Open Collections

UBC Theses and Dissertations

UBC Theses Logo

UBC Theses and Dissertations

Spins of the 5.03 Mev and 2.14 Mev States in B11 from angular correlation measurements in B10 (dp) B11 Whalen, Brian Austin 1965

Your browser doesn't seem to have a PDF viewer, please download the PDF to view this item.

Item Metadata

Download

Media
831-UBC_1966_A1 W5.pdf [ 5.45MB ]
Metadata
JSON: 831-1.0302535.json
JSON-LD: 831-1.0302535-ld.json
RDF/XML (Pretty): 831-1.0302535-rdf.xml
RDF/JSON: 831-1.0302535-rdf.json
Turtle: 831-1.0302535-turtle.txt
N-Triples: 831-1.0302535-rdf-ntriples.txt
Original Record: 831-1.0302535-source.json
Full Text
831-1.0302535-fulltext.txt
Citation
831-1.0302535.ris

Full Text

SPINS OF THE 5.03 Mev AND 2 . 1 4 Mev STATES IN B FROM ANGULAR CORRELATION MEASUREMENTS IN B10(dp)BU by BRIAN AUSTIN WHALEN B.Sc, The University of Washington, 1959 M.Se> The University of B r i t i s h Columbia, 1962 A THESIS- SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY i n the Department of PHYSICS; We accept t h i s thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA November 1965 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 it freely available for reference and study. I further agree that per-mission 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 publi-cation of this thesis for financial gain shall not be allowed without my written permission. Department of / / / ^ S / C 5 The University of British Columbia Vancouver 8, Canada Date V / 96 s -The U n i v e r s i t y of B r i t i s h Columbia FACULTY OF GRADUATE STUDIES PROGRAMME OF THE FINAL ORAL EXAMINATION FOR THE DEGREE OF DOCTOR OF PHILOSOPHY of BRIAN AUSTIN WHALEN B.Sc.(Hons„)The Uni v e r s i t y of Washington, 1959 M.Sc, The Univ e r s i t y of B r i t i s h Columbia, 1962 FRIDAY, DECEMBER 3, 1965, AT 3:30 P.M. IN.ROOM 304, HENNINGS BUILDING COMMITTEE IN CHARGE Chairman: Dr. B. N, Moyls F. K. Bowers K. L. Erdman M. K. Craddock G. Jones •B.-L. White Research Supervisor: B. L. White • External Examiner: A.. E. Li t h e r l a n d A.E.C.L. Chalk River, Ontario. SPINS OF THE 5.03 MeV. AND 2,14 MeV STATES IN B u EROM ANGULAR CORRELATION MEASUREMENTS IN (dp) B ABSTRACT An experimental i n v e s t i g a t i o n of the spins of the 2,14 ( J i ) and 5.03 (J) Mev Levels i n B ^ has been made using the B ^ (dp)B reaction to populate the 5.03 Mev l e v e l i n B ^ and then studying p H and p iT angular co r r e l a t i o n s to determine the values of J and J j , The th e o r e t i c a l analysis of the angular c o r r e l a t i o n data i s based on a method i n which the dp reaction mechanism i s represented by a r e l a t i v e l y small number of experimentally determined parameters and therefore the r e s u l t i n g spin assignments are not open to the usual, c r i t i c i s m s of the use of (sometimes doubtful) nuclear reaction theories for the p o s i t i v e determination of nuclear spins. Using the information gained from this experiment and previous experimental information on the s t a t i s t i c a l d i s t r i b u t i o n of Ml to E2 multipole mixing r a t i o s i t was possible to assign an overwhelming s t a t i s t i c a l p r o b a b i l i t y i n favour of the J = ^ , J i = j- spin assignment. These spin assignments are i n agreement with previous tentative ones and with the t h e o r e t i c a l s h e l l model c a l c u l a t i o n s of Cohen and Kurath. The parameters;, determined by th i s experiment;, describing the dp reaction are compared with those calculated using s t r i p p i n g theory and are shown to be i n disagreement with both the Butler Plane Wave and Distorted Wave Born approximation c a l c u l a t i o n s . GRADUATE STUDIES F i e l d of Study; Nuclear Physics Electromagnetic Theory Theory of Measurements Nuclear Physics Elementary Quantum Mechanics Group Theory Nuclear Reactions Theory of Solids Waves Theory of R e l a t i v i t y G. M. Volkoff J., Prescott Jo B.-Warren W. Opechowski W. Opechowski B. L. White R, Barrie P. Savage P. R a s t a l l Related Studies; Electromagnetic Theory E l e c t r o n i c Instrumentation G, B. Walker F. K. Bowers PUBLICATIONS E. Almqv.ist, D.A. Bromley, J.A. Kuehner and B. Whalen. "Alpha P a r t i c l e s From C 1 2 ( C l 2 }o<) Ne 20" Proc. Int. Conf. on,Nuclear Structure, Kingston (1960) J.A. Kuehner, B. Whalen, E. Almqvist and D.A, Bromley. "Quasi-Molecular Resonances i n + C12» Proc. Int. Conf. on Nuclear Structure, Kingston (I960) E. Almqvist, D:A. Bromley, J.A. Kuehner and B. Whalen. "Spins and P a r t i a l Widths of Quasimolecular Resonances i n C^ 2 + c^-2 Interactions" Phys. Rev. 130, 1140 (1963) B. Whalen and B.L. White. "Spins of the 2.14 and 5.03 Mev States i n B 1 1 From-Angular C o r r e l a t i o n Measurements i n B 1 0(dp) B 1 1 " B u l l . Can. Phys. Soc. Vol.2 (1965) - i i -ABSTRACT An experimental investigation of the spins of the 2.14- ( J j ) and 5.03 (J) Mev lev e l s i n B 1 1 has been made using the B ^ d p ) ^ 1 reaction to populate the 5.03 Mev l e v e l i n B^" and then studying p and pYjf angular correlations to determine the values of J and J . The th e o r e t i c a l analysis of the angular correlation data i s based on a method i n which the dp reaction mechanism i s represented by a r e l a t i v e l y small number of experimentally determined para-meters and therefore the resulting spin assignments are not open to the usual c r i t i c i s m s of the use of (sometimes doubtful) nuclear reaction theories f o r the positive determination of nuclear spins. Using the information gained from t h i s experiment and previous ex-perimental information on the s t a t i s t i c a l d i s t r i b u t i o n of Ml to E2 multipole mixing r a t i o s i t was possible to assign an overwhelming s t a t i s t i c a l probab-3 1 i l i t y i i n favour of the J =• —, J j «• — spin assignment. These spin assign-ments are i n agreement with previous tentative ones and with the th e o r e t i c a l s h e l l model calculations of Cohen and Kurath. The parameters, determined by t h i s experiment, describing the dp reaction are compared with those calculated using stripping theory and are shown to be i n disagreement with both the Butler Plane Wave and Distorted Wave Born approximation calculations. - i i i -TABLE OF CONTENTS Page CHAPTER I. INTRODUCTION 1 CHAPTER I I . RADIATION FROM ALIGNED NUCLEI 7 A. INTRODUCTION 7 B. THE DENSITY MATRIX AND STATISTICAL TENSORS 8 C. ALIGNMENT RESULTING FROM NUCLEAR REACTIONS 10 D. ANGULAR DISTRIBUTION OF GAMMA RAYS FROM CYLINDRICALLY SYMMETRIC STATES 12 i . ) DOUBLE ANGULAR CORRELATION 12 i i . ) TRIPLE ANGULAR CORRELATION U E. RESTRICTIONS ON MUETIPOLE MIXING L4 F. PRELIMINARY RESTRICTIONS ON SPIN ASSIGNMENTS FROM PREVIOUS MEASUREMENTS 15 i . ) LIFETIME MEASUREMENTS 16 i i . ) STRIPPING 17 i i i . ) BRANCHING RATIO MEASUREMENTS 18 G. ANGULAR CORRELATION TABLES 19 CHAPTER I I I . EXPERIMENTAL ARRANGEMENT FOR THE CORRELATION MEASUREMENTS 20 A. TARGET CHAMBER AND TARGET 20 B. COUNTERS 21 i . ) SOLID STATE COUNTER 21 i i . ) GAMMA COUNTERS 22 C. THE ANGULAR DISTRIBUTION TABLE 23 D. ELECTRONICS 24 i.) NUVISTOR PREAMP 25 - i v -i i . ) PHOTOMULTIPLIER EMITTER FOLLOWER 2 6 i i i . ) FAST COINCIDENCE CIRCUIT 26 i v . ) SLOW COINCIDENCE (RANDOM COUNT RATE MONITOR) CIRCUIT 27 CHAPTER IV. . ANGULAR CORRELATION MEASUREMENTS AND RESULTS 29 A. PROTON, GAMMA COINCIDENCE MEASUREMENTS " 30 i . ) p 3 y 3 ANGULAR CORRELATION MEASUREMENT 32 i i . ) p ^ and p ^ ANGULAR CORRELATION MEASUREMENT 34 B. TRIPLE COINCIDENCE MEASUREMENT 36 C. DISCUSSION OF ERRORS 37 i . ) ERRORS RELEVANT TO ALL ANGULAR DISTRIBUTION MEASUREMENTS 37 (a) Counter Mounts 37 (b) Target Box Absorption 38 (c) Non C y l i n d r i c a l Symmetry 38 i i . ) ERRORS RELEVANT TO INDIVIDUAL DOUBLE ANGULAR CORRELATION MEASUREMENT 39 (a) P3&^ Angular D i s t r i b u t i o n 39 (b) p3^2 A n g u l a r Distributions 39 i i i . ) CALCULATION OF ERRORS IN THE DOUBLE ANGULAR DISTRIBUTION COEFFICIENTS (b n) . 40 i v . ) ERRORS IN THE TRIPLE CORRELATION MEASUREMENT 41 CHAPTER V ANALYSIS OF CORRELATION RESULTS 4 3 A. METHOD OF FITTING CORRELATION RESULTS TO THEORY 43 B. DOUBLE CORRELATION ANALYSIS 4 6 C. TRIPLE CORRELATION ANALYSIS 48 D. DISCUSSION OF RESULTS 49 - V -E. CONCLUSIONS ON SPIN ASSIGNMENT BASED ON MULTIPOLE MIXTURE PROBABILITIES 50 F. POSSIBLE FUTURE EXPERIMENTAL IMPROVEMENTS TO VERIFY SPIN ASSIGNMENT 53 i . ) IMPROVEMENTS ON PRESENT EXPERIMENT 53 i i . ) EXPERIMENT TO DETERMINE MULTIPOLE MIXING 5A CHAPTER VI COMPARISON OF RESULTS WITH THEORY 55 A. COMPARISON OF SPIN ASSIGNMENTS WITH THE INDEPENDENT PARTICLE MODEL PREDICTIONS 55 B. CALCULATIONS OF THE DENSITY MATRIX FOR THE 5.03 MeV LEVEL AND COMPARISON WITH STRIPPING THEORY 56 APPENDIX I ANALYSIS OF THE RANDOM TRIPLE COINCIDENCE COUNT RATE &U APPENDIX I I DERIVATION OF ESTIMATE OF FITTED PARAMETERS AND ERRORS IN PARAMETERS 68 APPENDIX I I I SOLID ANGLE CORRECTION FACTOR FOR GAMMA COUNTERS 70 APPENDIX IV CALCULATION OF PROBABILITIES OF MULTIPOLE MIXING 73 REFERENCES 76 - v i -LIST OF FIGURES. To follow FIG, 1. TARGET CHAMBER , 2. SOLID STATE COUNTER RESPONSE , 3. RESPONSE OF 2" x 2" Nal CRYSTAL . 4 . INTERPOLATED GAMMA RAY SPECTRUM , 5. ANGULAR DISTRIBUTION TABLE 6 . CIRCUIT DIAGRAM „. 7. 8 . PHOTOMULTIPLIER EMITTER FOLLOWER 9. 10. 11. SLOT/ COINCIDENCE CIRCUIT 12. 13. u . 15. 16. 17. IB. 19. 20. - v i i -LIST OF TABLES To follow page 1. Double Correlation Table ( p ^ and p ^ ) 19 2. Double Angular Correlation Table (p^ #2) , 19 3. Triple Correlation Table 19 4 . tf^ Angular D i s t r i b u t i o n i n text 33 5 . P 3 t1 and ?^ ) f 2 Angular D i s t r i b u t i o n 35 6. Triple Coincidence Results 36 7. Double Correlation 48 8 . Summary of Correlation Analysis JQ - v i i i ACKNOVS LiSDGElviENTS . The author wishes to. express his gratitude to fi r , B„L„ White f o r his many helpful suggestions on both the experimental methods and the o r e t i c a l analysis used i n t h i s report. He also wishes to tiiank Messrs, E.W* Black-more and A.J, Kestelman f o r t h e i r assistance with the experimental measure-ments and Dr. 6 . Jonas f o r his suggestions on the design'of the electronics used during these measurements. The author i s also indebted to the National Research Council f o r the scholarships received during the course of the experiment. He also wishes to thank Mrs. K e l l y O'Malley f o r her many hours of t o i l spent typing t h i s thesis . - 1 -CHAPTER I INTRODUCTION A major part of the e f f o r t made i n th e o r e t i c a l low energy nuclear physics has been directed toward the production of models which approximate the many-body nuclear systems. The accuracy and usefulness of these models i s then investigated by comparing the predicted ch a r a c t e r i s t i c s of the nuclear states with those derived from experiment. The cha r a c t e r i s t i c s most ea s i l y determined by experiment are the spectroscopic quantities of the nuclear states such as t o t a l angular momentum, p a r i t y , energy and ground state magnetic moments. I t has been of primary interest to experimental nuclear physicists to make r e l i a b l e measurements of these nuclear properties since they may be calculated using various nuclear models. The existence of the very large body of spectroscopic information about the low l y i n g energy levels of l i g h t nuclei would seem to suggest that f o r a l l but a few unimportant exceptions, the chara c t e r i s t i c quantum numbers of a l l of these states have been determined. However, a closer investigation of the experimental methods used to determine the spin assignments of the excited states of various nuc l e i reveals that the existing assignments are not a l l completely conclusive. In a t y p i c a l spectroscopic experiment measurements are made of branching r a t i o s , l i f e t i m e s , angular d i s t r i b u t i o n s and angular correlations. The f i r s t two types of measurement have r e l i e d f o r t h e i r spec-troscopic interpretation upon the use of sp e c i f i c nuclear models and arguments based on the relationship between gamma mul t i p o l a r i t y and l i f e t i m e s which are at the same time imprecise and i n many cases c i r c u i t a l i n nature. Any spin assignments wich are based primarily on such arguments must be regarded as tentative only. The second two types of measurement can provide conclusive spin assignments but i n many instances where the spin assignments are of par t i c u l a r importance i n a c r i t i c a l analysis of the theory they have been based, on the f i r s t class of experiments rather than the second.. This thesis describes measurements made i n an attempt to provide the f i r s t conclusive spin assignments; to the f i r s t and t h i r d excited states of .-BL1. 1 1 The low- l y i n g excited states of B - have been studied extensively by previous works** f o r a variety of reasons: I.1) Reliable spectroscopic information on the states of B ^ i s essen-t i a l since- t h i s nucleus i s i n the middle of the l p s h e l l i n the, independent ' p a r t i c l e model and as Cohen and Kurath 1.) have shown i s therefore a p a r t i -c u l a r l y good nucleus on which to te s t that model. 2. ) The B ^ d p j B 1 1 reaction leading to the f i r s t excited state i n shows a t y p i c a l II ss 1 stripping angular d i s t r i b u t i o n . (Evans et a l 2.) That i s , the proton angular d i s t r i b u t i o n measured with respect to the i n c i -dent deuteron beam shows a strong forward peaking, the shape of which i s quite accurately predicted using stripping theory. Since B^ O has.a J ^ o f 3 +, the 0. ~ I stripping pattern would indicate that t h i s state has negative pa r i t y and a spin value in, the range ^  £ J ^  ^ . -However, other independent investigations of the f i r s t excited state such as the measurements of B^°(dplf)B^ angular corre l a t i o n (Gorodetzky et a l 3.) l i f e t i m e (Metzger et a l 4 . ) and branching r a t i o s (Donovan et a l 5.) a l l suggest.a spin assign-ment of To explain the discrepancy i n the spin assignment derived from these and the stripping measurement, the mechanism of "i r r e g u l a r " stripping was introduced (Wilkinson 6.). A positive determination of the spin of t h i s l e v e l would confirm the experimental evidence that such a mechanism must indeed e x i s t . 3. ) The concept of charge independence of nuclear forces was o r i -g i n a l l y established by comparing the corresponding states i n mirror n u c l e i f including and B^. (A study of the states i n C^'was conducted recently i n t h i s laboratory by A.S. Rupaal ;7.) i n which the spins of the low l y i n g OJ ^ <n l e v e l s i n "(J** were assigned,) This concept i s of auch fundamental impurt" ance to the theory of nuclear interactions, and to the theory of eltoehtaygr p a r t i c l e s , that the basic experimental results concerned in' demonstrating i t s v a l i d i t y must be established with the utmost rigour. For t h i s reason i t i s essential to determine the spectroscopic properties of the low l y i n g states of with certainty. On the basis of experimental evidence li m i t e d almost exclusively to branching r a t i o measurements such as those of Gove et a l 8.) and the 'above described measurements the following tentative J assignments have been made fo r the low l y i n g l e v e l s i n 5-05 \ 4 . 4 - 6 5 / £ * . ' 3 — — — y £ \ G.tb. As has been indicated a more conclusive method of determining the spins of these l e v e l s especially the I s * excited siate was required. Sincej, i n general, gamma correlat i o n measurements are only dependent on the angular momenta of the p a r t i c l e s involved i n a t r a n s i t i o n and the miiitipolapity of the gamma ray t r a n s i t i o n s , correlation measurements provide such a method. There are a number of correla t i o n measurements that Could be made on the BJ'J-excited states. In the past particle-gamma correlations have been measured i n such reactions as B 1 0(dpo')B 1 1 (Gorodetzky 3 . ) and B 1 I(p 1p'|j)B ; L 1 B l a i r et a l 9.), however, to interpret these results nuclear reaction theory had to be introduced and since the v a l i d i t y of the theory i t s e l f i s dddbtful',1-the spin assignmentswhich resulted were inconclusive. Therefore measurements, i n the interpretation of which nuclear models or reaction theories play no role or are represented by a small number of experimentally determined paya-= meters were required i f the co-relation measurements on the B- (dp)B 1 J* reaction were to be used f o r d e f i n i t e spia assignments„ A convenient method developed recently and outlined by Litherland and Ferguson 11.) has the re-quired c h a r a c t e r i s t i c s . They show that the (dptf) double and (dptfg) t r i p l e c o r r e l a t i o n can be expressed as a function of the spins of the states of,the dattghtef nucleus arid a limited number Nj of parameters, ca l l e d s t a t i s t i c a l tensors,, representing the excited states formed by the dp reaction. The. number of parameters required i s shown to be further l i m i t e d to i f the state populated i n B i s formed i n a c y l i n d r i c a l l y sym :..«*> \c fashion., I f enough information i s gained from the experiment, these parameters along with the spins of the l e v e l s i n the daughter nucleus involved i n the trans-i t i o n s can not only be determined but possibly overdetermined,, As an ind i c a t i o n of the amount of information that can be derived from double and t r i p l e c o r r e l a t i o n measurements consider the ground, f i r s t and t h i r d excited states Ct B .'It i s known from Donovan et. a l 5°) that the 5 . 0 3 M«v , state decays by gamma cascade through the f i r s t excited state to the ground state or by gamma emission d i r e c t l y to the ground state - 5,05 5 - 2 J 3 -G.S If the ( p ^ ) s (p a n (* ^ "^3^ a n g u ^ a r correlations are measuredp the results are usually f i t t e d to a sum of Legendre Polys .cml&lg ? i s e l -dom are terms higher than P^ necessary to f i t the r e s u l t s , e.g. If no interference between intermediate states Is considered only even A orders w i l l occur. Usually only the r a t i 6 s of the co e f f i c i e n t s ^ are s i g n i -fidant^so as a re s u l t of each of the measurements two r a t i o s have been deter= mined, making a t o t a l of s i x . I f the t r i p l e c o r r e l a t i o n Cp5^#,) i s measured", the results are f i t t e d to functions: of three angles, the angles' 1^ and c ^ ' Q make with a fixed; axis ( i n t h i s experiment t h i s axis corresponded to the i n c i -dent'.•> deuteron beam, direction) and the. azimuthal angle b etween^ and'&j. % h ^ ^ ) ^ N i A ^ P ^ e o s e i ) E|(c-ose 2)cos ; N</> Under the same conditions which: required ; terms up to P^ to: be: used i n the double* c o r r e l a t i o n , nineteen-coefficients. (Ag^) are required^ f o r the expan-sion of above^ With these c o e f f i c i e n t s eighteen r a t i o s K^MN c a n D e deter" A 0 0 0 mined from the t r i p l e c o r r e l a t i o n and;including the six-from the double corre= l a t i o n we have a t o t a l of twenty-four. A l l of these r a t i o s , however, are 0 not independent and when normalization i s allowed f o r i t i s found that'there are l e f t only eighteen independent r a t i o s . The number of adjustable theore-t i c a l parameters (spin values, s t a t i s t i c a l tensor ©lamentsand multipole-mixing ra t i o s ) available to f i t these experimental r a t i o s i s dependent on the spin of the i n i t i a l ( 5 . 0 3 H.«? " state) but usually, w i l l be no more than ten s therefore the determination of, thea® parameters s and i n part i c u l a r the spins of the nuclear states should be assured. The s t a t i s t i c a l tensor elements determined as described i n the pre~ ceeding paragraph are d i r e c t l y related to the population, parameters of the various magnetic substates of t h e . i n i t i a l state formed by the dp rsastion. These, population parameters can also be calculated using dp reaction theory,, By comparing, the calculated with the experimentally determined parameters, i t was possible to investigate the .va l i d i t y of the use of dp reaction theory t© diascribft the reaction inv«stigat@d i n t h i s experiment. An outline of the theory of cor r e l a t i o n measurements of radiations from a x i a l l y symmetric aligned nuclei appears i n Chapter I I . The experi-mental arrangement and apparatus f o r the correlation measurement are described i n Chapter I I I and the results of these measurements appear i n Chapter IV, In Chapter V the correl a t i o n results are analysed i n terms of the theory out-li n e d i n Chapter I I and the most probable value of spin assignment made. The parameters determined by the correl a t i o n measurement are then i n Chapter VI compared with those predicted by nuclear models and reaction theory. CHAPTER I I RADIATION FROM ALIGNED NUCLEI A.) INTRODUCTION This chapter describes the theoretical interpretation of c o r r e l a t i o n measurements made between protons populating levels i n B 1 1 i n the.reaction B3-^(dp)B3-3- and gamma rays r e s u l t i n g from the decay of these states. The theo-r e t i c a l treatment of angular correlations of p a r t i c l e s and gamma rays re s u l t i n g from nuclear reactions w i l l be discussed using the procedure developed by Fano 12.) and reviewed by Biedenharn and Rose 13.) and B l a t t and Biedenharn 1 4 . ) The t h e o r e t i c a l background and interpretation of t h i s experiment i s most clear-l y defined by Litherland and Ferguson 11.) In the past various forms of nuclear reaction theory have been employ-ed i n an attempt to describe the dp stripping reaction and i t s e f f e c t on p Y correlations. As was pointed out i n Chapter I , the use of t h i s type of reac-t i o n theory i n the determination of nuclear spins should be avoided i f possible since the a p p l i c a b i l i t y of the theory i t s e l f i s often i n doubt. In the follow-ing, no u n j u s t i f i a b l e assumptions are made concerning the dp reaction, (such assumptions are made when using the usual stripping formalism), only arguments based on symmetries present i n the dp reaction w i l l be used. The density matrix and s t a t i s t i c a l tensor formalism i s introduced i n Part B.) of t h i s chapter to f a c i l i t a t e the incorporation of these symmetries into the angular c o r r e l a t i o n theory. As w i l l be shown t h i s formalism presents a convenient parameterization of the states of B^ formed by the dp reaction and s i m p l i f i e s the process of reducing the number of parameters needed to des-cribe the state by appealing to the r o t a t i o n a l properties of the s t a t i s t i c a l tensors. In the remaining portion of the chapter the corre l a t i o n between pro-tons populating the 5.03 -Mev . l e v e l i n B 1 1 and gamma rays r e s u l t i n g from the de-excitation of these l e v e l s i s calculated i n terms of the parameters describ-ing the formation of the le v e l s . The correla t i o n functions w i l l be used i n Chapter IV i n the interpretation of (ptf) and (pUtf) c o r r e l a t i o n measurements made i n the reaction B 1 0(dpX)B i : L and B ^ d p U j B 1 3 . B.) THE DENSITY MATRIX AND STATISTICAL TENSORS The notation used i n t h i s section c l o s e l y follows that of Litherland et a l 11.). Let \|/ be the wavefunction representing a pure ( i . e . f u l l y deter-mined) state. In the following \|/ represents the wavefunction of the state formed i n B"'"^ (5.03 - Mew ,. le v e l ) and i s i n general a time dependent wavefunc-t i o n . Since AJ/" represents the i n i t i a l state i n the decay scheme investigated i t i s convenient to define the time (t) at which the state i s formed by dp re-action as t = 0. "y at time t ^  0 (ty0)  m & 7 D e expanded i n terms of a com-plete orthonormal set of stationary states U^. In t h i s representation the density matrix fi^t i s defined as: J^Yk* ~~ Ak^ £' — complex conjugate of A^t I t i s e a s i l y shown Beidenham et a l 13.) that i f Q i s an observable the expectation value of Q iss <Q> =• TR(pQ) I t may be noted that a l l the information on the s t a t e i s contained i n the elements J^y^t . The states considered i n t h i s report are ©igenfunctions of t o t a l angu-l a r momentum J and are expanded i n terms of the set of states V(Jm) which are eigenfunctions of J and J^p projection of J on the quantization (Z) axis. That iss, V o < j > - Z * » V o < j » > m The density matrix i s then written ass I t i s usuiful to enquir* what r e s ^ f f i t f t i o n s I f any-can b® placed on th© element^ of i f various r o t a t i p n a l symmetry properties are imposed ©n • the states. However^ transforms under rotations as a product representation 3 x 3 (Eeferance 33) and the e f f e c t on of demanding various r o t a t i o n a l sym-metry properties i s not obvious. This d i f f i c u l t y i s overcome i f the s t a t i s -t i c a l tensor R(kq % 33} i s introduced, where R i s defined as; R(kq s JJ)» H .(-l) J = m'(JJai-m"5kq) mm1 mm" 3 3 (JJm-m'^kq) the ClebsdrGord^n Coefficient, I t can be shown (e.g. W.T. Sharpe 21^ that the R1 s are tensors of rank k and have a l l the usual r o t a t i o n a l properties of any tensor, that i s , i n par-t i c u l a r that the R(kq)'s transform under rotations as the spherical harmonics Y(lm) (Defined i n Condon and Shortley 36). Frorni the d e f i n i t i o n of the R's i t can be seen that k takes values 0 ^ k ^ 23 and that q, the component of the tensor, takes values -k $ q ^ k. Therefore, the t o t a l number- of elements i n the s t a t i s t i c a l tensors i s ( 2 J + 1 ) . The term "tensor, parameters" i s usu-a l l y used f o r these elements and w i l l be used i n this, chapter. I f normali" zation i s included there, w i l l be i n genera^ {23 +v-l) i ' - 1 independent parameters. The inverse of the above expression i s e a s i l y found to be, (using the ortho-gonality conditions of the Clebs:S-Gordon c o e f f i c i e n t ) s P T J ' = L (-l)J"ffi'(JJm-mMkq.)R(kq s JJ) kq Any r o t a t i o n a l symmetry properties are now readily handled by appeal-ing to the w e l l known rot a t i o n a l properties of the tensors R(kq s J J ) . Ih par t i c u l a r i f the state i s a x i a l l y symmetric and this, axis i s chosen as: the Z a x i s , Litherland 11) shows that t h i s symmetry condition requires that only the R(ko s JJ)'s are non-zero. Further, i f the state has d e f i n i t e parity - 10 -and r e f l e c t i o n s through the o r i g i n are considered ( i n t h i s case t h i s i s equi-valent to considering rotations of 180° about the Y axis) i t i s shown that only the R(ko s JJ)'s with even k are non-zero. The number of tensor parameters needed to describe the state has been reduced by the condition of a x i a l sym-metry. In summary, the a x i a l symmetry of the state when applied to the st a -t i s t i c a l tensors representing the state requires? R(kq s JJ) = 0 q ^ 0 R(ko : JJ) =-0 k not even. The equivalent requirements on the density matrix are A l l i ' _ n n m f J 3 3 - ) 3 3 0 mm1 Qm m _ £)-m-m / 3 3 / 3 3 I t i s now possible to define the orientation, alignment, and p o l a r i -zation of a state i n terms of these tensor parameters representing the state. Steenland and Tolhoek 15.) show that f o r a random orientation a l l t h e ^ j j 1 s are equal and the only non-zero tensor parameter i s R(OOsJJ). I f a l l the J-)® j 1 s are not equal then the system i s said to be oriented. Further i f the state i s symmetric about a quantization (Z) axis the only non-zero tensor parameters w i l l be the oneswith q = 0 (as pointed out previously) i . e . ; R(kq ; JJ) = 0 q ^ 0 Then when R(ko s JJ) = 0 f o r k 0 we haves 1. alignment i f k = even 2. p o l a r i z a t i o n i f k = odd. C.) ALIGNMENT RESULTING FROM NUCLEAR REACTIONS As an example of how alignment can r e s u l t from nuclear reactions con-sider the case where a spinless p a r t i c l e i s incident on a spinless nucleus and only spinless p a r t i c l e s from the reaction are detected by a counter at - 1 1 -to the incident beam. I f the incident p a r t i c l e d i r e c t i o n i s defined as the Z axis (axis of quantization) only states with zer© projection of t o t a l angular momentum (m » 0) can be populated i n the residual nucleus, which must have non-zero spin f o r t h i s to be a meaningful statement. This i s clear since the only angular momentum carried by the incident and emergent p a r t i c l e s i s i n the form of o r b i t a l angular momentum which f o r a plane wave has by d e f i n i t i o n m JS 0t. Since there can be no Change i n 1 t o t a l magnetic quantum number during t h i s reaction, ... only m =• 0 states in'the residual nucleus w i l l be popu-lated o Any radiation from the residual nucleus i n coincidence with the par-t i c l e s detected at zero degrees to the incident beam w i l l have an angular d i s -t r i b u t i o n c h a r a c t e r i s t i c of the decay of an m .jfe 0 state. Also the density matrix f o r states of the residual nucleus formed i n t h i s manner w i l l have only one non-zero element, which correspond to a high degree of alignment. I f we consider instead an i n i t i a l nucleus and bombarding p a r t i c l e both having non-zero spins,, i t i s possible f o r m •= 0 states to bo populated. However, the population of the magnetic substates of the residual nucleus w i l l b® a f f -ected by the non-random d i s t r i b u t i o n of the o r b i t a l angular momenta Ji of the incoming and. outgoing p a r t i c l e s (m. ^ 0) 'and a non-isotropic alignment of the • i. ' residual nucleus may re s u l t . For a more detailed discussion of thes® effects see Litherland et a l 11,,). I t was pointed out i n the previous paragraph that f o r the B^^(dp)B^" reaction^since a l l the p a r t i c l e s involved i n the reaction hav® non-zero spins, no d e f i n i t e statements as to the value of the magnetic substate populations 11 of the residual B nucleus can be made on the basis of these simple arguments. I f however, the dsuteron beam and glO target are unpolarized and the protons from the reaction are detected by a non-polarization sensitive counter i n an a x i a l l y symmetric stay, the axis of symmetry (Z axis) being the incidsnt beam di r e c t i o n , the B"^ states w i l l be for ed i n an a l l y symmetric fashion. In - 12 -th i s s i t u a t i o n the conditions on the s t a t i s t i c a l tensors representing a c y l i n -d r i c a l l y symmetric state outlined i n Part B . ) may be applied to the states i n b U . To reta i n the a x i a l symmetry the protons were detected by a c i r c u l a r semiconductor counter whose face was centered at 0° to the beam. Any gamma rays detected i n coincidence with these protons w i l l then have an angular d i s -t r i b u t i o n characteristic of the decay of a c y l i n d r i c a l l y symmetric state. The calculation of these angular d i s t r i b u t i o n s i s outlined i n Part D.) f o r both (ptf) and (ptftf) coincidence measurements. D.) ANGULAR DISTRIBUTION OF GAMMA RAYS FROM CYLINDRICALLY SYMMETRIC STATES i . ) DOUBLE ANGULAR CORRELATION. The term double angular c o r r e l a t i o n w i l l refer to the measurement of the coincidence count rates between protons populating a nuclear state of spin J and gamma rays from the decay of t h i s state, as a function of the an-gle between the rays and the Z axis. The Z axis i s the axis of symme-try and as pointed out i n Part C.) corresponds to both the deuteron beam d i r -ection and the d i r e c t i o n of the average momentum of the protons detected by the s o l i d state counter fixed at 0° to the beam. The figure below shows an B 1 0 + d energy l e v e l diagram of the reaction studied i n t h i s experiment. 5.03 Mev The correl a t i o n function W can be expressed i n terms of the tensor parameters R (kOsJJ) f o r the 5.03 Mev l e v e l formed by the dp reaction (Litherland 1 1 . ) . W^ -e-) - R(kO : JJ) A k ( j J i L i L i ) x J P k(cos -&) 1 k L ^ p i take values 1 or 3 Jj_= t o t a l angular momentum of the f i n a l state involved i n the tf^ t r a n s i t i o n . L^L^ — the m u l t i p o l a r i t i e s of the }f ^ t r a n s i t i o n Xj_ = multipole mixing r a t i o s between and Lj_ U e - x _ < J i l | I I + l l J> l~~ < J i l l % || J > p takes the values of 0, 1 , 2 f o r dipole, interference and quadrupole terms i n the correlation: A k ( J W l ) = R e ( i - L i + 1 T -TT' ~ 2 k + 2 + 2 J i " 2 J ) ^ L ^ L j I ^ - l l J k O M J L j j L ^ J ^ ) a * (2a t- l)i jf •=. p a r i t y of the radiation (LjLj, - llJkO) = Clebsh - Gordon Coefficient W = Racah Coefficient as defined i n W.T. Sharp 2 4 . ) and tabulated by Sharp et a l 37.) A convenient tabulation of the A k may be found i n Kaye et a l 32.) I f the excited state represented i n the c o r r e l a t i o n function by the R(k0 s JJ)'s (the 5.03 Mev level) decays through the intermediate state of spin J j by the emission of gamma ray (tf]_), then J j decays to the f i n a l state of spin J ^ by emission of ^ a n <* only the ^ angular d i s t r i b u t i o n i s recorded, the ( p ^ ) d i s t r i b u t i o n may be calculated (Litherland 1 1 . ) and takes the form; = 21 R(kO f: J J ) C k ( J J I J f L 1 L 2 L 2 ) x]> P k(cos «e) 2 kLjLjLgp C k = ( - 1 ) I ' J f - P - ^ 2 " J 1 / i ; i ( L i L i - l l I 00)W( J J j J J z : L xk) Z^( ^ J j L ^ J j s Jfk) - 14 -Z^ i s the c o e f f i c i e n t defined by B l a t t and Biedeharn 14) and tabulated i n Reference 37.) i i . ) TRIPLE ANGULAR CORRELATION. The t r i p l e angular correlation measurement w i l l refer to the t r i p l e coincidence measurement made between the protons populating the 5.03 Mev. .-l e v e l i n th« manner outlined i n Part C.) and the two cascade gamma rays ^ and y^ (See previous f i g u r e ) . The t r i p l e correlation function w i l l depend on the angles -©^  and -Q^ "that gamma rays Y-^ and Y 2 m a k e w i t h t h 8 z ax*-s> and on the azimuthal angle (J) between a n d ^2* Writing the t r i p l e c o r r e l a t i o n func-t i o n W(6=j62<|>) using the same notation as Kay® ©t a l 32), W(e1-e2(J))=Z R(k0 J J j A ^ j j j j y L i L ^ L ^ k J x f l x / 2 The summation i s over KMNL^L^L,^ and k. The A^j are tabulated by Kaye et a l 32) and are defined ass k A K M ( j J I J f L l L l L 2 L 2 k ) = ( - D J l " J r P 2 " ^  | L^(k-M0NKN) (3 L x Jj> J L ^ j J W I L I S ' G and Z1ar® coe f f i c i e n t s defined and tabulated by^Ferguson and Rut-ledg© 38) X N <tl - r(2K + l)(K-N)j, (ai-H)(M-N)<7 fc Pjj(cos -e-2) - 0 0 3 N <|i. x^ and x 2 are the multipola mixing r a t i o s of t^ and JT2 a s defined i n D.) i . ) E„) RESTRICTIONS ON MULTIPOLE MIXING In Chapter I evidence was presented which set the parity of a l l three states under consideration as negative, therefor® the gamma ray t r a n s i t i o n s ^ %2 a n c* Y-j must be of the type Ml, E2, M3 and E4 or some mixture of these. The t r a n s i t i o n p r o b a b i l i t i e s (which are proportional to the radiation widthsP) f o r each type of decay were estimated (assuming on average gamma ray energy E ^ = 3 Mev); using the Weisskopf extreme single p a r t i c l e model (Wilkinson 16.) and on t h i s basis a r e s t r i c t i o n was set as to the extent of multipole mixing. r w ( M l ) = 0.6 ev P W(E2) - A K T 4 ev . . rjj(M3) = 3.5 10""1G ev , fJ(E4) - U 10" 1 4 ev ••. I t i s seen that the lowest m u l t i p o l a r i t y L allowed by the rule JjL+ 3^ Jjj+,Jf where and 3f are the angular momenta of the i n i t i a l and f i n a l states, should also be the dominant component of the t r a n s i t i o n - i . e . L = J i - J f f o r - J f = 0 L = 1 f or 3i - 3f - 0 , J i - 0 Experimentally i t has been found that rE2 i s enhanced over the Weiss-kop estimate (Wilkinson 16.) i n many instances and the multipole mixture 2 IE2 x(where xr=> L—• ) can become non-negligible (x 1 s up to 0.2 or 0.3). There ' T mE fEA have been no experimental measurements of mixing r a t i o s p*jj<2 or which how-ever are not expected to be much greater than the values predicted by the Weisskopf l i f e t i m e s ( i . e . -|| * 10"6, - J | ^10"^ with re s u l t i n g x's of 10"3 and 10"2), Multipole mixing of this amount would have a negligible e f f e c t on E2 the angular c o r r e l a t i o n , therefore only -gg mixing was included i n the c o r r e l -ation formalism. That i s , i t was assumed that L = - J f f o r Jj_ - 3£ > 2 L = 1 and 2 f o r - J f = 1 or 0 F.) PRELIMINARY RESTRICTIONS ON SPIN ASSIGNMENTS FROM PREVIOUS MEASUREMENTS. The number of possible spin assignments f o r J and J j (spins of 5.03 and 2.L4 Mev states i n that would have to be allowed i n the c a l c u l a t i o n of angular d i s t r i b u t i o n s would b« very large indeed i f no previous knowledge were available on these states. However as has been pointed out a large body of data i s available i n the l i t e r a t u r e from which preliminary r e s t r i c t i o n s on the possible range of spin values f o r J and J j may be made. The r e s t r i c t i o n s made i n the following are based on l i f e t i m e measurements and are compared f o r consistency with r e s t r i c t i o n s that may be made on other less conclusive meas-urements. Note that the l i f e t i m e measurements are being used only to rule out certain m u l t i p o l a r i t i e s j they are not being used to choose a single multipole assignment from several nearly equally probable assignments, i . ) LIFETIME MEASUREMENTS. The l i f e t i m e of the f i r s t excited state ( 2 . 1 4 Mev) has been measured by Wilkinson 6 . ) and Metzger -4.),'and was found to be approximately 5 1 0 " ^ sec. This value may be compared with the Weisskopf extreme s i n g l e - p a r t i c l e l i f e t i m e estimate to put a l i m i t on the maximum mu l t i p o l a r i t y of the X decay to the ground state. I f L i s the angular momentum of the multipole radia-t i o n emitted, the Weisskopf estimate of the l i f e t i m e Tg^ of t h i s state i f i t decays by e l e c t r i c multipole (EL) i s (Reference 16.) TE1 — 2.4 10" 1 6 sec. TE2 1.2 1 0 " U sec. T -6 1.2 lCf° sec. TE4 •= 0.17 sec. The s t a t i s t i c a l factor(S) i n Reference 16«) whose magnitude Is f o r a l l cases of the order of unity, has been taken to be unity f o r t h i s estimate. Wilkinson 16.) shows that the extreme single p a r t i c l e estimate i s quite often out by a factor of 10 3 - 10^ " from the experimentally determined l i f e t i m e and i n some extreme cases the estimate deviates from the measured 5 value by a factor of approximately 10 .Therefore i f i t i s assumed that the Weisskopf estimate i s nctiprong by factors exceeding 10^, E3 and E4 t r a n s i -tions may be ruled out with certainty f o r ^ ( 2 . 1 4 M e v s* 5 1* 9 t o ground state decay gamma ray). I t i s unlikely that ^ 2 ^ s E2 t r a n s i t i o n but L = 2 tran-s i t i o n s cannot be d e f i n i t e l y ruled out by the l i f e t i m e measurement. On t h i s basis a l i m i t on the possible range of spins f o r the 2 . 1 4 Mev state ( J j ) may be set since J I = J F + L 3 where Jj . = spin of ground state = L = maximum allowed L value f o r the ^ t r a n s i t i o n = - 2 . That i s 2 1 2 No measurements on the l i f e t i m e T of the 5.03 Mev l e v e l has been found i n the l i t e r a t u r e by the author, however an upper l i m i t was set i n t h i s experiment of T $10 sec. by using a f a s t coincidence (~10ns resolution time) c i r c u i t (Refer to Chapter IV Part B.). I t i s known (Donovan et a l 5 . ) that t h i s state decays mostly (~9Q£) by gamma emission d i r e c t l y to the ground state. The single p a r t i c l e l i f e t i m e estimate f o r an E4 t r a n s i t i o n from t h i s state to the ground state i s found to be (Reference 10.) TE4 = 0 , 1 s e c * Therefore, using the same reasoning as before L -k t r a n s i t i o n s may be ruled out and the range of allowed spin values of J(spin of 5.03 Mev state) is s 1 < / 9 i i . ) STRIPPING. The angular d i s t r i b u t i o n s of the protons from the B ^ d p j B 1 1 reaction leading to the f i r s t three excited states has been studied extensively (e.g. N. Evans et a l 2 . ) and a l l the proton groups exhibit j£= 1 stripping patterns. I f - 18 -the reaction follows the regular stripping mechanism J and J j should have values which s a t i s f y ; J ^ I = 3Bl°+ J 3 1 or, since JglO — 3 However, as pointed out i n the Introduction (Chapter I.) there i s strong evidence i n the l i t e r a t u r e that J j = This spin value can be reached by the stripping process i f the "spin f l i p " mechanism (Wilkinson 6.) Is introduced. A spin of therefore should not be excluded on the basis of the stripping data and the range of spin values should be extended to include That i s This range of values f o r J and J j , derived from the stripping data, i s consistent with that derived from l i f e t i m e measurements i . ) i i i . ) BRANCHING RATIO MEASUREMENTS. The branching r a t i o s f o r the gamma ray de-excitation of the f i r s t three excited states of B^" has been measured by Donovan et a l 5.). Using the rather imprecise arguments usually necessary f o r the interpretation of 3 1 such measurements the spin assignments J = - j , J j = ^  were made. Obviously these values l i e i n the range of spins allowed by the l i f e t i m e measurements. In summary the range of values f o r the f i r s t and t h i r d excited states i n B"^ ", allowed by l i f e t i m e measurements i s 2 ^ J I ^ 2 2 " J " 2 This range of values i s consistent with that predicted from s t r i p -ping data and from gamma branching r a t i o measurements. I t was assumed therefore i n the cal c u l a t i o n of the correlation functions that only those values of spins which l i e i n t h i s range need be considered. <o JO, *" Go) ANGULAR CORRELATION TABLES The t h e o r e t i c a l p*^ and. angular cprr,alati©n functions Wjf (•©•) and (•©) were calculated f o r a l l the possible choices of J and J j consistent with the r e s t r i c t i o n s imposed i n Part E.) and F,) of t h i s chapter and the resul t s of t h i s calculation appears i n Table .1, Listed here are the values of B^Of^) f o r each choice of i n i t i a l and/final state spins (J,Jj_) of the $ ^  decay, where the B^ ' s are the theoreticallyLoalculated c o e f f i c i e n t s i n the expansion. W (6)tr Z j ^ B j ^ J J ^ P ^ cos *@) 1 k The a.^'s are the-te'nsor parameters ; a.k H(k0?JJ) and appear i n a l l the calculated c o r r e l a t i o n functions. S i m i l a r l y Table 2. displays the results of evaluating the double c o r r e l a t i o n function Wy (•$») ^defined i n D„) I , ) , Listed here are the values G2 • • • of Cjj.CiTg) f o r each choice of spin of the 5.03 Mev^-state (J) and 2,14, Mev state ( J j ) where the C j ^ ^ ) ' 3 are-the c o e f f i c i e n t s i n the'expansion, . (-&) = £a^Cj^jJPj^cos •&) 2 k ,; The theoretical t r i p l e correlation function W("©][©2$) w a a calculated f o r the s i x possible choices of>spin assignment which were found i n Chapter V to be consistent with the three double co r r e l a t i o n measurements. The values of the multipole mixture parameters ^ determined from the double correl a t i o n measurements (See Chapter V) have beetfl substituted into the t r i p l e c orrela-t i o n function. Table 3.; l i s t s the th e o r e t i c a l values of the t r i p l e c o r r e l a -t i o n function f o r the f i v e points at which the correlation was measured. TABLE 1. DOUBLE CORRELATION TABLE ( B / r and J J i k a 0 k a 2 • k = 4 k = 6 1 3 1 2 T (1+ x j 2 ) (0.5000+ 1.7360xi - 0.5000xi2) 3 3 2 2 (1 + X i 2 ) (-0.4000 •+• 1.5491xi) 3 5 2 2 (0.1000 - 1.1833xi + 0.3571xi2) 3 7 2 2 1 - 0.1429 5 1 2 2 1 - 0.0534 ' -0.6171 5 3 2 2 (1 + Xi 2) (0.3741 + 1.8972xi - 0.1910xi2) 0.7053xi2 5 5 2 2 (1 + Xi 2) (-0.4275 -+ 1.0l42xi+0.1910xi 2) -0.4951X12 5 7 2 2 (1 + X i 2 ) (0.1337 - 1.3887xi+ 0.3244xi2) 0.1176xi 2 7 1 2 2 1 0.1237 -0.1066 -0.0151 7 3 2 2 1 -0.4686 -0.3582 7 5 15 "2 U + Xi 2) (0.3274 + 1.8899xi - 0.0780xi2) 0.6366 7 7 2 2 (1 + X i2 ) I (-0.4365+ 0.7560xi+ 0.2493xi 2) -0.3560xi2 9 3 a an 2 2 1 -0.2522 0.1281 0.0031 9 5 2,2 1 -0.4324 -0.2685 9 7 2 2 (1+ X?) (0.3027 + 1.8709x + 0.0197x2) 0.4365X2 TABLE 2. DOUBLE ANGULAR CORRELATION TABLE (%£) J J j k = 0 k = 2 k = 4 4 1 3 3 2 2 ( I f * 2 2 ) (-0.0800 + 0.3099x2) 3 5 2 2 (0.2799+ 1.4l99x 2 - 0 . l 4 2 9 x 2 2 ) 3 7 2 "2 1 - 0 . 3 0 6 1 5 3 2 2 (1 + x 2 2 ) (- 0.2994 + 1.1594x2) 5 5 2 2 U + X2 2) (0.2460 +1.2468x 2 - 0.1254x 2 2) -0.1008x 22 5 7 2 2 1 - 0.4090 -0.2079 7 3 2 2 (1 + x / ) (- 0.2618 + 1.0l42x 2) 7 5 2 2 (1 + x 2 2 ) (0.3274 + 1.6598x2+ 0.l659x 2 2) 0.4092x 2 2 22 2 2 1 - 0.3785 -0.1308 9 3 2 2 (1 + x 2 2 ) (-0.0121+ 1.5223x 2+ 0 . 3 0 1 1 x 2 2 ) 9 5 (1 + x 2 2 ) (0.3028 + 1.5352x2 - 0.1545x 2 2) 0.3069x 2 2 2 2 1 - 0.4324 -0.2685 s TABLE 3 . TRIPLE CORRELATION TABLE X J l 3 1 2 2 3 3 2 2 3 5 11 2 2 5 5 2 2 90 9 0 0 0.891a„ 0 0 , 8 4 6 a Q a 2 0.700a a 0 2 0.894a 0a 2 0 . 7 3 3 a Q a 2 - 0 . 3 5 9 a 2 2 -0.571ao 2 - 0 . 0 4 8 a Q - 0 . 1 4 3 a 2 0 . 5 1 5 a Q 2 -0.0286a 2 - 0 . 0 3 6 a | 90 90 4 5 0.891a G 1 . 3 2 9 a Q a 2 0.836a Qa 2 0 . 7 6 8 a 0 a 2 0 . 8 5 0 a Q a 2 - 0 . 4 7 9 a 2 2 0.113a 2 0 0 . 0 2 4 a 2 2 -0.035a Q 0 . 3 9 9 a Q 2 -0.028a 2 2 0.018a 2 2 0 . 9 7 3 a Q 2 0.972a 0a 2 0.462a Qa 2 0 . 9 6 7 a 0 a 2 90 9 0 9 0 0.891a0 0 . 9 7 3 a Q a 2 0 . 7 9 7 a 2 0 . 0 9 6 a 2 0.073a 0 2 - 0.198a 2 2 -0.028a 2 2 0.072a 2 2 0.283a 2 2 0 . 8 3 6 a 0 a 2 0.893a Qa 2 0.731a 0a 2 90 0 0 0.891a 0 0.892a Qa 2 0.113a 2 - 0 . 0 4 7 a 2 0.071a 2 - 0 . l 6 0 a 2 2 - 0 . 0 1 4 a 2 2 - 0 . 0 3 6 a 2 2 - 0.077a 0 2 1 . 0 6 4 a 0 a 2 1 . 4 6 3 a a 0 2 1.301a 0a 2 0 90 0 1.220a 0 1 . 1 4 3 a Q a 2 0.113a 2 -0.047a 2 0.071a 2 0 - 0.077a 0 2 - 0 . 0 1 4 a 2 2 - 0 . 0 3 6 a 2 2 FIGURE 1. TARGET CHAMBER - 20 -CHAPTER I I I EXPERIMENTAL ARRANGEMENT FOR THE CORRELATION MEASUREMENTS The schematic diagram below shows the placement of the target, c o l l i -mators, gamma counters and s o l i d state counter with respect to the incident deuteron beam d i r e c t i o n . The angles ^ j j O v j and (j) , defined i n Chapter I I Part D i i . ) are also indicated i n t h i s figure. Collimators Deuteron Beam Au backed B 1 0 target The p a r t i c u l a r s of the ind i v i d u a l parts of the experimental appara-tus are discussed i n the following Chapter, including the design and cons-truction specifications of the electronics and mechanical components. Also included i s a description of the alignment and operation of the mechanical com-ponents. A.) TARGET CHAMBER AND TARGET. The target chamber including the placement of the s o l i d state counter (S)and beam collimators ( A ) i s shown i n Figure 1. Considerable e f f o r t was made during the design and construction of t h i s chamber to ensure that the arrangement of collimators, target (5) and s o l i d state counter was symmetric about the Z axis (K ) . Also, the mass of gamma absorber i n the region between the source target) and the gamma counters was kept to a minimum. Refer-rin g to Figure 1. t h i s i s the region above the plane which passes through the l i n e (L) and i s perpendicular to the plane i n which the crossection of the target box was taken. To t h i s end the sphere (B) forming the majority of the target box enclosure was made of aluminium and the mass of the s o l i d state counter mount i n the above region was kept to a minimum. As a check on the effect produced by gamma absorption i n the target chamber, the intensity of gammas from 60 a Co source placed at the target point, with the chamber removed, was recorded over the half sphere described previously. The gamma in t e n s i t y was then record-ed with the source i n the same pos i t i o n but with the target chamber and s o l i d state counter i n place. I t was found that to within experimental error ( - 1 $ ) no f l u c t u a t i o n i n gamma in t e n s i t y was introduced by the target chamber and s o l i d state counter. The effect due to gamma ray absorption i n the Au target backing was calculated using the known shape of the backing and was found to be negligible (< O.tf). The i s o t o p i c a l l y pure ( 9 9 $ purity) B 1 0 targets, 5 0 0 micrograms per om2 i n thickness, were obtained from A.E.R.E. Harwell. Gold was chosen f o r the target backing because of i t s low (dp) reaction crossection. A backing t h i c k -ness of 0 . 0 0 0 5 inches was selected which j u s t stopped the 1 .5 '.Mev?., incident deuterons, but did not introduce a large amount of energy straggle i n the r e -action protons which passed through the f o i l . This target was supported by a brass ring of 0 . 0 0 4 inch thickness and 0 . 0 1 0 inch width which produced neg-l i g i b l e gamma absorption. B.) COUNTERS i . ) SOLID STATE COUNTER. The Q value f o r the B ^ d p j B 1 1 reaction i s 9 . 2 4 Mev , therefore the s o l i d state counter used to detect the protons from t h i s reaction had to stop approximately 10 Mev. protons i n i t s active region to ensure that the res-ponse of the counter was l i n e a r f o r a l l the protons r e s u l t i n g from the reac-t i o n . For t h i s reason a deep depletion depth (r-'lmm) Nuclear Diodes (PH 8 -2 5 - 1 0 ) counter was chosen. The response of t h i s counter to a t h i n Am 2^ alpha i r - 4 7 / M E v 3 " . 4 - 3 5 M E V X 24-1 A M SOURCE 5 . 5 7 8 M E V ,-' i5o 2.00""' 2S"o 3 o o CHANNEL NUMBER FIGURE 2» SOLID STATE COUNTER RESPONSE CHANNEL NUMBER FIGURE 3. RESPONSE OF 2" x 2" Nal CRYSTAL - 22 -source i s shown i n Figure 2. This spectrum was obtained using the ORTEC. Low Noise Preamp. Model #101 and Ortec. Low Noise Biased Amp. ( t o t a l electronic Noise ~ 6 Kev). Indicated on t h i s curve are the alpha energies corresponding to each peak. The resolution of th i s counter at 300 v o l t s applied bias was found to be approximately 30 Kev. During the angular corre l a t i o n measurements i t was found that deuterons scattered from the beam were reaching the s o l i d state counter. To prevent t h i s from happening an aluminum f o i l thick enough to stop deuterons of the beam energy (approximately 0.0008 inches) was placed d i r e c t l y i n front of the s o l i d state counter. Referring to Figure 1., the f o i l (d) i s indicated. Also, during these measurements i t was found that a f t e r 10^ to 1 0 ^ proton counts of average energy 5 > Mev \, the resolution of the counter decreased and the leak-age current increased rapidly with increasing counts. This counter degenera-ti o n was consistent with the normal l i f e t i m e of s o l i d state counters as out-l i n e d by Dearnaley and Northrop 35.) i i . ) GAMMA COUNTERS. Harshaw 2" x 2" Nal (Tl) c r y s t a l s mounted on 6810A photomultipliers were used f o r the gamma detectors. The 6810A photomultiplier was chosen be-cause of i t s f a s t response, high resolution, and low delay time between anode pulses a r i s i n g from electrons emitted simultaneously at d i f f e r e n t points on the photocathode. This low delay time i s desireable f o r use with the f a s t coincidence c i r c u i t . The response of t h i s system to various gamma ray c a l i -bration sources i s shown i n Figure 3. Indicated i n t h i s figure are the spec-t r a from two radioactive sources Cc£® (2 gammas 1.33 MeV. • and "1117 Mev.) and Rath (2.61 Mev'.)« To determine the response of the gamma counter to higher energy gammas an i s o t o p i c a l l y pure B 1 1 target was bombarded with a 163 Kev proton beam from the U.B.C low energy accelerator. The 4.43 Mev. GAMMA BAY ENERGY IN Mev < FIGURE 4. INTERPOLATED GAMMA RAY SPECTRUM FIGURE 5 . ANGULAR DISTRIBUTION TABLE gamma component from the reaction i s displayed i n Figure 3. To interpret the gamma spectra obtained from the B3-^(dp)B-^ reaction i t i s necessary to know the response of this counter to 2.89 M«r,"\ gamma rays. The relative heights of the three peaks, the f u l l energy and the one and two photon escape peaks, were obtained by linear interpolation between the 4.43 Mev. and the 2.61 Mev . peaks. Figure 4. displays the interpolation method. The relative peak heights of the f u l l energy peaks were determined from e f f i -ciency curves and peak-to-total ratios, (References 17 and 18) for a 2" x 2" Nal crystal. The heights of the peaks i n the spectra were assumed to change linearly from 2.61 i Mev , to 4.4.3 i-Mev"'-''.. gamma energies. Since the interpo-lation i s only carried over a small energy interval (approximately 300 KevO above the 2.61 ;. Mev:V, gamma, no large errors i n spectrum shape should be i n -troduced by this procedure. The same procedure was used in approximating the valley depths and other features of the spectrum. The outline of the expected 2.89 Mev.'. gamma spectrum shape resulting from this linear interpolation i s indicated i n Figure 4. by the dashed l i n e . C.) THE ANGULAR DISTRIBUTION TABLE The requirements on the table were that the distance between the tar-get point and both gamma counter faces would be held fixed, independent of the position of the counters and that the faces of both counters would be held perpendicular to a line joining the target point and the geometric centre of the counter face. A drawing of the table i s shown i n Figure 5. Gamma counter number 1 i s mounted on a T beam arm (T) which i s fixed by means of two MJ2 R & M r o l l e r bearings 3 to a 3" diameter steel shaft (2) tapered at the top to f i t the bearings. The shaft was tapped at the lower end and screwed into the gun mount base (4). This structure allowed gamma counter 1 to be moved in the horizontal plane at a fixed distance from the target point which . . lay on the centre line of the shaft. Gamma counter 2 was mounted on an alum-5 . S . P R E A M F NQ.I X T A L PHoronuLT l £ E M I T T E R M 6 I X F A L AHOTOMOLT 2 d EM (TTEfi V A R . D E L A T VAR. DELAY C O S M I C O , D . L . L I N . A M R 1 OD5M/C S C A 1 + v e o u r - vaour C 0 6 H I C O.O.L . . U N / . A M P 2 h V A R . D E L A Y C O S M I C UD.L. UN. A M P 3 . 5 C A 2 D E L A Y E D O U T P U T C O S M / Q S C A 5 NUCLEAR D / V T A _ _ 1 2 0 •' I N T E R N A L I G A T E . F A S T SliDW COtMC. COjNC. G G C V l T SCALER 1 S C A L E R . 2 . S C A L E R 4-FIGURE 6. CIRCUIT DIAGRAM "* 24 -inium arm ($) attached to a cart (6) which moved along an arc described by th© curved 6" steel I beam (?), ..... The I beam was_ f i x e d to the''gun mount tabl®;(S) which rotates i n the horizontal plane on the gun mount bearing ( 9 ) . - ''By- ao^ipg the cart and the gun mount table, i t was possible to locate gamma counter 2 oh any po^nt o f " a h a l f sphere centered at the target point. This system was designed such that a load of 3 0 0 pounds of shielding, when placed at the extremities of either of the two arms of mounts ( l ) and ( 5 ) , would oause less than a deflection of the arm ends from the no load position. This chaapf acteristic was checked under load - no load conditions and found to be satis* fied. -;-The centres of rotation of both gamma counter mounts were aligi^ ®fi such that they coincided with the target point which was set at 44£" above the floor of the Van de Graaff laboratory. This i s approximately the distance that a horizontal particle beam from the Van de Graaff emerges from the beam 8ele%£or box. The alignment was carried out by mounting pointed brass rods from the centre of, and perpendicular to, the positions of the gamma counter faces on each counter mount. The mounts were then adjusted such that the ends of both brass pointers coincided with the target position, independent of the posit-ion of either mount. The target chamber was then set i n place and the proton counter, target and collimators were centered visually by sighting down th® Van de Graaff beam direction with a WILD T2 theodolite. A l l three components. 1" ' • 1 " were centered to within an estimated ^ 7 , s D„) ^ELECTRONICS A1 schematic diagram of the circuitry used for the correlation measure-ments i s shown i n Figure 6. A current pulse generated in the solid stats . counter'by an impinging charged particle was fed into th© solid state. pr«ara»..,• p l i f i e r and a pulse proportional to the tine integral of th© current f x m counter appeared at the preamplifier output. This pulse pass@d FIGURE 7 . , Nuvistor Preamplifier Circuit » 25 ~ 500 nano second variable delay and was amplified by a COSMIC DOUBLE DELAY LINE LINEAR AMPLIFIER 1 (DDL LIN. AMP 1). The amplified and shaped pulse was fed into COSMIC SINGLE CHABNEL ANALYSER 1 (S.C.A. l ) and i f the pulse was of the amplitude range selected by the analyser, a pulse was generated at both outputs when the delay l i n e shaped pulse passed through the zero crossover point. The pulse from the p o s i t i v e output went to SCALER 1 and the slow coincidence c i r -c u i t . The pulse from the negative output went d i r e c t l y to the f a s t coincid-ence c i r c u i t . A s i m i l a r process was followed by pulses from the two Nal -photomultiplier - emitter follower combinations which were recording gamma rays. When a coincidence event occurred i n the f a s t coincidence c i r c u i t , a pulse was generated, opening the i n t e r n a l gate i n the NUCLEAR DATA 120 KICKSORTER.(K.S.) allowing the coincident gamma ray pulse from the delayed output of the amplifier to be analysed by the kicksorter. As indicated i n the diagram scalers^ and 5 re-corded the number of f a s t and slow coincidence events which occurred during each run. i . ) NUVISTOR PREAMPLIFIER. The c i r c u i t diagram of the nuvistor preamplifier appears i n Figure 7. The c i r c u i t design, b a s i c a l l y that of Alexander 33.) was modified by Dr. G. Jones 34-.) f o r use i n t h i s experiment. The two nuvistors served as a high gain amplifier and the output was'fed back through capacitor C3(5p£), making the system a current integrator. A pulse proportional to the t o t a l charge i n -jected at the input by the s o l i d state counter appeared at the emitter of transi s t o r (T2). The combination of T 3 a n d T 4 s e r v e d a s a l o w output im-pedance emitter follower. This amplifier combined the desireable character-i s t i c s of low noise (approximately 25 Kev) and f a s t risetime (approximately 15 ns)„ (These values of noise and risetime refer to the case of zero input capacitance.) The combination of R 2 and C-j was chosen to give the required pulse decay time (100 ns) compatible with the Cosmic amplifiers. o+39 V. [I o) O U T P U T FIGURE 8. PHOTOMULTIPLIER EMITTER FOLLOWER FIGURE 9i PULSE GENERATOR ' - 2 6 ° i i 0 ) PHOTOMULTIPLIER EMITTER FOLLOWER. An emitter follower was used i n conjunction with the output of the 6810A. photomultiplier and the c i r c u i t diagram i s given i n Figure 8. A negative cur-rent pulse from the anode of th® photomultiplier was fed into the input (base of Tl) of the emitter follower. This pulse turned on a small amount of current through T l which turned T2 hard on. The pulse which appeared at the output of the emitter follower was proportional to the time i n t e g r a l of the current pulse ( t o t a l charge) from the photomultiplier which was integrated on the 3000pf capacitor. The lOylif condensor between the emitter of T l and point PI "boot-straped" t h i s point and had the desired effect of lengthening the decay time of the pulse to make i t compatible with the Cosmic amplifiers. i i i , ) FAST COINCIDENCE CIRCUIT. The Fast Coincidence C i r c u i t consisted of three 10ns wide pulse gener-ators, triggered by the negative outputs of the Cosmic S„C„A, and a f a s t d i s c r i -minator. The c i r c u i t diagram of the pulse generators appears i n Figure 9o A negative 6 v o l t (20 ns risetime) pulse from the Cosmic S„C„A. appeared at the input of the pulse generator. This pulse was shaped by diode T l 72 such that the negative going portion of the pulse turned on enough current i n t r a n s i s t o r T l to trigger the 20 ma tunnel diode. The positive going portion turned trans-i s t o r T l completely off and reset the tunnel diode. The f a s t r i s i n g pulse (approx, 3ns risetime) generated by the tunnel diode turned the 10 ma standing current i n transistor. T2 completely off. The re s u l t i n g 10 ma pulse appeared at the c o l l e c t o r of T2 where i t was clipped by the shorted delay l i n e to th© required length (5 ns). This pulse then passed through t r a n s i s t o r T3 to the output point. The net ef f e c t of t h i s c i r c u i t was that a short (^5 ns i n width) f a s t (3 ns r i s e and f a l l times) 5 ma pulse was generated at the output when a pulse from the S„C„A, arrived, A small negative pulse also appeared at the output caused by the resetting of the c i r c u i t . FIGURE JO*";. Coincidence Circuit - 5 v + 3 0 v INPUTS i cHf IMIOO — H — W \ A — r IN'OO - J — V \ A 3.CHr IN/oo f.ZK "°-*-6V B A T T E R Y IK O. I y U f O U T P U T T D 104-(to m a ) He FIGURE 11. SLOW COINCIDENCE CIRCUIT - 27 -The coincidence c i r c u i t diagram appears i n Figure 10, Pulses a r r i v i n g from the pulse generators at the three inputs passed through t r a n s i s t o r s T l , T2 and T3 and then through a 20 ma tunnel diode 0 The amount of steady state cur-rent passing through the tunnel diode was controlled by the "NUMBER OF COINCI-DENCES" switch. When the switch was i n i t s "singles" mode, approximately 17 ma was flowing through the tunnel diode. Therefor® i f a pulse (5 ma) arrived at any of the three inputs the 20 ma TD switched into i t s high voltage state, turn-ing on tran s i s t o r T 5 , The voltage at the c o l l e c t o r of T5 f e l l to approximately - 5 v o l t s turning tra n s i s t o r T6 off f o r the length of time determined by r e s i s t -ance (R) and capacitance (C), After a time RC (approximately 2yus) t r a n s i s t o r T6 again came into conduction and i t s c o l l e c t o r voltage f e l l to - 5 v o l t s . The 6 8 p f capacitor connected to the c o l l e c t o r of T6 drew enough current at th i s time to reset the tunnel diode. The pulse generated was designed to open the kick-sorter gate f o r 1 ^is„ The difference between kicksorter "ON" time (l/As) and T6 off time (2yus) resulted from the slow r i s e time (lyus) of the gating pulse. By setting the "NUMBER OF COINCIDENCES" switch to "doubles" or " t r i p l e s " mode a 12 ma or 7 ma standing current flowed through the tunnel diode. Thus when the switch was set to "doubles" (or " t r i p l e s " ) two (or three) pulses had to ar r i v e simultaneously ( i , e , within twice the pulse width) at the inputs to cause the tunnel diode to trigge r an output pulse, iv. ) SLOW COINCIDENCE (RANDOM COUNT RATE MONITOR) CIRCUIT. During the T r i p l e Coincidence measurement i t was found that the correct-ion f o r random coincidences was s u f f i c i e n t l y large to require the continued moni-toring of the t r i p l e random rate. Therefore a c i r c u i t to monitor the random coincidence rates was incorporated, A ™Slow Coincidence C i r c u i t " which d u p l i -cated the response of the f a s t t r i p l e coincidence c i r c u i t but which had a longer resolution time was b u i l t for t h i s purpose and i t s c i r c u i t diagram i s displayed i n Figure 11. Pulses from the positive outputs of the S„CA„'s arrived at the •* 28 *• three inputs and only when a l l three pulses arrived simultaneously (within 100 of each other) did the tunnel diode trigger into i t s high voltage state. This tura®d the tra n s i s t o r on producing a negative 6 v o l t puis© a t the output,, NUMBER OF COUNTS XIO" 3 PARTICLE ENERGY IN Mev. - 29 -CHAPTER 17 ANGULAR CORRELATION MEASUREMENTS AND RESULTS The target^ collimators and s o l i d state counter were aligned as outlined i n Chapter I I I Part Co) and a 1.5 Mev deuteron beam from the U.B.C. Van de Graaff accelerator was directed into the target system (refer to Figure 1.) along the collimation axis. The machine energy was maintained at 1.5 Mev to an accuracy of about 5 Ker throughout a l l the measurements made during the experiment. P a r t i c l e s from the glO target emerging i n a ^ o cone at an angle of less than 5 from the incident beam were detected by the s o l i d atate counter and t h i s p a r t i c l e spectrum i s displayed i n Figure 12.). The sources of the various peaks i n the spectrum are indicated above each peak along with t h e i r corresponding energies. Effects which produce most of the broadening of the proton peaks, along with an estimate of the magnitude of each effect f o r 5 Mev protons ares a) B"^ target thickness (25kv) b) Straggling of protons i n target, gold target backing, and aluminum f o i l i n front of the proton counter. (50kv) c) Varying length of t r a v e l of protons i n backing and f o i l due to di f f e r e n t angles of incidence of protons i n f o i l s . (20kv) d) Kinematics, i . e . proton energy difference due to change of energy with angle of emission (50kv) e) Electronic noise i n counter and amplifier. (4-Okv). The t o t a l peak broadening to be expected from the above mentioned effects i s approximately 90 Kev which compares favourably with the measured value of 110 Kev. This small discrepancy can be accounted f o r i n terms of the beam energy spread and the apparent increase i n counter noise from the high f l u x of beta p a r t i c l e s (see below). Alpha p a r t i c l e s from c£)B©^ leading to the ground and f i r s t CHANNEL NUMBER . FIGURE 13. DEUTERON = EXCITED GAMMA SPECTRUM 7 5 - 6 - 3 o 3 6 ? IZ DELAY TIME IN nsec FIGURE Uo COINCIDENCE CIRCUIT RESPONSE - 30 -excited states also appear i n Figure 12. . The energy c a l i b r a t i o n f o r the alphas i s not the same as that indicated f o r the protons, because of the much higher energy loss of the alphas i n the gold and aluminum f o i l s . Most of the a alpha peak width i s due to the large width of the states populated i n the Be° nucleus. The gamma spectrum induced by the deuteron beam i s displayed i n Figure 13o This spectrum i s devoid of any prominant peaks bscauss of the exceeding-l y large number of states populated by the (dp) and(dn) reactions with the target and surroundings. A sharp r i s e at the lower end of th© spectrum (ap-prox„ channel 14) r e s u l t s from f$ a n n i h i l a t i o n radiation from the reaction B 1 0 ( d n ) C n ( B 1 ] L y 2 + ) o That i s th© dn reaction on B 1 0 produces C n which has a +• yShalf l i f e of approximately 20 minutes„ At the high energy end of the spec-trum (approx. channel 120) an apparent peak, results from the saturation of the Cosmic Linear Amplifier. A.) PROTON9 GAMMA COINCIDENCE MEASUREMENTS To obtain measurements of (p^Tf) coincidences the base l i n e and window width of the Cosmic Single Channel Analy s®@ c i r c u i t diagram) j, Figure 6. was adjusted to select protons populating the 5«03 Mev l e v e l i n B^„ Referring to Figure 12. th i s corresponded to selecting only those pulses whose heights lay between the two arrows on either side of p^ « S 0C 0A„ 2 was set to select those gamma pulses whos® height was above the point indicated i n Figure 13. by tho arrow marked 1. This corresponded to choosing only those gammas whose energy was greater than 0.7 Mev. The coincidence c i r c u i t was switched to the double coincidence mode and the timing adjusted by varying delays 1 and 2 (see c i r c u i t diagram) u n t i l a maximum coincidence count rate had been obtained. (For the relativ® delay time versus coincidence count rat© curve see Figure l 4 . ) o In t h i s figure the background random coincidence count rat© has been subtracted. The region inside the two v e r t i c a l dotted l i n e s cor-GAMMA ENERGY IN Mrav, - 31 -responds to the resolution time of the coincidence circuit., From t h i s the eff i c i e n c y of the coincidence network was estimated as being the r a t i o of the area inside the v e r t i c a l l i n e s to the t o t a l area 0 In t h i s estimate i t was assumed that a l l the "true" coincidences lay inside the region shown i n Figure 1 4 . and that the response of the coincidence c i r c u i t did not d i s t o r t the shape of the actual time versus count rate curve,. The ef f i c i e n c y was also calculated as the r a t i o of the observed coincidence count rate to the t h e o r e t i c a l coincidence count rate calculated from the single count rates i n the proton and gamma detectors (allowing f o r the counting efficiences of the proton and gamma counters), and was found to agree with t h i s estimate. This loss i n e f f i c i e n c y which arose from time " j i t t e r " i n the c i r c u i t , can be attributed almost exclusively to the Cosmic Single Channel Analysers, each of which has a time " j i t t e r " of 8 - 9 n s . This j i t t e r r e s u l t s from the f a c t that d i f f e r e n t pulse heights from the Cosmic Amplifier cause the out-put pulses from the S.CA. to be generated at times that vary by 8 - 9 n s a f t e r the amplifier pulses pass through t h e i r zero crossover points. With the timing set at the maximum count rate position, the pulses from the gamma counter were gated into the kicksorter when a coincidence ©vent was recorded. The coincidence gamma ray spectrum (p^tf) i s shown i n Figure 15. Indicated on t h i s spectrum are the f u l l energy ^ ( F ) and single and double escape ^ ( S ) , X^(D) peaks of the three gamma rays r e s u l t i n g from the two dif f e r e n t decay schemes shown. In the analysis of the 5.03 Mev l e v e l decay scheme i t was necessary to know the sfe&pe of a 5 Mev gamma ray spectrum below the pair production peaks. An approximation to t h i s spectrum was obtained by recording the coin-cidence gamma ray spectrum f o r the decay of the 4 . 4 6 Mev l e v e l which has been shown (Donovan et a l 5.) to decay by gamma emission d i r e c t to the ground state only. This spectrum was obtained by adjusting S„C<,Ao 1 to select the protons CHANNEL NUMBER FIGURE 16. p 2£ COINCIDENCE SPECTRUM - 32 -(p£) populating the 4-«4-6 le v l e v e l and proceeding with the p^tf coincidence measurement as outlined f o r the p^Jf measurement. Figure 16 0 shows the spec-trum of the gamma rays depopulating the 4.4.6 Mev state, i . ) P3&3 ANGULAR CORRELATION MEASUREMENT The (p^&j) c o r r e l a t i o n measurement was obtained by setting S.C.A. 1 so that protons populating the 5.03 Mev l e v e l (p^) were selected as before,, S.C.A. 2 was then adjusted to select only those gamma pulses whose height was above the point indicated i n Figure 15„ by the arrow marked 2. This corres-ponded to selecting gamma energies above 3.3 Mev» The coincidence count rate w a s * n e n recorded f o r various angles -©•, where i s the angle between the incident beam and the gamma propagation d i r e c t i o n , (see Table To ensure that no time dependent effects would introduce fluctuations i n the an-gular d i s t r i b u t i o n , the angle -©-was chosen randomly and the count rate at each angle recorded numerous times. Th© number of coincidence counts at each point was normalised to the number of protons (p^) recorded by scaler 1« Throughout the double correlation measurement, readings of the random coincidence count rate were recorded. These were obtained by setting the pro-ton c i r c u i t timing (delay l ) approximately 200 nano seconds off coincidence and recording the coincidence count rate. The angular d i s t r i b u t i o n of random co-incidences was found to b® i s o t r o p i c , and had an average magnitude of 28 counts at each angle l i s t e d below. The random coincidence rate was also calculated using the usual equation. N , » 2N N y T rand p 0 ^ ' ^rand = random count rate Np = count rate at coincidence c i r c u i t input from proton counter Njr = count rate at coincidence c i r c u i t input from 3 - 33 -counter '"}"' ~ width at one half maximum of the pulses from the timing pulse generators =* 5ns The calculated and measured random count rates w@rs found to agree very d o s e l y (within 10$) <, TABLE 4 c p-X ANGULAR DISTRIBUTION CoinCoCounts "ft $f F i t t e d Curv® 0 184 183 10 171 183 20 181 .185 30 208 1S9 40 200 194 50 1S6 201 60 199 209 70 218 217 80 221 223 90 234 225 100 232 223 110 211 217 120 219 209 * Coincid®nc© Counts per 4 proton counts Defined and calculated i n Chapter Y An upp®r l i m i t on the l i f e t i m e of the 5,03 Mev stats was measured during t h i s measurement f o r the purpose outlined i n Chapter I I Part F.) This l i f e t i m e l i m i t was determined by i n j e c t i n g timing pulsss at the inputs to the s o l i d stata counter preamplifier and gamma counter emitter follower i n time - 3 4 -coincidence (within 1 0 ns of each other) which closely approximated the actual pulses produced by these counters. The timing of the coincidence c i r c u i t was then adjusted to the maximum coincidence count rate position by varying the time delays f o r the two counters and the timing noted. The P ^ - j coincidence measurement was then made and the coincidence count rate optimised as described at the beginning of t h i s section and the timing noted once again. The timing f o r the P ^ ^ coinci= dence measurement was found to agree to within 10 nswiththe timing found using the simulated pulses which represented a aero l i f e t i m e state ( i , e . l i f e t i m e short" er than mesureable with the coincidence c i r c u i t , ) , Also the half width of the P^ft-j coincidence count rate versus time delay curve (refer to Figure 1 4 . ) was found to be approximately 15 ns. Therefore, an upper l i m i t on the l i f e t i m e of the state may be set at approximately 35 ns and f o r the purpose of Chapter I I Part F 0) a l i m i t of 1 0 0 ns may be set with certainty, i i , ) p 3« 1 and p 3 * 2 ANGULAR CORRELATION MEASUREMENT. Spectra s i m i l a r to the one displayed i n Figure 1 5 . ) were obtained f o r var-ious angles -6- using the method outlined previously and the P-J^J A N < * ?3^2 A N 6 U X A R d i s t r i b u t i o n s were extracted from these spectra by a method of spectrum stripping. Spectrum stripping i s applied to cases where i t i s necessary to determine the contribution of i n d i v i d u a l gamma rays i n a spectrum which i s the sum of the spec-t r a of several d i f f e r e n t gamma rays. The spectra to be analysed are sum spectra of ^ 2 and ^  and the contributions of each of these spectra are determined by f i t t i n g the known shapes of the in d i v i d u a l gamma ray spectra to the sum. The shape of the in d i v i d u a l gamma ray spectra are.usually derived from separate ex-periments and i n th i s case the shapes of and if were determined using sources (Chapter I I I Part B.) and the shape of was approximated by the shape of the pure 4 o 4 6 Mev gamma ray from the B ^ ( d p ) B ^ reaction. One advantage of using the 4 . 4 6 Mev (p 0^) coincidence gamma rays from the B^(dp 0)B"^* reaction i s that the pjf coincidence spectrum i s recorded under the same conditions 1*4 M Q 300 200 CO E-* C5 8 o ss M KJ 100 o -100 • E X P E R I M E N T A L . x R E M I N D E R 3 1 2 H 1 40 CHANNEL NUMBER 3 63 FIGURE 17. SUMMED AND p 3 SPECTRUM - 3 5 -as the p^ }f spectrum, therefore the background (random coincidences) should be approximately" the same f o r both spectra since they both took of the order of the same length of time to be recorded. The p a r t i c u l a r method used i n the analysis of the p^ ^ coincidence spectrum was f i r s t to s t r i p the portion of the spectrum (refer to Figure 1 5 o ) from the sum spectrum, the remainder was then analysed into i t s two components ^ and The f u l l energy and two pair production peaks of the 4 o 4 6 Mev gamma ray (Figure 1 6„) and ^  (Figure 1 5 „ ) spectra were caused to f a l l i n the same channels by s h i f t i n g the whole p^ft" spectrum up 9 channels. The p^ X spectrum was normalized to give the same t o t a l number of counts as the Y>2 ^  spectrum between channels 6 0 and 9 0 . The shape of the 5 . 0 3 Mev (#^) spectrum below channel 6 0 as obtained by t h i s process i s displayed i n Figure 1 5 . as a dotted l i n e . The shifted ( p 0 X ) spectrum was then subtract-ed from the normalised (p-^X) spectrum leaving only the summed (p^ Xj) a n d (p<2 spectrum. The •©• - 0 ° spectrum resulting from t h i s subtraction i s displayed i n Figure 1 7 . Three curves are indicated i n t h i s f i g u r e , the con-tinuous l i n e represents the gym of the and ^  spectra. The dashed l i n e s t a r t i n g at channel 28 represents the expected shape of the X-^ spectrum i n the region from channel 28 to channel 5 1 obtained by the method outlined i n Chapter I I . The dashed curve has been normalised to the experimental points by making the area under the curve betvieen channels 5 1 and 6 0 equal to the to-t a l number of counts i n these ten channels. The dotted curve i s the remain-der when the 5^ spectrum ( i . e . dashed curve) has been subtracted from the summed spectrum and should represent a pure %^ radiation. Referring to F i g -ure 1 7 . the t o t a l number of counts i n the subtracted spectrum l y i n g between channel 5 1 and 6 1 was recorded f o r each angle •©- at which ( p ^ ) coincidence spectra were taken and are l i s t e d i n Table 5 . i n column 1 „ Also l i s t e d i n th i s table i n column 2 i s the t o t a l number of counts i n the remainder spectrum TABLE 5o ) Y AND p ANGULAR DISTRIBUTION 3°1 *3 2 •e- 1. 2 o Norm. Factor P 3 Y 1 A.D. P 3 tf2 A» D« O 546 2002 1 546 2002 15 538 2470 1.01 543 2495 30 325 1576 1.03 335 1623 45 486 1713 1.08 525 1850 60 338 1828 1.15 389 2100 75 279 1208 1.21 338 1461 90 217 1869 1.23 267 2299 ~ 36 ~ between channels 30 and 45. As described i n the preceeding paragraph the areas l i s t e d i n Table 5. are normalised to give a constant number of counts i n the (p ft ) peaks so, 3 3 to normalise the (p # ) and (p y ) spectra to a constant reaction rate, the 3 1 3 2 P3 0*3 angular d i s t r i b u t i o n must be removed. The (p^ K^) angular d i s t r i b u t i o n i s l i s t e d i n Table 5., under the column marked 11 Normalisation f a c t o r 1 and was taken d i r e c t l y from the f i t t e d p ft angular d i s t r i b u t i o n (Chapter V Part A , ) i . ) 3 3 To obtain the normalised (p^ and (p^ angular d i s t r i b u t i o n columns 1 and 2 i n Table 5o were multiplied by the normalisation factor and the r e s u l t i s l i s t e d i n the remaining two columns,, Bo) TRIPLE COINCIDENCE MEASUREMENT Cosmic Single Channel Analyser 1 was adjusted to select only the p^ protons as outlined i n Part A.) S.C.A. 2 and 3 were set to select only 0 ^ and #2 respectively, when the coincidence c i r c u i t was operated i n the t r i p l e coin-cidence mode0 This was done by setting the baseline of S.C.A. Z at a l e v e l which corresponded to selecting those gamma pulses which lay above the point indicated on Figure 17 D by the arrow marked 3> ( i . e . Eygreater than 2„4» Mev). S„CoA0 3 was then set up to select gamma pulses below t h i s l e v e l and above the ^ ^ a n n i h i l a t i o n r a d i a t i o n peak (above channel 15 i n Figure 17t), The t r i p l e coincidence count rate was maximised by adjusting delays 2 and 3 f o r maximum p^o' double coincidence count rates. With the coincidence c i r c u i t set on the t r i p l e coincidence mode, f i v e quantities were recorded by the scalers (see c i r c u i t diagram) f o r each p a r t i c u l a r angular configurations a) The t o t a l number of counts i n each of the three counters. b) The number of f a s t t r i p l e coincidence counts. ?) Slow coincidence (random coincidence monitor) counts. A sampling of the random coincidence count rate was taken at i n t e r v a l s throughout the measurement by setting the p^ timing (delay l ) approximately TABLE 6. TRIPLE CORRELATION RESULTS « i * 2 • N coinc N p i o - 7 ^coinc Np10"5 90 90 0 206 2.61 0.79 90 90 45 81 0.87 0.93 90 90 90 14-3 1.63 0.87 90 0 0 144 1.66 0.87 0 90 0 139 1.29 1.08 - 37 -300 nano seconds off coincidence and recording the coincidence count rate. The random coincidence counts were subtracted from the measured t r i p l e coin-cidences and the r e s u l t i s displayed i n the Table 6 C (For a discussion of random rates and subtraction see Appendix I i ,. The angles, defined i n Chapter I I Part D„) were chosen to give a maximum fluct u a t i o n i n the t r i p l e coincidence count rate 0 C.) DISCUSSION OF ERRORS io) ERRORS RELEVANT TO ALL ANGULAR DISTRIBUTION MEASUREMENTS. (a) Counter Mounts An attempt was made during the design of the target chamber and gam-ma counter mounts to ensure spherical symmetry about the target spot. I t was found that the centre of rotation of both counters was coincident with the target spot to — inch f o r a l l counter orientations, therefore the d i s -tance of the counter face ( r ) to the target spot was f i x e d to t h i s order of magnitude. A f l u c t u a t i o n i n distance dr between counter and source at the target spot would introduce a fl u c t u a t i o n i n the count rate due to the change i n s o l i d angle dQ-1 r e f f r e f f ~~ "effec t i v e distance" of source from detector (See J.L, Leigh 39.) r e f f = r e x + e (e i s > 0) depends;1/ on the gamma ray energy and cry-s t a l geometry. From above i t i s found that d£2 _C1 dr, r e f f e f f The experimental conditions are such that - 38 -e « r ex rex = 3 ' 5 i n c h dr . i . 32 Therefore 1 d.m - 2 ( 3 2 ) _ %ax. 3.5 That i s , t h i s would introduce a maximum apparent f l u c t u a t i o n i n gamma int e n s i t y of about 2%, (b) Target Box Absorption The absorption of the target box was checked with a gamma source as outlined i n Chapter I I and was found to be spherically symmetric to within experimental error (approx. 2%). Therefore the maximum f l u c t u a t i o n i n t r o -duced by geometrical and attenuation effects was less than 3 $ . (c) Non C y l i n d r i c a l Symmetry I t was pointed out i n Chapter I I I that a great deal of care was taken i n the construction and alignment of the target box and s o l i d state counter mount to ensure that the condition of a x i a l symmetry about the deu-teron beam axis was s a t i s f i e d . As an experimental check on how well t h i s condition was s a t i s f i e d a reading of the p 0)T coincidence count rate was re-corded i n both the plane of the beam (as a function of -6-) and i n the plane perpendicular to the beam and passing through the target point (as a function of (J)). I t was found that the coincidence count rate; 1. ) varied by approximately 10$ In the •©• plane and was symmetric about the -6-= 0 point (•& = 0 being the deuteron beam direction) and 2. ) was is o t r o p i c i n the § plane to within experimental accuracy (~1%). Although t h i s was not an absolute test i t does give a strong i n d i c a -t i o n that the condition of a x i a l symmetry was s a t i s f i e d to the degree required f o r the experiment. <=. 39 -ii„) ERRORS RELEVANT TO INDIVIDUAL DOUBLE ANGULAR CORRELATION MEASUREMENT (a) y ^  Angular Distribution Care was taken during t h i s measurement to ensure that no time depend-ent effects would introduce fluctuations i n the angular distribution,, Also the amount of background due to random coincidences that had to be subtracted from the angular d i s t r i b u t i o n was small, and adds very l i t t l e to the s t a t i s -t i c a l error. Therefore, most of the error, outside the s t a t i s t i c a l error, should arise from those effects outlined i n i . ) (b) $1 a n d P3 y 2 A n S u l a r Distributions 1) Normalization. As outlined i n Part B.) i i . ) of t h i s chapter these two angular d i s -tributions were normalized to the p„ }f 0 angular d i s t r i b u t i o n and since there i s an uncertainty i n the p^ ^ d i s t r i b u t i o n , t h i s w i l l introduce a correspond-ing error i n the p^ ft" ^ and p^ ^ angular d i s t r i b u t i o n s . The angular d i s t r i -butions p„ were f i t t e d to functions of the form (-©) = ^ b n ( ^ ) P n ( c o s - 0 2 ) i n -(See Chapter V Part A.) and the s t a t i s t i c a l errors on the c o e f f i c i e n t s b n ( ^ ) i n the normalized ( i . e . b Q= l ) p^ angular d i s t r i b u t i o n were found to be; W error = * °' < B This w i l l introduce a corresponding error i n the p_ and p. ' c o e f f i c i e n t s of the same order of magnitude. 2 ) Spectrum Stripping. The p^ a n d P3 ^ 2 a n S u l a r d i s t r i b u t i o n s were obtained by spectrum stripping. This involves the subtraction of one s t a t i s t i c a l l y uncertain spec-trum shape from another which also has a s t a t i s t i c a l uncertainty. As an example consider the -©- = 0° case (refer to Figure 15.). The number assigned to the ,6- = 0 o point i n the P ^ ^ angular d i s t r i b u t i o n table (Table 5.) i s the value of the area A^ which i s the r e s u l t of subtracting the area A^ below the dashed l i n e i n the region between channels 51 and 61 from the t o t a l area under the continuous curve i n t h i s region A^+ Ag. The s t a t i s t i c a l error i n each point on the angular d i s t r i b u t i o n curve i s notVA^ but i s a c t u a l l y /A ^ + ~ ~ A ^ Here and i n the following, the quantities A^ when calculating errors refer to the numerical values of A^ which correspond to the sum of the number of counts i n the region defining A^a A s i m i l a r argument leads to the assignment of an error to the error i n the -0" - 0° p^ t^ angular d i s t r i b u t i o n point, where A^ i s the area i n Figure 15. under the continuous curve between channels 30 and 45, An additional error i n the p^ X2 angular d i s t r i b u t i o n arises from the uncertainty i n the amount of p^ $ ^  spectrum (A^ i n Figure 17,) that should be subtracted'from the summed P^tf-^ and p^ spectrum. The s t a t i s t i c a l uncertainty i n A. i s proportional to the uncertainty i n the value of the p0^T, 4 3 J-angular d i s t r i b u t i o n at this point (-0- - 0°) which i s \f k^^- k'^e That i s , the \ / A i + A' Ay uncertainty due to the p- If, subtraction w i l l be equal to A " ,1 2 = 4 I A X + A 2 \fktfl2 In summary, the errors CT introduced by spectrum stripping i n t o the •©• = 0° point of the P 3 Y ^  a Q d p^ % 2 angular d i s t r i b u t i o n s by spectrum s t r i p -ping were; Similar reasoning was used i n the assignment of(Fp #,0^) ^ o r the other 3 i angles "9-at which the double correlations were measured, i i i . ) CALCULATION OF ERRORS IN THE DOUBLE ANGULAR DISTRIBUTION COEFFICIENTS (b ) The measured p^ tf^ d i s t r i b u t i o n s were f i t t e d to functions of the form - 41 -WX (^)=]T b (£. )P (cos -6-.) i n Chapter V D The errors i n the c o e f f i c i e n t s i n ^ n ( ^ ) were calculated using the s t a t i s t i c a l errors Ol „ (-©•) i n each point x P^ 5 ^ •8- as defined i n ii„) (a) and (b)„ That i s Ay (-6) p 3 * l CTp3 * 3 « » = ±-N(-O-) = number of counts at angle -6- i n the' ]?3 ^  c orrelation measurements,, To the calculated errors i n the b j ^ ' g , based on these s t a t i s t i c a l errors ( f o r c a l c u l a t i o n see Appendix I I ) , were added the errors outlined i n i„) (a)j (c) and i i 0 ) (b) 1 which are expected to produce systematic fluctuations i n the b 1 s„ n i v . ) ERRORS IN THE TRIPLE CORRELATION MEASUREMENT The errors outlined i n i . ) apply to the t r i p l e c o r r e l a t i o n as wel l as the double correl a t i o n measurements. The only additional error involved i n t h i s measurement r e s u l t from the background count rate subtraction outlined i n Appendix I. The random background amounted to about 25% of the t o t a l t r i p l e coincidence counts N(P) at each point P„ The s t a t i s t i c a l uncertain-ty (XN»(P) i n the actual number of t r i p l e coincidence counts N ° ( P ) ( i . e . with randoms subtracted) was Typically W (P) N(P) = 200 counts N°(P) - 150 counts — 50 counts «*• 4 2 ~ and The t o t a l error i n each t r i p l e coincidence measurement was taken to be equal to t h i s error since the errors from i„) are negligible when compared with the s t a t i s t i c a l errorOJJB (pj ( t y p i c a l l y 10$)o - 43 -CHAPTER V ANALYSIS OF CORRELATION RESULTS The double and t r i p l e correlation measurements made between protons (Po) populating the 5o03 Mev l e v e l i n IT*"1 and gamma rays If 9 K and from J 1 2 3 the decay of t h i s state are outlined i n Chapter IV along with the results of these measurements,. These results are analysed i n t h i s chapter i n terms of the correlation theory developed i n Chapter I I . A.) METHOD OF FITTING CORRELATION RESULTS TO THEORY One conclusion can be drawn immediately from inspection of the p^lf^ angular d i s t r i b u t i o n s and that i s that since there i s a non-isotropic angu-L 1 l a r d i s t r i b u t i o n from the decay of the 5.03 Mev state, therefore J p A 2 more detailed analysis of the results i s necessary before further statements can be made of the spin3 of the levels concerned. The results of the p^ #\ ( i - 1, 2 and 3) double co r r e l a t i o n measure-ments Wj, (3.-©: .) appear i n Tables 4, and 5. f o r each angle ••©; . a t which W j ^ i j •fcj was measured. The theo r e t i c a l expression f o r the double co r r e l a t i o n function, expressed i n terms of the spin J J j , the multipole mixing r a t i o s x^ and the tensor parameters a Q , wass W T.(Aifl^JJjXjan) = W . ( J J j c j G j j ) (Chapter I I Part C.) - Z a n % ( J J l ^ i ) E n ( c o s n For the t r i p l e c o r r e l a t i o n measurements, the experimental r e s u l t s , and the theoretical c o r r e l a t i o n functions were (when the two gamma counters were at angles -6^, -0^, ^ ) Wg O ^ j - e ^ j ) - ( l i s t e d i n Table 6.) and n ^ ^ V ~ Z l a ^ A ^ l i J J j X ^ k J X ^ e - ^ ^ ^  )-(See Chapter II Part D«) The Maximum Likelihood estimates ( J , J j , a^, x^) of the values of the - u -parameters J 5 J T S , a were those values which maximized the l i k e l i h o o d a x n function L, NI K W ^ J J j a ^ i ) = rp f j ( W E ^ J J I a n x i ) P ( J J I ) P ( a n ) f • ( X i) with respect to the variables J , J j S a & , x^ „ f (Wg <7^  JJja nXj_) i s the nor= malized d i s t r i b u t i o n function describing the pr o b a b i l i t y of obtaining the measured value of Wg a t the angular positions f o r "the N d i f f e r e n t sets of angular orientations) i f the true value of W was that obtained by equating the set ( J , J p a f l, x^) to the set ( J , J p &&s x^) and i f the meas-ured value of had an error OZ, associated with i t . I f we assume that the errors are distributed i n a Gaussian fashion, with variances CT^  3 then Y The f u s i o n s PCJJj), P(a n) and £ m (x.^) are pr o b a b i l i t y functions introduced to take account of previous knowledge of the p r o b a b i l i t i e s that given values of these variables should occur, P(JJj) - 1 2 ^ J ^ 2 / Follows from 1 / ,.7 f Chapter I I Part F.) 2 ^ J I ^ 2 J — 0 f o r a l l other values of JJ^. P( an) - 0 i f the value of a predicts a negative (unphysical) diagonal density matrix eleraento = 1 otherwise. Two methods of dealing with the p r o b a b i l i t y f w ( x ^ ) have been used i n t h i s chapter. The f i r s t was to assume that a l l values of are equally probable-(This i s not consistent with the experimental evidence (WiUkinson]6.) regarding - 45 -r e l a t i v e p r o b a b i l i t i e s f o r Ml and E2 t r a n s i t i o n s , but i s i n accord with the desire to t r y and make a spin assignment without recourse to arguments i n -volving t r a n s i t i o n p r o b a b i l i t i e s ) . Allowing x^ to take any value leads to the problem of having to solve sets of non-linear equations. The equations were l i n e a r i z e d by assuming a l l the x^ » s were small compared with unity (which i s a physically r e a l i s t i c assumption) and then solved. The x^' s so obtained were i n f a c t small enough to j u s t i f y the o r i g i n a l assumption. I t turned out that t h i s method gave r i s e to f i v e sets of spins J J j which could not be rejected on the basis of the \ test (see Part D.). However, a l l but one of these spin assignments require values of x^ which are considerably larger than those expected on the basis of the observed r a t i o s of Ml and E2 t r a n s i -t i o n p r o b a b i l i t i e s . The second method (Part E.) uses values of f n ( x ^ ) e s t i -mated as described i n Appendix IV. The r e s u l t of including these probabil-i t i e s i n the Maximum Likelihood c a l c u l a t i o n i s that only one spin assignment then remains'. The evaluation of and f o r each set of J J ^ allowed by P ( J J j ) as outlined above would be an extremely tedious calculation but i s i n p r i n c i p l e possible. However, the evaluation of the maximum L was s i m p l i f i e d by factor-; ing L into two sections, one corresponding to the frequency functions f o r the three double c o r r e l a t i o n measurements and the other corresponding to the frequency functions of the t r i p l e c orrelation measurements. (At t h i s point i t i s assumed that P ( & n ) — 1 f o r a l l a n and P(x^) = 1 f o r a l l x^ „ The cor-rect, form f o r P(& n) w i l l be introduced i n Part D.), and f o r P(x^) i n Part F.) L = L« • L" N« where L1 = -JT f - 4-6 -N' = number of double correlation measurements (W„ ) N -N' — number of t r i p l e c o r r e l a t i o n measurements (W ) E r -The reason f o r factoring L i n t h i s way i s that experimentally the errors i n the measured di s t r i b u t i o n s were much smaller f o r the double cor-relations than f o r the t r i p l e correlation ( i , e , (J « ( T * 1 ) therefore the point of maximum l i k e l i h o o d was to a f i r s t approximation independent of the t r i p l e c o r r e l a t i o n measurement. The procedure used i n the analysis was to maximize L 1 (Part B.) then calculate the second order effects of w"EJ ( t r i p l e c o r r e l a t i o n measurements) on L (Part D.). B.) DOUBLE CORRELATION ANALYSIS The experimentally measured points W^-©.) of the double co r r e l a t i o n function (Chapter IV Part A.) were least square f i t t e d to functions of the form V" 9") T. bn(^i)pn(cos-©-) n = 0,2,4 (Chapter I I ) n and the normalized b n ( ^ ) c o e f f i c i e n t s and the errors i n these c o e f f i c i e n t s (as calculated i n Appendix II) were found to bes bg(^) — 1 (normalization) =0.38 ±0.07 b 2 ( ^ ) = 0.00 ±0.06 \ ( \ ) - 0.06 ± 0.09 = 0 « 1 3 ± 0 , 1 2 h2 ( % 3 ) = -0.18 ± 0.03 V 3^ = 0.04 ± 0.05. Figures 18., 19. and 20. show the experimental points and the l e a s t squares f i t t e d curves f o r the P^^ d i s t r i b u t i o n s . The t h e o r e t i c a l correlation functions (Chapter II) have the form (for a pa r t i c u l a r choice of JJ^) W TW = W^(-e)= ^ a ^ J J ^ P j c o s - e - ) The maximum L' estimate of the a^ and x^ f o r a given J J j are those I i I i t i t 0 15 3 0 A5 6 0 75 90 ANGLE =& IN DEGREES FIGURE 19. p 3 y 2 ANGULAR DISTRIBUTION FIGURE 20. p 3T 3 ANGULAR DISTRIBUTION - 47 -values which maximize N» " " i - i f J ( W J l V i ) I f the form of f (described i n Part A.) i s substituted into L' , the maximum L" estimate leads to the Least Squares estimate of the a and x.. n i * That i s a n and x^ minimize S where N« (%• " W T j ) 2 i ? i ° i 2 The values of a and "x. were derived from the values of the b ' s f o r n i n the following three cases,, a„) Those assignments of J and which gave r i s e to a number of unknowns ( a Q and x^) which exactly equalled the. number of equations a v a i l -able to determine them. b.) Those assignments of J and J j which gave r i s e to a number of unknowns which was more than the number of equations. Co) Those assignments of J and J j which gave r i s e to a number of unknowns which was less than the number of equations. For the f i r s t two classes the maximum L 8 requirement Is s a t i s f i e d by the following equations In the f i r s t case equation ( l ) gives a d e f i n i t e set of values f o r a^ and x 4 * the second case gives relations between the "a and x*., which are n x n x used i n the treatment of the t r i p l e c o r r e l a t i o n , to get values of "a and x^. In the t h i r d class the maximum L' values of a and x,, could not be n i derived from the system of equations ( l ) since the a"n and x^ values were overdetermined. In pr i n c i p l e the values of a*n and "x^ f o r various J J j assignments could be obtained by maximizing L" with respect to the a n and - 43 -x^, one would then accept or reject these assignments on the basis of a"Xj tes t . I t was possible however f o r these cases to reject a l l these assign-ments by dividing the sets of equations ( l ) into subsets (each f a l l i n g into class a,) and then checking f o r consistency (to within experimental error) i and x. n X subset of equations represented on independent determination of the unknowns a^ and x^, therefore as an example i f the results of calculating a Q from subset 1 was a 1 i (J"(al) and subset 2 was a -±"(7"(a2), the assignment was nx n2 rejected i f | a ^ - a ^ j > 2^CT(a2)+- CT(nl)] I t was found that f i v e values of J J j f e l l into class a,) and only 3 1 the assignment JJ t = — — f e l l i n t o class b.); these assignments are l i s t e d 2 2 _ i n Table 7. along with the maximum L' estimate of ^ ( x ^ ) . In some cases two values of x^ were allowed and i n these cases the smallest value of x^ was chosen. The reason f o r choosing t h i s value i s outlined i n Section D,) of t h i s chapter. The values of the maximum L' f o r the s i x J J j ' s l i s t e d i n Table 7. are a l l equal, therefore any further d i f f e r e n t i a t i o n between the J J j ' s as to which i s the best assignment must come from the inc l u s i o n of the t r i p l e c o r r e l a t i o n measurements into the l i k e l i h o o d function. C.) TRIPLE CORRELATION ANALYSIS a n The values of x. and which maximized L' f o r each of the s i x re-0 a 2 maining possible values of J J were related to — by equations of the forms - , a2, P ) *0 tt0 * a 0 These equations were substituted into L" then L" was maximized with respect to a2 TABLE 7. DOUBLE CORRELATION J J j x 2 3 1 * N i l ; 2 2 J 2, 0.261 0.18 2 2 3 5 -0.20 ± 0.05 2 2 5 1 N.I. n mat 2 2 5 3 2 2 0.26+ 0.05 5 5 0.20 ± 0.05 2 2 i. ^-N.I. means no information was gained from the experiment on these parameters. 49 -j=N' J * j % l a Q and QIL" a 2 Then with the re s u l t i n g values of the values of x. were determined, a _ Table 8. l i s t s the parameter estimates and x. opposite each of the s i x a 0 remaining choices of J J j , Do) DISCUSSION OF RESULTS On the basis of the three double correl a t i o n measurements ( p ^ ^ ) and the assumptions outlined i n Chapter I I , i t has been possible i n the previous sections of t h i s chapter to l i m i t the spin assignments f o r the 5.03 (J) and 2.14 ( J j ) Mev states to the s i x values l i s t e d i n Table 8. The X 1 s (Hoel 29.) f o r the f i t of the experimental results to the t h e o r e t i c a l t r i p l e c o r r e l a t i o n functions W^Triple) obtained f o r each set of parameters (J,Jp"a n,x^) were calculated and are l i s t e d i n Table 8, opposite the s i x possible choices of J J . The only assignment which might be rejected on 2 3 5 2 the basis of its * X 7 value i s - - =10). The prob a b i l i t y of t h i s value 2 3 5 of\or larger occuring i s approximately 0„03 i f 2 2 ^ s * n e correct assign-2 ment. The remaining possible values of J J ^ have a 7^ assignment of three which corresponds to the most probably value (the prob a b i l i t y of a value of 2 X 3 or larger occuring i s 0.60). The values of the parameters — which were also determined i n the a 0 experiment have not as yet been discussed and, without referring to any reaction theory, there i s no way of knowing; i f the values found are reas-a onable. However, there i s one requirement on the values of and that i s a 0 that they must predict a density matrix which has positive values f o r a l l diagonal elements ( i . e . population parameters). The tensor parameters f o r TABLE 8. 3 1 2 2 SUMMARY OF CORRELATION ANALYSIS - % w - H P ( $ 2 ) % P ( % ) 5 3 * P , ( J J l ) N.I. 0.60 N.I. 0.85 N.I. 0.75 N.I. 3 1 0.38 1 11 0.00 0.30 0.26 0.02 0.37 0.05 -1.0 3 1 3 10'^ 10° 3 -2 J -0.21 0.10 -0.20 0.03 0.21 0.08 -1.5 10 0.15 3.6 10" 5 10"4 , 2 2 •|-2 N.I. 0.85 -0.19 0.10 -7.1 -0.1 3 1 | | -0.16 0.20 0.26 0.02 0.1B 0.13 0.5 0.0 3 1 5.2X0*4- 1.3 10" 3 11 0.00 0.30 -0.20 0.03 0.28 0.07 -0.9 0.0 3 1 6.3 10"4" 1.5 10~3 N.I. means that ne information on the absolute values of these parameters was gained from these experiments. P(x\) i n t h i s case i s chosen as the value corresponding to the most probable value of x^. The places which are l e f t blank i n the table correspond to instances where the parameter i s not present i n the formalism. - 50 = 5 1 J J ^ sa> ~ » were corrected f o r the f i n i t e s o l i d angle of the gamma counters (refer to Appendix I I I ) and found to be ~ -7. 6 a2 a0 T" = -0.15 o These values were then substituted into the equation r e l a t i n g the density matrix and s t a t i s t i c a l tensors (Chapter I I Part B.) and i t was found 5 that the population parameter f o r the m — 2 magnetic substate was negative. Since t h i s does not correspond to any r e a l i s t i c physical s i t u a t i o n , t h i s a^ assignment was rejected, that i s P(—- — -7.6) » 0, No si m i l a r inconsistences a0 were found f o r the remaining f i v e assignments. On the basis of the experraental evidence presented so f a r the pos-s i b l e values of J J ^ have been reduced from sixteen to a maximum of f i v e pos-s i b l e assignments with one assignment (^  4|) being approximately seven times less probable than the other four. To determine which of the JJ^'s i s the most probable assignment further"experimental information must be introduced. In Part F.) suggestions are made as to further experimental investigations which can be made to absolutely determine J J ^ and i n Part E.) arguments based oa previous measurements of t r a n s i t i o n p r o b a b i l i t i e s and l i f e t i m e s are i n t r o -duced to weight the possible J J ^ assignments by the known p r o b a b i l i t i e s of getting values of x^ as large as those needed to maximize L (refer to Table 8.)« E.) CONCLUSIONS ON SPIN ASSIGNMENT BASED ON MULTIPOLE MIXTURE PROBABILITIES Wilkinson 16 0) has compiled the experimental data on the l i f e t i m e s and radiation widths P of Ml and E2 transitions i n l i g h t nuclei (A<20) and has compared t h i s data with the the o r e t i c a l estimates as calculated using the Weisskopf extreme single p a r t i c l e model. From t h i s data the frequency functions of ["111, pE2 and x.2 (where x 2 « ^ J S^A were estimated using the 3- «L ' 51 -method outlined i n Appendix IV. The p r o b a b i l i t i e s of p a r t i c u l a r values of x ^ ; a r i s i n g were incorporated i n t o the,Likelshood estimates of a^ and x^ i n Part A., h = L n , ( ^ i / J i V i ) f , , ( x i ) where N L " ' ^ f J ( ^ J J i V i > p < J J i ) p ( ^ In the proceeding sections L"1 was maximized under the assumption f " ( x 1 ) = 1 f o r a l l values of x.^ . However, the effect of f M ( x •) on the max-imum L must be considered i f f"(x^) i s not equal to a constant. The maxi-mum of L under t h i s condition i s more e a s i l y determined by evaluating the maximum i n W where W = In L W •» i n L = I n f ' U ^ + I n L n l Only the equations maximizing W with respect to the x^ » s w i l l change from those used to maximize L"'| The remaining equations were unaffected by the inclu s i o n of f"(x^) since a = 0 n The new term . [ j f B ( x j J | " °^ xi) introduced into the above equations was small compared to the second term ["since L M I = C exp=W(anx^)J as long G> f" as t — was small 0 The condition wa3 s a t i s f i e d by the p a r t i c u l a r form of f"(x^) derived i n Appendix IV. The otheroonditionin which the new term became large was when f n ( x ^ ) was small, but th i s corresponded to a minimum i n L and t h i s solution was ignored. We were then j u s t i f i e d i n ignoring the effect of f n ( x . ) on the e s t i -mates of the maximum l i k e l i h o o d values a and x,. This means that the e s t i -n i mates derived i n the previous- section are unchanged to any extent by the inclusion of f«(i: ) in,the L i k e l hood function, Of course, ta(x^) does have a very important effect i n estimating the r e l a t i v e p r o b a b i l i t i e s to be associated with each spin assignment J J ^ . These p r o b a b i l i t i e s now become p» ( p ( * £ i J J j TT ±H^J PCX 2) max W " i l Z k i J o I = the r a t i o of (the probability of getting PCX2) max a thi s value A. ) to (the pro b a b i l i t y of .. getting the most probable value of/(. ) PCx^) — probability ef getting a value of j-x^/ i n the range j x 4 ± A x ^ j the magnitude of 4 x ^ i s chosen small ( 0,05) compared with l j - i t s actual magnitude i s not s i g n i f i c a n t siueethe quantities of int«= . erest i n Table 8, are ratios of probab-P" ( J * T ) i l l t i e s =»--=«-«^=--^~ The p r o b a b i l i t i e s PCs:,,) f o r each value of j f j p calculated as outlined i n Appendix I?„ are l i s t e d i n Table 8. along with ' 1 and P ( J J T ) c 3 x P ( ^ ) mas: 1 The assigaeent of JJ„= ^ =s was found to hat?© .a m l u s of r c (<JJr) a factor of 1000-larger than any other assignment, and i s therefore at least one thousand times more probable than any of the other four p o s s i b i l i t i e s , a 2 The tensor parameter r a t i o "SZ f o r the 5 . 0 3 M o t l e v e l was not deter- . °0 3 1 mined f o r the spin assignment J J j — — - i f any a r b i t r a r y value of >;:'i.xz.po.L& mixing was allowed iany value of x„8s)o However on the basis of pluvious experiments i t i s apparent that the value:of the mixing parameters (See . Appendix IV f o r most probable values of ) should be quite small and i n - 54 -creasing both the e f f i c i e n c y of the counting system and the experimental running time. With the system as i t i s i t would require approximately hours of running time to acquire s u f f i c i e n t s t a t i s t i c s on the t r i p l e cor-r e l a t i o n function to d i f f e r e n t i a t e between the f i v e possible spin assignments. I f the size of the Nal cr y s t a l s used were increased from the present 2" X 2" to 5" X 4", the improvement i n counting e f f i c i e n c y of each gamma counter could be increased by a factor of seven producing a t o t a l r i s e i n e f f i c i e n c y of approximately f i f t y . Under these conditions the required s t a t i s t i c s could be gathered i n approximately twenty running hours, i i . ) EXPERIMENT TO DETERMINE MULTIPOLE MIXING G.J. McCallum 20.) describes a technique f o r measuring the l i n e a r p o l a r i z a t i o n of the gamma rays from which the multipole mixing r a t i o can be determined. As he points out t h i s method i s p a r t i c u l a r l y applicable to the sit u a t i o n outlined i n t h i s thesis. A rough cal c u l a t i o n of the counting rate expected f o r such a measurement using the same experimental arrangement as McCallum 1s and reaction rates s i m i l a r to those encountered i n t h i s experiment indicates a running time of ten hours to get s t a t i s t i c s good enough to deter-mine the multipole mixing r a t i o s . When used i n conjunction with the data from the correl a t i o n measurements outlined i n t h i s t h e s i s , the multipole mixing r a t i o should determine the values of J and J T absolutely. - 55 -CHAPTER VI COMPARISON OF RESULTS WITH THEORY In t h i s chapter the spin assignments made i n Chapter V are compared with those predicted by the Independent P a r t i c l e Model (Part A.). The pre-sent assignments agree with those made by previous workers. In Part B.) the r e l a t i v e populations of the magnetic substates of the 5.03 Mev l e v e l as mea-sured i n Chapter V are compared with those calculated assuming the Butler Plane Wave (B.P.W.) stripping mechanism and Distorted Wave Born approxima-ti o n (D.W.B.). A.) COMPARISON OF SPIN ASSIGNMENTS WITH THE INDEPENDENT PARTICLE MODEL PREDICTIONS The experimentally determined l e v e l scheme f o r the ground and f i r s t three excited states of including the re s u l t s i n t h i s thesis, has the following appearance. (For the spin assignment to the ground and second excited state see Lauritsen et a l 21.) 3 2 5 2 5.03 Mev 4.46 Mev 1 2.14 Mev 2 3 2 G.S, B ^ Using the Independent P a r t i c l e Model Cohen and Kurath 22.) have c a l -culated the l e v e l scheme for B 1 1 using f i r s t l y the Intermediate Coupling Mod-e l ( I n g l i s 23.) and secondly the many parameter Two-Body Matrix Element Model. Cohen and Kurath show that both models give good agreement between the ex-perimental l e v e l scheme and theory. - 56 -B.) CALCULATION OF TEE DENSITY MATRIX FOR THE 5*03 Mev LEVEL AND COMPARISON WITH STRIPPING THEORY The r a t i o of the tensor parameters f o r the 5.03 Mev l e v e l i n B^- pop-ulated by the reaction was determined i n Chapter V Part F.) on the basis of the reasonable assumption that the |xjj *s were le s s than 0.05. The value of the r a t i o (of the attenuated tensor parameters, see Appendix I I I ) was found to be; *2 - 0.53 1 0.07 ^ 0 The s o l i d angle corrected r a t i o of tensor parameters •=?— i s c a l -H00 di l a t e d i n Appendix XII to beg R 20 l00 - 0.59 * 0.07 The equation r e l a t i n g these tensor parameters to the density matrix was shown i n Chapter I I to have the form; f = X. ( - l ) J " H \ j J a - f f i ^ > % k < J J > ? J J Using t h i s equation the density matrix corresponding to the observed R 20 r a t i o of the tensor parameters °sr= may be calculated *00 Since only the Rg^, k- 0 and K even are non-zero (Refer to Chapter I I ) , J-^ i s diagonal and sym-metric and 1there are only two independent non-aero matrix elements. ' 2 2 / 2 2 -1 R00 * 0.21 3 3 -3 ~3 1 Tl Tl Tl J 3 3 ~ Pi 3 2 + a ( 0 « 5 9 ) - 0.79 R, 2 2 / 2 2 The normalization of the wavefunction describing t h i s 3tate requires - 57 -T R A C E ^ = 1 (Sharp 24.) which when applied to t h i s density matrix gives RQQ= =. The density matrix then takes the form; m in / 2 2 I 0 0.40 Q 0.10 0 .10 0.40 P' mm j j (Refer to Chapter I I Part B.) represent the prob a b i l i t y of finding the state i n the par-t i c u l a r magnetic s u b s t a t e ^ j ^ d n •= projection of J on the quantization axis which i n t h i s case corresponds to the incident deuteron beam d i r e c t i o n . ) . I t i s seen then that i n the B 1 0(dp)B^ 1 reaction leading to the 5.03 Mev l e v e l 3 1 the m = — states are more favourably populated than the m = states by a factor of four. I t i s now of interest to enquire how the states would be populated i f some p a r t i c u l a r nuclear reaction mechanism i s postulated. The B"^(dp)F^ reaction has been studied at a deuteron bombarding energy of 2 Mev (Marion et a l 25.) and the protons (p^) populating the 5 .03 Mev l e v e l were found to ex-h i b i t a very we l l developed stripping peak, that i s , the proton angular d i s -t r i b u t i o n with respect to the deuteron beam showed a strong peak at 25°, I t i s reasonable to expect that the application of stripping theory to t h i s re-action should predict a density matrix similar to the one found experiment-a l l y . The simplest version of stripping theory and one which can be solved a n a l y t i c a l l y i s the Plane Wave Butler (P.W.B.) theory. To apply P.W.B. stripping theory to the dp reaction, the deuteron i s considered as a very loosely bound proton - neutron system. When the deu-teron i s incident on the target nucleus X the neutron i s captured i n t o the target nucleus producing the resultant nucleus Y and the proton continues on past without reacting strongly with either X or Y. These assumptions are - 58 -formulated s p e c i f i c a l l y by f i r s t w riting the complete Schroedinger equation f o r the system as: < H x + V T n + T s p + »xn+ W ( r n r p f ) = Y B and then introducing the P.W.B. approximation by deleting various terms from the Hamiltonian. In t h i s equation: H x = Hamiltonian f o r the target nucleus i n the laboratory system. T & « Kinetic energy of nucleon (a) i n the laboratory system. V^j = inte r a c t i o n potential between p a r t i c l e s i and j . = i n t e r n a l coordinates of the target, r =. proton s p a t i a l coordinates. P r n =• neutron s p a t i a l coordinates. The f i r s t approximation used f o r plane wave Butler stripping i s to set V Xp = 0 (the proton does not interact with the target nucleus). Second-l y V n p = 0 when neutron n i s within the boundary of the target nucleus. The f i n a l major assumption and approximation i s that the only part of the t o t a l wavefunction \|/" which i s considered to contribute s i g n i f i c a n t l y to the ma-t r i x element i s the incident plane wave deuteron wavefunction. A more com-plete description of the assumptions made i n P.VLB. calculations may be found i n Butler 26.) Satchler 27.) has calculated the th e o r e t i c a l s t a t i s t i c a l tensors R ^ J J ) assuming P.W.B. stripping theory. Quantities proportional to the s t a t i s t i c a l tensors appear i n t h i s a r t i c l e labeled Y)^ (See equation 2 Reference 27.) In t h i s a r t i c l e he showed that using P.W.B. stripping assump-tions the capture process i n dp stripping i s formally the same as that of a plane wave neutron captured along the target nucleus r e c o i l axis which i n t h i s 59 - . experiment corresponds t o the -Z d i r e c t i o n ( i . e . the protons are detected i n the Z d i r e c t i o n and the target nuclei r e c o i l i n the -Z di r e c t i o n ) . The tensor parameters describing the state formed by t h i s reaction are shown i n Reference 27.) to be: R^CJJ) = 0 k ^ o **KQ(JJ)OC ^ ( J J J ^ J ) where the constant of proportionality; independent of K and k. j =• t o t a l angular momentum of captured neutron i . e . f = £ + S N .= - | ~ spin of i n i t i a l nucleus = spin of B 1 0= 3 3 J = spin of 5.03 Mev l e v e l = -2 From Table 4 a.) Reference 27.) B 0^(i|) _ Vi-^l) ' t 3 3 1 • ~ = -0.200. R00<2 2* \ I f now the actual experimental conditions are considered, where the protons are detected i n a f i n i t e s o l i d angle (•©-^  5 ° ) instead of j u s t at -9- = 0° and an average of the. tensor parameters due to protons emitted at angles 0*^-0- $ 5 ° i s taken symmetrically about - 0°, i t i s expected that the cy-l i n d r i c a l symmetry conditions on the resultant average R^ 1 s w i l l be applic-able and only two R^ 1 s need be defined. This averaging process should cause the alignment to tend toward istropy but only by a small amount since the average i s taken over only 5 ° . This tendancy toward isotropy when re-ferred to the tensor parameter R ^ means that R^q should tend toward zero, that, i s to be reduced i n magnitude by a small amount. I t w i l l be assumed i n the following that the tensor parameters resulting from protons being emitted at -9- = 0° are a good approximation to those that would r e s u l t i f an - 60 -average over -6- up to 5° were taken and that t h i s c e r t a i n l y i s a good estimate of the upper l i m i t of the R^ 1 s resulting from t h i s t h e o r e t i c a l (P.W.B.) re-action. Following the same procedure as with the experimental tensor para-meters, the theoretical density matrixj3*(P.W.B.) calculated from the Plane Wave Butler assumptions i s found to take the form: P i / mm ' ( P . W . B . ) 22 -2 2 That i s , plane wave stripping theory predicts that the M-~ states 3 should be more favourably populated than the m - "". states. This r e s u l t may be interpreted from a semi-classical viewpoint by considering only those neutrons which are captured into the B ^ nucleus with o r b i t a l angular mom-entum jt — 1 and with t h e i r l i n e a r momentum i n the -Z di r e c t i o n . This s i t -uation i s arranged by selecting only those protons which come from dp reac-t i o n i n the+Z(-0- =0°) d i r e c t i o n ( i . e . along the deuteron beam a x i s . ) . Using the d e f i n i t i o n of o r b i t a l angular momentum and the Law of Conservation of 1 angular momentum, we now show that only m = — states can be populated. The law of the conservation of angular momentum states that: T = \ +• I + T 3 Since J = -j and = 3 the only way angular momentum can be conserved i s f o r both Z and S to be a n t i p a r a l l e l to (since ~j[ — l ) . But by de-f i n i t i o n ( SI OC VX r ), I i s perpendicular to the d i r e c t i o n of motion (V) of the captured neutron and therefore m^ — 0, where i s the projection of >£on the Z axis . Also as was j u s t pointed out only those nuclei with angu-l a r momentum oriented a n t i p a r a l l e l to JL contribute to t h i s reaction, there-fore the projection of on the Z axis must also be zero (Mj^= 0). Only - 61 -the i n t r i n s i c spin of the neutron (S) may have a non-zero projection on the Z .1 a x i s , and must take the values m = The only magnetic substates of the s 2 5.03 Mev l e v e l (spin J) which are populated have values m - M j ^ rn^-r m3 = ma = ± £. Based on t h i s very r e s t r i c t e d semiclassical model the density matrix f o r the 5.03 Mev l e v e l takes the forms r\, .mm' / ° n c ^ H{ semiclassical) 3 3 = 2 2 I O 0.5 0 In quantum mechanical terms the question of which of the magnetic substates can be populated by the coupling scheme: I + T = 7 i s equivalent to asking f o r what value of m i s the product of the rector coup-l i n g c o e f f i c i e n t s : (fiS0m s|jm s)(jJ l f f i. M j i| S m ) „ non zero In t h i s case we have I = 1 J i = 3 S - X , . 3 3 2 By referring to my table of Clebsh-Gordon c o e f f i c i e n t s i t can be seen that no value of in ( f i n a l state magnetic quantum number) i s excluded by t h i s coupling scheme, however from the previous semiclassical argument one would expect that the in = states would be more favourably populated. This effect indeed occurs i n the plane wave stripping calculation. I t can be seen that the P.W.B. calc u l a t i o n of the alignment re s u l t i n g from the dp reaction and that found experimentally do not agree either i n magnitude or i n sign, hence, - 62 -the need f o r a more accurate theory to describe t h i s reaction i s indicated. Such a theory has been developed and i s referred to as the Distorted Wave Born Approximation. (D.W.B.) The D.W.B. uses o p t i c a l model e l a s t i c scattering wavefunctions i n place of plane waves to represent the r e l a t i v e motion of the incident and target nuclei and of the outgoing residual n u c l e i , thereby approximating the effect of both the coulomb and nuclear forces on the incoming deuteron and outgoing proton. Also a spin-orbit interaction i s sometimes introduced when calculations of the p o l a r i z a t i o n of the resultant protons i s made. (Newns and Refai 28.). Goldfarb 29.) ha3 calculated, using the D.W.B., the tensor parameters of the state formed by the dp reaction where the protons are de-tected at zero degrees to the incident deuteron beam. He shows that the calculated tensor parameters are independent of the form of the o p t i c a l potentials used and depend only on the spin o r b i t d i s t o r t i n g potential i n the following manner; R^D.W.B.) R £ 0 R00(D.W.B.) Rj 00 4k(k+ 1) X (2j-i)f23+3) i+;^_ where R^Q =• Plane Wave Butler s t a t i s t i c a l tensors calculated previously and where ^  i s referred to as the spin d i s t o r t i o n parameter and i s positive and real. This factor arises from the D.W.B. formalism when a spin dependent effect i s included i n the d i s t o r t i n g potential and i s a measure of the r a t i o 3 1 of the populations of the ^^"^ *° m n = 2 s P ^ n s * a t e s °? t n e transferred neutron, which takes into the nucleus a t o t a l angular momentum j with magne-t i c quantum number m^  (n^refers to the projection of j on the quantization axis which i n t h i s case corresponds to the deuteron beam direction.) I f no spin dependent d i s t o r t i o n i s included Goldfarb 29.) shows that ^ = 0 - 63 -By choosing values of between zero and i n f i n i t y ( i . e . allowing any magnitude of spin d i s t o r t i o n ) the predicted value of the tensor parameter varies over the range ^ 0 RkO< D° W' B-) Rk0 ~ R£O " R00(D„1.B.) ^ %7 or, using the value of —^2 =, -0.20 derived e a r l i e r . R00 Y R , . 0 (D.W.B.) s -0.20 ^ -r~ r ^ 0.20 R 0 0 (D.W.B.) Apparently the introduction of a spin dependence i n the potential can change the sign of the tensor parameter r a t i o but no amount of d i s -t o r t i o n of the deuteron wave function, as introduced by D.W.B., can produce the required agreement between experiment and theory. The i n a b i l i t y of the D.W.B. to predict the correct tensor para-meters i s usually attributed to the interference of such reaction mechan-isms as compound nucleus formation, heavy p a r t i c l e stripping and stripping associated with coulomb excitation. - 6 4 -APPENDIX 1 ANALYSIS OF THE RANDOM TRIPLE GOINGIDENCE COUNT RATE The random t r i p l e coincidence monitor (Slow Coincidence C i r c u i t - See Chapter I I I , Part F.) was operated i n p a r a l l e l with the fa s t coincidence c i r c u i t to record the random coincidence and t r i p l e coincidence rates simul-taneously. In order to measure the random coincidence rates with adequate pre-c i s i o n , the resolving time i n the random coincidence monitoring c i r c u i t was made larger, by a fact o r of ten, than the resolving time i n the main c o i n c i -dence c i r c u i t . This resulted i n the random monitor counting many more random coincidences than those detected by the main coincidence c i r c u i t . There are three processes by which a t r i p l e coincidence event occurs i n either coincidence c i r c u i t s i g n i f y i n g a time overlap of the coincidence pulses. These occur when there i s an a 1. ) Overlap due to three uncorrelated events. 2. ) Overlap due to two correlated ("true") and one uncorrelated event. 3. ) Overlap due to a "true" t r i p l e coincidence. The number of events occuring i n time T from 1.) i s * Nrand 1.) 8 3N l N2N 3T 2 w h e r Q T s resolving time of c i r c u i t Nj_ • Count rate i n i^h channel The contribution of 2.) w i l l be: N. rand 2.) B H^^T 4 N 2 3 N-LTf-NyNpT (a) (b) (c) N = number of "true" double coincidences between channels i and j - 65 -The contribution of 3.) to the coincidence count rate recorded i n the T 15ns c i r c u i t i s the quantity required f o r the t r i p l e c o r r e l a t i o n measurement. Typical measured count rates during the experiment were; *= 2.1 10 3 protons/sec H 2 - s 3 IQ4 gamma rays/sec N12 = 4 sec" N13 = N12 N23 = 10 sec' »1 In a time t the t o t a l numbers of counts recorded i n the slow N r a n^^ 0^. ) and fast N C Q ^ n c coincidence c i r c u i t were t y p i c a l l y ; t = 2.4 10 3 sec Nrand(tot.) = 511 N c o i n c = 2 6 The t h e o r e t i c a l number of counts i n the random (slow) coincidence c i r -c u i t ((T » 1.5 10"J<t.sec) iss N(slow) = 3(2.1 103)(3 104) 2(1.5 10^)2 rand 1.) = 14.1 102 sec"! •- •• \--N(slow) , = 2(4)(3..104)(l;5 io ° 7 ) 1 0 ( 2 . 1 1()3)(1.5 10"?) rand 2.) =. 4 10~2 sec -1 N(slow) - 2 N c o i n c (The factor of 2 arises from the 50$ rand 3.) e f f i c i e n c y of the f a s t coincidence c i r c u i t ) therefore N(slow) = fN(slow) . + ; N(slow) 1 t + 2 N e o i n e rand(toto) L rand 1.) rand 2.)j c o i n c = (338 +• 91 +• 5) + 52 sr486 counts (theoretical) - which i s i n reasonably good agreement'with the experimental value of 511. When taking a random reading to determine the r e l a t i o n between the two coincidence c i r c u i t s , the timing; on the•'p'rotbn^ounter'was changed by about 300 n sec. thereby destroying a l l the "true" triple.coincidences i n both - 66 -c i r c u i t s . This also destroyed a l l the "true" double coincidences produced i n parts (a) and (c) i n N r a n c j 2 . ) 0 Therefore the reading f o r the same beam condi-tions when the c i r c u i t i s off coincidence should be: N ' r I n T ) * N r | n 2 w ) " N t r u e " Nrand 1.) (a) and (c) That iss N ' ( | l | w ) = 486 - 91 - 52 =343. The actual experimental reading off coincidence was 366 which on®.again i s good agreement between theory and experiment. The number of random t r i p l e coincidences i n the fa s t coincidence was recorded at the same time as N1 (slow) and t h i s rate was found to be: rand H' (fast) =. 5 counts ( t h i s i s an average'over 5 readings taken rand with constant beam conditions) The t h e o r e t i c a l estimate of N' (fast) i s -rand N« (fast) r fN(fast) -+- N(fast) r a n d rand 1.) rand 2. [_ ) part (bj) Substituting i n the appropriate valuess N'Jfast) * (13 i o - 4 + 3 10-4). 2 . 4 1 0 3 ? 4 counts - which compares quite favourably with the measured value of 5 counts i n 2,4 10^ sec. The th e o r e t i c a l r a t i o of: N'(slow) . i~ , -a p / • \ p rand i . ) „ fr(slow)l A / 150nsyv - 100 1A - (italarJI 2 i) " [T(fast)j N«(fast) [T(fast)j \ 15ns j rand 1.) and i t was found that (experimentally) N(slow) = 3 3 6 rand 1.) N(fast) - 3 rand 1.) which gave good agreement between the the o r e t i c a l and experimental performance of - 67 -t h i s system, With t h i s i n t e r n a l consistency i t was possible to convert the ran-dom count rate as measured by the slow coincidence c i r c u i t to the equivalent random rate i n the fa s t coincidence c i r c u i t . In t h i s way i t was possible to monitor the random t r i p l e coincidence count rate as the t r i p l e coincidence mea-surement was made. The method used to calculate the random count rate N(fast) wass rand 1. ) Subtract from the measured N£|low) the calculated effects of N r ^ 2 . ) and3.) l e a V i n g N k M . ) 2. ) Calculate N(fast) . from the known r e l a t i o n rand 1.; N(slow) rand 1.) _ IQO N(fast) rand 1.) 3. ) Add to t h i s the calculated smaller effect N(fast) v , „ N to rand 2.) and 3.) get the t o t a l number of random events. This number was then subtracted from the measured t r i p l e coincidence to give the corrected number of coincidences. - 68 -APPENDIX I I DERIVATION OF ESTIMATE OF FITTED PARAMETERS AND ERRORS IN PARAMETERS The resu l t s of f i t t i n g the angular d i s t r i b u t i o n measurement PyX^ to functions of th© forms n are reported In Chaptsr V along with an estimate of the accuracy of the deter-mination of th® b n's. The method known as Maximum Likelihood (Cramer 30.) was used to estimate the b 's and t h e i r associated e r r o r s 0 a Let x be the experimentally measured value of W(-e-) at '©•='6-.. I f x D i s a reasonably large number (larger than 50) i t can be shown (Cramer 30.) that the Maximum Likelyhood estimate of the b »s leads to the commonly used lea s t squares estimate, that i s the best estimate of the b n*s i s one which 2 maximizes the function F(b_x a) where F <Va )* f - LVw(*aj) (Ja 4 r - i a ~l (1) and CXa - estimate of th© error i n x&. N = number of samples of W(-6-). Cramer also shows that the Maximum Likeljhood technique also gives the estimate of the srror i n the determination of the b n f s ass W(«a) F n"(oos ^ a ) 2 Usings x = ^(*%) ( s a t i s f i e d to a good approximation f o r a l l x ) •2 _ and c r x = (Cramer) i t i s found 7 p n 2 (cos-e- a ) i 2 (2) - 69 -In summary th© procedure used to estimate the b a's and the associated errors i n these estimates ^ Ib n)^was to least squares f i t the data to functions of the form of W(-d) following th© same format outlined i n Cramer 3<X) to deter-mine the b n»s, then calculate theCT(b ) using (2), APPENDIX I I I SOLID ANGLE CORRECTION FACTOR FOR GAMMA COUNTERS The th e o r e t i c a l angular correl a t i o n functions W(@J (See Chapter II) describe the expected f l u x of gamma rays from the source (B i C" target) through an element of s o l i d angle dfl at an angle©. Measurements of W(©) (Chapter IV) were made using 2" x 2" Nal cr y s t a l s which subtended a large s o l i d angle at the target and which were not 100$ e f f i c i e n t . The effect of the f i n i t e s o l i d angle of these counters on the measured d i s t r i b u t i o n Is outlined i n the following. The gamma counters i n the experiment were r i g h t c i r c u l a r cylinders whose base was oriented toward the o r i g i n . The source at the o r i g i n was on the i n -tersection of the axis of the cylinders and the z axis (deuteron beam axis) t Rose 31o) shows that the measured angular co r r e l a t i o n function W0) i s obtained from the t h e o r e t i c a l c o r r e l a t i o n function W(^ where W(0) = Z-CkPj^cos-e-) k by multiplying each c o e f f i c i e n t Cjj. by calculable correction factors Qjj, so that w(e) = Z -CfcQ^cose) k The factors Qjj are functions-of the counter geometry. (See above) Qk - Qk (h,t,r,T) T = gamma ray absorption c o e f f i c i e n t In Chapter IV the Statement was made that the spin assignments based on the double c o r r e l a t i o n measurements were independent o f , t h i s solid'angle corr-ection factor 0^, In that chapter, the theo r e t i c a l W (©) were shown to have the forms - 71 -\ ( e ) =Z k RkC-( JJ)A kP k(cosO-) A k •= known function of the angular momenta involved. R k0- the tensor parameters f o r the i n i t i a l state of the decay. I f now the s o l i d angle correction i s added, the experimental c o r r e l a t i o n function i s obtained. \ & QfcRkcAkRkCcosO-) aad i f the substitution QjjRjjO = a ' k ^ s ma<*Q a l l "the double c o r r e l a t i o n equations l i s t e d i n Chapter I I are obtained,, The a k referred to i n Chapter I I , IV and V as tensor parameters should more corre c t l y be call e d attenuated tensor parameters. The attenuated tensor parameters contribute no more undetermined para-meters than the tensor parameters themselves (other than the f i n i t e experimen-t a l l y l i m i t e d accuracy of cal c u l a t i o n of the quantities Q k), so that no addi-t i o n a l indeterminacy Is introduced into the spin assignments by the geometri-c a l corrections to the double c o r r e l a t i o n measurements. The s o l i d angle correction to the t r i p l e c o r r e l a t i o n measurement i s a l i t t l e more complicated with the function W^jO^) (refer to Chapter I I Part D) taking the corrected form: W^j©#) = JZ R k 0 AKM%% pK(cose i)PM(cos-e2)cos N(|) KMNk (See Kaye et a l 32) The determination i of t h i s function required the c a l c u l a t i o n of the RJ^Q's from the a k derived from the double corre l a t i o n measurement. The func-t i o n 1 ( 6 ^ ) was then calculated using the Qg» s derived i n Rose 31.) f o r the dimensions used experimentally: 1 r = 1 i n . h = 3.5 i n . t = 2 i n . T was assEBnSd to have the sa&e value i'or each of the three gamma raceme*" gies measured. The effect of ignoring the change i n due to the change i n T ever the gamma ray energy ranfS considered was to introduce a possible error i n Q £ of less than $6. Xfsipg these values the cors?ec tiqa-• factors whoa QQ i s normalized to unity (sines th©..geometry'makes no difference f o r i s o t r o p i c radiation), were found to bes Q 2 s» 0.96 0^ = 0.91 The spin:assignments of the f i r s t and third excited states of were made i n Chapter ¥ and the r a t i o of the attenuated tensor parameters -g^ were determined, for the s p i n assignment J 3 j = £ a s a 5 ~ •= 0.53 -F 0 . 0 7 * a 0 l00 •The corrected r a t i o s used i n Chapter ¥1 ares i f L 0o59 ± 0.07 B©0 Q2 ^0 a 0 * The estimate of -the value of and error i n this estimate was determined ^ a ? from the two independent estimates of _£ derived d i r e c t l y from the r a t i o s A 0 ^ 2 ^ j - ^ ( f o r i = i and 3) using the assumption that x. =0 fsee table i ) , B T T ^ T 1 J - 73 -APPENDIX IV CALCULATION OF PROBABILITIES OF MULTIPOLE MIXING In the analysis of the co r r e l a t i o n r e s u l t s i n Chapter V the probab-i l i t i e s of obtaining p a r t i c u l a r values of x^, the multipole mixing r a t i o were introduced. These p r o b a b i l i t i e s were estimated from the experiment-a l data compiled by Wilkinson 16.) on the pro b a b i l i t y d i s t r i b u t i o n describ-ing the 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 Ml(fTn.) and E2(f~E2) t r a n s i t i o n s . The frequency d i s t r i b u t i o n of the data i s shown i n the figures below as the crosshatched portion and the heavy dark l i n e outlines the shape assumed f o r s i m p l i c i t y i n the pro b a b i l i t y calculations. - 4 - 3 -2 -1 0 1 In both figures the symbol f~|j refers to the Weisskopf width estimate (16,) The assumed frequency function can be seen to have the form; 74 W — 0 elsewhere fmW>\ = b 5 io-3< & <:5 fw = 0 elsewhere Converting these to l i n e a r functions of thel""1' s and using the requirement thats oo ^ j " f ( i D d r = 1 (normalized probability) 0 i t was found that the frequency functions f o r pE2 and P M I have the forms f(f~E2) = , Q * v o 1 < £ E 2 $ 50 lw CD elsewhere w = 0 elsewhere The f a c t that these approximate d i s t r i b u t i o n s are i d e n t i c a l l y zero outside ce r t a i n i n t e r v a l s of P d o e s not a f f e c t our arguments, since the values of r32 andTMl we wish to discuss always l i e inside those i n t e r v a l s . The p r o b a b i l i t y of getting a p a r t i c u l a r value of P&2 bet?/een f^E2 and /^ B2 dfl52 i 3 aimply2 P(P 0E2) = £(P^Z)d PE2 (3) The values of P ( x 2 ) were calculated (Chapter V Part E«) using equations (1) and (3), To evaluate P ^ ) and P(x 3 ) the frequency function of 2 E2 x — —3? had to be calculated. I f i t i s assumed that the measurements of PE2 and P M I are uncorrela-ted, the frequency function f"(, E2, Ml) w i l l bes f(pE2,rMi) = f(rE2)f«(rki). 0 fE2 0 The frequency function f(fE2,rMl) such that = xr i s s - 75 -f ' ^ x ^ M l j f l l ) = £(xTklJf» ( f l l ) The t o t a l frequency function f ( x ^ ) w i l l be t h i s function summed over a l l the allowed values o f p M l i . e . ; f t x 2 ) = j f ( x 2 n i ) f ( r M i ) d r M i u ) 0 The frequency function f"(x2) were evaluated numerically using equa-tions ( 1 ) , ( 2 ) and ( 4 ) f o r the values of x ^ and x 2 which appear i n Chapter V Table 8 . From t h i s , the p r o b a b i l i t y of these values of x± f p ^ )] were c a l -culated using: P ( X i ) = f ( X i )d{xi) where the d(x^ j ) was determined by the range of x^ _ allowed i n Chapter V Part E.) The i n t e r v a l d(x^ r ) was treated as a constant factor occuring i n the calcu-l a t i o n of a l l the P U j ; ) 3 a 0 that i t s value does not a f f e c t the values of the ra t i o s of the P(x^ J * s which are the important quantities. » 76 -REFERENCES l o ) 5. Coh6n aad D. Kurath. Argonn© national Laboratory, to ba published, 2. ) 9. ivans, rroc. Pnys. See. A67, 684. (1954). 3. ) .3. Gpsodstsky. Compt. ReAd. 242, 2545 (1956). 4*) 5 F.R. Metsger, C0P. Sisann and V.K. Raamussen. Phys, Rev. 110, 906 (1953), 5. ) P.F. Donovan, J.V. Kane. R. E. Pixley and D,H. Wilkinson. Phys. Rovr , 123, 589 (1961). 6. ) D.H. l i l k i a s o n . Phys. Rev. i©5, 666 (1957). 7. ) A.S. Rnpaal. Ph.D. Thesis U.B.C. (1964). 8. ) H.R. Gove, J.A. Ibohms, A.E. Litherland, E. A^quiat Ard D.A. Bromlay, v ,. ., Bull Am. PhyB. Soc. 2 Ho. 1, 51 RA2 (1957). 9. ) J.K. Blair, J.D. Kington and J.B. Willard. Phys. Rev. 100, 21 (1955)* 11. ) ,, A.E. Lltherland and A.J. Ferguson. Can. and Phys. 39, 788 (1962). 12. ) , ,tT. Fano. Rev. Mod. Phys. 29, 74 (1957). 13. ) T L.Q., M^danharsj and M.S. Rose. Rev. Mod. Phys. 25, 729 (1953). 24c) J.M. Blatt and L. C. Biedenharn. Rev. Mod. Phys. 24, 258 (1952). 15c) ,.. M.J. Steenland and H.A. Tolhoek. Prog, i n Low Temp. Phys. Vol. 2. 16. ) D.H. Wilkinson. "Analysis of Gamma Decay Data. Nuclear. Sp@ctroa6ory Part B. 17. ) S.H. TSegors, L.L. Marsden and R.L. Heath. Calculated iffieian.ci©s of Cylindrical Radiation Deteetors. A.E.C. Report. I.D.0. -. ... „ 16370. 18. ) Sarshaw Chemical Company Cataloguo 0 19o) ., P. G. Qo©lc Introduc tion to Mathsmatieal St a t i s t i c s , v20.) G.J. McOallnm. Phys. Rev. 123, 568 (1961). •» -ry L:v; ' •; ' . .... ,21.). ft Lauritsen aad F. aenb©rg-Selo.u©. Enorgy Levels of '.Light R a c i a l (1962 -". . J • .22.-) S. Cohan axvi i), Ku.?ath. Etftctiva Interactions for tn® lp'Shell. (To be published). 23.)F: D, R. Iaglis. Rov. Kod. Phys. 25 i 390 (1953). 24«> W.T. Sharp, She Quantum Theory of Angular Momentum. A.E.C.L. Ltd* C.R.L. — 43. - 77 ~ 25.) J.B, Marion and G. Weber* Phys. RevV 103, H08 (1956), 26») S.T. Butler, "Nuclear Stripping Reactions", 27. ) G.R. Satchler. Proc, Phys. Soc. 66^, 1031 < 1953 28. ) H.C. Newns and Refal. Proc. Phys, Soc, A71, 627 (1958) 29. ) L.J.B. Goldfarb, Nuo. Phys, 57, A (1964). 30„) H„ Cramer, "Mathematical Methods of Statistics,". 31. ) M.S. Rose. Phys. Rev. 91, 510 (1953) 32. ) G, Kaye, E.J.'C. Read and J.C. Willmott. "Table of Coefficients for Analysis of Triple Angular Correlations from Aligned Nuclei" (Chadwick Laboratory 1963). 33. ) T.K. Alexander. Private Communication, 34*) G. Jones, Private Communication, 35. ) G. Dearnaly and D.'C. Northrop. "Semiconductor Counters for Nuclear Radiation" (Wiley), 36. ) E . U . Condon and G.kShortley. "Theory of Atomic Spectra" (Rambridge 1951) 37. ) W.T. Sharp , J,M. Kennedy, B.J. Sears and H.G. Hoylo, "Tables of Coeff* i c i e n t s f o r Angular Distribution Analysis" C.R.T. - . 556 . 38. ) A.J. Ferguson and'A.R. Rutledge. "Coefficients for Triple angular Correlation Analysis in Nuclear Bombardment Experiments" ' . C.R.P. - 615. 39. ) J.L. Leigh. M.Sc, Thesis "The"Efficiency of Scintillation Counters fpr Gamma Ray Detection" (1964), 

Cite

Citation Scheme:

        

Citations by CSL (citeproc-js)

Usage Statistics

Share

Embed

Customize your widget with the following options, then copy and paste the code below into the HTML of your page to embed this item in your website.
                        
                            <div id="ubcOpenCollectionsWidgetDisplay">
                            <script id="ubcOpenCollectionsWidget"
                            src="{[{embed.src}]}"
                            data-item="{[{embed.item}]}"
                            data-collection="{[{embed.collection}]}"
                            data-metadata="{[{embed.showMetadata}]}"
                            data-width="{[{embed.width}]}"
                            async >
                            </script>
                            </div>
                        
                    
IIIF logo Our image viewer uses the IIIF 2.0 standard. To load this item in other compatible viewers, use this url:
http://iiif.library.ubc.ca/presentation/dsp.831.1-0302535/manifest

Comment

Related Items