c . <. r..r.7.\ . .w . a. a, 5.... .2.: kind...” 4.... J3... 9 2 J. v .3 . . 5. o. .. a . .LNawucxtu . . . .. a: . fixavén . . .. r“ . ~ . r... .2. n ' t ~ . O . m... Vd..........,.. , .5?” I I t: . .a. GB?!) 9 a. . .hummwmrr. .c , flux“. «Ln 1 11 0E, ff an '6 '1'. ’ ssefia r “.15. ”fly—Or: i . f . mun-n. “Jun... figgawg gaming... .7 . .3... .2.. .2 .5. 2 ....§«§§£&§§¢ rd This is to certify that the thesis entitled ~ C a. i . ANOMALOUS e/B+ DECAY BRANCHING RATIOS presented by RICHARD BRIAN Fl RESTONE has been accepted towards fulfillment of the requirements for Ph .D . degreé in Chemistgy LIBRARY thkifiquaficau: l1ndve:sflgy E W Major professor Date June 18, 1974 0-7639 ABSTRACT + ANOMALOUS e/B DECAY BRANCHING RATIOS By Richard Brian Firestone 8/B+decay branching ratios were measured for decays from 1HEu, luagsm, and luang. Yi-Y triple coincidence experiments were performed to get relative positron feedings to respective daughter levels. KAY coincidence measurements were done for 1“596d decay to obtain relative electron capture feedings and these feedings were inferred from singles intensities for 1“3Eu and 1“3gSm decay. The decay schemes of 1“3Eu and 1”98111 were also studied, and 12 levels were reported in 1“Pm as well as 16 levels in 1“3Sm. Data were also gathered relevant to 1"MPm decay and a 8+ endpoint for 66Ca decay is reported. deecay theory is discussed in some detail, especially concerning the implications of e/B+ decay branching ratios. It is shown that anomalous ratios can only occur for higher order terms in allowed transitions or if a pseudoscalar force is present. Neither possibilities were hitherto eliminated experimentally. Among 15 measured €/B+ branching ratios, two were significantly anomalous. The transition to the 808.5 keV level in lusEu was anomalous by a factor of 24 and the transitionto the 1173.1 keV level in 1“3Pm was anomalous by a factor of 4.9. Each case involved a similar hindered I 4 ‘J h a", {m Richard Brian Firestone 0/ u (’16 K) allowed transition and it was suggested that the higher order allowed , terms might be dominant in those instances. ‘ l ANOMALOUS 8/8+ DECAY BRANCHING RATIOS By Richard Brian Firestone A DISSERTATION Submitted to Michigan State University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Department of Chemistry 1974 To Chuang Tzu ii ACKNOWLEDGMENTS I wish to thank Dr. Wm. C. McHarris for suggesting this very interesting and fruitful project, and for his valuable help and friendship during its completion. I would also like to thank Dr. W. H. Kelly and Dr. F. M. Bernthal for their willing assistance throughout this work. ' Very special thanks are extended to Dr. R. A. Warner for his in- valuable experimental assistance and to Mr. C. B. Morgan for his extensive aid in the preparation of many of the figures. To all the other members of the nuclear spectroscopy research group I offer my thanks not only for the innumerable times you assisted me in my work, but especially for the great kindness you showed me during my illness. I gratefully acknowledgetthe aid of the cyclotron staff in the completion of this work, and the financial assistance of the National Science Foundation, U.S. Atomic Energy Commission, and Mflchigan State University. Finally, a special thanks goes to my wife, Mary, who married me for better or worse and got a little of both, and to my parents without whom this work could not have been possible. l iii DEDICATION. . . . . . ACKNOWLEDGMENTS . . . LIST OF TABLES. . . . LIST OF FIGURES . . . Chapter I. II. INTRODUCTION. TABLE OF CONTENTS EXPERIMENTAL APPARATUS AND METHODS. 2.1. 2.1.1. 2.1.2. 2.1.3. 2.1.4. 2.2. 2.2.1. 2.2.2. 2.3. 2.3.1. 2.3.2. 2.3.3. 2.4. 2.4.1. 2.4.2. y—ray Spectrometer Systems. . . . . Ge(Li) Singles Spectrometer Ge(Li) - Ge(Li) Coincidence Spectrometer. . Ge(Li) - 8X8-in. NaI(Tl) Split Annulus Pair Spectrometer . Ge(Li) - LEPS Electron Capture- feedings Spectrometer . . Pilot B Plastic Beta Spectrometer. Beta Spectrometer Systems . . . . Si(Li) Beta Spectrometer. Target Preparation. . . MSU Sector Focused Cyclotron. The Rabbit System . The He - Jet Thermalizer. POLYPHEMUS. TOOTSIE iv Data Acquisition Programs . . . . Page ii iii ix xi 16 17 17 19 23 23 23 24 24 25 25 Chapter 2.4.3. 2.4.4. 2.4.5. EVENT and IIEVENT . . . . . HYDRA . . . . . . . O . O . PDP-g FHA Programs 0 o o o 0 III. DATA ANALYSIS . . . . . . . . . . . . . . 3.1. y-ray Energy and Intensity Analysis 3.1.1. 3.1.2. 3.1.3. y-ray Intensity Calibration y-ray Energy Calibration. . Data Analysis Programs. . . 3.1.3A. MOD7 . O . . . . .~ 3.1.3B. SMO. . . . . . . 3.1.3C. EVENT RECOVERY . . 3.2. Analysis of Beta Spectra. . . . . . 3.2.1. 3.2.2. 3.2.3. 3.2.4. Principles of Beta Analysis FEMLOT . . . . . . O . . . Preparation of Fermi—Kurie Plots. Energy Calibration of Scintillation Beta Spectra. . . . . . . . . . IV. THE DECAY OF 1‘3Eu. . . . . . . . . . . 4.1. Introduction. . . . . . . . . . . . 4.2 Source Preparation . . . . . . . . . 4.3 Experimental Data. . . . . . . . . . l4.3.1.. 4.3.2. Singles y-ray Spectra Coincidence Spectra . . . . 4.3.2A. Megachannel Coincidence Spectra. . . . . . 4.3.2.8 Pair Spectra . . . Page 26 26 26 28 28 28 31 31 31 32 32 33 33 34 34 34 37 37 38 39 39 40 40 56 Chapter V. VI. VII. 4.4. Proposed Decay'Scheme . . . . . . . . . . . . . 4.5. Discussionj . f . . . . . . . . . . . . . . ... 4.5.1. Single-particle States. . . . . . . . . 4.5.2 Other States . . . . . . . . . . . . . . 4.5.3. The 143E“ Ground State. . . . . . . . . 4.5.4. e/B+ Ratios . . . . . . . . . . . . . . THE DECAY 0F 1 u 38m. . . . . . 0 . . O . . . . . . O . 5.1. 5.2. 5.3. 5.4. 5.5.4 Introduction. . . . . . . . . . . . . . . . . . Source Preparation. . . . . . . . . . . . . . . Experimental Data . . . . . . . . . . . . . . . 5.3.1. Singles Ybray Spectra . . . . . . . . . 5.3.2. Coincidence Spectra . . . . . . . . . . 5.3.2A. Megachannel Coincidence SPeCtra O O O O O O O O O O O 0 5.3.23. Pair Spectra . . . . . . . . . Proposed Decay Scheme . . . . . . . . . . . . . Discussion. . .i. . . . . . . . . . . . . . . . 5.5.1. Single-Particle States. . . . . . . . . 5.5.2. 3-Quasipartic1e States. . . . . . . . . 5 . 5 . 3 . Remaining states . . . . . . . . . . . . 5.5.4. e/B‘+ Ratios . . . . . . . . . . . . . . THE DECAY OF 1 u oum . . . . . . . . . . . . O O O . . ‘6.1. 6.2. 6.3. 6.4. IntrOdUCtion O . I O . O O O O C O O O O O O . O Source Preparation. . . . . . . . . . . . . . . Experimental Da ta 0 0 O O O O O O O O O O O O 0 Results and Conclusions . . . . . . . . . . . . MEASUREMENT OF THE 6* ENDPOINT or °3ca DECAY. . . . . v1 Page 56 63 63 64 64 64 68 68 69 7o 70 74 74 75 75 83 84 85 88 88 90 9O 90 90 93 99 Chapter VIII. e/B+ BRANCHING RATIOS . . . . . . . . . . . 8.1. Introduction to Beta Decay Theory. . 8.1.1. 8.1.2. 8.1.3. 8.1.4. 8.1.5. 8.1.6. 8.2. Mathematical Derivation of Allowed Beta The Neutrino . . . . . . . . The Electron . . . . . . Angular Distribution of Beta About Polarized Nuclei . . . Electron-Neutrino Angular Correlations . . . . . . . . Neutron Decay. . . . . . . . General Theory of Beta Decay Decay Theory . . . . . . . . . . . . 8.2.1. 8.2.2. 8.2.3. 8.2.4. 8.2.5. 8.2.6. Transition Probabilities . . Lepton Wave Functions. . . . The B-InteraCtion. o o o o Particles H - The Beta Interaction Current. 8 The Neutrino Wave Function . E/B+ Branching Ratios. . . . . . 8.3. IImplications of the Allowed Assumptions. 8.4. Experimental Measurement of e/B+ Ratios. 8.4.1. 8.4.2. ”5966........... 8.4.1A. The Decay of 1“ngd . 8.4.13. 8/8+ Branching Ratios " 59Gd . the Decay of InagSm and 143E“ . . . . . . 8.4.2A. The Decay of 1“3Sm and “aEu . . . O . 8.4.23. E/B+ Branching Ratios in the Decays of 1“398m and 1”3Eu. . vii Page 102 102 107 107 109 110 115 119 119 119 125 128 135 144 150 155 158 158 158 162 168 168 170 Chapter 8.5. 8.6. 8.7. BIBLIOGRAPHY. Discussion of Anomalous €/B+ Ratios . Possible Explanations of Anomalous e/B+ Ratios . . . . . . . . . . . . . Further WOrk on Anomalous EIB+ Ratios viii Page 170 175 177 180 LIST OF TABLES Table Page 2.1 Ge(Li) Detector Specifications. . . . . . . . . . . . 4 2.2 Canberra 1464 Restorer-Rejector Performance . . . . . 6 2.3 Calibration Energies from 66Ga. . . . . . . . . . . . 20 3.1 Y Ray Energy and Intensity Standards. . . . . . . . . 29 3.2 Calibration Sources for the Energy Calibration of Scintillation Detectors. . . . . . . . . . . . . . 36 4.1 Energies and Relative Intensities of y-rays from the Decay of 143E“. . . . . . . . . . . . . . . . . . 42 4.2 1“3Eu Coincidence Summary . . . . . . . . . . . . . . 55 4.3 Levels in 1“38m From Beta Decay and Reaction Studies . . . . . . . . . . . . . . . . . . . . . . . 59 4.4 Single Particle States of N=81 Nuclei . . . . . . . . 65 4.5 Comparison of Experimental and Theoretical §£§°t)/B+ Ratios for the Decay to States in S O . . . . . . . . . . . . . . . . . . O . . 0 . 66 5.1 Energies and Relative Intensities of Y-rays from the Decay of luasm . . . . . . . . . . . . . . . 73 5.2 luamSm Coincidence Summary . - - . . - . - - . . - . 79 5-3 Levels in luaPm From Beta Decay and Reaction Studies. 82 5.4 Theoretical and Experimental €(totMB+ Branching Ratios for the Decay of 38m . . . . . . . 89 6.1 Half-life Data for the Decay of luoum. . . . . . . . 91 6.2 Relative Singles and Coincidence Intensities for the B+/€ Decay of luman. . . . . . . . . . . . . 94 8.1 Matrix Elements Contributing to Allowed Transitions . . . . . . . . . . . . . . . . . . . . . 157 8.2 Experimental Positron and Electron Capture Relative Y—ray Intensities for luscd Decay. . . . . . 160 8.3 Experimental Positron and Electron Capture Relative B-Feeding Intensities for 1“ Gd Decay. . . . 161 'Iv Table ' Page 8.4 Theoretical and Experimental e/B+ Branching Ratios for luscd Decay. . . . . . . . . . . . . . . . 165 8.5 Z B-Feeding to Levels in 145E“ from 1‘5Gd Decay Singles and €/B+-Feeding Coincidence Experiments. . . 167 8.6 B-Feedings from Bees 3 of 1“38m and 1‘3Eu to Levels in 113Pm and ”38m Respectively. . . . . . . . 169 8.7 Theoretical and Experimental €(tot)/B+ Branching Ratios for the Decays of 143E“ and 11.38m . . . . . . . . . . . . O . O . . O . . O . 171 Figure 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 4.1 4.2 4.3 4.4 4.5 5.1 5.2 5.4 LIST OF FIGURES y-y Coincidence block diagram . . . . . Yi-y Coincidence block diagram. . . . . yi-y NaI(T1)-A, NaI(T1)-B, and Ge(Li) yi—y TAC spectra for 22Na decay . . . . + y"-y Coincidence, singles, and chance coincidence spectra for 22Na decay. . . . . . . . . . . . . . 13703 B—spectra taken with pilot-B plastic B-detector; without absorber (bottom) . . . . . . 556a Double-escape spectrum . . . . . . . . . . . Si(Li) Energy calibration curve; standard deviation of linear fit 2.6 keV . . . . . . ... . 1“3Eu singles spectrum taken with 10.4% detector. . 1“3Eu y-Y coincidence spectra gating on the and displaying the coincident X axis. . . . 1“3Eu y-y coincidence spectra gating on the and displaying the coincident Y axis. . . . 1“3Eu yi—y pair coincidence spectrum. . . . Y axis 1“3Eu decay scheme. A11 energies are given in keV. and transition intensities are given in terms of percent per disintegration of the parent. The 107.7-keV transition is corrected for internal conversion. The logft values are calculated from tables by Cove (G071) . . . . . . . . . . . . . . . 1“3Sm singles spectrum taken with 10.4% detector. . 1“3Sm y-y coincidence spectra gating on the Y axis and displaying the coincident X axis. . . . . . . . 1“38m y-y coincidence spectra gating on the X axis and displaying the coincident Y axis. . . . . . . . 1“3Sm decay scheme. A11 energies are given in REV: and transition intensities are given in terms of percent per.disintegration of the parent. keV transition is corrected for internal conversion. The logft values are calculated from the tables by Cove (6071).. . . . . . . . . . . . . . The 272.2- Page 11 13 14 15 18 21 22 41 44 50 57 58 72 76 77 80 Figure Page 5.5 Stylized model of 1138m decay to single-particle and 3-quasiparticle states in 113Pm . . . . . . . . . 87 6.1 nong singles spectrum taken with 10.42 detector . . 92 6.2 ll”’97Pm y-y coincidence spectra taken with 10.4% (X) and 7 . oz (Y) detectors . . . . . . . . . . . . . . . . 95 6.3 1‘”gm: decay scheme. All energies are given in keV, and transition intensities are given in terms of percent per disintegration of the parent. The 6+, 3000 keV level is postulated without firm experimental evidence. Logf$ values are calculated from tables by Gove (G071). . . . . . . . . . . . . . . . . . . . 98 7.1 Adjusted Kurie plot of betas from 63Ga and 6"Ca decay measured with a large Pilot B plastic scintillator detector. Nonrlinearity at lower energies is due to lower energy beta branches which were not resolvable in this plot. . . . . . . . 100 8.1 Possible orientations of lepton spin S to nuclear spin 1. ... . . . . . . . . . . . . . . . . . . . . . 111 8.2 e+¥v spin orientations. . . . . . . . . . . . . . . . 111 8.3 Mixing of correlation coefficients. . . . . . . . . . 114 8.4 Time reversal . . . . . . . . . . . . . . . . . . . . 114 8.5 Screw sense of the proton recoil. . . . . . . . . . . 118 8.6 luSng Yi-Y positron feeding coincidence spectrum.. . 159 8.7 1“59Gd XFY integral and coincidence electron capture feeding spectra . . . . . . . . . . . . . . . 163 8.8 lksgcd XrY coincidence TAC spectra. . . . . . . . . . 164 8.9 Stylized model of lungd decay to single-particle and 3-quasiparticle states in 1“5Eu . . . . . . . . . 173 xii CHAPTER I INTRODUCTION A major purpose of this thesis is to investigate the anomalous 8/8+ decay branching ratios first reported by Eppley and McHarris (Ep7l). These ratios are an important test of allowed 8- decay theory, and large discrepancies, such as those Eppley saw in luSng decay, are not easily explained. It was shown for pure Fermi (0+ +'0+) decay (Ar53) and for 22Na Gamow-Teller decay (Sh53) that the interference terms were small or zero, and many measurements of allowed decay have since confirmed this. Unique forbidden decays were shown to yield much larger ratios than allowed decays, but this seemed adequately ex- plained by CVCA interference terms (Pe58). Thus, instances of small deviations (mx «mma was now seem .60.. mo amen >6: NNH was “on seams n o aouw 80 mm um ooom mo xmon >63 Nmma on» ou uooamou nuazo souaaou ou 300m oouuo Ammmqv nosoam nIIII o>m own N av movofin “moaoaz Hmwxoou many H ou mm n>ox H.~ Nq.oa uuuoaamo Hmaxmoo onus - H on an n>mx m.H No.5 oouuo Hmwxmoo onus H ou SN n>ox a.H No.¢ oouuo Hmwxooo many H on «N n>ox o.~ No.m ,.nmA souuoaomu pm: Hmvaoumemua H on o n>mx.~.m No.an nousuoomaomz manna ofiumm soausaomom mxoaouowmmm msoaumofimaoomm uouoouoa Aquoo HIN DHQMH were not especially elaborate. The FET preamps were all supplied by the detector manufacturers, and the most commonly used amplifiers were the ORTEC 451, 452, and 453 spectroscopy amplifiers. Those spec- tra involving large count rates (25000/sec) were taken with a Canberra 1464 Restorer/Rejector following the amplifier. This module included a high quality base line restorer as well as pile-up rejection. The pile-up rejection system eliminated counts that would create false "Compton" background below a peak and thus maintained good peak-to- Compton ratios at high count rates. Table 2-2 shows typical perfor- mance of the Restorer/Rejector with the 4.6% detector. Data were collected with three separate multichannel analyzer systems depending on the experimental convenience and the "up" status of the system in question. Some data were acquired with a Nuclear Data 2200 ADC with a hard wired 4096 channel memory and a 16 MHz digitizing rate. This system had the advantage of being quite por- table but the only means of output at the times it was used was a Tally punch. This output was quite slow, and the means of interpret- ing the ensuant IBM-format tape was very tedious. A second system which was employed was a Northern Scientific 625 ADC interfaced to a DEC PDP—9 computer and with a digitizing rate of 40 MHz. The ADC was capable of taking two 4096 channel spectra but our use was limited to only a single spectra. The disadvantages of taking data at the PDP-9 were the distance from the cyclotron which made counting short- lived activities into a foot race, and also some output slowdowns because of the fact that an interface to the Sigma-7 cyclotron computer was not yet completed. Nevertheless, the PDP-9 was usually our most reliable system and its isolation from many sources of electronic , 563 «m2 3. 63866:; .zm3h >63 o.~ I A>63 mmmav acauadom63 “H"NN I sounaou ou 366m um v6u6u uouo6u6v Nm.¢6 >63 m.~ Hua.mH >63 m.~ Hum.MH >63 o.m Hum.MH oooo¢ >63 o.~ Huo.H~ >63 o.~ .-. Hum.oa >63 o.~ Hum.oa oooom >63 o.N Hun.H~ >63 o.~ -- Hu¢.mH >63 m.~ Hum.o~ cocoa >63 o.~ Hum.H~ >63 c.~ aua.wa >63 m.~ Huo.ma ooom souaaou .. sOumaoo scumaoo Amaov soauaaom63 0» 366m sofiuaaom6m cu 366m aofiusaom63 0» 366m 6u63 usaoo .wouo6fi63 6:6 u6uoummm n >Hco u6uoum63 n 666cham 6uawsuomu6m nouo6fi6mlu6uou663 qoqfl <33mmz 2 source .______JL______ .JL V ' Linear Timing Timing Linear Amplifier Filter Filter Amplifier Amplifier Amplifier Constant Time to Constant Fraction Pulse Fraction SCA Height SCA Converter \ . SCA SCA SCA 1 A B C Slow Coincidence - _ \ -LJ' Coinc. Synchronous Mode Coinc. ‘\J' 4096 X-side ““31d8 4096 811112]. 0—6v 0-6V Channel ADC ADC Magnetic Tape Channel .Channel No. of W o. of x- Event vent . F” Figure 2—1. .y—Y "FAST-SLOW" coincidence block diagram. Time to Pulse Height Converter. An identical system was established for both detectors to start and stop the TAC whose output was sent through an ORTEC 418 Slow Coincidence box along with the output from ORTEC 451 Spectroscopy Amplifiers corresponding to the energy signals from each detector. The three signals in the slow coincidence were gated by ORTEC 455 Single Channel Analyzers to allow choice in the dynamic energy range observed as well as the ability to restrict thev .time range accepted. The slow coincidence output guaranteed pairs of real coincident events so noise triggering was mostly eliminated. This output was used to gate the ABC's which operated in synchronous mode to collect both gated energy legs under program EVENT or IIEVENT where data were written in pairs on magnetic tape. A description of EVENT follows later in this chapter. Timing obtainable using this system was better than 20 nsec FWHM for a TAC peak, but at high count rates (~l0,000/sec) and large dynamic range the realized timing was sometimes as low as 80 nsec. This meant that at worst a count rate of ”SOOO/sec in each detector would allow 2 chance events per second; yet real rates were generally 2100 coincidences per second. The problem of chance coincidences is further reduced by background subtraction in the data recovery which is discussed in a later chapter. 2.1.3 Ge(Li) vs 8X8-in. NaI(Tl) Split-Annulus Pair Spectrometer A large split-annulus with optically isolated halves was used in these experiments. The energy resolution of each half of this annulus was nearly 10% for 137Cs (662 keV), which was sufficient for gating annihilation Yi radiation. The absolute singles efficiency 10 for a central source in the annulus was about 60%, allowing reasonable efficiency for coincidence spectrometry. The annulus was open at both ends allowing the snout of a Ge(Li) detector cryostat to project to the center of the annulus. In positron-feedings experiments the annulus was used to verify positron events. The source was enveloped in a teflon absorber such that all B+'s annihilated near the source. Coincidences between the Yi annihilation photons in the two halves of the annulus signified a positron event. Having established a positron event, a third y-ray coincidence in the Ge(Li) detector was sought to label the level which was positron fed. The spectrum obtained in the Ge(Li) detector then corresponded strictly to the positron decay of the source and provided relative feedings to all levels of the daughter nucleus with the exception of the ground state, which, of course, doesn't deexcite. An electronics block diagram for this spectrometer is shown in Figure 2-2. Each half of the annulus' phototube outputs was sent through a cathode follower to a Canberra 1411 DDL Spectroscopy Amplifier where the DDL output was sent to an ORTEC 455 Timing Single Channel Analyzer gated on the Sll-keV Yi annihilation radiation. The crossover timing TSCA output was then sent to an ORTEC 414A Fast Coincidence box. Timing signals from the Ge(Li) detector were sent through an ORTEC 454 Timing Filter Amplifier and an ORTEC 453 Constant Fraction Discriminator, where the constant fraction timing output was also sent to the fast coincidence box. Requiring coincidences from both halves of the annulus as well as the Ge(Li) detector within a preset time of up to 110 nsec, the resultant logic signals ll .Emuwmao 36032 msumpmmam 66:6wfiocfioo 6HQH~H xnaamuaam umwmaamud amoumouuumam .N-N .tswat umnmaaar. mmoomonuumam uo6uaa. u6fimw3m8< aaoowouuo6am 1‘ oa< 36cc6£o omoq .oafloo u0umzwsauomaa cowuomum unnumcoo II: dowH 666; on I pN I 66:6vaoawoo H6MWHHQE< umom u6uafim wcHEfiH 63 33m Amvaz - x7 ‘ Aagvmu > e)\v\ . .o >63 Han Aomwzm comm 8.3 8.8 8.8 _ 89 moom ZO._.n__>_Oo + zo_._._mz916 O m>m30 ZO_._.0mwzm :1: _m 1 00m I 009 r 009 I OOON HBEIWHN ‘EINNVHO 23 energies. Collimation was difficult, and many of the lost events involved scattering from the detector. The response problem is well known in the literature (Gr66,Wi67,Be68), and, although complex cor- rection functions are given, it would be difficult or impossible to obtain good Fermi-Kurie plots. With these problems in mind, the measurement of B endpoints was abandoned for the présent. 2.3 Target Preparation 1 2.3.1 MSU Sector Focused Cyclotron All of the activities discussed in this thesis were prepared at the Michigan State University Sector Focused Cyclotron. This machine allowed a variable energy proton beam to over 50 MeV deuteron beams to 28 MeV, 3He beams to over 70 MeV, and l'He beams to over 50 MeV. Generally activations with currents considerably above 1 uA were pos- sible, and during long experiments the machine could be expected to operate quite reliably. 2.3.2 The Rabbit System In order to facilitate the handling of short lived activities, a rabbit system was constructed to transport samples to the bombard— ment area. Transport time was typically about 3 seconds, and the system was fully automated to allow efficient operation. Most irradiation samples were in powder form so that it was neces- sary to use aluminum foil packets to contain the material. Separation of the activity from the packet, after bombardment, took about 30 seconds additional time. Those samples which were foils could be counted more quickly, but, in general, it was desirable to allow 24 several seconds for short activities to die away. Whenever possible, bombardments were done at the energy desired and no degraders were needed. When necessary, aluminum degraders were placed in front of the target to obtain required energies. 2.3.3 The He-Jet Thermalizer In order to study very short activities (250 msec) a gas transport system was developed (K0873). This involved thermalizing nuclear recoils from the target, transporting them by He-gas flow through a narrow capillary at near-sonic velocities, and collecting them on a . tape-transport system. It was possible to collect the recoils directly in front of a detector or elsewhere on a movable tape which could be stepped in any desired fashion. . Generally, metal foil targets were preferred although the use of fragile rare earth targets, prepared in this laboratory, was unsuc- cessful because of the short (:1 hour) usable life of such targets in intense beams. Attempts were made with powder-epoxy emulsions with some promising results. 2.4 Data Acquisition Programs The cyclotron computer staff provided a number of valuable data- taking programs to ease our experiments considerably. These programs utilized NS629 ADC's, routing electronics, storage scopes, and their associated switches. Input was by means of teletype or scope switches, and output was available through line printer, plotter, or card punch as well as live display on the scopes. Data taking programs for the PDP-9 were provided by DEC and altered 25 by Giesler, Bradford, Howard, and others (PDP73). These were all, essentially singles routines and are explained later. Display at the PDP-9 was limited to an oscilloscope for all but one program which used a storage scope. Input was generally through the teletype, and output was via teletype, paper tape, DECTAPE, or magnetic tape. A brief discussion of each of the major programs used follows. No attempt will be made to explain the inner workings of these programs. 2.4.1 POLYPHEMUS POLYPHEMUS (Au70) is a program which allows the use of up to four . 8192-channel ADC's independently. This program was most useful for taking singles spectra and setting up for coincidence experiments. 2.4.2 TOOTSIE TOOTSIE (Ba7l) is a program designed for multiparameter data ac- quisition. A variety of options are available to perform various calculations on the data before they are stored. This program operates in two modes, SETUP and RUN. In SETUP mode the data are stored in two-dimensional arrays of dimension (64,64), (64,256), 128,128), or (256,64). The data are displayed on a scope as a series of bands across the screen. Polynomial fits to a set of points obtained by accepting the coordinates of a movable cross define the lower and upper bound for each band. Up to eight independent detector systems may be used at one time with three routing bits supplying detector identification. In SETUP mode data from one detector are stored, all others being discarded, permitting bands for each detector to be set. 26 Having set all bands, the program is switched to RUN. Tables of 256 points per fit are generated and stored for each detector. Events are then checked against the tables corresponding to the detector indicated by the routing bits and the aypropriate channel of the spectrum in which the match was found is incremented. 2.4.3 EVENT and IIEVENT EVENT (Ba70) is a program for taking up to four parameter coinci- dence data. Each multiplet of coincidence data is written on magnetic tape for off-line analysis. In the instance of data appearing in only some of the ABC's, a channel zero conversion is given for the barren ADC's and recorded on tape. IIEVENT (Au73) is a more recent version of the program with the ability to monitor the input from each ADC and set one gate for observa- tion simultaneous to the data acquisition. This program runs in two modes, SETUP where no data is recorded on magnetic tape and RUN where data is recorded on tape. 2.4.4 HYDRA HYDRA (Au69) is a program.which can use routing to store up to eight spectra in a given ADC. This is useful for half-life determina- tions where consecutive spectra from the same ADC are desired. This program is limited to only half-words (65,535 counts) requiring fre- quent punching of data in some instances. 2.4.5 PDP-9 PHA Programs A package of Pulse Height Analysis (PHA) programs was available from DEC(PDP73) and modified here. These programs allowed singles 27 spectra to be taken. A total of 262,143 counts were obtainable for each channel upon which data taking was either disabled, or a channel overflow message was printed out and data taking resumed with the over— flowed channel reset to zero. CHAPTER III DATA ANALYSIS 3.1 YbRay Energy and IntensitypAnalysis 3.1.1 41%Ray Intensity Calibration The Ge(Li) detectors uSed in these experiments were calibrated for relative efficiency by myself and others using standards listed in Table 3-1. As these standards varied in their energy ranges, their relative intensities were normalized and fit by least squares analysis to give an efficiency function. This function was usually fitted in two parts, a lower energy region where efficiency dropped off rapidly from its maximum and a higher energy region. Generally, efficiency curves were measured at contact, 2 inches, and 10 inches from the detector. The differences in these curves were generally minor until one looked at lower energies, but, in general, data were taken at sufficient distance to justify using the calibration for 10 inches. While most standards were internally consistent, deviations between different standards were often con- siderable. It was general practice to weed out all "bad" points and take the best fit of the remaining points. It is this experimenter's opinion that an efficiency curve cannot be trusted to better than 5-lOZ below 300 keV. Below this energy detector efficiency is changing very rapidly and source thickness and geometry become increasingly important. It is doubtful if efficiencies in the low energy range can be believed to better than 20% and some very low values may not be believed at all. 28 Y Ray Energy and Intensity Standards 29 Table 3-1 E I E I E I 2“141(16068) 13703(Led68) l°°mH£(Led68) 1368.53 100. 32.1 6.85 93.3 16.8 2753.92 100. 36.5 1.54 215.3 80.6 661.6 100. 332.5 94.8 56CO(Ca71) - 444. 83.0 846.79 100,000 152Eu(Mow70) 501. 14.2 1037.91 143021170 186.18 8.2010.12 1175.13 2302125 242.00 16.1 10.21 1“Ta(cu69) 1238.30, 676381680 295.20 41.4510.56 100.10 0.119 1360.22 4340145 351.76 79.7 11.1 152.44 0.071- 1771.41 157781160 609.19 100. 156.39 0.0272 2015.36 3095131 1120.4 34.9 10.7 179.39 0.0317 2598.58 168511170 222.11 0.0798 3202.30 3030130 16°Tb(cu69) 229.32 0.038 3253.62 7392174 86.79 0.209 264.07 ‘0.0376 3273.26 1756118 197.04 0.065 1121.28 0.370 3451.56 87519 215.62 0.050 1189.03 0.171 298.54 0.350 1221.28 0.289 57Co 309.49 0.011 1230.93 0.121 14.41 114101500 337.30 0.0054 ' 122.06 100000 392.43 0.019 192Ir(ceh73) 136.47 130001400 765.20 0.017 205.77 3.8610.08 6°Co(Led68) 879.31 0.40 295.95 34.6410.35 1173.23 100. 962.46 0.14 308.45 ‘ 35.7710.36 1332.48 100. 966.17 0.344 316.50 100. ' 1002.90 0.0163 468.06 58.0 10.9 88Y(Led68) 1115.16 0.0216 588.57 5.5210.10 898.0 93. 1177.98 0.206 604.40 10.0410.26 1836.1 99. 1199.92 0.033 612.45 6.5510.13 1251.30 0.0017 ll°’"Ag(Lav71) 1271.90 0.103 2°3Eg(Led68) 657.70 100. 1312.17 0.040 * 72. 11.9 677.57 11.9310.41 82. 3.4 686.71 7.2510.33177mLu(8ern69) 279.2 100. 706.78 17.1510.85 105.3 100. 763.81 23.7310.72 113.0 184. 226Ra(Me70) 818.25 7.8110.39 128.5 131. 53.24 0.123 884.22 80.2814.01 153.3 144. 186.21 0.032 937.31 37.3111.42 204.1 117. 241.98 0.079 1383.85 28.2611.42 208.3 512. 295.24 0.202 Table 3—1 (Cont'd.) 30 E I E E I 1506.65 15.19 0.49 228.4 310. 351.93 0.401 ‘ 281.8 118. 609.27 0.484 13386(0069) 327.7 152. 665.40 0.0165 81.01 0.360 378.5 240. 768.45 0.0532 .160.60 0.0076 413.6 135. 785.80 0.0121 276.29 0.075 418.5 172. 806.16 0.0131 302.71 0.196 466.0 20. 934.06 0.0334 355.86 0.670 ' 1120.28 0.160 383.70 0.094 1155.17 0.0182 1238.13 0.062 1280.98 0.0156 1377.64 0.0418 1509.22 0.0230 1661.24 0.0121 1729.55 0.0307 1764.99 0.166 1842.44 0.022 2118.52 0.0123 2204.14 0.0530 2293.21 0.0034 2447.63 0.0165 31 3.1.2 Y-Ray Energy_Calibration Energy calibration of Ge(Li) detectors could be performed quite precisely although sometimes not with comparable accuraCy. This is because the centroids of peaks would tend to shift with count rate or source position. For accurate energy calibration, Y ray standards were counted simultaneously with the source of interest. A list of energy standards is also presented in Table 3—1. Using the appropriate internal standards, the more prominent transitions of interest could be accurately measured for use as secondary standards for weaker transitions. Additionally, decay scheme information on cascades with crossover transitions allowed further refinement of energy accuracy. If only a rough energy calibration was desired, external standards were run separately from the source of interest. These calibrations were generally done solely to orient oneself when going over data. 3.1.3 Data-Analysis Programs A series of analysis programs were available to automate the recovery of spectral energies, intensities, and coincidence informa- tion. A brief description of each program is presented with no explana- tion of the finer workings or input parameters inherent in their use. Such information can be obtained through the references given below. 3.1.3A MOD-7 MOD-7 (Ba70) is a program to generate peak intensities and centroids in an interactive fashion. Raw spectral data are displayed on a storage scope where a set of switches allow the operator to expand portions of his data in any manner. He can then visually make up to a ninth 32 order polynomial fit of the background under the peaks of interest and then either generate peak areas and centroids or start over with a new background fit. This program was particularly useful for analyzing peaks with unusual backgrounds. 3.1.38 SAMPO SAMPO (R069) is a program to automatically find peaks in a spectra, analyze them for energy and intensity, and tabulate the results. The necessary input included only peak-shape intensity, and energy- calibration information. The program would automatically generate shape calibrations for peaks of the users preference and could search out peaks to analyze automatically if desired. An additional feature 'of SAMPO is its ability to strip out multiplets of up to six peaks. SAMPO was heavily used for data analysis in this work because of its great speed and consistency in obtaining good fits. 3.1.3C EVENT RECOVERY EVENT RECOVERY (Mo73) is a program to analyze data taken under EVENT or IIEVENT mentioned earlier. This program allows the setting of gates on up to three ADC's and displaying coincidences in the remaining ADC. In addition to merely gating a region under a transi- tion, background regions near the transition can also be gated and the background under the data of interest effectively subtracted out. Background gates were automatically normalized to the size of the peak gate so that the correct subtraction was made. 33 3.2 Analysis of Beta-Spectra 3.2.1 Principles of Beta Analysis The energy distribution of electrons in a beta spectrum is given in general form by N(E)dE = ——1—— gZIMI2 pE(E.,--E)2 dE (3-1) 3 5 7 20 c K where E6 = total decay energy E = electron energy p = electron momentum IMI2 - matrix element of the interaction Eo-E = neutrino energy (neglecting recoil energy). Correcting for Coulomb interaction between the escaping beta and the recoil atom, the term F(Z,E) is incorporated. The Kurie function K(E) can then be written K(E) = [fifilz zconst. (Ea-E) (3-2) where a plot of K(E) vs E should yield a linear plot intersecting the energy axis at E.. The function K(E) is available in tables of Wapstra et a1. (Wa59), Nuclear Data Sheets (0071) and elsewhere. Additionally, such plots can be prepared through computer program FERMPLOT discussed in the following section. 34 3.2.2 FERMPLOT FERMPLOT (31372) is a program for calculating the Kurie function K(E) (3-2) and fitting the best possible least squared fit straight line to the data plotting K(E) versus E. The calculation is done as outlined in Wapstra, and output is in the form of binary spectra for direct plotting on the Calcomp plotters. 3.2.3 Preparation of Fermi Plots Although the calculations outlined in the previous sections are straightforward, actual preparation of Fermi plots were quite difficult. This is because the actual data collected in both scintillation and Si(Li) experiments do not directly correspond to actual beta spectra. An additional efficiency correction must be folded in for detector response as a function of energy. This arises from scattering of betas out of the detector as well as resolution corrections. Such correc- tions were difficult to do and, as exact beta shapes were not desired, they were dispensed with. The data are thus reasonably analyzable only in the region near the endpoint. In the case of multiple end- points FERMPLOT was-not used and data were compiled by hand. In this case tabulated values of K(E) were used and as many endpoints as practical stripped out by hand. 3.2.4 Energy Calibration of Scintillation Spectra In the case of scintillation spectra, calibration was done primarily by means of measuring known beta endpoints. As has been seen frequently before, it was observed that such calibrations were quite linear to pulse height over a wide energy range. This meant that two points technically supplied enough information to define a calibration .°z_g anEL u; HEATS axe pasn spiepuens sq; °Aosinoos Jannaq 10; uesoqo exam snurod notaslqrteo IBlaABS asn {snaps uI °aA1no SE 36 Table 3-2 Calibration Sources for the Energy Calibration of Scintillation Detectors Source E (keV) Transition 22Na 340.7 Compton edge 137Cs 624.2 Conversion electron 228 Th 2381.7 Compton edge 62Cu 2930. Beta endpoint 1““Pr 2990. Beta endpoint 6"Ga 6050. Beta endpoint CHAPTER IV THE e/B+ DECAY 0F 1‘3Eu 4.1 Introduction 1' I This investigation continues our overall studies of the E/8+ decay of N580 odd-Z isotopes. The decays of 1“1Pm and 139Pr were reported previously (Ya73, Bee69). Very little has yet been reported on ll’E'Eu decay. Kotajima (K065) measured the half-life of 173Eu as 2.3 min and determined a 8+ end- point of 4.0 MeV using a plastic scintillator. Investigation with a NaI(Tl) detector indicated only Sll-keV annihilation radiation. Malan (Ma66) measured lllO—keV and 1550-keV Y rays with NaI(Tl) detectors; these were assigned to luaEu and found to decay with a 2.61 min half-life. To the best of our knowledge, no conversion-electron studies have been made on 143E“, although, indeed, its short half-life and preponderance of high energy transitions would make such measurements quite difficult. In addition, no high resolution Y-ray studies have been made. The lack of electron data makes it more difficult to assemble a complete decay scheme, for we have to work without direct information on the multipolarities of the transitions. The spins and parities of a number of states in the daughter 1“3Sm were determined through analysis of the ll”'Sm.(p,,d)“‘33m reaction studies by Kashy and Jolly (Jo7l) and Chaumeaux et al. (Ch7l) as well as the 1'."'Sm('r,a)- 1“3Sm reaction study by Woolam et a1. (W071). These reactions tend to favor simple states such as the single particle, low-lying states 37 38 predicted in shell model theory. The reaction studies were generally of sufficient energy resolution to compare unambiguously with levels found in Y-ray spectroscopy, and in this manner the spins of a number of levels were placed. 4.2 Source Preparation 143E“ sources were prepared predominantly by the l""Sm(p,2n)“’3Eu reaction, which has a Q value of ~16.5 MeV (M071). 28 MeV proton beams (threshold for 1“2Eu production) furnished by the Michigan State University (MSU) Sector-Focused Cyclotron.were used to bombard enriched targets of l""Sm203 (95.10% 1““Sm, obtained from the Isotopes Division, Oak Ridge National Laboratory). Typically, 25-ug targets were bombarded for l min with 0.5 0A of beam current. It is interesting to note that several competing reactions can accompany (p,xn) reactions. The 1""Sm(p,d)”WSm reaction competed quite well producing l-min (K060) luamSm. In addition, the low binding energy of 0 particles below the Ns82 closed shell lead to significant 1""Sm(p,cxxn) competition. Most noticeably, the 1“Sm-(19,0173) was seen to produce the 5.8 min (At66) 1|"”"l’m rather abundantly. Other impurities were also produced to a lesser degree. Our bombardments did produce predominantly 143E“ sources, and what- ever impurities were produced could easily be eliminated by half- life comparisons. In addition, most impurities have already been well characterized. A somewhat more difficult problem involved the 8.8 min (3168) 173Sm daughter activity. Stronger transitions could be assigned on the basis of half-life, but the weaker transitions were 39 assigned, in part, on the basis of a detailed study of 1“3Sm decay at this laboratory (Fi74a). Each source was retrieved from the bombardment area within several seconds by a rabbit retrieval system (K0870) and counting was initiated from 1-2 min after the end of the bombardment. Counting continued up to a maximum of 10 min, i.e. approximately four half-lives. The Y rays attributed to 173Eu all retained their constant relative inten- sities over this period. 4.3 Experimental Data 4.3.1 Singles Y-Ray Spectra Two separate Ge(Li) detectors were used to obtain the 1“3Eu Y-ray spectra. One was a 4.51 efficient (relative to 3X3-in NaI(Tl) detector at 25 cm) detector manufactured by ORTEC, Inc., and the other was a 10.4% efficient detector manufactured by Nuclear Diodes, Inc. The best resolution we obtained was 2.1 keV full width at half maximum for the 5°06 1333 keV peak. The Y-ray energies were determined by counting the spectra simul- taneously with 56Co, llomAg, 152E“, and 225Ra standards. The larger peaks, in turn, were used to determine the energies of the weaker peaks in spectra taken without standards. The centroids and net peak areas were determined with the aid of the computer code SAMPO (R069). The backgrounds were first subtracted and the centroids were then determined by fitting the peaks to Gaussian functions having exponen- tial tails on both the upper and lower sides of the peaks. The specific peak shapes were determined by comparisons with reference peaks 40 specified at intervals throughout the spectrum. The energies were then determined by fitting the centroids to a quadratic calibration equation. Peak areas were then converted to Y ray intensities through curves previously determined in this laboratory (G171) for each detector.. These curves were obtained by using a set of standard Y—ray sources whose relative intensities had been carefully measured with a 3X3-in NaI(Tl) detector. A caution about the energies of the higher-energy Y rays (Eh > 1600 keV): Because of the weakness of these peaks, internal energy calibrations could not be run. In these cases, both external standards and linear extrapolation were used to determine the energies of the Y rays. One should be somewhat wary, however, of possible systematic errors in the energies of these Y rays. After taking spectra and following the decay of at least 6 sets of H3Eu sources at widely varying times, we have identified 33 Y rays as resulting from thee/B+ decay of 1“3Eu. A singles spectrum taken with the 10.4% detector is shown in Fig. 4-1. A list of these Y rays and their relative intensities is given in Table 4-1. All values from our work are averages from many determinations, with the quoted errors reflecting the statistical fluctuations found among the different runs and the quoted errors on the standards used. 4.3.2 Coincidence Spectra 4.3.2A Megachannel Coincidence Spectra Our two-dimensional "megachannel" coincidence experiment utilized two Ge(Li) detectors, the Nuclear Diodes 10.4% detector and an ORTEC 7.0% detector. A block diagram of the electronics was shown earlier 41 r GLES SPECTRUM r "3E0 SIN . ram-2... Em 012:7. Em Juan—o .1. mm mm 8 3.! 8.18.. El 1. .\ 09. E 8.3m ~38: 628.. SB: ,, I- 8.37 Indmr rem-3H Swazi madam... mafia T - L b - 7. .6 . «v u. no nu no no 3mzz<1m 1011mezmmo 3 10 - 2000 1000 500 1500’ 2.36le amdurml mmdrrw1 - r rm.2m~l m fiommml :afiwwml : mmflmnml - oo.~o_~l mo 58... r rwdmom.‘ om.NmmT 8&3 T. nmfom T. w 7195.. madam? ANNOQT. . nmfimnnl mmémn? Em SK , . 6.01m - N37 . FIE-2U: . . . 1m 9.88.. 3.82.. 3500 H000 CHANNEL NUMBER 3000 2500 u. 3 2 0 0 0 3mzz '\. I\ 3 O. I H 13' I 10' < : ml - 1000 2000 3000 L1000 CHANNEL NUMBER 52 Figure 4-3 (cont'd.) 1538.7 m7 GATE 0N Sc-AXIS 10 _" . r51].0 l 0 10 1607.3 keV GATE 0N )E-AXIS r‘ 102 . “. r\ O '1" Q 1 2 10 0;) I L L '— 10 1715.1 keV GATE ON xixxrs 10 ’ d -511.0 1000 2000 3000 4000 CHANNEL NUMBER 53 Figure 4-3 (cont'd.) 1” 1305.0 berGATE 0N xT-AXIs 3021.“: 1 '23. =1 . :: ‘9 10‘ ~ - 1312.8 mi GATE 0N X—AXIs - ' 102 > . a. ”P 101 - 1° 1332.11 1.17 GATE. on i-Axrs A i 10’ . ' ' 1 A. . I\ a '1" 10 1000 2000 3000 9000 CHANNEL NUMBER . .54 Figure 4-3 (cont'd.) 2070.0 m7 GATE ON xTAxxs F 10» ' « '51”) llll . 0 1° 2102.0 mi GATE 0N X—AXIs ; 10- "51 1.0 1 >2000 m? GATE 0N i-AXIG r ". 102 gr 3 “2 “I" g I 10' 11111111 - 1000 2 00 3000 L1000 CHANNEL NUMBER [ l ’1. I . ll ‘1 ‘11.! J 1|. ‘ 'III 55 Table 4-2 1“3Eu Coincidence Summary Gate (keV) Coincident Y Rays 107.7(Y) 999.3, 1428.9, 1458.2, 1607.3, 1805.0, 1962.4, 2449.7?, 2478.8? 458.9(Y) 1107.2 607.6(X) 1107.2 805.3(Y) 1107.2 1107.2(X) 203.3, 429.5, 458.9, 805.3 1107.2(Y) 429.5, 458.9, 607.6, 805.3 1369.0(X) 732.9 1428.9(X) 107.7 1458.2(X) 107.7 1536.7(Y) 691.1 1607.3(X) 107.7 1805.0(X) 107.7 1962.4(X) 107.7 > 2000 (X) 107.7, 203.3, 1107.2 > 2000 (Y) 1107.2, 1369.0, 1536.7 56 4.3.23 Pair Spectra The two halves of an 8X8-in NaI(Tl) split annulus were-used‘in 1conjunction with the 10.42 Ge(Li) detector to determine the relative amounts of 8+ feeding to the various levels in 1“3Sm. Each half of the annulus was gated on the 511-keV Yt peak and a triple coincidence (resolving time, 21 - 100 nsec) was required among these and the Ge(Li) detector. The resulting spectrum is shown in Figure 4-4. A discus-' sion of the 8+ feedings extracted from this experiment is deferred until Section 4.5 where theyfare presented in Table 4-5. 4.4 Proposed Decay Scheme Our proposed decay scheme for 1“3Eu is shown in Figure 4-5. 0f the 33 Y rays listed in Table 4-1, 29 have been placed in the decay scheme, accounting for over 98% of the total Y ray intensity. It is entirely possible that many of the remaining Y rays proceed from levels that decay via a single transition. These Y rays are all too weak to have been seen in any of our coincidence work, so, with no further evidence for their placement, we have omitted them entirely. The assigned spins and parities, discussed in Section 4-5, represent a combination of deductions from our work and conclusions from reaction studies. The agreement between our work and reaction studies are quite good. The results of the reaction studies are presented in Table 4-3. We used the total e-decay energy measured by Kotajima (K065) of 5.0 MeV. The feeding to the ground state of 1“3Sm was inferred from the intensity of the 511 keV annihilation peak corrected for contribution Figure 4—4. 11°3Eu Yi-Y pair coincidence spectrum. 57 H". 3000 2 D 0: |-_ Q E O'ZOIZ\ (J) 001.03" — . a: UJ 93l6l— —~—-———0 1.. 0 8% E *— NE a [9291 6.999" ‘ S: g ms 09191” ‘ __ d 0 ms 1'20171— -:=_= z ._ — Z >~ <1: I ms 2'22"- 5 $1 22011- +- UJS Z'QQOI“ o _ 3 ..o LIJ O m .. 3. E (I) {5.3 0119— v0 ”0 No .9 09 13111111115 113d "31.10000 Ave. .11d .1 - 58 >m_>_ Own .0 5&qu 0m nm 5E in §¢OQ mw NQOOQW wowNmm 3 vm mm gvg 359V E $011... 0 383 1J_..1_N_'G 59 Table 4-3 Levels in 1“3Sm From Beta Decay and Reaction Studies lhusmw’d) (Ch71) Energy Assignment S lbusmw’d) (Jo7l) Energy Assignment S This Work Energy Assignment 0.0 107.7 754.1 1107.2 1369.0 1536.7 1565.9 1715.1 1912.6 2070.0 2102.0 2167.3 2227.4 2270.1 + + 3/2 1/2 11/2‘ 5/2+ 7/2+ 5/2+ 5/2+ 5/2 5/2 5/2 <7/2+) 7/2+ (7/2+) 7/2+ + + 4. 2311.8 (3/2,5/2+) 2556.8 (3/2,5/2+) 2586.5 (3/2,5/2+) 0.0 110. 760. 1110. 1360. 1530. 1720. 2060. 2160. 2290. 2470. 2590. 3040. 3170. + + 3/2 1/2 11/2- 5/2+ 7/2+ 5/2+ 5/2+ 5/2+ 7/2+ 7/2+ 11/2+ (11/2‘) (7/2+) (5/2+) 4.0 1.4 7.4 3.7 3.1 0.34 0.8 0.6 2.7 1.8 1.4 0.84 2.1 0.7 0.0 107. 748. 1100. 1360. 1532. 1697. 2040. 2440. + 4. 3/2 1/2 11/2- 5/2+ 11/2'] 5/2+ 5/2+ 5/2+ 7/2 4.0 1.6 9.86 2.57 1.04 0.1 0.65 60 from 173Sm decay. The contribution of electron capture was calculated yielding the percentage ground state feeding. The intensities in Fig. 4-5 are given as percent of decays of 1“3Eu. Specific evidence for the placement of levels and transitions in the decay scheme is given as follows: Ground, 107.73, 754.06, 1107.15, and 1369.03-keV states. These states were all papulated strongly by the 1|"’Sm(p,ci.’) reaction and ap- pear to be essentially single particle states, viz., the da/z, 81/2, h11/2, ds/z, and 97/2 in that order. We, too, see specific evidence for all but the 754.06-keV state which is the h11/2 metastable state and would not be expected to be populated by low spin 1‘3Eu. The 107.73 keV transition appears to deexcite the level of the same energy. Coincidence data indicate the 999.33-, 1428.92-, 1458.24-, 1607.27-, 1804.95-, and 1962.36-keV y-rays in certain coincidence with this transition, and very weakly suggest that the 2449.69- and 2478.80- keV transitions may also be in coincidence with it. Thus, all the transitions shown through the 107.73-keV level are at least weakly indicated in coincidence data. Clearly the unadjusted intensities. into the 107.73-keV level greatly exceed the intensity out of this level. This can be accounted for by calculating the y—ray intensity loss due to internal conversion. The 107.73-keV y ray should-be an E2,Ml mixture where the calculated conversion coefficients are 1.1 and 1.4, respectively (R058). Assuming no 8 feeding, the conversion coefficient necessary to get an even intensity balance is l.810.4. This is within error for a pure M1 transition and if other losses are taken into account might allow some E2 mixing. Beta feeding to this level appears quite unlikely in accordance with the chosen spin of 61 5/2+ for the ground state of 1”Eu. Indeed spins l/2+ or 3/2+ are very unlikelywithout significant feeding to the 107.73-keV state in 1“3Sm. Such low spin states are not expected to lie so low in energy. The 1107.15-keV y-ray transition depopulating the level of the same energy is the single most intense transition in the spectrum. y-rays at 429.54, 458.85, 607.46 and 805.28 keV are seen strongly in coinci- dence with this transition. 'The log ft of 5.5 is well within the range of allowed transitions and supports the spin 5/2+ assigned to 1”3Eu ground state which is discussed later. The 1369.03-keV transition is in coincidence with the 732.86 Y ray. This state was assigned as 7/2+ by Kashy and Jolly (Jo7l) but as 11/2- by Chaumeaux (Ch71). The log ft of 6.2 is somewhat high for an allowed transition but still unreasonably low for the third- forbidden transition. This state will be discussed further in Section 4.5. 1536.69-, 1565.85-, 1715.14-, and 1912.60-keV states. These levels are certainly quite similar in that they all deexcite through the ground, first, and second excited states. In addition, the 1536.69- keV Y is in coincidence with a 691.12 keV y-ray deexciting the higher lying 2227.41—keV state. The other states in this group evidence no corresponding transitions, but such transitions may easily be too weak to be observed. Kashy and Jolly found levels at 1530 and 1720 keV both of spin 5/2+. We have thus chosen spin 5/2+ for all four levels on the basis of their similarities, although the spin-parity assignments for the 1565.85- and 1912.60-keV levels have no basis in reaction studies. The log fi measurements for feeding to these levels are 5.5, 5.8, 5.8, and 5.2 respectively indicative both of their 62 similarities and that they are allowed transitions. Their nature will further be discussed in Section 4.5. 2069.99-, 2311.84-, 2556.76-, and 2586.45-keV states. These states are all similar in that they deexcite only through the ground and first excited states. The log ft values to these levels are 5.9, 6.7, 6.8, and 6.4, respectively, indicative of rather hindered allowed transitions. Angular momentum and 8 selection rules constrain these levels to spins 3/2+ or 5/2+. The only level in this group reported in reaction studies is the 5/2+ level at 2060 keV. The 2069.99-keV state appears to correspond to this level and is assigned spin 5/2+. The other states cannot be interpreted any more unambiguously. 2101.00-, 2167.28-, 2227.41-, and 2270.13-keV states. These states are all alike in that they all feed the ground state but not the first excited state. The log ft values to these levels are 5.8, 7.0, 6.0, and 6.6, respectively. These are again indicative of slow allowed transitions. The 8 selection rules indicate these levels can be 3/2+, +, or 7/2+, but the lack of feeding to the 1/2+ first excited state 5/2 strongly suggests that all these levels may be (5/2+) or 7/2+. Kashy and Jolly report 7/2+ levels at 2160 and 2290 keV so we will make definite assignments of 7/2+ to the 2167.78- and 2270.13-keV states. The 2102.00- and 2227.4l-keV states will also be assigned spin 7/2+ or (5/2+) but with somewhat lesser certainty. The ground state spin of the parent 173Eu is now clearly established. Allowed transitions proceed to 3/2+, 5/2+ , and 7/2+ states which can only be true if the ground state of 1“3Eu is 5/2+. This is also pre- dictable from the systematics of N-80 nuclei as will be shown in the following section. 63 4.5 Discussion Some 16 states have now been placed, with varying degrees of con- fidence, in Inasm. All five major neutron orbits between NESO and 82 lie reasonably close together, resulting in relatively lowblying single particle states which are not terribly fragmented. 4.5.1 Single-Particle States The five states at 0, 107.73, 454.06, 1107.15 and 1369.03 keV compromise the major components of the single-neutron orbits between N550 and 82, viz., dalz’ 81/2, h11/2’ ds/z’ and 97/2 respectively. The first four states appear to be well documented in reaction studies and confirmed by their decay properties. The 11/2- and 1/2+ states are not fed at all by the B decay of 1“fin as would be expected but the 11/2- is made directly. The fifth state 1369.0 keV appears to be wrongly assigned by Chaumeaux (Ch7l) as an 11/2- state which is not expected so low in energy. This state is weakly fed in l""Sm.(p,d) leading to significant statistical uncertainty. In addition, the cross sections for the i=4 transitions were so much smaller than for i=5 transitions (experimental (Jo7l) and DWBA) that Kashy and Jolly et al. claim that this state is definitely the 7/2+ single particle state . The systematics of these single-particle states in the N=81 nuclei 139Ce, 1“Nd, and 1”Sm is most notable for its monotony. The ordering of states is the same in all cases, and the 1/2+ first excited state decreases from 254.7- to 107.73-keV in the interval relative to the 3/2+ ground state. Similarly, the 11/2- state stays at virtually the same energy and the 5/2+ state decreases only from 1320.0 to 1107.15 keV. 64 The 7/2+ state increases only slightly from 1347.4 to 1369.03 keV in the same interval. Clearly the addition of proton pairs to the Nk81 nuclei does little to disturb the single particle neutron states. These systematics are shown in Table 4-4. 4.5.2 Other States Assuredly, the higher lying states in 173Sm are more complex. Although little quantitatively can be said about these states, it is clear that at least some of these states must arise from couplings of the lowest 2+ or 3- core excitations to the single neutron hole states. Indeed, Kashy and Jolly show that the 1536.69-, and 1715.14-keV states probably have an appreciable (d3/2)v-1><2+ component. In addition, other states of possible multiparticle nature are expected to be present, although little can be said about them at this time except that they should not be expected to be seen in ll"'Sm(p,d) studies. 4.5.3 The 1“Eu Ground State The decay properties of 173Eu clearly indicate the 5/2+ nature of this ground state. This spin is strongly confirmed by the systematics of Né80 nuclei where the 5/2+ state first falls below the 7/2+ state in 139Pr and continues to drop rapidly at 1“Pm. Although no direct measurement of the spin of the 1“3Eu ground state has been made, we are quite certain of its accuracy. 4.5.4 E/B+ Ratios + From our y" gated spectrum (Figure 4-4) the relative 8+ feedings to seven of the 1“3Sm states were measured. The deduced e/B+ ratios for transitions to these states are listed in Table 4-5. We normalized 65 Table 4-4 Single Particle States of N=81 Nuclei Energy (keV) State 139Ce 1“1Nd 1‘3Sm 3/2+ o. o. o. 1/2+ 254.7 193.8 107.7 11/2' 628.6 756.7 754.1 5/2+ 1320.0 1223.3 1107.2 + 7/2 1347.4 1345.8 1369.0 66 Table 4-5 Comparison of Experimental and Theoretical €(tot)/B+ Ratios for the Decay to States in lkasm Energya €(tot)/B+ b (keV) Experimental Theoretical 1107.15 ' 0.51:0.O6C 0.46 1536.69 0.62:0.06 0.69 1565.85 0.69:0.15 - 0.72 1715.14 0.75:0.17 0.83 1912.60 1.0710.11 1.02 ‘ + aThese are the only states in luasm that are measurably fed by B decay, as determined from that Yi gated coincidence spectrum. bThese values are from Nuclear Data Tables 19_(1971). Q8 = 5.5 MeV is chosen to get these values. CThe experimental ratios were normalized to the theoretical ratios by assuming that the transition to the 1107.15-keV state is allowed and unhindered, presumably yielding the expected ratio. I]- I Al" I ll.“ In} 67 our experimental values to the theoretical values for the transition to the 1107.15-keV state which we considered to be a fast, straight- forward transition with good statistical information. lndeed, the measured values compare quite well with the theoretical values. 68 Chapter V 8/8+ Decay of 173Sm 5.1 Introduction: This investigation continues our over-all studies of the N381 isotones. The decay of 1“Nd and 1“5Gd have been presented previously (Bee68,Ep7l). 4 Although several papers have characterized 1“3Sm decay properties (Bl68,Be166,Dew72,DeF68,DeF70), the most definitive work published to date is by D. DeFrenne et al. (DeF70). They produced 1“3Sm by the 1“"Sm(y,n) reaction and presented a decay scheme based on very limited coincidence data and energy sums. To the best of our knowledge, no conversion electron studies have been done on 173Sm decay, so the multipolarities of ybray transitions are undetermined. This makes the assignment of spins to levels in luaPm more difficult, but the problem is partially obviated by the 172Nd(T,d) and 1""Sm(d,'r) reaction studies of Wildenthal et al. (Wil7l). It is often possible to associate levels found in reaction studies with the same levels seen in decay scheme studies. The 173Sm decay study presented here was in part an offshoot of the study of ll”Eu decay (Fi74d). In order to characterize 1“3Eu decay, a knowledge of its luasm daughter decay was necessary. In addition to that, however, the 173Sm decay offers particularly interest- ing information in light of our studies of its neighboring isotones. Of particular interest are the apparent 8 transitions from the 3/2+ and 1/2+ ground states of 1“Nd and luscd to the 7/2+ first excited 69 states in their respective daughters, 1“Pr and 1“5Eu. In the first case the second forbidden transition which should proceed with a log ft in the range of 10 to 14 is seen tvoccur with a logft of 28.8. The second case involves a second forbidden unique transition which should proceed with a logft of ”14. This transition occurs with a logft of 27.5. It seems likely that a similarly enhanced, second for- bidden transition will exist to 1“3Pm. D- DeFrenne (DeF70) indicates a small feeding to the corresponding level and one of the goals of these experiments was to measure this feeding more precisely. 5.2 Source Preparation H3Sm sources were prepared primarily by the 1""Sm(p,2n)1'°:"Eu (B+/€)1“38m reaction which has a Q value of -l6.5 MeV (M071). 28 MeV proton beams (the threshold for 172Eu production) furnished by the Michigan State University (MSU) Sector-Focused Cyclotron were used to bombard enriched targets of 1““Sm203(95.102 1""8111, obtained from the Isotopes Division, Oak Ridge National Laboratory). Typically, 25-ug targets were bombarded for ~1 min with 0.5 uA of beam current. We also attempted the 1“2Nd(T,2n)1“3Sm reaction (Q - -7.7 MeV) with little Success. The major product of this reaction appeared to be 1""Pm (tl/2 - 6.8 min) although the (T,axn) reactions were also evidenced. Clearly the T reactions are quite complex in this region where protons and alpha particles are rather weakly bound. The l""Sm(p,2n) reaction was considerably cleaner, although again the major impurity was ll”’Pm produced presumably by the 1""Sm(p,0m) reaction. Other impurities included 1“3mSm (63 sec), 1“28m (72.5 min), 70 and 1“Sm (20.9 min) and were of low intensity and easily resolved by ‘half—life from the 8.83 min 173Sm. The initial 1“3Eu (2.61 min) quickly decayed to HaSm and sources were counted starting from 10-15 min following bombardment, for intervals up to a maximum of 40 minutes following bombardment the y-rays attributed to 1“’Sm decay all retained their constant relative intensities over that period. Transitions emanating from 1”3Eu were easily distinguished by the study of 173Eu decay presented in Chapter 4. 5.3 Experimental Data 5.3.1 Singles y—Ray Spectra Several separate Ge(Li) detectors of different manufacturer were used at different times to obtain the 1"33m spectra. They were all of coaxial design and varied from 4.5% to 10.4% in efficiency (relative to a 3X3-in. NaI(Tl) detector at 25 cm). The best resolution we obtained was 2.0 keV full width at half maximum for the ”Co 1332-keV Y ray. In most cases these detectors were used with baseline restora- tion and pileup rejection which decreased resolution slightly but allowed large count rates (upwards of 60,000 counts per second). The y-ray energies under 1600 keV were determined by counting the spectra simultaneously with 56Co, 152Eu, 11°mAg and 226Ra standards. The larger peaks in the spectrum were calculated using these standards. These calibrated peaks, in turn, were used to determine the energies of the weaker peaks in the spectra taken without standards. The centroids and net peak areas were taken with the aid of the computer code SAMPO (R069). The backgrounds were first subtracted and the centroids then 71 determined by fitting the peaks to Gaussian functions having exponential tails on both the upper and lower sides of the peaks. The peak shapes were determined by comparisons with reference peaks specified at inter- vals throughout the spectrum. The energies were then determined by fitting the centroids to a quadratic calibration equation. Peak areas were then converted to Y-ray intensities through curves previously determined in this laboratory (D069) for each detector. These curves were obtained by using a set of standard Y-ray sources whose relative intensities had been carefully measured with a 3X3-in. NaI(Tl) detector. Transitions above 1600 keV were all too weak to be calibrated in the foregoing manner. In these cases a combination of methods was used to find the centroids. External standards were run before or after the singles spectra to give a fairly close-calibration. These are prone to systematic error, so, in addition, well known energy differences for transitions from a state to levels of a better measured energy difference were used in an interactive manner to improve the energy calibrations. It must be emphasized, however, that there may be systematic errors in the energies of the higher transitions. After taking spectra from and following the decay of at least 6 different sets of 1738m sources prepared at widely separated times we have identified 23 Y rays resulting from the 6/8+ decay of 1“3Sm. A singles spectrum is shown in Figure 5-1. A list of these Y rays and their relative intensities is given in Table 5-1, where they are compared with the results of DeFrenne et al. (DeF70). All values from our work are the averages from many determinations, with the quoted errors reflecting the statistical fluctuations found in different runs and the quoted errors on the standards used. 72 '"Sm SINGLES SPECTRUM 2000 1000 91.2. 92...... «66...? 92.5? D 8000 3000 o o a l amzzoo %n moHnou scum wouoaao lama mum mosam> ekwoq .uoouoa onu mo soaumuwousfloav use uaouuom mo mauou 0a 00>«w one mofiuamoouoa coauamaouu was >03 ca ao>fiw one meanness HH< .050000 50000 amsosa .mlo ounmam . on 8 O 4.80 82oz o.o — +o fin: u e +~ mt I. U 0 m s42: m _ 1 +8 2 .8 mo Q83 92%” W w A 0 6 0 o I\ 9 ( m o; $8 033 0 n +8 . 6 .7. 9 m. on He 00H a mg... L fiHE m.n ..uw Chapter VII MEASUREMENT 01' 8+ ENDPOINT 01‘ 6 30a The decay-of 32.4-sec 630a has been studied in great detail by Giesler et al. in this laboratory (Gi74). An important missing piece of data at this point is the 8+ endpoint energy for decay to the ground state of 63Zn. From the systematics of this region, the 8+ endpoint has been predicted to be 4.6 MeV. In order to measure this endpoint, apilot-B plastic scintillator was constructed 4.5 cm in diameter and 9 cm deep. This scintillator was mounted on an EMI 95308 photomultiplier tube., The tube was operated at 1000 V and the resolution for 662-keV conversion electrons was 20% (FWHM). This was acceptable because the detector was designed primarily for high energy particles and even then only to measure endpoints. All sources were measured both with and without absorbers in front of the detector such that correction could be made for the contribution of Y-ray Compton electrons to the spectra. Sources of 63Ga were produced using 30-MeV protons from the Michigan State University Sector Focused Cyclotron on a natural Zn target. Because of the short half-life (32.4 sec) of 63Ga, recoils from the target were collected by a He—thermalizer jet transport system and deposited in front of the detector on a moving tape transport so as not to allow a build up of longer lived activities. The energy cali- bration of the system was performed using 6"Ga and 1""Pr standards. 2 A modified Fermi-Kurie plot for 63Ca decay is shown in Figure 7-1. Cramer et a2. (Cr62) indicate that a plot of the form 99 100 .uoHa mfisu 0H 0H00>Hom0u 000 0003 scans m0£oo0un 0000 mwu0c0 u030~.0nu 00 030 we m0fiwu000 0030H 00 muau00oHH 10oz .00000000 Hou0HHHuCHow 0H000Hm mluoawm 0wu0H 0 :uHB 00050005 m0000 0osm 000 00mm Scum 00000 no uoaa 0Husx 00umsfi0< rzleagass .e-a assess LmMn can 03.. can . - OOH oolfl DON OQN ONN OON 0(111140.’ « a 11 ( Z q q u A .. access so: 3.. “must >0: moé . 1 O l N 0 p./ ... + 71/ ... J/ - i c / ,7, m» + 83 + . . -.. 7 . . . cono+uosuo -. . A 1.. .--.+ . o . m. - . ... - - 4 v o e . .... . 1 a . - .. -- - - s:.1 - -a.1-... -. . . ... .... -.. I . - 1 . 03023510 33 . .N. .- . ... .. . H. .2 V a. , . - ..- 1 - e - .1. -........1 6. 1 S. .o . 11 1 g. a _ 1 _ ... _ . . _ ... . . ..u .-.-1 1. p .. .... . . .1.....1r...o. . 1 .r .. _.. . i... ...1....:..p.;.....,; in}... 101 const C 1/2 (13;) ,3 E N(C) number of counts in channel C 0 ll channel number will give a fairly good endpoint. Since Ezc, we made the substitution, and the best value of the constant was 1. An endpoint of 4.5:0.1 MeV was measured after stripping out the higher energy 6"Ga endpoint. This is in good agreement with the predicted endpoint, however, transitions to other states in 63Zn were not resolvable. CHAPTER VIII e/B+ BRANCHING RATIOS 8-1 Introduction to Beta Decay Theory B—decay is a process by which energetically unstable nuclei may decay to more stable nuclei. Most generally, 8 decay appears as simply the transformation of a proton to a neutron or vice versa accompanied by creation or annihilation of an electron such as to conserve charge. Although the nucleons should be considered as bound particles, the three forms of B decay most commonly seen can be abbreviated as B- decay n +'e- + 3 + p+ : a) 8+ decay p+ +e+ + v + n b) (8-1) e. capture p+ + e- + V + n' I C) where the neutrino v and the antineutrino 5 are leptons originally postu- lated to conserve energy. Studies of numerous examples of B decays have shown that the elec- trons are always emitted in a continuous spectrum of energy and momentum.‘ Because the transition involves a known constant decay energy, W it 0’ was apparent that the energy and momentum were not being conserved. Even considering the recoil energy, it was evident that momentum would not be conserved. In order to remedy this situation it was postulated by Pauli (Pa30) that a previously undetected particle called a neutrino was emitted, statistically sharing energy and momentum with the electron. This particle must be neutral and have half integral spin in order to fit in with what was known of B decay. 102 103 Strong indication that this approach was correct was found in its prediction of the B—energy spectrum. The basic assumption of statistical sharing of energy between the electron and neutrino is that the probability of a specific subgroup of states to emerge is proportional simply to the number of such states. Thus, if dA/A is the fraction of decays involving momentum ranges (d3) - dpxdpydpz for the electron and (d3) for the neutrino, dA d d '—7T ° ... (d (d . (3'2) where the denominator encompasses the total momentum space. This can' easily be simplified by substitution to (K066) d1 _ .jL 2 _ 2 4 1/2 __ afi-(wawo-m [w me] ‘ (3—3) [(wb _ W)2 _ u2c4]1/2/(mc2)5 where P f0(po)~= J[ o dppqu/(mc)2c (8-4) 0 and W = c(p2+tn2c2)1/2 is the electron energy 2 1/2 7‘: l — c(q2+p2c is the neutrino energy w = w + K. o 104 .The recoil energy has been neglected above. An additional correction must be included for the Coulomb interaction between the electrons and the nucleus. This correction can be generalized as w F(Z,W) = l—Elz ~dA (8—5) 4’0 where [wolz is the free particle spacial density and Iwclz is the spacial density of the electron at the nucleus. By simple manipulation one can write 7% = CpW(Wo-W)2F(Z:w)/ f (2.910) (mc2)5 (8-6> where f(Z w ) - w dW 2 2 5 . o - - epW(Wo-W) F(Z,W)/(mc ) (8-7) mCZ This equation can be used to accurately predict the shape of many 8 spectra. For convenience in discussion total decay rates (8-6) can be written 2 S 2 dA = C-F(Z,W)qu dW/(mc) c (8-8) where integration gives = Cf(Z,W0) (8-9) 105 or ft = £n2/C (8-10) Thus, reduced decay rates (ft) can be calculated for all 8 decays from analogy to well understood electromagnetic theory and this simple calcula- tion indicates that the ft's should be identical. In reality, the reduced rates range widely from 103 4'21018 sec. These large deviations can be explained by such things as: I l) Angular momentum change - the half-spin leptons can carry off a total of only one unit of angular momentum unless they are ejected off-center from the nucleus. Such occurrences have a lower probability and, indeed, decays involving large angular momentum changes invariably proceed with larger ft values. 2) Parity mismatches - transitions between states of differing parity are seen to be greatly hindered. 3) Nuclear states - decays between very similar initial and final nuclear states are generally quite fast. Mirror and some O+-+ 0+ decays are particularly fast whereas other modes may be much slower. For the continuation of this text I will discuss only "allowed" decays; that is, decays involving angular momentum transfer of 0 or 1 unit with no parity change. The discussion of "forbidden" decays is beyond the interest of this discussion although a thorough discussion is given by Konopinski (K066). The emission of the leptons in allowed 8 decay may be such that their spins are antiparallel (singlet) or parallel (triplet). These general modes of decay are frequently called Fermi (F) and Gamowaeller (GT) 106 decay, respectively. Their interaction strengths will be proportional to C2 and C2 F GT' Clearly decays involving a change of one unit of nuclear angular momentum result from triplet decay, but decays involving no angular momentum change can be either singlet or triplet (an exception is 0+ +'0+ which can only be singlet emission). Thus, in neutron decay (l/2+ +l/2+) the reduced transition rate might be proportional to -1 ~ 2 2 (ft)n ~CF + 3CGT (8 11) where each of the three possible triplet spin projections to the nuclear axis are given equal weight to the singlet projection. Similarly, for ll'0 (0+ +'0+) where either of the two protons outside the core might B decay -1 2: 2 _ (mo 2c:F (8 12) The measured ft values for n and ll‘0 decay are 1180:35 sec ($059) and 3060:10 sec (He6l) respectively. Thus, 2 C (ft) 53 = l [2 —°— -1] = 1.40:0.05 (8-13) C2 3 (ft) n F gives an estimate of these relative strengths. It should be noted that it was assumed that the core nucleons in 1"0 did not contribute significantly to the decay rate. Here an assumption has been made that CéT and c; are basic constants irrespective of the decaying nucleus. If this approach sufficiently describes the B-decay force, a constant interaction strength is to be expected. 107 8.1.1 The Neutrino Although this particle is almost impossible to detect, the neutrino's existence has been verified by Reines and Cowan (Re59) through anti- neutrino capture in protons. Coincident positrons and neutron reaction ‘Y-rays were observed proving the neutrino's existence. It is also shown that more than one kind of neutrino exists as the above reaction is not observed from neutrinos created during positron decay (DaSS). Further information is obtained about the neutrino from carefully measured statistical electron spectra. The higher energy data are very dependent on the neutrino rest mass p. All experiments to date have indicated that the neutrino is probably massless, with an upper limit of 250 eV predicted by the 3H spectrum (La52). Actually recent speculation on solar neutrino fluxes suggest that the neutrino might have a very small rest mass (Bah72), but this would not seriously affect any of the discussion here. Finally, in an ingenious experiment by Goldhaber et al. (6058) it was indicated that the neutrinos emitted in 152Eu electron capture had 100% negative helicity; that is they were all left-handed. This means that the neutrino's spin is lined up antiparallel to its direction of motion. 8.1.2 - The Electron Because the neutrino appears to be fully left-handed, it is of interest to investigate the polarizations inherent to the electron. Most generally, the polarization of the emitted electron is defined as (1)."L — dA1k ._). P = (S 0") = -————- (8_14) e dA~$ + dA‘.L where the average projection of electron spin in the direction of motion 108 is proportional to the difference in decay rates producing parallel and antiparallel spins and motions. Clearly a positive polarization indicates dominance of right-handed particles while negative polarization leads to left-handed electrOns. Measurements of electron polarizations for both positron and negatron decay have shown that (Wu57).' + P a i v/c for e- (8'15) This indicates that the positron is the mirror image of the electron and indeed suggests that a good explanation of all antiparticles may be as mirror particles to the normal particles. Thus, it suggests that the antineutrino is fully a right handed particle. The electron-does not appear to be fully polarized, however, so it is necessary at this point for a short diversion to explain this partial polarization.' The instantaneous measurement of the electron velocity can give only the two values ic. This is because the instantaneous velocity u = AX/At, but in the limit At +0 the uncertainty principle tells us that the energy AE.Z_h/At +-m. Infinite energy corresponds to u a ic. If the electron has a precise momentum 3'= (W/cz) $, then.3 must be. defined as the average of point motions alternating between states u = ic. If a is the probability amplitude for forward motion (+c) |a|2 (+c) + (1 - |a|2)(-c) = v - lal = 1/2(l+v/c) and (1 - |a|2 = 1/2(1 - v/c) 109 Therefore, if the negatron is fully left-handed, its spin must flip as it alternates between opposite velocity states. Thus, P = (-1) 1/2 (1+ %) + (+1) 1/2(1 - i5) =-% (8-17) indicating that beta-negatrons are indeed fully left-handed. Similarly, P = +v/c is predicted for positrons. 8.1.3 - Angular Distribution of B Particles About Polarized Nuclei The polarization of the negatron in the direction of nuclear orienta- tion is written as V a dx+ - dA+ P (11+ + an A '+ . XL: _.X. - I -I‘ c, (c)cose (8 18) for e- where dA+,+ represents decay rates for spin up and spin down respectively relative to the nuclear spin I. 6 is simply the angle to initial nuclear spin of the emitted electron. Clearly the two decay rates dA+ and dl+ can be written dM,+ :11 @1039 (8-19) as seen by substitution into (8-18). For spin up, the projection of total lepton spin S2 = S511 a M8 = +1 and for spin down MS - -1 so one can write (11048) = 1 - MS (1c!) c086 (8-20) For transitions involving MS = 0 there are equal numbers of spin up and 110 spin down decays washing out any anisotropy as predicted in (8e20). For singlet transitions 8 - MS - 0 and only isotropic electron distribu- tions are observed. In the case of triplet radiation S -.1 so MS a O, 11, and in addition more than one orientation may arise in each form of angular momentum transfer as indicated in Figure 8-1. Generalizing (8-20) to e+-decay for all transitions. dMG) = 13- (%)cose (8-21) For triplet transitions it can be shown that (K066) a) ' b) c) If = Ii - 1 I1 Ii + 1 (8—22) (MS) a 1 1/(Ii+1) -Ii/(Ii+1) 8.1.4 - Electron-Neutrino Angular Correlations In this casethe projection of electron polarization in the neutrino direction 3337c is the quantity of interest. Analogously to section 8.1.3 for positron decay v zi— - dl+,+ 1 (c)cosBev (8 23) where 98v is the angle between the positron and neutrino emission direc- tions. This will lead to v d (eev) ~l + (c)coseev (8-24) lll Figure 8—1. Possible orientations of lepton spin S to the nuclear. spin 1. FERMI (SINGLET) RADIATION s, S. 3:8. y «~495351n-c» c#%:> :26+ 0 ........ fig? (1 v a 5.3%: ~l-a-v/c - ~l+a°V/C_ SCALAR VECTOR Figure 8-2. Electron-neutrino spin orientations. 112 where is the average projection of electron spin on the neutrino momentum. Singlet radiation can be inferred to proceed in two general forms diagrammed in Figure 8—2. The "scalar" arrangement involves only right-handed particles while the "vector" involves left-handed particles. If CS and C give the scalar and vector coupling strengths, then for Fermi V transitions 2 ' L!’ - dAF(6eV) l + aF(c)coseev (8 25) where 2 2 a BCV—CS F 2 2 Cv + C8 allowing for the possible mixture of both terms. Measurements on 35A mirror decay (He57, A159) which proceeds largely by Fermi emission gives aF = +0.97i0.14 or CE/Cs <0.09, indicating that the vector contribution is predominant. For triplet radiation a similar picture of the emitted leptons is obtained by reversing the neutrino spin directions in Figure 842. Here the three orientations of the triplet radiation tend to wash out the correlation and one gets a "tensor" contribution ~ .1 _ dAT(eev) l + 1/3 (c)cosfiev (8 26) yielding right-handed neutrinos and a "axial vector" contribution, ~ _ L! _ dAA(Gev) ~1 1/3 (c)c086ev (8 27) 113 yielding left-handed neutrinos. Analogous to the Fermi transitions, for Gamow-Teller transitions we write N X — dAGT(Bev) ~l + aGT(C)coseev (8 28) where _ l_ 2 _ 2 2 2 aGT ' 3(C'r CA), (CT + CA) Measurement of a for 6He (Jo63) and 23Ne (A159) yield GT a(6He) = -o.334:o.oo3 a(23Ne) = -o.37:o.04 (8'29) suggesting that Cg/C: <0.004. This again strongly indicates participa- tion of left-handed neutrinos with right-handed positrons. When a mixture of singlet and triplet radiation is seen, the correlation coefficient + (l - x) a = xa Plotting known values of x versus a in Figure 8—3, F aGT ' where lines corresponding to pure 8 or V and pure T or A are drawn, suggest dominance of A—V interactions or participation of left-handed particles and right-handed antiparticles. The mathematical reasoning for the names S, V, T, and A will be indicated in later sections. Thus, apparently, (4x - l) 2 (8-30) 0 n UHP‘ where aF = l and aGT = -l/3. 114 O Figure 8—3. Mixing of correlation coefficient. 'I I I I I I I 1 Figure 8-4. Time Reversal. 115 8.1.5 - Neutron Decay The simplest form of B decay should be the neutron decay (1/2+ + l/2+) in which only the single nucleon participates. If all possible correlations are to be considered between the nucleon and the two leptons, ol<+ d). z1+an2io +I-[A +Bq+D€iX] (8-31) The last term concerns time invariance, and reversing all motions (t + -t) as in Figure 8-4 should give equally valid results where a +. component 1%1(-I )'[(-q)X(--E)] would exist. If both cases are equally probable, the two parts will cancel. Measurements of the decay of polarized neutrons in a given arrangement as well as oppositely polarized neutrons in the same experimental arrangement will clearly offer a test of time invariance. Such an experiment (Bu58a) has shown that DD = 0.04:0.07, suggesting that B decay is indeed time invariant. From here on it is assumed that Dn = 0. Looking at unpolarized neutrino beams yields average coefficients An = Bn = in = 0, since the associated anisotropies average to zero. This leaves only an for the neutron and the fraction of decay resulting from the vector interaction is clearly C: x = -—-———-—- (8—32) n 2 2 CV + 3CA so 1 03 ‘ C: a —-—(4x - l) = (8—33) “ 3 n 2 2 116 As shown previously from 1‘*0 data Ci/Cs = 1.40:0.05, yielding xn = 0.192i0.007 (8~34) For pure axial vector decay it was previously shown (8-22) that for I = i If = 1/2, i__ __'._l___; .. AA ‘ ‘ Ii+l ‘ 3 (8 35) so for neutron decay the triplet contribution to A is 2 ‘ZCA A = -- (l-x ) = --—*—-— = -O.539i0.005 (8-36) A 3 n 2 2 CV +30A Since the antineutrino is right-handed, clearly Bn = -An = = 2/3, so B = —A = 0.539 (8-37) From recoil proton-electron angular distributions it was found (Bu58b) that An = -0.1110.02, Bn = +0.88:0.15 (8-38) Because vector radiation is isotropic, the only other contribution to An and BH could be the interference term /2 _ _ 2 2 _ 1. _ 1 _ A — v — zchcVI/(cV + 3CA) — 2[3(1 xn)xn] (8 39) AV BA 117 such that An = AAiAAV and Bn = BAiBAV (8-40) ‘where we can calculate from 8—39 AAV = BAC = i0.46i0.02 (8-41) If one chooses the positive sign, An = -O.54+0.46 = -0.08i0.02 Bn = +0.54+O.46 = l.OOi0.02 (8—42) agreement with experiment is well within the experimental errors. The experimental implications of the preceding section are indicated in Figure 8-5. There are four possible orientations, corresponding to M8 = 0,:1. In polarized neutron decay the M = -l orientation cannot be constructed. 8 The MS = 1 orientation does not specify the recoil proton direction because the electronand neutrino alone can conserve momentum. This orientation does, however, indicate the sign of BA 31°; and AA zi'v. The two cases involving MS - O differ only in the net polarization predicted for the recoil proton. Clearly, the first case in Figure 8-5 predicts the signs of A ~i-3 and B ”1°; as being negative, while the last case indicates AV AV both signs are positive. The experimental results indicate that the last case is correct so we can thus predict that the recoil proton will have a net left-handed polarization. Of course, this polarization is too AIS=O (S=03nd l) P 11 1 (1’ 1 <1? 118 M,-=0 [S=0 and 1) mv q 1 +0: 10 1.1}. <15 mv q mv AAV‘BAvhzv-‘Cu AAs-BA=-2%x AAV" Av" +-1' x “315‘ A~-1. 3+0 (#1) A—-OTB--+1 Figure 8-5. Screw sense of the proton recoil. 119 difficult to detect, since P = vp/c is quite small, but a general statement of the form of B decay may be made. 8.1.6 - General Theory of Beta-Decay All of the data so far have indicated that in the B decay process all particles participate fully through their left-handed states and all antiparticles appear in their right-handed states. These decays proceed through two forms, a singlet or Fermi decay of strength proportional to C2 and a triplet or Gamow-Teller decay proportional to C2 Two other V A' forms involving tensor forces proportional to 02 and scalar forces propor- T tional to c: are not measurably observed in any studies to date. 8.2 Mathematical Derivation of Allowed B-Decay Theory 8.2.1 Transition Probabilities For the general decay type, n + v:: p + e-, the B-interaction energy density can be given by (K066) h = 81/2gJ(pn)J(ev) + c.c. (8-43) B where g, the "Fermi coupling constant" gives the strength of the inter- action, and J(pn) = w+TxT w (8-44a) p +n J(ev) = quxwv (8—44b) represent the transition currents for the transformations n'+ p and 120 v +'ef. WP, Wu, We and wv represent the wave functions for the cor- responding particles and FX represents the form of the 8 interaction. 1+ is the isospin operator converting a neutron into a proton. It is apparent that the 8 interaction operator, H8’ in the Hamiltonian of the nucleon can be inferred from = hB to be HB = 81/ng'x‘r+J(ev) + h.c. (8-45) I This may be expanded to an entire nucleus of A nucleons to get 1/2 A rxa a u (A) = 3 g 2 r [J(ev)}—>a + h.c. (8-46) B ‘+ r a=l where the electron-neutrino current is now evaluated at the position of +a the transforming nucleon r . Expressing the variables of the nuclear system in the complete set of eigenstates Ik>exp(-iEkt/M) where <£|k> = 62k; if HA is the energy Hamiltonian for a nucleus, HA|k>=Eka>, then HB can be treated as a perturbation to the isolated nuclear system. Then any state in the perturbed system may be represented by |(t)> =Z|k>ak(t)e-1Ekt/“ (8—47) . k 4 ' where I(t)> is not of the set |k> but a time dependent state. Iak(t)l2 is the probability of finding the nucleus in state |k> at time t. The time dependent equation of motion, (in 5%) I (t>> -- (HA + HB>|> (8-48) 121 must apply, so upon substitution of(8-42) into the above and projection on some state, ak(t)e1(Ef-Ek)t/“ (3-49) k These transition equations can be used to find the growth in time of various nuclear states from a specific initial state. Obviously, ak(t - -“0 - ski’ and before any new states grow in appreciably, iuéf(t) - e‘iwbt/“ (8-50) where W0 - E1 - Ef. This takes into account only first order processes, i.e. dflrect i'+ f transitions. Second order transitions like i +-k +1f, represented by , would not be expected to be significant as they are proportional to first-order effects by the factor g (8 coupling strength) which is small. , To analyze transition rates further, knowledge of the time dependence of 8 coupling is required. Through straightforward Fourier analysis, on iwt - ' I 8'51 “IdeIHB'Dwe ( ) where w dt -iwt m - mf('27)e (8-52) Time integration of (8-50) gives 122 co co 1“8f(co) . I at! dwwe'i(“"wo/mt (8-53a) -- 211 f.” dww6(0-Wo/h) _ (8-53b) - 2flwo/“ (8-530) This is consistent with (8-52) and 2 00 laf(°°)I2 - u“ | f dteiwotIn|2 (8-54) -00 which is the final probability fraction. The oscillations in the B-coupling theory come from the neutrino- electron current in J(ev). In general, the neutrino wave function can be expressed in free spherical and plane waves while the electron is ade- quately described by Coulomb-distorted spherical and plane waves. The wave field should be composed of a superposition of all energy eigenstates, including "negative energy" states. Thus, it is convenient to represent we or Wv as w=f dwx -1w:/M ce+1Wt/M _ W>O k [uke + uk ] (8 55) where uk and u: (charge conjugate) are time dependent Dirac spinors of arbitrary normalizations and phases. Thus, Fourier analysis of the J(ev) trivally gives frequency relations 0 - (1w icq)/h for electron energies w and neutrino energies cq. Time dependence in J(Ev) comes out in form J(E§) = (wirxwg)t_oe1("+CQ)t/“ (8-56a) 123 . x -i(W+cq)t/h - J(e+v) = (¢E+F wv)o e (8 56b) J(e-v) = (w:wav)oi(w-CQ)Uu (8-560) J(e+ 5) . 4.211%)0 e‘1<""°q"/“ (8-56d) where proton-to-neutron transformations are proportional to T; and can be written by exchanging wv== we. The probability of reaching a given final state If> can now be written for p +~n + e+ + v as 2 -2 2 °° -1(w+c+w)/K2 Iaf(°°)| =11 |<£|n8|1>| If dt e ‘1 o t _| a (8-57) The first of the two conjugate time integrals is simply Znfl6(W'+ cq + W0), corresponding to conservation of energy. The second integral exists for W + cq 8 -WO; hence the second time integral becomes I - I: dt + 00. Thus, if the decay rate per nucleus is d1 8 Iaf(w)I2/T d). = (211/11) ||26(W + cq + wo) (8-58) Assuming both lepton waves are normalized to unity in V, the desired rate is obtained by multiplying (8-58) by -+ +1 2 6 (dP)(dq)V /(20K) (3'59) where we can substitute d; a pzdpdp and d; = qqudq to get 2 2 2 all; 3 RA (dfingmv ||2 ' (8'60) 9 (Zn) h c ' ' 124 for decays in which the electron and the neutrino momenta end in the specific ranges (d3) and (d3). The G-function is eliminated by integrat- ing over the infinitesimal range of neutrino energy, remembering that .[cdq 5(W + cq + W0) = 1. (5) and (a) give the proton and neutrino solid angles. Integration over all solid angles simplifies (8-60) to 2 2 2 .31 .. LL33 ||2 (8-61) p Zfl K c The above equation cannot be trivially integrated to obtain the rate constant explicitly without knowledge of momentum (energy) dependence of I|2. Clearly, Coulomb interactions of the electron with the nucleus may be significant, so the matrix element is generally factored, giving ||2 = F(z,wo)g2 (8-62) where F(Z,WO) contains the Coulombic interaction between the electron and the nucleus. Integrating (8-61) 2 ___§__ po 22 dp p o - q F(Z,W) 2fl3fl7c A = (2n3n7c)’1f(z,wo)g2 (8—63) where f(Z,Wo) is generally evaluated numerically. This result should be. correct for all forms of B decay, both induced and spontaneous. Although this form is essentially exact, f(Z,Wo) and 52 need to be further 125 elaborated before actual calculations may be performed. Nevertheless, a preliminary investigation of the case of competing positron decay and electron capture decay indicates some interesting possibilities. The - 1 rates of these forms of decay in a given nucleus are 3 7 -1 . A6 = (2n n c) 5.423(0)};E (8-64a) and AB+ a (2n3fi7c)-1f8+(z,wo)€8+ (85641)) Assuming that the B-interaction force is identical in all forms of B- decay (i.e. there is only one 8 force), then to first order 58 8 58+ and A8 f8 . r— ' ”f— (8-65) 8+ 6+ ' This quantity is experimentally measurable and, because f(Z,Wo) is composed strictly of electromagnetic effects, such values are straightforwardly calculated. Thus, such calculations should be easily compared to experi- ment to test their accuracy. In addition, the assumption £8 ' 58+ can be tested to confirm the B theory presented here. 8.2.2 Lepton Wave Functions Since the neutrino is presumably an essentially non-interacting particle, its description as a free spherical wave will be sufficient. Any wave function chosen must satisfy the Dirac equations w<1> = 56-3 15 + mcz <1 (seas) 126 w e s c3-3$ - mcze (8-67) where W 3 (021)2 + mzc4 1/2 is the energy of the particle in question. Q and G are Pauli spinors representing forward and retrograde motion and composing the four-component spinor, ¢(r) w n ( ) e-iw': . (8'68) ¢(r) .+ In order to obtain eigenstates of total angular momentum 3 -‘I + a one must consider superpositions of spin eigenfunctions §+ll2 and angular momentum eigenfunctions §£m defined such that ++ ++ Xjuxj'u' a ajj'du'u (8'69) In the case of orbital angular momentum, iim'+ Y£m(6,¢), the familiar spherical harmonic function. Thus, an eigenstate of I can be expressed as X2u(6’¢) ' DZ§1,,;bY,u_p<2§

l:> (8-70) where K = 11,12,13, ... u a i1/2,i3/2,0009ij j IKI—l/Z 2 j+l/2 = K for K>0 R = jél/Z = -(K+l) for K<0. 127 The orthonormality condition +1 4 ' fdnxKuxK'IJ' ' GKK' 61.11." . (8‘71) applies, and the coupling term in 8070 is simply the vector addition co- efficient connecting the two eigenstates. If we then introduce a radial dependence, gK(r), the spinor ¢(?) - gK(r)§ku(f), and the Dirac equation, becomes 66?) = (W + 1)‘1 (Ed; + £35) 3.5L“ (8-72) so the Dirac four-component spinor takes the form, . wise .001 w (r) - e (8-73) K“ + A if_K(r))cKu(r)) where n f_K(r) = (w + 1)“1 (5‘11,- + 135) gK(r) -1 <1 1+K 8K(r) -(W-1) (dz + r) f_K(r) gK(r) and f_K(r) can be chosen proportional to spherical Hankel functions h£(pr). giving _ 1/2 - _ gK =- NwhuK) + £_K - 2Nw[(w—1)/(w+1)] h£(_K) (8 74) where Nw is a normalization constant and 2 = K/IKI. Since the neutrino 128 is emitted in a continuous spectrum of energies, the wave function should represent a superposition of eigenstates, writ) = fdw :4; awKquu(¥.t) (8-75) where aWKu is defined such thatIE:IawKuI2dW is the probability of finding KU a particle in the range dW and + 2 2 _ §(dr)ltpl [aw glam“ 1 (8 76) The eigenfunction for the emitted electron is necessarily more complicated by the Coulomb potential V(r), so Coulomb distortion can be created by substituting W - V(r) in the above equations. 8.2.3 The 8 Interaction Presumably, the internal variables of any particle can be described by four-component Dirac spinors resulting from intrinsic spins :1/2 and intrinsic velocity tc. Thus, the radial part of the particle wave- function is assumed separable from the intrinsic parts. The total particle wavefunction can be written as w = 1216511“ ,2<+c) + wen,1 ”(+0 + 123n+1,2<-c) + + W4u_1/2(-c) ' (8-77) which can be written in matrix form as 129 [01(3) + I 11:20:) (‘1’ ‘ l”1X1/2 + l"2’L1/2) + _ + = g + -+ (8-78) W(r) ' 03(r) / ‘ 9 w3X1/2 + th-llz \wfi) / where Xil/Z are eigenspinors represented by + l +- 0 X+1/2 ' (o) X41/2 ' (1) (3‘79) Similarly, the four-component eigenspinors can be constructed as _ 1 + = a 1 + 1 “1/2(+C) ‘ o x1/2 g “-1/2(+°) (o)X-1/2 = (o) 0 O 0 0 O 0‘+ 0'+ “1/2(’C) = (1)X1/2 g (3) “-1/2('°) g (1)’L1/2 ' ((1)) (8-80) Eigenvectors can be constructed to project the eigenvalues for the various spin projections, giving the equations, Ozu+l/2(ic) = +u+1/2(ic); ozu_1/2(ic) = -u_1/2(ic) .(8-81) resulting'in CO?l'-' COD-‘0 HOOD .. 0; 0 (8-82) 0 0; 130 where O; are the Pauli spinors. Similarly, an eigenvector can be created to project intrinsic velocity giving ‘ 1 o = (0 _) (8-83) u C>C>C>hi rc>a>hac> e>hic>c> P‘C>C>C> Also, an eigenvector can be constructed to project out chirality such that the eigenvalue +1 refers to left-handed states and -1 for right-handed states. Thus Y59;1/2(*°) = +“;1/2(*°)‘ Y5“11/2(i°) ’ '“11/2‘ic) (8'84) where -1 o o o o 1 o o -o; o y = O 0 l 0 E ( . = -o a (8'85) 5 o o 0-1 0 0z z 2 Actually a total of 16 independent 4-dimensional matrices may be con- structed, including =01 _o-1-*1 d'_1o D1 1 o ’ p2 1-1 0 3“ p3 0-1 Also, 0 0 Z '(0 0,); 912’ 02 z, 03 Z; 00 = (3 2) (8—86) 131 where Oz =23, 0‘2 = Q3, and Y5 = -p3 Z3. Also, historically, p3 = 0‘2 It is clearly desirable that solutions to the Dirac equation ww = (c333'+ Bmc2)w (8-87) lead to real physical quantities and that 0+0 be an invariant scalar. ' If this is to include relativistic invariance, it is necessary to treat . the time coordinate as the fourth component of a four-vector mu(w,icw+w) on equal footing with the three special vectors. For this to be true, 301 + «Hm/dc a 2 d0/ dX = o (8-88> u u 11 Where it turns out that $’= cw+3¢ (8-89) Since wu(cw+am,icw+w) is a 4-vector, 0+0 is not a scalar and cw+aw represents the special components. The scalar expression, ¢+H¢ = c(¢+a¢) + (WTBw)mc2 = wwfw (8-90) shows that the multiplication of E = 0+3 into 0, not 0+, produces an invariant scalar product. Since 82 = 1, denote the current density 4- vector as wu(c $Ba¢,ic$80), where the internal dynamical variable components are Yu = ('18018) (8-91) 132 with properties of a 4-vector. The current 4-vector is simply 0 = icx-IJY a; (8-92) and the Dirac equation becomes (Yud/qu + mc/M)w = 0 (8-93) Thus, the entire dependence on inner variables is expressed as a function of Y“. Indeed, all possible internal motions can be expressed in terms of Y“, which can be used to generate the sixteen independent 4-dimensional matrices. If all possible internal motions can be represented by four 4- component spinors, then it follows that any internal variable must be. representable by a 4X4 matrix. Thus, an arbitrary function can be written generating all sixteen possible spinors using the anticommutation property, 1/2(Yqu + yvyu) = Guv (8-94) and constructing all possible products, f=f+2fy+zfyy+ 0 U U U “#V UV U V + if yvv+fyvyv (8-95) u#v#p uvp u v p 5 x y z 4 W0 where f can coincide with an arbitrary matrix having sixteen parameters. Note that only five distinct types of products can be created. The first 133 two include a scalar and a 4-component vector whose properties are already quite clear. The six products Yqu = —Yvyu form components of a second-rank tensor. This tensor may be represented as = - 21 = -i - 8-96 0 (YUYV Yvyu)/ Yqu (ufx) ( ) 11V where the factor 1 makes the component hermitian. The spacial components form the circular relationship oxoy - (nyy - yyyx)/2i - (§x§)z/21 _=. (Exaz/u (8—97) and symbolically o = (Baa/21 (8-98) The space—time component has the form 0x4 8 -1YxY4 = 0‘x (8-99) so the total array of tensor components is \ 0 oz .0y ax . -oz 0 0x 1.1 (auv) =1 oy 0x 0 oz ' ouv(o,a) (8-100) \rax-ay-az 0// 134 The last term becomes simply Y5 - YnyYzY4 (8-101) As was shown previously, this undergoes changes under right-t0 left— ~ + -+ handed frame inversions (r +-r). In all other changes of reference frame this behaves like an invariant scalar hence this quantity is termed a pseudoscalar. Finally, the four components YpYqu can be written as - l mu iysyu (8- 02) which behaves like a 4-vector under all operations except space inversion where 3'- iY V does not change signs. For this reason wu is termed an axial vector. Thus, the possible inner variables can be written,' f I f + +‘ + f' + f 8-103 0 Z qulJ iZfWoW Z umIJ 5Y5 ( ) in terms of the five covariants scalar, 1 vector, Yu(-1BG,B) +1+ tensor, ouv(o,a)- axial vector, wu(BU.'iBY5) pseudoscalar, Y5 Each has the covariance shown only under the operation 135 ‘1!)er 2 0+8wa (8-104) where TX is the covariant in question. 8.2.4 HB , The 8 interaction current between nuclear states can be written as where it has been shown that most generally (8-46) A a a BB = 81/2g E: :2: Barx TiW:Barx wv + h.c. (8-105) X a=l ' The interaction is summed over all nucleons as well as all interactions. Two interactions that are quite similar are the vector (V) and axial vector (A) interactions. These are written in the form V : Y A : YaYS , (8-106) Each consists of four components, which can be taken as I F * (Ya + YaYS) z Ya(l + Y5) . * (8-107) Assuming each form contributes proportionally to l/ZCV, -l/2CA, respectively, V+A _ _ _ Fa - l/zYa(Cv cAyS) (8 108) so that HVA = 21/2 2A; (35‘1 81(c -c a) an- (ev)]->a + h c (8—109) 8 g Ya v AYS T: a r ' ° a=l 136 Ba and ?a = -133? and the lepton current Ja(ev) is composed + a of components J2,J , it is easy to rearrange the above to HI Also, since Y: \éA = 21/28 ZT:{J:(CV_CAYS)_1J (ova + CA 0 +65} + h. c. (3'110) a where -y53’= 3. Since nuclei are considerably smaller than lepton wave- lengths, the lepton current J0 need only be evaluated to first order at the nuclear center, r = 0. Thus, as an approximation, J:(ra = 0) E J: leading to 1/2 «In;A |1> = 2 g{JZ[CV fl-CA [751-1J°[0Af3+ Cv [01]} + h.c. (8-111) where '3 = ~ 2 g{ch4 f1 1CA3° fa} (8 113) Several interesting results can be inferred from the above B-transition amplitude. Most interestingly, the Vector and axial vector interactions 137 separate into different terms. Transition selection rules may be obtained for each interaction form. For the vector interaction, a z fi'M'ilM - 61.161,”I (8—114) a a + + where xIM’ XI'M' represent the angular;momentum eigenstates ”YLM' Similarly, for the axial vector interaction I _ +3 +- ++ = z XI'M'OXIM (8-115) 8 where 3 projects out states I' - I, I i 1. In addition, terms I' = I = O vanish. Also, the parity operation, Pw1(r1,...,rA) E w1(-r1,...,—rA) (8-116) leads to another selection rule. Since P2 - l, P2 - (8-117) This can only be true if the initial and final states have identical parities. Thus, selection rules for the Vector and axial vector inter- actions are Vector (V): .fl - 0 unless AI - O, ninf - +1 (Fermi rules) Axial Vector (A): .fo - 0 unless AI = O or 1, Hiflf = +1 (Gamow-Teller rules) - 0 when I - I' - O 138 ' These rules, better known As Fermi and Gamow-Teller rules, indicate that Vector and axial vector interactions can both contribute to'B transi- tions in which the nucleons do notchange in angular momentum, except for 0+ +-O+ transitions when only vector-interactions can occur. Transitions involving a change of one unit of angular momentum can only contain axial vector contributions. In addition, these rules indicate that no transi- tions can take place between states Pf opposite parity. 1 Actually, the non-existence of higher-order transitions involving parity change is a result of two approximations made previously, i.e., r8 = 0 and nonrelativistic nucleon motion. This is termed the "allowed approximation" and the above rules are actually for "allowed" transitions. Indeed, "forbidden" transitions do exist, but their transition rates are assumed to be considerably slower than allowed transitions as they involve higher order terms inrr and v/c and will be ignored for the present. Similarly, the Scalar (S), Tensor (T), and Pseudoscalar (P) cone tributions to the interaction are ' STP z (1 + oW + Y5) = (1-1(3 + 3) + ys) (8-118) P Discarding the relativistic terms as before, PSTP z (1 - 13) (8-119) and proportionality constants C CV lead to S, STP r = (c - 1cT3) (8-120) S 139 Thus, as before HZTP = 21/28 : 83(05-136CT)T:[Ja(ev)];a 21/ 2g 2 T:Ba{J2(cS)-133(CT33) (8-121) a ‘ This form is quite similar to that for V-A terms except that the terms are proportional to Ba BYa. Thus the B-interaction current contains the terms and . In the case of non relativistic particles, Bw~w, so for the nucleons the above terms can be simplified. It must be remembered, however, that this simplification is unwarranted for leptons. The 8 interaction becomes 2: 1/2 0 -*O + _ 2 g{(Cv4£S)J4jl-1(CA+CT)J for} (8 122) The terms [I and [3 can be simplified using the Wigner-Eckart relation <6'I'M' |sjm|5m> = <1' (M')J(m) |I(M)> (8-123) where is the reduced matrix element which is independent of angular momentum orientations. Thus, .. v I = _ 11 - <1> 61,16M,M<1> (8 124a) fr; =- <1' (M')1(m)II(M)> (8-124b) where is the singlet (Fermi) B moment and is the triplet (Gamow- 140 Teller) moment. Note that 2_Z 2_ 2 2 2 _ m [/65] - | 0*I + | 0&1 + | ozl (a 125) The three spacial components of the transition current can be written 3" - f3 5 Z; 111118; = ;Jm (8-126) The selection rules suggest that the allowed transition current terms can be divided into those leading to no angular momentum change and those involving a change of one unit of angular momentum between the initial and final nuclear states. This leads to 2 2 2 || = 2g {l(cs+cv)J4f1-1(CT+CA)Jo/;xz| + 2 2 2 2 where the leptons carry away angular momentum equal but opposite in sign to the nucleons. This can be separated into a purely Scalar-Vector part, a Tensor-Axial vector part, and a factor containing all four interactions as follows 2 2 2 2 2 II sv = 611,23 (cs+cv) <1> |J4I (8-128a) |<£|HB|1>|§A = 2g2(CT+CA)22 E , IJmI22 (8-128b) m 2 1/2 STVA= ‘5 l| 2g2<1>[I/1 II' (M/I)(iJ:J4 + c.c.) (8-128c) 141 For randomly oriented nuclei it is necessary to average overall orienta- tions -IfoI by the operator (ZI+l)-1'2 :. The S-V term is not a function M of orientation, so (21+1)‘1 2M: ||§V = 51'I282(CS+CV)2<1>ZIJ4IZ (8-129) The cross term in STVA is linear in M, so (21+1)’1 Z ||§TVA = 0 (8—130) M : Finally, the TA term requires the average vector addition coefficient (21+l)--1 Z [(2I+l)/3]2 E 1/3 (8-131) M where the initial coefficient has been reordered by use of symmetry relations. This leads to -l 2 2 2 2 l 2 (21+1) EM: IITA = 2g (cA+cT) . 3 Em ; IJmI (8-132) The B-interaction forms were proportional to ST. VA :8 1/2ya(1+ys) r z 1/2ya(1-y5) . (8-133) F The terms l/2(liY5) are informative when one writes 12 E 1/2(1+Y5)w + 1/2(1-ys) (8-134) 142 where the two parts are orthogonal eigenstates of Y5 with eigenvalues +1 and -1, respectively. Remembering that Y5 is the chirality operator, lepton states 95 = 1/2(1175)w (8-135) can be created in which the property 75¢ - :0 holds for l/2(liYS). Thus, l/2(liYS) selects out intrinsically left- and right-handed components from w, where l/2(1iY5)¢ = ¢ ' (8-136) befitting a projection. Therefore, the essential difference in the two pairs of interactions is that the V-A form will lead to intrinsically‘ left-handed particles while the ST form involves only right-handed particles. Thus, the lepton wave functions involved in the Brinteraction need only the forms «>800 = 1/2<1t15>we(:z); ¢v = 1/2(1i75)w§ (3-137) The lepton current may now be written in a form inclusive of all first order interactions . J = ¢+BY ¢- (8-138) 01 eav It is desirable to evaluate the following terms in the 8 interaction. 143 IJalz = ¢:BY0¢5¢$BYa¢e (8-139) The neutrino wave function product (113112;!) can be evaluated using the free particle wave function 1 . A + = .3; 1/2 +n i(q°r-cqt)/K _ wv (2V) (Oq xve (8 140) where 1 2 " + A 1 - + A 2219?) = 5? (111(5) (1 + o-q) = W (1i75)(1 o-q) (8-141) v . . Then 2 'I' 2 1' 2 Zia! 2.8 me 4*. — —1- +(1+y )(1 1 3-“)<1> 4v e “ 5 ‘1 e _ ;L, + +;A — 2V ¢e(l I o q)e (3 142) Similarly, 1 2.._1_ 12 + ‘22“ 3; lJmI - 2V (be 3 m [Om(l+o (Donne = — <1> (li3°'<1‘)e (8-143) 144 8.2.5 The Neutrino Wave Function As was shown previously, a free particle can be described as the four-component spinor w =(§>ei(3;;;Wt)/H (8-144a) which must obey the relativistic equation of motion i“ %% =Hw = (~1hc3-V + Bmc2)¢ = (ca-3 + Bmc2)w (8-144b) Substitution gives the pair of equations W$ = cE-Eg + mc2¢ (8-145a) wg = c3431 — mc2§ (8—145b) which can be solved to give 8 = (w + mc ) co-p (8-146a) ¢ = (w - mcz)‘1c3'fi 6 (8-146b) If w>0, then 6<¢ and in the nonrelativistic limit where p< 0 are considered "normal particles". Alternately, "antiparticles" with W < 0 give 9 as the large component in the nonrelativistic limit. In either case, one 145 can choose one of the Pauli spinors as a linear combination of the Pauli spin eigenstates, §+ll2 mentioned previously. Thus, for +W choose 0 ~ xil/Z and normalization such that (8-147) + ~1~ ‘ (dr)wuwu' Guu' giving 2 + 1 + *;+; wil/Z - (22337)1/2 (LO i;/(wumléj)xw:1/2‘31(p r wt)/“ (8'148) Similarly for W < 0 choose 9 z §+1/2. The antiparticle (W < 0) wave- function WC can then be written 2 ++ ++ c = - W+mc co'p/(Whmc% ._ 1(p-r-Wt)/K _ wil/Z + (zwv )1/2 ( 1 )§+1/2e - (8 149) where WC describes a normal particle of —W, -3 and -uh spin projection equivalent to an antiparticle of +W, +9: and +M. A charge conjugation operator relating particles and antiparticles by $3 - cm: can be formulated, 81V1n8 C 5 Y2. The leading signs are added merely to allow an arbitrary phase. The term 3'3 = i1 (8 = p/p) is of particular interest in that it gives particle polarizations for spin parallel (+1) or antiparallel (-l) to 3. The polarization of particles participating in B decay is a measur- able quantity, and 3-5 is termed the helicity operator. wil/Z will be eigenstates of the helicity operator if the i¥1l2 are defined relative + to p as a quantization axis. Unfortunately, it is not the helicity, but chirality, which will 146 determine the B strengths. Chirality eigenstates will, nevertheless, lead to predictions of polarizations. As an example, the left-handed helicity state le/Z is an eigenstate of both 3 and 35$ = -1. The "intrinsically" left-handed chirality states u;1/2(ic) are eigenstates of intrinsic velocity c(a;$) a ic and chirality + A A 15 = -(o-p)(&’-p)l= +1 (8-150) as referred to the axis 3. A general way to project out the components. of each chirality is through the identity 1 s 1/2(1 + vs>w + 1/2<1 - 15>w <8-151) where, since Y; = +1, the two parts are orthogonal eigenstates of Y5 with eigenvalues +1 and -1, respectively. The leftehanded projection is denoted ¢ = 1/2(1 + vs)w = +Y5¢ (8-152) and l/2(1 + Y5)¢ = ¢ as befits a projection. The left-handed projection of w can be most conveniently found by using the unitary transforma- il/2 tion -1/2 1 -1 U = 2 (1 1) (8-153) giving the representation, Y5 +'p3, 8 +9. Thus, 147 L + 811/2 = 1/2(1 + Y5)” wil/ZU 1_ w+mc2 1/2 1-1 +_+ 1 2 1 1 + e1(p-r-Wc)/fi 2 2wv 0 0 CO'p/(W+mc ) -1 1 X:1/2 2 1/2 + + -> +.+— %_(W+mc ) (éi)(1_Co.p/(w+mcz))xil/2ei(p r Wt)/h W+mc2 1/2[ 1 0 + 1 O ] 1—1 + + 1 2 + . 81(P'r-Wt)/“ 2wv o 1 0—1 1 1 CO'p/(W+mc ) X:1/2 . w 1 u (2v>‘1/2e1(31?‘Wt)/“ 1 (3—154) = _1 :1/2 where 2 1/2 +-+ L _ W+mc _ c0 2 +' _ U:1/2 1/2( w ) (1 ("I C2))X11/2 (8 155) Normalization at unit density (V=l) gives + +1 ¢i1/2¢:1/2 = U:1/2U11/2 = 1/2(l I sz/W) 2 since (3P3)2 = p . For right-handed projections, l/2(l - Y5), R - 1+ , -1/21<‘5-?-Wt>/n ¢:1/2 ' (1)“:1/2 (2V) 9 and 2 1/2 w+mc ) [1+c3‘3/(W+m°2)]§11/2 W +R Uil/Z = 1/2( (8-156) (8-157) (8-158) 148 where normalization gives R +R R _ R _ ¢il/2¢il/2 - uil/Zu 1/2 — l/2(licpz/W) (8-159) If fl is the quantization axis, cpZ/W = v/c, and the relative inten- sities of the two helicities is _ 2 2 _ _ _ dA+/dA+-— |¢1/2| /|¢_1/2| - (1 V/C)/(l+v/c) (8 160) yielding a polarization, -v/c. The projection 1/2(1 + 75) on the anti- particle state yields CL - * - I * 8 161 ¢u - 1/2(1 + 75)cwu - c[1/2(1 - Y5)w“] ( - > since c = Y2, Y5 anticommute. Thus, antiparticle states are generated in their net right-handed polarizations by 1/2(1 + 75), so L +-+ c c 1 -1 2 -1 - -w I“ ¢:1/2 = l/2(1+Y5)“’:1/2 = i(-l)u$1/2(2V) I e (p r t) (8'162) and ¢C+ CL — i = 1/2(1+ /W) (8-163) :1/2¢:1/2 ‘ u+1/2“-'+1/2 ’sz yielding a net polarization v/c. Similarly, 1/2(l—Y5) will generate net left-handed polarizations of antiparticle states where 149 R _ 1 + -1/2 -i(3¥¥¥Wt)/M ¢11/2 ' :(1)“11/2(2V) e (8‘164) and RT R - ¢:1/2¢:1/2 = 1/2(1 + cpz/W) (8—165) Thus, states generated by l/2(liY5) can be identified by the net polariza- tion of the emitted particle. This formalism is given for a free particle such as the neutrino. -+ '+ Indeed, if the neutrino is a massless particle and W +'c3, p +q ¢t = (_i)3§(2V)1/2ei(q'r'Ct)q/“ (8-166a) ¢R = :(l)3 (2v)’1/2e1(a'r'Ct)Q/“ (8-166b) v 1 v . c A.» ¢L = :( 1)3'62v)'1/2e‘1(q'r’Ct)Q/“ (8—166c) v -1 - c A. ¢R = 1(1): (2v)’1/2e'i(q'r'°t)Q/“ (8-166d) V l 'V where 3 = 1/2(1-3-'+)‘+ (8-167) \) \) distorted spherical waves. These solutions lead to more complicated equations, yet yield no further implications of physical significance and won't be discussed here. 150 8.2.6 E/B+ Branching Ratios The 8 interaction Hf1 can be written as 1/2 11 ==2 g[ 1w:(cv+ecs)¢v-i 3w: o, . (8—171) = 0 if W < O 151 Such an operator can be constructed as +-+ 2 2 +-+ + = caflp +gBmc + 1w] 1 Bmc a-V _ Ae ZTWT 2 (l +- W -+ c ) (8 172) Similarly, for the antineutrino (massless) + + - = ca-q - 1°91=.l _ +.A _ Av 'ZTCqT 2(1 0 q) (8 173) where Av¢v = ¢v if cq < 0 a 0 if cq > 0 ' (8‘174) Since only positive energy electron states and negative energy neutrino states are produced, +=++ + wewe Aewewe = Ae (8-175a) ¢v¢: = A5¢v¢3 = A; (8-17Sb) applying the closure properties of wave functions. The matrix element simplifies to +-+ 2 meg/1:) = Tr[%8(1 + 51%} + 0‘—;—‘1>(1 - 341)] 2 + = 111%(8 + fir + %§)(1 - 2.2-61)] (8-176) 152 To interpret these terms it is necessary to realize that the terms linear + in B and a vanish since TrB = O; Tra = 0 (8-177) and terms proportional to the 4X4 unit matrix are easily evaluated, since Trl = 4. Thus, for e- decay, 2 me Cicjwzs¢v¢iwe = Tr(BA;A :) = Cicj'TfiT (8-178) Positron decay yields negative energy electrons and positive energy neutrinos, so the analogous result is Tr(BA+A-) = - “2 (8-179) v e cicj IWI ' Similarly for terms proportional to .fb we get terms like f + T + +-+ + - we8a¢v¢v80we Tr(BOAvBOAe) (8-180) + ... Tr(oAver) 9+ T lam + 8ch + '3) (112)] T “4 M c 0 “‘1 Tr 1%? (1- (951) (3- aha] Tr[%3°3] 153 where (E-w'r’xa-o = (E-Q’HE-m- M + 1(3G’xé‘) (ts-181) + Tr (o) = 0 and v 2 3 % cos eev = 0 averaging over all electron and neutrino c orientations. Also, for terms such as gamma - Tr < auto/g) Tr[%-Bo(1—3°€)G(l + Ema °‘ V) )1 2 - Tr[1 : o] - “2 (8—182) W 'for negatron decay. The matrix element in full form thereby reduces to 2, .Ymc . 2 0 +2 2 - 2 |Hfi| 2g21{|f|(C2 +0: +2cvcs w )+| or|(cA+cT yomc2 iZCACT w )} .. (8—183) for e+ decay a2 22)1/2 is included from the Coulomb analysis of the where y. - (1- problem. Of particular interest are the cross terms containing cicJ which have an energy dependence. The nature of these terms will be shown 154 to be important as their existance can be tested experimentally. ,The above matrix element will be written more economically as |Hfi|2 = €[liyob(m—§3)] for e; (8-184) where g = 1/2[(c$ + c§)<1>2 (c: + c§>2] (8—185) and bg = [cvcs<1>2 + CACT2] (8-186) The transition rate shown in (8—64) can be written as A = (2n3u7c>‘l|nfi|2 2 - 3 7 -l mc2 ; = 8 F(TZ,W)(2" H c) ElliYob(Zfi;)] for e decay (8-187) where is the main electron energy necessitated by integration to get A. For electron capture the same general equation occurs where 2 mc A = g2(2n3u7c>‘1F€(z,w>g[1 + y.b<>1 6 (8-186) and F€(Z,W) contains the Coulomb contributions to electron capture. Both FB+ and F8 are calculated functions, so the relative branching ratios when both forms compete is given by 155 __ As __ Fe 1 +yob “TI-FM -11 B B l-Yob 1+Job = R0 [ _1 ] (8‘189) l‘Yobm > for m = c = 1. Also, since A = 52.n2/tl/2 f c - 2 2[g(1 - b) '1 (8-190) 8+ " n Yo , ] ft= 2n2[g(1 + y°b)]-l (8-191) The ft values differ in that the energy dependence cannot be completely separated from the matrix element. If b = 0, both terms are equivalent and only CA + CV terms of CS + CT terms may contribute. If b # 0, then all four forms may contribute, at least as far as cross terms. 8.3 Implications of the Allowed Assumptions fAlthough the allowed assumptions are generally valid for numerous allowed decays, there are also examples of allowed spectral shapes that are not adequately explained by the theory (Dan68). Indeed the overlap between the allowed and forbidden logft values is great enough to imply that by analogy higher order corrections to allowed decay may not be as small as initially assumed. Any breakdown in the allowed assumption will be most sensitively measured in e/B+ decay branching ratios. No thorough Ill-ill!!! 156 investigation of the consequences of higher order corrections will be presented here, but the more salient aspects will be discussed. When we include off—center (r # 0) decay terms, we can obtain an infinite number of correction terms. Those matrix elements most likely to contribute are of order 2(qR)2 or (qR)-(v/c) and are presented in Table 8—1. There are up to ten such elements which can contribute to simple allowed decay. These terms can cause large affects on €/3+ ratios through VA interference which appears quite analogously to Fierz- type interference. The similar case of second-forbidden unique transitions has been shown to offer slB+ ratios six times as large as those calculated for allowed decay. Unfortunately, no such calculations are available for the allowed corrections so we must only assume they are of a similar magnitude. When we consider higher order matrix elements, it is important that we also include Pseudoscalar terms. Although Scalar and Tensor forces may be eliminated, the lower limit of'CP < 90CA (K066) leaves plenty of room for such an affect. Keeping all the terms in the transition amplitude, 1/2 _ o -o - 2 g J4[(CS+CV) 1—(CA+CP) Y5] - iJ [(CAfCT) O + + (CV+CT) 5] (8—192) where the complete allowed matrix element now contains the Fierz terms 2 _ 2 2 bE — 2[cscv<1> + c + -+ 2 ACT + CACP + CSCT<0L> ] (8—193) Of particular interest would be the CACP interference which could lead to 157 Table 8-1 Matrix Elements Contributing to Allowed Transitions (Sc66) Matrix Element Selection Rules Power of Type of I 1111 Forbidden qR.v /c Transition 1 f N f1 0 (+) —-- 0 0 + Allowed f0 0,1 (+) 0-0 0 O Inf/102 0 (+) --- 2 o [1 (SJ-W11 o (+) --- 1 1 /§ fiysflR 0,1 (+) 0-0 1 1 Corrections to + + Allowed Transi- /3/2f(axr)/R 0,1 (+) 0-0 1 l tions f3(r/R)2 0,1 (+) 0-0 2 o _ (+.+)+ 1 +2 3//2f O r r 3 r 0,1 (+) 0-0 2 0 R2 /lS/2fRij/R2 0,1,2 (+) 0-0 2 o /3/2fiAi /R 0,1,2 (+) 1/2-1/2 l 1 2nd Forbidden j 2 (AI=2,3) and /5/2fiTi./R 0,1,2 (+) 0-1 2 o corrections to J 2 allowed transi— 6/15/2fBi.k/R 0,1,2,3 (+) o-o,1/2—1/2, 2 o tions (AI=O,1) J o-1,o-2,101, 1/2-3/2 . _ 2 = = _ 2 +3+ With Rij — (rirj 1/351jr ) Aij YSBij {airj+ajri 3613(a r)}etc. _ 2 +.+ = —> + + + Bij - {oirj+0jri— 361j(o r)} Tij (OXr)1rj + (oxr)jri 158 significant anomalies in SIB+ ratios. Of course these terms would appear to be of the same order as the other allowed correction terms in Table 8-1, but their implications would lead to the advent of beta decay through right-handed helicity states. Pseudoscalar forces have been eliminated in pion decay studies so if one accepts a unified weak interaction theory, they should not appear in beta decay. Nevertheless, this factor should be investigated because any such correction would be intriguing. 8.4 Experimental Measurement of e/B+ Ratios 8.4.1 1“596d 8.4.lA Decay of l“sGd - Relative positron feedings to states in 1“5Eu were measured in y-Y triple coincidence experiments described in Chapter 2. A spectrum of y-ray transitions in 1“5Eu in coincidence with annihilation radiation is presented in Figure 8-6. The relative positron fed intensities inferred from this spectrum are shown in Table 8-2. In Table 8-3 the net positron feeding to known levels in 145E“, calculated from the gross intensities, is presented. It is important to remember that the high efficiency of the 8X8 in. NaI(Tl) annulus causes coincident summing which lessens the coincidence intensity of cascade transitions. This is evident in the 1072.0-keV deexcitation of the 1880.6-keV level where the intensity is only 40% of that seen in singles spectra. The intensities in Table 8-3 take into account the above phenomena although, fortunately, the vast majority of decay is through direct ground state transitions. The electron capture decay of 1“SGd was studied through x-Yray coincidence experiments as described in Chapter 2. The integral 159 .ssuuoomm ooaovfioaaou >I+m cummsfi .onm shaman mmmzzz JMZZ9. NNvmm . } L >2 mNmt - mum—>52 JMZZ63. mama. O 3.. be. 09 .mmm com . >2 269. i 4 04... .2momez. . 00m OmN O 4 h _N . >mx 0.0mm. 1 O OO_ >2 prom. O . ..o. x S no. X m._ ....o. x mum m..o. x on m...“ . Om. . mNN oom do LO 0 E52 0 O 8 Q 'IEINNVHC) 83d SanOO .l l.‘ l]...\lf]l. ll i i. El 1. ll 165 Table 8-4 Theoretical and Experimental SIB+ Branching Ratios for lusGd Decay * (€(tot)/B+)+ Energy (keV) theoretical experimental 808.5 0.45 10.7 i 2.0 1041.9 0.60 ‘ 0.72i0.05 1757.8 1.17 El.l7i0.05 1880.6. 1.37 1.34:0.05 2494.8 3.39 3.07:0.3 2642.6 4.41 4.70:0.5 * Values from Nuclear Data Tables_lg, 205-317 (1971) +Values from data in Table 8-3. li‘illll]‘|ovllllfllll . 166 normalizations are made to the theoretical ratio for the 1757.8-keV level. This is a level where good statistics exist and the yhtransition seems straightforward and fast. If the theoretical ratios are indeed correct, the only difference in normalization would be statistical. Even if the theory is sometimes incorrect, previous data suggest that "normal allowed" transitions are correctly predicted. In any case the final intensities should correctly predict the total singles intensity. Comparison of the theoretical and experimental €/B+ values show generally good agreement for all of the strong transitions except the transition to the 808.5-keV level. This level is well placed experimentally (Wil7l) and its feeding is quite well understood (Ep7l). The anomaly in this value is well beyond statistical significance. Table 8—5 presents a comparison of the inferred singles intensities from coincidence to levels in ll'SEu with measured singles intensities. For the higher levels agreement is quite good but some discrepancies are seen for the lower levels particularly the 808.5? and 1041.9-keV levels. Those differences are still small enough to not change any arguments given previously, but they are, nonetheless, interesting. Explanations of the missing intensity cannot be made from experimental arguments because most of the transitions fit the experimental singles intensities quite reasonably. Some error may involve the choice of normalization to the 1757.8-keV level 8/8+ feeding ratio, but this is clearly not a great source of error. Also, errors in the decay scheme should not be significant. Feedings from above to the lower levels are expected, and we have already seen evidence of misplaced transitions. This source of error would tend to lower singles intensities below their present values without as great an affect on coincidence feeding intensities. 167 Table 8-5 % B—Feeding to Levels in ll‘5Eu from ll“r’Gd Decay Singles and €/B+—Feeding Coincidence Experiments Energy (keV) * % B(Singles) Z B(coincidence)+ 808.5 1041.9 1757.8 1880.6 2494.8 2642.2 5.13 9.93 35.7 36.3 1.33 1.98 3.30 8.74 35.7 335.2 1.14 1.93 * From Eppley et al., Phys. Rev. C, 3, p. 282 (1971) 1‘Inferred from relative B-feedings normalized to Eppley's singles intensity at 1757.8-keV. 4T.I'll|ll!|lllil '1 11 168 Finally, a potentially exciting cause of these discrepancies is anomalies in K/L capture ratios. If some e/B+ ratios are anomalous, it corresponds that K/L capture ratios might also be errant. If L-capture is enhanced, the intensities can all fall into line. ‘The significance of these results and other results discussed later will be explored in Section 8.5. 8.4.2 luagSm and 173Eu 8.4.2A Decay of luagSm and 173Eu - The decays of HaEu and ll‘3Sm were discussed in considerable detail earlier in this thesis. Positron feeding ratios were measured through a Yi-Y experiment similar to that described in the previous section. The activities for M3Eu and ll‘S’Sm were produced simultaneously and coincidences all measured in one spectrum. The differences in energies between transitions in the two spectra were sufficiently large to give clean results. A spectrum of positron fed transitions for 173Eu and 1738m was shown in Figure 4-4. The total positron fed y-ray intensities are shown in Table 8-5 which are the corrected intensities for total feeding into and out of each level. Because no corresponding x-Y coincidence information is avail- able, it is necessary to infer relative electron capture feedings from the singles intensities and the relative positron feedings. In order to do this, the €/B+ ratio for a "fast", well understood transition must be taken as the theoretical value. This was done for feeding to the 1056.65-keV level in 1“33m and the 1107.15-keV level in 1“3Eu. The results are shown in Table 8-6. 169 Table 8-6 B-Feedings from Decays of 1‘3Sm and 1“3Eu to _ Levels in ll”Pm and 1”Sm Respectively E(keV) IB+(total)a I€(total)b 1“3sm 1056.69 99.6 966. 7 1173.11 3.8 237. 1403.11 5.6 196. 1514.92 16.8 509. 1“3Eu .1107.15 92.8 47.7 1536.69 51.1 31.9 1565.85 24.2 16.8 1715.14 18.3 13.7 1912.60 ‘ 52.3 56.0 + aFrom y‘-Y triple-coincidence experiments. bInferred from singles intensities and theoretical value of €(tot)/B+ of 9.7 to the 1056.69-keV level of 173Pm and the value 0.51 to the 1107.15-keV level of 1“’Sm. 170 8.4.2B €/B+ Branching Ratios in the Decays of H3Eu and H3Sm - Theoretical E(tot)/B+ branching ratios for ll*3Eu and 1”Sm are presented in Table 8—7. These values are calculated as discussed in Section 8.4.1B. Next to these values in Table 8—7 are the calculated experimental values for €(tot)/B+. In this case the statistical certainty is not nearly as good as the 175Gd data and larger errors are expected. The €(tot)/B+ branching ratios measured for 173Sm agree quite well with theory for the transitions to the 1056.69-, 1403.11-, and 1514.93- keV levels. The value to the 1173.11-keV level shows yet another anomaly where the theory does not predict the correct €(tot)/B+ branching ratios. The lu3Eu ratios all agree quite well with theory. These results and their implications will be discussed later in Section 8.5. 8.5 Discussion of Anomalous e/B+ Ratios The results of the previous section indicate two concrete examples of anomalous €/B+ ratios. Since most of the data agrees with theory, these anomalies stand out quite sharply. It is imperative, nevertheless, that all experimental doubt be removed. One concern is that a long-lived Y—ray transition might lead to erroneous results. There is no reason to expect such a transition as the anomalous levels have been observed in charged particle spectroscopy and are well understood (Jo7l,Ch71,Wo7l, and Wil7l). These levels de- excite via high energy, low multipole transitions which would not likely lead to significantly delayed states. The 808.5-keV state of 11’SEu has been measured to deexcite promptly and the other cases should likewise 171 Table 8—7 Theoretical and Experimental €(tot)/8+ Branching Ratios for the Decays of ll”Eu and 173Sm + + a [E(tot)/B ]b [€(tot)/B ] Energy (keV) theoretical experimental 1“3Eu 1107.15 0.46C 0.51:0.06d 1536.69 0.69 0.62:0.06 1565.85 0.72 0.69:0.15 1715.14 0.83 0.75:0.17 1912.60 1.02 l.07i0.ll 1“3Sm 1056.58 9.7 9.7 10.7 1173.18 13. 63. :10. 1403.06 29. 35. i5. 1514.98 49. 30. 17.3 8Values from Chapters 4 and 5. bValues from Nuclear Data Tables 19, p. 206 (1971). c . . . Q€ value chosen as 5.5 MeV to get best comparison w1th experiment. dData normalized to this value for best fit with theory. 8This point will fall in line if an error of 100 keV is assumed in the QE value. 'Ill’..l."‘ln I ll'll‘llll l 172 be fast. Another possible flaw could be an error in the decay scheme. This is not likely as each decay is quite simple with most of the ybray strength accounted for. Any unplaced transitions are too weak to significantly affect the results. Nevertheless, the measured ratios could vary several percent and care should be taken in using these numbers. Finally, choice of normalization to theory and errors in the decay energies can cause considerable deviations in the results but not individual anomalies. Correction for L-capture does not significantly alter rela- tive electron capture feedings, and adjustments for deexcitations through cascades are generally small and always straightforward. Corrections were made for feedings from above, where necessary, using experimental coinci- dence efficiencies, however these changes were quite small. This leaves little alternative but to believe that the €/B+ ratio anomalies are real. Other interesting features have already been published about these decays (Mc69,Ep7l), so an attempt will be made to tie this information together. Clearly these anomalies can offer new insights into beta decay theory. A pictorial description of 1"5ng decay is given in Figure 8-9. The 1757.8— and 1880.6—keV levels in HSEu are 3-quasiparticle states of the general form (flh11/2)(vh9/2)(vsl/2)-l as described by Eppley et a1. (Ep7l). Such states are seen elsewhere and described by McHarris et a1. (Mc69) and in Chapter 5 of this thesis. The 808.5- and 1041.9-keV states are essentially the (081/2) and (fldalz) single particle states respectively. Transitions to the 3—quasiparticle states are fast (logft = 5.8) because they involve primarily the breaking of a (flh / )2 pair and the formation 11 2 4 - » Figure 8—9. Stylized model of lusng decay to single-particle and 3-quasiparticle states in MSEu. .“n .._————.1_ --— "m—H—“—~.- _‘.—-—- .._.___ . 173 h9/2 SIIZL danger; hI l/Ztr h 1 v2 ' d5/2»._jd5/2 tr; :. 97/2 97/2“.— I4ngd " 64 8| log ft “II/2vL “we! (is/2:: 751/2: l757.8+l880.6 kev/ 5.8 C‘s/2% 1041.9 keV / 6.7 swi 808.5 keV / 7.1 h9/2 5 I /2=: d 3/2: :2 hll/Z d5/Zc=“ g 7'“ yam..- dS/ZOO-OO-oo— g mh— 9 mo...— 7/2 p 7/2 n I45 Eu 63 82 174 of a (Nhg/z) particle. Decay to the single particle states is not so straightforward. Here a (Nd )2 pair is broken, a (vs / )2 is formed, 5 1 2 /2 and a (Nd ) particle is promoted to a (Nd ) or (N8 ) state. Al- 5/2 3 2 1/2 / though this is considered to be a single-step process, it will certainly proceed slowly. That is borne out by the large logf% values of 7.1 and 6.7 to the 808.5- and 1041.9-keV states respectively. A similar case exists for 1338m decay. The ground state of luasm is essentially the (Vd ) single particle. The fastest transition (logf% = 3 2 / 4.9) is to the ground state of 1“3Pm. This involves the breaking of a (Nd )2 pair and formation of a (vd )2 5/2 3/2 forward. Almost as fast (long = 5.8) are transitions to the (N7111 )- /2 2) 3—quasipartic1e states at 1056.6- and 1515.0-keV described pair which is quite straight— (Vh )(vd 9 2 3 / / in Chapter 5. Transitions to the (Nd ) and (N81 ) single particle 3 2 2 / / states at 1173.2- and 1403.1—keV in luaPm are slower (logfi = 6.4). Again the transition rates parallel those in usGd for similar reasons. The 1“Sm transitions to single particle states are faster than those corresponding transitions from ll’sGd. This may be because the 11”Sm case involves essentially (NdS/Z) + (vd3/2)’ AI = 1 while the ll’SGd case involves (Nd5 ) +(vs1 /2 / necessarily more complex. Anomalies occur for the hindered transitions ), AI = 2. Such a transition is 2 from InsGd and ll'3Sm to the single particle states in the daughters. A small anomaly of 315% may exist in transitions to the (Nd3 2) states. / These transitions are somewhat simpler, involving the promotion (Nds/z) + (Nd ), A1 = 0 rather than the more complex (Nd ) + (N8 ), A2 = 2 3/2 5/2 1/2 transition. Although the allowed decay of near N=82 nuclei to single particle states can be strongly hindered, forbidden transitions to the single 175 particle states are seen to be enhanced (logfi = 7.2-7.8). Despite the fact that there may be some uncertainty in these numbers, the trend isv unmistakable. These transitions should be straightforward, and it is also evident here that forbidden transitions may proceed more rapidly with larger A. There is no obvious reason why the forbidden transitions might be enhanced, but this appears to be the case. These decay peculiarities must be involved in the anomalous ratios. The most complex transitions from ll’5Gd and l”38m decay give the greatest anomalies. The decay to the 808.5-keV, (Nsl/z) state of 1“5Eu deviates by a factor of 24 from the theoretical e/B+ value, and decay to the 1173.1- keV state of lkaPm varies by a factor of 5. The implications of this chapter will be explored more directly in the following section. 8.6 Possible Implications of Anomalous E/B+ Ratios Anomalous €/B+ ratios will occur when interference terms appear in the matrix element. They may contain an energy dependence and occur with opposite signs for E or 8+ decay. More radical explanations of the anomalies such as fundamental differences in electron capture and positron emission seem to contradict too much of the established evidence to be considered at this time. Interference can take several forms. The allowed assumption permits terms of the form CSCV and CACT. Measurements of €/B+ ratios for allowed decays such as 22Na (Sh54) indicate these terms are small or are zero. Thus, it appears that scalar and tensor terms may be ruled out. Dispensing with the allowed assumption, the terms CSCT and CACP arise. The CSCT term should be quite small from previous experiments, but the CACP term needs 176 further investigation. This term occurs with the nuclear matrix element .{YS which changes parity for a given first order transition. Thus, to contribute to allowed decay it appears in second order proportional to + .sti. Clearly, the CACP term should be small if it exists at all. Nevertheless, finding such a contribution would be quite exciting. The pseudoscalar force allows the participation of right—handed particles in beta decay, a fact never yet observed. This might jeopardize two component neutrino theory. Pion decay studies indicate that N—e/N—u decay ratios (An59) indicate the lack of pseudoscalar contribution. A unified weak interaction theory suggests that CP = O. Experimentally this is difficult to determine as measurements of 0 + 0, AN = yes beta spectra (Bh60) indicate only C < 9OCA. Thus, pseudoscalar forces are not P entirely ruled out of beta decay. A small pseudoscalar contribution should exist with CP 2 (1/20) CA (Bh60), but this would not be expected to lead to a significant effect. The question arises as to why the pseudoscalar term could possibly be significant in the anomalous E/B+ ratios. Although the transitions are allowed, their large long's indicate that the allowed matrix elements must be small. Thus, higher order terms like the pseudoscalar may not be small relative to the allowed terms. This is suggested by the fast forbidden transitions observed in these nuclei. If the pseudoscalar term is indeed possible, certainly numerous other higher order terms may be present. Indeed, at least ten matrix elements may contribute to allowed decay in second order and these are listed in Table 8—1. These terms can introduce an interference of the form C C which does not exist if the r = 0 assumption is made. Studies A V of forbidden decays indicate that large deviations from allowed theory 177 for €/B+ ratios are to be expected (Ber63,Ber68,Dan68, and Pe58) for higher order matrix elements. Calculated predictions are not possible for cases where several matrix elements may contribute, because each term affects the e/B+ ratio differently and no assumptions can be made without the relative contributions. For unique forbidden cases only one matrix element is important so the ratio is predicted by the equation (Be63) A A ._E =.Z$E£ill _§. (8-194) A+unique (Wo-l) A+ Allowed forbidde For the ll’SGd anomaly this would predict a ratio 2.6 times the allowed ratio and, for the ll'3Sm anomaly, a ratio 3.8 times as large. Other matrix elements may give much higher values. Of particular interest may be the J = 2 matrix elements which may promote (Nd5 ) + (vs 2) transitions /2 1/ strongly. Clearly (Vd3 2 + Nf7/2) transitions are fast so a strong / similarity may be observed. In conclusion, one must lean towards the contribution of normal higher order terms to explain the anomalies. This does not basically alter the current theory and pulls together several observed facts. Pseudoscalar forces do not seem so likely, but it is important that this prospect be further investigated. + 8.7 Further Work on Anomalous E/B Ratios + In order to better understand the anomalous E/B ratios a series of difficult experiments should be performed. lkscd with its longer half— life and documented anomaly is a prime target. I“3Sm and lusEu are more 178 difficult due to short half-lives and the preponderance of decay to the ground state of the daughter. Of primary importance are the measurements of the beta spectra. The endpoints for these nuclei are not well measured, and the spectral shapes should be sensitive to the various forbidden matrix elements. Unfortunately, these experiments must be B—Y coincidence measurements to simplify the problems because of the complexity of the decay scheme. The anomalous 8+ branches are weak so one is forced to demand a very small chance to true coincidence ratio. The use of magnetic spectrometers is probably difficult or impossible, and the inherent problems with Si(Li) leave only plastic detectors as likely 8+-spectrometers for this experiment. This may be marginally sufficient to get gross beta shapes, but will severely limit minute analysis of the contribution of different matrix elements. ‘In addition, B-Y (directional) and B—Y (circular polarized) coinci- dence experiments will also give valuable data concerning the matrix elements. Unfortunately, the directional experiment gives an isotropic distribution of betas for intermediate states of spin 0 or 1/2. The luscd and ll”Sm cases involve spin 1/2 states so they cannot yield useful information. Their circular polarization correlation would be useful, but this experiment may be impossible to perform. Direct measurements of positron or neutrino helicity would be of great value in searching for pseudoscalar forces, but these experiments are also difficult. It is certainly also valuable to search for new anomalous cases. 1‘+3 11+? Gd and Dy which should also decay Good prospects would be similarly to lusGd. Also certain allowed transitions with large logft values should be good candidates. It might be very valuable to pursue 179 cases where beta spectrum shape information is available. One such 56C case is o. The B+—spectrum gives a shape factor of the form -1 l + b with b = 0.2 — 0.3. An interference term of this size would give an E/B+ ratio 50% larger than the allowed value. Work should also proceed on the theoretical aspects of this problem. Calculations are needed to estimate the size of the higher order matrix elements as well as the allowed matrix elements. 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IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII ll][I[MHLIIZIIMNH[WINNINIWIV[llellll 3056 480