61 corresponds to the spectrum in Figure 2.3(c). An additional amount of 2.86 was estimated to be in the background not shown in Figure 2.3(c) [And 85]. The theoretical and experimental shapes are qualitatively the same as well as the B(GT‘) strength values themselves (see Table 2.1). But quantitatively there are some interesting differences which indicate a preference for the MSOBEP over the MH interaction. In the pure j-j coupling model, the first 1+ excited state in 485c can be understood as a (71' f7/21/ f7/2") particle-hole configuration. The theoretical calculation based on the MSOBEP interaction and the experimental data are both in good agreement with this simple picture. But for the MH interaction, this particle- hole state is the second l+ excited state located at 3.13 MeV. The first 11' of 48Sc in the MH calculation has a negligibly small B(GT‘) value. This state, however, 25 lo ,_ r r r r i 1 r r I h I f j— , (a) r 4 5 5... “Ca -> “Sc 3 = a 4 E— __ 2 E— U —§ (b) s 5 :— N _i r‘ 4 :- z' "I “E S 3 i" " 1, . j = .q‘ ' t, = ‘ E 2 5 i i ., T 4 a 1 g. 't‘ ”' t I ‘\ ..‘ ‘ i l. :1 i .2 4 Experiment _= 3 “E f ‘ r24 __ o -- ‘g 1 1 l 1 1 1 - I A 1 E 0 5 15 . 20 10 22,,I (MeV) Figure 2.3: The B(GT‘) spectra for “sCa—i4BSc. The spectrum with FWHM 2 100 keV obtained with the MSOBEP interaction is shown in (a). The spectra with FWHM 2 400 keV obtained with the MSOBEP (solid line) and MH interaction (dashed line ) are shown in (b). The effective operator defined in Eq.(2.1) is employed in our calculations. The experimental B(GT") from [And 85] and ED. Anderson (private communication) is presented in (c). The hatched area indicates the uncertainty re- sulting from subtracting the Fermi strength in the 0"” (T24) state at 6.8 MeV. 26 0.30fi T fir [ l r 'T 1T f l_fi 'T l I 1T 7‘ 3 0.25 MH -— 020 2 B(GT‘) a 0.76 —= 0.15 - 0.10 —: 3:: 11 L1 Ii [[11 .L_ E 0.25 MSOBEP — :‘ 0.20 1: B(G‘r’) a 0.60 —-3 S 0.15 -= E 0.10 -; 0.05 l —3 0.00 L 24 e 3 0.25 Experiment -: 0.20 z: B(GT") . 0.50 -; 0.15 ' -3 0.10 -; 0.05 I | —: 0'00! 1 1 1 L m L L L l 1 1 1 1 I n L L 1 l m r 1 ‘l 0 2 4 . 5 s 10 B, (MeV) Figure 2.4: The B(GT1') values for 48Ti -»‘“Sc. The experimental values from [Alf 91] are compared to the results obtained with the MSOBEP and MH interactions. The effective operator defined in Eq. (2.1) is employed in our calculations. 27 has a relatively large overlap with 46Ca plus a deuteron-cluster configuration, which explains why the state comes low in energy. The total B(GT‘) strengths in T=4 states are 0.78 and 0.77 for the MSOBEP and MH interactions, respectively. For the MSOBEP interaction, only B(GT‘)=0.42 contributes to the single state at 16.1 MeV, the rest is spread between 15 ~ 20 MeV. ( see Figure 2a ). But for the MH interaction, most of the B(GT‘) strength (0.72) is in a single state at 15.4 MeV. Thus comparison with experiment again favors the MSOBEP interaction ( see Table 2.1 ). The [3+ strength distribution and total strengths for theory and experiment [Alf 91] are compared in Figure 2.4 and in Table 2.1. There is the possibility for B(GT+) strength above 6 MeV in the data [Alf 91] not shown in Figure 2.4. We see that the 13+ spectrum and Z B(GT+) are strongly dependent on the effective interact-ions. The spectrum for the MSOBEP interaction is in best agreement with the experiment, especially for the first state (see Table 2.1). The calculated Mgfi values are presented in Table 2.2, and compared with previ- ous calculations. We have modified the results from previous calculation to take into account the effective operator of Eq. (2.2). We note that the value of < Em >=5.86 MeV assumed by Haxton is too large in agreement with the conclusion of [Tsu 84]. We also note the excellent agreement between our result with the MH interaction and the result obtained with the new method of Ogawa and Horie [Oga 89] who also used the MH interaction. This new method implicitly takes into account the spectrum of intermediate states exactly in the full basis. But it does not produce the explicit intermediate state spectrum which was important for the fl‘ and 5+ comparisons made above. Also we note that the results obtained with the MSOBEP and MH interactions are not very different, indicating the relative stability of the calculation 28 Table 2.2: Comparison of the nuclear matrix elements B(cls) and Mg}, the average excited energy < Em > and half life T1 /2. The shell-model space configurations are described by fflg"(p3/2f5/2p1/2)" with n=0 to mm“, for the fp shell referring to the initial(i), intermediate(m) and final(f) states. Reference Interaction nm” B(cls) Mé‘r’r < Em > T1 /2 i m f (MeV)‘1 (MeV) (1019yr) Experiment > 3.6 present MSOBEP 4 5 4 0.204 0.070 1.06 1.9 present MH 4 5 4 0.213 0.055 2.01 3.0 Oga89 ‘3 MH 8 8 8 0.053 3.3 Hax84 8 KB 8 4 0.266 7.2b(1.1°) Tsu84 '3 MH 2 2 2 0.278 0.073 1.94 1.7 Zam82 "’ MBZ 0 0 0.216 Sk083 “ KB 0 0 0.150 a) Modified by taking into account the effective operator in Eq.(2). b) Based on an assumed < Em >= 5.86 MeV. C) Based on the exact < Em >= 1.06 MeV. with respect to reasonable variations in the interaction. There are several reasons why Mé‘ir in the 2uflfl decay of 48Ca. is relatively small. The energy region of the strongest B(GT‘) strength (6~10 MeV) is mismatched from the region of strongest B(GT+) strength (2.52~6 MeV ). Also there is a systematic cancellation between the M8111 in the low and the high energy part (see Figure 2.2). The qualitative reason for this behaviour can be understood as follows. In the simple j-j coupling model where the initial and final states are pure f7/2 configurations, the only partitions for the intermediate 1+ states which can be reached by 6‘ and fl+ transition are A(7rf7/2uf7‘/12), B(Tl'f5/2Vf7712) and C(Wf7/2Vf5/2Vf773). 6' transitions can go to A or B and ,8+ transitions can go to A or C. Thus the [M transition can only go through A. These partitions will be mixed in the physical system, and in particular 29 Table 2.3: The first ten B(GT‘), B(GT+) and MET values obtained with the MSOBEP interaction E... << lillét‘HOf >12 (<1;H&t+II0}‘>>2 —’——<°+”“””1§.§:;3“5‘_"°‘> 2.520 1.102 0.065 0.061 2.759 0.022 0.163 -0.013 3.122 0.180 0.120 0.030 3.620 0.010 0.000 0.000 3.789 0.037 0.146 0.013 4.257 0.053 0.015 0.005 4.425 0.048 0.000 0.000 4.934 0.002 0.014 -0.001 5.104 0.305 0.001 -0.002 5.568 0.006 0.006 0.001 mixing of B and C will lead to two states |1f >= a|B > +fl|C > and |1§F>= ,BIB > —a|C >, which can both be reached by fl‘ and fl+ transitions. The numerator of the flfl matrix element will then have the form < 0}'||c7t‘||11+ >< 11+||at‘||0,+ > + < Ojillat‘lllrf >< |12+||at-H0;t >= 06 < 0}“Ilat'llC >< Bllat‘||0,-+ > —afl < OFIIUt‘IIC >< BHUt‘HO:r >. Thus we find two 33 routes each of which is nonzero but differing in sign so that they cancel. Mixing of B and C into A is important in modifying the 5,6 strength through the lowest 1+ state relative to pure j-j coupling. This aspect of the BB strength function shows up qualitatively in all of our calculations (see Figure 2.2). And it is remarkable in our most complete calculations with the MSOBEP interaction that the total Mé‘lr matrix element(0.070) is nearly exactly equal to the contribution from the first state alone(0.061). We give the B(GT‘), B(GT+) and MGT(Em) values for the first ten eigenstates in Table 2.3 obtained with the MSOBEP interaction. They are the main positive contributions to Mg"? The states with small B(GT‘) and B(GT’“) strengths will be missed in the experiment because of the finite resolution. Consequently some states 30 will be seen in (p,n) and not (n,p) and visa versa. Nevertheless, the results given in Table 2.3 are in excellent agreement with the analysis of [Alf 91] based entirely on experimental data. To study the effects of truncation, now we discuss the several cases of interest shown in Table 2.4. The Mé‘ir for 48Ca in more highly truncated fp-shell spaces are presented, where only the MSOBEP interaction is used. One of them is obtained from the truncation (nma, :2) for 48Ca, 48Sc and 48Ti used by Tsuboi et a1 with the MH interaction. (We note that at this level of truncation the MH interaction gives the lowest 1+ state with a structure as expected in the simple picture discussed above.) The B(GT) strengths from this space will not give the sum rule (Eq. (2.1)) because the intermediate state is incomplete. However, The M34 is changed very little when nma, =3 is allowed for 48Sc. This indicates that the sum rule violation is not so important for Ma"? From the Table 2.4, we find that the Mé‘fi: in the highly truncated spaces differ significantly from the one in our expanded basis. The MGT(Em) spectra in Figure 2.2(b) show these differences in detail. To test the accuracy of our truncation, we compare the calculations for M3} values in the space we used and in the full—basis for 22O in the sd shell and 46Ca in the fp shell. These comparisons indicate that the truncation we used is a good approximation to the full space results. We may expect that the present Mé‘r’r value of 48Ca will be reduced a further 5~10% if the full—basis in the fp shell is employed. (Compared with these more complete calculations, a previous estimate [BAB 85] of the extrapolation from the nm” =2 space to the full-space value for M341 is found to be in error by about a factor of two.) Beyond the fp shell model space there are several processes which we should consider. The role of A-isobar admixtures have been investigated in previous work 31 Table 2.4: Comparison of Mé‘fi- in different truncations. The shell-model space con- figurations are described by f?fi"(p3/2f5/2p1/2)" for the fp shell and dg7zn(sl/2d3/2)" for the sd shell referring to the initial(i), intermediate(m) and final(f) states with n=0 to mm”. The full-basis means nmu=8 in the fp shell or nmfl=6 in the sd shell. The MSOBEP interaction was used for 48Ca and 46Ca and the interaction [Wil 84] was used for 220. 1,... Ma”. i m f 0 0 0 0.124 0 1 0 0.143 “Ca —> 4’sTi 1 2 1 0.049 2 2 2 0.086 2 3 2 0.088 4 5 4 0.070 “Ca —+ 46"Ti 4 5 4 0.134 Full Full Full 0.127 2 2 2 0.077 22O —> 22Ne 4 5 4 0.041 Full Full Full 0.039 [G00 81, Bob 81, Ari 87, Tow 87], The contribution from the direct excitation of the A-isobar nucleon-hole configuration, for which the excitation energy is about 300 MeV, is negligible [Zam 82, Gro 86] because of the cancellation between 5+ and fl" and because of the large energy denominator in Eq. (2.3). The A-isobar admixtures in the low-lying states are already approximately taken into account in our calculation in the effective operator 6t of Eq. (2.2) as well as in the effective interaction. In addition, 2p2h admixtures beyond the fp shell can lead to B(GT) strength at higher excitation [Ber 82]. The possible strength seen experimentally in the background above 6 MeV in 0" and 15 MeV in [3‘ may be due to these 2p2h admixtures. The effect of these 2p2h admixtures in the low-lying states are also approximately taken account in the effective operator and effective interaction. The contribution from the direct excitation of the 2p2h configurations may again be small because of cancellation 32 and large energy denominator but should be investigated further. 2.4 Summary In summary, we have studied the 2146B decay of 48Ca in a large basis shell-model space. An effective Gamow-Teller operator (”It is employed, which well describes B (GT‘) and B(GT+) behaviour in the energy region (2.5~15.0 MeV). Of two effective interactions we have employed, new MSOBEP interaction seems to be a better interaction for the fl‘ and fl+ spectra. With this interaction we predict the 2uflfl decay matrix element of 48Ca is Mé3r=0070 giving a half life T1/2= 1.9X1019yr, which differs by nearly a factor of two from the experimental limit [Bar 70] of T1/2 >3.6x1019 yr. We believe that the most important aspect of these calculations which cannot be directly tested by the (p,n) and (n,p) experiments is the amount of strength in the (n,p) 6+ spectrum above 5 MeV in excitation. This is because there is a large uncertainty in the amount of Gamow-Teller strength in the background above this energy. The Gamow-Teller strength in this region may be sensitive to further refinements in the effective interaction as well as to direct excitation of 2p2h states, and should be studied further. In addition, we believe that it is important to confirm and improve upon the present experimental limit. Chapter 3 Comparison between anRPA and Shell Model I: 3"“ Decay 3. 1 Introduction The Gamow—Teller transitions are of interest in their own right in addition to their role as virtual transitions in 66 decay. The transition strengths of 6+ decay in heavy nuclei are understood poorly [Wap 71, Kle 85]. In recent years, the anRPA theory has been employed to calculate the ,8 Gamow- Teller transitions in heavy nuclei [Cha 83, Sub 88]. In the anRPA, the proton- neutron correlation plays an important role. This equation contains two types of interactions, particle-particle and particle-hole, but the former was neglected in early calculations [Hal 67]. Recently several authors have investigated the particle-particle interaction term which was reintroduced by Cha [Cha 83], and found that the 6+ transition matrix elements are sensitive to this term [Cha 83, Sub 88]. As mentioned in Chapter 1, the anRPA is an approximation model, so there may be some correlations that could be important in 6+ decay which are not included in the anRPA. Therefore it is very important to examine the validity and accuracy of the anRPA approach. These tests can be achieved by making a comparison of the 33 34 anRPA and full-basis shell-model calculations in the nuclear mass regions where the full-basis shell-model calculations are possible. The full-basis shell-model calculations include all types of the correlations within a major shell. We believe that such a comparison is meaningful if and only if the anRPA and shell-model calculations are performed in the same model space and use the same effective interaction. This kind of comparison has been made by Lauritzen [Lau 88] for several sd shell nuclei, Brown and Zhao [Brz 89] for 28Mg and Civitarese, et.al. [Civ 91] for 2GMg. Lauritzen and Brown and Zhao have concluded that the anRPA does not include some important correlations in fl+ decay and fails to reproduce the shell-model results. Civitarese contradicts this conclusion. However, their comparisons only concentrate on total B(GT+) and strength distributions, and relative contributions of the various single-particle state in Gamow-Teller transitions are not considered. In this Chapter, a self—consistent BCS-anRPA is developed in sections 3.2.1 and 3.2.2. In section 3.2.4, we introduce the coherent one-body transition density (COBTD) and coherent transition matrix element (CTME), which can describe the single-particle state effects in one—body transitions. Our comparison between the pn- QRPA and shell model is presented in section 3.3, where we investigate total B(GT+) strengths, strength distributions and COBTD and CTME. In our study, 46Ti ,6” de- cay is the example. The model space is the full fp shell and the effective interaction used is MSOBEP [Ric 91] (see Chapter 2). We give the summary and conclusions in section 3.4. 35 3.2 Formalism 3.2.1 anRPA equations We start with the equation of motion [Row 70, Rin 80], where the excited eigenstates [u > are constructed from the phonon creation operator Qt which is defined by [V >= QZIO >, and QVIO >= 0, for all u (3.1) where [u > and [0 > are the excited eigenstates and the physical ground state. They satisfy the Schrédinger equation, Hlu >= EL,[V > and HIO >= EoIO > . (3.2) Then one obtains the following equation of motion from the above relations; [H1Qlllo >= (Eu — Eo)Ql.|0 >- (3.3) Multiplying from the left with an arbitrary state of the form < 0|6Qy, we get < 0|l5Qu,lH1Qllll0 >= hw < 0|[5QWQIIIO >» (3-4) where hw : Eu — E0. In order to obtain the anRPA equation, the phonon creation operators in the angular momentum coupled representation are written as [Hal 67, Cha 83, Lau 88], QanM) = ZfXEMLJPnJMFYumfipnfpnaJMD, (3-5) Pun A;n(pn,JM) = 2 chPc§nmn, (3.6) mpymn ~ Apn(pn,JM) = (—1)J+M.Apn(pn,J——M), (3.7) 36 where cl”) is the quasiproton (quasineutron) creation operator. In terms of the spherical shell-model states, the particle and quasiparticle creation and annihilation operators are related by the Bogoliubov transformation, e.g. for proton Gimp = “10“sz 'l’ (—)Jp+mpvpajp-mp’ (3'8) where 21,2, + v; = 1, and a}p(a,-p) is the proton creation (destruction) operator for the single-particle state. of, turns out to be the occupation probability. The eigenvalue ha: and the forward- and backward- going amplitudes X and Y are obtained by solving the anRPA equation (:2. mm =10), with the orthogonality and normalization relation axgnxg’r‘ — Yuri/3”] = 5.... (3.10) I)“ The matrix elements A and B are explicitly given by [Hal 67, Cha 83, Lau 88, Sub 88] Agnp’n’ = < QRPAllAPn(pn’ JM)’ [H’ A'fm(p’n’7 JM)lllQRPA > 2 < BCSll‘APn(pni JM), [H7 A;n(p’n’a JM)lllBCS > = (E,+E,)5,,,5m.+(H;:)g,,,,,, (3-11) 3:... = < QRPAIM..(pn.JM).[H. fl..(p'n'.JM)]1IQRPA> y. < BCS|[Apn(pn,JM), [H.lenfp’n’JMHHBCS > = —(G‘°")Z...I..~ (3-12) with (H5: = gv< + 1.0.1.) +91”; Wilnplnl (upvnuplvfl + vpunvp1un1), (3.13) 37 J J (Gpn)pnp,n, = gppignp,n,(upunvplvnl+vpvnup/unl) —gphWPan,n,(vpunuplvnl + upvnvp/unl). (3.14) where IQRPA > is the QRPA ground state defined in Appendix A. The relation between the QRPA and anRPA is discussed in Appendix B. The quasiproton and quasineutron energies E, and E,1 and occupation factors u, v are obtained by solving the BCS equation in section 3.2.2 or Chapter 5. The matrix elements of the particle- particle (V) and particle-hole (W) interaction are related by Pandya transformation, . . ' ‘ . .n J ' Wlfnyp'n’ : —(_1)JP+Jn+JPI+JnI ;(2J’ + 1){ j: 31’: J’ }VP{I’,P’n' (315) In order to discuss the results as a function of the strength associated with each part of the interaction, the multiplicative factors gph and gpp are conventionally introduced for the particle-particle and particle-hole, respectively. They are both equal to one in the standard anRPA theory. The charge-exchange transition matrix elements of the Gamow-Teller operator between the ground state |0:F > and the excited state l1: > are given by l 2Jg+l B(GT) = (< ltllotIIOZ‘ >)2. (3.16) where J,- = 0. We denote M,(GT)=<1j[lot||0,-+>, (3.17) where 0' is the Pauli spin operator. The isospin operator t can be the raising or lowering operator, t+ or t‘, which are corresponding to 3* and fl‘ Gamow-Teller transitions. In the anRPA, we can obtain Mu(GT’) Z < Pllalln > (Xfimupvn + Yvapun), (3-18) pn M,,(GT+) — Z < p||a||n > (anvpun + Yupnupvn). (3.19) pn 38 The above equations for M‘,(GT) in the anRPA can be rewritten as, MU(GT) = Z < pllalln > OBTD(p,n, V)QRPA, (3.20) pn where the OBTD(p,n, V)QRPA is a one-body transition density of the anRPA given by the Eq. (3.18) or Eq. (3.19) explicitly. 3.2.2 BCS equations The BCS theory will be reviewed and extended in Chapter 5. Here we just present the standard BCS formalism for the proton-neutron system. In the BCS, the quasiproton energies Ep, occupation probabilities of, and pairing gaps AP are given by E, (5,, — A”)2 + A3,, (3.21) 5,, — A, (5,, - Ar)? + A: '23" +1 A = — _p—u I‘U IVJ=9 ,, (3.23) P 2p; 2JP+1 p p PP1PP where A, is the proton Fermi energy, 5,, and Vppvpzp: are the single proton energy and 2 _ l _ up _ 2(1 ), (3.22) proton two-body interaction, respectively. The above equations can be solved using the constraint for the total proton number N,r = 232]}, +1)vf,, (3.24) p to determine the constant A”. A similar set of the equations can be solved for neutrons. The single proton energies 6,, are related to the bare single—particle proton energies 52 at the closed shell by addition of the rearrangement terms Fr, 5,, = 52 + Pp, (3.25) 39 1 Pp = -.—— 2(1 + 6pp’)v:’ 2(2‘] + 1)‘/P‘II’/sPPI 2],, + 1 p, J 1 2 J + (2].? +1); vn, zJ:(2J +1)i/,,,,,,m,. (3.26) The first term refers to like particle correction and the second term to unlike particle correction. The rearrangement term for 6,, has the same form but with the p / n indices interchange. Eq. (3.26) can be verified by simple shell-model calculations. The BCS equations (3.21-3.24) plus the rearrangement terms can be solved iteratively. The Eqs. (3.9 —- 3.14,3.21 — 3.26) are called the self-consistent BCS-anRPA equations. The self-consistent means that the input ingredients in the BCS-anRPA are consistent with those in the shell model, namely, the bare single particle energies E? at the closed shell and two-body interaction matrix elements Vii-kl. 3.2.3 B(GT) in shell-model calculations In the shell-model calculation, the Gamow—Teller strength is equal to the product of one-body transition density and single particle matrix element [Bru 77], and the matrix element ML, in Eq. (3.17) is expressed by Mu(GT) = Z < PHUH" > OBTD(P,n, V)SMa (3-27) pn where [a;' (8 (”MM OBTD(P7n,l/)SM =< HINT“ HO?“ > , (328) where AJ = 1. The one-body transition density describes the shell configuration effects when a proton in jp orbit transfers to a neutron in jn orbit. In this work, 40 OBTD(p,n,u)SM is calculated from the OXBASH shell-model code [Etc 85]. The proton (neutron) occupation probabilities in the shell model are given by < SMlaan)ap(n)|SM > . 3.29 where |SM > denotes the shell-model wave function. 3.2.4 The coherent one-body transition density The B(GT) spectrum itself lacks information about the single-particle state contri- butions in the charge-exchange process, because we sum over all single-particle state components p, n in Eqs.(3.18—3.20) for the anRPA and Eq.(3.27) for the shell model. Therefore, we introduce two quantities which can describe such single-particle state effects in the Gamow-Teller transition. We define the coherent one-body transition density (COBTD) as COBTD(p, n E MV(GT)OBTD(p, n, u), (3.30) ,_ _1__ M2,, B(GT) u where ML,(GT) is given by Eq. (3.17). The OBTD is given by Eq. (3.28) and (3.20) for the shell model and the anRPA, respectively. Also the coherent transition matrix element (CTME) is defined by CTME(p,n) =< pllolln > COBTD(p,n). (3.31) The COBTD and CTME are a function of the single-particle state components. All final states(V) are summed up in Eq. (3.30). The relation between the B(GT) and CTME is given by {)3 CTME(p,n)}2 = Z B(GT). (3.32) pm In Appendix D, we will discuss the COBTD in detail. 41 Table 3.1: 46Ti: Proton and neutron: single particle energies 5,- (MeV), gap parame- ters A.- (MeV) and occupation probabilities v? from BCS and the shell-model E,(SM) and v,-2(SM) . The Fermi energies obtained from the BCS calculation are -12.968 MeV and -10.868 MeV for proton and neutron, respectively. level 5; A.- v? E,(BCS) E,(SM) v,-2(SM) 7rf7/2 -l2.234 1.206 0.239 1.412 1.412 0.187 7rp3/2 —8.194 0.911 0.009 4.861 4.305 0.078 7rf5/2 -5.739 1.314 0.008 7.347 6.845 0.022 «pl/2 —5.994 0.874 0.004 7.028 6.431 0.031 Vf7/2 — 10.809 1.447 0.480 1.448 1.414 0.404 Vp3/2 —6.852 1.104 0.018 4.164 3.336 0.092 Vf5/2 —4.044 1.574 0.013 7.003 6.099 0.054 Vpllg —4.553 1.069 0.007 6.405 5.564 0.040 3.3 Calculations and Discussions 3.3.1 Comparison of anRPA and shell model We have presented the self-consistent BCS-anRPA equations in section 3.2.1 and 3.2.2. In the BCS equation, the pairing gaps are state dependent and are self- consistently calculated. But in practice, when the anRPA equation is used in heavier nuclei or in the simple BCS calculations, the quasiparticle energies and occupation probabilities are not obtained in this way, but rather based on some empirical value for the pairing gap A ( often one which is independent of single-particle state ) and on some interpolated values for the effective single-particle energies. The single-particle energies with addition of rearrangement terms, quasiparticle energies, pairing gaps and occupation probabilities of 46Ti are given in Table 3.1. The proton pairing gaps are almost 40 % less than those obtained from the empirical formula AP = 1224-1”, where A is total nucleon number in the nucleus [Rin 80]. 42 4 T r r I T T T f F r fl I j ' PW" «n-» “s. 1 - llrnodol 4 3_ she _l A _ 1 * Cl E's ' --_ 2— F‘" — E I. I «4 DJ _ ___| q . | .. "’ : I l~‘¢- 1— | —1 . I . i «It .. I ..... ‘ . , q o'n.'...1.. Laa.2l..n. -5 0 5 10 15 E Figure 3.1: Summed Gamow-Teller strength for “Ti -)‘°Sc. The solid line is the shell-model result while the dashed line is the anRPA result. 43 4- I I ‘ r r f I I r r r I r f .- ' 48Tl --> 463c: 4| .. .- ----- 4 .. "" 8,,-0.0* F, 1, ..... 8 _05 3— ” P-- - ""‘" 3,-1.4 , _.; .................. _j — shell modolnIm; i . : ' q f‘ ' : , J ‘55 ' : , — -—I BE 2_ , . a ' I ‘ . . , , I . i 4 I ——I ' d . L . . 1 , 1 1° 15 Figure 3.2: Summed Gamow-Teller strength for 46Ti —>“6Sc with various particle- particle strength 9,, where 9,). = 1. 44 We put these BCS parameters into the anRPA equation and obtain the anRPA fl+-decay spectrum. The full-basis shell-model calculation is carried out with the OXBASH code. The J-dimensions are 1514 and 2042 for the “Ti ground state and the 46Sc 1““ excitations, respectively. Figure 3.1 presents the running sum 2 B(GT+) as a function of the 46Sc 1+ excitation energy Ex corresponding to the 46Ti ground state. The running sum 2 B(GT+) is defined by z: B(GT“) = :V:(< 1j||0t+||0f >)2. (3.33) The anRPA and shell-model results are shown by dashed and solid lines, respec- tively. The Coulomb shift 7.586 MeV is taken into account [Bro 79]. We find that the anRPA calculation does not give sufficient suppression for the total B(GT+). It overshoots by about 50 % compared to the shell-model result. The shapes of the two models are also different. The energies of the first excited state in the two models differs by about 6 MeV. The coherent one-body transition densities and the coherent transition matrix elements defined by section 3.2.4 are given in Table 3.2 and Table 3.3 for the anRPA and shell model, (see columns (A) and (E)). There are significant difference between the two models. For example, the COBTD and CTME values in the anRPA and shell model have the opposite sign for the f7); -> f7); and the f5); —; f5/2. Since the B(GT+) strength are more sensitive to gm, than gph, we will set gph = 1 and discuss the dependence on gpp. In Figure 3.2, the B ( GT") spectra with various 9“, values are presented. One can find the anRPA results are very sensitive to g,,,, values, which is in the agreement with the previous conclusions [Cha 83, Lau 88, Sub 88]. Especially, the strengths in the low-lying states decrease rapidly around the value 9sz = 1. The total shell-model B (GT+) is reproduced by the anRPA with gpp = 1.4, 45 Table 3.2: Comparison of the coherent one-body transition density (COBTD) ob- tained in the anRPA, modified anRPA and shell-model calculations of 46Ti. (A): COBTD in the anRPA with BCS occupation and quasiparticle energies; (B): COBTD in the anRPA with BCS occupation and SM quasiparticle energies; (C): COBTD in the anRPA with SM occupation and BCS quasiparticle energies; (D): COBTD in the anRPA with SM occupation and quasiparticle energies; (E): COBTD in SM (shell-model). jp '9 in A B C D E f7/2 —> f7). 0.034 0.019 -0005 -0031 -0.096 f7/2 -* fs/g 0.443 0.448 0.310 0.315 0.386 173/2 -> 193/2 0.002 —0.001 0.075 0.072 0.006 193/2 —> f5/2 -0.001 -0.001 -0.006 -0.010 0.017 p3/2 -* pl/z 0.005 0.003 0.087 0.088 0.060 f5/2 -* f7/2 0.088 0.099 0.091 0.109 0.086 f5/2 -> [23/2 0.001 0.002 0.006 0.009 0.010 fs/z —+ f5/2 0.001 0.003 -0.007 -0.005 -0.011 191/2 -* 123/2 0.003 0.005 -0.012 -0.006 0.009 P1/2 —+ 191/2 -0.001 0.000 -0.012 -0.013 -0.003 46 Table 3.3: Comparison of the coherent transition matrix elements (CTME) obtained in the anRPA, modified anRPA and shell-model calculations of 46Ti. Labels (A), (B), (C), (D) and (E) are given by Table 3.2 ip-ein A B C D E f7,2-+f7,2 0.108 0.062 -0017 -0099 -0.308 fm—>f5,2 1.641 1.661 1.149 1.167 1.431 p3/2-+ 173/2 0.005 -0002 0.195 0.186 0.014 p3/2—> f5/2 0.000 0.000 0.000 0.000 0.000 pan-+1)”; 0.011 0.006 0.200 0.202 0.139 far» fm -0325 -0.366 -0.338 -0403 -0319 fag—>103). 0.000 0.000 0.000 0.000 0.000 fag—45,2 -0002 -0005 0.014 0.010 0.024 pug—4123,. -0.006 -0011 0.027 0.013 -0020 Pug—>131” 0.000 0.000 0.010 0.011 0.002 but the shape of the strength distributions of two models are totally different. When gm, increases up to a certain value, the lowest eigenvalue becomes the imag- inary and the anRPA equation collapses. It means that the anRPA theory is no longer a valid model. Around this gpp, the equation gives unrealistic large amplitudes X and Y, and consequently presents an unphysical B(GT+) strength. 3.3.2 Phenomenological improvements of anRPA We now investigate various ways to understand and then improve the agreement be- tween the anRPA and the full-basis shell model. First, we introduce shell-model quasiparticle energies. Since the BCS quasiparticle energies can be understood as the lowest excited energies of the odd nucleus [Row 70, Law 80], one may appropriately analyse the odd nucleus energy spectra in the shell-model calculations, and find those excited states which are qualitatively equivalent to the single quasiparticle E, excita- tions. The overlap method is employed to find these states [Etc 85]. For example, the 47 shell-model quasineutron energies of 46Ti can be obtained as follows: the one-particle transfer amplitudes are calculated between the 46Ti ground state and the 47Ti excited states J = 7/2’ (or 3/2“, 5/2" and 1/2'). The eigenvalue of the state which has the largest overlap in one-particle transfer is considered as the shell-model quasineutron energy E ”/2 (or Ep3/2, E ,5/2 and Epl/z). Possible improvements of the anRPA may be obtained by replacing the quasi- particle energies and occupation probabilities of the shell model to those of the BCS in the anRPA. The shell-model occupations of 46Ti ground state are evaluated with Eq. (3.29), where |SM > is the ground state wave function. The shell-model quasipro— ton and quasineutron energies are obtained by analysis of 46Ti isotopes and isotone in the shell-model calculations. These parameters are given in the Table 3.1. The modified models are called “ hybrid ” anRPA. The calculations for three types of “ hybrid ” anRPA are shown in Figure 3.3. The dashed line is the anRPA with BCS occupations and shell-model quasipar- ticle energies. The dotted line is the anRPA with shell-model occupations and BCS quasiparticle energies. The dot-dashed line is the anRPA with shell-model occupations and quasiparticle energies. One finds that the total B(GT+) strength is suppressed in the “ hybrid ” anRPA. That is 2.053 in the anRPA, 1.811 in the “ hybrid ” anRPA with the shell-model quasiparticle energies and the BCS occu- pation probabilities, 1.536 in the “ hybrid ” anRPA with the BCS quasiparticle energies and the shell-model occupation probabilities, and 1.181 in the “ hybrid ” pn- QRPA with the shell-model quasiparticle energies and the occupation probabilities. It is close to 0.928, the total B(GT+) in the full-basis shell model. The “ hybrid ” models have not improved the strength distribution. But com- paring to the shape of the spectrum of the anRPA with gm, 2 1.4 in Figure 3.2, 48 the “ hybrid ” anRPA with the shell-model parameters (the dot-dashed line in F ig- ure 3.3) still keeps a reasonable shape. We know both of them almost reproduce the shell-model total B(GT+) value. In the “ hybrid” anRPA, the position of the 1+ state almost remains the same, i.e., still differs by 6 MeV to the energy from the shell-model calculation. The COBTD and CTME values of the “ hybrid ” anRPA are presented in the columns (B), (C)and (D) of Table 3.2 and Table 3.3. The COBTD and CTME in the column (B) are obtained by using the shell-model quasiparticle energies and the BCS occupation probabilities in the anRPA. We find that the difference between the columns (B) and (E) is decreased only for the transitions f7); -—> f7/2, but the COBTD and CTME in other transitions become worse or remain the same. The COBTD and CTME values in the column (C), using the BCS quasiparticle energies and the shell-model occupation probabilities, and the column (D), using the shell-model quasiparticle energies and the occupation probabilities, present similar behaviour. The COBTD and CTME in the transition f7/2 —+ f7); now have the same sign as those in column (B). The COBTD and CTME for the transition p3/2 —> p1/2 are and f5); —9 f5); have been improved over those in the anRPA. But other COBTD and CTME values become worse compared to those in the anRPA. We compare the occupation factors between the BCS and shell-model in Table 3.1 and find in order to reproduce shell-model occupation factors, the pairing gap in the BCS equation should be unrealistically increased to around 4 Mev [Lau 88]. It requires a very strong and unrealistical effective interaction. The present BCS gaps are around 1 MeV. We have made several similar comparisons for some nuclei in sd shell and 48Ti in fp shell. Similar conclusions are obtained. The detailed calculations for 48Ti have 49 been shown in ref. [Brz 90]. 3.4 Summary and Conclusions We have investigated the anRPA as a model to study [3+ decay. The Gamow- Teller transition strength B(GT+) for 46Ti —r46Sc, asian example, is calculated by the anRPA and full-basis shell model. The formulation of the self-consistent BCS- anRPA is given in the section 3.2. The coherent one-body transition densities (COBTD) and coherent transition matrix elements (CTME) are introduced and ap- plied for analysis of the single-particle state contributions in 6+ decay. The compar- ison of the anRPA and shell model is made, including the total B(GT+), shape of the strength distribution, COBTD and CTME values. Our comparison shows that the anRPA cannot reproduce the shell—model results. The large disagreements im- ply that there are some correlations which are important to 3+ decay but have not been taken into account in the BCS-anRPA equations. Our results agree with those of [Lau 88] and [Brz 89] who made similar comparisons for some sd shell nuclei. We confirm that the particle-particle interactions in the anRPA provide the suppression mechanism in 6* decay in agreement with previous studies. The total B(GT+) is decreased when the parameter gm, is increased. But we note that the shape of the spectra is poor when gm, is increased beyond unity. The shell-model quasiparticle energies and occupations are defined in section 3.3.2 and Eq. (3.29). Empirical improvements for the anRPA, namely, the “ hybrid ” anRPA, have been introduced by replacing the shell-model quasiparticle energies and/or occupation factors in the anRPA equation. Suppression of total B(GT+) and some improvement in the COBTD and CTME are found in our study. The calculations show that the main suppression mechanism is from the shell-model occu- 50 4 - Y T r T ‘l’ I r T— I— l I T . If , ‘i I - - - - - shell model B. BCS 11.17 43 . 45 4 - --------- BCS 8. shell model u,v T1 --> SC 4 r- - - -- -- shell model E.u.v J 3 1— -— shell model fl " 1 A _ «I + 8 - .. E5 2 :- _ _ _ ‘ . r - d '4 N . ' ---------- .f _ ,---' ............ . - I , _ - _ . _ ..... l... .‘ " _ 1 ,. .. -I r _ _ " I ..... ¢". “ " — .. - Jl .. I I " '#....-....... ........ ‘1 I- I: -I o 1 1L 1_ l L L g L L g 1 1 l 1 1 1 l 1 -5 0 5 10 15 E: Figure 3.3: The anRPA result with the shell-model quasiparticle energies and/or occupation probabilities ( see Table 3.1 ). 51 pation factors. Thus we conclude it may be possible to put together “hybrid” models of this type that are more reliable than the conventional anRPA. Also we note that a solid theoretical study is necessary to generalize the anRPA model, and this will be discussed in Chapters 5 and 6. Chapter 4 Comparison between anRPA and Shell Model II: 21266 Decay 4.1 Introduction Study of 21166 decay is an important test of our understanding of nuclear structure properties since the decay process occurs within the standard weak interaction model. But until 1986, there have been large discrepancies between the experimental results and the simple shell model and/or other model’s predictions. The theoretical half- lives of the double-beta decay nuclei are almost 5 ~ 100 less than the experimental ones, i.e., the experimental 21166 decay matrix elements are strongly suppressed (see Eq. (2.4)) [Hax 84, Doi 85]. In recent years, the anRPA has been applied to calculate 66 decay matrix ele- ments [Vol 86, Civ 87, Eng 88, Mut 89]. Several calculations indicate that 21166 decay matrix elements are suppressed if the particle-particle interaction term is included in the anRPA equation. They decrease rapidly when parameter gpp is increased (g,,, is introduced as a multiplicative factor to the particle-particle interaction, see Eqs. (3.13,3.14)). Also previous studies have shown that the anRPA equation tends to be unstable and collapse when gm, is larger than unity [V0186, Civ 87, Eng 88, 52 53 Mut 89]. In Chapter 3, we concluded that the anRPA has not taken into account some correlations but which are important in 6+ decay. Therefore it is necessary to test the validity of the anRPA in 66 decay. In 1989, Brown and Zhao [Brz 89] made a comparison of the anRPA and full-basis shell-model calculations for hypothetical 66 decay of 28‘Mg in the sd shell. In their work, the BCS equation was solved by assuming a state-independent gap value which was obtained by analysing experimental data. Later Muto et. al. [Mut 91] presented a similar comparison for the 66 decay of 48Ca. But in their work, the model spaces of the anRPA and the shell model are not the same: the former is the (sd + fp + g) shells, where the g shell means (199/2, 1g7/2), and another is the fp shell only. Therefore the comparison may be meaningless because a meaningful comparison requires that the calculations of these models are performed in the same model space. On the other hand, the state-independent gap in the BCS was used in their work as well. In this Chapter, the comparison of the anRPA and the full-basis shell-model calculations is made for 2116 6decay of 46Ca. The model space and effective interaction are the fp shell and MSOBEP interaction, respectively. The self-consistent BCS- anRPA equations given in Chapter 3 are used. Two types of 21166 decay matrix elements are discussed here, the energy dependent and the closure 66 decay matrix elements. The first gives the exact 2116 6 decay matrix element, and the second relates an approximate method, namely, the closure approximation ( see Chapter 2 ). The energy diagram for mass A=46 is presented in Figure 4.1. It is obvious that the 66 decay is the only decay mode for 46Ca ground state because all single 6 decays are forbidden. In section 4.2, the formulas for 21166 decay matrix elements are presented. The 54 calculation results are given in section 4.3. The summary and conclusions are given in the final section. 4.2 Formalism of 21166 Decay In the shell-model calculations, the (intermediate state) energy dependent matrix element is defined by M (E )_ E": < Oilldt‘llli. ><1:;||at-Ho;+> GT m — Em-Ei'i‘TO/z-l'mccz , m=1 (4.1) where Em are the 1+ excitation energies of the intermediate states, E,- is the intial state energy, To/ 2 is Q-value of 66 decay, for 46Ca, To = 0.986 MeV. m..c2 is electron mass. The total matrix element for 21166 is given by MGT = MGT(Em = 00). The closure matrix element is defined by (see Chapter 2) Em BCL5(Em) = Z < Div-Hat-III; >< 1;“0t7H0?’ > . (4.2) m=l The total matrix element is given by BCLS = BCL5(Em = 00). In the anRPA calculations, the 66 formulas become more complicated because the summation in Eqs. (4.1 — 4.2) involves the product of two transition matrix elements and each one contains the intermediate states of the intermediate nucleus. Of course, in the shell model, the intermediate states in the two transition matrix elements are the same. But in the anRPA, we recognize that the intermediate states in the two transition matrix elements are different. They depend on which parent nucleus is being considered. Thus a problem arises that the intermediate states resulting from the two different anRPA calculations are not orthogonal. Of course, they should be the same physically. In order to solve such a mismatch problem, we 55 4+ 0+ 463c 46 ‘45 - 46Ti —46 ._ Mass excess (MeV) Figure 4.1: Mass spectrum for A=46 nuclei, where the double-beta decay is the only possible decay mode for the 46Ca. ground state 56 introduce an overlap matrix element between any two intermediate states J; and J,’,',. < ngJg. >= Z(X,f,’}’flX,€,"'J" — Ygr'flygzn’”) (4.3) pn where m, m’ denote two states with eigenvectors (X, Y) and (X, Y), which are con- structed from intial and final states, respectively [Civ 87, Gro 86, Mut 89]. Obviously, if (X, Y) and (X, Y) are identical, then we have < J,’,',IJ,’,',. >= 6mm: (see Eq. (3.10)). With the overlap matrix element, Eq. (4.1 - 4.3) can be rewritten by < 0}||at-||1;; >< 1.1.0.1: >< 1;.||at-||0,+ > M E... = , . GT( ) 7;”, Em — E.- + To/2 + mec2 (4 4) BCL5(E.,.) = Z < 0}||at’||1j, ><1,T,|1;.><1;.||at‘||0§'>. (4.5) m,m' 4.3 Results and Discussions In this section, we compare the calculation results of the anRPA and full-basis shell model for the 21166 of 46Ca. In this case, the initial (final) nucleus has 0 (2) protons and 6 (4) neutrons in the fp shell. The double-beta decay matrix elements MGT and/or BCLS consist of the virtual decay routes, 6’ and 6+ decays. The 6* Gamow-Teller transition strength in the pn- QRPA was presented and discussed in Chapter 3. Here the running sums Z B(GT‘) of the 6" decay of 46Ca are shown in Figure 4.2. The dashed line is obtained from the BCS-anRPA and solid line from the full-basis shell model. The Coulomb shift 7.173 MeV is included in the calculations [Bro 79]. The matrix B in the anRPA is zero because 1),, = 0. Then the anRPA reduces to the anTDA. The amplitudes Y are consequently equal to zero. The 6+ transition strength of “Ca vanishes because there are no valence protons in the fp shell. So the total B(GT‘) strength is equal 57 to 18 because of the sum rule Eq. (2.1). In Figure 4.2, the first 1+ eigenvalue of the anRPA differs by about 5 MeV to that of the shell model. In Figure 4.3 and 4.4, we present the running sum of matrix element MGT(Em) and BCL3(E'm) as a function of the 1+ excitation energies corresponding to the 46Ca ground state. The excitation energies are obtained by the anRPA calculation for 46Ca -—?4SSC, and are employed to evaluate the energy denominator in Eq. (4.4). Qualitatively the anRPA and shell-model show a similar behaviour. There is a cancellation between the matrix elements in the low- and high-lying states. The relative shapes qualitatively agree but quantitatively disagree with each other. We find the anRPA does not provide enough suppression for total MGT and BCLs, which are about three times larger than those in the full-basis shell-model calculation. In detail, the matrix elements in the low-lying states are rather different between the two models, for example, the first BCL5(E1) is almost 8 times larger than the corresponding shell—model result. On the other hand, in the anRPA, the first B(GT‘) and B(GT+) are dominated by the transition f7/2 —> f7/2, but in the shell—model, this transition contributes only about 50 %. We present the running sum of BCLs(Em) with respect to various gm, values in Figure 4.5 to 4.7, where we keep gph = 1. The particle-particle interaction suppresses the total matrix elements in agreement with previous studies [Vol 86, Civ 87, Eng 88, Mut 89]. If the particle-particle channel is shut off (gmD = 0 ), the cancellation of BCLS(Em) between the low- and high-lying excitations disappears as shown in Fig- ure 4.5. However, the cancellation emerges and becomes stronger and stronger as gm, is increased. At gm, = 1.28 (Figure 4.7), the anRPA agrees with the total shell-model BCLS but fails to reproduces the relevant shape. At gpp = 1.20 (Figure 4.8) the anRPA 58 20 T I I T I T I I T r T I Y T7 I I T ‘1 - - - - - anRPA “Ca --> “SC ' .. I 1 - shell model r 4 15 - L. r‘ " . [:5 .- 10 —- ‘ ES _ . («I - 1 . '1 5 _ —I . . 1 oo 1 J— l l 20 Figure 4.2: Summed Gamow-Teller strength B(GT') of “Ca ->“68c. The dashed and solid lines are the anRPA and shell-model results, respectively 59 0.5 - I I I T I r T I1 r r T If f I I’ ' I l I ‘ . i - - - - anRPA “Ca --> “T1 3 l. 0.4 )— —— shell model d A 0.3 1" 4 .P - -| — a I- - - " l '4 fa} .. I a v d 5 ' ‘ - i - _ - < z - — 0.2 l— ‘ I- 1 0.1 #— J¥L l L L_‘L L 1 1_ L 1 L L L .L 0'00 15 20 25 Em (MeV) Figure 4.3: Summed energy dependent 21166 decay matrix element MGT(E,,,) as the function of the excitation energies, the solid line and dashed line are the shell-model and anRPA results, respectively 60 5 I F T T I'— I I r I I I I I I I - - - man “Ca --> “Ti -— shell model 4 801303111) 4 1 1 l l l l 1 l 1 l L |_-1_L_1 l l J A l_L_ I N IUIIIIUITTTITIIIIIIIIIII N (I Figure 4.4: Summed closure 21166 decay matrix element BCLS(E,,,) as the function of the excitation energies, the solid line and dashed line are the shell-model and anRPA results, respectively 61 5- I I I I I r fir IT T I IT r I I I— T I - I - - - 3,, - 0.0 “Ca --> “r1 1 - — shell model . 4— — .. p-—-4 1 - 1 A 3 -— —I B . . El ‘3 " a an ' . 2 - _. C q 1_ —-I - 4 - 4 o L l L L L L I O 20 25 Figure 4.5: Summed closure matrix element BCL3(E,,,), the solid line is from the shell- model calculation, the dashed line is anRPA result with 9,, = 0.0 and 9,), = 1.0 “bu-unf- 62 5 I T I I rfr IT IT rTT’ IT f file I 7’ I' f ‘ .. 45 - r. _ _-- a" . 05 Ca -‘-> T1 q‘ 1' — shell model .. - ." ‘1 ‘. 4— I L —I .. | , .1 .- . , j P ‘ l-- , " I ‘ " ‘ 1 "a 3 "' I ‘1 - -I 1:: _ ' . j I . I , m I. ' «1 2*- _. ~ 1 : 1- 1— j C I i o .L 1 L l L 1 L L L 1 1 0 20 25 Figure 4.6: Summed closure matrix element BCL5(E,,,), the solid line is from the shell- model calculation, the dashed line is anRPA result with g”, = 0.5 and g,;. = 1.0 63 5 _fiil' I I I l I T I I I I l I | i 4‘ : - - - 5,, . 1.28 “Ca --> “Ti : . —— shell model - 4 — _ A . 3 — —.( E . . Ed - V - a _ . U . m . 2 _ - 1 _ _ , .I I 00 25 Figure 4.7: Summed closure matrix element BCLS(Em), the solid line is from the shell- model calculation, the dashed line is anRPA result with g,,, = 1.28 and 9,}, = 1.0. The anRPA reproduces shell-model total BCLS value. 64 0.5 p f I T T I T T I I I I I I IT IT I I T IT I 4: : - - - 3,, - 1.20 “Ca —-> “11 3 - --— shell model ‘ 0.4 - " _ -I .. ‘I ._( A 0.8 I— q 3 ' . m I- v _ -I 0.2 I— '2 k d : 4 0.1 — '1 0.00 L ‘ ‘ 25 E, (MeV) Figure 4.8: Summed energy dependent matrix element MGT(E,,,), the solid line is from the shell-model calculation, the dashed line is anRPA result with 9,, = 1.2 and gph = 1.0. The anRPA reproduces shell-model total MGT value. 65 Table 4.1: 46Ca: Neutron single particle energies 5,- (MeV), gap parameters A,- (MeV) and occupation probabilities v? and quasineutron energies E,- (MeV) from BCS, the shell-model E,(SM), v:-"(SM) defined in Chapter 3. The neutron Fermi energy is —8.712 MeV. level 5; A; v? E.- E,-(SM) v,-2(SM) 11f7/2 — 9.395 1.347 0.726 1.510 1.700 0.703 11p3/2 — 5.507 1.042 0.025 3.370 3.343 0.049 11f5/2 - 3.354 1.455 0.013 5.552 6.377 0.025 11p112 — 3.107 1.026 0.008 5.698 5.830 0.016 agrees with the total shell-model MGT but fails to reproduce the relevant shape as well. Encouraged by the successes of the improvements of the anRPA in Chapter 3, we also consider the use of “ hybrid ” models for the 66 decay matrix elements. The shell-model quasiparticle energies and occupation probabilities for the initial and final nuclei are given in Table 3.1 and Table 4.1. In the following, we compare the following three curves, the dashed line is obtained by the anRPA, the dotted line by the “hybrid” model, and the solid line by the shell model. In Figure 4.9, one uses the shell-model quasiparticle energies and finds that BCLS is suppressed. In Figure 4.10, one uses the shell-model occupation numbers and finds that BCLS is unfortunately enhanced. In Figure 4.11, one uses all shell-model parameters and finds that BCLS is suppressed. But suppressions in Figure 4.9 and 4.11 are not enough to reproduce the shell-model results. One may conclude the “ hybrid ” models do not work well for 2116 6 decay. Finally, we notice that there is a some arbitrariness to the choice of the energy denominator in Eq. (4.4) for calculating MGT in the anRPA. This is because the intermediate states constructed from the initial and final states are mismatched. The 66 5 r I I I I T I I I T I I T T I IT I r T 1| C - - - acs E.u.v I - 45 45 . ------- shell model B. BCS u,v Ca --> T1 * 4, _— shell model _ _ ., _1 I- I I -I .. l _ _ _ 3.. ...... l ..... ., . l l “ 3 .- J .......... l " a . i . . BI . \é " . ..... I I < m - I s ‘ _ _ . z — I; _. |- I: """" 1 I- IS -I . I; . .. I: . 1 — '5 ~— . I; , I- I: 4 -- Ii 4 r- '3 I o 4 l L L l l l L L LIL l. 1 IL 1 l L 1 l l I. l l l L 0 5 10 15 20 25 Em (MeV) Figure 4.9: Summed closure matrix element BCL5(E,,,), the dashed line obtained by the anRPA with E, u,v from BCS and the dotted line with E from shell model and u,v from BCS, respectively. 67 5 p r f f r I r I r r 1 I F r I f I l .- _ " " " BCS 8.11.7 " - -------- scs 1:. shell model u,v “Ca --> “Ti 4 _ J 4, _ —— shell model _ _ q __, .. ' I * 1- ‘ _ _ _ l- ...... l q _ I, .......... , . - I: I ‘ A 3 - IE I -‘ a I- l: l 1 CI! : p .— _ a: l'.'. 'g' ......... " m ,_ | ' I. I. _ _ -I 2 _ I ....... _- . l " . I . . I , .- I .1 1 I ‘ C- . 4 D l 1 o L l L L L L L L IL L L L IL JL IL 0 5 10 15 20 25 E. (MeV) Figure 4.10: Summed closure matrix element BCL5(E,,.), the dashed line obtained by the anRPA with E, u,v from BCS and the dotted line with E from BCS and u,v from shell model, respectively. 68 5 .- I r T r I T F F W fit r T I z . I . I c‘ - - - BCS E.u.v . r 45 45 l - -------- shell model E.u.v C3 ——> T1 4 4 f_ —— shell model _ _JI - l , ' I- l __ - - J- I 1 D I I 4 . I I J A 3 _ t ................ I ..... v - I l 4' p a- - J I _ ‘ -I m9 P l _L ‘- - - «I 2 — ': """ : . — _ : ....... 1 . J 1 — _ o b 1 l L L L l l 1 L IL 1 LIL L l 1 .1 O 15 20 25 En (MeV) Figure 4.11: Summed closure matrix element BCLs(E'm), the dashed line obtained by the anRPA with E, u,v from BCS and the dotted line with u,v and E from shell model, respectively. 69 simplest way is to let Em—E; be the excitation energies from 46Ca ->4GSC calculation as we used before. Another approach is using the relation Em — E,- = Em — Ef +AM and calculating Em — E, from 46Ti —+4GSC, where AM = —0.986 MeV is mass difference between 460a and 46Ti. In fact, there is no big difference between the two methods in our calculations. 4.4 Summary and Conclusions In this Chapter, we have made a comparison of the anRPA and shell-model calcula- tions for the 2Vflfl decay matrix elements of 46Ca in the fp shell. The comparison not only gives insight into the total matrix elements as in previous studies, but also inves- tigates the relevant matrix element distributions. The self-consistent BCS-anRPA equations given in Chapter 3 are used to calculate fl‘ and 5+ Gamow-Teller compo- nents involved in the 2Vflfl decay formulas. Since the intermediate states constructed from the initial and final states are mismatched mathematically in the anRPA, the overlap matrix is introduced and used to match fl‘ and fl+ virtual decay routes. Two types of matrix element are investigated, the energy dependent MGT(Em) which is the exact 2Vflfl decay matrix element, and closure BCLS(Em) which is re- lated to the closure approximation. In our work, we confirm that the suppression mechanism of the 2uflfl matrix elements is due to the particle-particle interaction. MOT and HOT decrease when gpp value increases. Our calculations show the dis- crepancies between the anRPA and shell—model are not only in the total MGT and BCLS values but also in the shape of the matrix element distribution. The disagree- ments between the two models mainly come from the low-lying excitations, where the anRPA gives relatively large fifl matrix elements compared to the shell model. The 2Vflfl decay matrix elements are also calculated by “ hybrid ” anRPA mod- 70 els, i.e., using the shell-model quasiparticle energies and/or occupations in the pn- QRPA. Suppressions in the “ hybrid ” are found, but are not enough to reproduce the shell-model calculations. Chapter 5 BCS Theory and Extension 5.1 Introduction The BCS theory developed by Bardeen, Cooper and Schrieffer [BCS 57] has suc- cessfully explained the superconductivity of the superconducting metals at very low temperature. The adoption of the BCS theory into nuclear physics followed the sug- gestions of Bohr, Mottelson and Pines and the exploratory work of Belyaev [BMP 58, Bel 59]. The first application is to even number semi-magic nuclei, where the ground states are considered to be constructed by pairing configurations. This model is a useful tool to explain a large variety of nuclear properties[Law 80, Rin 80]. The best known form of this theory is obtained by means of the Bogoliubov transformation and Ritz variational principle. The BCS is an independent quasiparticle theory and its ground state is the quasiparticle vacuum. The disadvantage of BCS is that particle number is not conserved. The application of the BCS theory to a proton-neutron (pn) system, i.e., open proton and neutron shells, is complicated because we have two types of interacting particles. Of course, these particles can been rewritten as identical particles if another quantum number—isospin— is introduced. But we will not use isospin formalism here 71 72 because the anRPA equation is more easily derived keeping protons and neutrons separate and it is usually applied in heavy nuclei with a large neutron excess where the pn formalism is more appropriate. Several theories were suggested in order to describe pn system. The simplest method provides separate Bogoliubov transformations for protons and neutrons, re- spectively. Thus the quasiprotons and quasineutrons are well defined, and the ground state is the vacuum corresponding to both types of quasiparticles. In this formalism, the BCS equations for protons and neutrons retain the same form as those obtained from the semi-magic nuclei case, and they are just coupled through the mean field. In this model, only 1111— and nn— pairing are taken into account. Another for- mulation that has been proposed by Lane [Lan 64] and considered by a number of authors[Row 70, G00 70, G00 79] is to give up the distinction between the protons and neutrons by using a generalized Bogoliubov transformation that mixes protons and neutrons to obtain two kinds of new mixed quasiparticles. BCS types of equations can be derived. They include all kinds of pairing, i.e. pp—, 1111— and 1972— pairing, where complex mean field and potentials are required [G00 79]. In this Chapter, we concentrate on the first method because well defined quasipro- tons and quasineutrons are necessary when one develops anRPA theory. In section 5.2, we re-derive the BCS equation in angular momentum coupling space for the proton-neutron system. Also a correlated BCS wave function is introduced through perturbation theory and the extended BCS equation are given in section 5.3. The spurious states are discussed as well in this section. The summary and conclusions are given in section 5.4. 73 5.2 BCS Theory for Proton-Neutron System 5.2.1 Nuclear Hamiltonian in quasiparticle space In the shell-model basis, the nuclear Hamiltonian in the second quantization formal- ism can be written as H = T + V 1 = Eegataa + E Z < aflIV|76 > aiagagam (5.1) 0' 0&5 where 52, is the single-particle energy and < afllVI'yd > is the antisymmetric two- body interaction matrix element. The labels (0676) refer to the particular state (n,l,j,m,tz) in the shell-model basis, where n is principle quantum number, I is orbital momentum, j and m are angular momentum and its third component, and t, is isospin third component, where isospin t = 1 / 2 for proton and neutron. For simplicity, (n, l,j,m,t,) is denoted as (jmtz) and then the Hamiltonian can be rewritten as _ 0 + . H — Z Ejtzajmtzajmti + jmtx l . . . . _ Z < Jamatzaymebtzblvljcmctzcajdmdtzd > jamai"'{dmd zai‘” 2d (1+ jamntza + . . ajbmbt,baJdmdtxdaJcmctlc' (5'2) Eq. (5.2) is called the m-scheme formalism. The above Hamiltonian can be expressed in proton-neutron formalism with the definition of isospin of proton (t,tz) = (1/2, —1/2) and neutron (t, tz) = (1/2, 1/2), _ 0 + . Z 0 + . H _ Z: EjpajpmpaJPmP + ‘ EJnaJnmnaJnm" mep Jnmn +12 <'m'm|V]'m'm>a-+ (1+ a- a~ 4 ’ _ 191 Mb: p2 .7123 paJpI p4 Jplmp, 3mmp2 Jmmm Jpampa JPIV'I'JPA Mpl-"mp‘ 74 +l§ <'m'm|V|'m'm>a*—' (17 a- - 4 . . Jul 111an 112 Jim 713.7114 714 Jnlmnl Jngmnz JnImNaJ'Iamns Jn y"'Jn . I . . + + ' . + Z < .7101 mpiJnI mm lVlJm msznz mnz > ajp, mp1 “in, mu1 “an m", “392mm j," ..._,',,2 "‘Pi """nz (5.3) where p,n label proton and neutron. We are reminded that additional terms which contain matrix elements such as < pPIVIpn >,< pnIVInn > and < pp|V|nn > are dropped since the nuclear interactions conserve charge numbers. We introduce the two-proton, two-neutron and proton-neutron creation operators defined by I A;P(p1p2’ JM) = 2 pl \71 :5 p2 ujPImPlajP2mP2 (5‘4) mleP'z mp: AI...(nIn2.JM) = Z ‘ 1+”, ” a§.....,a.-..m., (5.5) ”film": 711712 ALn(P1"1IJM) = Z < jp.mp.jn.mn.IJM > a},,..,,,a,..m.,. (5.6) mplmnl The Hermitian adjoint operators are defined by App = (ALPVI (5.7) Ann = (Afln)f$ (5-8) Apn = (AM (5.9) Thus the Hamiltonian given by Eq. (5,3) can been written as H = ZegI/ij + 1(aJ-+P ® éjP)J=0.M=0 + 252 [2].” +101; ® djn)J=o'M=o p 71 1 +1 2 (1 + 6mm)“ + (Spam/11mm. ALp(P1P2a JM)App(P3P4I JM) PI'NPI JM 75 1 +1 2 (1+ (Sni'MX1 + 6n3n4)Vnin2n3n4 Afm(n1n2I JM)Ann(n3n4a JM) "1,...“ JM + Z: Vp‘impznz A;n(p1n1, JM)Apn(P2n2, JM), (5.10) "'2 where VJ“ =< ilelkl >J and the label p and n represent proton’s and neutron’s (nlj). The Hamiltonian in Eq. (5.10) is called the J-scheme coupling in proton- neutron formalism. The quasiprotons and quasineutrons are introduced by the Bogoliubov transfor- mations, in the spherical shell-model basis, Gimp = 1117(1sz + (—1)jp+mpvpajp—mp? (5.11) CL... = unafim. + (-1)j"+m"vnaj.—m., (5.12) with ui+v2=ui+vZ=L (5.13) Eq. (5.13) is required if the quasiparticles are assumed to be fermions[Row 70, Law 80, Rin 80]. In order to derive the BCS equation, we relax the fixed proton and neutron number restriction and introduce the extra terms to the Hamiltonian. H = H — m7. — m7), (514) where N, and N, are proton and neutron number operators. The Lagrange multipliers A, and A” turn out to have the physical interpretation of proton and neutron Fermi energies. They are chosen to ensure that mean proton and neutron numbers are correct. 76 After carrying out the Bogoliubov transformation, the Hamiltonian can be rewrit- ten as I? = HBCS + H... (5.15) where H305 = H0+Hf1p+H{‘{‘+(H§§+h.,c)+(H“3‘+h.c), (5.16) H... = H?” + HP” + H?“ (5.17) ml: int int i where the numerical subscripts indicate the number of quasiparticle creation and annihilation operators in each piece of the Hamiltonian. We also require that the above equations should be normal ordered, that is, destruction operators c stand to the right of creation operators cl. Him will be discussed in section 5.3.1. 5.2.2 BCS equations ( independent quasiparticle ) The BCS equations only depend on H303 in Eq. (5.16). The terms in H1305 are given by . 1 . 1 H0 = 2(2JP + 1)(51) — )‘Ir “ Err)”; + 21237: +1)(3n ‘ Av _ Eran)”: p n. 1 —- ;Z( (23', +1 )A pupvp— 2 -Zn:(2jn +1)A,,unv,,, (5.18) Hf," = 2“ (-—6p M301 — v :)+ Zapv,p Ap}cjpmpcjpmp, (5.19) pimp Hf,“ = 2m», (in — v 3,)+ zunvnnnyjnmncjnmn, (5.20) . m 1 H55) = Z(_1)JP+ P{(5p - A”up”? " 50‘: “ v;)Ap}C},mPC}p—mpa (5°21) 7’ H33 = E(—1)j"+m"{(e:n — A ,,)u,.v,. _ %(u2 -v 3,.)A }c at (5.22) Jnmn Jn-mn’ n 77 with e, = 53+1‘,,, (5.23) 5,, = 52+ 1‘“, (5.24) 1 I‘, = 2],? +1[Z;(1+ 6”,)(2J + 1) va/pf, pp, + 21(2J+1)v ELI/p, W], (5.25) P’, ""9 1 1“,. = . [Z( 1+6nnl)(2J+1) v§.Vn{,. n,+2(2J+1)v ).VJ,, .,,,] (5.26) 2Jn+1nIJ 1,:va p”p 2],, +1 A = — ,]——u upm .VJ=.°., (5.27) p 2p; ij'l'lp p P??? An = —; Zj" +1unlvanJ=0, (5.28) Pp and In the are proton and neutron single-particle energy rearrangements. AP and An are proton and neutron gaps. The r.h.s. of the gap equations ( Eq. (527,528)) only depend on J =0 interaction matrix elements. The J 715 0 terms are automatically equal to zero due to Clebsch-Gordan coefficient coupling. So the BCS is an S-pair theory. Also one finds only pp— or nn— paring is involved in the proton or neutron gap equation. In order to derive the BCS equations, we will drop quasiparticle interaction terms Him, and let the BCS ground state be the vacuum corresponding to quasiprotons and quasineutrons introduced in Eqs. (511,512), i.e., CijPIBCS > 0, (5.29) cjnmnlscs > = 0. (5.30) where IBCS > is the BCS ground state. The BCS equations can be obtained from the Ritz variation principle, in which the expectation value of 1:1 is minimized [Row 70, Rin 80, Zha 92]. It requires the 78 coefficients of H53) and H33 in Eqs. (521,522) to be equal to zero, i.e., 1 (510 “' A«)upvp - 50‘: - v:)Ap = 0, (6,, — Au)unvn — $01: — v:)An = 0. We introduce the quasiproton and quasineutron energies defined by E, = \/(5,,—A,.)2+A§, E, = (flan-maria. Then one solves Eqs. (531,532,513) and obtains 2 1 Ep—Aqf v = —(1 — ), p 2 Ep 1 e —A 2 _ _ _ n u 1),, — 2(1 En ). The BCS wave function has a. trial form in the particle basis [Law 80], [BCS > = H (up + vpagmeEILmPXun + vna}nmn&}nmn)| > ime>° Jnmn>0 = II{uEPeXpI—:—:s+(j.>1u2"expl—{3354.091II > . jpjn where aim = ("1)j+ma.i-m? 2j +1 9 = —— 2 5.0) = XI IrmaEmaLm jm>0 : Z ”2]. +1 < jmj — m]00 > aimai—m’ jm>0 (5.31) (5.32) (5.33) (5.34) (5.35) (5.36) (5.37) (5.38) (5.39) (5.40) where l > is the particle vacuum, i.e., 0ij >= 0. Eqs. (537,540) imply that the BCS wave function has seniority-zero and consequently represent a 0+ state in an 79 even-even nucleus. Since the BCS equation is obtained by dropping quasiparticle interaction Him terms, the BCS is an independent quasiparticle theory. The BCS wave function is required to have correct mean particle numbers. Thus the proton and neutron Fermi energies A, and AV are chosen so that the mean proton and neutron numbers in BCS satisfy < BCS]N,|BCS > = 2(2),, +1)v; = 17,, (5.41) P < BCSINVIBCS > 2(2). +1)v,2, = N., (5.42) n where N, and N, are the proton and neutron numbers in the nuclear system. At this 2 p and 12?, introduced in Bogoliubov transformations turn out to stage, parameters 1) have the physical meaning of proton and neutron occupation probabilities. Eqs. (5.23-5.28,5.33-536,541,542) are called the BCS equation. They can be solved iteratively. In the BCS equations, we can find that the equations for protons and neutrons are coupled only through the proton-neutron interaction terms in the rearrangement I}, and 1‘”. We reiterate that the BCS theory ceases to conserve particle numbers. The particle-number uncertainty can be calculated as (AN)2 =< BCSINZIBCS > —N2 = 2(21‘ + 1M5}. (5.43) J The fractional uncertainty in particle number is defined by (AN) T' (5.44) When 222 and 71,2, satisfy Eqs. (535,536), the BCS Hamiltonian in Eq. (5.16) P becomes HBCS = H0 + Z magma-pm, + Z Encgnmncjnmn. (5.45) 80 and the Hamiltonian in Eq. (5.15) H: H0 '1' E E PcipmpchmP + Z Enc cjn'mncjnmn + Him, (546) p,mp nmn Then Hamiltonian given by Eq. (5.1) is expressed by H = HBCS + Hint + AWN‘K + AUNV' (547) 5.2.3 BCS energy spectrum The BCS ground state energy is given by EBCS = < BCSIHBCS + /\,N, + AVNVIBCS > = H0 '1' AwNw + AVNV . l . 1 = 2(217 +1)(5p - Err)”: + 29371 + 1)(571 " 51171)”: p 1 . 1 . —§ 2Q]? +1)Apupv,, — -2- 2(2J, +1)A,,u,v,,. (5.48) P n The BCS excitations, sometimes called unperturbed states ( which refers to drop- ping the Hmh) are defined as l m m IBCS >. (5.49) j—l CJPI' p CJI'AJ n ":5:- The excitation energy is thptqn = Eacs + 2 Ep,‘ + 2 En, (5.50) i=1 The configuration corresponding to this excitation is obtained by creating h quasipro- tons and I quasineutrons with respect to the ground state IBCS >. 81 5.3 Extended BCS Theory 5.3.1 Interactions between quasiparticles As we mentioned before, the Him terms describe the interactions between the quasi- particles, consisting of like particle interaction H53 and H,“ n, and unlike particle 1n- teraction H p .Since H-1111 m, is identical with H p t if p / n labels are interchanged, we will give H5}: here. Now the expressions of each term in Him are given by HE: = 1171);!») + (H31!) + h. c) + (Hf: + h. c) (551) H5: = H5: + (H2: + h.c) + (Pg:n + h.c) +(N§{‘ + h.c) + Pf: + N552 (5.52) where the numerical subscripts again indicate the number of quasiparticle creation and annihilation operators in each piece of Him. 1 H20) : —Z Z (1 + 6P1P2)(1 + 6P3P4 )uPluP2vP3vP4Vp‘1p2p3p4 “.1: A1,.(p1p2. J M 141,114.10. JM ). (5.53) Hgf) = "' 1:( 1+ 6p1p2)( )(1 + 6P3P4)‘/p{p2p3p4 2P1J’ P4 (“m um um UPI — vpivpzvpsum)AIi>p(p1p21JM)fi1i>p(p3p4a JM): (5'54) 1 Hi); = " Z {(1 + 6mm)“ + 6p3p4)(up1umupaup4 + vPlvP2vP3vP4)‘/p{p2p3p4 4 P1 ”.P‘l JM “Hum ”paupavm Wimp”, \/(1 + 610110le + 6703704)(1 + 6721703“1 + 6702704)} A;p(p1p2, JM)A:)p(p3p4i JM), (5.55) Hf: = _ Z nfnlpgng up! ”"1 vP2vn2'A11in(p1n1$ JM)'A]1;n(p2n21 JM)? (5'56) P1035302. [33,1n = — Z Vpimpgng {uplunl vP2un2ALn(p1n1$ JM)’pgn(p2n2$ JM) Pl ”192"? JM pn N31 pn H22 pn P22 pn N22 82 +vP1 ”mum ”n2 4:1(19177'1, JM)73;n(P2n2, JM)}’ - Z vhinipznz {UPI ”“1 ”P2 U712 A;n(p1n1, JM)NJn(P2n2, JM) Pl”1"2”2v JM +vP1vn1uP2un2‘Zan(plnl a JM)fl/Jn(p2n25 “IA/1)}, J 2 {(umuniumunz + ”pivnivpzvnz)vp1n1p2n2 P1";Kl2n2i J + (UPI vnl uP2 vn2 + vpl uni UP? “”2 ) Wpl 711132112} A;n(plnliJM)APn(p2n21 JM): — Z VP‘inlpznz up! Univ” un2fign(p1nli JM)pgn(p2n25 JM): P1"1”2”2! JM _ Z ‘é‘fnlpgng vPlunl uPz vn2NJn(p1n1’ JM)NIIn(p2n2’ JM)‘ P1"1P2"2o JM The notations used in the above equations are defined by with < j m j m |JM > 1 _ p p p p f T A PP(p1p2’ JM) _ Z 1 /11 2 6 2 jpimPichszz’ mm mm + P1P: Dpp(p1p2y JM) : m: < jPl mpljpz mp2 |JM >C}P1mP1 éjszP? , 1’1sz A;n(plnla JM) : m 2’; cj'mmp1 Cjnlmnl’ P1 ''1 ppn(p1nl’ JM) = 27;; < jpl mpljnl mnl |JM > é3:1,1‘1'71p1 Cjnl mu] 5 mm “1 an(p1n13 JM) = Z < jPimpijm mm |JM > 6}!)1771131 ij1 771,61 a mp1 mnl A(j1j2,JM) = (A*(j1j2,JM))l, 210.52. JM) = (-1)’+MA(515..J — M). Ejm = (—l)j+ij_m. (5.57) (5.58) (5.59) (5.60) (5.61) (5.62) (5.63) (5.64) (5.65) (5.66) (5.67) (5.68) (5.69) 83 WJ is particle-hole interaction defined by the Pandy transformation given in Eq. (3.15). 5.3.2 Correlated BCS wave function Since the quasiparticle interaction term Him is dropped in the BCS, we can incor- porate its effects approximately by using it to improve our lowest-order results with the Rayleigh-Schrodinger perturbation theory. The first-order correction terms are rather easily evaluated if we are only interested in the BCS ground state. H31 and H 22 both contain an annihilation operator c as the rightmost operator and hence give zero when operating on the BCS ground state (quasiparticle vacuum). This means that, in this order, the corrections mix the unperturbed four quasiparticle excitations with the quasiparticle vacuum via H40. The unperturbed four quasiparticle excitations are given by section 5.2.3. Since H40 consists of three terms H53), H 25‘ and H53, the unperturbed quasiparticle excitations connected to the ground state corrections contains three kinds of config- uration, four-quasiproton, four-quasineutron and two-quasiproton two-quasineutron excitations. They are denoted as |4qp >, |4qn > and |2qpn >, and corresponding to (h = 4, l = 0), (h = 0, I =2 4) and (h = 2, l = 2) in Eq. (5.49), respectively. Therefore excitation energies are given by Eq. (5.50). The correlated BCS ground state wave function can be represented as < k|H3m|BCS > EBCS - Ek |CBCS > N(|BCS > + 2 Us >) k < 4qp|Hf§|BCS > Enos - E4qp N(|BCS> +2 |4qp> 4C1? + Z < 4qn|H40 IBCS > |4qn > 84 < 2 Hpm BCS > + Z qpnl 40' l2qpn >). (5.70) 2mm EBCS - E2qpn where N is the normalization factor, and II: > denotes any unperturbed excitations. However, this correction is expected to be small so that the gap properties of the pairing solution are not destroyed. In order to simplify our calculation, the interactions between like particle are assumed to be relatively small and dropped. Thus Eq. (5.70) becomes pn |CBCS >2 Muses > + Z < 2C11511le IBCS > 2 11>. 5.71 2m 31305 _ Equn Mi) ) ( ) The word correlated implies including the quasiparticle correlation terms in Eqs.(5.70,5.71). 5.3.3 Construction of two-quasiproton two-quasineutron ex- citations in the J-scheme The two-proton and two-neutron doublet excitations |2qpn > are simply obtained by choosing h = l = 2 in Eq. (5.49) in the m-scheme, where one normalized factor may be introduced. But if we are only interested in some special J values, the J-scheme representation is useful because of rather smaller dimensions involved. For example, only J =1 excitations are needed in Gamow-Teller transitions. Thus we define a two- quasiproton two-quasineutron doublet creation operator in the J-scheme, B31Ju(p1n1p2n2, JM) = NB(At(p1n1, J’Ml) ® At(p2n2, JHM”))JM = N3 2 < J’M’J”M”|JM > MIMI! xAl(p1n1,J'M')./1l(p2n2, J’IM”), (5.72) where N3 is the normalization factor of 3“”. The two—quasiproton two-quasineutron excitations are given by I2qpn >= 337J17(p1n1p2n2, JM)|BCS > . (5.73) 85 However, when one constructs the |2qpn > excitations, identical states must be ex- cluded in order to avoid double counting. The normalization factor N B is determined by < 2qpn|2qpn > = < BCSlBJIJu(p1n1p2n2, JM)B}.J"(p1n1p2n2, JM)|BCS > = 1. (5.74) Then we obtain, . . Jpn jni J, N52 = 1+ (—1)J"1+J”2+J(2J +1)(2J’ + l)(2J" + l)6p,p2{ jug jpz J” } J" J' J . . Jul jPi J, +(-1)JPI+JW+J(2J+1)(2J’+1)(2J”+1)6n,n2 j.2 j.2 J” J” J’ J +6J’J”6p1p26n1n2- (5.75) Since the BCS ground state has spin J = 0 (see Eq. (5.37)), only J = 0 two- quasiproton two-quasineutron excitations in Eq. (5.73) contribute to the ground state correction. Therefore we have J’ = J” and M’ = —M”. Then two-quasiproton two-quasineutron excitations used in the correlated BCS wave function are restricted to |2qpn> = NBBS,J.(p1n1p2n2,OO)IBCS > l(P1n1)J’(P2n2)J' > - (5-75) The last notation will be frequently used in later discussions. According to perturbation theory, all unperburbed states involved in Eq. (5.71) must form an orthogonal basis. But the excitations created by Eq. (5.73) or (5.76) 86 are not orthogonal each other. Let’s consider any two states, lk > 11' |2qpn >= NBB}J(p1n1p2n2,OO)IBCS > , lk' > = |2qpn’ >= NBIB}.J.(p’1n'1p'2ng,00)]BCS > . (5.77) (5.78) The orthogonal condition requires < klk’ >= 6W. But the overlap between them is ka' = < BCSIBJJ(P1n1P2n2,00)B‘1}1J1(p’1n’1p'2n'2, 00)|BCS > = NBNBI{6P1P’16711"'16P2P£ Mn; 61-” . . , J1». +(_1)Jn1+an+J+J (2J +1)(2J’ +1){ jnz J! . . , in. +(-1)”"+””+J+J (N + 1)(2J' + 1) { jpz J! +6P1p§6n1n5 510w; 6712711 6JJ' l1 J P2 J1 11°. in. J! ORR } 67511036711711 6102?; 67:27:; ORR. } 67311016711 n; 6702135671271; (5.79) where N3 and NB, are the normalization factors for states |2qpn > and |2qpn’ >. The kal is nonzero only for the following cases, A)P1=P'1n1=niP2=P§n2=néJ=fl B)p1=p’2n1=n’2p2=p’1n2=n’1J=J’ ( ( (C)p1=p’2 n1=n’. P2=p'1 n2=n'2 J, J’ ( For case (A) or (B), it is easy to obtain x = 1. It indicates that states |2qpn > and |2qpn’ > are identical. On the other hand, we can exclude all double counted |2qpn > states through this overlap procedure. For cases (C) and (D), 0 < x < 1 implies that two states are not orthogonal. 87 We have to use the Gram-Schmidt orthogonalization procedure to obtain normal- ized orthogonal states from the normalized nonorthogonal states |2qpn > ( in fact, they are not linearly independent ). Finally, we obtain lPlnIP’znz, ’9 >= 2 aikl(P1n1)J‘(P2n2)J‘ > - (5.80) The coefficients a are obtained from orthogonalization which relate to X~ For the new state Iplnlpgng, k >, we have < 191711112712, klPlnipznz, 19' >= 5w. (5.81) These states will be employed to calculate the first-order perturbation. The excitation energy for state |p1n1p2n2,k > is given by E). = (EBCS + Ep, + Ep2 + En, + EM). (5.82) So the energy denominator in Eq. (5.71) is EBCS - E1: = —(Ep1 + E... + En1 + Enz)’ (583) 5.3.4 Formalism of extended BCS theory Based on the discussions in the last subsection, we can start to work on the formalism of the extended BCS theory. In this subsection, we will derive the correlated BCS wave function expression, occupation probability and energy shift, and so on. One inserts Iplmpgng, k > into Eq. (5.71) instead of |2qpn >, the correction terms become 2 < 2qpn|Hf§|BCS > gqpn Escs - E2qpn |2qpn > < p1n1p2n2a lefngCS > = " lp1n1p2n2a k > 171%": Em + Em + Em + EM 88 2.01.. < BCSIBJ.J.-(P1n1p2n2,00)H§’5‘IBCS > Pl”l:’2"2 Epl + En; + Epg + Eng X Emu-831.1,. (171721122722, 00)|BCS > J =-: $1.25 akgakj < BCSIBJiJ_.(p1n1p2n2,00)Hf§|BCS > PlnleQ'W EPI + E“! + EP2 + EnZ XB}jJJ(p1n1p2n2,00)IBCS > , (5.84) where the matrix element in above equation is < BCSIBJiJ,(p1n1p2n2,00)HE:IBCS >= -N3.-‘/2J.- + nglnlpznz’ (5.85) with Gianna; = fompgn2(umunlvpzvn2 + UPI ’Unl up: ”712) —Wp]:n1p2n2(vplunl uP2vn2 + UPI vnlvpzun2)' (5'86) Therefore the correlated BCS wave function in Eq. (5.71) is given by F CBCS >= N 1 “”1”” Bl. . ,00 BCS > , 5.87 l { +52%. Em + 15., + 53., + E... ’5’5lp‘mmnz )}| ( ) Fpimmnz = 22 Nagakgakfl/QJ; + lijnlpm. (5.88) k i One finds F is a function of j where k and i are already summed over. The normal- iza.tion factors in Eq. (5.87) can be calculated from, < CBCSICBCS >= 1. (5.89) 89 Then we obtain F n n Fin! InIX n n In! In! N—2 =1+ P1 1P2 2 P1 1P2 2 Pl 1?? 21P1 1P2 2 , 5.90 l 1 2 2 where X is the overlap between states |(p1n1)J‘(p2n2)J‘ > and |(p’1n’1)J."(p’2n’2)J{ >, which is presented in Eq. (5.79). The correlated BCS wave function is required to have correct mean proton and neutron numbers, and we have < CBCS|N,.|CBCS> = . (5.91) < CBCSINVICBCS > = N. (5.92) Where NAN.) is the proton ( neutron ) number operator. We evaluate the left side of these equations and obtain, F N... = 2]“ +1).:2 + N7 “mm“ 2, 5.93 ;( p p 1311;327:3(EP1 + Enl + E? + E712) ( ) N. = 2(2),. + 1).): +N2 2 ( F “"155" )2. (5.94) n P1192711“ EPI + E": + Em 'l' En Eqs- (5.23-5.28,5.33-5.36,5.93,5.94 ) are called the extended BCS equations. They can be solved iteratively as well. We point out that parameters 1): and 123, have lost the p113rsical interpretations of occupation probabilities, because of the existing additional ternns in Eq. (593,594). The correlated BCS ground state energy Econ. is obtained by second-order pertur- b at ion theory, 13..., = EBCS + E“) + E”). (5.95) 90 where E“) — o, (5.96) Eu) = |< lefngCS > '2 Plfllkpzflz EBCS - El: _ _ (ZiakiNBiV2Ji + 16:;,.,,,,,.,)2 (5 97) _ Pinisznz Em + E711 + Em + E112 . ° EBCS is given by Eq. (5.48). 5 -4 Application We have applied the extended BCS theory to study the ground state properties of 46'1“ i. The occupation probabilities and the quasiparticle energies are calculated from the extended BCS equation Eqs. (5.23-5.28,5.33-5.36,5.93,5.94). They are solved i teratively with constrained proton and neutron numbers to be 2 and 4, respectively. In our calculation, the spurious states discussed in Appendix F are not projected out Yet . The occupation probabilities obtained from the extended BCS < CBCSIaj-ajICBCS > / (23' + 1) are given in Table 5.1 and compared to the BCS and the full-basis shell- I17l(> /(2j + 1) for protons and neutrons obtained from the BCS, extended BCS (EBCS) and shell-model calculations, respectively. level BCS EBCS Shell model 1rf7/2 0.239 0.222 0.187 1rp3/2 0.009 0.022 0.078 1rf5/2 0.008 0.020 0.022 1rp1/2 0.004 0.008 0.031 Vp3/2 0.018 0.036 0.092 up”; 0.007 0.011 0.040 Table 5.2: The extended BCS parameters, u?, v}, gap parameters A,- (MeV) and quasiparticle energies E,- (MeV) for 46Ti level U2 v2 A; E; 1rf7/2 7r133/2 1rf5/2 7r131/2 Vf7/2 l’P3/2 l/fs/g l’1’1/2 0.817 0.992 0.993 0.996 0.546 0.983 0.987 0.993 0.183 0.008 0.007 0.004 0.454 0.017 0.013 0.007 1.203 0.911 1.314 0.874 1.447 1.104 1.574 1.069 1.556 5.105 7.592 7.275 1.453 4.236 7.076 6.478 92 5.5 Summary In this Chapter, we re-derived the BCS equations for proton-neutron systems in an angular momentum coupling space. The correlated BCS wave function is intro- duced through first-order perturbation theory, where the quasiparticle interactions Him dropped in BCS theory are incorporated approximately. In order to simplify our calculations, we only consider the proton-neutron interaction term in Him which is be- lieved to be more important than the like particle interactions. The two-quasiproton two-quasineutron doublet excitations in the correlated BCS ground state are intro- duced and discussed in the J-scheme. Since the excitations may be not orthogonal to each other, the Gram-Schmidt orthogonalization method is employed to obtain a new orthogonal basis. We also present the extended BCS equations based on the correlated BCS wave function. We applied the extended BCS equation to the study of the ground state properties of 46Ti and found that the the occupation probabilities have been improved compared to the standard BCS. The correlated BCS wave function will be employed to develop the extended anRPA equation in the next Chapter. The extended BCS equations in angular momentum coupled space is given in Appendix E. The spurious states due to the violation of particle-number conservation will be discussed in Appendix F. Chapter 6 Extended anRPA Theory and Applications 6.1 Introduction In Chapter 3 and Chapter 4, we tested the validity and accuracy of the anRPA as a model to study fl+ and BB decays. The tests were carried out by making the com- parisons of the anRPA and the full-basis shell-model calculations. The comparison consisted of (1) for 3+ decay: the total B (GT+) value, strength distributions, COBTD and CTME, (2) for 21166 decay: the energy-dependent and the closure double-beta decay matrix elements (MOT and BCLS), and their distributions with respect to the excitations. We found that the anRPA does not give a sufficient suppression for transition strength and presents a large discrepancy between the anRPA and shell model in the shape of the strength distributions. We concluded that maybe some correlations which are important to [3+ and 36 decay have not been included in the anRPA. The “ hybrid ” models, where the shell-model parameters are used in the anRPA, improve the calculations in 3" decay but fail in 21163 decay. On the other hand, the “ hybrid ” models are purely empirical improvements, which lack any solid theoretical background. Thus it is important and may be possible to develop a new 93 94 kind of equation, in which more correlations are taken into account from the theory of many-body problem. In Chapter 5, the correlated BCS wave function was introduced and the relevant extended BCS equation was derived as well. In this Chapter, we will derive an extended anRPA equation based on using the correlated BCS wave function to calculate the matrix elements in the equation. In section 6.2 and 6.3, the derivations and applications of the extended anRPA are presented. 6+ decay of 46Ti and £6 decay of 46Ca are employed again as examples. In section 6.4, we give a summary and conclusion. 6.2 Extended anRPA Equations We start with the equation of motion which we discussed in Chapter 3, < QRPAII5QV,[H,Q111|QRPA >= hw < QRPAIMQW QIIIQRPA > (6.1) where hw = E, — E0 and Q: is phonon creation operator, 010,210 == Eur/then. JM) - YNpupn, JM)). (62) P9" The excitations are given by |u >= QIIQRPA > . (6.3) The operators A11,“(pn, J M ) are two quasiparticle creation and destruction operators defined by Eq. (3.5) and Apn(pn,.] M ) is defined by Eq. (3.7). The Hamiltonian is given by Eq. (5.45). IQRPA > is the QRPA ground state defined in Appendix A. 95 By choosing variational operators 6Q, as Apn(pn, JM) and .xlilm(pn, JM), we ob- (:2. 22m =h~(if)- It appears the same as the anRPA form (Eq. (3.9)). But the matrix elements A tain and B derived by using the |CBCS > rather than the IBCS > as follows are given by Agnp,n. < QRPA|[Apn(pn, JM), [11, A;n(p'n', JM)]] |QRPA > 2 < CBCS|[Apn(pn,JM), [H,A;n(p’n',JM)]]ICBcs > I? (E? + En)6PP'6nn' + (H§;):np’n’ + (AA):np’n’ (65) Bgm. = 2 < CBCS|[Apn(pn,JM), [H,Ji,,,.(p'n', JM)]]|CBCS > 2 40%,... + may“), (6.6) where (Hi): inp’n’ = gPP‘é‘fzp’n’(uPunuP'un' '1' ”pvnvp’vn’) +gphWJ,p,n,(upvnuplvnr + vpunvptunl) (6.7) (Gpn):np’n’ = gppnip'n'Wpunvp’vn’ + ”pvnup’un’) —gphWI;{,p,n,(vpunup:vnl + upvnvpmnt) (6.8) In Eqs. (6.5,6.6), we approximate IQRPA > by |CBCS >. ( In the anRPA, we approximate IQRPA > to IBCS >, see Eqs. (3.11,3.12)). |CBCS > is the correlated BCS wave function introduced in Chapter 5. The corrections are expressed by (AA), , , = 26 ,5 ,{ Z Grinpznz Frinmnz (VQJ’ +1) Pnpn PP M: “133112 E?! ‘1' En + Ep2 + En2 2.7a +1 + Gigimm FWHM": (VZJ' + 1)} (6.9) 9231;”? E? + En} + Em + Eng 2jp + 1 96 . . J (H§;)J" F n ’n ABJ ., = 4 «2.1' 1 3.? J." ,} PM“! “I? ( t... z + in J“, J {3..+E..+E.+E. Jl (Hggy’ Fpnnipn' p’npl m + E... + E... + 13,. + Eat} (6'10) The quasiparticle energies E and parameters u, v are obtained by solving the extended BCS equation in subsection 5.3.4. The quantities Fplnlpzng are given by Eq. (5.88). And (V) and (W) are the particle-particle and particle-hole interaction matrix ele- ments, respectively. In AA, all off-diagonal matrix elements are equal to zero because of Clebsch-Gordan coefficient coupling. However, they are not zeros in m-scheme or uncoupled space. Eqs. (6.4—6.10) are called the extended anRPA equations. Now we calculate the transition matrix elements between excited states and ground state for a one-body operator 0, < ulOlQRPA > < QRPA|[QV,0]|QRPA > l2 < cscsnomo‘ncscs > = N’{< BCS|[Q.,0]|BCS > Z < 2qpn|Hf§|BCS >< 2qpn’leg‘IBCS > 2qpn (EBCS - E2qpn)(EBCS "' E2qpn’) 2qpn’ x < 2qpn’|[Qu,0]|2qpn > (6.11) g < Bcsl[Q.,,0‘]|Bcs > . (6.12) In Eq. (6.12), we keep the zero-order term and drop the second-order terms. The first-order term is zero. Therefore for Gamow-Teller operator, the transition matrix elements are given by B(GT') {MAGTW}2 ’2 {Z < pIIO'Hn > (Xtmupvn + Yulmvpun.)}2 (6-13) pfl B(GT+) = {M,,(GT+)}2 2 {— Z < pllalln > (XLmvpun + Yu’m‘upvnfl2 (6.14) pn 97 where 0‘ is the Pauli spin operator. Although above equations have the same expres- sion as Eqs.(3.18,3.19) in Chapter 3, the amplitudes X and Y are obtained by the extended anRPA rather than the anRPA, and u,v factors are obtained by the extended BCS in Chapter 5 rather than the standard BCS. 6.3 Application and Discussions We apply the extended anRPA equation to study the 6+ Gamow-Teller transition of “Ti —>‘SSC. The running sum B(GT+) in the extended anRPA is presented in Figure 6.1 as a function of the excitation energy and compared with those in the anRPA and shell model. We find that total B(GT+) is suppressed. It goes about half way between the anRPA and shell-model results. Actually, there is a competition between a suppression and an enhancement mechanism in our case. The suppression mechanisms are mainly from the off—diagonal terms in AB and the new u,v parameters. The enhancement mechanisms are due to the positive AA values in the extended anRPA. In our calculation, almost all strengths are suppressed except the first one. The relatively strong enhancement in the B(GT+) at the first excited state arises in the AA term from the (1rf7/2Vf7/2, 77f7/2Vf7/2) four quasiparticle configuration, which corresponds to the smallest energy denominator in Eq. (6.9). We know the first B(GT+) is dominated by this configuration. Our calculation shows the enhanced strength is almost equal to the suppressed strength so that the first B(GT+) is unchanged. The energy of the lowest state is shifted up about 0.1 MeV. The calculation has shown that the B(GT+) strength in the second state has the largest suppression, which decreases 30 %. The COBTD and CTME values are given in Table 6.1 and 6.2. Comparing to the anRPA and shell-model calculations, the COBTD and the CTME for f7); —+ f5/2 98 Table 6.1: Comparison of the coherent one-body transition density (COBTD) ob- tained for the anRPA, extended anRPA and shell model calculations of 46Ti. The labels represent (A): COBTD in the anRPA ; (B): COBTD in the extended anRPA ; (C): COBTD in the shell model. jp _’ in A B C f7/2 —2 fm 0.034 0.048 -0.096 f7/2 —2 f5/2 0.443 0.375 0.386 123/2 —2 123/2 0.002 0.001 0.006 123/.» —2 f5/2 -0001 -0001 0.017 123/2 —2 191/2 0.005 0.005 0.060 f5/2 —2 f7,2 0.088 0.081 0.086 f5/2 —2 123/2 0.001 0.001 0.010 f5/2 —» f5/2 0.001 -0002 -0.011 121/2 —2 123/2 0.003 0.003 0.009 P1/2 —’ P1/2 -0.001 0.000 -0.003 is improved but the f7/2 —> f7/2 and f5/2 —2 f7/2 terms become worse, and the p —2 p terms are almost the same as those in the anRPA. We note the sum rule in Eq. (2.1) is violated in our calculations. This is because we have dropped the second-order terms in Eqs. (6.13, 6.14). (There are about twenty terms in the second-order corrections.) The violation is around 10 % in our example. We now apply the extended anRPA to calculate the double-beta decay matrix elements of 46Ca. Figure 6.2 and 6.3 show the running sum of the closure matrix element BCLS and energy dependent matrix element MGT as the function of the excitation energy. Suppression of the transition is found but not enough to reproduce the shell-model result. For 46Ca, since the proton occupations are zero, we have 1),, = 0 and Gimp?” = 0, so consequently all the corrections are zero, the extended anRPA reduces to the anRPA. Thus the excitation energies in the extended anRPA shown in Figure 6.2 are identical to those of the anRPA. 99 1- f I r r T If r T I T f I r 1 3 ‘ " ““““ anRPA ‘ : .............. US$034“ anRPA 4°Ti - > ‘OSC j 2.5 L— .. ' shell model 1 2.0 _ 'r ------- _«1 A g . I + . 9* " 1 ‘ U - I ‘ ' 1 : E 1.5 1:- . _ N E 1.0 — 0.5 '- 0.0 L -5 Figure 6.1: Summed Gamow-Teller strength for 46Ti —>“°Sc. The solid line, the dashed line and the dotted line are obtained by the shell model, anRPA and ex- tended anRPA, respectively. 100 0.5 l’ I I 1' T T 17 I f r l l - - - - anRPA i " ........ 45 46 - '4 0'4 __ extended anRPA Ca --> T1 _ - shell model < A 0.3 '_ -' a _ a kl . v 1- - u . 2 _ . 0.2 —' _. 0.1 _- 0.0 1 1 1 1 1 1 1 0 25 Figure 6.2: Summed energy dependent 2uflfl decay matrix element MGT(E,,.) as the function of the excitation energies, the solid line, the dashed line and the dotted line are obtained by the shell model, anRPA and extended anRPA, respectively 101 5 p T 1 T I fi fi r r f r I r l 1 l fi . [ T q .. ‘ " " anRPA “Ca __> “T1 41 - -------- extended anRPA 7 - 1' 4 _ shell model |_ _ ., _i 1- | q - 1 _ __ _ ,- I 11 ' I ,. """ I 1' . 1 .......... I a In I l 1 E33 , . v ' | I a - --+ !-- ‘ m " ! """ b ‘ - - .4 2 — ! ............. q I- I a I I d . 1 4 1- | .4 1 — I _ ' d o r 1 L l l L L l L Lil. l L L L l P I; L l I. 1 1 l L S O 5 10 15 20 25 8,, (MeV) Figure 6.3: Summed closure 21236 decay matrix element BCL3(E,,,) as the function of the excitation energies, the solid line, the dashed line and the dotted line are obtained by the shell model, anRPA and extended anRPA, respectively 102 Table 6.2: Comparison of the coherent transition matrix elements (CTME) obtained in the anRPA, extended anRPA and shell model calculations of 46Ti. Labels (A), (B) and (C) are given by Table 6.1 .lp _’ in A B C f7/2 -2 fm 0.108 0.155 -O.308 f7/2 —2 f5,2 1.641 1.387 1.431 123/2 —2 1123/2 0.005 0.003 0.014 123/2 —2 f5,2 0.000 0.000 0.000 123/. —+ 121/2 0.011 0.011 0.139 f5/2 —2 fm -0325 -0299 -0319 f5/2 —2 [23/2 0.000 0.000 0.000 fs/z —> f5/2 -0.002 0.003 0.024 121/2 —2 123,2 -0.006 -0.006 -0020 101/2 —+ 121/2 0.000 0.000 0.002 6.4 Summary and Conclusions We have derived an extended anRPA equation, in which the correlated BCS ground state is used to calculate the extended anRPA matrix elements. The equation has the anRPA form but the matrix elements include additional first—order correction terms which are due to the correlated BCS ground state. The equation is applied to study the 5+ decay of “Ti. Around 40 % suppression of total B (GT+) has been found. But the B(GT”) strength distribution in the extended anRPA still differs with the shell-model results. We also discuss the suppression and enhancement mechanisms in the 46Ti —>4GSc transition. We find the sum rule is violated around 10 %. This is because the second-order terms are dropped in the transition matrix element formula. The extended anRPA has also been applied to calculate flfl decay of 46Ca. The suppression of the total double-beta decay matrix elements is found but not enough to reproduce the shell-model results. Like the application of the extended anRPA 103 in 6” decay, the [3,8 matrix element distribution with respect to the excitations is not yet improved. Chapter 7 Further Improvements and Considerations 7 .1 Introduction In Chapter 6, we derived the extended anRPA equation in which the correlated BCS ground state is used. Our calculation showed that the total B(GT+) and 2Vflfl decay matrix elements are suppressed but not enough to reproduce the shell-model results. Also the shapes of the strength distributions are still in poor agreement with those obtained in shell-model calculations. In this Chapter, we explore another possible technique in theory of many-body problem to improve the anRPA, namely, the second anRPA theory. In this model, we concentrate on the extensions of the phonon creation operator, which will be expanded up to four quasiparticle creation and destruction operators. But the BCS ground state is still used to calculate the matrix elements in the equation. Also we can combine the extended and second anRPA, i.e., the phonon oper- ators include two and four quasiparticle creation and destruction operators and the correlated BCS ground state is used to calculate the relevant matrix elements. This extension is called the extended second anRPA equation. 104 105 We will derive the second anRPA equation in section 7.2 and the extended second anRPA in section 7.3. The derivations are both obtained in angular momentum uncoupled space where a REDUCE [Hea 88] algebra code is used to calculate the expression of the operator in the second quantization scheme. The summary is given in section 7.3. 7 .2 Second anRPA Equations In order to include higher-order correlation terms in charge—exchange mode, we ex- tend the phonon creation operator Q), to four quasiparticle creation and destruction operators which can contribute to charge exchange processes, 12> -—- QLISQRPA >={Z(X"c*cT 4:444.) 1 P1 111 Pin! + Z (X;C;1 6;; c133 CI“ _ quc‘nl CPS CP2CP1) P1 - (7-1) In this section, the labels on X and Y are 1 = (plnl), 2 = (p1p2p3n1), 3 = (721712713111), 4 = (pInlpgng). The ground state is |SQRPA>, which is discussed in Appendix D. To derive the second anRPA, we insert the above phonon creation operator and appropriate variational operators cmcm, cllcfn, 4210;142:1411 , -~, and c,,,c,,,c,,1 cn1 into the equation of motion (Eq. (6.1)). The BCS ground state is used when we calculate the matrix elements in the second anRPA. Then the second anRPA is obtained 106 with the following matrix form, ( Aw —Bf1. A21' A312 B... 41.2 11,3. 1 (2m (Xq - 1'12 -Aizv 4113' Y12 Y1 1422' A23' X2’ X2 —A;.2 -A;.: —A*. 14. 16 A32: A337 2 X3: = 7202 X3 .(7.2 —A31' -A§22 -A§32 Y3’ Y3 A442 X42 X4 —A;.2/ \nJ The matrix elements are given by A11' = z 311' = z A12' :- 2 A132 = I? 6n1n§~A(n’21n3)(N < SQRPA|[cn,cp,, [H,c; .c ,]]|SQRPA > < BCSI[c,,,cp,,[H, c; ,c .]]|BCS > (Ep, + En1)6p1p’l6n1n'l + (H§;)p1n1p'ln’l (7-3) — < SQRPA|[c,,,c,,,[H, cnrlcpllHISQRPA > — < BCS|[c,.,c,,, [H,c,,21c,,;]]|BCS > —(Hfg)p1n1pin’l +(Hf:)p1n'lp;m < SQRPAHcmcm, (H, clicgclg " (H261)pin’1p1n1 +(Hfd1)pin1mn’1 5;; ]]|SQRPA > < BCS|[c,,,cp,, [H,c;;c;3c;3c;.l]]|BCS > 12112' A(1321103)“) )pgngpgn. + 5p1p;A(PnP3)(Pf3")pgn;pgm +5p1pg-A(P2a Pi)(Pf3n)p',n;p;m (M<%<%) < SQRPA|[c..,cp,, [H,c;.c 3c;.]]|SQRPA > < BCSI[c.,.,cp,,[H, cl .c:t 'cné 2c19.3]]IBCS > pn I I pn 13 )p’lnapmé + 6n1n;A(n12 n3)(N13 )pingpm’l +6n1n3-A(n’21 ”'1 )(N pn 13517212124 A22’ A331 Aw I? 107 (n’. < n; < 223) (7.6) A},- for i¢j (7.7) < SQRPAHCMC,..,c/,,c,p1 , [H, 3;; 6;; 61:36:21 ]] ISQRPA > < BCS|[cn,cp3cmc,,, [H,ciiclacléclifllBCS > (E1121 '1' EP2 + E93 + Em )612112’1612212561231’3611172’1 ‘1' 621112; [6121121 (Highs/22123;); + 6P1P§(H§§)P2P3P3P; ‘1' 6p1p5(H§’§)psp2p$pi +6921”, (Hi); )121123123125 + 6102124115; )pipapipé + 612210;.(Hi); )pwspépi +6P3pi (H§;)p1mp§pg + 6Pap§(H§i))p1p2p3pi + 6p3p3(Hi>2p)mp2pip§l +6P1PQ6P2PQ(H§;)p3nipgni + 610273; P3P3(Hgl)mmpini +5p3p35p1p;(H§$’)p2mpan; - 5mg 12.141755914314144 “6123105 6121121 (1.153122221123711 - 6101125 6pap§(Hi);)p2mpin’1 ‘61021216103103 (H§;)Plnlpan’1 + 6122123 6p1p5(H§;)p3n1pini +6P3P36P2P'1(H§:)P3n1p’ln’l (121 < 122 < pa. #1 < 123 < 123) (78) < SQRPAHcmcnacmcm , [H, 61110;; clsclimSQRPA > (exchange p, n label in (A22) ) (7.9) < SQRPAHcMcmcmcm,[H,cgsclichciamSQRPA > < BCS|[c,.,c,p,c,,l C1,, , [H, 6:214“, czacLamBCS > (E31 + En, + Em + En,)5p,p; 611.111.; 5,6»; Mn; +6n1ni6ngné(H§:)P1P2PiP§ + 512112; p2p3(H?§)n,n2n;n; +6p1p'16n1n’1(Hg;)P2n2P£n’2 '1' 6121721 6n2n§(H§§)p2nipéni +6pzpé6n1n’1(Hi)i))p1n2p',ng + 6112p; 5n2n;(H§2p)p,n1p’lng PP PP +5161»; 5m; (H22 )mmpgn; + 51411245211214sz )Pznzp'ln'l 108 +6mp'16n2n’1(H§§)p1mp§n§ '1' 6122121 6n1n£(H§;)P1n2P$ni ‘6121121622222’1 (Hgghzmpén; - 61211215221225 (H§§)pznzp£n’1 -5121125 222223 (H§§)pzn1pini _ 61211256221221 (Hgghznzp’lné ‘6p21216n2n3(H§;)p1n1123n1 — 61221215221221 (13312121272512; ‘6122123671222’1 (Hgghmipiné — 6P2P§6n1n5(H§§)p1n2P£ni (121 < 122. 221 < 222. 12'. < 123. n’. < 223) (7-10) The operators associated with the X1, Y1 in Eq. (7.1) describe states in the (N :1:1 , Zzlzl) nuclei; X2, Y2 describe (Nil , Z:l:l,3) nuclei; X3, Y3 describe (Nil,3 , Z:l:1) nuclei; and X4, Y4 describe (N 21:0,2,4 , Z 21:0,2,4) nuclei. Hence, the X4, Y4 amplitudes decouple from the others and cannot describe beta decay states. We will not consider them in further calculations. The closure relation for the second anRPA becomes Z = < SQRPA|[Q.,, 011130111214 > 2 < BCS|[Q.,,I3‘]]|BCS > = (expressions of Eqs. (318 — 3-19))- (7'14) Although Eq. (7.14) has the same form as Eqs. (3.18-3.19), the coefficients X1 and Y1 here are obtained by solving Eq. (7.2) or Eq. (7.12) rather than the anRPA. Thus, the four quasiparticle excitations and the ground state correlations are included in the coefficients. 7 .3 Extended Second anRPA Equations The extended second anRPA can be obtained by replacing IBCS > by |CBCS > in Eqs. (7.3,-,7.10). Here we are interested in the low-lying excitations where the coefficients X.- and Y.- ( 2' Z 2) are assumed to be small quantities and have the same magnitude of the corrections in the correlated BCS wave function. Then we drop the second—order terms and obtain the following results, (1) the matrix element A112 and B“: in Eqs. (7.3,7.4) have the corrections AA and AB, which are given by Eqs. (69,610) in angular momentum space, (2) other matrix elements have the same forms as Eqs. (7.5 — 7.10). 7 .4 Summary We have derived the second anRPA equation based on the extension of the phonon operator up to four quasiparticle creation and destruction operators. In the equation, the BCS ground state is still used. We also develop the extended second anRPA 110 equation, in which the phonon operators are expanded up to four quasiparticle cre- ation and destruction operators and the correlated BCS ground state is employed to calculated the matrix elements. The numerical evaluation of the second anRPA and the extended anRPA will provide a formidable challenge and we plan to proceed along the lines developed for the extension of the RPA [Warn 88, Dro 90]. Chapter 8 Summary and Conclusions The nuclear shell model and the anRPA equation are very important nuclear struc- ture theories for studying B and BB decays. But the approach of these two models is different. Many or all types of many-body correlations are taken into account in the large- or full-basis shell-model calculations. Thus the full-basis shell model is the exact calculation when it is feasible to carry out. The anRPA is an approximate model, which includes only some special classes of the correlations. The goals of this thesis consisted of three aspects, (1) study of B and the BB decay of mass A = 48 nuclei with the large-basis shell-model calculations, (2) examinations of the validity and accuracy of the anRPA for 3+ and BB processes, and (3) improvements and extensions of the anRPA model. In Chapter 2, the large-basis shell-model calculations are employed to calculate fl" and 6+ Gamow-Teller spectra of 48Ca and 48Ti, and 2ufl/B decay matrix element of 48Ca. The B(GT‘) and B(GT+) behavior in the energy region (2.5~15.0 MeV) are well described by the effective Gamow-Teller operator 6t = 0.7701 and a new effective interaction MSOBEP. With this interaction and effective operator, our shell- model calculation predicts the 21166 decay matrix element of 48Ca to be Mé‘ir=0.070, giving a half life T1/2= 1.9x1019yr, which differs by nearly a factor of two from the 111 112 experimental limit [Bar 70] of T1/2 >3.6x1019 yr. We believe that the most important aspect of these calculations, which cannot be directly tested by the (p,n) and (n,p) experiments, is the amount of strength in the (n,p) [3+ spectrum above 5 MeV in the excitation. This is because there is a large uncertainty in the amount of Gamow- Teller strength in the background above this energy. Also further refinements of the effective interaction are necessary. Since the anRPA is an approximate model, in Chapter 3 and 4 we have investi- gated the accuracy of the anRPA approach to W“ and BB decays. Comparisons of the anRPA and full-basis shell-model calculations have been made. In our study, a self-consistent BCS-anRPA have been developed. “ Self-consistent ” means that the input ingredients in the BCS-anRPA are the same as those in the shell model, namely, the bare single particle energies at the closed shell and two-body interaction matrix elements. Therefore there is no free parameter for both models in our com- parison. The coherent one-body transition density (COBTD) and coherent transition matrix elements (CTME) are introduced for analysis of the single-particle state con- tributions in [3+ decay. Our comparisons have shown the anRPA overestimates the total B(GT+) and ,Bfl decay matrix elements, and there are large discrepancies in the shapes of the strength distributions between the anRPA and shell-model calcula- tions. The COBTD and CTME of the anRPA are also in poor agreement with those of the shell-model. Thus we may conclude there must be some correlations which are important to fl+ and ,Bfl decay but have not been included in the anRPA. Empirical improvements for the anRPA, namely, the “ hybrid ” anRPA, which are obtained by using the shell-model occupation probabilities and the shell-model quasiparticle energies in the anRPA, are discussed. About 50 % suppressions are found for the total B(GT+) strength. The shape of the B(GT+) distribution has not 113 been improved but it is more reasonable than that in the anRPA with larger gm, values. In Chapter 4, we find the “ hybrid ” anRPA does not work for BB decay. The theoretical improvements of the anRPA equation start with the introduc- tion of the correlated BCS theory. The quasiparticle interaction dropped in BCS is incorporated by first-order perturbation theory. The extended BCS equation has been derived based on the correlated BCS and has shown improvement over the stan- dard BCS for the occupation probabilities. In our study, only the proton-neutron interaction terms are considered. We have derived an extended anRPA equation, in which the correlated BCS is used to calculate the matrix elements. The equation has the anRPA form but the matrix elements include additional first-order correction terms. The equation has been applied to 3+ decay and gives an additional suppression of about 40 % in the total B(GT+). Around 10 % of the sum rule is violated because of dropping the second-order terms. Also the double—beta decay matrix elements are suprressed but not enough to reproduce the shell-model result. The disagreements in the shape of the 6+ and flfl spectra have not yet been improved compared to the shell-model results. The extended BCS and anRPA equations were derived in the J-scheme coupling space. In Chapter 7, we considered the development of the second anRPA equation based on the extension of phonon operator up to four quasiparticle creation and destruction operators. In the equations, the BCS ground state is still used. We also derived an extended second anRPA equation, in which not only the phonon operators are expanded up to four quasiparticle creation and destruction operators, but also the correlated BCS ground state is employed to calculate the matrix elements. These equations should provide more accurate methods for studying the transition 114 of one- and/or two-body charge exchange modes. The numerical evaluation of the second anRPA and the extended anRPA will present a formidable challenge and we plan to proceed along the lines developed for the extension of the RPA. In Appendix A, the QRPA equations have been derived. In Appendix B, we have discussed the relations between the QRPA and anRPA equations. In Appendix C, the second QRPA equation has been derived with the phonon creation expanding to four quasiparticle creation and destruction operators. In Appendix D, the coherent one-body transition density and coherent transition matrix element are discussed in detail. In Appendix E, the BCS and the extended BCS theories are derived in angular momentum uncoupled space. Finally in Appendix F, the spurious states in the BCS and the extended BCS are discussed. Appendix A QRPA equations In this appendix, we review the derivation of the QRPA equation from the equa- tion of motion method. In the equation of motion, the excited eigenstates IV > are constructed from the phonon creation operator Q1 which is defined by |u >= QL|0 >, and Q.,|0 >= 0, for any (A.1) where |1/ > and I0 > are the excited eigenstate and the physical ground state. They satisfy the Schr6dinger equations, HIV >= EVIV > and H|0 )2 E010 >. (A2) Then one obtains the following equation of motion from the above relations; [H,Qlllo >= (Ev — EolQIIO >- (A3) Multiplying from the left with an arbitrary state of the form < 0(6Qy, we get < 0115492, [H.Qllllo >= M < 0|[5Qlell0 > (A-4) where 71w = E, — E0. 115 116 In order to derive the usual QRPA equation, we assume that the excited states are obtained by creating or destroying two quasiparticles from the QRPA ground state |QRPA >, (.2 >= QIIQRPA >= 2 (163,5ch — n1,ck.c,.)|QRPA > . (A.5) k= 0. (A6) Inserting Q: into the Eq. (A4), and choosing 6Q to be c/clc/c and chL, one obtains the QRPA equation ([113. _i.)(§)=hw(§). (A.7) The matrix elements are given by (k < k’,l < l’) A...” = < QRPA|[c/./c/,, [H,cIc),]]|QRPA > z < BCSHc/c/ck, [17,431,]“1308 > = (El: + Ek'l5kl5k'l' + (H22lkk’ll' (A.8) Bkw/ = — < QRPAHc/c/ck, [H,c/nc/]]|QRPA > z — < BCS|[c/c:ck, [H,c/Ic/HlBCS > = (H4olkk’ll' + (H4o)ulkkl - (H40)klk’l’ — (H4o)k'l’kl —(H40)I’kk’l - (H40)lk'kl’ (A.9) where we approximate the QRPA ground state as the BCS state in Eqs. (A.8-A.9). Appendix B Relation between QRPA and anRPA Equations When we consider Eq. (A.5) including the proton and neutron components, it can be expanded as, Ql = 2(XEkICICI2-Yki20k'6k) k < BCS|[c,,/c,,, [H,chLHIBCS > (P§2“)pnp2..l (3.3) — < QRPAHcp/cp, [H,cnlcanRPA > — < BCS|[cp:cp, [H,CnICntCS > —(Hfé‘)pnplni — (H5395... +(H40)p’npn’ + (H40)1m’P’n (BA) < QRPA|[c,,Ic,,, [H,c;c;,)]|QRPA > < BCSch/cn, [H, c;c;,]]|Bcs > (N;;)pnp’n’- (8.5) From Eq. (B.2), we can find the sub-matrix in low-right corner is the anRPA equa- tion which is decoupled from the proton-proton and neutron-neutron QRPA matrix located in up-left corner. The reason for the decoupling is that the operators asso- ciated with X”, Y” in Eq. (B.1) describe the states in the (N,Z :t 0,2) nuclei; and X“, Ynn describe the (N :t 0,2, Z) nuclei whereas and X”, Y” describe the (N :1: 1, Z :1: 1) nuclei. Hence, the X P“, Y” amplitudes decouple from the others. But X W, Ypp and X n", 12"“ both can describe the (N, Z) nucleus and consequently couple together. Appendix C Second QRPA Equations In this Appendix, we will generalize the QRPA to the second QRPA equation. The phonon creation operator is expanded up to four quasiparticle creation and annihila- tion operators, i.e., |u > = QEISQRPA > = {Z (Xi’CICII - Yf’Ck'Ck) k, (c.1) k(k’ is the second QRPA’S ground state defined by QVISQRPA >= 0. (02) When Q; is put in the equation of motion, we obtain the second QRPA equations, A11! B11' A12! X11 X1 ~81. —A:.2 —A1.2 Y1, Y1 =77. C.3 A212 42.2 X22 .5 X. ( l -1131, —A;.2 Y4 Y2 The matrix elements are given by A112 = 119 120 z < BCS|[c/¢:c/., [H,c}c{,]]|Bcs >= (E. + Ek:)6/,/6/,:/I + (H22),.,,,,,, (k a: — < BCS|[ck:ck, [H,c/2c/]]|BCS > = (H40)kk’ll' + (H40)Il’kk' - (H40)klk’l’ - (H4o)lc’l’kl -(H4o)1'kklt - (H4olwk12 (k < k’,l < 1’) (OS) A12: = < SQRPA|[c/..c/.,[H, clef, or, 5311180111211 > z < BCS|[ckIC/.,[H, arc/151,, 5;,1111308 >) = $331,340,343, I'>2.2..21A z < BCS|[c/./c/., [H, c{c{,c;,c* #1111308 > = (El: + Ek’ + Em + Em’)6kl6k’l’6mn6m’n’ + 5kl5k'1'(H22)mmInnl +6mn6m’n’(H22)kk’ll' + 6kn6k’n’(H22)mm’ll’ + 6ln6l’n’(H22)kk’nn’ +4405, [Cl/1“,, [MW/1"HAW,”Mal.n')5w6m2n2(H22)km1n +.A( k’, k)A(l’, l).A(n’, n)5,,./6/.../(H22),,,m,,,, +A(k’, k)A(l', I)A(m', m)6k:/25/m2(H22)m/mr,, +A(k', k).A(m', m).A(n', n)6m2n26/¢,,2(H22)/m/2/ +A(l’, l).A(m', m).A(n’, n)6(lm16mln(H22)kklnl (kaLam (D.1) 026 where Ia > and (B > are the single particle states. The transition matrix element between the initial state Ii > and the final state |f > is given by M). =< f|F|2' >= 2 < a|F|fl >< flaLa/gli >. (D2) afi < alFIfl > is called the single-particle matrix element (SPME) which is only related to the single-particle states Ia > and Ifl >. < fIaLa/gli > is called the one-body transition density (OBTD), which is the function of the initial and final states as well as the single-particle states la > and [B >. Then Eq. (D.2) can be rewritten as M,. = Z SPME(o)OBTD(o, f, i) = 2 TME(o, f, 1'), (D3) where 0 represents the single-particle states. Now we introduce a coherent state |C > which is defined by |c >= NcFlz' >, (0.4) 122 123 where Nc is the normalization factor which is determined by < C|C > — N5 < 1|F1F|i > Néz< f|F|i>. (D5) 1' Then we have NC = 4 (D6) V2} < leli >2. The transition matrix element between the coherent state and initial state is = No NC: < ilFllf >< f|F|i > f NC 2 M;,—SPME(o)OBTD(o, f, i) 0.! — ESPME(0)COBTD(0,1) = gamma”). (D-7) where the COBTD and CTME are the coherent one-body transition density and co- herent transition matrix element which are defined by COBTD(0,2') = No ZM;,~OBTD(0,f,i) f CTME(o, i) _ SPME(o)COBTD(o, f, 2'). (D8) The COBTD and CTME are a function of the single-particle state and initial state. It represents the single-particle state effects in the total transition strength. Appendix E BCS and Extended BCS in Uncoupled Representation In this Appendix, we will discuss the formalism of the BCS and the extended BCS in angular momentum uncoupled space. The BCS equations are given by a lot of textbooks [Row 70, Rin 80]. For the protononeutron system, the quasiproton energies Ep, the parameters of, introduced by Bogoliubov transformation and pairing gaps AP are given by E12 = (512 "’ A”)? + A?” (El) 1 e —/\,, v: = 5(1- ——"—,—>. (23.2) (512 — Ar) '1' A: 1 AP 2 -§ Euplvplvpp'p/Ip, (E.3) pl where A, is the proton Fermi energy, 5,, and Vpp'pvpr are the single proton energy and proton two-body interaction, respectively. The above equations can be solved under the constraint for the total proton number N,r = E < BCSIa£a|BCS > , (EA) P which determines the constant A,“ A similar set of the equations can be solved for neutrons. In the BCS, 122 turns out the physical meaning — the occupation probability. 124 125 The derivation of the extended BCS equations starts with introducing the corre- lated BCS wave function in section 5.3.2, which can be written as GP“ CBCS >= N BCS > — “"1"“ c, ,, BCS > , E.5 I (I P1222 Epl'I'E-n,‘ +Ep2+En2 PC CP2C712I )( ) n1 n2 where the restrictions p1 < p; and n1 < n; are introduced in order to avoid double . . n . . countmg the states. The expressmn of G'glnlmnz lS given by n _ (GP )mnxpznz “ ninimnzwmumvmvnz ‘I’ ”1210721 umum) —WP1n1P2n2(vP1un1uP2vn2 + uplvnivpzun2)7 (E6) where Wplnlpzn, is the particle-hole interaction to be equal to —Vp,,,,p,n,. The normalization factor N in Eq. (E.5) can be determined by < CBCSICBCS >= 1. (13.7) Then we obtain Grpn N-2 = 1 + w}? 13.8) 2:; { E. + En. + E122 + En. ( "l The occupation probabilities of protons and neutrons are given by G CBCS I CB _ 2+ 2_ 21V2 2: P711P2n2 2 < Iapapl CS > up (up up) < (E? I Em +Ep2 +En2) "1 "2 02 + (“2 _ v2)N2 z ( Grimm”: 2 < CBCSIaLanICBCS > . px= N, (13.10) P 2 < CBCSlaLan|CBCS >= NV. (13.11) The set of equations Eq. (E.l,—,E.3,E.5,——, 13.11) are called the extended BCS equation. They can be solved iteratively. From Eqs. (E9), we find that v2 is no longer the occupation probability. The ground state energy of the correlated BCS is given by E“... = EBCS + E“) + Em, (13.12) where E“) = 0 (13.13) E9) = _ 4(3me (13,14) 1211:1422 Em + Em + EP2 + EH2 . "1<"'z EBCS is given by Eq. (5.48). Appendix F Spurious States in the BCS and extended BCS. In Chapter 5, we constructed the first—order corrections for the BCS ground state. The correlated BCS ground state mixes two-quasiproton two-quasineutron excitations with the quasiparticle vacuum. But these excitations may contain some unphysical states - spurious states, which should be projected out in principle. Spurious excitations can occur whenever the approximation used breaks the sym- metries of the Hamiltonian or violates the conservation laws. Since the BCS violates the particle-number conservation, spurious states may occur when we construct the unperturbed excitations from the BCS ground state in subsect. 5.2.3. For example, if we consider the two-quasiproton excitations, the state (3p, > = N,,,2(N, — N,)|BCS > — 3.23742]. + 022.22.411.61, 00)IBCS > (El) p is an excitation but it is spurious i.e., if the wave function is the eigenstate of N, this state does not exist because we have Isp2 >= 0. The four-quasiparticle (like particle) spurious states were discussed by several authors before[0tt 67, Pal 67, Gmi 68] when they studied the excitations in quasi- 127 128 particle TDA equation. Their method requires an explicit construction of the most important spurious states to be eliminated. Following this method, we construct the spurious states in the two-quasiproton two-quasineutron doublet configurations, which should be removed when calculating the correlated BCS wave function. The spurious states can be written as Isp7r >= N,,,,.(N,, — N,)A:m(n1n1,00)|Bcs > , (R2) 01' Ispu >= N,,,,,(N,, — N,)A;P(p.p.,00)|Bcs > , (F.3) where Np and N, are the proton and neutron number operators, N, and N, are the proton and neutron mean numbers given by the nuclear system. In fact, the Eqs. (F.2,F.3) are equivalent to each other. In order to discuss the properties of spurious state and understand the projection procedure, we now consider a special spurious state, it is the combination of the states in Eq. (F2) or Eq. (F.3). lsp4 >= N,,,(N,, — N,)(N,, — N,,)|BCS >, (F4) where N3p is the normalization factor. In the quasiparticle basis, it can be expressed as __ 7 +171 +j +m I 1’ I 1’ [SP4 > — Nap: Z (—1)” P " "upvpunvncjmecJ-P_mpcjnmncjn_,,,nIBCS > . pn mpmn (F.5) N,p can be determined by < smlsp. >= 1. (F-6) 129 so we obtain N3” = 42(2). + 1x21. + 1)(..,.,..,,.,,)2. (F7) pn Isp4 > may be expanded in terms of our orthogonal complete basis obtained by Eq. 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