Abstract. Microscopic Study of Enhanced El Transitions in. Anna Catherine Hayes. Yale University, 1986

Transcription

1 Abstract Microscopic Study of Enhanced El Transitions in 18 O Anna Catherine Hayes Yale University, 1986 It is well known that light nuclei exhibit the coexistence of collective and single particle degrees of freedom within the same state. Several enhanced El transitions have been observed in 180, and of considerable current interest is how the different configurations and configuration mixing contribute to the strength of these transitions. We have carried out a microscopic theoretical study of the structure of the low-lying states in 180 to address these questions in detail. Strong E2 transitions have been associated with collective degrees of freedom and are usually an indication that the states involved are of similar intrinsic structure and often members of the same band..the strong El transitions in 180 have raised the question as to whether they also connect states of similar intrinsic structure, and if El transitions, like E2 transitions, can be used to determine band structure in light nuclei. Until now, all theoretical attempts to understand these transitions have been confined to ct+^c cluster models and, not taking other configurations into account, these calculations have only met with partial success. We have carried out a large basis shell model calculation, which

2 includes single particle, collective quadrupole, and cluster degrees of freedom. The basis includes up to 3tfw of excitation, with complete elimination of any spurious centre of mass motion. The model is quite successful in describing the known properties of the states in 180. Both the enhanced and hindered El transitions are reasonably well reproduced. The source of enhancement is found to arise from constructive interference between quite different components in the wave functions, and to be strongly dependent on configuration mixing. However, the positive and negative parity states connected by strong El transitions are not found to have similar structure. The El transitions are very sensitive to the details of the wave functions, and do not appear as suitable signatures for band structure in light nuclei.

3 MICROSCOPIC STUDY OF ENHANCED E1 TRANSITIONS A Dissertation Presented to the Faculty of the Graduate School of Yale University in Candidacy for the Degree of Doctor of Philosophy Anna Catherine Hayes June 1986

4 This thesis is dedicated to the loving memories of My Father and Mother Roger and Ann Hayes

5 ACKNOWLEDGEMENTS This thesis would not have been possible if it had not been for the help and efforts of many people, and I take this opportunity to express my gratitude to them. I would like to thank my advisor, Professor. D. Allan Bromley, for his help, encouragement and support throughout my stay at Yale. His ability to focus on the essential physics behind a problem has been a great help in this work and will continue to influence my approach to physics. I was very fortunate to have a second advisor, Dr. D. John Millener of Brookhaven National Laboratory. I am most grateful to him for his excellent and patient teaching, and for his very substantial input into this thesis. John's vast and detailed knowledge of nuclear structure remains a source of amazement and inspiration. My sincere thanks to the members of my thesis committee: Professors Tom Appelquist, Jolie Cizewski, Peter Parker, Onno van Roosmalen, D.Allan Bromley, Dr. Millener and my external reader, Professor Paul Ellis of the University of Minnesota. Their comments and suggestions have been very helpful. I would like to thank Professor Rex Keddy of the University of Witwatersrand, Johannesburg for his help and encouragement during my early years at WNSL - I shall always value his friendship. I would also like to thank Drs. John Olness and Ernie Warburton of Brookhaven National Laboratory for very helpful discussions and for the interest they have shown in this project.

6 I wish to thank all the members of the WNSL support staff for maintaining the laboratory and for their efforts in providing a pleasant atmosphere for research. I would like to say a special thanks to Susan Anderson, Rita Bonito, Clair Buckley, Marvin Curtis, Karen DeFelice, Annalee Jacunsky, Sandy Sicignano and Mary Anne Schulz for their professional help, kindness and friendship over the years. I am very grateful to the computer support staff Niel Bauer, Nitza Hauser and John Baris for their expert advise and help on many occasions. I thank Sara Batter for her willingness to go out of her way to help, no matter what the problem. I am happy to acknowledge my fellow graduate students and friends in physics: Tzu Fang Wang, Jonathan Harold Engel, John O'Connor, Helene Nadeau, B. Shivakumar, Mark Drigert, Bronek Dichter, Patty Blauner, Laurie Baumel, Fabio Dellagiacoma, John Manoyan, Paul Magnus, Kevin Hubbard, Erhan Gulmez, Paul Cottle, B.S. Frumpton and Dr. Nick Tsoupas. My special thanks to 'Big Foot' (i.e. P.M.) who, despite his insistence on walking over my desk and papers in his dusty sneakers, was a great help to me in ironing out many problems and difficulties I encountered during this work. I owe a special thanks to Mrs. Elizabeth and the late Hr. Walter Miniter for the hospitality they have shown me in welcoming me into their family since my first arrival in the U.S.. They have acted as 'adoptive parents' to many foreign students and we shall always remember them for their kindness and friendship and their sincere interest in the studies of all their adoptees. My compatriots at Yale have always been good for a game of cards, a

7 row, or a few pints, but most of all I thank them for their warm friendship. I thank my in-laws, the Sterbenz family, for their warm hospitality and support. My deepest gratitude, however, must go to my family. My three brothers and my sister, Ruadhan, Conall, Odran and Derval showed me their greatest support when the going seemed the toughest. Without their love and friendship this work could not have been completed. They are also responsible for introducing me to many concepts beyond the scope of this thesis! But it is to my dear parents, that I owe the most, for they made so many sacrifices toward the education of their children. Sadly, they died before seeing the rewards of their efforts and it is to them that this thesis is dedicated. Finally and most important of all, I thank my office mate, best friend, and husband, Steve, for his love and constant encouragement. Steve helped me with many aspects of this work and it is his interest that makes it all worth while. I can not find enough words to express all he has done for me and means to me.

8 Table of Contents CHAPTER 1 Introduction 1 Glossary of technical terms 3 CHAPTER 2 Motivation for a Study of the Mass 18 Nuclei 6 2.A The Mass 18 Nuclear System 6 2.B New Data 10. _ 2.C.Systematic.Behaviour in Light Nuclei 13 Neon The 4p-xh bands A=16 to A=20 16 E2 transitions in light nuclei 18 _E1 transitions in light nuclei 19 CHAPTER 3 Theoretical Framework 22 3.A The U(4) Molecular Model 22 Application to oxygen The El sum rule 27 The transition region from U(3) to 0(4) 30 Comparison of the U(3)-limit to the microscopic SU(3) model 30 3.B The Nuclear Shell Model 36 The SU(3) shell model 37 Symmetry properties of the wave function 38

9 The physical importance of the SU(3) classification scheme 40 The truncation scheme 44 The Millener shell model 44 Spurious centre of mass motion 51 3.C Application to Mass 18 Nuclei 54 The basis 54 The positive parity states 62 The band structures 63 The negative parity states 65 The electromagnetic transitions rates 65 CHAPTER 4 Extensions of the Model The (60) representation 67 3hw excitations 68 6p-4h excitations 70 Are higher core-excitations important 70 Extensions of the shell codes 73 Testing the codes 76 The transition rate codes 76 What can the codes handle 78 CHAPTER 5 The Nuclear Cluster Model 80 The relation between cluster and shell model wave functions 81 Alpha particle cluster states in oxygen 18 85

10 Triton clustering in oxygen 18 El matrix elements between cluster states The Cosh potential CHAPTER 6 The Full Shell Model Calculations 101 The positive parity wave functions 101 The negative parity wave functions 110 The El transition strengths 116 The overlap of the shall model wave functions with those for pure cluster states 118 A revisit to the predictions of the cluster model 121 The K^=0+ bands in 20Ne and the 4p-2h states in The K^=0" bands in 20Me and 160 and the negative parity states in CHAPTER7 Conclusions 132 References 135

11 CHAPTER 1 Introduction. The principle subject of this dissertation is the application of a nuclear shell model to the structure of the 180 nucleus. The mass 18 nuclei, 180, 18F and 18Ne are of particular interest in that they are known to be many-body systems which exhibit different degrees of freedom simultaneously within the same nucleus. There are states which are well described as comprising two nucleons outside of a closed 160 core. Coexisting with these single particle states are those in v/hich this 160 core is itself excited. The properties of these core excited states indicate coherent, collective participation of several nucleons. Included in such collective states are those which can be thought of as an alpha particle, triton or 3He cluster coupled to a core as well as those which exhibit quadrupole deformation. In addition, it has been demonstrated that many of these states in mass 18 involve configuration mixing in which the wavefunctions of individual states involve admixtures of different degrees of freedom. Recent experimental studies on 180 have identified several strong dipole and quadrupole electromagnetic transitions. The quadrupole transitions are ten to twenty times larger than single particle estimates, suggesting a dominant collective component in the wave functions of the states involved. The dipole transitions, although quite weak compared to single particle estimates, are strong compared to normal El transitions in light nuclei. In this study we attempt to understand these data in 180 from a consistent microscopic point of view.

12 - 2- In a nuclear shell model, the nucleons are treated as individual particles. In this way, any single nucleon degrees of freedom within the system can be readily understood. However, when core excitations are also taken into account, i.e. when nucleons are promoted out of the closed core into previously unoccupied orbitals, the shell model has also had substantial success in describing collective effects in light nuclei. The SU(3) shell model takes advantage of the symmetry properties of the nuclear wave function in introducing a convenient labeling scheme for shell model states. In this approach such collective features of nuclei as rotational bands and enhanced quadrupole transitions are clearly understood within the same framework as are single particle degrees of freedom. Furthermore, there is a simple correspondence between SU(3) shell model wave functions and wave functions describing a system in which an alpha particle cluster is coupled to a core. An SU(3) shell model is therefore well adapted for our studies of 180. In chapter 2 we examine what is known of mass 18 nuclei from both an experimental and a theoretical point of view. In addition, we try to put these facts in perspective by looking at the systematic behaviour of light nuclei. Chapter 3 is concerned with the SU(3) shell model and its application to mass 18. Our first calculations were only partially successful, and in order to explain all the observed data we found it necessary to extend the model which we had originally used. Extension of the model involved a substantial extension of the existing SU(3) shell model codes. A description of this undertaking is given in chapter 4. Before going on to apply our extended shell codes to 180, we consider the simpler cluster model, in which states are described in terms of a

13 - 3- cluster coupled to a residual core. In chapter 5 we examine the predictions of this cluster model and the relation between its states and those of the SU(3) shell model. Chapter 6 is devoted to the results obtained with our extended shell model codes. We compare the predicted levels, electromagnetic transition rates, spectroscopic factors and band structures with the experimental data. Finally, we present the conclusions we draw from these studies and discuss interesting calculations which remain for the future. We conclude this chapter by giving a brief explanation of terms and concepts used throughout this work. Glossary of Technical Terms. 1. Nuclear shell model configurations. (A) Major shells: In a nuclear shell model, where the average central potential is taken to be the harmonic oscillator potential, the major shells are determined by the oscillator quantum number N, with N=0,1,.... These major shells are separated in energy by hu and are H(M+l)(N+2) -fold degenerate. The allowed L values for a given shell are given by N=2n+L, where n is the number of nodes in the wave function, so that L= N, N-2,.. The L degeneracy is lifted by adding T.s and l 2 terms to the

14 - 4 - one-body potential, without affecting the major shell structure, and throughout this work we refer to these shells as the OS-shell (N=0), the P-shell (N=l), the SD-shell (N=2), the PF-shell (K=3), etc. (B) Htfu configurations: Shell model configurations are usually classified by the number of oscillator quanta Minvolved, relative to the lowest possible configuration allowed by the Pauli exclusion principle, and are then referred to as Mhw configurations. For example, in 180 the lowest configuration involves two neutrons in the SD-shell with closed OS- and P- shells, and all possible (sd)2 configurations make up a Ctffu basis. We can then add an additional quantum of excitation to the system forming negative parity lffto configurations by either promoting a particle from the P-shell up to the SD- shell (denoted by p_1(sd)3) or by promoting a neutron from the SD-shell up to the PF-shell (denoted by (sd)(pf) ). 2. Relative motion quanta associated with cluster states. In a cluster model we assume that the wave function can be written as an antisymmetrized product of the internal wave functions of the core and cluster and their relative motion function. If the cluster and the core have zero internal excitation, then all the quanta of excitation (Q) are associated with the relative motion wave function. If we use harmonic oscillator wave functions, then there is a clear relationship between these relative motion quanta Q and quanta of excitation in the shell model. As an example, let us consider a+14c cluster states in 180. The O ffw states for A=18 contain two neutrons in the SD-shell. Removing

15 - 5- these two neutrons, and two protons from the P-shell leaves us with a Ohw configuration for 14C. There are six quanta associated with the four particles removed. Thus, the Pauli exclusion principle requires the minimum number of quanta carried by the relative motion wave function to be Q= 2N+L =6, where Mis the number of nodes in the wave function and L is the relative angular momentum. If the four nucleons were removed from a lhu> configuration in 180 to give us a Ohw for 14C, there would be seven quanta associated with the four particles. Thus, a Q=7 cluster configuration corresponds to lhw configurations in the shell model. In a similar way, a 0=8 a+14c cluster configuration corresponds to 2Rw in the shell model, and so on. 3. Weisskopf Units (W.u.) It is traditional in studies of electromagnetic transitions between nuclear states to express transition strengths in terms of so called, single particle Weisskopf units. The Weisskopf estimate is obtained by evaluating the transition strength expected for a single particle described by a square wave function, which is constant inside the nucleus and zero elsewhere. Although based upon an extremely crude model, these units have the great advantage of focusing on those aspects of the transition strength independent of the nuclear mass and charge.

16 CHAPTER 2 Motivation for a Study of the Mass 18 Nuclei. 2.A The Mass 18 Nuclear System The mass 18 nuclear system has been the subject of much investigation (Aj82) both theoretical and experimental. The great interest in these nuclei is mainly because in their simplest configuration 180, 18F, and 18Ne can be considered as comprising two particles outside of a closed 160 core, and hence present themselves as good candidates for the simplest non-trivial shell model calculation possible within the SDshell. In such a calculation we treat these nuclei as two active nucleons in the SD-shell, with the 160 core as an inert closed shell contributing only to the total binding energy and to the establishment of the single particle energies. We take these single particle energies from the experimental spectrum of 170 relative to 160 as being: E(d5/2)="4'143 ; e(s1/2>="3-272 E(d3/2)=0'937 HeV and we use, for example, the Kuo-Brown(Ku66) values for the two body matrix elements required to describe the residual interaction between the two valence nucleons. This simple picture reproduces the structure of most low-lying levels of these mass 18 nuclei quite well and is supported by the strong population of these states in two-particle transfer reactions, wherein the two valence nucleons are added directly to an unexcited 160 core. However, the energy of the single particle -*-evel rather high, and it can easily be shown (La76) that such a simple model only

17 - 7- predicts two 0+ and two 2+ states in 180 at excitation below 6-MeV. This prediction is in disagreement with experimental observations, since there are known to be three low-lying 0+ levels (the ground state, the 3.63-MeV, and the 5.33-MeV states) and three low-lying 2+ levels (the 1.98, 3.92,and 5.26-MeV states) in 180. If we apply a similar (sd)2 shell model calculation to 18F and 18Ne again we find that we cannot reproduce some of the observed low-lying positive states. The unexplained states in 18F are the T=0 1.7-MeV(0+), 2.52-MeV(2+) and 3.36-MeV(3+) states. In 18Ne there are three 0+ states at 0,3.57, and 4.59 MeV, again a situation that is not reproduced by the model. If one is to explain these "intruder" states our description of these nuclei must include core-excitations. Let us first consider the three low-lying J*5 = 0+ T=1 states found in these nuclei. The 0+ ground states of 180 and 18Ne and the T=1 0+(1.041 MeV) analog state in 18F are all strongly excited in two particle transfer reactions (Co81,Ro73), and are associated with the (d^2)2 confi<3urati n- 0n the other hand, the T=1 O^ states in 180, 18Ne and 18F at 3.63, 3.55 and 4.75 MeV, respectively, are associated with four particle-two hole (4p-2h) configurations in view of their weak population via two particle transfer reactions and their strong population in alpha particle transfer or capture reactions. Furthermore, these T=l, 0+2 states show very little Coulomb energy shift relative to the lowest T=1 0+ state unlike the T=1 0+^ states which show a very large Thomas Ehrman shift in going from 180 to 18F and from 18F to 13Ne. The T=1 0+^ states are populated strongly in two particle transfer reactions and are considered to be dominantly (s1/2)2 in character. Similar evidence and arguments(la76) have led to

18 - 8- the identification of a (4p-2h) dominant structure in the 2+(5.26MeV) and 4+(7.11MeV) states in 180. The three low-lying T=0,l+,2+ and 3+ states in 18F which are unexplained by (sd)2 models(ro73) are strongly populated in alpha particle transfer(et83,br79) and are thought to be predominantly of (4p-2h) nature. They have been identified (Co77) as part of a KP = 1+ (4p-2h) rotational band in 18F, the members of the band being 1+(1.7), 2+(2.523), 3+(3.358), 4+(5.298), 5+(6.567), 6+(9.58) and7+(11.22) MeV. Therefore it is clear that even at low energies, positive parity states in mass 18 nuclei can exhibit relatively complex structure involving core-excitations alongside states of a more simple structure, where the 160 core remains inert. Of course, to explain all the experimental data we must consider mixing between 4p-2h configurations and 2p configurations as well as excitations involving higher shells. The simplest way to form negative parity states in mass 18 nuclei is to promote a particle from the 160 core into the SD-shell forming a three particle-one hole state, or to promote a particle from the SDshell into the PF-shell forming a two particle state. Most of the lowlying negative parity states in 18F and 180 are thought (Co81,Ro73,Ha68) to be 3p-lh states as they are strongly populated in three-particle transfer reactions(ha78). These low-lying negative parity states are weakly populated in two-particle transfer reactions on 160 (Co81) and one neutron transfer reactions on 170 (Li76), suggesting that two-particle sd-pf configurations play only a small role in these states. In 18F the low-lying T=0 and T=1 negative parity states exhibit (Ro73) similar structure and a well-defined Y? = 0" (3p-lh) rotational band has been

19 - 9- identified. However, some of the negative parity states in 180 and 18F are also strongly populated in alpha particle transfer, and some negative parity states above the alpha particle threshold have relatively large alpha particle reduced widths. This would suggest contributions from configurations such as 31Tuj may be important for these states. On the other hand, 3p-lh states have a sizeable alpha particle spectroscopic factor and can be seen strongly in alpha particle transfer. Many of the negative parity states in 180 and 18F are populated in alpha particle transfer reaction with strengths consistent with the addition of p(sd)3 and p2(sd)(pf) configurations to the A=14 p10 target to form l1tw states of A=18. The alpha particle spectroscopic factors are, however, relatively small, being of the order Above the alpha particle threshold in 18F some negative parity levels have large alpha particle widths and it has been suggested that they possess a weakcoupling structure of the type 14N(g.s)x20Ne(5.78HeV, 1"). In the shell model such a structure involves 3tuo configurations of the form p 2(sd)3(pf) + p'3(sd) (The K^=0" band in 20Ne built on this l*(5.78mev) level is described as a linear combination of (sd)3(pf) and p_1(sd)5 configurations with (Xy)= (90) ). The admixture of these 3tTw configurations into the basic TKw configurations of the negative parity states in A=18 could have an important effect on the alpha particle spectroscopic factors.

20 B. New Data Having taken a brief look at the general properties of these nuclei we are in a position to appreciate the value of new studies and data on mass 18 nuclei. In particular, information on the electromagnetic properties of the states is most important in bettering our understanding of the detailed structure of these nuclei The lowest l'(4.45 HeV) level in 180 deexcites (En79) via a strong El transition [ (2.8±.6)xl0"2 single particle units] to the excited 0+ state at 3.63 MeV. As we have noted, this 0+ state is populated strongly in alpha particle transfer reactions and is believed to have a large overlap with a 14C plus alpha particle cluster structure. In chapter 5 we shall see how such cluster states, when expanded in terms of shell model states, have a large 4p-2h component. Recent experiments at Yale and at Brookhaven (Ga83,Ru84,Wa86) have measured the electromagnetic properties of the higher lying natural parity states in 180, and have established several additional enhanced El and E2 transitions. *4C, levels in 180 above the Using the radiative capture of alpha particles by a+14c threshold (6.23 MeV) were studied for bombarding energies 1.1< E <2.7 MeV. A helium beam from the Brookhact ven 3.5 MV Van de Graaff accelerator was used with an enriched 14C target and the reaction 14C(a, 2f)180 was studied at resonant energies corresponding to six levels in 180 at excitation energies 7.86(5"), 7.62(1"), 8.04(1"), 8.12(5"), 8.214(2+) and 3.28(3") HeV. The deexcitation modes of these levels were studied, and electromagnetic deexcitation branching ratios were obtained. These results are reported in Ga83; however, recently a more detailed analysis of data from this experiment has been

21 carried out (Wa86), and these more up to date results are given in table These experiments were further extended to levels in 180 below alpha particle threshold using the 14C(7Li,ty)180 reaction. A 7Li beam from the Yale HP tandem accelerator was used with an enriched 14C target. All states of 180 between the 1.98-Hev(2+) and 7.11-MeV(4+) states were populated in this reaction, and the measured electromagnetic deexcitation branching ratios are given in table 2.2. The results show several enhanced and interesting transitions. From tables 2.1 and 2.2 we find a number of E2 transitions with strengths between Weisskopf units (W.u.). Such enhancement of E2 transitions over single particle estimates would suggest that these transitions are of a collective nature and that they connect states having similar structure. The 2+^(5.26-MeV) state decays by an E2 transition having 27±16 W.u. of strength to the 0+2(3.63-I-IeV) state. This strong transition confirms the suggestion that these two states are of a similar nature. However, it is noted that the well known 4p-2h state at 7.11(4+2) was not observed to deexcite to the 2+ state, and an upper limit of 18 W.u. was established for this E2 transition. E2 transitions are also observed connecting negative parity states. The 3_2(6.4MeV) state was observed to deexcite to the 1'^ state with transition strength of 10.4±6.2 W.u., and the l ^(8.04MeV) state deexcites to the 3"^(5.1MeV) state with a strength of 4.7±3.8 W.u. Other strong E2 transitions between negative parity states are reported in Ga83; however, the more recent analysis of the 14C(a,2f)180 experiment does not confirm the observation of these branches, and only upper lim-

22 Table 2.1 Electromagnetic Transitions in 180 from 14C(a,?f)180 E. 1 JP Ef Mult. Br.Ratio El El 6213 M2 6= " 4456 Ml 811 E2 6= El < El Ml, E2 < HI, E El > El El El El < Ml E El El " 5530 Ml, E2 < Ml, E2 <2 B(E\) W.u. 5.9±.13x ±.8xl ± ± ±.9 <3.1xl ±.15xl0-3 < x x10" x x10-3 <5.1x10" x10' x10" <0.1 < 0.26

23 El >95 7.7±.15xl E2 <2 < El <2 <1.7xl E ± Ml,E2 29± Ml,E2 3± E2 <3 < Ml,E2 3±1 9.5i.38xl El 29± xl0-3 3" 5098 El 17± x10-3 3_ E3 <7 < El <3 <1.8xl El 63± xl El <3 <5.5xl0~ E2 <10 <38 3" 5098 Ml,E2 <8 <8.5xl El 37± x El <4 <2.5xl0-3 2" 5530 Ml,E2 <8 < HI, E2 <5 <0.26

24 Table 2.2 Electromagnetic Transitions in *80 from *4C(7Li,ty) and from AJ82 ± + ± ± Ea Ji Ef Jf Branching Mult BCtoX:^ -* Jf ) W.u. MeV MeV Ratio From AJ E ± E ± E ±.5 E ± ±.5 Ml o» <.2 El < 5. x ±.5 El 2.6 x ±.8 X ±.5 El 2.8 x 10"2 (2.8 ±.8) X 10" ± 2 El 2.33 x 10" 3 (4.8 ± 1.1) X ±.6 El 6 x 10-4 (5.7 ± 2.3) X El 3.47 x 10"4 (5.0 ± 2) X 10~ ±.4 El 2.5 x 10-3 ( ) X r <.14 E2 < ±.9 E ± ±.1 Ml (5.8 ±.3) X E E ± ±.3 El 9 x 10-3 (8.2 ± 1) X 10" ± 5 E < 12 E2 < r 54.8 ± 5 El 5.7 x 10"3 (4.2 ±.9) X 10"3

25 (Table 2.2 Continued) ± ± ± ± E^ J Branching Mult BCtoXrJ^ - Jf ) W.u. MeV MeV Ratio From AJ El 1.4 x 10"3 (1.4 ±.2) X < ]L.3 El < 6.67 x El 7.5 x <.9 El < 2.83 x IQ" ±.4 Ml (9 ± 3.6) X " <.7 E2 < El 2.17 x El 8.4 x 10~ El 3.6 x 10-4 ( ) X IQ' El 1.29 x 10" El 2 x 10-4 (3.1 ± 2.2) X r E " El 1.9 x S E E El 3 x <.6 E2 < <.2 El < 1 X 10 3

26 - 12- its could be determined (Wa86). These experiments also revealed several strong El transitions in 180 in the lo'2 W.u. range. The 3^(8.29 HeV) and' 1'2(6.2 MeV) states deexcite to the 2+^(5.26 MeV) state by El transitions of (16±4)xl0~3 and (13±2.4)xl0-3 W.u. respectively. The 1"2 state also deexcites to the 0+2(5.33HeV) state v/ith strength 8.4±lxl0"3 W.u.. Although hindered in comparison to the strength expected of a single particle transition, these El transitions, together with the known l - to 0+2 transition, are enhanced compared to usual El strengths in light nuclei. The fact that these El matrix elements are rather small compared to a single particle estimate,however,implies the presence of substantial cancellation among different single particle transitions. Hence it is difficult to conclude much about the structure of states connected by such strong transitions a priori, since the El matrix elements are very sensitive to the details of the v/ave functions involved. However, it is worth noting that both the enhanced E2 transitions and some of the El transitions in 180 involve collective states and that most of these states are seen in alpha particle transfer reactions. This has led Gai et al.(ga83,a182) to suggest that this combination of El and E2 transitions is indicative of an a+14cmolecular structure in 180. In particular they suggest that the 0+(3.63 HeV), 1"(4.45 MeV), 2+(5.26 MeV), 3 (8.29 MeV) and 4+(10.29 MeV) states are part of a molecular a+14cband, and furthermore that the alpha particle cluster structure is fragmented over many states in 180. We leave detailed discussion of this molecular model of 180 until a later chapter and for now try to put these electromagnetic transitions in 180 in perspective by examining

27 -13- the known data and the structure of neighbouring nuclei. 2.C. Systematic Behaviour in Light Nuclei. As in the case of mass 18 nuclei, the structure of neighbouring light nuclei is dominated by the doubly closed shell at 160 and the existence of low-lying correlated four nucleon configurations. That two neutrons and two protons give rise to an energetically favoured fourbody correlation in nuclei is well known and is a reflection of the strong central attractive force involving nucleons in a relative S-state and of the Pauli exclusion principle. Two nucleons in a relative S-state are described by a spatially symmetric wave function, and such states of high spatial symmetry are preferred by the short range interaction. However, the Pauli exclusion principle restricts fully symmetric spatial wave functions to at most four nucleons since the spin-isospin part of the wave function cannot be fully antisymmetric for more than four nucleons - each nucleon can have spin up or down and isospin up or down. Such an alpha cluster correlation, involving four nucleons with properly aligned spins and isospins, is found to lie very low in energy throughout light nuclei. However, it is important to note that since the total wave function describing the nucleus must be antisymmetric with respect to all the nucleons we cannot think of this correlated substructure as an alpha particle within the nucleus. The existence of alpha particle cluster structure in these nuclei

28 -14- is reflected, however, by the presence of rotational bands with members of the band exhibiting similar alpha particle spectroscopic factors and alpha particle reduced widths and being connected' by strong E2 transitions. We now turn to explicit examination of several nuclei in the mass region of interest to us. Neon 20 : Several well defined band structures have been identified in 20Ne. There are four relatively low-lying KP = 0+ bands based on the 0+(g.s.), 0+2(6.72 MeV), 0+3(7.19 MeV) and 0+4 (3.3 MeV) states and KP = 2',0",1- bands based on the 2'(4.97 MeV), l'(5.78 MeV) and 1"(8.84 MeV) states. In its lowest energy configuration 20Ne consists of two neutrons and two protons outside a closed 160 core, and the ground state band is thought to have predominantly an (sd) 4 configuration with maximum spatial symmetry and corresponding SU(3) quantum numbers (X,p) = (8,0). This (sd) 4 shell configuration is identical with an (8,0 ) a cluster state, and in fact the members of this band (0+(g.s), 2+(1.634 MeV), 4+(4.248 MeV), 6+(8.77 MeV) and 8+(11.95 MeV)) are selectively populated in alpha particle transfer reactions (Co74,Sa77,Br79). Detailed calculations on 20Ne have been carried out by Tomoda and Arima (To78) and these confirmed the dominant (8,0) nature of the ground state band. The number of relative excitation quanta between the 160 core and the alpha particle cluster is Q = 2N+L = 8 and is the lowest allowed by the Pauli exclusion principle This is reflected in the fact that the predicted mean separation between the core and the cluster is only R^ s =3.5 fm for this band, which is much less than would be expected from a simple

29 -15- picture of two touching spheres. In contrast to the ground state band, that band based on the 0+4 (8.3 MeV) state is predicted to have a mean separation between the clusters of Rr s =4.6fm and to have large alpha particle cluster components (= 65%) with \ > 8. The members of this band have very large alpha particle widths T kev. The moment of inertia of this 0+. a 4 band is much larger than that of the ground state band reflecting the higher degree of localization of the clusters in the 0 +4 case. The 0 +2 band is strongly populated in the 19F(3He,d)2 Ne reaction and is thought to have predominantly an (sd) 4 (\,vi) = (4,2) shell configuration. That members of this band have sizeable alpha particle widths is explained by mixing with the 0+4 band. The members of the 0*^ band have small reduced alpha particle widths but are strongly populated in 8Be transfer reactions (Hi80) and are considered (Ar71b) to be mainly 8p-4h states. The K^3 = O' band based on the l'(5.78 MeV) level has been identified as a well developed negative parity a cluster band. The members of this band have large reduced alpha particle widths and are predicted (To78) to have a large rms separation between the alpha particle cluster and the 160 core. In contrast the K*3 = 2' and K*3 = 1' bands have- small reduced alpha particle widths and are described by lffw shell model configurations. The structure of 20Ne can be described by a coexistence of shell and cluster configurations. The ground state band can be identically described in terms of (sd) 4 shell model configurations or an a cluster configuration with 8 quanta of relative motion. On the other

30 -16- hand, the more developed a cluster bands KP = 0+4 and KP = O' involve excitations from many shells and are more easily understood in terms of a cluster picture. The 4p-Xh Bands A = 16 to A = 20 : The 4p-0h ground state band of 20Ne has analog 4p-xh bands in many neighbouring nuclei. Rotational bands corresponding to four correlated nucleons in the SD-shell are found to lie at very low excitation in 19F, 19Ne, 18F, 180, 170 and 160. The energies of the known states in these bands are shown in Fig 2.1. The 4p-lh KP = 1/2' band in 19F is based on the 1/2"(110 kev) state while in 19Ne the lowest member of the band is the 1/2'(275 kev) state. As we have already noted, the 4p-2h bands in mass 18 start with the 1.7 MeV 1+ state in 18F and the 3.63 MeV 0+ state in 180, and its analogues in 18F and 18Ne. In 160 the rotational band built on the 6.05 MeV 0+ state is now well established as a 4p-4h band, being mainly described as four particles in the SD-shell outside a 12C core. In 170 the band structure is somewhat more complicated as the 4p-3h states mix strongly with 2p-lh configurations. Arima and Hamamoto (Ar71) have attempted to describe these four particle bands from A = 16 to A = 20 in terms of n h les weakly coupled to the ground state band of 201!e. Taking the strength of the interaction between the P-shell holes and the four SD-shell particles from the spectrum of 19F, they predict the position of (sd) 4 - n hole states as follows ; (1) a 1+ T=0 state at 1.7 HeV in 18F and a 0+ T=1 state at 3 HeV in 180, (2) a 1/2' T=l/2 state at 4 MeV in 170 and (3) a 0+ state at 6 HeV in 160. These predictions agree very well with experiment, and the weak coupling of

31 Figure 2.1 Experimental energy spec tra of the 4p-xh bands A=16 to A=18 The states in 180 are possible mem bers of a similar 4p-2h band.

32 15 4Particle-X hole states A= 16 to A=20 6+ EXCITATION ENERGY (MeV) 10 8 * / 2 ' 4+ 9 /2 7/2-7/ 2 9 / 2" + 3 / / 2 " = 5/ 2 5 / / q + 5 / 2 2+ i + 1/ /2, / 2 ' 2 0 Ne l 9 F l 9 N e 4p-0h 4p-lh 4 p -lh l8 0 l70 l6 0 4p-2h 4p_3h 2p-lh 4p -4h

33 -17- P-shell holes to 20Ne becomes particularly obvious in the comparison of the spectra of mass 19 nuclei with that of 20Ne. In a similar weakcoupling calculation, Lawson et al. (La76) have taken the particle-hole interaction from the energy of the K*5 = 0+ band in 160 and predicted the 4p-2h band to occur in 180 at excitation energies 3.3 HeV 0+, 5.0 MeV 2+ and 8 MeV 4+. As already pointed out, the 0+(3.63 MeV), 2+(5.26 MeV), 4+(7.11 MeV) and 6+(11.69 MeV) states in 180 are usually considered to be part of a 4p-2h rotational band and these states are plotted as functions of J(J+1) together with the well established (Co77) 4p-2h band of 18F in Fig 2.2. The slopes of these proposed bands are roughly similar. Gai et al.(ga83) have suggested the 4+(10.29 HeV) state as a more probable candidate for the 4+ member of this band, as the 4+2 (7.11 MeV) state has not been observed to deexcite to the 2+(5.26 HeV) state. However, as shown from Fig2.2, this assignment would require the slope of the 4p-2h band in 180 to be much larger than that in 18F in disagreement with weak coupling predictions, and with the suggested well developed alpha particle cluster structure, and would necessitate a strong isospin dependence in the structure of these bands. Fig 2.3 compares the slopes of 4p-Xh bands in 160, 180 and 20Ne; here again coupling of particles to holes does not appear to cause any substantial effect in going from one nucleus to another. The K*3 = O' band in 20Ne based on the l'(5.78 HeV) state has an analog negative parity band in 160 based on the l'(9.632 MeV) state. As in the case of 20Ne, the K^5 = O' band in 160 is considered to be an excited alpha particle cluster band with nine or more quanta of relative excitation between the core and cluster. Corresponding bands have not

34 Figure 2.2 J(J+1) dependence of the established 4p-2h band in 18F and the possible 4p-2h band in 180. The 18F band is a T=0 K=l+ one, allowing a spin sequence 1+, 2+, 3+,.... The corresponding band in 180 is a T=1 K=0+ band giving rise to spin sequence 0+, 2+,... The moments of inertia associated with these bands are h2/2i=.180(18f),.186(180)

35 J ( J + l ) EXCITATION ENERGY (M ev)

36 Figure 2.3 J(J+l) dependence of the well established 20Ne 4p-0h and 160 4p-4h bands and the possible 4p-2h band in 180. The moments of inertia for these bands are h2/2i=.248(160),.186(180),.205(20ne)

37 EXCITATION ENERGY (MeV) J (J-H)

38 -18- been identified in neighbouring nuclei, although Buck et al.(bu77) have suggested that the positive parity doublets in 19F (l/2+(5.34 MeV), 3/2+(5.50 MeV); 5/2+(6.28 MeV), 7/2+(6.33MeV)) are suitable candidates for membership in a Q=9 alpha particle cluster band. In any case, on the assumption of weak coupling, we would expect such an alpha particle cluster band to lie about 5 MeV above the lowest member of the 4p-xh band in these nuclei. The low-lying negative parity states in 18F and 180 are expected to be mainly 3p-lh in nature as suggested by three particle transfer reaction data. E2 Transitions in Light Nuclei E2 transitions in light nuclei range in strength from 10-2W.u. to 60W.U. depending on the structure of the states involved. E2 strengths between single particle states are usually of the order of a Weisskopf unit, a typical example being the decay of the l/2+(0.87mev) state to the 5/2+(g.s.) in 170 with B(E2:l/2+ - 5/2+) = 2.4 W.u.. On the other hand, E2 transitions connecting members of collective rotational bands are very enhanced and are of the order of 20 W.u. or more. Table 2.3 collects the known intraband transition strengths for the 4p-xh bands we have been discussing. They are clearly of collective character, connecting states of a similar intrinsic nature. The transition strengths in 160 are exceptionally strong, reflecting the increased deformation in this band resulting from the presence of the four P-shell holes. A more complete compilation of gamma-ray strengths in light nuclei is given by Endt (En79).

39 Table 2.3 Nucleus Exi -> E _ xf 'I -V Jf B(E2) W. u 1 6 q > » ± » > ± 8 CO » ± » > ± > > ± ± 6 o CO y > ± p y /2 -y 1 / 2 " 2 0 ± y / 2 " y 1 / 2" 25 ± y /2" y 5/2" 28 ± 6 l^ne 1.51 y /2" y 1 / 2 " 58 ± Ne 1.63 y 0 2+ y ± y y ± y y ± y y 6+ 9 ± " -> 1" 50 ± 8

40 -19- El Transitions in Light Nuclei El transitions in light nuclei range in strength from 10~6W.u. to.33 W.u. and are generally very hindered with respect to a single particle unit. This weakness of El transitions between low-lying nuclear levels is explained by the fact that most of the El strength to a given level is exhausted by the giant dipole resonance located at much higher energy. Fig 2.4 shows a histogram of known isospin-allowed El transitions for nuclei up to A = 25. Most El transitions in light nuclei lie in the 10"3 to 10"4 W.u. range. The strong El transitions in 180 (of the order of 10~2 W.u.) lie in the upper range of this histogram, although a considerable number of even stronger El transitions have been measured in light nuclei. A table of the known dipole transition strengths (A < 25), taken from (En79), is shown in Table 2.4, with the kind permission of P.M.Endt. The largest known El transition between discrete states occurs in 11Be where the l/2'(.32 MeV) state deexcites to the l/2+(g.s.) with strength B(E1) =.33 W.u.. This transition, together with the enhanced dipole transitions in 9Be and 13C, are a reflection of weak-coupling, particle-core structure (Mi83). Several strong El transitions are found, in 13N and in mass 14 nuclei, and these have been studied (La84) in terms of the weak coupling of a particle to the core. The electromagnetic transitions in mass 14 have been examined in detail in a shell model context by Millener (Ko81), and the El strengths there were found to be very sensitive to the details of the wave functions involved. Buck et al. (Bu77) have attempted to explain strong El transitions in 19F in

41 Figure 2.4 Histogram of all known isospin allowed El strengths between discrete states A<25. The transitions in 180 on the 1 0_2W.u. scale lie in the upper quarter of this histogram. El transitions in even-even nuclei with AT=0 make up a very small and restrictive class. The only known values occur in 10Be, 180 and 22Ne, and the transition strengths in 180 appear typical (Table 2.4). Theoretical predictions for AT=0 El transitions in other even-even nuclei are also on the 10-2 scale (La84, K0 8I).

42 NUMBER OF TRANSITIONS <IO'5 io'5 10 "r 10 w 10 El - STRENGTH IN WEISSKOPF UNITS

43 terms of triton and alpha particle clustering; however, their predictions overestimate the experimental values by as much as two orders of magnitude in some cases. Similarly, a cluster model has been applied to 180 (As84, Sa84, DeS5) and again the El strengths are greatly overestimated. These models will be discussed in further detail in a later chapter. States connected by strong E2 transitions (>10W.u.) are usually of similar intrinsic structure, and out-of-band E2 transitions are generally quite small. In contrast, states in light nuclei are often seen to deexcite by strong El tarnsitions to several different levels. The GDR, which consumes most of the El strength to the g.s. of a nucleus, can be described in terms of the protons as a whole oscillating against the neutrons as a whole. Thus, the g.s. and the GDR state have quite different intrinsic structure, even though they are connected by a very strong El transition. Unfortunately, a systematic study of the origin of relatively strong El transitions between discrete in light nuclei has not been carried out to date. However, detailed calculations on many of these nuclei i show that a number of configurations are usually involved in dipole transitions and that the amplitudes of these configurations may often be small Having taken a brief look at the systematic behaviour in light nuclei, we can now try to understand the enhanced electromagnetic transitions in 180 within a consistent model. In this chapter we have looked

44 -21- at interesting data on 180 and have shown how mass 18 nuclei fit into the overall context of light nuclei. The experimental data all suggest that 180, or mass 18 nuclei in general, exhibit interesting structure. To understand this structure, we need a theoretical model which allows for the coexistence of different types of nuclear excitations. In particular, our model must include single particle, collective and cluster degrees of freedom and must lead to a consistent explanation of the observed data. The remaining chapters of this thesis are devoted to a theoretical study of the mass 18 nuclei and are an attempt to understand the structure of these nuclei in detail.

45 Table 2.4 Table of all known isospin allowed El transitions A<25; (taken from En79.)

46 P. M. E N D T G u m m ; i - R ; i > S t r e n g t h s. A = TABLE III. Strengths of Isospin-AIIowcd El Transitions (Cl 1V) Nucleus L i * L f t or r 111 Branching I S (MeV) (*) (W.u.) SBe /2* - 3 /2 ' from B (tl) (2. 2 ± 0.9H -1 ) CD O n * p /2*;3/2-5/2* ' 17 i 2 ev 7.4 * 1.8 (0 ) ( K -3) /2* 5.2 i 1.1 [0 ] ( )(-3 ) l/2 '; 3 / 2-1/ 2* 16 2 ev 8.3 * ( )(-3 ) /2* 9.1 * ] ( )(-3) * - r PS' 24 * 2 0 ( }(-2) Be / 2' - 1/ 2* fs ( ) ( - 1) U B (3 /2.5 /2 )* - 3 /2 ' 1.2 i 0.3 ev 87 * 2 1 0] ( M -3) 12c 13c 13m M» 15h /2 ' 5.5 i 1.0 (0 ] ( «2 )(-3) * /2" 7.5 X ] ( K -2 ) /2* - 5 /2 ' 300 mev 85.8 t 0.4 [ 0 J ( 7 2 )(-3) /2* 13.3 * ] ( 8 3 )(-3 ) /2* - 3 /2 ' 2.3 ev 19.7 * 1.0 (0 1 ( )(-3 ) /2 ' 67.5 t 1.0 [0 ] ( )( 2 ) * 7 / 2 ' t 0.7 ( 0 J (5.3 l-b )(-2) *; 1 * 3 ' 12 i 2 ev 2.9 t 0.4 (0 ] ( )(-3 ) *; 1 * 2* 8. 0 ev (0 ] ( )(-2 ) l ' i 1-0 * 49 ev 90 0 ( ) ( - 2 ) * * 1 0 ( 0 ) ( 7 2 )(-3 ) 3.09 * 0 1/ 2* * 1/ 2 * 1.55 * 0.15 fs ( )(-2 ) /2 ' - 1/2* 1.59 i 0.13 fs 1.1 * 0.5. [0 ] ( 6 2 )(*3) 3.85 * /2* - 3 /2 ' 12.7 t 0.3 PS 36.0 * 0.7 ( 0 ) ( H - 2 ) /2 ';3 /2-1/2* ev 9.1 * 1.6 [0 ] ( H -3) 2.37 * 0 1/ 2* - 1/ 2 * 510 * 90 mev too 0 ( ) ( - l) 3.51 * /2 ' * 1/2* 660 mev 8 * 1 [0 ] ( )(-2) S * - 0 * * 2 fs 69 0 ( }(-3) *; 1 * 1* 12.3 ev 79 [0 ] ( H -2 ) * ] ( ) ( - 2 ) * 0. 6 ( 0 ] ( ) ( - 2 ) * 2* [0 ] ( K - 2 ) 8.62 * * ; 1 * 1* 5.2 ev 12 0 ( 6 2 )C-2 ) 8.91 * *; 1 * 3* 410 mev 5.2 [0 1 ( )(-3 ) * 2* 2. 8 (0 ] ( )(-3 ) 9.17 * * ; 1 * 1 ' 11.9 ev 0.49 * 0.10 (0 ] ( )( 3) ' 0.61 * 0.08 ( 0 ] ( )(-3 ) 9.51 * *; 1 * 1* 4.9 ev 6 t 1 ( 0 ] ( )( 3J *; 1 * * 1.5 ev 2.4 (0 ] ( )( 3) ' 1.9 [ 0 ] (5.4 l-8 )(-3 ) S / 2* 1/ 2 ' 25 t 7 fs ( X -4 ) /2* «1 /2 ' 250 * 100 as 99.2 t 0.2 «0 ( )( 2 ) /2* - 5/2* fs 8 * 2 [0 ] ( )( 4) * 7.16 * 5/2* 2.3 * 0.5 [0 ] ( 8 3 )(-4 ) * 7.57 * 7/2* 5.0 * 0.6 [0 ] ( B )(-3) /2* - 5/2* 370 i 70 mev 37.4 * 0.6» 0 ( )(-3 ) * 3/2* 2.3 * ( )(-4 ) / 2* * 1/ 2* 4.2 * 0.7 ev ( 6. 8 l.l ) ( - 3 ) / 2*;3/2-1/ 2 " 54 i 20 ev 91 * 3 0 ( 7 3 )(-2 ) /2*;3/2 * 3/2* 4.6 t 0.7 ev 5.8 * 0. 6 «0 ( )(-3 ) /2* - 1 /2 ' 3.0 * 0.9 ev 100 * 0 ( ){-3) /2* - 1/2' 8.2 * 1.5 fs ( )(-3 ) 7.56 * 0 1/2* * 1/2* 42 mev 3.5 * ( 8 3 ) ( - 6 ) * 3 /2 ' 57.4 c 0.6 (0] (2.3 O.BK-2) /2* - 1 /2 ' 1.43 ev 53.8 * 0.3 roj ( K -3) * 3 /2 ' 2. 2 * 0. 6 (0) ( 8 3 H -3) /2* * 3 /2 ' 480 nev 33 [0] ( H -2 ) A to m ic D ata an d N uclear Data Tables. Vot 23. No I. January 1979

47 P. M. E N D T Gamma-Ray Strengths. A = 6-44 TABLE III. Strengths of Isospin-Allov.ed El Transitions (El,v) hjcleus - E, f (HeVJ * T 1 * ;--t r T o r r n y BrancMns (*) I s (V.u.) I5n 16 17u 18«18c 19c /2" 5/2* 2.4 ev 6.5 [0 ) (4.S 1.6K -3) /2* (1.1 t 0.4 )(-2 ) /2* 5.1 [01 ( K /2* 5/2* 5.1 ev 19 [0 1 ( J(-3 ) l/2 * ;(3 /2 ) 1/ 2 ' 32 5 ev ( ) ( - 2 ) /2 ' 22 8 [01 ( ) ( - l ) /2* 1/ 2 * 7.4 t 0.6 ev 74» 5 [0 1 ( )(-3 ) /2* 5/2* 1.9 t 0.3 ev 63 9 [01 ( ) ( - 2 ) /2* 3/2* 10 * 2 ev 53 7 [0 1 ( )(-2J /2 ' 5/2* ev 100 [0 1 ( }( 2) *; l 0 * 45 8 ev SS ( H -2 ) * ( ){-3) / 2* 1/ 2 * «: \ p* ( ){-6) /2* 14 ± 3 * 0 (3.0 ± 1.4){-5) /2* 3 /2 ' 33 3 ps ( )(-5 ) /2* 43 4» 0 ( )( 5) / 2 * 1/ 2* 12 * 2 mev ( H -3 ) /2* 5/2* mev 100 [01 ( )(-3 ) * 2* 65 t 15 fs ( )(-4 ) * ( ) ( - 2 ) * 2* 62 ± 25 fs 76 2 *> 0 ( 6 t 2 )(-4 ) * a 2 [01 ( 5 ± 2 }(-4) * 16 2 [0 1 ( )(-3 ) * 1' 200 t 40 fs ( )(-3 ) * 0* fs ( K -3 ) * 0* 340 mev ( )(-4 ) * 62 3 [0 1 ( )(-3 ) * < 1 0 ( )(-3 ) * 0 * 890 r.ev 16 I 0 ( ){-4) * [0 1 ( }( 3) * ( )(-3 ) * 5 I [0 ] (4.5 l.s )(-3 ) * 4* 220 oev 100 [ 0 ] ( )(-5 ) r 0 * ;1 310 t 70 fs 34 ± 2 0 ( )(-4 ) * 2* fs 30 3 * 0 ( )(-3 ) i* 0*51 10 i 2 rtev ( K -4 ) * ( 9 * 3 )(-4 ) * 4*;1 51 * 10 mev [0 1 ( ) ( 3) *;1 3* 1.5 ± 0.3 ev [0 1 ( 9 * 2 )(-4 ) * [01 (1.4 * 0.5 )( 3) * [0 ] (2.5 * 0.7 )( 3) * [01 (3.1 * 1.0 )( 3) / 2* 1/ 2* 853 t 10 ps ( X -3 ) /2 ' 5/2* ps [01 ( )(-6 ) /2 ' 1/ 2* fs » 0 ( ){-3) /2* ( ){-4) /2* 1/ 2' 3.5 t 1.3 fs [01 ( 6 2 ) ( 3) /2 ' 1/ 2* fs 35 4 [01 (1.8 * 0.8 )(-3 ) /2 ' 5/2* 41 * 8 mev 6 1 [01 [ )(-S ) /2 ' 5/2* rev 80 [01 ( 5 * 2 )(-4 ) /2* 1/ 2' ev 25 [01 { 7 2 )(-3 ) /2 ' 15 [01 ( 1. 0 * 0.3 )(-2 ) /2* 5 /2 ' mev ( )(-3 ) /2 ' 26 2 " 0 ( >(-3) /2* - 5 /2 ' mev ( )(-4 ) A to m ic O a ta a n d N u c le a r O a ta T a ilc s V o l 2 3. N o. 1. J a n u a r y 19~9