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The decay of ¹⁵²Eu Walton, Thomas George 1971

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THE DECAY OF 1 5 2EU THOMAS G. WALTON B.Sc. The University of British Columbia, 1965 M.Sc. The University of British Columbia, 1967 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY in the Department of PHYSICS We accept this thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA September, 1971 In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make i t freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the Head of my Department or by his representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Department of pM-/SsCW The University of British Columbia Vancouver 8, Canada Date ABSTRACT The excited states of "^Sm and -^Gd obtained from the decay of 152 Eu have been investigated. Precise measurements of the energies and intensities.of•the gamma-ray transitions have been made using Ge(Li) de-tectors. The placement of transitions in the decay schemes was aided by Ge(Li) - Ge(Li) coincidence measurements. 5 8 gamma-rays were observed 152 in the decay of Eu and 37 coincident transitions. Sixteen of the co-incident transitions were observed for the fi r s t time in this work. The three lowest energy members of the ground state, ]3 and y—vibrational bands were observed in "*"^ Sm. Two Kn~ O levels were identified in l^Sm having B(EL) ratios agreeing with adiabatic symmetric rotation model predictions. An alternate K assignment is suggested for the I = 2~ and 3~ levels at 1 5 3 0 keV and 1579 keV. The effect of including band mixing in the calculations of the B(E2) ratios by the symmetric rotation model was investigated for transitions from the j3 and y- vibrational bands of 152 Sm. Simple y band mixing was found to work, within experimental error for transitions from the / band, however the inclusion of p , y band mixing could not bring agreement with experiment. For transitions from the p -band, both simple p band mixing and p , y band mixing could not make the calculated ratios agree with experiment. The asymmetric rotation model of Davidson was also compared with experiment and the value of the defor-mation parameter y and the softness parameter^/* was found to be 11.1* 1 5 2 and 0 . 3 9 6 respectively for Sm. The asymmetric model was able to pre-dict correctly the B(E2) ratios from the y band but not from the p band. Monopole contributions in transitions from levels in the p band to levels of the same spin in the ground state were estimated and compared with the model predictions via "x" ratios. It was found that both models over-estimated the amount of the EO contribution in these transitions. Some 152 evidence has been found for a new 1293 keV level in Sm. Eleven 1^ 2 levels in < Gd were identified, seven of which could be identified to quasi-rotational bands. TABLE OF CONTENTS Page CHAPTER I INTRODUCTION 1 CHAPTER I I THEORY: THE COLLECTIVE MODEL OF EVEN—EVEN NUCLEI 4 1. The Hydrodynaniic Surface 4 2. Symmetry Considerations 9 3. The A x i a l l y Symmetric Model 13 4. The A x i a l l y Asymmetric Model 17 5. Electromagnetic T r a n s i t i o n P r o b a b i l i t i e s 21 CHAPTER I I I THE GAMMA SINGLES SPECTROSCOPY 32 1. General Considerations 32 2. The Lithium D r i f t e d Germanium Detector 33 3. The Ge(Li) Gamma-Ray Spectrometer 36 4. Experimental Procedures 36 5. Peak F i t t i n g 42 6. Results 49 CHAPTER IV GAMMA-GAMMA COINCIDENCE SPECTROSCOPY 62 1. General Considerations 62 2. The Gamma-gamma Coincidence Spectrometer 63 3. Experimental Procedure 65 4. Analysis and Results 71 CHAPTER V DECAY SCHEMES AND MODEL FITTING 87 1. The Decay Schemes 87 2. Model Comparison 97 CHAPTER VI CONCLUSIONS 107 1. I d e n t i f i c a t i o n of Bands 107 2. Model Comparison 109 REFERENCES 111 LIST OF FIGURES Figure Page 3-1 Gamma-ray Spectrum from Monoenergetic Photons 34 3-2 Schematic Diagram o f G e ( L i ) Gamma-ray Spectrometer 37 3-3 P a r t o f 6 0 C o Spectrum 38 3-4 Detector E f f i c i e n c y Versus Energy f o r Sources 20 43 cm from Detector 3-5 Least Square F i t o f a 1332 keV Peak o f 6°Co 45 3-6a Graph o f Ranges o f "R" Parameter Values Obtained from 47 Least Square F i t o f C a l i b r a t i o n Peaks 3-6b Ranges o f Values o f Parameters R T and Rg Obtained from 48 Least Square F i t o f C a l i b r a t i o n Peaks 3-7a P o r t i o n o f 1 ^ 2 E u y -ray Spectrum 100 keV t o 420 keV 54 3-7b P o r t i o n o f 1 5 2 E u y -ray Spectrum 200 keV t o 520 keV 55 3-7c P o r t i o n o f 1 5 2 E u y -ray Spectrum 500 keV t o 680 keV 56 3-7d P o r t i o n o f ^ E u y -ray Spectrum 650 keV to 900 keV 57 3-7e P o r t i o n o f ^ E u y -ray Spectrum 900 keV to 1050 keV 58 3-7f P o r t i o n o f 1 5 2 E u y -ray Spectrum 1050 keV t o 1200 keV 59 3-7g P o r t i o n o f 1 5 2 E u y -ray Spectrum 1200 keV t o 1300 keV 60 3- 7h P o r t i o n o f 1 5 2 E u y -ray Spectrum 1300 keV t o 1800 keV 61 4- 1 Schematic o f a T y p i c a l Cascade 62 4-2 Coincidence Detector Assembly 64 4-3 Schematic o f Coincidence Apparatus 66 4-4 Time R e s o l u t i o n Spectrum o f Coincidence System 68 4-5 Gate Detector Spectra 70 4-6 Decay Scheme o f ^ S m 76 4-7 Decay Scheme o f 1 5 2 G d 77 4-8 Schematic o f T y p i c a l Decay Cascades 78 LIST OF FIGURES (continued) Figure Page 4- 9 High Energy ON and OFF gate Spectra for 121.7 keV 79 Gate 5- 1 Comparison of Experimental Quadrupole Energy Levels 99 with Asymmetric Model Predictions for LIST OF TABLES Table Page I I I I I I IV V V i a VIb V i c VId V i e V i l a V l l b V I I I IX XI X I I X I I I Number o f Allowed L Values f o r each Representation 11 Band Mixing C o r r e c t i o n F a c t o r s 29 Peaks Used f o r Energy C a l i b r a t i o n 40 Peaks Used f o r I n t e n s i t y C a l i b r a t i o n 41 Energies and R e l a t i v e I n t e n s i t i e s o f the y -rays 50 Emitted i n the Decay o f ^ E u Gamma Rays Observed i n Coincidence w i t h 121.7 keV 72 T r a n s i t i o n Gamma Rays Observed i n Coincidence w i t h 244,6 keV 73 T r a n s i t i o n Gamma Rays Observed i n Coincidence w i t h 295.9 keV 73 T r a n s i t i o n Gamma Rays Observed i n Coincidence w i t h JkhtJ keV 74 T r a n s i t i o n Gamma Rays Observed i n Coincidence w i t h 444,0 keV 75 T r a n s i t i o n Comparison Between C a l c u l a t e d and Experimental 81 Coincidence Rates Comparison Between C a l c u l a t e d and Experimental 82 Coincidence Rates Experimental and T h e o r e t i c a l K - S h e l l I n t e r n a l Con- 89 v e r s i o n C o e f f i c i e n t s f o r T r a n s i t i o n s i n the Decay of 1 5 2 E U Comparison Between Experimental and Asymmetric Model 100 Branching Ratios i n 1 5 2 E U Comparison o f Mixing Parameters f o r T r a n s i t i o n s from 102 the y -band t o the Ground S t a t e Band i n Comparison o f Mixing Parameters f o r T r a n s i t i o n s from 103 the p -band t o the Ground St a t e Band i n 1^ 2Sm R e l a t i v e B(EL) Values f o r Negative P a r i t y Levels i n 105 -^Sm T h e o r e t i c a l and Experimental Values o f the "X" Ratios 106 ACKNOWLEDGEMENTS I wish to express my gratitude to Dr. K, C, Mann for his guidance, encouragement, and help throughout the work. I am also indebted to Dr. G, Jones, Dr. G. M. Griffith, and Mr. G. Lee for allowing me the use of some of their equipment; and to Mr, J. Johnson for his assistance, particularly with the data processing. Technical assistance by Mr, A. Frazer, Mr. B. Haynes, Mr. P. Tammin-ga and Mr. J. Lees is highly appreciated. The present project was supported by the National Research Council through Grants-in-Aid of Research to Dr. K, C, Mann and a Post Graduate Bursary to myself. 1 CHAPTER I I n t r o d u c t i o n The 12,k year isomer o f "^ 2Eu decays i n t o two daughter n u c l e i , 15*2 15*2 £^Gd via p~ -decay and £2^ m v~^a "decay and e l e c t r o n capture. There are many reasons f o r stu d y i n g the decay o f "^ 2Eu, The daughter s t a t e s , ^ 2 S r a and 1^ 2Gd, are i n the t r a n s i t i o n a l r e g i o n be-tween s p h e r i c a l n u c l e i which e x h i b i t v i b r a t i o n energy l e v e l s , and s t r o n g l y deformed n u c l e i , which e x h i b i t r o t a t i o n - l i k e energy l e v e l s 152 as w e l l . Indeed, J Gd appears t o be n e a r l y s p h e r i c a l i n shape, and has p r i m a r i l y v i b r a t i o n a l l e v e l s a t low energy, although q u a s i r o t a -1 152 t i o n a l l e v e l s have r e c e n t l y been suggested, J Sm, on the other hand, has r o t a t i o n s t a t e s b u i l t upon low energy v i b r a t i o n l e v e l s , which i n d i c a t e i t i s not s t r o n g l y deformed, as v i b r a t i o n s t a t e s u s u a l l y appear a t h i g h e r energies i n s t r o n g l y deformed n u c l e i . The t r a n s i t i o n a l nature o f ^ 2 S m poses an i n t e r e s t i n g problem f o r models to e x p l a i n s p h e r i c a l o r deformed n u c l e i , as i s a h y b r i d between the two i d e a l i z a t i o n s . E x p e r i m e n t a l l y , the study o f the complex "^ 2Eu decay i s i n t e r e s -t i n g because w i t h the advent o f the modern l i t h i u m d r i f t e d germanium d e t e c t o r s and t h e i r s o p h i s t i c a t e d a s s o c i a t e d e l e c t r o n i c s , there i s a l -ways a p o s s i b i l i t y o f f i n d i n g much new gamma-ray t r a n s i t i o n data. The Ge(L i ) has an energy r e s o l u t i o n a f a c t o r o f ten o r more b e t t e r than the Nai s c i n t i l l a t o r s used p r e v i o u s l y . The f i r s t comprehensive s t u d i e s were c a r r i e d out by Grodzins and 2 Ke n d a l l u s i n g sodium i o d i d e s c i n t i l l a t o r s and a s i n g l e l e n s magnetic spectrometer, and Bobykin and Novik^ us i n g a 180 degree magnetic spec-trometer. The e a r l i e r i n v e s t i g a t o r s worked with sources t h a t unavoidab-2 l y contained a l a r g e contaminant. ^^E\i has a very s i m i l a r spec-152 trum t o Eu and hence complicates the a n a l y s i s . The f i r s t i n v e s t i g a -152 t i o n c a r r i e d out w i t h a source s u f f i c i e n t l y enriched i n ^ Eu t t o r e -154 duce the Eu contamination t o n e g l i g i b l e amounts was by Cork, B r l c e , h Helmer, and Sarason who used a magnetic spectrometer. A f t e r a decade of study, culminated by the work of Dzelepov and h i s coworkers, 5*6,7,8 ^almsten, N i l s s o n and Anderson^ u s i n g magnetic spectrometers, and Larsen, S k i l b r e i d , and V i s i t e n u s i n g e a r l y G e ( L i ) d e t e c t o r s ^ , there were s t i l l c o n t r a d i c t i o n s and i n c o n s i s t e n c i e s i n the p u b l i s h e d r e s u l t s . I n a d d i t i o n , some workers s t u d i e d only segments o f the d e c a y^ » H » ^ which meant a piece-meal c o l l e c t i o n o f d a t a t h a t has proved t o be d i f f i c u l t t o c o r r e l a t e and u n i f y . At t h i s time, we f e l t t h a t there was a need f o r a thorough study of the gamma-ray spectrum o f encompassing the complete energy range of emission. The aim o f the i n v e s t i g a t i o n was to measure as ac-c u r a t e l y as p o s s i b l e the photon t r a n s i t i o n s i n ^^Eu ^ D determine i t s decay scheme, and t o compare two models which are used t o represent the ^525 m daughter nucleus. The two models compared were a c t u a l l y two d i f f e r e n t approximations 13 14 o f Bohr's Hydrodynamic Model J* , and w i l l be d i s c u s s e d i n d e t a i l i n the f o l l o w i n g chapter. The Hydrodynamic Model c o n s i d e r s the nucleus as an incompressible f l u i d i n which the allowed energy s t a t e s o f the nucleus are those o f n u c l e a r s u r f a c e v i b r a t i o n s and o f n u c l e a r f l u i d r o t a t i o n s . Approximating the n u c l e a r s u r f a c e i n a s e r i e s o f s p h e r i c a l harmonics a l l o w s two degrees o f freedom, f and y , f o r the s u r f a c e t o v i b r a t e . The p v i b r a t i o n i s an a x i a l l y symmetric s t r e t c h i n g and c o n t r a c t i n g of the s u r f a c e and the f v i b r a t i o n i s a n o n - a x i a l l y symmetric d i s -3 t o r t i o n o f the s u r f a c e . This model p r e d i c t s a nucleus v i b r a t i n g w i t h d i s t o r t i o n s d e s c r i b e d by quadrupole expansions f o r the s u r f a c e and w i l l have p o s i t i v e p a r i t y s t a t e s . The symmetric model assumes no i n t e r a c t i o n between the v i b r a t i o n and r o t a t i o n s t a t e s and b u i l d s r o t a t i o n bands upon the two v i b r a t i o n s t a t e s . The asymmetric model assumes the nucleus i s a r i g i d r o t a t o r which i s softened t o a l l o w v i b r a t i o n s i n the p degree o f freedom. Both models a l l o w the nucleus t o have octupole v i b r a t i o n s (negative p a r i t y ) as w e l l as quadrupole v i b r a t i o n s and p r e d i c t the same s p i n s and p a r i t i e s o f the allowed s t a t e s . While t h i s i n v e s t i g a t i o n was i n progress , Riedinger , Johnson, and H a m i l t o n 1 ^ and V a r n e l l , Bowman, and T r i s c h u k 1 ^ p u b l i s h e d s t u d i e s on the decay o f 1^ 2Eu. Then d u r i n g the w r i t i n g o f t h i s t h e s i s , R e i d i n -ger, Johnson and H a m i l t o n ^ p u b l i s h e d f u r t h e r s t u d i e s on the same i s o -tope, and t h i s t h e s i s p r e s e n t a t i o n has been modified t o i n c l u d e t h e i r l a t e s t r e s u l t s f o r comparison. 4 CHAPTER I I Theory: The C o l l e c t i v e Models o f Even-Even N u c l e i 1 The Hydrodynamic Surface The nucleus i s considered t o be a drop o f charged l i q u i d w i t h the a) i t i s incompressible and hence has constant volume, b) the s u r f a c e i s held i n e q u i l i b r i u m by two f o r c e s , coulomb r e p u l s i o n and sur f a c e t e n s i o n , and c) these two r e s t o r i n g f o r c e s a l l o w the s u r f a c e to undergo s m a l l o s c i l l a t i o n s o f simple harmonic motion about some e q u i l i b r i u m shape. The p o s i t i o n o f any p o i n t o f the l i q u i d drop s u r f a c e i s g i v e n as an expansion o f s p h e r i c a l harmonics. The co-ordinate system can be e i t h e r w i t h r e s p e c t t o the drop I t s e l f , c a l l e d the body f i x e d system w i t h c o - o r d i n a t e s ( r'y > <P' ) or w i t h r e s p e c t t o a l a b o r a t o r y frame, c a l l e d the l a b o r a t o r y f i x e d system w i t h co-ordinates ( fj &j <P ). The p o s i t i o n o f a s u r f a c e p o i n t i n the body f i x e d system i s The l a b o r a t o r y and the body f i x e d systems are r e l a t e d by the t r a n s -formations f o l l o w i n g assumed p r o p e r t i e s t (1) and i n the l a b o r a t o r y f i x e d system i t i s J a. - £ D (»c) cC (3) A where the J) (&c) are the dimensional r e p r e s e n t a t i o n s o f the 18 r o t a t i o n group d e f i n e d by Rose , which are f u n c t i o n s o f the E u l e r angles r e l a t i n g the o r i e n t a t i o n o f the body f i x e d system w i t h r e s p e c t to the l a b o r a t o r y system, w i t h & I j 2 ) ? - oQ-j/?'j y • To i n s u r e t h a t the surf a c e i s r e a l r e q u i r e s C. ^ - (->)'"'<%* and cC - (-t)^aC * . Furthermore, the terms ^ / / t c and °Ccorrespond t o t r a n s l a t i o n s o f the whole nucleus and v a n i s h i f the o r i g i n o f both systems i s taken a t the c e n t e r o f mass, A A expansion o f the s u r f a c e t h e r e f o r e has ( 2 A *• I ) independent parameters, three o f which s p e c i f y the o r i e n t a t i o n o f the drop and/2/^-^)are a s s o c i a t e d w i t h the shape. The nature o f the nuc l e a r matter must now be considered. One simple p o s t u l a t e i s t h a t i t i s s o l i d and the nucleus undergoes r i g i d r o t a t i o n 1 ^ . Another p o s t u l a t e i s t h a t i t i s a f l u i d undergoing i r r o -t a t i o n a l flow. I n t h i s case a v e l o c i t y p o t e n t i a l "<p" can be d e f i n e d a s V = -^ 4>, which combined w i t h the assumption o f nuc l e a r f l u i d incom-p r e s s i b i l i t y , becomes a s o l u t i o n o f Laplace's equation. For s m a l l d i s -20 t o r t i o n s about a s p h e r i c a l shape I t must be noted, however, t h a t the assumption o f i r r o t a t i o n a l flow o f the nuc l e a r f l u i d i m p l i e s t h a t the mean f r e e path o f the nucleon be very s m a l l compared t o the dimensions o f the nucleus. I n r e a l i t y , the nucleons must have a mean f r e e path o f the order o f the nuclear r a d i u s because o f the P a u l ! P r i n c i p l e , The r i g i d r o t a t i o n model assumes the nucleonic mean f r e e path i s much g r e a t e r than the nuclear dimensions, hence r e a l i t y i s somewhere between the two ex-tremes. 6 The k i n e t i c energy o f the system i s 2 S/*. * As* } (5) where - Ra f and fa = nu c l e a r d e n s i t y . From s m a l l o s c i l l a t i o n 21 theory , t h i s i s the ge n e r a l form o f the energy v i b r a t i o n i n g e n e r a l -i z e d c o - o r d i n a t e s . Thus the cC can be considered g e n e r a l i z e d co-or-AM-d i n a t e s f o r s m a l l o s c i l l a t i o n s . The p o t e n t i a l energy can a l s o be w r i t -14 22 ten i n terms of these g e n e r a l i z e d c o - o r d i n a t e s as * Z A^ A*- ' \°J where r*f*- A J * Z77(ZA + l) RQ (?) Here y i s the c o e f f i c i e n t o f s u r f a c e t e n s i o n . The p o t e n t i a l i s c o n s e r v a t i v e because there i s no v i s c o s i t y term and t h e r e f o r e a g e n e r a l i z e d momentum can be d e f i n e d as 7 7 d_L - 8 cC , , The Harailtonian f o r s m a l l o s c i l l a t i o n s about a s p h e r i c a l shape then becomes W - " i f f i / F | ^ C|<< | ' ) , ( 8 ) Z ^ A**- A AM > which i s the same as the Hamlltonian f o r a system o f uncoupled har-monic o s c i l l a t i o n s w i t h frequencies U-^Z= CA/8a , The angular mo-mentum i s d e f i n e d c l a s s i c a l l y as 1 = Jf0 h X V(lr) dz (9) where k e V ) - - 9 ha. d Q(.fr) , P u t t i n g the ex p r e s s i o n f o r ^ and the expansion f o r <p(Vj i n t o equation (9) and i n t e g r a t i n g over the nucl e a r r a d i u s one gets JL **w A' D e f i n i n g an operator su A/* 7 which i s d i a g o n a l i n A a l l o w s the angular momentum t o be w r i t t e n as z = c- £ B^OC < <<*^ i £ M ^' > . (10) /W*' A AM' The system can be quantized by demanding I f the deformation energy becomes g r e a t compared t o the v i b r a t i o n e n e r g i e s , the system can be t r e a t e d as a s t a b l e deformed r i g i d r o t a t o r . This occurs where there are a con s i d e r a b l e number o f nucleons ( o r holes) o u t s i d e o f c l o s e d s h e l l s as f o r example, i n the r a r e - e a r t h and a c t i n i d e r e g i o n s . I n these regions the nucleus s t a b i l i z e s on the average about some n o n - s p h e r i c a l shape so t h a t a frame o f r e f e r e n c e , a body f r a a e , f i x e d t o the e q u i l i b r i u m shape can be d e f i n e d . Uhe k i n e t i c energy, d e f i n e d i n the l a b o r a t o r y frame, can be t r a n s -formed, t o the body f i x e d frame u s i n g the t r a n s f o r m a t i o n o f equation (3) and the time d e r i v a t i v e s o f the r o t a t i o n matrices iL%;)= &* - c £ f L <LT\L,{L\<> f*(•!.(*,. (12) where the u»k are the body-fixed components of the angular v e l o c i t y . 21 3 The u\ are , <^ k = £ cr_ (&£) O. S-i h J where ^ _ i ^ s c H e>t si'hOj <-°s ef cos &t The k i n e t i c energy then becomes A (13) T* = JL £ B ( £ \d <Ao'\L / \*S>tv,v, The f i r s t term represents the v i b r a t i o n energy, the second term r e p r e -sents the r o t a t i o n energy and the l a s t term represents the r o t a t i o n -v i b r a t i o n c r oss terms. The body f i x e d system i s then s p e c i f i e d i n 8 order to d i a g o n a l i z e the r o t a t i o n energy term. A s u f f i c i e n t c o n d i t i o n i s t h a t a. must be non zero, only f o r even o r odd values o f o *31 which a l s o makes the r o t a t i o n - v i b r a t i o n c ross term v a n i s h . The p o t e n t i a l energy i n the body f i x e d r e f e r e n c e system becomes ' > z (15) (16) (17) V - l £ c \ a IAs" As" For convenience, a. may be re d e f i n e d as cZ - g e where The k i n e t i c energy then becomes TB= J L / B(iU f £ C ) + i j ' A ) L u Z A A, £ *A Ss-A ^ **/ A < (18) where l"l= Z?= BAPZ£ € . € < ^ ' ^ ^ = ^ P T T ! (A) (19) The p o t e n t i a l energy i s ]/• -' j_ £ C gz - [/ (A) . (20) 2 A A 'A To quantise the system we note f i r s t t h a t the motion o f d i f f e r e n t degrees o f A do not i n t e r f e r e . The k i n e t i c energy can be expressed i n terms o f the time r a t e o f change o f the l i n e element i n a space o f g e n e r a l i z e d co-ordinates , T~ - j . / af_s ) . (21) where (ds) - £ G- <4xecJxJ and cr. - i s the c o - v a r i a n t m e t r i c C J ' J tensor. By comparing equation (21) w i t h equations (18 and 19) the me t r i c t e n s o r &. & J i s seen t o be o f the form D(A) o R (A) where 0 (A) i s the d i a g o n a l m a t r i x (22) 8. 1 o o 1 (23) and i s n x n where n =, ^-d.) + / ^  i f A even = (-*±L) ; i f A odd. R ( ^ ) i s the m a t r i x a s s o c i a t e d w i t h the k i n e t i c energy o f r o t a t i o n and u -J Oi < = / ^ AJ ' -* The determinant o f (G(A )) i s /(?£>/= £ = 8 p I, - I 7 -13 sc* &z . (25) c j . . The c o n t r a v a r i a n t m e t r i c tensor G ( A ) i s d e f i n e d by 0- & C^) - J~.L and the L a p l a c i a n operator becomes i n JA c* t h i s system V* =• £ A. fiTf1 ci'cZ) i__ ) *<~ veT* *<j i t ( 2 6 ) Hence, the quantum mechanical Hamiltonian H, = - J l y + V . " 2 becomes H ^ = _ *L a £ J _ l_ (lit? J_ )+ IsOll . • (27) 2 Symmetry Con s i d e r a t i o n s The Schrodinger equation can be separated i n t o r o t a t i o n a l and v i b r a t i o n a l p a r t s . The r o t a t i o n a l s o l u t i o n s can be t r e a t e d most e a s i l y by u s i n g an angular momentum r e p r e s e n t a t i o n . The t o t a l angular momen-tum L i s d e f i n e d to have the u s u a l commutation r e l a t i o n s Etx,tJ*-c-f3 (28a) c y c l i c a l l y ^ w h i l e the body f i x e d components have the commutation r e l a -t i o n s l L l i L j - L a t 3 (28b) 22 c y c l i c a l l y , fc The s t a t e f u n c t i o n s i n t h i s r e p r e s e n t a t i o n can be denoted by I LMK> where 112 /LMK> = L(L + l ) I LMK > L_ | LMK > = M ( L M O (29) L3 I LMK> = K I LMK > 10 K i s the p r o j e c t i o n of L on the body f i x e d a x i s and M i s the p r o j e c -t i o n o f L on the Lj, l a b o r a t o r y f i x e d a x i s . The s t a t e v e c t o r s f LMK > are the e i g e n - f u n c t i o n s o f the symmetric top and are the r o t a t i o n mat-^ * 18 r i c e s D ) > (see Rose ), The r o t a t i o n Hamiltonian becomes, i n the body f i x e d system, . 2 z r ^ -A2- 1 ^ . . . . . . . (30) The operators can be r e l a t e d t o r a i s i n g and lowering operators o f the angular momentum s t a t e s , the r e s u l t being t h a t T connects s t a t e s i n which A K~Q + 2 , Thus, f o r the most g e n e r a l r i g i d top, K must be a l l even or a l l odd. The most gene r a l case i s f o r a non a x i a l l y symmetric nucleus i n which K i s not a good quantum number. Hence, the s t a t e v e c t o r s o f the nucleus have t o be mixtures o f the b a s i c v e c t o r s o f the form. c U M> = £ A^lL/v\K> . (31) The assignment o f the body-fixed a x i s l a b e l s 1, 2, 3 i s a r b i -t r a r y and can not affect any p h y s i c a l q u a n t i t y . Therefore, some symmetry p r o p e r t i e s must be i n v o l v e d . There are twenty-four p o s s i b l e ways o f l a b e l i n g the a x i s f o r each o f the r i g h t handed and l e f t handed co-ordinate systems. However, the d i s c u s s i o n w i l l be l i m i t e d t o the r i g h t handed cases. Four c l a s s e s o f r i g i d r o t a t o r s t a t e f u n c t i o n s a r i s e from the l a b e l i n g p o s s i b i l i t i e s which belong to the f o u r r e p r e -mi> .13 25 s e n t a t i o n s o f the p o i n t group , and can not be mixed together. The symmetry r e l a t i o n s f o r each r e p r e s e n t a t i o n a r e f a) f o r K even; ( i ) f o r the A r e p r e s e n t a t i o n A_ K = (-l ) ^ A ^ ( i i ) f o r the r e p r e s e n t a t i o n A _ ^ = - ( - l ) * ^ 11 b) f o r K odd; ( i i i ) f o r the B 2 r e p r e s e n t a t i o n A_£ = - ( - l ) L A K ( i v ) f o r the r e p r e s e n t a t i o n A . K = (-l)LAj£. These r e l a t i o n s r e s t r i c t the number o f allowed L values f o r each 22 r e p r e s e n t a t i o n , TABLE I Number o f Allowed L Values f o r Each Representation Representati on B even even odd odd K -K -(-D L -(-D L (-DL Number o f allowed L values L even ( L + 2) 2 L 2 L odd ( 1 - 1 } 2 > ( L + 1)  r 2 An o r d i n a l quantum number N i s d e f i n e d to l a b e l the s t a t e s t h a t can have more than the same value o f L„ For example i n the A r e p r e s e n t a t i o n f o r ; L = 0; K = 0, A o= A 0, | LMN> = A e/ 000 > , N = 1 L = 1; K = 1. A 0= -A, = 0, I LMN>= 0, N = 0 L = 2 j K = 0,2 A a= A a= A o f | LMN > = A 0 | LM0> + ( i LM2 >+ |LM -2>) N = 1.2 L = 3l k = 0,2 A 0= -A Q= 0, A, = -A |LHN>= t l (|3H2>- I 3M - 2 >), N = 1. ^2 = ^ ° J 1 2 The energy eigen-values ( L N , A ) aret £ ( 0 1 , A ) = o l (31.A) = * * ' ( f t - ^ f ) - (32) The r e l a t i o n £(21,A) + £(22,A) = £ ( 3 1 , A ) (33) can be used to check i f experimental e x c i t a t i o n s are p u r e l y r o t a t i o n a l i n nature. In g e n e r a l , the eigen-values cT(LN ,A) are c a l c u l a t e d and sub-s t i t u t e d back i n t o the equations T ^ j L M N ^ £(LN,A) ( LHN> t o f i n d the values o f the c o e f f i c i e n t s . The p a r t i c u l a r r e p r e s e n t a t i o n t o be used must be chosen from 22 experiment, Davidson has shown t h a t f o r the p a r i t y operator P, P fLM> = + ILM> f o r A and r e p r e s e n t a t i o n s , P(LM/ >= - [ LM/> f o r and B^ r e p r e s e n t a t i o n s . For even-even n u c l e i the ground s t a t e had L = 0, w i t h l e v e l s o f even p a r i t y . Hence, only the A r e p r e s e n t a t i o n i s p o s s i b l e . For negative p a r i t y s t a t e s i n e v e n - n u c l e i , the B-^  o r B-j r e p r e s e n t a t i o n s can be used. However, as experimental evidence i n d i c a t e s t h a t l e v e l s have predominantly K even, the B 1 r e p r e s e n t a t i o n i s used only when stud y i n g the octupole (^ = 3) l e v e l s . The theory s p l i t s i n t o two main branches a t t h i s p o i n t . The 13 22 f i r s t theory i s the symmetric model of Bohr and Mottelson J % , which was the e a r l i e s t . I t t r e a t s the n u c l e a r s u r f a c e as a x i a l l y symmetric, on the average, and then a l l o w s v i b r a t i o n s i n the /? and y g e n e r a l i z e d co-ordinates upon which r o t a t i o n bands are b u i l t . The second i s the asymmetric model of Davydov and c o - w o r k e r s 2 ^ ' 2 ^ ' 2 ^ , 2 9 , which t r e a t s the nucleus as an asymmetric r i g i d r o t a t o r which i s softened to a l l o w 13 v i b r a t i o n s i n the p degree o f freedom. 3 The A - x i a l l y Symmetric Model The n u c l e a r s u r f a c e i s r e s t r i c t e d to A = 2 and the c AM. are d e f i n e d as €Z l o = c o * y e , = A- (34) S e t t i n g p2 - p the moments o f i n e r t i a become l ^ = B 2 p * £ € € , <AK'I £ \Au> - tiF2*7s,H (y-zjTl). (35) From equation (30) the r o t a t i o n Hamiltonian becomes p%p* An " " ( r - ( 3 6 a ) Using the met r i c t e n s o r o f equation (22) Z82 y _JL i _ [ I ^ ( ?m j L . \ - - ft* f I . i _ s 9 ^ +. —L A. (sCh?y A.)l . (37a) ~2.82\ *P ' p's<:*?>- 6 y s if'! The p o t e n t i a l energy i s The complete Hamiltonian becomes H = -*2f±. L p^A. + ' i (SLK syj. \\ + £ — ^ r - -f- J- C,B2 . 2^^ 2 4I=I i ^ Y / - ^ 2 (39a) T only has p* as a f a c t o r which allow s the t o t a l wave f u n c t i o n r o t t o be separated i n t o The v i b r a t i o n s are uncoupled from the r o t a t i o n s by assuming t h a t the p and Y o s c i l l a t i o n s are s m a l l i n amplitude, high i n frequency, and o s c i l l a t e about the mean g e n e r a l i z e d c o - o r d i n a t e s p -Pe> , and y - o°. 14 Expanding the raoaents o f i n e r t i a i n the lowest orders o f and y y i e l d s This approximate uncoupling of the v i b r a t i o n s and r o t a t i o n s i s c a l l e d the " a d i a b a t i c approximation". A consequence o f <s, —J2 i s th a t K i s a good quantum number which a l l o w s a r o t a t i o n s t a t e to be descr i b e d by a s i n g l e b a s i c v e c t o r | LMK/*. The determinant o f the metric tensor now becomes &• =36 Bf jff y z Jth* ez and where g ^ g # (37b) and the v i b r a t o r p o t e n t i a l term i s taken as ' I/ - ± (p-faY + L C r yz j (38b) where Cp and C^, are parameters to be f i t t e d . I f the ^Tz Kz p a r t o f the r o t a t i o n Hamiltonian i s put i n t o the v i b r a t i o n p a r t , the t o t a l wave f u n c t i o n can be separated i n t o ^CfjiTjO,^ £»•})- f(p)<j(ir)I*-^K>, w i t h the Hamiltonian s e p a r a t i n g i n t o three d i f f e r e n t equations. [-Al _2L! <- _ L C, (p-fiof\f(r) - f t f f O r J 3 (^2a) J |LMK> = J l _ ^a f/)~K 2 J (LMK> . (42c) Equation 42a i s the Schrodinger equation f o r a one dimensional har-monic o s c i l l a t o r o f mass IL, o s c i l l a t i n g about p0 . The s o l u t i o n s 22 c o n t a i n Hermite polynomials and have energy eigen-values efi « T o / A r >y v M ) hp - o} - • • (43) 15 where ^p*' ' cp/&p . I f K i s r e s t r i c t e d t o even v a l u e s , equation 42b i s the equation o f the two dimensional harmonic o s c i l l a t o r and the s o l u t i o n s c o n t a i n c o n f l u e n t hyper-op geometric f u n c t i o n s . The allowed energy values a r e " £r= 1\ u>y (h^i-i ) ? (44) where UJ ^ = C y / S y and h r = z A/ + (J< I A/- ot i ? • • • 2 ' The t o t a l energy E = } L (L+l)- K (^ 5) which means t h a t upon each v i b r a t i o n energy a r o t a t i o n band i s b u i l t . For example, the ground s t a t e band, w i t h hy=hp-ot t< - o and L - £7,2, H •• f i s a s e r i e s o f s t a t e s each w i t h t o t a l v i b r a t i o n a l energy o f ^ y + ^P a n £ i r o t a t i o n a l energy o f The p band i s c h a r a c t e r i z e d by hp-/ , hy-o and as p v i b r a t i o n s preserve the a x i a l symmetry, IKI = 0. The s p i n sequence i s aga i n L = <V . The f i r s t ex-c i t e d gamma band has hp - °j*>f=' and hence IK I = 2 (see equation (46b)), The s p i n sequence i s L - 2}3, H ' ' * K f / , Higher energy bands o f more than one quantum o f e x c i t a t i o n w i l l not be d i s c u s s e d because o f com p l i c a t i o n s from e x c i t a t i o n o f p a i r s from the core. The s t a t e s o f odd p a r i t y can be de s c r i b e d by c o n s i d e r i n g the octupole 30 expansion o f the nuc l e a r s u r f a c e . The development f o r the octupole case f o l l o w s the quadrupole case very c l o s e l y . The moments o f i n e r t i a f o r a sur f a c e can be w r i t t e n i n terms o f the body f i x e d , octupole expansion + 30 parameters : y/3^ 83 (6 *j0 + 2 in? aJO a32 +%af2 ) 720>= B3(6a]Q -z\TJJ <zlo a J z +8*1,) These have been d i a g o n a l i z e d to e l i m i n a t e v i b r a t i o n - r o t a t i o n c ross terms by s e t t i n g <%3± l j 3 =o* The i n e r t i a l t e n sor can a l s o be d i a g o n a l i z e d by s e t t i n g CX3* -o , ^ - & , z , but S o l o v i e v ^ has shown 16 t h a t s t a t e s u s i n g these degrees o f freedom are not o f a c o l l e c t i v e nature. The octupole l n e r t i a l tensors are added t o the quadrupole moments of i n e r t i a . Assuming the octupole o s c i l l a t i o n s preserve a x i a l symmetry y & 3 z m a-30'^c> hence J, - J 2 ~ o , hence the t o t a l moments o f i n e r t i a become J3°lve2 pj>2 + 8BSG,1 = v ff?/?/ (y^y) } (4?) where 3 1 - 2. ff? . The c l a s s i c a l v i b r a t i o n energy i s the sum o f the octupole p l u s quadrupole v i o r a t i o n energy. TV,, , x *t y ') * ± Bs ( a l +z a,\ ) . m Once a g a i n , the a d i a b a t i c approximation i s used and i n v o l v e s r e p l a c -i n g the f a c t o r p by pa and n e g l e c t i n g other v a r i a b l e s i n comparison w i t h po . The q u a n t i z a t i o n procedure used p r e v i o u s l y i s repeated and p l a c i n g the r o t a t i o n a l term A «/Zjj i n t o the v i b r a t i o n operator y i e l d s T W A R - A. + J _ £ + J L [A? + _____ L + A. Z 1 32 bp1 B3 U 2 By \Jt _ 4 ___-^____ +l/(p>bJfJ}) , (49) 2 ^ f - j y 2er(r*+i*) where b= 4.3o . A harmonic form o f the p o t e n t i a l i s assumed and i n a s i m i l a r manner to the quadrupole case, two harmonic o s c i l l a t o r equations which are s i m i l a r t o equation (42a), separate, one f o r the fi degree o f freedom and another f o r the b degree o f freedom. The energy o f each i s g i v e n as € hp = * ^ h f i 4 i ) *V " °) 'J z •' • J a n d e*6 - * ( h b + { > hb - °j ', * ' • • ' The p and 6 v i b r a t i o n s are a x i a l l y symmetric, hence \k\-om R o t a t i o n a l l e v e l s of energy €. - A* [rfr+if] are b u i l t on each v i b r a t i o n -rr a l l e v e l . The p band, which i s a quadrupole band, has l e v e l s o f X -» a n d t n e ^ band, which i s an octupole band, has l e v e l s o f % n~ I . 3 ~, 5" • • • , 17 The y and <y degrees o f freedom are separated by the a p p r o x i -mations y-Pcaicr f ^- r Sc^cr and - Cy4 <f w i t h 6" s m a l l , A p e r t u r b a t i o n s o l u t i o n i s found f o r <S ~ o , which i s the s o l u t i o n o f the r a d i a l equation o f a three dimensional o s c i l l a t o r , and i s doubly degenerate,' The energy o f both s t a t e s o f one phonon of e x c i t a t i o n i s ^r-MlSi: J? . The f i r s t order p e r t u r b a t i o n g i v e s a s e p a r a t i o n o f the y and 3. v i b r a t i o n band heads o f MF| r ( / I n ifs^Cy1 , Both the y and y v i b r a t i o n s have IKI - 2 , and r o t a t i o n a l energy l e v e l s o f c\.a, = _*H. I^ r (r+< ) - k 2 J are added t o the band head ° IT + + + energies. The quadrupole y v i b r a t i o n l e v e l s have 1 - 2 }3 } H and the octupole c^. v i b r a t i o n s t a t e s have X n - 2 ~} 3~j , 4 The A x i a l l y Asymmetric Model The asymmetric model r e l a x e s the r e s t r i c t i o n o f a x i a l symmetry, but assumes the r o t a t o r i s r i g i d , t h a t i s y - 0 , which reduces the k i n e t i c energy t o T s±8/p*)* -LB? £ V / ? W ( > - Z 7 r * ) g f* . (50) This development i s only f o r A-Z s u r f a c e s , w i t h f$i=p , There are now only f o u r g e n e r a l i z e d co-ordinates ( pj 62 ) , the me t r i c tensor being reduced t o G- ~ B2 O j o z IS o o 0 \ r2 £ £ O o ( I o \ o o o T 3 f f u / A 5 D The p o t e n t i a l er.- rgy i s assumed to be of the form o f yg v i b r a t i o n s about an e q u i l i b r i u m displacement /?«, , and i s g i v e n as V(p) - ± C2 Cp- . The r o t a t i o n a l energy i s g i v e n by equation (38a), Using the m e t r i c tensor o f equation (53)• the t o t a l Hamiltonian becomes \zB2[fJ Jf\ dfi/ HP Mr, Jc-^Ci-^V^J 2 J^u - 4 ^ ^ ^ ; ^ ) * ( 5 2 ) 18 Because ths operators LA only operate on the angles ^ J ^ - J , the s o l u t i o n i s separable i n t o (p, , P 3 ) - ^  (&•{) pL C/i) • Equation (54) separates i n t o 7 ? sc**a>-z*r**> '"J*L ' (53a) and ^ 8 2 ^ dp T dp J 2 2 p r ^Jz ^ L ^ (53b) where ^ ^ gi v e s the r o t a t i o n a l energies o f the asymmetric r o t a t o r i n u n i t s o f _____ , and N i s an o r d i n a l number. H6ZP2 The s o l u t i o n s o f equation (53a) are those o f a deformed asymmetric r i g i d r o t a t o r and have been c a l c u l a t e d f o r v a r i o u s values of y t L, and N . 3 2 , 3 3 Equation (53b) cannot be solved i n a c l o s e d form, and t h e r e f o r e a t r a n s f o r m a t i o n o f v a r i a b l e made, which reduces i t t o \ZB2 dp' </V ^ / < > ^ J C ; where ^1JA/ * s a g e n e r a l i z e d p o t e n t i a l , W[t) =-*(?L„ + 3.) +X(p-fi)Z • (54) C e n t r i f u g a l s t r e t c h i n g w i l l g i v e r i s e t o a new e q u i l i b r i u m p o s i t i o n , which i s found by minimi z i n g ^2 Jfi^ w i t h r e s p e c t t o /? . Hence The d i f f e r e n c e between the two e q u i l i b r i u m p o s i t i o n s , . i s due to the r o t a t i o n - v i b r a t i o n i n t e r a c t i o n . D e f i n i n g a "non a d i a b a t i c i t y o r s t i f f n e s s parameter'j^a and 2? - A' , transforms equation (55 a) i n t o Z f f *-.__) (55b) To s o l v e equation (53b) a value o f i s chosen, then equation (55b) i s s o l v e d ( t a k i n g p o s i t i v e r e a l r o o t s only) to g i v e a va l u e f o r 2, . The p o t e n t i a l energy, equation (54) i s expanded about p/'and only the 19 lowest non-vanishing term i s kept. A change o f v a r i a b l e i s made y- 2t (p-pj)) - j * £ y f - o C , which reduces equation (53b) i n t o the form * j|L* + 2o + / - -y j pf,*) = . ( 5 6 ) # The s o l u t i o n s P (p) become Djffi')') and are Weber's p a r a b o l i c c y l i n d e r funct ions3\ and J i s determined by the boundary c o n d i t i o n s t>v(-lf? Bt ) = O j (57) and i n g e n e r a l i s not an i n t e g e r . The energy eigen-values are (58) 2 where A ^  i s an o v e r a l l energy s c a l e parameter and the a d d i t i o n a l quantum number h , l a b e l s the p a r t i c u l a r v i b r a t i o n band. h - I f o r the ground s t a t e band and h - Z f o r the beta band. Values o f f o r h - / , and »-> - Z have been computed and t a b u l a t e d as 18 f u n c t i o n s o f Z . The ground s t a t e band head i s l a b e l l e d by L~ oJ A/='} h- ( • the beta-band i s l a b e l l e d by L-<^A/=/;h-Z , and the gamma- band head by L-ZjA/rZjh-' . Equation (53 a) g i v e s the l e v e l spacing o f a r i g i d r o t a t o r , which a u t o m a t i c a l l y Includes the ground s t a t e and Y band o f the symmetric model. The p band l e v e l s a r i s e from equation (57) when y «* becomes non zero. G e n e r a l l y the model i s employed by f i n d i n g a value o f y which w i l l g i v e the c o r r e c t r e l a t i v e spacing o f the 2 + ground and 2 + i ^ - l e v e l s from equation (53a), and then f i n d i n g a value o f w h i c h g i v e s the c o r r e c t energy o f the ° + and Z"+ l e v e l s o f the band from equation (57). A parameter search i s c a r r i e d out over a sm a l l range o f Y and y- to minimize the ro o t mean square d e v i a t i o n of the experimental energy l e v e l s from the t h e o r e t i c a l l e v e l s . 20 The negative p a r i t y s t a t e s are a l s o d e s c r i b e d i n t h i s model by c o n s i d e r i n g the h- 3 , or octupole s u r f a c e . The moments of i n e r t i a are those of equation (46a) as the same d i a g o n a l i z a t i o n was performed. The development now f o l l o w s almost completely the treatment o f the quad-rupole case. Two shape parameters f and are d e f i n e d b y ^ - SCcs1j ^ - f**"*^ , 3 5 which makes the moments o f i n e r t i a become: J (J> ~ <V 5 j1 (scty + ±f?s<:hj- cos y cos^y ) The Hamiltonian f o r the corresponding r i g i d r o t a t i o n s o f t h i s system i s The b a s i c d i f f e r e n c e between the octupole and the quadrupole system i s th a t the former does not have f r o n t to back symmetry, hence the 8t r e p r e s e n t a t i o n i s used, 77 I*-AA> - - ILI^O , Using the 8, r e p r e s e n t a t i o n means there w i l l be no l e v e l s o f L - o , one l e v e l f o r I- - ' , one f o r L - 2 , and two l e v e l s each f o r L-3 and H , The 1 v i b r a t i o n s are added to the r i g i d r o t a t o r i n the same way as the p v i b r a t i o n s were added t o the quadrupole r o t a t o r . ( 5 9 ) (60) 2 J 2 ( 6 1 ) The system i s quantized g i v i n g (62) The r o t a t i o n a l p o r t i o n separates i n t o ( 6 3 ) M 21 whose s o l u t i o n s are g i v e n i n reference 33. The s o l u t i o n of the v i b r a t i o n equation f o l l o w s e x a c t l y the steps o f the quadrupole case, equations (53) to (58) by r e p l a c i n g by £ , /< b y = ; . 7 --_ . The r e s u l t i s t o produce a band o f energy l e v e l s due to the ^  v i b r a t i o n s which separate from the r i g i d octupole l e v e l s as the r i g i d r o t a t o r i s softened by making non zero. 5 Electromagnetic T r a n s i t i o n P r o b a b i l i t i e s Electromagnetic t r a n s i t i o n p r o b a b i l i t i e s are a b e t t e r t e s t o f a model than the s t a t i o n a r y s o l u t i o n s o f the Schrodinger equation because they t e s t the model s t a t e f u n c t i o n s d i r e c t l y r a t h e r than j u s t the energy eigen-values. The t r a n s i t i o n p r o b a b i l i t y f o r the e m i s s i o n o f a photon o f energy ~hu; - ficA and angular momentum A i s ^f^o. 'O 2 * ( 6 5 ) w h e r e 6 Y V c ^ ) = _ L _ £ 9E '^*>| 2 • (66) B ( ^ ) i s c a l l e d the reduced t r a n s i t i o n p r o b a b i l i t y o f m u l t i p o l e o f order A , For an u n p o l a r i z e d system the reduced t r a n s i t i o n p r o b a b i l i t y i s taken as an average over the i n i t i a l s t a t e s and as a sum over the f i n a l s t a t e s . The y are the m u l t i p o l e operators o f the p a r t i c u -l a r electromagnetic t r a n s i t i o n . The type o f m u l t i p o l e operator i s determined by the s e l e c t i o n r u l e s , \l--t-f( 7/A7/ Ui | t and the p a r i t y c o n s i d e r a t i o n s 4»rr-C-M A f o r e l e c t r i c multipoles, O rr - - (-<\A f o r magnetic m u l t i p o l e s . The lowest m u l t i p o l e i s the e l e c t r i c monopole ^ o r E ( o ) , 22 Selection rules however, require that A L = 0. There can not be photon emission because a photon carries away one unit of angular momentum. Therefore, electric ?nonopole transitions only occur through the inter-nal conversion process. The interaction Hamiltonian for the internal conversion process 36 is given as, cf. o) =-e ffh<ef\± _ _±_ I e.> cl , J *"A re (67) where j>H is the nuclear density and \ec> and I e^> are the i n i t i a l and final electron state functions respectively. The transition probability is generally written as T(o) = SL fz where _TL is an atomic factor and a function of energy,-and f is the nuclear monopole strength parameter and i s , 37 9c <*T* (68) Q T - and Q are the i n i t i a l and final nuclear states functions respec-tively. CT is usually ~ 0.1 so only the fi r s t term of f> is taken. Converting the sum over the protons into an integration over the 38 nuclear volume and defining To as, j> - <f I T0\C> , gi> Lves, T0 " 5" M A ^ (69a) where -A = 2 for quadrupole surfaces and A = 3 for octupole surfaces. Transforming into the body co-ordinate system gives T 0 * - - HI (HT[ • ... j (69b) 23 Only t r a n s i t i o n s i n the quadrupole s u r f a c e s w i l l be considered, hence For the symmetric model J> i s g i v e n as 3>B K hp h L MX \ T0S'ihJft'i'M'K') , where the primes denote the i n i t i a l s t a t e s . (70) The f i r s t terra o f T0 does not make a c o n t r i b u t i o n because the angular momentum i s not the same i n the i n i t i a l and f i n a l s t a t e s . The second term corresponds to t r a n s i t i o n s between the beta-band and the ground s t a t e band and has a value f - ?2- e (2 pa)( ft I (71) For the asymmetric model J> becomes / - <<P CP) \ Tc\<MPV , > because 71 does not a c t upon the angular momentum s t a t e s . The term i s again the only term which c o n t r i b u t e s t o lowest order. Hence f becomes o As the only allowed t r a n s i t i o n s are between the beta and ground s t a t e (72) band heads, Z(f -^>c ) f?»f • D e f i n i n g dp } (73) 39 which can be c a l c u l a t e d n u m e r i c a l l y , ^ 7 and d e f i n i n g o where X - z?, p/p/, a l l o w s the monopole n u c l e a r s t r e n g t h parameter t o be g i v e n as (74) 24 (75) 40 and has been calculated numerically for several values of L. 41 Magnetic dipole M(l) transitions were shown by Davydov to 42 43 be forbidden in the symmetric model and Tamura et al, * J shows that they are also forbidden for the asymmetric model. The basic reason is that the M(l) operator is proportional to L which when acting on a state function LMK connects states of the same L and AK + 1, The collective models only have states of the same L with AK. - 0 or + 2, hence the M(l) transitions are forbidden. Electric dipole transitions, E(l), are described by an operator proportional to the electric dipole moment, which corresponds to the center of charge. As the charge is considered uniformly distributed throughout the nucleus, the center of charge coincides with the center of mass, which is taken at the origin and at rest. Therefore, the electric dipole moment and the E(l) transitions probability vanish. The most favorable transitions are the electric quadrupole tran-sitions, simply because the nucleus is itself basically an oscillating quadrupole. The electric quadrupole operator Q. is given as in the body fixed co-ordinate system, where the nuclear charge is (76) o assumed to have constant density fh - 3 1 * , This transforms into the laboratory frame of reference by 9 = Z P (&j <f>) 9 The matrix element (77a) 25 The angular momentum p o r t i o n o f the matrix element i s , f >-v dr'dj^' . (77b) An e x p l i c i t e x p r e s s i o n f o r the angular momentum s t a t e f u n c t i o n s o f the P (&;) t C-i) D (t>:) M-K 22 quadrupole s u r f a c e , I I ZL+I 1 £ A which i s v a l i d f o r both the symmetric and asymmetric models, y i e l d s , L H K->° ' < 4 ' K ' -I t ( - ^ 4 £ A C A4 [cU;2tfj KcZKf)+C(Qk;-2kf)J where 2o (77c) . 22 ) and the upper s i g n i s f o r t r a n s i t i o n s between s t a t e s o f p o s i t i v e p a r i t y and the lower s i g n f o r s t a t e s o f negative p a r i t y . Therefore 9j*U±*?az. and 92~- U**? + «_„ ) • 8 rr 8TT For A = 2 s u r f a c e s the quadrupole tensors become, (78a) 90u - 22____T p C o * y and 9 = U±*I pfesihy. (78b) For s i m p l i c i t y equation (77c) can be w r i t t e n as (79) where {" } i s the term i n s i d e the bracket •(" } i n equation (77c), The matri x element <-f/ 4o F 11 > - < < f r f f y r ) K ^ * V / - j M ^ I <J>;(p,*i> can now be w r i t t e n as <jlCQeli>= (-'UfzuTi1 C(LiZLi--f+i-*-M,)<$Ap,*)\{'}\$.(?)rt > (80) 26 18 Rose has shown t h a t which s i m p l i f i e s the reduced t r a n s i t i o n p r o b a b i l i t y t o and only 9o0 a n<i 9zo °^ "C } a c ^ upon the i n t r i n s i c s t a t e f u n c t i o n s lp(p,ir)> . The i n t r i n s i c s t a t e f u n c t i o n s \<t>(pJ¥)> , which are s o l u t i o n s o f the d i f f e r e n t i a l equations i n the v i b r a t i o n a l degrees of freedom o f the v a r i o u s models, can be checked w i t h experiment by comparing the branching r a t i o s o f t r a n s i t i o n s from the same i n i t i a l s t a t e . From equation (65) i t can be seen t h a t the r e l a t i v e i n t e n s i t y o f two photon t r a n s i t i o n s o f energy E and E ' from the same i n i t i a l s t a t e to the f i n a l s t a t e s L+ and L^/ i s ZA + I T(*)Li->L* = (k\ . d(A]L.c~>L<) . T ^ t - ^ WW B(*}^->^) ( 8 2 ) ZA+l 2A + 1 The r a t i o - (—) a r K* * s ' c n o w n *" r 0 R l the energy of the t r a n s i t i o n s , hence B(A) -» i f ) can be d i r e c t l y compared w i t h experiment. For the symmetric model one can assume p and y are constants pa and o° , then only t r a n s i t i o n s w i t h i n bands are allowed. For example, the reduced t r a n s i t i o n s p r o b a b i l i t y f o r t r a n s i t i o n s w i t h i n the ground s t a t e band becomes B(E2- Zf j ) -9o* A»f £ Ci*(-^30*(-')Li]cUiz^j ooo)J (83) 27 and the reduced branching r a t i o becomes simply as C (LC2L± j o oo) (84) = / However, t h i s s i m p l i f i e d approach does not agree w e l l w i t h experiment, except i n a few t r u l y a d i a b a t i c cases. One use f o r t h i s s i m p l i f i e d approach i s t o c a l c u l a t e the r a t i o , 38 Using the approximation o f equation (77)• (85a) B(E2j op -> o 3 j =• 3i ez * 'i7TZ\f^c~1 (86) and hence Xfop -> o}) = v p*. The r a t i o X(ofi-> o^) c a n be r e l a t e d t o experimental q u a n t i t i e s by s u b s t i t u t i n g -j2- T(BQ\L-^L) a n d 8TT (A w) kZA" i n t o equation (86a) which then becomes >; o -? o ) \ (85b) where Fy i s the gamma t r a n s i t i o n energy i n Mev. T r e a t i n g the i n t e r a c t i o n s between r o t a t i o n and v i b r a t i o n bands 13 as p e r t u r b a t i o n s shows t h a t there i s some band mixing. To c o r r e c t the t r a n s i t i o n p r o b a b i l i t i e s the "pure" v i b r a t i o n s t a t e f u n c t i o n s o f the symmetric model are modified w i t h admixtures o f other band s t a t e 44 f u n c t i o n s , Lipas shows t h a t the modified s t a t e f u n c t i o n s would r e s u l t i n the reduced t r a n s i t i o n p r o b a b i l i t y being where 5 0foji T^ (-5'4 fK f) i s the reduced t r a n s i t i o n p r o b a b i l i t y between "pure" bands u s i n g the approximation o f equation (83) 28 and "ffcfij 2^ 2-p^£-CjL+) i s a c o r r e c t i v e f a c t o r t h a t i s a f u n c t i o n of the admixtures o f the s t a t e f u n c t i o n s . 15 Hamilton and Marshalek J % put the admixture i n t o the form loo L i^\o>'~ \ oo Lr*o> - £p fp(c) \ \oLt^ay - £y-fy(c) fo z > (88a) f o r the ground s t a t e band and \ \ O L M O > ' - e/,fpfr)looLMo> + l<oLMo> +Znfr(i-)\o{LrAz> (88b) f o r the beta band, and | 0 / i M 2 > / = J_[l + (-<)LJ f^-fJO l o o i M o ) Z ~± ll+(-<)L] fpy olMo> f [ o ( i M Z > . (88c) The ~fpfL) and iy(-L) are the s p i n dependent terms o f the p e r t u r b a t i o n i n t e r a c t i o n between the beta band and the ground s t a t e band and the 13 gamma band and the ground s t a t e band r e s p e c t i v e l y , and x L(Li-l)- f r C L \ ^ l z ( L ' i ) ( L ) ( L t ^ ( L f Z ) Y . (88d) The^j , ty , and £py are the p a r t s o f the p e r t u r b a t i o n amplitude de-pending on the i n t r i n s i c v a r i a b l e s \^>(?,^y o n l y , and should De constant f o r a l l changes o f L, A mixing parameter f o r each band i s d e f i n e d as ZP " " ^ 3 <( hp - O t hy-o { <y*oo I hp-Q)^y-°> j < l o ( <?oa l a o > $23 €y <OOf QCQ / Oo> } and (89) _ <IQI 9 o » | O Q > m The assumption i s made t h a t <°o I 9 o o | oo> - < I o I 9 o J / o> - <"©( f 9 ^  | o / > ; Thus a l l i n t r a - b a n d t r a n s i t i o n s have the same I n t r i n s i c f a c t o r , and the o v erlap between the y v i b r a t i o n band and the beta v i b r a t i o n band i s n e g l i g i b l e , i e . [<lo[ 9 ^ / ( 0 > [ ^ {<°( | 9 2 o [o o> / . 29 Hamilton u s i n g the assumptions o f equations (88) and (89) has produced a t a b l e f o r -f ( _ ^ ? ^ L £ j cf) r S(ez; U k j - . L ^ ) f o r a wide range o f t r a n s f o r m a t i o n s . Table 2-2 Using experimental branching r a t i o s , g (£2; Lt- Ay —>Lfkf) ^ Ba (FZ) lc fry -»lf kf) • {(fyiPj 2-py j L c t f ) t and Table 2-2 one can c a l c u l a t e R, (eZj L{ k- -9ly^f).-f(?tl*P, 2-0 fj <+' V ) values o f F^j , , and -y$>- which should be constant f o r a l l the quadrupole t r a n s i t i o n s v i t h i n the three v i b r a t i o n bands. TABLE I I Band Mixing C o r r e c t i o n F a c t o r s K - 0 o - 2 i ( E 2 J ic}z - ^ / ^ O Bo(e2j PJ^'JO 0 ; L-Z L - / L L+ 1 L*z L L L L L r ' 2 [ 1 + (L*2)?^]Z f 1+ 2 UU + t} ?pyJZ [l-(ZL + l) Z}.+(L+l)(L + 2 ) Z p J 1^ 142 (ZL-i) Ze~ (L-Z)U-ft J J Z f , + l(L-i)(L + Z) _ 7 ^ J ? [l-2(zL+3)?fr-(i. + 3)(i. + *)JAi. J Vfhere 7., - < \<oi\ oo>| Z ? 1 B0 (E2; 0* -•> 22) £ p r 6 Kiot q^l °o?l 6 Bo(ez) o--702) For the asymmetric model < <fy 0?,>-J / 9<>o ( 4st" > " 3 %e<*s)r<i%(fi)\p\<p-(p)> a n d 07T * 30 The f a c t o r < " I <Pt-0»J > , which can be c a l c u l a t e d n u m e r i c a l l y i s the o v e r l a p o f t h e v i b r a t i o n a l wave f u n c t i o n , \%W>-y^D@y))A/i and J4>.^J> = / '*(j>(lf? j) J ( s e e equation (62) ) . The overlap becomes ab S e t t i n g J ^ , i 7 and ^ - -y _y /»_ ( 90a ) , the overlap be w r i t t e n as _ 2 ( 90b ) where ( 9 0 c ) and must be c a l c u l a t e d n u m e r i c a l l y . Remembering the d e f i n i t i o n o f N j , the square o f the overlap can be g i v e n as ( 91 ) 46 T c and has been c a l c u l a t e d f o r v a r i o u s t r a n s i t i o n s . Equation ( 7 7 c ) becomes </|'9 F lc> ^ f<L,t^J DZ | _ - M ; > f o r i -V + -Mi X ^ < ^ M , i D ^ J ) 2 K c ^ - > ) [ 11_L__!) 5* 2 Because the v i b r a t i o n o verlap i n t e g r a l does not depend on AV , o r y<x the reduced t r a n s i t i o n p r o b a b i l i t y can be w r i t t e n as ( 92 ) B(-9^*^11+)- B^(BZ)v< h t - - ? _ f /v^  * f ) } ( 93 ) 31 where B^i^Zj^-;K/--^if^f) a r i s e s from the summation of the squares o f the f i r s t two f a c t o r s o f equation (92) overM (., , a n d , and i s dependent on the r o t a t i o n s t a t e s only. A constant fair g &o ) \ V TT J can be f a c t o r e d out by d e f i n i n g 6 ( < • AA ~->Lf Ay }= f B(E2 • * t- Vt- ~ * L f H f ) -Isj-etf) (94) The reduced t r a n s i t i o n r a t i o s then become B(E2\Zj~?L+ ) _ o f f ? ; £<• V,- -~>l+A/* ) , which can be checked w i t h experimental v a l u e s . L The r a t i o becomes (95) • - ? i t / l « > • b C e ^ j t-pHp-'pt-^/v^) (96) and can be compared w i t h the experimental e x p r e s s i o n 3 2 CHAPTER III The Gamma Singles Spectroscopy 1 General Considerations In the previous chapter, two specific models were discussed in detail. To check the validity of these models for the daughter nuclei 152 of Eu, the decay scheme and the transition mechanisms must be deter-mined. This can be most efficiently carried out by measuring the gamma-ray emissions from the two nuclei. Precise values of the gamma-ray and internal conversion inten-sities of a transition can often enable one to determine the multi-polarity of the transition. The spin and parity values of a state can be deduced via selection rules from knowledge of the multipolar!ty of the transitions to and from the state. A few selection rules are listed below, a) The angular momentum of the emitted photon, L , must be greater than the difference between the total angular momentum of the i n i t i a l and final states, and less than the sum of the total angular momentum of the i n i t i a l and final states, or / Zi -Xf I - L 6 I rt- -»• r.^  I . (98) b) Parity must be conserved. Therefore, only even parity emission is possible between states of like parity, and only odd parity emission is possible between two states of disimilar parity. The parity of electric raultipole radia-tion of order, L , is T7 F- (~[)L , and for magnetic multipole radiation is n * = - ( - i i L . 33 c) For emission of energy up to 2 Mev, the p r o b a b i l i t y o f emis-s i o n i s only s i g n i f i c a n t f o r the lowest order o f L , w i t h the exception t h a t E2, Ml mixtures are f a i r l y common. d) The magnetic r a d i a t i o n p r o b a b i l i t y i s l e s s than the e l e c t r i c r a d i a t i o n p r o b a b i l i t y o f the same L , by a f a c t o r o f between 0.4 and 0.001. e) For c o l l e c t i v e models, another s e l e c t i o n r u l e i n v o l v i n g the r o t a t i o n a l quantum number, K, e x i s t s , i . e . L>, Ikf-kfl . (99) As t o t a l c o l l e c t i v e model s t a t e s c o n t a i n mixtures o f r o t a t i o n -a l s t a t e s w i t h v a r i o u s K v a l u e s , t h i s s e l e c t i o n r u l e h i n d e r s , r a t h e r than f o r b i d s c e r t a i n t r a n s i t i o n s . A measure o f the K hindrance i s d e f i n e d as J - IKt- - l<4 I - L . (100) Besides t r a n s i t i o n s which are hindered o r f o r b i d d e n , there are t r a n s i t i o n s which are enhanced due to the c o n f i g u r a t i o n o f the nucleus. 152 For deformed n u c l e i , l i k e J Sm, E2 r a d i a t i o n s are expected t o be en-hanced because p a r t o f the nuclear matter, and hence the e l e c t r i c charge, are o s c i l l a t i n g quadrupoles, 2 The Lit h i u m D r i f t e d Germanium Detector The measurement o f the energies and i n t e n s i t i e s o f the photon emission i s obtained best by a l i t h i u m d r i f t e d germanium G e ( L i ) d e t e c t o r , as i t s energy r e s o l u t i o n i s an order o f magnitude b e t t e r than a s c i n t i l l a t i o n p h o t o m u l t i p l i e r d e t e c t o r . A spectrum from monoenergetic y -rays i s shown i n Figure 3-1. The sharp peak i s c a l l e d the photo peak, r e s u l t i n g from complete t r a n s -f e r o f energy to the d e t e c t o r by p h o t o e l e c t r i c a b s o r p t i o n . The low 34 1 0 64 6-k o C/2 1 .8 .6 . U o o . 2 .1. 8 0 5U Mn Photo Peak 8 3 5 keV Conrpton Edge 6h0 keV 1 2 0 1 6 0 CHANNEL NUMBER 2 0 0 iho 3 2 0 FIGURE 3 - 1 Gamma-ray Spectrum Of Monoenergetic Photons From ^Mn 360 35 energy background i s due to Compton s c a t t e r i n g i n which only p a r t o f the energy o f a photon i s t r a n s f e r e d t o the d e t e c t o r . The energy t r a n s f e r e d t o the d e t e c t o r i n a s i n g l e i n t e r a c t i o n w i t h the c r y s t a l i s g i v e n as E = / . (101) where -m0 i s the r e s t mass o f the e l e c t r o n , C ^ - i s the i n c i d e n t photon energy, and &• i s the angle through which the photon s c a t t e r s . High energy gamma-rays can a l s o produce peaks called, double and s i n g l e escape peaks as w e l l as photo peaks, by the jjrod u c t i o n o f a p o s i t r o n -negatron p a i r . The s i n g l e escape peak i s .511 Mev l e s s than the photo peak, due t o the escape from the d e t e c t o r o f one of the two ,511 Mev photons produced i n the a n n i h i l a t i o n o f the p o s i t r o n w i t h an e l e c t r o n . The double escape peak i s 1.022 M-.-w l e s s than the photo peak where both a n n i h i l a t i o n photons escape the d e t e c t o r . The response o f the d e t e c t o r t o a f l u x o f ^ - r a y s i s the sum of a l l three e f f e c t s . Besides the Compton background i n t e r f e r i n g and masking weak peaks, peaks must a l s o be checked f o r s i n g l e or double escape peaks. In l a r g e r d e t e c t o r s , some o f the Y -rays from Compton s c a t t e r i n g and p a i r p r o d u c t i o n do not escape, but g i v e t h e i r energy p h o t o e l e c t r i c a l l y to the d e t e c t o r and produce c o n t r i b u t i o n s to the photo peak. For y - s i n g l e s spectroscopy, the l a r g e s t d e t e c t o r a v a i l a b l e i s used because i t has more e f f i c i e n c y f o r a l l purposes, and because i t i n t e r a c t s w i t h a g r e a t e r p r o p o r t i o n o f Compton and a n n i h i l a t i o n pho-tons. Hence, the r e l a t i v e s i z e o f the photo peak to Compton background i s i n c r e a s e d . 36 3 The G e ( L l ) Gamma-ray Spectrometer The Ge(Li) gamma-ray spectrometer c o n s i s t s o f three b a s i c u n i t s t (a) the d e t e c t o r , (b) the s i g n a l e l e c t r o n i c s , and (c) the pulse height a n a l y z e r . The d e t e c t o r converts the energy o f the detected photons i n t o charge. I n t h i s i n v e s t i g a t i o n a Nuclear Diode CR -30 Ge(LO) t r a p e z o i d a l d e t e c t o r o f 30 c c , volume and 20 p f capacitance w i t h an op e r a t i n g b i a s o f -1250 v o l t s was employed. The s i g n a l e l e c t r o n i c s convert the charge pulses from the detec-t o r i n t o v o l t a g e p u l s e s , which are subsequently shaped and a m p l i f i e d f o r the pul s e height a n a l y z e r , A Tennelec TC 135 M p r e a m p l i f i e r con-v e r t s the charge pulses from the d e t e c t o r i n t o v o l t a g e pulses which are shaped and a m p l i f i e d by a Tennelec TC 203 BLR a m p l i f i e r . An Ortec 408 b i a s e d a m p l i f i e r s e l e c t s the range o f pulses t o be analyzed, and an Ortec 411 pulse s t r e t c h e r lengthens the pulses to g i v e the pulse h e i g h t a n a l y z e r s u f f i c i e n t time t o analyze. The p u l s e height a n a l y z e r , o r raulti channel a n a l y z e r , MCA, s o r t s the incoming pulses i n t o v o l t a g e ranges and counts the number o f pulses f a l l i n g w i t h i n each range, A voltage range, corresponding to a range of photon energies, results. The MCA used i n t h i s i n v e s t i - " g a t i o n was a V i c t o r e e n PIP, 400 channel a n a l y z e r , F i g u r e 3-2 i s a blo c k diagram o f the spectrometer, 60 F i g u r e 3 -3 shows a p o r t i o n o f the spectrum o f Co wi t h the 1332 keV peak. The r e s o l u t i o n o f the spectrometer was 5 ,0 keV FWHM on the 1332 keV peak o f ^ C o , and the peak t o Corapton r a t i o was 9.1. Experimental Procedure The 1 5 2 ^ s o u r c e K a s prepared from two m i l l i g r a m s o f Europium Detector Preamplifier Amplifier Nuclear Diode Tennelec — Tennelec CR - 30 TC 13£M TC 2 0 3 BCR \ Pulse Height Analyser Pulse Stretcher Ortec 411 Biased Amplifier Ortec 1+08 Victoreen Pip 400 FIGURE 3 - 2 Schematic Diagram of Ge(Li) Gamma-ray Spectrometer 38 1173 keV 1332 keV FWHM 5 .0 keY PWHM 5.O kgv] 80 120 160 CHANNEL NUMBER 200 2i;0 280 320 FIGURE 3-3 Part of 6 oCo Spectrum 39 oxide, which was enriched t o 99# ^ I E U , The Europium oxide was subjected t o neutron i r r a d i a t i o n by the Union Carbide Corporation to produce ^^Eu. oxide which was d i s s o l v e d i n IN HCI to produce EuCl^. The Europium c h l o r i d e s o l u t i o n was concentrated by evaporating some o f the water and "source c a r d s " were prepared p l a c i n g drops o f v a r i o u s s i z e s and concen-t r a t i o n s o f the source s o l u t i o n onto one i n c h diameter c i r c l e s o f computer cards. A f t e r evaporating the s o l u t i o n , a l a y e r o f Scotch tape was placed over the card to prevent source m a t e r i a l from escaping. The source cards were easy t o mount and v i r t u a l l y t r ansparent to the photons o f the energy measured. Because o f the s m a l l c a p a c i t y o f the MCA and the l a r g e energy range to be surveyed, the spectrum was taken i n s e v e r a l o v e r l a p p i n g stages. Each energy segment was approximately 200 keV and was arranged to i n c l u d e a t l e a s t three c a l i b r a t i o n energies. Each run was set up by p l a c i n g c a l i b r a t i o n sources o f known energy, from the I n t e r n a t i o n a l Atomic Energy Agency, before the detec-t o r and a d j u s t i n g the a m p l i f i e r gains so the lowest c a l i b r a t i o n peak appeared approximately a t channel number 40 and the highest a p p r o x i -mately a t channel number 360, This process ensured t h a t the most l i n e a r r e g i o n of the MCA was employed. The Europium source was placed before the d e t e c t o r , and a spectrum was taken of the Europium and the c a l i b r a t i o n sources simultaneously. The more in t e n s e Europium peaks became secondary standards by l i n e a r i n t e r p o l a t i o n between the primary standards. Table I I I l i s t s the c a l i b r a t i o n sources used. The l i n e a r i t y o f the system allowed i n t e r p o l a t i o n between two peaks o f known energy to w i t h i n .1 keV. Once the energy of the more i n t e n s e pe?.ks was determined, longer 40 Table JU Peaks Used f o r Energy C a l i b r a t i o n R adionuclide Energy (keV) Radionuclide Energy (keV) 241, a An 59.5^3 + 0.015 207 B i b 569.63 + 0.08 203,, a kg 72.873 + 0.001 R d T h ( 2 0 8 T i ) b 583.139 + 0.023 20\ a 82.5 + 0.2 R d T h ( 2 U B i ) b 609.37 + 0.016 5 7 C o a 121.97 + 0.03 ^ C s a 661.635 + 0.076 5 7 C o a 136.33 + 0.03 a 834.81 + 0.03 R d T h ( 2 1 2 P b ) b 238.61 + 0.01 88Y a 898.04 + 0.04 R d T h ( 2 l 4 P b ) b 241.924 + 0.030 2 0 7 B i b 1063.58 + 0.06 1 1 3sm a 255.06 + 0.08 6 0 C o a 1173.23 + 0.04 279.191 + 0.008 2 2 N a a 1274.55 + 0.04 R d T h ( 2 l 4 P b ) b 295.217 + 0.039 6 0 C o . a 1332.49 + 0.05 R d T h ( 2 l 4 P b ) b 351.992 + 0.062 R d T h ( 2 2 8 T h ) b 1592.46 + 0 . 1 n 3 m m a 391.70 + 0.05 R d T h ( 2 l 4 B i ) b 1764.45 + 0.22 2 2 „ a Na 511.006 + 0.002 2 0 7 B i ° 1770.00 + 0.07 85^ a 511.006 + 0.002 88y a 1836.13 + 0.04 Energy va l u e s obtained from I n t e r n a t i o n a l Atomic Energy Agency D Energy values obtained from Table o f Isotopes c S. H. Brahmavar and J . H. Hamilton, N.I.M. 69(1969) 353. 41 Table IT Peaks Used for Intensity Calibration (From International Atomic Energy Agency) Isotope Strength a Half-Life Energy % Per (uc) (keV) Disintegration 24LAm 10.38 + 0.0? 432.9 + Years 0.8 59.5 35.9 + 0 .6 n 3 S n b 115.2 + Days 0.8 255.1 2 . 0 + 0.2 n 3 l n c 1.657 + Hours 0.002 391.7 64.2 + 0 .6 5?Co 11.43 + 0.10 271.6 + Days 0.5 122.0 136.3 85.0 11.4 + + 1.7 1.3 2° 3Hg 20.25 + 0.20 46 .8 + Days 0.2 72.9 82.5 279.2 9.7 2 . 8 81.55 + + 0.5 0.2 0.15 2 2Na 9.16 + 0.09 2.602 + Years 0.005 511.0 1274.6 181.1 99.95 + + 0.2 0.02 10.35 + 0 o18 30.5 + Years 0 . 3 661.6 85.I + 0 . 4 10.96 + 0.07 312.6 + Days 0.3 834.8 100.00 6 0Co 10.57 + 0.06 5.28 + Years 0.01 1173.2 1332.5 99.87 + 99.999 + 0.05 0.00: 8 8 y 10.85 + 0.13 107.4 + Days 0.8 511.0 898.O I836.I 0.40 9 1 . 4 9 9 . 4 + + 0.02 0 .7 0 .1 a Strength on January 1, 1970, 00.00 Universal Time 0 In equilibrium, the disintegration rates of and are probably equal. 0 The number of 393 keV^-rays emitted was (4.22 + 0.06) x lO^ per second. 42 runs of approximately twenty-four hours were taken with Europium only to observe the weaker peaks. As the source cards were taped to an aluminum frame at a fixed distance from the detector, the detector geometry was constant for a l l runs. After the long runs, short runs with calibration sources were repeated to check for drift in the electronics. Background runs with a l l known sources removed, were taken to check for stray background peaks which could show up in the long runs. Peaks at 1592 keV and 182? keV were found to be background peaks, and. the intensities of the peaks at 1458 keV and 1770 keV had to be reduced because of a background component. The relative efficiency of the detector was calibrated by timed runs with calibration sources of known intensity in the same position as the Europium source (see Table IV..). As the Intensity calibration sources were used within two weeks of calibration, uncertainty in in-tensities due to source decay was negligible. An efficiency-vs-energy relationship was determined by fitting a polynomial by-least-square f i t to the efficiency data. The fitted function was within the 3% uncer-tainty of the calibration data (see Figure 3=4). 5 Peak Fitting The peaks in a y -ray spectrum contain the relevant gamma-ray data. The energy of a gamma transition is determined by the position of the peak, and the intensity by the area of the peak. Hence, the accuracy of energy and intensity measurements depend upon the accuracy of measuring peak positions and areas. Positions and areas of peaks can be measured manually, simply by plotting the appropriate peaks, However, in a complicated spectra, 88 Y 44 there are o f t e n many peaks f o l d e d i n t o one another. The u n f o l d i n g o f peaks r e q u i r e s a machine c a l c u l a t i o n w i t h a known shape f o r each peak. I d e a l l y the shape of a peak should be t h a t o f a P o l s s o n d i s t r i b u t i o n , but i n p r a c t i s e , the peak shapes t u r n out t o be t h a t o f a Gaussian d i s t r i b u t i o n w i t h a low energy t a i l and a step i n the background, (See f i g u r e 3-5). A method u s e f u l f o r peak p o s i t i o n i s to assume a Gaussian shape and t o ignore the low energy t a i l . However, when u n f o l d i n g two peaks, the low energy t a i l o f the h i g h e r energy peak can be i n c l u d e d i n the area o f the low energy peak. Thus, the lower energy peak's area i s over estimated. In t h i s i n v e s t i g a t i o n , the peak shape was approximated more c l o s e l y by i n c l u d i n g two s m a l l e r s a t e l l i t e Gaussians on the low energy s i d e o f the main Gaussian and an arctangent step f u n c t i o n on a l i n e a r background. The equation o f the f i t t i n g f u n c t i o n was where Y i s the number of counts and X i s the channel number, to P^ are v a r i a b l e s t o be f i t t e d by a l e a s t - s q u a r e f i t , P 1 i s the i n t e r c e p t v a l u e and i s the slope o f the l i n e a r background, P^ i s the height o f the step i n the background. The height o f the main Gaussian peak i s g i v e n by P^, and the p o s i t i o n by P^, P 5 i s the width parameter of the main peak, R^ t o R^ a r e f u n c t i o n s o f energy. The s i z e o f each s a t e l l i t e peak, r e l a t i v e to the main peak i s g i v e n toy R^ and R^and and R^ determine the s e p a r a t i o n o f each s a t e l l i t e peak from the main peak. A, B, and C are constant?. The procedure used to determine the optimum parameters was as r[—<n Ij i n P6 I (102) 45 CHANNEL NUMBER FIGURE 3-5 Least Square Fit Of A 1332 keV Peak Of D UCo The Circles Indicate The Experimental Counts. The Solid Line Is The Total Fitted Function And The Broken Lines Are The Components Of The Fitted Function. 46 f o l l o w s . F i r s t estimates of the constants A, B, and C were made, then peaks over a wide energy range were f i t t e d , a l l o w i n g the R parameters to optimize f o r best f i t s . Then A, B, and C were re a d j u s t e d and the procedure repeated. This procedure was continued u n t i l no f u r t h e r improvement i n peak f i t t i n g was obtained. The values f i n a l l y adopted were A = 5 . 0 , B = 3 . 0 , and C = 1 . 5 . With these values the f r e e - f l o a t i n g parameter R-j was found t o have values which showed no d e t e c t a b l e energy dependence, while R^ showed a l i n e a r dependency. These dependencies are i l l u s t r a t e d i n Figure 3 - 6 a . The f u n c t i o n s of R- and R^ deduced were R_ = 0 ,300 , and (103c) % - 0.344 + 1.43 x 1 0 ~ 4 x E, (103d) where E i s the approximate energy o f the peak i n keV. R^ and R^  showed no rec o g n i z a b l e energy dependent trend because o f s c a t t e r . However, when the values o f R^ and R^ were taken from the s t r a i g h t l i n e v alues o f equation (103), and the peaks r e f i t t e d , the energy dependence o f R^ and R- became more evident as shown i n f i g u r e 3 - 6 b f where the l i n e a r f u n c t i o n s are R 1 = 0 .0 + 9 x 1 0 " 5 x E, and (103a) R 2 = 1.80 - 4 .50 x 1 0 " 4 x E. (103b) The peaks were r e f i t t e d w i t h the R parameters f i x e d by equations 103 and the f i t t e d peaks superimposed on the experimental peaks to ensure acceptable f i t s . F i g u re 3-5 shows a f i t o f a c a l i b r a t i o n . peak. The lea s t - s q u a r e program was capable o f f i t t i n g up to s i x separate com-pound peaks a t a time. However, i n p r a c t i s e , we were unable to f i t more than f o u r peaks a t once because o f background u n c e r t a i n t i e s . The p o s i t i o n o f a f i t t e d peak was taken as the p o s i t i o n , P^ of the main Gaussian, and the area was taken as the sura o f the areas 47 .6 .8 .2 I T-.6 .4 .4 1.2 1.4 1.6 1.8 .6 .8 1.0 ENERGY (Mev) Figure 3-6a Graph Of Ranges of "R" Parameter Values Obtained From Least Square Fit of Calibration Peaks. Straight Line is Linear Function Chosen 48 Function Chosen I Function Chosen I .8 1.0 1.2 1.1+ 1.6 1.8 FIGURE 3-6h Ranges of Yalues of Parameters R_^  and Obtained From Least Square Fit of Calibration Peaks. R-i and R^  Yalues Fixed by Equations lOi+c and lOl+b. 49 of the three Gaussians, The uncertainty of the position was taken as the uncertainty in fitting the position parameter which was calculated "by the least-square f i t . The area uncertainty was taken as the sum of the uncertainties in the peak height, P^ , the peak width, and in the background parameters, P^ , P2, and ?y The background error was usually the greatest contributor to the uncertainty of the area, 6 Results A summary of the gamma-ray singles data obtained in this work is given in Table ? , taken from spectra of which some examples are 17 shown in Figures 3 -7a to 3 -7h . The data of Riedinger et al ' and Varnel et a l 1 ^ are both included in Table V for comparison. The intensities of 121.73 keV peak in Figure 3-72 has been re-duced by one percent and that of the 1004.9 keV peak in figure 3-7e by 215S to correct for the 123,07 keV, and 1004,75 keV peaks of the contami-nant 1^i|Bu, The transitions of 2 5 1 . 5 , 315 .2 , 3 2 4 . 7 , 329.4, 4 1 6 . 0 , 5 2 0 . 7 , 5 3 4 . 9 , 7 9 4 . 6 , 9 2 6 , 5 , 1348.6, and 1363.9 keV were found by both Riedinger et al and this investigation, but not by Varnel et a l . However, the peaks of 2 1 2 . 4 , 6 7 1 . 3 , 7 6 9 . 3 , 9 8 9 . 8 , 1122.9 , 1171.0 , 1233.5 , 1292.6 , 1433.7, 1537.4, 1608.2 and 1643.4 keV proposed by Riedinger et al could not be observed here. The 352.0 keV peak (see Figure 307b) has not been reported elsewhere. Table V 152 Energies and Relative Intensities of the tf-Rays Emitted in the Decay of Eu ENERGY (keV) RELATIVE INTENSITIES Present Work Varnell a et al"^ Riedinger et a l ^ Present Work Varnell et al^* Riedinger et al 121.73 + 0.05 121 .8 121.77 + 0.08 102 + 3.7 116 +4 108 + 5 212.4 + 0.6 .067+ 0.029 244.60 + 0.05 24.4.7 244.69 + 0.08 28.8 1.0 29.0 + 0.9 28.2 + 1.4 251.5 + 0.3 251.7 + 0.6 0.18 + 0.07 0.26 + 0.04 271.2 + 0.1 271.0 271.1 + 0.6 0.33 + 0.03 0.27 + 0.02 0.28 0.04 275.2 + 0.3 275.4 275.6 + 0.6 0.15 + 0.03 0.12 + 0.01 0.11 + 0 .03 295.9 + 0.1 295.9 296.0 + 0.3 1.65 + 0.07 1.60 + 0.04 1 .51 + 0.09 315.2 + 0.1 315.1 0.3 0.23 + 0.09 0.17 + 0.03 324.7 + 0.7 325.0 + 0.3 0.24 + 0.15 0.26 + 0.03 329.4 + 0.1 329.4 + 0.3 0.41 + 0.41 0.44 + 0.04 344.34 + 0.05 344.2 344.22 + 0.08 100 100 100 352.0 + 0.1 367.6 0.18 + 0.03 367.75 + 0.05 367.7 + 0.2 3.14 + 0.13 3.16 + 0.09 3.15 + 0.18 411.1 + 0.1 410.9 411.11 + 0.08 8.09 + 0.30 8.25 + 0.16 8.05 + 0.4 416.0 ± 0.1 416.2 + 0.3 0.39 + 0.06 0.38 + 0.04 443.8 + 0.1 b 443.8 b 443.95 + 0.08 b 10.1 + 0.4 10.4 +0.6 10.3. + 0.6 444.1 + 0.1 b 444.1 b 444.0 b 1.3 ± 0.4 1.14 ± 0.12 0.9 + 0.3 488.7 + 0.1 488.4 488.7 + 0.2 1.53 + 0.09 1.49 + 0.04 1.48 + 0.09 494.2 + 0.2 493.6 493.7 + 0.4 0.12 + 0.08 0.11 + 0 . 0 3 0.09 + 0.04 Table V (continued) ENERGY (keV) RELATIVE INTENSITIES Present Work Varnell et al Riedinger et al Present Work Varnell et al Riedinger et al 503.7 + 0.1 503.3 503.5 + 0.2 0.58 +_ 0.09 0.55 + 0.02 0.55 + 0.03 520.7 520.2 + 0.2 0.22 + 0.07 0.18 + 0.04 534.9 + 0.2 534.2 + 0.2 0.15 + 0.04 0.16 + 0.04 564.1 + 0.1 563.8 564.0 + 0.2 1.93 + 0.14 1.79 + 0.05 1.87 + 0.15 566.6 + 0.2 566.3 566.8 + 0.4 0.42 + 1.1 0.45 + 0.03 0.41 + 0.1 586.5 + 0.1 586.0 586.3 + 0.3 1.89 + 0.14 1.72 + 0.05 1.62 + 0.21 656.5 + 0.1 656.5 656.5 + 0.2 0.63 + 0.11 0.55 + 0.03 0.46 + 0.05 671.3 + 0.7 0.046+ 0.021 675.05 + 0.1 b 674.7 674.7 + 0.2 b 0.42 + 0.10 0.65 + 0.06 0.30 + 0.08 675.1 b 674.7 + 0.2 b 0.38 + 0.10 0.36 + 0.09 678.7 + 0.1 678.6 678.6 + 0.2 1.41 + 0.12 1.71 + 0.11 1.61 + 0.12 688.9 + 0.1 688.7 688.6 + 0.2 3.15 + 0.17 3.03 + 0.06 3.03 + 0.17 713.5 + 0.3 712.9 713.4 + 0.2 0.38 + 0.08 0.33 + 0.03 0.27 + 0.07 719.8 + 0.1 719.4 719.3 + 0.2 1.25 + 0.10 1.19 + 0.04 1.11 + 0.10 765.1 + 0.3 765.0 + 0.3 0.47 + 0.40 0.64 + 0.11 769.2 769.3 + 0.3 0.26 + 0.02 0.29 + 0.08 778.8 + 0.1 779.1 778.84 + 0.09 48.6 + 1.6 46.5 + 0.4 46.6 + 2.3 794.9 + 0.4 794.6 + 0.6 0.08 + 0.08 0.15 + 0.04 810.6 + 0.1 810. 810.4 + 0.2 1.20 + 0.10 1.17 + 0.04 1.08 + 0.09 840.9 + 0.2 841.8 841.4 + 0.2 0.73 + 0.09 0.58 + 0.03 0.59 + 0.08 867.38 + 0.05 867.7 867.32 + 0.09 16.2 + 0.7 14.9 + 0.2 15.0 + 0.7 Table V (continued) ENERGY (keV) RELATIVE INTENSITIES a 16 17 16 Present Work Varnell et al Riedinger et al Present Work Varnell et al Riedinger et al 902.3 + 0.7 901 .0 901.2 + 0.3 0.36 + 0.17 0.32 + 0.16 0.23 + 0.7 919.6 + 0.2 919.7 919.3 + 0.2 1.42 + 0.13 1.49 + 0.04 1.47 + 0.11 926.5 + 0.4 926.2 + 0.2 0.80 + 0.09 0.91 + 0.08 930.4 + 0.1 930.8 930.7 + 0.3 . 0.27 + 0.07 0.28 + 0.03 0.24 + 0.06 964.2 + 0.1 964.4 964.03 0.10 56.0 + 1.7 52.6 + 1.1 52.6 + 2.6 989.8 + 0.3 0.13 + 0.04 1004.9 + 0.1 1005.0 1005.0 + 0.3 1.93 + 0.14 2.31 + 0.12 2.37 + 0.24 1005.8 + 0.1 1086.0 1085.79 + 0.1 40.1 + 1.7 36.7 + 0.50 37.2 + 2.2 1090.2 + 0.2 1089.8 1089.8 + 0.2 5.28 + 0.53 6.09 + 0.21 6.24 + 0.50 1111.9 + 0.1 1112.2 1112.05 + 0.1 52.7 + 1.9 49.1 + 0.5 49.6 + 2.5 1122.9 + 0.4 c 0.074+ 0.028 1171.0 + 0.4 0.13 + 0.03 1213.1 + 0.2 1212.8 1212.8 + 0.3 n 5.32 + 0.26 5.14 + 0.18 5.11 + 0.27 1233.5 + 0.4 0.095+ 0.020 1250.1 + 0.1 1249.7 1249.8 + 0.3 0.66 + 0.06 0.65 + 0.04 0.62 + 0.07 1261.2 + 0.3 1260.9 0.5 0.17 + 0.05 0.12 + 0.03 1292.6 + 0.4 0.38 + 0.07 1299.2 + 0.1 1299.2 1298.9 + 0.3 6.07 + 0.30 5.90 + 0.45 6.01 + 0.31 1348.6 + 0.2 1347.9 + 0.5 0.09 + 0.02 0.45 + 0.018 1363.9 + 0.2 1363.6 + 0.4 0.1 + 0.03 0.084+ 0.024 1407.9 + 0.1 1408.1 1408.04 + 0.12 78.1 + 2.4 75 . 9 1.8 77.6 + 3.9 1433.7 + Is 0.5 0.015+ 0.007 Table V (continued) ENERGY (keV) RELATIVE INTENSITIES Present Work Varnell3 et a l ^ Riedinger et a l ^ Present Work Varnell et a l ^ Riedinger et al 1447.3 + 0.4 c 1457.9 + 0.2 1458.3 1457.6 + 0.3 1527.0 + 0.3 1529.8 1528.0 ± 0.3 1537.4 ± 0.6 c 1606.1 + 0.3 1606.0 + 0.7 1608.2 + 0.7 1643.4 + 0.7° 1647.0 + 0.2 1647.5 ± 0.7 1769.7 + 1.1 1769.3 + 0.7 0.15 + 0.02 1.60 + 0.16 1.81 + 0.09 1.91 + 0.10 0.99 + 0.13 1.2 +0.1 1.30 + 0.07 0.0054 + 0.0022 0.47 ± 0.008 0.027 + 0.006 0.023 + 0.005 0.019 + 0.004 0.03 ± 0.01 0.026 + 0.005 0.03 + 0.01 0.033 + 0.003 ft Uncertainty of every energy reported to be +0.1 keV b Complex peak components estimated from coincidence data c Believed to be sum peaks FIGURE 3-7a Portion of 1^ 2Eu Y-Ray Spectrum 100 keV to 1+20 keV in o o o 30 20 h 10 \— 6 r-1 h-.6 h .2 CBAMEL NUMBER FIGURE 3-71> Portion of 1^ 2Eu y -Ray Spectrum 200 keV to 520 keV CM CM » • r H "UA CM CM -4-CT\ • O N CM CM c—-d; • • • rH CM CM 00 r O f O / CM ro 1A ro H O H V O 3 3 C— CM C— O • • • • 0O_cT CO rH CD C^i O H _3"_H; XA lr\ 100 2 0 0 3 0 0 4 0 0 CHANNEL NUMBER FIGURE 3-7c P o r t i o n o f Ea f-Rzy Spectrum .100 l:oV to 680 keV 564.1 14 •586.5 o r-i o 10 •503.7 •511.0 520.7 'v.-/ 534.9 r— 5>>6»S • v 675.1 -678.7 8 7.5 100 2C0 ' CHANNEL NUMBER 300 i+CO FIGURE 3-7d Portion of ^ E i i y ^ Ray Spectrum 650keV to 900 keV 778.8 •656.5 r—675.1 A 678,7 I—6 8 8 . 9 r—713.5 719.8 765.1 r A. r—8 1 0 . 6 81+0.9 ...... . . A * 867.38 _L A. 100 200 C H A N N E L N U M B E R 300 J+00 FIGURE 3-7e Portion of 1 ^ 2 E u y-R&y Spectrum 900 keY to 1050 keY 60 50 uo 30 20 10 9 8 7-6 \ . r 902.3 -919.6 926.5 4—964.2 r - 1004.9 100 200 CHANNEL NUMBER 300 i+00 FIGURE 3 - 7 f Portion of 1^ 2Eu r-Ray Spectrum 1,05*0 keV to 1200 keV vO CBAMEL MMBER 1IGURE 3-7g Portion of 1^ 2Eu y-Ray Spectrum 1200 keV to 1300 keV 7, 5 t •1213*1 1250.1 1261.2 1274.2 -A.. 100 200 CHANNEL NUMBER FIGURE 3-7h Portion of """^Eii r-Ray Spectrum 1,300 keY to 1,800 k e Y 600 r i " 1+03 h 200 100 60 1*0 20 <3 10 H X ro 6 I 4 1 .6 .4 •2 1299.2 131+8.6 — , 1363-9 1407.9 r 1457.9 •15:17.0 1— B^ G, 1—I6O6.I 1 — B»G. •1769.7 B«G« B . G T 100 200 300 400 CHANNEL NIMBI R CHAPTER IV Gamma-Gamma Coincidence Spectroscopy 1 General C o n s i d e r a t i o n s More i n f o r m a t i o n i s r e q u i r e d on the t r a n s i t i o n s found by the s i n g l e s spectroscopy i n order t o c o n s t r u c t complete decay schemes, which i n d i c a t e the s t a t e s from which these y -rays o r i g i n a t e . Coincidence spectroscopy i s simply the d e t e c t i o n o f two c o i n c i -dent t r a n s i t i o n s . For example, i f there i s a photon t r a n s i t i o n , y{ , from a s t a t e A, to a s h o r t l i v e d s t a t e B, and i f B decays by the gamma-ray, Yz » *° ^ he -tat.- C, the r e s u l t w i l l be the emission o f a second gamma r a y , y2 , very soon a f t e r the emission o f yx . F i g u r e 4-1 shows the schematic o f the y- yz cascade. Figure 4-1 Schematic of a Typical Cascade . A 1 , B h 1 C I f a d e t e c t o r i s set up t o produce a pulse only when the photon yz i s detected, and only the pulses o f a second d e t e c t o r , c o i n c i d e n t w i t h the f i r s t d e t e c t o r p u l s e , are recorded, the r e s u l t w i l l be a c o i n -cidence spectrum o f gamma-rays which feed l e v e l B. The f i r s t d e t e c t o r i s c a l l e d the gate d e t e c t o r , and the second, the spectrum d e t e c t o r . The angular d i s t r i b u t i o n o f yz w i t h respect t o ys can provide i n -formation on the m u l t i p o l a r l t y o f y{ and y2 . As angular c o r r e l a t i o n experiments could not be performed w i t h the a v a i l a b l e apparatus, the coincidence measurements were performed a t a f i x e d angle. I f the t o t a l e f f i c i e n c y o f the gate and spectrum d e t e c t o r s are 63 and (UJOZ respectively, and i f there are N 0 cascades per second, the number of coincident events detected per second are, Nc = N0 (cui)t (LU£ ) Z . (104) Here UJ refers to the geometrical efficiency, i.e. the fraction of the total solid angle subtended by the detector, and £ is the in t r i n s i c efficiency of the crystal. If the resolving time of the system i s ~C , the number of random events per second is N R = 2T h/0(u>£)iN0(Uj£ )z . (105) For the measurement of coincident events to be unambiguous the ratio of true coincidence events to random events must be as large as possible, therefore the source strength one can use i s limited by the resolving time of the system as the signal-to-noise ratio i s given by M r = / . 2XN0 (106) General practice has been to employ a Nal(T* ) crystal as the gate detector and a Ge(Li) detector as a spectrum analyzer. As sodium iodide detectors have poor energy resolution, typically 40 keV FWHM, then gating on a single transition i s impossible i f many other transi-tions occur within the resolution range of the desired gate. Because Europium has many transitions with small energy separations, a Ge(Li) detector was also employed as a gate detector to insure isolation of the gate transition. 2 The Gamma-gamma Coincidence Spectrometer The coincidence spectrometer consists of two Ge(Li) detectors coupled to a fast-slow coincidence network. One detector was the 30 c.c. detector employed i n the singles spectroscopy and the second de-tector was a 15 c.c. detector fabricated by Mr, P. Taraminga at U.B.C. The 30 c.c. crystal was used as the spectrum analyzer and the 15 c.c, detector with a resolution of 7.5> keV FWHM on the 1332 keV peak of ^Co, was used as the gate detector, Spectrum Detector ( 3 0 c.c.) 4 . 5 cm.-»J-<-3«4 cm i n 4 Liquid Nitrogen Vacuum Chamber Detector (15 c . c ) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ W FIGURE 1L-2 Coincidence Detector Assembly 65 The geometry of ihe system is shown in Figure 4-2. The source card was sandwiched between the two inch lead, baffles that separated the detector assemblies. The lead baffles absorb photons which back scatter from one detector into the other and which can be recorded as coincidence events. The source was approximately 4.5 cm, from the center of the gate detector and Jem. from the center of the 30 c.c. detector. The coincidence network, shown in Figure 4 -3 , was a typical fast-slow system, using cross-over timing. The output from each de-tector was amplified by a preamplifier and amplifier, and the timing pulses were generated from the zero level cross over of the amplifier outputs. An Ortec 420 timing single channel analyzer (TSCA) which produced variably-delayed timing pulses for inputs within a certain voltage, hence energy range, generated the timing pulses. The output from each timing single channel analyzer was fed into a fast coincidence analyzer with the resolving time set to 50 (nsec, If both inputs of the fast coincidence analyzer received pulses within the resolving time, a delayed gate pulse was generated. The gate pulse was fed into the MCA which analyzed only pulses received in coincidence with a pulse at its gate input. The slow coincidence channel started with delayed signal from the 30 c.c, detector amplifier. Once stretched, the slow-pulse was fed into the multi-channel analyzers for analysis i f a gate pulse from the fast channel was received. 3 Experimental Procedure Because pulses travelled through each channel of the' coincidence system at different rates, time delays were employed to make pulses from coincident events arrive simultaneously at the appropriate Main Amp. Pre Amp. Spectram Tennelec Tennelec Detector 203 BLR TC 135 M ( 30 o.c.) Past Slow Biased Amp. Ortec 1+08 T 3 C A Ortec 1+20 A MCA Victoreen PIP 1+00 FIGURE 1+-3 Schematic Of Coincidence Apparatus Gate Pre Amp. Main Amp. Detector Tennelec -> Ortec ( 1$ CC.) TC 135 L 435 A Source Fast — Coincidence TSCA Analyzer Ortec RIDL 1+20 A 1 Gate and Scalar Delay Tennelec ^ Generator Ortec HI6 TC $62 67 inputs. Pulses produced by a pulser were fed into both detector pre-amplifiers simultaneously and the delays on the timing single channel analyzer (TSCA) were adjusted for simultaneous arrival of the timing pulses at the coincidence unit. The gate pulse delay was varied until the pulses reached the MCA at the same time as the signal from the slow channel. The time interval, when pulses are considered coincident, is twice the resolving time T. , and could be varied on the coincidence analyzer. Hence, the smaller the resolving timet:, the smaller the num-ber of chance coincidences. However, the detectors and the timing singles channel analyzers have a finite time resolutioni making r shorter than this time resolution reduces the number of true coincidences. The optimum value for x is approximately equal to half the time resolution. The time resolution of the detectors and the timing singles chan-nel analyzers was found by measuring the difference in arrival time of timing pulses for coincident gamma-ray emission with a time to amplitude converter (TAC), A time to amplitude converter produces an output voltage pulse proportional to the time between the arrival of a pulse in the start input and a pulse in the stop input. Then, the output pulse from TAC is analyzed by a MCA to produce a spectrum of the time between pulses, A ^Co source, which emits 1173 keV and 1332 keV gamma-rays in coincidence, was placed between the detectors. The TSCA connected to the gate dotector was adjusted to produce timing pulses whenever a 1332 keV gamma-ray was detected. Thus, one channel was gated on the 1332 photons. Upon detection of a 1173 keV photon, the TSCA connected to the spectrum detector produced a timing pulse. The pulses from the gate TSCA were fed into the start of the TAC, while the pulses from the spectrum TSCA were fed into the stop of the TAC. The resulting spectrum, shown in Figure 4 - 4 , was a peak corresponding to coincidences, •Spectrum Channel Decayed 300 nseo. 500 o o 300 80 h sec.-*" 200 100 h • • • « • 0 . « 100 200 300 TIME ( n sec.) 0 •% 0 a -o» • »<> 0 » » J+OO 500 CD FIGURE li-U Time Resolution Spectrum of Coincidence System 69 upon a l i n e a r background t h a t corresponded to random p u l s e s . Time reso-l u t i o n was 82 usee, FVHM. We chose to s e t f a t 50 n s e c . The ^ C o source was rep l a c e d by a ^^Eu source, w i t h one detec-t o r gated on an a p p r o p r i a t e t r a n s i t i o n energy. The spectrum TSCA was adjusted to accept pulses w i t h i n the s p e c t r a l range t o be i n v e s t i g a t e d . F i n a l d e l a y matching on the f a s t c oincidence channels was accomplished by v a r y i n g the delay o f the t i m i n g pulse from the gate TSCA u n t i l the coincidence r a t e was a maximum. Various source s t r e n g t h s were t r i e d u n t i l the chance t o coincidence r a t i o was approximately The chance r a t e was c a l c u l a t e d from the formula 7 (107) where /vj, and Ng are the numbers of t i m i n g pulses per second from the gate TSCA and the spectrum TSCA r e s p e c t i v e l y . Once the coin c i d e n c e network was s e t up, a s i n g l e s spectrum was taken by the spectrum d e t e c t o r . The peaks i n the s i n g l e s spectrum were used f o r the energy c a l i b r a t i o n o f the coincidence s p e c t r a . A two day coin c i d e n c e run, the ON gate run, was taken w i t h the gate channel s e t on an a p p r o p r i a t e peak. Another s i n g l e s spectrum was taken to check d r i f t by the spectrum channel. A spectrum was a l s o taken by the gate d e t e c t o r to check f o r d r i f t . The gate TSCA was then s e t t o gate on a s e c t i o n o f background a t s l i g h t l y h i g h er energy than the previous gate peak. Next, a two day OFF gate spectrum was taken to determine the coin c i d e n c e r a t e w i t h the Compton background, which must be s u b t r a c t e d from the ON gate spectrum. F i n a l l y , another s i n g l e spectrum was taken t o check f o r d r i f t . F i g u r e 4-5 shows s p e c t r a o f a gate r e g i o n taken by the gate detector,, F i g u r e 4-5b i s a spectrum o f the gamma-rays t h a t produce gate photons f o r the ON gate and OFF gate runs. h k 2 U (a) Ungated Speotrara 310+.3U 367.75 • l i l l . l 20 30 T+o 50 T o 70 80 (b) Gated Spectra ON gate OFF gate 10 20 30 1+0 50 CHA1TREL KDMBSR 60 70 80 FIGURE 1+-5 Gato Detector Spectra. Energy range of gate photons i n CN gate and OFF gate run3 for the 344 keV coincidence run3« 71 Analv"is and Results The analysis of coincidence data is a more complex process than the analysis of singles data. First the area of the peaks in the ON gate and OFF gate run must be corrected for chance using equation 107. Next, the true coincidence rate for each gamma-ray is found by subtrac-ting the photons in coincidence with the Compton background. The true coincidence rate, N/^  y , of the gamma ray yx in coincidence with the gate gamma-ray, y , is obtained by the formula A / ^ ^ / V - A / V (108) where Nc is ON gate coincidence rate, /Vc is OFF gate coincidence rate, and is the ratio of the Compton background intensities b of the ON gate to the OFF gate. The ratio Nra y-x is taken, where fry is the singles rate of yx , and is to allow a preliminary survey to establish i f yx is in coincidence with y„ , This ratio is usually at least ten times larger for gammas in coincidence with the gate than for non coincident gammas, and was used to identify coincidences. The results of this analysis are shown in Tables VI and as an example, the ON gate and OFF gate spectra from which part of 'Table Via was 'deduced are shown in Figure 4-9. The coincidence results in Tables VE marked with an asterisk * are those which have been detected for the f i r s t time. On the basis of these results the greater part of the decay schemes shown in Figure 4-6 and Figure 4-7 could bo constructed. More quantitative results require the postulation of these tenta-tive decay schemes and a knowledge of the absolute efficiency of the coincidence system. Figure 4-6 is the decay scheme of ^-*2Eu to "^-^Em Table Via Gamma Rays Observed in Coincidence with 121.7 keV Transition Energy (keV) N Y g y x 10 + 5 c/s N Y o y x 10 + 3 Coincident = N° - a/b NC,_ n. With Gate? on off ' 121.73 6.4 + 1.6 0.014 + 0.003 No 244.60 185. I 1 -3 2.58 + 0.02 Yes 271 .2 1 .2 + 1.2 1.6 + 1 .6 ? 295.9 6.3 + 0.3 1.94 + 0.09 Yes * 344.34 -2.5 + 0.8 — No 367.75 -0.85 " 0.17 — No 411.1 -0.06 + 0.04 No 444.0 20.0 + 0.5 2.00 + 0.05 Yes 488.7 2.42 + 0.15 1 .58 + 0.10 Yes * 675.1 0.63 + 0.13 1.21 + 0.13 Yes * 688.9 3.40 + 0.20 1 .72 + 0.11 Yes 719.8 1.17 + 0.23 1.56 + 0.29 Yes * 778.8 -2.83 + 1.0 No 810.6 0.34 ~ 0.21' 0.55 + 0.34 No 840.9 1 .38 + 0.46 3.83 + 1.3 Yes 867.38 18.4 + 1.0 2.39 + 0.13 Yes 919.6 1.12 + 0.25 1.77 + 0.39 Yes * 964.2 59.4 + 1.0 2.52 + 0.04 Yes 1004.9 4.84 + 0.37 6.22 ± 0.38 Yes * 1085.8 -0.23 + 0.29 No .1090.2 No 1111.9 36.2 + 1.0 1 .91 + 0.05 Yes 1213.1 2.69 + 0.26 1 .52 + 0.15 Yes * 1250.1 0.18 + 0.1 0.9 + 0.5 Yes * 1274.5 0.18 + 0.1 0.66 . ± 0^ 37 Yes 1299.2 -0.10 + 0.15 No 1407.9 63.5 + 0.8 2.87 + 0.04 Yes 1457.9 0.47 + 0.07 1.09 + 0.16 Yes * 1577.0 0.71 + 0.07 2.82 + 0.28 Yes * Previously unobserved coincidences Table VIb Gamma Rays Observed in Coincidence '. '• th 244.6 keV Transitions Energy (keV) x 10+^ c/s ^yay x Coincident = N - a/b N ,_ n v With Gate? on off r 244.60 1.1 + 0.7 0.008 + 0.005 No 295.9 1.5 + 0.2 0.30 + 0.03 Yes * 344.34 1 .18 + 0.90 0.004 + 0.002 No 367.75 No 411.1 1.0 + 0.4 0.05 + 0.02 No 444.0 4.8 + 0.5 0.20 + 0.02 Yes 867.38 21 .6 + 1.1 1 .57 + 0,08 Yes 919.6 0.0 + 0.2 No 926.5 1 .41 + 0.20 2.24 " + 0.32 Yes * 964.2 -0.58 + 0.50 No 1004.9 2.81 + 0.30 1 .38 + 0.15 Yes 1085.8 0.12 + 0.40 0.004 + 0.001 No 1090.2 0.06 + 0.10 0.02 + 0.02 No 1111.9 -1.60 + 0.50 No Table Vic Gasma Rays Observed in Coincidence with 244.6 keV Transitions Energy (keV) N Y o ^  x 10 c/s N Y o y x 10 Coincident = NC on - ^ Noff n r With Gate 121.73 19.6 + 6.2 2.0 + 0.6 Yes * 244.60 2.6 + 1.2 1 .72 + 0.79 Yes * 344.34 1.5 + 1.2 0.44 + 0.35 No 367.75 0.19 + 0.26 1.9 + 2.6 No 411 .1 0.11 + 0.26 0.45 + 1.07 No 444.0 -0.27 + 0.26 — No 675.1 -0.15 + 0.21 No 719.3 0.00 4- 0.17 No 840.9 0.09 + 0.06 12 + 8 No 867.38 0.62 + 0.09 3.82 + 0.56 Yes * 919.6 -0.03 + 0.10 No 964.2 0.07 + 0.16 0.14 + 0.31 No 1085.8 -0.07 + 0.14 No 1090.2 No 1111 .9 1 .82 + 0.12 4.55 + 0.30 Yes * 1299.2 0.03 + 0.05 0.8 I 1-3 No * Previously unobserved coincidences 74 Table VId Gamma Rays Observed in Coincidence with 344.3 keV Transitions Energy (keV) +3 x 1 0 c/ s = NC - a/b N C„. on off i o + 3 Coincident With Gate? 121.73 17.1 + 2.2 0 .017 + 0.002 No 244.60 5.12 + 1.3 0.032 + 0.008 No 271 .2 1.70 + 0.65 1.07 + 0.41 Yes * 315.2 1.41 + 0.11 0.95 + 0.07 Yes * 34^ .34 -12.5 + 2 . 2 No 367.75 8.61 + 1.0 0.84 + 0.07 Yes 411.1 32.3 + 1.0 1.43 + 0.04 Yes 444 .0 - 2 . 4 + 0.5 No 488.7 - 0 . 5 + 0,07 No 564.1 -0.12 + 0.04 No 586.5 , 1.57 + 0.38 0.49 + 0.12 Yes 6 7 5 . 1 \ D 1.93 + 0.47 0.60 + 0.15 Yes 678.7j 778.8 82.1 + 1.2 1.43 + 0.02 Yes 840.9 0.09 + 0.43 No 867.38 1.14 + 0.40 0.07 + 0.02 No 964.2 -0.16 + 0.26 No 1085.8 No 1090.2 6.42 + 0.46 1.53 + 0.11 Yes 1299.2 6.47 + 0.40 1.54 + 0.10 Yes 1407.9 0.28 + 0.12 0.01 + 0.005 No Area of both peaks taken together * Previously unobserved coincidences 75 Table Vie Gaoma Rays Observed in Coincidence with 44+.0 keV Transitions' Energy (keV) N^r x 10 + 5 c/s Coincident on - a/b N Q f f With Gate? 121.13 30.4 + 4.1 3.00 + 0.40 Yes 244.60 5.5 + 1 .6 3.5 + 1.0 Yes 271.2 — — — No 275.2 0.41 + 0.15 26 + 9 Yes 295.9 -0.02 + 0.26 No 344.34 -5.1 + 1.6 • No 367.75 0.07 + 0.26 0.69 + 2.55 .No 411.1 -0.29 + 0.42 .... • No 444.0 -0.19 + 0.46 No 488.7 -0.06 ± 0.39 No 564.1 0.12 + 0.42 3.3 + 12 No 688.9 No 719,8 0.11 + 0.09 6.7 + 5.5 Yes 867.38 0.00 + 0.04 No 919.6 -0.19 + 0.09 No 964.2 4.98 + 0.15 9.65 + 0.15 Yes 1085.8 4.55 + 0.15 1.08 + 0.5 Yes 1090.2 No 1111 .9 0.12 + 0.06 0.29 +0.15 No 1407.9 -0.08 + 0.03 No 2 ~5 or 3~ 2" + 2 2 ^ 0 . 0 + 0 4 + 0 c-CM co CO C\J VO _OI t-2+0 5 -a- a CM rt O vo IT r- l I T 1^ 1085.8 1022.9 810.5 685.8 to o o cr vo c~ a* CT? VC d1 i CT C-< C-<Mi ir LO, - 3 M i x CO cr r-o IT CP tO to VO C\J cr 3~0 i_0_ to LP vc r--VO VC IT CO d L T v CVI d CVJ ^ 1292.7 t o cr d CO VO cn cr» 1041.2 962.8 4+2 CX> to o VO L O VO CTi L O C T i CVI 2 + 2 VO L O CVJ VO o> cr. va r o VCJ cr a — r •r-i cr 6 4 m o c\» o co to • S t - ' -3-V t OJ to cc ir 0 C C 1768.7 1730.0 1649.7 1579.7 15P9.fi 137L3 1233.7 1085.8 121 .7 o!o 0 I W K 152 6 ^ 9 0 152, Figure 4-6 Decay scheme of Sm. Numbers in brackets indicate relative gamma-ray intensities. Dashed lines energy fits outside error limits. E(keV) —j CT\ 77 2 ' (5) 1643.6 1606.1 1468.7 1434.5 1123.1 1109.2 930.8 755.4 615-5 -O y * , s s—V , -V CO CO CO to CM LO . • LO o s—' LO CM c— . * to O co to CJ-i c— o o VO LO CO *- to * • s. ° CM T -• • r- LO VO C— CM MD CM LO CM -r-3 ° c— cn cn CM to CM LO LO O vo CM r— co to r~ cr-v to CM CO CM O d" «3- LO CM ^ O IC IA cn IA cO ^ O CTi LO to -CO LO VO CO CO CO t> f - VD C— to tO T -to . -3-CM t— CM OJ d CM to 344.3 to to 0 0 .0 E(keV) 1 5 2Gd 64 M88 Figure 4-7 The decay scheme of Gd. The numbers in the brackets denote the relative gamma-ray intensities. 78 while Figure 4-7 shows the decay of ^ E u to 1^2Gd. If the number of photons of a given energy in coincidence with a gate transition is known, i t is possible to identify transitions that cannot be resolved by the singles data. The true coincidence rates for gamma rays y, and yz , which are fed by the gate transition y , as in Figure 4-8a, are given by • o A Nr *. = *v k v (ujiK (> •* *}~J , and . r*r, ii y, r, y ' (109a) A ff —/ ^ v. - b b Cot) C/t«) *>vt *° y, » * i n } (109b) where fi., is the counting rate of the gate peak, io A 0_ is the branching ratio from level A by means of transi-* i 47 tion y, etc. fivO^, is the efficiency of the spectrum detector, and O+flcf'is the probability of the transition taking place by photon emission. Figure 4-8 Schematic of Typical Decay Cascades B c A B (a) (b) 79 6,000 5,000 -4,000 -3,000 " 2,000 -1,000 0 1 , 0 0 0 r-00 CTi O CTi r-l co CM c-vo r- r- co rH ON A. 8 ON GATE CM C M O r - l LO C M C M ML cr> C M J _ co o LO CM LO 100 200 CHANNEL NO. 300 cr> c- OFF GATE cn C T i C M JL 0 100 300 200 CHANNEL NO. Figure 4.9 High Energy CN and OFF Gate Spectra for 121.7 keV Gate 400 80 For gates which are fed by previous photons, as in Figure 4-8b, the true coincidence rates of y 2 and are fort (110a) (110b) where is the fraction of the level B fed by y . The efficiency can be split into two factors, one energy dependent where I is the relative intensity of the y emission found from the singles data. Using equations ( 1 0 9) and (110) together with f.. and b factors for well known levels, and transitions found from the singles data of Table V , the value of k can be found. For a fixed geometry k is constant and can be used with experimental coincidence rates to calcul-ate and confirm the intensities cf other coincident transitions. The values of k for the 122 keV gate runs were calculated for the well established cascade transitions with the 244, 86?, 964, 1112, and 1408 keV gamma rays. The values of k found were (4.46, 4.14, 4.00, 3 . 0 3 , and 4.56) x 10 respectively and the average was taken as ( 4 + l ) x 10"^, Using this average value for k, coincident rates were calculated for other coincident photons, which are compared with experi-ment in Table VII , The general agreement, usually within a factor of two, confirms the placement of the transitions in the decay scheme (see Figure 4-6), There was an indication that the 271.2 keV gamma-factor h y / z y » and a geometric factor, k, by the equation <*£) - A ( n r / Z r ) > (HI) 81 Tablo Vila A Comparison Between Calculated and Experimental Coincidence Rates Gate Energy Energy of ^ H Y o yx x 10~3 c/s (keV) (keV) Experimental Calculated 121.73 a 271.2 1.6 +1.6 c 295 .9 6.3 +0.3 8. + 2. 444.0 20.6 +0.5 23.8 + 6.0 488.7 2.42 + 0.15 2.07 + 0.50 688.9 3.40 + 0.20 4.99 + 1.20 719.8 1.17 + 0.23 1.88 + 0.48 840.9 1.38 + 0.46 0.9 + 0.2 919.6 1.12 + 0.25 1.59 + 0.40 1004.9 4.84 + 0.37 2.31 + 0.58 1213.1 2.69 + 0.26 4.1 + 1.0 1250.1 0.18 + 0.1 0.53 + 0.14 '1457.9 0.47 + 0.07 1.03 + 0.25 1527.0 0.71 + 0.07 0.63 + 0.C8 244.60 b 295.9 1.5 +0.2 1.86 + 0.37 926.5 1.41 + 0.2 1.0 +0.2 1004.9 . 2.81 + 0.3 3.2 +0.6 a Geometric factor k = (4.0 + 1.0) x 10 b Geometric factor k = (1.25 + 0.25) x 10~5 c ~152 This gamma-ray not assigned Sm. See text. 82 T a b l e Y l l b A Compar i son Betvreen C a l c u l a t e d and E x p e r i m e n t a l C o i n c i d e n c e R a t e s Gate Ene rgy Energy o f yx N Y o Y k x 1 0 "3 c/s (keV) (keV) E x p e r i m e n t a l C a l c u l a t e d 295.9 121.73 19.6 + 6 . 2 14.5 + 0 . 8 244.60 2.6 + 1 . 2 3.8 + 0 . 2 867 .33 0.62 + 0 . 09 0 . 79 + 0.04 1111 .9 1.82 + 0.12 1.95 + 0.1 344.34 271.2 1.70 + 0.65 1.37 + 0 .07 315.2 1 .41 + 0.11 1 .23 + 0.07 367.75 8.6 + 1 . 0 14.9 + 0 . 8 586.5 1.57 + 0.38 4 .70 + 0.25 675.1 \ C 1.9 + 0 . 5 3.5 + 0 . 6 6 7 3 . 7 / 6.47 + U.40 6 . 15 + u.;; 444 .0 b 121.73 . 0 . 4 + 4.1 31 . 8 + 1 . 7 275.2 0.41 + 0.15 0.15 + 0.01 719 . 8 0.11 + 0 .09 0 .13 + 0.01 964.2 4 . 98 + 0 .29 4.08 + 0.22 1085.8 4.55 + 0.15 2.52 + 0.14 Geometric factor k = (2.24 + 0.12) x 10 for a l l gates Gate includes both components of the 444.0 keV doublet A r e a of both peaks taken together 83 ray was i n coincidence w i t h the 122 keV gamma-ray (see Table V i l a ), but there i s strong evidence, which w i l l be d i s c u s s e d i n the next chapter, t h a t t h i s was a spurious r e s u l t . U n f o r t u n a t e l y the o r i g i n a l 244 keV gate runs proved to be i n v a l i d , because of an instrument d r i f t d u r i n g the OFF gate run. The runs were repeated a t a l a t e r date, but because of equipment commitments i t was impossible to achieve the same geometry and counting time as the e a r l i e r runs, The l a t e r runs were only able to d e t e c t a few o f the most i n -tense cascades. The valu e of k was found t o be (1,25 + 0.25) x 10 from the coincidence r a t e o f the 867 keV t r a n s i t i o n . Table V i l a compares the c a l c u l a t e d coincidence r a t e s w i t h the observed r a t e s f o r the 244 keV gate runs. The geometry and source were constant f o r the 296, 3^4, and 444 keV gate runs. The valu e o f k was c a l c u l a t e d from the coincidence r a t e s o f the 411, 778, and 1299 keV photons i n coincidence w i t h the 344 keV gate and the values (2.19, 2.19, and 2.24) x 10 -- 3 were found. The average val u e o f (2,24+0.12) x 10 was used t o c a l c u l a t e the expected coincidence r a t e s f o r the remaining t r a n s i t i o n . Comparison between the c a l c u l a t e d and experimental r a t e s i s made i n Table vilb. I t has been known f o r some time t h a t the 444,0 keV t r a n s i t i o n i s a c t u a l l y a doublet. R i e d i n g e r et a l ^ deduce from t h e i r c o incidence experiments t h a t the bulk o f i t connects the 2~ l e v e l a t 1529.6 keV and the 2 + l e v e l a t I O 8 5 . 8 keV i n 1 5 2Sm. The remainder was i d e n t i f i e d as a 2 + 4 + t r a n s i t i o n between the 810.5 keV and 366.3 keV l e v e l s . Various estimates of the r e l a t i v e i n t e n s i t i e s of the two components have been made by d i f f e r e n t autho-rs on the b a s i s o f coined'..^nce a n a l y s i s . We i d e n t i f y the E l component as 443.8 keV and 8 4 the E2 component as 444,1 keV based on the energy d i f f e r e n c e s between the a p p r o p r i a t e s t a t e s . We have used two ways to c a l c u l a t e the i n t e n -s i t y r a t i o s o f the components from our c o i n c i d e n t data. The i n t e n s i t y o f the 444.1 keV t r a n s i t i o n can be c a l c u l a t e d from the measured i n t e n s i t y of the 244 keV gamma-ray which i s i n coincidence wi t h both 444 keV components, using the doublet peak as a gate. The expression f o r the i n t e n s i t y o f the 244.6 kaV peak i s g i v e n as N„„ - y\ b6& / * 2 ^ P * « c ) - ' A foX'zijt, ^ 1 1 ZU<i I -r- / 2 l / f where h i s t o t a l 444 gate r a t e = ^ -* H , and k i s (2.24 + 0.12) x 10"3. The r e s u l t was = (11.6 + 3.9)% o f n ^  . Using the coincidence i n t e n s i t i e s o f the 444 keV doublet found i n coincidence w i t h a 244 keV gate, (12 + 3)% of the doublet was found t o belong to the 444.1 t r a n s i t i o n . The average value o f the 444,1 keV t r a n s i t i o n then becomes (11,8 + 3.5)% which i s i n good agreement w i t h Varnel's value of 9.9%,1^ somewhat higher than Reidinger's value o f (8 + 3)%^ and much higher than a l a t e r Baker's value o f (4.5 + 1.3)% (see ref e r e n c e 48), The r e l a t i v e i n t e n s i t i e s o f the 443,8 and 444,1 are then deduced to be 10,1 +0,40 and 1,3 + 0.4 r e s p e c t i v e l y , where the i n t e n s i t y o f the 344 keV gamma-ray i s 100, 17 49 There has been p r i o r evidence 7 t h a t the 675.1 keV t r a n s i t i o n i s a l s o a doublet, p a r t o f which de~excites the 1606.1 keV l e v e l i n •^ 2Gd and the r e s t connects the 1041,4 keV and 366.3 keV l e v e l s i n 17 samarium. The only estimate to date has been made by Riedinger e t a l from t h e i r coincidence data. We can make an estimate from our d a t a s i n c e we have a d i r f ;-.t measure o f the coincidences between the 675.1 85 keV and 121,7 keV gamma-rays. The r a t e o f 6?5 keV photons i n c o i n c i -dence w i t h the 122 keV gate was used to f i n d the i n t e n s i t y o f the 152 t r a n s i t i o n i n ' Sm, v i a the r e l a t i o n The v a l u e found was (52 + o f the t o t a l i n t e n s i t y , which y i e l d s r e l a t i v e i n t e n s i t i e s o f the 675.05 and 675.1 t o be 0.42 + 0 .10 and 0 .38 + 0 .10 r e s p e c t i v e l y . Riedinger's 1 7 values o f 0 .30 and O.36 con-f i r m the calculation There has never been any doubt that the 271.1 keV transition be-152 longs in the Gd side as an E2 transition between the 615.5 - and 3UU.3 1,12 keV states. It shows up unambiguously in a l l coincidence runs in-Q e l u d i n g our own (see Table VJd). I n a d d i t i o n , Malmsten e t a l f i n d K and L s h e l l c o n v e r s i o n l i n e s belonging to a 271.1 keV t r a n s i t i o n i n gadolinium and not samarium. However the conversion c o e f f i c i e n t measurements, both o f Rie d i n g e r e t a l 1 ? and of t h i s i n v e s t i g a t i o n are much too low t o be c o n s i s t e n t w i t h an E2 assignment. Both o f us have used Malmsten*s conversion d a t a , Malmsten comments i n h i s paper t h a t the K-conversion l i n e "was not d i f f i c u l t t o measure" although the L - l i n e was. He quotes an u n c e r t a i n t y i n the K l i n e i n t e n s i t y o f +40?S, Even i f we assume the extreme e r r o r quoted, the value o f «<K 17 obtained by Riedinger et a l i s s t i l l l e s s than the t h e o r e t i c a l E2 v a l u e . Our own values would only be s l i g h t l y b e t t e r . T his i s the only incorisistercy found from the use o f the Malmsten e t a l con v e r s i o n e l e c t r o n data. On the b a s i s o f an energy f i t , the 271,2 keV transition could j u s t as w e l l go between the 1233,7 keV and 962,6 keV l e v e l s i n ^ S r a , ( I t c o u l d a l s o be f i t t e d here u s i n g the v a l u e s o f Riedinger e t a l . ) I f so then i t should show up i n coincidence w i t h the 121,7 keV gamma-ray, and Table Via seems t o confirm i t . Yet i f p a r t o f t h i s t r a n s i t i o n i s i n samarium, i t i s even more i n c o n s i s t e n t w i t h the s p i n assignments (3+—*1~). We next examined the coincidence runs which are shown i n Fi g u r e s 4-9(a) and 4-9(b) and d i s c o v e r e d t h a t the OFF gate s e t t i n g was uncomfortably c l o s e t o the Compton edge of the 3^- keV gamma-ray, which i s i n coincidence w i t h a 271.2 keV t r a n s i t i o n i n the 1 ^ 2 G d decay. There i s the p o s s i b i l i t y t h a t the OFF gate s e t t i n g missed p a r t of the 344 Compton continuum. Then the Compton coincidences would not be compensated by our c a l c u l a t i o n s (see equation (108)), I f so then spurious 121,7 - 271,2 keV coincidences would be recorded. However we could a l s o expect to observe f a l s e coincidences between the 121,7 keV gate and the i n t e n s e 411,1 keV and 778,8 keV gamma-rays, and we found none (see Table Via ). We are unable t o e x p l a i n t h i s anomaly and 152 so have omitted the placement of the 271.2 keV t r a n s i t i o n i n the Sm decay. The u t i l i z a t i o n o f the coincidence data i n c o n f i r m i n g the t e n t a -t i v e decay schemes w i l l be d i s c u s s e d i n the f o l l o w i n g chapter. 87 CHAPTER V Decay Schemes and Model F i t t i n g 1 The Decay Schemes The decay schemes o f Eu to Sm and Eu t o ^ Gd deduced from the s i n g l e s and coincidence s p e c t r a were presented i n the previous chapter i n F i g u r e s 4 - 6 and 4-7. They are more complete than those based 9 10 15 16 49 on e a r l i e r i n v e s t i g a t i o n s ' ' * and are i n e x c e l l e n t agreement 17 w i t h the most re c e n t . In what f o l l o w s we w i l l attempt t o i n t e r p r e t the schemes i n terms of c u r r e n t theory, i n p a r t i c u l a r the l e v e l s t r u c t u r e 152 o f Sm which has the t y p i c a l r o t a t i o n band s t r u c t u r e of a deformed nucleus. "^ 2Gd on the other hand appears t o be almost s p h e r i c a l , although j u s t r e c e n t l y a q u a s i - r o t a t i o n a l s t r u c t u r e has been observed. a) The 1^ 2Sm S t r u c t u r e The ground s t a t e band: This i s the r o t a t i o n band b u i l t upon the ground s t a t e c o n f i g u r a t i o n I n K 0 +0. There i s a 2 + member a t 121.7 keV and a 4 + member a t 366.3 keV, each of which i s e x c i t e d d i r e c t l y by the decay o f 1 ^ 2 E u . There i s a l s o a 6 + l e v e l a t 712 keV which has only been reached by Coulomb e x c i t a t i o n " ^ . This was the f i r s t such 152 band t o be i d e n t i f i e d i n Sm and presents no problem of i n t e r p r e t a t i o n . The E2 m u l t l p o l e c h a r a c t e r o f the i n t r a b a n d t r a n s i t i o n s o f 121.7 keV and 244 , 6 keV confirm the quadrupole shape of the nucleus i n t h i s con-f i g u r a t i o n . The l e v e l spacing f o l l o w s approximately the l ( l + l ) r u l e p r e d i c t e d f o r pure r i g i d r o t a t o r s . The d e v i a t i o n s i n the l e v e l spacing are due t o c e n t r i f u g a l s t r e t c h i n g and i n t e r a c t i o n w i t h low l y i n g v i b r a -t i o n a l s t a t e s . 88 The b e t a - v i b r a t l o n a l band: The second band p r e d i c t e d by theory i s a hand b u i l t upon the f i r s t v i b r a t i o n a l e x c i t a t i o n o f the nucleus. The p r e d i c t e d c o n f i g u r a t i o n s o f t h i s band are the same as f o r the ground s t a t e band, namely 0 +0, 2 + 0, 4 +0, e t c . No l e v e l s corresponding to these 152 are excited d i r e c t l y from the 3 ground state of Eu, however Marklund e t a l - 5 1 found a 0 +0 s t a t e a t 685 keV populated by the 0" s t a t e o f the 152 9.3 hour metastable Eu which he i d e n t i f i e d as the beta band head. The 2 + member a t 810.5 keV and the 4 + l e v e l a t 1371.3 keV were i d e n t i -f i e d by having the p r e d i c t e d s p i n s and p a r i t i e s and because the l e v e l spacing obeyed the l ( l + l ) r u l e w i t h the same moment o f i n e r t i a as the ground s t a t e band. The theory p r e d i c t e d the nucleus would have the same moment o f i n e r t i a w i t h o r without one quantum o f v i b r a t i o n a l ex-c i t a t i o n as the e q u i l i b r i u m deformation p0 remains the same. The band head decays d i r e c t l y t o ground by an EO i n t e r n a l con-v e r s i o n o n l y , and t o the 121,7 keV l e v e l by the 563 keV gamma-ray which i s obscured by a more i n t e n s e 564.0 t r a n s i t i o n from the 1649.8 keV l e v e l . The 2 + l e v e l i s the only member o f t h i s band t h a t has been confirmed by co i n c i d e n c e . Our measurements show the 688.9 keV t r a n s i -t i o n t o be i n coincidence w i t h the 121.7 keV gamma-ray, and our 444 doub-l e t a n a l y s i s lends support t o t h i s placement. The 4 + s t a t e depopulates by a 902 keV t r a n s i t i o n t o the 2 + ground s t a t e , and v i a a 656,5 keV t r a n s i t i o n to the 4 + ground s t a t e . The theory p r e d i c t s a l a r g e EO component f o r interband t r a n s i t i o n s o f the same I ^ K , which i s indeed observed t h r o u g h l a r g e Eo values f o r the 656.5 and 688,9 keV t r a n -s i t i o n s (see Table VIIJ). The a n a l y s i s o f the EO components w i l l be dis c u s s e d l a t e r . Table VIII Experimental and Theoretical K-Sholi Internal Conversion Coefficients for Transitions in the Decay of Eu n i - • ' J. -i Theoretical <*• xio . , Ey experimental K Assignee., hultipole, (keV) E1 E2 M1 Nucleus, and Transition 121.7 6570 470 1350 6600 244.6 790 + 50 210 810 271 .5 270 + 65 176 618 295.9 91 + 21 128 455 315.2 210 + 210 120 410 344.3 303 + 11 96 308 ^o7.o 63 + 10 83 260 411.1 185 -r 22 63 185 443.8 34 T 10° 48 142 /IOC] •/ rj •*a 1 i ? 503.7 110 4- 4 39 110 564.1 34 + 17 28 78 566.6 121 + 65 28 76 586.5 249 + 59 28 76 656.5 401 + 120 20 54 678.7 52 + 12 21 54 688.9 353 + 44 18 48 719.8 41 + 19 17 43 765.1 81 + 52 16 41 778.8 16.7 + 0.2 16 40 810.6 32 + 16 13 33 867.4 26 + 4 12 23 964.2 22 + 2 9.4. 23 8200 E2 Sin 0 + 1200 E2 Sra 4'-^  JL 2' 1070 E2 Gd 0 + - 2 + 730 E1 S3 3~ 720 E2 Gd 4-2"-*- 0 + 565 E2 Gd 2 + - 0 + 480 E1 Gd 3~-^ ,+ 358 E2 Gd ,+ 2 + 250 E1 / + "T 1 QA E2 213 E2 Gd 2' 135 E1 Sm 2~->-135 E2 or M1 Sra 2~-*- 1" 149 EO + E2 Gd 2 + - 2 + 94 EO .+ E2 Sm 4 + - 4 + 102 E2 Gs 2 + - 2 + a4 EO + E2 Sm 2 + - 2 + 76 E2 Sm 2 + - v 4 + 76 E2 Gd 2' 2 + 73 E1 Gd 3~ — 2 + 57 E2 Sin 0 + 48 E2 Sm JL 3 ' — « + 37 E2 Sm 2 + - 2 + 90 Table VTII (Continued) „ . . . Theoretical , . , Ey Experimental K Assigned Kultxpole, (keV) E1 E2 M1 Nucleus, and Transition 1O04.9 33 + 13 8.7 21 34 E2 Sm 4+_>2+ 1035.8 18 + 3 7.6 18 28 E2 Sm 2 + - 0 + 1111.9 16 + 2 7.2 17 27 E2 Sm 3 + - 2 + 1213,1 6 + 0.4 6.4 14 22 E1 S H 3 " - 4 + 1250.1 18 + 6.7 6.0 -13.6 20 E2 Sa 4 +-2~ 1299.2 6.3 7.5 6.1 14 22 E1 Gd 2~-2 + 1407.9 4.5 + 3.4 4.9 12 15 El Sa 2 " - v 2 + 1457.9 5.7 4- 1.4 4.6 10 14 E1 Sm 3 - - 2 + 1527.0 4.0 + 0.3 4.2 9.1 12 E1 Sa 3~or 2*2+ lt-06.1 j.t- + 1 .4 y.} E2 Gd 2 + - ~ 0 + 1647.0 8 4 3.7 8.0 11 E2 Sa •*- + 2 - v 2 1769.7 9.2 + 3.6 3.4 7.0 9.2 E2 Sa + + 2 — 0 Kalmsten, Nilsson and Anderson's conversion electron intensities were used up to the 1111.9 keV transition. The intensities v?ere normalized to the 3 4 4 . 3 keV transition which was assumed to be pure E2. Larsen, Skilbreid and Visetsens conversion electron intensities used after 1111.9 keV. Conversion electron intensity reduced to account for calculated contri-bution fron 444.1 keV transition v;hich is assucod to be pure E2. 91 The gamma-vlbratlonal band: The level sequence 2+2, 3+2, 4+2, etc., characterizes the gamma band. The symmetric model attributes this band to a vibration in the y mode, whereas this band can be thought of as a result of the asymmetric rotator and not the result of another mode of vibration in the asymmetric model. All members of this band are fed 152 directly by Eu and the resulting strong decays from these levels enables us to confirm each member by coincidence results. The 2+, 3 + , and 4 + levels of this band have been identified at 1085,8 keV, 1233.7 keV and 1371.3 keV respectively. The three transitions of 1085.8 keV, 964.2 keV and 719.8 keV from the band head to the 0+, 2 + and 4 + ground state levels a l l show l i t t l e or no E0 or Ml admixtures (see Table VII), whicn is consistent with the theoretical prediction of quadrupole levels. The 275.2 keV gamma-ray is placed between the band head and the 2 + beta band level solely on the basis of an energy f i t , as i t is too weak to be observed in coin-cidence spectra. The 400,5 keV gam;na-ray which Varnell et a l ^ places between the y a n ( ^ A5 D a n c i heads, was not observed. The 3 + y level decays via the 1111.9 keV and 867.4 keV transitions + + to the 2 and 4 ground states, and their values indicate pure E2 type transitions, which is also expected by the theory The transitions from the 4 + level of 1250.1 keV and 1004.9 keV to the 2 + and 4 + ground states have oc^ values larger than expected for pure E2 transitions, however the error limits are very large. The intensity of the 1004,9 keV gamma-ray had to be reduced by 21% to 154 52 correct for an Eu contaminant, and Aubin et al suggest a 10% Ml admixture in the 1250.1 keV gamma-ray. The 329.4 keV transition is 92 placed from t h i s l e v e l . On the b a s i s of energy f i t a l o n e , t h i s t r a n s i t i o n can a l s o be placed between the 1292.7 keV and 962.8. keV l e v e l s . However t h i s second placement i s outside Riedinger e t a l * s e r r o r l i m i t s . We th e r e f o r e a s s i g n t h i s t r a n s i t i o n between the 1371.3 keV and 1041.4 keV l e v e l s . Negative p a r i t y (octupole)bands> The lowest energy octupole band has K 7 7 = 0". This band i s not fed d i r e c t l y by ^ 2 E u , but the band head i s s t r o n g l y f e d by Eu. The s p i n s , p a r i t i e s , and K values o f t h i s band are those g i v e n to an octupole band by the theory. The 1~0 l e v e l a t 962.8 keV was i d e n t i f i e d by Marklund e t a l ^ who observed a 963.3 keV t r a n s i t i o n i n the study o f 1 5 2 m E u . We place the 840 .9 keV t r a n s i t i o n between the 1~0 band head and the 2 + ground s t a t e on the b a s i s o f energy f i t alone. The K value o f t h i s l e v e l was deduced by 8 , x Dzhelepov e t a l by comparing the experimental B(E1) branching r a t i o s w i t h those p r e d i c t e d by the a d i a b a t i c symmetric model, (See Table X I I ), The 3*0 l e v e l was i d e n t i f i e d a t 1041,4 keV by v a r i o u s workers through Coulomb e x c i t a t i o n - ' - ^ ' 4 ' ^ ^ The 919.6 keV decay from t h i s l e v e l t o the 2 + ground s t a t e was confirmed by coinci d e n c e w i t h the 121.7 keV gamma ray as was the 675.1 keV t r a n s i t i o n t o the 4 + ground s t a t e which was r e s o l v e d from the 675 keV doublet and di s c u s s e d i n the pre-vious chapter. The K = 0 assignment f o r t h i s l e v e l was a l s o made v i a comparison o f B ( E l ) r a t i o s w i t h the simple theory. An octupole band o f l e v e l s 2~1, and 3~1 bas been i d e n t i f i e d a t 1529,6 keV, 1579.7 keV r e s p e c t i v e l y because o f the p r e d i c t e d s p i n 5,11 sequence , However the l e v e l spacing does not f o l l o w even to a f a i r approximation the l ( l + l ) r u l e and the B(E2) r a t i o s do not agree w i t h the symmetric model p r e d i c t i o n s (see Table X I I ) , Evidence f o r the 1511 keV 93 17 l e v e l could not be observed i n t h i s work or i n that o f Riedinger e t a l as i t i s not fed by 1 5 2 E u but by 1 5 2 m E u . The 295.9 keV and 488.7 keV t r a n s i t i o n s from the 2" l e v e l have been confirmed here v i a coincidence with the 121.7 keV gamma-ray f o r the f i r s t time. The 1407.9 keV and 444.0 keV t r a n s i t i o n s from t h i s l e v e l were a l s o confirmed by coincidence data. The 5 6 6 . 6 keV t r a n s i t i o n was placed between t h i s l e v e l and the 1~0 l e v e l on the b a s i s of energy f i t and because the value i s c o n s i s t e n t w i t h t h i s placement (see Table V I I l ) . The i n t e r n a l conversion r a t i o s are a l s o c o n s i s t e n t f o r a l l our observed t r a n s i t i o n s from t h i s l e v e l . We have confirmed two decays from the j" l e v e l , o f 1457,9 keV and 1212.1 keV v i a coincidence w i t h the 121.7 keV gamma-ray f o r the f i r s t time. The <*K values f o r these t r a n s i t i o n s are c o n s i s t e n t w i t h the expected E l m u l t i p o l a r ! t y . The 494.2 keV gamma ray has been placed from t h i s l e v e l to the IO85.8 keV l e v e l only on the b a s i s of an energy f i t . The 16 769.2 keV t r a n s i t i o n from t h i s l e v e l reported by V a r n e l l e t a l and 17 Riedinger e t a l was not observed i n t h i s work. A 5~1 l e v e l i s t e n t a t i v e l y i d e n t i f i e d w i t h the 1730.0 keV l e v e l by Veje e t a l " ^ who place the energy a t 1726 keV. Rie d i n g e r et a l l ? and ourselves observe a weak 1363.9 keV t r a n s i t i o n i n our s i n g l e data that f i t s between t h i s l e v e l and the 4 + ground s t a t e although there i s no d e f i n i t e c o i n c i d e n c e c o n f i r m a t i o n . Levels of no i d e n t i f i a b l e bands; A new l e v e l a t 1292.7 keV has been r e c e n t l y suggested by S c h i c k ^ who g i v e s i t an assignment o f I = 2+0. Of the s i x t r a n s i t i o n s R i edinger e t a l place from t h i s l e v e l , we observe only the 926.2 keV and the 329.4 keV t r a n s i t i o n s which we place to the 4 + ground s t a t e and the 1~0 octupole s t a t e r e s p e c t i v e l y . These placements 94 are made solely on the basis of energy fits as are those of Riedinger et 17 al, Riedinger has extracted a K conversion coefficient value for a 482,8 keV transition, from Halmstens data which indicates an EO com-ponent supporting placement of this transition to the 2 + beta level. The 1649.8 keV level has been assigned three decays, one of which, the 1527.0 keV, has been observed in coincidence with the 121.7 keV gamma-ray. The 564,1 keV transition to the 2 + gamma level, and the 4l6,0 keV decay to the 3 + level have been placed by energy fits only. The negative parity assignment is consistent with the internal conversion coefficient values for the 1527.0 keV and 564.1 keV gamma rays which indicate El transitions (see Table VLTf), As no transitions are seen to 0 or 4 state, the spin assignment of 2 seems the most appropriate. Following Larsen et a l ' s 1 0 placement, a level at 1769.7 keV is 152 assigned to Sm with a transition of 1769.7 keV to the ground state and one of 1647.0 keV to the 2 + ground state. The values for these transitions agree with an E2 assignment which indicates that this level is a 2 + state. b) The 1^2Gd Structure 1 59 Bowman et al and Gono et a l ^ identify some of the positive 152 parity levels of Gd on the basis of the quasirotational bands sug-gested by Sakai. However only seven of the eleven levels observed by us have so far been identified as quasirotational states. These states are the ground, 344.3 keV and 755.4 keV levels which are assigned to the quasiground-state, the 615.5 keV and 930.8 keV levels identified as the lowest members of the quasi-^-band, and 1123.1 keV and 1468,7 keV states assigned to the K = 0 octupole band. The 344.3 keV level has been well established 9'3-5.17,49 a n d h a s the I n assignment of 2+, The 344.3 keV transition from this level 95 to ground has been used as the gate to observe and i d e n t i f y t r a n s i t i o n s i n ^ G d . The 615.5 keV l e v e l i s confirmed by the ob s e r v a t i o n of the 271.2 keV gamma-ray i n coincidence w i t h the 3^.3 keV gate. However the value i s l e s s than expected as i t i s c l o s e r to an E l m u l t i p o l e value than an E2 v a l u e and was dis c u s s e d i n the previous chapter. The E2 m u l t i p o l a r ! t y was expected because o f the assigned 1 1 7 = 2 + . Coincidence between the 411.1 keV and 344.3 keV gamma-rays con-f i r m s the placement of the 755.4 keV l e v e l and the E2 m u l t i p o l a r ! t y i s c o n s i s t e n t w i t h the 4 + s p i n p a r i t y assignment. The 930.8 keV l e v e l i s confirmed by the ob s e r v a t i o n o f a 586.5 keV- 344.3 keV coi n c i d e n c e . The 930.4 keV decay to ground i s placed from t h i s l e v e l on the b a s i s o f energy f i t . The presence o f an EO r a d i a t i o n i s i n d i c a t e d by the l a r g e <*K r a t i o s upporting the 2 + s p i n p a r i t y assignment. The E2 m u l t i p o l a r i t y o f the 315.2 keV t r a n s i t i o n t o the 0 + l e v e l a t 615.5 keV i s a l s o c o n s i s t e n t w i t h i " " = 2 + , The only support f o r the e x i s t e n c e o f a 2 + l e v e l a t 1109.2 keV, reported by Gromov et a l ^ and Adam et a l ^ from the decay o f ^ -^Tb, i s the energy f i t o f a 765.1 keV gamma ray t o the 344.3 keV s t a t e and E2 or Ml m u l t i p o l a r i t y suggested w i t h i n the l a r g e u n c e r t a i n t y o f the r a t i o . The 534.9 keV gamma-ray can a l s o be energy f i t t e d from the 1643.6 keV l e v e l , however the e r r o r l i m i t s are l a r g e i n t h i s case. The 3" l e v e l a t 1123.1 keV i s w e l l confirmed by the o b s e r v a t i o n o f the 778.8 keV and 367.8 keV gamma-rays i n coincidence w i t h the 344.3 keV gamma-ray. The 778,8 keV gamma-ray i s placed to the 344,3 l e v e l ana the 367.8 keV emission i s placed to the 755.5 keV l e v e l o f I = 4 + , Both have oCR r a t i o s i n d i c a t i n g E l m u l t i p o l a r i t y which i s c o n s i s t e n t w i t h these placements. 96 Transitions of 1090.2 keV, 678.7 keV and 503.7 keV are placed from the 3+, 1434.5 keV state to the 344.3 keV, 755.4 keV, and 930. 8 keV levels respectively. The 1090.2 keV transition was confirmed by coincidence with the 344.3 keV transition. Although the 678.7 keV and 503,7 keV transitions have been assigned from this level just on the basis of energy f i t s , their E2 multipolar!ties support the 3 + assign-ment of this state. Block et a l ^ 1 have observed a 5~ level at 1467 keV, We can only tentatively identify a level at 1468,7 keV by placing the 713.9 keV rr + transition from this level to the 755.4 keV state of I = 4 , The level at 1606.1 keV has been assigned I = 2 + by Gromov 49 + et al . Transitions to the ground and the 2 state at 3^.3 keV can be assigned by energy fi t s . The expected E2 multipolarity of the 1606,1 keV emission is confirmed in Table VIII CB t noconversion coefficient data ' is available for the 1261.2 keV transition. The existence of 675.1 keV decay to the 930,8 keV level is confirmed by coincidence measurement and has been discussed in-the previous chapter. The 989.8 keV gamma-+ 17 ray from this level to the 0 excited state observed by Riedinger et al , was not observed in this investigation. The 1299.2 keV transition, which was observed in coincidence with the 344,3 keV gamma-ray, confirms a state at 1643.6 keV, Angular 6l 62 63 correlation measurements ' ' have assigned a spin and parity of 2 for this level and the observed El multipolarity of the 1299.2 keV transition is consistent with this assignment. The 520.7 keV transition to the 1123.1 keV level is placed because of an energy f i t . Only our 352.0 keV and 1348,6 keV gamma-rays have not been placed in the decay schemes, Adam et a l ^ have observed gamma-rays of energies 97 351.5 *eV and 1 3 U 8 . 6 keV i n the decay o f 1 5 2 T b , but they p l a c e the 1 3 U 8 . 6 keV t r a n s i t i o n from a 266? keV l e v e l which could not be fed by the decay o f 1 ^ 2 E u . They p l a c e the 351.5 keV t r a n s i t i o n from a 1282.7 keV l e v e l o f I 7 7 = 4 + to the 930.8 keV s t a t e but no other worker study-i n g the decay o f 1 5 2 E U has seen t h i s gamma-ray. A search through the l i t e r a t u r e t o f i n d a p o s s i b l e contaminant w i t h s t r o n g emissions a t these energies has been u n s u c c e s s f u l . 17 The suggestion o f Rie d i n g e r e t a l o f a s s i g n i n g t h e i r u n f i t t e d 79^.6 keV, 1347.9 keV, and 1447 .3 keV gamma-rays t o 1^ 2Gd i s c o n t r a d i c -t o r y to the work o f Adam et a l ^ on the decay o f ^ ""^To, They do not observe the 794,6 keV t r a n s i t i o n ; p l a c e the 1347.9 keV gamma-ray from 152 a l e v e l Eu could not populate; and cannot place the 1447 .3 keV t r a n s i -152 t i o n i n the Gd decay scheme, 2 Model Comparison The theory f o r c a l c u l a t i n g the energy s t a t e s and reduced t r a n s i -t i o n p r o b a b i l i t i e s f o r the Symmetric and Asymmetric Rotator Model was giv e n i n Chapter 2 , The Asymmetric Model c a l c u l a t i o n s were made w i t h the use o f f i v e 64 b a s i c programs w r i t t e n by Davidson, The f i r s t program c a l c u l a t e d the m a t r i x elements o f the r o t a t i o n a l Hamiltonian, f o r a g i v e n y , by c a l -c u l a t i n g the m a t r i x elements o f the angular momentum components, (see Equation 53 ) . Then, the Hamiltonian i s d i a g o n a l i z e d by usin g the Jacobian method which c a l c u l a t e d the eigenvalues t(t-) and the c o e f f i -c i e n t s , The second program s o l v e s the r o t a t i o n - v i b r a t i o n i n t e r a c t i o n o f Equation 55 f o r a g i v e n s t i f f n e s s parameter,yu. , and c a l c u l a t e s ? , . Using t a b u l a t e d values o f the quantum number J ( , the r a t i o s o f energy 98 of each s t a t e i n the ground s t a t e r o t a t i o n band to the f i r s t 2 energy l e v e l (S 211), are c a l c u l a t e d by means o f Equation 58. Using the quantum numbers Jz , the t h i r d program, which i s the same as the second, c a l c u l a t e d the energies o f the f i r s t e x c i t e d beta band. The r o t a t i o n a l p a r t of the E2 t r a n s i t i o n p r o b a b i l i t i e s are c a l -c u l a t e d by the f o u r t h program, which uses equation (77c), a Clebsh-L N Gordon C o e f f i c i e n t s u b r o u t i n e , the c o e f f i c i e n t s , and the deforma-t i o n parameter y , The l a s t program c a l c u l a t e d the v i b r a t i o n a l c o n t r i b u t i o n to the E2 branching r a t i o s by s o l v i n g equations (77c) and (90c), With the a i d o f the f i r s t three programs, a parameter search over values o f y and was c a r r i e d out u n t i l the quadrupole energy l e v e l s o f the model f i t t e d as c l o s e l y as p o s s i b l e the experimental l e v e l s . The optimum values were y = 11.1 andy - = 0.396, i n e x c e l l e n t agreement with o liA Davidson's values o f y = 11,3 andy"- = 0,396, and Aisenberg's r e s u l t s of Y - 11,0 andy^ = 0,4 , Of course t h i s agreement i s not unexpected as the energy o f the l e v e l s f i t t e d are w e l l known. Figure 5-1 shows the comparison between the experimental and t h e o r e t i c a l l e v e l s . The experimental E2 branching r a t i o s are c a l c u l a t e d by means o f Equation (82) and becomes .B(ez\U- ^*y) .(Ey\5 y e f l 8(ez;Lf ~ U * J J where I (£2) and I (E2) are the E2 i n t e n s i t i e s o f the t r a n s i -t i o n s Y and yz r e s p e c t i v e l y . The Asymmetric Model branching r a t i o s , c a l c u l a t e d by the l a s t two of Davidson's Programs, can be compared d i r e c t l y w i t h the e x p e r i -mental r a t i o s , which i s i l l u s t r a t e d i n Table IX f o r t r a n s i t i o n s from oo J J GROUND STATE BAND /3_ BAND r - BAND EXP. THEORY EXP. • THEORY EXP. THEORY 16 I^ K IN n IN n 612 I^ K IN n 14 4+2 3+2 511 421 12 311 412 2+2 221 4+0 10 -8 -611 2+0 0+0 212 012 6 -4 411 2 -?+0 211 n 0+0 011 FIGURE 5-1 Comparison of Expe-imental Quadrupole Energy Levels with 152 Asymmetric "odel Predictions for Sm. 1 0 0 Table IX Comparison Between Experimental and Asymmetric Model Branching Ratios in ^-^Sa L N n i T T f LNn+j. Experimental Theoretical LNni LNnf' E7£ BE2 Ratio BE2 Ratio 421 -* 4 l l b 421 — 211 1004.9 1250.1 10.1 + 1.2 (9 + 2) a 8.20 221 —* 211 221 -* 411 964.2 719.8 10.3 + 0.9 9.42 221 211 221 —» Oil 964.2 1085.8 2.54 + 1.2 2.20 221 —* Oil 221 —•> 411 1085.8 719.8 4.10 + 0.36 4.29 311 —•» '11 311 —*211 867.4 1111.9 1.08 + 0.05 0.86 212 4 l l c 212 — 2 1 1 444.1 688.9 3.9 + 1.4 4.26 212 —* 411 212 — 211 810.6 686.9 0.168 + 0.015 0.379 212 —•» 411 212 -» Oil 444.1 810.6 22 + 9 11.0 412 — 211 412 -* 411 902.3 656.5 0.12 + 0.06 0.246 a Value obtained when assuming a 10% JQ. admixture in the 1005 keV transi-tion, as suggested by Aubin et al . 101 the positive parity bands to the ground state band. The theoretical values from the y band are in excellent agreement with experiment. The model ratios for transitions from the p band are not in such fine agreement, although they do agree to within a factor of two. The comparison of the symmetric model predictions with experiment is not as straightforward as with the asymmetric model. Fitting the energy levels is not a valid test, because there are enough parameters to ensure an exact f i t . One must compare mixing parameters. For each branching ratio, a mixing parameter is calculated, via equation (84) and Table II , which will bring the model value into agreement with experiment. If the band mixing is valid, then one would expect a constant mixing parameter for a l l transitions from one band to the ground state band, as the parameter is spin independent. Column 3 in Table X contains the mixing parameter found when including only Y band and ground state band mixing. The mixing parameter is not constant, so the possible admixture between the p and y bands must be considered. The 3 + level of the y band cannot contain any admixture from the band, as the p band does not have a level with a spin of 3. Thus, the i^,value found from the 3*"Zr-? H+f- branching ratio should not need any 2 p correction. Using this value of ^y in the other branching ratios, the p, y mixing parameter 2 ^ , was calculated from the other ratios. Once again, one would expect consistent values of Zpy , how-ever, this is not the case. Column 4 in Table X shows the various 15 values of ?p r found, which are in disagreement with Riedinger*s more consistent values shown in Column 7, The p -ground mixing parameter Z~p , was not constant for transi-Table X 152 Comparison of Mixing Parameters for Transitions fr>a the y -Band to the Ground State Band in Sm ^ y - v . ^ 3 - FRESENT INVESTIGATION L nr L BE2 Ratio Z Y x 10 +2 Zprx 10 RIEDINGER ET AL 1 5 + ? b +2 BE2 Ratio Z r x 10 Z, yx 10 10.1 + 1.2 7.5 + 0.6 0.4 + 0 .4 11.2 + 1 . 9 8.1 + 0 . 3 - ( 0 . 2 + .0 .4) ^ + ?- (9 + 2 ) a (6.8 + 0.8) E 0.6 ± 0.6 10 .3 + 0 .9 7 .3 + 1.9 - ( 0 . 3 ± 0 .3) 10 .0 + 1 . 4 6.7 + 1 .8 - ( 0 . 4 + 0 .8 ) r + f 2.54 + 0 .12 9.9 + 0.6 -(1.0 + 1.0) 2.33 + 0.18 8.8 + 1.4 -(0.8 1.0) •z * y 0*7. ?•* y 3.10 + 0 .36 7.9 + 0 .3 -(1.9 ± 0 .9) 4 .19 + 0.61 7.6 + 1.1 - ( 0 . 1 + 1.6) z *-'<r 3*-y 1.08 + 0 .05 8.5 + 0.5 1.00 + 0 .05 7.7 a. 0 .5 ^ - ground s tate band; y ~- gamma band Value assuming 10$ Ml admixture in 1004.9 keV transition ' 2. y found by assuming p - y band mixing negligible o Comparison of Mixing Parameters for Tran i.itions from tho /3 Band to the Ground State Band PRESENT INVESTIGATION L n f i LP- fi BE2 Ratio Without Mixing x 10 Vit'i Mixing BE2 Ratio RIEDINGER ET AI. Z /? x 10~5 Without Mixing 15 With Mixing 2 + fl t- 't+<3- 3.9 ± 1.4 3.2 + 1.9 3.1 r i 1-9 2.8 + 0.87 1.8 + 1.4 1.7 + 1.4 z* p z+ 9-2 +fi 0.168 + 0.15 8.5 + 0.4 3.7 - 0.4 0.15 + 0.02 8.3 + 0.6 9.1 + 0.6 2 <- p 2 + ft V 22 + 9 5.8 + 1.3 5.9 :-. 1 . 3 18.3 + 6.0 5.5 + 1.0 5.6 + 1.0 z + e w (ft-fl y. 2<">- 0.12 + 0.06 4.9 + 0.6 5.2 /; °-6 0.05 + 0.03 5.4 + 0.4 5.7 + 0.5 Hi-ft >-j-= ground state band; p = beta band 104 tions from the /? band to the ground state band, when only /3 -ground b~nd mixing was considered, (see Table XI ). The p, y- mixing was calculated by the calculation of the parameter 6 e0(ez) o ? —? a where the value of ffJe2jQ9-^ ? ir) _ 1 > i + 2 f r o m C o u l o m D 53 56 ^ " i * * - ' ^ excitation data, ' and & was taken as Riedinger*s value of -0.29 x 10-2.39 The 2p parameter was recalculated, with the incorporation of mixing, using Table II , and the results are shown in Table XI , The inclusion of py mixing s t i l l could not produce a consistent set of Bp values, which casts doubt on the validity of the band mixing approach for transitions from the /3 -band. Relative B(EL) values were calculated for transitions from the negative parity states. Table XII shows the comparison of the experi-mental values with the theoretical predictions using the adiabatic-symmetric model. The relative B(EL) values for this model are just squares of Clebsh-Gordon Coefficients,^ and are given by 2 The assignment of the K values from this table was discussed previously. Using Malmsten's conversion electron intensities normalized to the present investigations 344,6 keV gamma-ray intensity, with the theoretical E2 conversion electron intensity subtracted, the ratios were calculated using equation 97. The experimental values are 40 compared with the asymmetric model predictions in Table /.III. The symmetric model predicts a value of V paz for the . 40 ratio, which becomes 0,39 as /?„ is given as 0,314, 105 Relative E ( E L ) Values for Negative Parity Levels in ' *"~Sn Energy of Initial Level (ke.7} Relative B(EL) Relative B(EL) Tneoretical (keV) Experimental K. =0 i K. = 1 K. = 0. 9 6 3 . 3 * 0 . 5 5 + 0 . 0 5 0 . 5 2 . 0 0 8 4 0 .9 1.00 1 .00 1.00 9 1 9 . 6 1 . 3 4 + 0 . 4 1.33 0 . 7 5 6 7 5 . 1 1 . 0 0 1 . 0 0 1 . 0 0 1 4 0 7 . 9 0 . 2 4 0 . 0 2 2 9 5 . 9 0 . 5 5 + 0 . 0 6 2 . 0 0 0 . 5 0 4 4 3 . 8 1 .00 1.00 1 . 0 0 4 8 3 . 7 7 . 6 + 2 . 4 4 . 0 0 0 . 2 5 5 6 6 . 6 1 . 0 0 1 . 0 0 1 . 0 0 4 9 4 . 2 1 . 9 + 1.5 1 2 1 3 . 1 5 . 7 + 0 . 9 1 . 3 3 0 . 7 5 1 .00 1 .00 *\ . OG 1 5 2 7 . 0 0 . 2 6 + 0 . 0 0 5 4 1 6 . 0 0 . 5 0 + 0 . 1 1 2 . C O 0 . 5 0 5 6 4 . 1 1 . 0 0 1 . 0 0 1 . 0 0 9 6 4 . 2 1 0 4 1 . 4 1 5 2 9 . 6 1579.7 1649.7 3 2 2 2 2~ 3" \ 2 " 2 2 " 0 + 0 2 + 0 2 0 <0 1 0 2> 2+0 2 2 Value measured by Dzhelepov et al (Ref. 8). E 2 Transitions assumed. 106 Table XIII Theoretical and '.cperimental Values of the "X" Ratios Isotope XCI^—• L f) X ratio x 10 Experimental Theoretical Sm a Ofi —* Of Op — 2^ (7.2 + 1.1) x IO - 1 3.0 Sm 2 A — 2? 4.2 + 0.6 16.4 Sm 4 A? - * 4 f-V —• 4g. 4 .0+1 .5 21.4 Calculated using the K-electron/photon ratio from reference 67. The much larger theoretical "X" ratios than experimental ratios in Table XH Illustrate that both models greatly overestimate the EO component in transitions between the /? -band and ground state band. 107 CHAPTER VI Conclusions 1 Identification of Bands i 52 The existence of low lying quadrupole bands in " J Sm is now well established, but the evidence presented so far for some of the postu-lated octupole bands is not conclusive. The ground state band is easily Identified bacause its members' obey the fundamental rotation band criteria. The spin and parity sequence, 0+, 2+, 4+, etc. is right, since the rotation band members are based on a 0*0 band head. The level spacing obeys the l(l+l) rule quite well, and reduced branching ratios to this band are pretty much as predicted by theory for rotation bands. The existence of another band, built upon a ^ -vibration level (the second 0+0 configuration) is also generally accepted with its band head at 685.8 keV, The same spin and parity sequence and level spacing as the ground state band is found. Furthermore, the predicted branching ratios from this band to the ground state band are within a factor of two of experimental values. Qualitative support comes from the evidence of E0 interband transitions between states of the same l " K values, which the theory also predicts, A third band built upon a ^-vibration level I^K = 2+2 at 1086 keV is also well estab-lished. Levels of the predicted 2+, 3+, 4 + spin and parity sequence have been found and they obey the level spacing rule. In addition the theoretical B(E2) ratios for transitions from these levels to the ground state band agree with the observed ratios. The quadrupole character of these bands is further suggested by the pure E2 multi-108 polarity of transitions between band members. The case for negative parity bands is not so strong. We observe 1" and 3" levels at 963 and 1041 keV and this, is the spin sequence asso-ciated with a KTr= 0" band. The B(El) branching ratios from these levels agree with the adiabatic rotator predictions (see Table XII) . Veje^^ finds a 5" state at 1222 keV from inelastic scattering experi-ments which adds supporting evidence to the band identification. In the last chapter the 2", 3" levels of 1529.6 keV and 1579.7 keV had been placed in a K u = 1" band after the suggestion of Veje et a l ^ who found the 1511. keV level with I T T= 1~, However the only evidence for this identification is the I^sequence 1~, 2", 3" expec-T T — ted for a K = 1 band and the theoretical expectation that a K =1 band should appear somewhere above the Krr= 0~ band. There is no experimental evidence to support this conclusion. Wo cannot observe 1 ^  the 1 level at 1511 keV because i t is not populated by the -'"Eu decay. The branching ratios from the 2~ state do not agree with those predicted by the adiabatic symmetric rotator model for this choice of K value. Indeed the B(El) ratio of the 2~—> 3" and 2~—> 2 + tran-sitions agree better with a K = 2 assignment for the 2" level (see Table XII), In addition the branching ratios from the 3~ to 4 + ground and 3 io 2 + ground transitions do not agree with any K assignment but they are closer to ratios for K = 0 than K = 1, It is to be noted 17 that Riedinger et al s t i l l identify these states as memoers of a Krr= 1 band in spite of these difficulties and suggest the presence of strong coupling between this and other negative parity bands is responsible for the non-agreement. The lack of clear evidence for assigning K = 1 to these levels allows another postulate which agrees somewhat better with experimen-tal evidence. The alternative is to assign the 1~ and 3~ levels to a second K = 0 band as suggested by transition intensities. The 2 level could be considered as the head of another 2~2 band. This choice is consistent with both the models discussed earlier, which alio.'? only states of even K. Clearly there is a need for more theoretical wcrk to produce estimates of the expected energies of these levels and their branching ratios, as well as for better experimental data. 11 Model Comparison Both the symmetric and asymmetric models predict the gross band structure and level sequence described in the previous section. The IHIAICIO ttOxk Insol o i i l i t e t h r e e Ion' e n e r g y quauxupoie uands wnere sysune-tric model fits the energy levels very well, especially i f the ccriolis 12 (l+l) 2 term is introduced to the level spacing. The asymmetric model does not f i t the energies as well, being almost 1 2 ^ high for the level, However, i t only uses three parameters, a scale parameter, the stiffness parameter, and the deformation parameter, and in this respect is more satisfying than the symmetric model which uses nine parameters,, The asymmetric model predicts the B (E2) ratios for transitions from the /'-band to the ground state band within our error limits. We find that a constant ^-band mixing parameter Zy can be found which will bring symmetric model B (E2 ) predictions within our error limits of the measured values. This result is in disagreement with the findings of Varnell et a l 1 ^ and Riedinger et a l . 1 ^ Both Riedinger 110 et al ^  and ourselves have no success in finding a consistent mixing parameter to f i t the data, although our mixing parameters are less consistent than those of Riedinger et a l . ^ Better experi-mental branching ratios are needed to clear up the question of mixing than are available at present. Both models f a i l in predicting the correct branching ratios of transitions from the p -band. The asymmetric model is within a factor of and i t is impossible to find a consistent set of mixing parameters for the symmetric model, with either just p mixing, or p, y mixing. 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