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Structure dependent asymmetry in sequential breakup from the reaction Li⁶(He³,pα)He⁴ Reimann, Michael Andrew, 1967

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The University of British Columbia FACULTY OF GRADUATE STUDIES PROGRAMME OF THE FINAL ORAL EXAMINATION FOR THE DEGREE OF DOCTOR OF PHILOSOPHY of MICHAEL ANDREW REIMANN B.A.Sc , University of British Columbia, 1959 M.A.Sc., University of British Columbia, 196U MONDAY, MARCH 6, I967 AT 2:30 P.M. IN ROOM 301, HENNINGS BUILDING' Chairman: I. McT. Cowan G. Jones P. 'W. Martin M. McMillan R. Stewart G. M. Griffiths F. K„ Bowers External Examiner: H. D. Holmgren Department of Physics University of Maryland College Park Maryland U. S. A. Research Supervisor: P. W. Martin STRUCTURE DEPENDENT ASYMMETRY IN SEQUENTIAL f. o k BREAKUP FROM THE REACTION Lx (He ,pc()He ABSTRACT An a x i a l asymmetry about the d i r e c t i o n of motion of 5 L i has been seen i n the breakup of the ground s t a t e of t h i s nucleus as an intermediate s t a t e i n the r e a c t i o n L i ^ ( H e ^ ,pof)He l f 0 The t o t a l cross s e c t i o n f o r t h i s process was found to be a few hundred m i l l i b a r n s , an order of magnitude g r e a t e r than p r e d i c t e d by a s t a t i s t i c a l model c a l c u l a t i o n s and the asymmetry amounts t o a f a c t o r of 2,0 ± 0,3 i n the double d i f f e r e n t i a l cross s e c t i o n . I t i s shown t h a t t h i s asymmetry i n the secondary decay of the se q u e n t i a l breakup process can a r i s e from the memory r e -ta i n e d by the intermediate s t a t e of i t s s t r u c t u r e d u r i n g the primary r e a c t i o n , i n which a neutron i s t r a n s f e r r e d from L i ^ to He 3. I t was found t h a t the s t r e n g t h of the asymmetry i s s h a r p l y dependent upon the d u r a t i o n of the f i n a l s t a t e i n t e r a c t i o n . This confirms the proposed i n t e r p r e t a t i o n , and suggests t h a t f u r t h e r data could l e a d t o u s e f u l i n f o r m a t i o n r e g a r d i n g c o r r e l a t i o n of the p - s h e l l nucleons i n the L i ground s t a t e . AWARDS 1955 Physics Prize, Royal Roads 1958 Engineering Physics Prize, RoM„Cc 1962=66 National Research Council of Canada Bursary and Studentships 1967 National Research Council of Canada Overseas Postdoctoral Fellowship ^GRADUATE STUDIES F i e l d of Study. Experimental Nuclear P h y s i c s Nuclear P h y s i c s T h e o r e t i c a l Nuclear P h y s i c s Cosmic Rays and High Energy P h y s i c s P h y s i c s of Nuclear Reactions Elementary Quantum Mechanics E l e c t r o n i c Instrumentation Electromagnetic Theory Theory of Measurements Group Theory Methods i n Quantum G„ V o l k o f f A. M. Crookei .B. L., White W. Opechowski F. K. Bowers J . Bo Warren M. McMillan Bo Warren Mechanics W. Opechowski PUBLICATIONS AND PAPERS M.A. Reimann, J.R. MacDonald and J.B. Warren, "Charged P h o t o p a r t i c l e s from Argon". Nuclear Phys. 66, U65 (1965)O M.Ao Reimann. "Simple Flow Switch". R e v . S c i . I n s t r . 37, 681 (1966)0 M.A. Reimann, P.W. M a r t i n and E.W. Vogt. " S t r u c t u r e Dependent Asymmetry i n Se q u e n t i a l Breakup from the xn STRUCTURE DEPENDENT ASYMMETRY IN SEQUENTIAL BREAKUP FROM THE REACTION L i 6 (He 3 ,poC)He4 by MICHAEL ANDREW REIMANN B.A.Sc., U n i v e r s i t y of B r i t i s h Columbia, 1959 M.A.Sc., U n i v e r s i t y 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 o f PHYSICS We accept t h i s t h e s i s as conforming t o the requ i r e d standard THE UNIVERSITY OF BRITISH COLUMBIA February 1967 In presenting t h i s thesis i n p a r t i a l f u l f i l m e n t of the requirements f o r an advanced degree at the U n i v e r s i t y of B r i t i s h Columbia, I agree that c h e L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r reference and study. 1 f u r t h e r agree that permission f o r extensive copying of t h i s t h e s i s f o r s c h o l a r l y purposes may be granted by the Head of my Department or by his representatives, I t i s understood that copying or p u b l i c a t i o n of t h i s t h e s i s f o r f i n a n c i a l gain s h a l l not be allowed without my written permission,, The U n i v e r s i t y of B r i t i s h Columbia Vancouver 8 , Canada ABSTRACT An axial asymmetry about, the direction of motion of Li^ has been seen in the breakup of the ground state of this nucleus as an intdrtoediate 6 3 4 state in the reaction Li (He ,po()He ... The total cross section for this process was found to be a few hundred.millibarns, an order of magnitude greater than predicted by a statistical model calculation, and the asymmetry amounts to a factor of 2.0 ± 0.3 in the double, differential cross section. It is shown that this asymmetry in the secondary decay of the sequential breakup process can arise from the memory retained .by the intermediate state of its structure during the primary reaction, in which a neutron is transferrjfed from Li^ to 3 He ; It was found that the strength of the asymmetry is sharply dependent upon the duration of the final state interaction. This confirms the proposed interpretation^ and suggests that further data could lead to useful iriformat^an regarding correlation of the p-shell nucleons in the Li^ ground state. TABLE OF CONTENTS ABSTRACT i i LIST OF FIGURES i i i LIST OF TABLES i v ACKNOWLEDGEMENTS v PUBLICATIONS v i PREFACE 1 Chapter I INTRODUCTION 3 General Introduction 3 F i n a l State Interactions 4 Chapter II REVIEW OF PREVIOUS WORK 9 Chapter III i THE KINEMATICS OF THE Li 6(He 3,po()He 4 REACTION 14 Reaction Channels 14 Three P a r t i c l e F i n a l State Kinematics 14 Choice of Detector Positions 16 Other Detector Positions 21 Residual Ambiguity 21 Chapter IV EXPERIMENTAL TECHNIQUE 22 Introduction 22 Target Preparation 23 Cross Section Measurement 26 The Target Chamber 29 The Detectors 30 E l e c t r o n i c s 31 Chapter V EXPERIMENTAL PROCEDURE AND RESULTS 34 ' Procedure 34 Phase Space Corrections g 38 The Asymmetry i n the Breakup of L i 40 The Total Cross Section 42 Chapter VI DISCUSSION 44 The T o t a l Cross Section 44 The Structure Dependent Asymmetry 45 V a r i a t i o n of the Asymmetry with Proton Energy 49 Future Work . - 5 2 Appendix A STATISTICAL MODEL CALCULATION 54 Appendix B PHASE SPACE CALCULATION 59 Appendix C THE ASYMMETRY AS A FUNCTION OF PROTON ENERGY 63 Appendix D REVERSE CONTRIBUTION CALCULATION 66 - i i i -LIST ' OF FIGURES To f o l l o w page Figure 1. Schematic diagram of open channels l e a d i n g t o the p,<*,<* f i n a l s t a t e f o r the bombardment of L i 6 by He 3 at very low energy. 14 6 3 2. Kinematic contours c a l c u l a t e d f o r L i (He ,p<f^) f o r bombarding energy 1.25 MeV. 15 6 3 3. Kinematic contours c a l c u l a t e d f o r L i (He ,p«o{) f o r bombarding energy 1.25 MeV. 15 4. Kinematic phase diagram f o r <X1 at 95° and a bombarding energy of 1.25 MeV. 19 5. P r e a m p l i f i e r noise as a f u n c t i o n of detector capacitance. 32 i 6. Block diagram of experimental arrangement. j 32 7. Singl e p a r t i c l e energy spectrum taken with the detector at a l a b o r a t o r y angle of 50 p and a bombarding energy of 1.00 MeV. 34 8. E l a s t i c s c a t t e r i n g energy spectrum.. 35 9. E x c i t a t i o n f u n c t i o n f o r alpha p a r t i c l e s from the reac t i o n " Li 6(He 3,pew) at a l a b o r a t o r y angle of 95°. 35 10. Singl e p a r t i c l e energy spectrum taken with l a r g e detector (<X.2) at -81° i n the l a b o r a t o r y and at bombarding energy 1.25 MeV. 35 11. Experimental p o i n t s shown w i t h p r e d i c t i o n of ' s t a t i s t i c a l model. 36 12. Two-dimensional energy spectrum taken with the t r i p l e coincidence gate. 37 13. Two-dimensional energy spectrum taken w i t h the t r i p l e coincident gate. 37 14. Schematic diagram of detector positions. 37 15. Two-dimensional energy spectrum. 37 16. Two-dimensional energy spectrum. 37 17. The asymmetry as a f u n c t i o n of proton energy. 42 18. Schematic diagram of primary r e a c t i o n . 46. - i v -LIST OF TABLES To f o l l o w page Table 1. Thresholds f o r s e q u e n t i a l processes induced by the bombardment of L i 6 w i t h He . 14 2 2. Energy l o s s of p a r t i c l e s i n 10 ug/cm carbon f o i l . 24 3. D e t a i l s of detector, geometry. ; 30 4. Commercial e l e c t r o n i c units'used i n the experimental arrangements. 32 5. D i v i s i o n of proton energy sp e c t r a i n t o energy bins. 41 6. Bin t o b i n y i e l d comparison. 41 - V -ACKNOWLEDGEMENTS The author is deeply grateful to Drs. K.L. Erdman and P.W. Martin for their kind supervision of this work. In particular the conscientious assistance rendered by Dr. Martin during the experimental measurements at a l l hours of day and night is greatly appreciated. The contribution of Professor Erich Vogt to the project was invaluable. His kindness and approachability together with his clear understanding of the problems involved have been a tremendous help and an inspiration to progress. The willing assistance in times of need of both faculty and student members of the Van de Graaff group is thankfully acknowledged. The author is indebted to the National Research Council of Canada for a Bursary and three studentships awarded him as a graduate student, and for their continued support in the form of a postdoctoral fellowship. Finally, the author wishes to thank his parents for their patience, understanding and loyal support during the course of this work. - v i -PUBLICATIONS Charged Photoparticles from Argon M.A. Reimann, J.R. MacDonald and J.B. Warren, Nuclear Phys. 66,, 465 (1965). Simple Flow Switch M.A. Reimann, Rev. Sci. Instr. 37, 681 (1966). Structure Dependent Asymmetry in Sequential Breakup from the Reaction Li6(He3,poc)He4. M.A. Reimann, P.W. Martin and E.W. Vogt, Phys. Rev. Letters (in press). - 1 -PREFACE Helium -3 ions have become available i n p a r t i c l e accelerators only i n recent years. Their use has been and continues to be a great asset to experimental nuclear physics because they o f f e r several advantages as bombarding 3 p a r t i c l e s (1). As a r e s u l t of i t s high mass excess, He induced reactions t y p i c a l l y have large Q-values, thereby providing access to compound systems of high e x c i t a t i o n , even at low bombarding energies. These systems tend to have many open channels f o r decay, and often lead to m u l t i p a r t i c l e f i n a l states. 3 A d d i t i o n a l advantages of He as a bombarding p a r t i c l e are i t s high charge to mass r a t i o (2/3), allowing i t to be used f o r the study of proton r i c h compound systems, and i t s larger radius and mass compared to a nucleon, allowing i t to impart more angular momentum to the compound system than can a nucleon of the same energy. The low bombarding energies h i t h e r t o available from the University of B r i t i s h Columbia Van de Graaff accelerator permit only l i g h t n u c l e i to be used as targets, owing to the height of the Coulomb b a r r i e r . A;search of the l i t e r a t u r e revealed l i t t l e work reported on the i n t e r a c t i o n of He^ with L i ^ . 3 The work up to 1959 i s summarized i n the Chalk River report,on[He induced reactions (1). Most i n t e r e s t i n g to the author was the work of Almqvist and his coworkers (2,3). They investigated the s i n g l e - p a r t i c l e alpha and proton 6 3 spectra from the bombardment of L i with He . The alpha spectra showed a peak which could be at t r i b u t e d to the existence of a l e v e l at ^1.3 MeV e x c i t a t i o n . T .5 • 1 i n L i . 6 3 8 The L i (He,p)Be reaction was investigated by Erskine and Browne i n 1961 (4), again by looking at s i n g l e - p a r t i c l e energy spectra. They i d e n t i f i e d g the 16.62 ," . 16.92 and 17.64 MeV l e v e l s i n Be but t h e i r spectra did not allow - 2 -them to draw any conclusions about states i n L i . It was therefore decided to 6 3 5 re-examine the reaction L i (He ,oc)Li and to attempt to measure the d i f f e r e n t i a l cross section and the angular d i s t r i b u t i o n of the alpha p a r t i c l e leaving L i ^ i n i t s ground state. I t was hoped that Almqvist's possible l e v e l at 1.3 MeV i n L i ^ could be v e r i f i e d or otherwise explained. By early 1965 an angular d i s t r i b u t i o n chamber had been constructed, and preliminary runs f o r t e s t i n g targets (see Chapter IV) were completed. Single-p a r t i c l e spectra taken with a surface b a r r i e r detector reproduced the alpha spectrum of Almqvist. It was then decided to u t i l i z e the dual parameter pulse height analyser acquired by the laboratory i n that year to look at spectra of two p a r t i c l e s i n coincidence. In the summer of 1965 the work of Young et a l (5) at the University 6 3 4 of Maryland was published. This group investigated the L i (He ,poc)He reaction using a two-particle coincidence technique. They were able to conclude that the reaction achieved the t h r e e - p a r t i c l e f i n a l state predominantly by way of 8 5 sequential processes through states i n Be and L i . Their findings are discussed further i n Chapter I I . It was then decided to revise the experiment i n order to attempt to extract more quantitative information than was obtained by the Maryland group. This required a very c a r e f u l analysis of the kinematics and led to the adoption of the t r i p l e coincidence,technique described below. As the experimental work progressed, the asymmetry which i s the subject of t h i s ttiesis was discovered. To the best knowledge of the author the e f f e c t has not been observed before, and i t was at f i r s t thought to be a quirk of the experimental geometry. The i n t e r e s t aroused by t h i s discovery superseded the o r i g i n a l aims of the experiment. The subsequent work i s reported here and has lead to pu b l i c a t i o n (6). - 3 -Chapter I INTRODUCTION General I n t r o d u c t i o n With the sturdy of the atomic nucleus we approach lower l i m i t s on the p h y s i c a l scales of s i z e and time needed t o describe the phenomena of Nature. Nuclear dimensions are commonly mea*sured i n u n i t s of l O - " ^ meter (fm), and the time taken f o r a siibnuclear p a r t i c l e of moderate energy t o cross the —9 9 nucleus i s of the order of 10 second, the " c h a r a c t e r i s t i c nuclear time". Nuclear physics must a l s o deal with the "strong interaction'.', an enormous 39 f o r c e , 10 times stronger than g r a v i t y , which acts at short range and i n a most complicated way between the p a r t i c l e s i nvolved i n nuclear s t r u c t u r e . These extremes cannot e a s i l y be v i s u a l i z e d i n l a b o r a t o r y terms, and the a p p l i c a b l e laws of physics are g r e a t l y complicated by quantal e f f e c t s u n f a m i l i a r i n a macroscopic world. The s c a l e s of time and space f o r b i d the observation of i n d i v i d u a l nuclear events without the help of i n d i r e c t means, of t e n i n v o l v i n g s o p h i s t i c a t e d instrumentation. In the face of these d i f f i c u l t i e s a .framework of nuclear theory has been e s t a b l i s h e d , attended by pohslderable success at f i t t i n g vast amounts of experimental data. I t i s not s u r p r i s i n g , howeverr t h a t the t h e o r e t i c a l approach r e l i e s h e a v i l y upon i n t u i t i v e and phenomenological models v i s u a l i z e d on a more f a m i l i a r s c a l e . Where these models succeed i n f i t t i n g v a r i o us groups of data, an e f f o r t must be made to j u s t i f y them by a ri g o r o u s quantal treatment. This exact treatment i s oft e n mathematically formidable, but the s e n s i b l e use of approximations has l e d t o a s a t i s f y i n g u n i f i c a t i o n of many models i n t o a s e l f - c o n s i s t e n t nuclear theory. - 4 -The ultimate aim of nuclear physics i s to achieve a thorough understanding of the strong i n t e r a c t i o n and i t s e f f e c t on a nuclear system. One approach i s through the study of the properties of nuclear energy l e v e l s l a r g e l y through information obtained from nuclear reactions. A complete treatment of nuclear structure would require s o l u t i o n of the many-body problem with a l i t t l e known force, and .so s i m p l i f y i n g approximations must be made from the s t a r t . These have led i n an obvious way to various nuclear models. Although these models may d i f f e r i n many respects, depending upon which nuclear property they e f f e c t i v e l y describe, they must ultimately converge as our understanding of the physics grows. The study of the r e l a t i o n s h i p between these models i s therefore important,. The other approach to an understanding of the nuclear force i s through study of the mechanisms of nuclear reactions. By causing the i n t e r a c t i o n between n u c l e i and t h e i r constituents to take place in,a controlled manner, the p h y s i c i s t can single out the e f f e c t of several parameters by varying them i n a known way. T h i s work describes the i n v e s t i g a t i o n of a nuclear reaction, and demonstrates how information leading to a possible explanation of the reaction mechanism can be obtained using standard experimental techniques. F i n a l State Interactions The simplest of a l l nuclear reactions to observe i s that between two nucleons. For t h i s reason the study of nucleon-nucleon s c a t t e r i n g and the states of the deuteron have received exhaustive attention. When larger numbers of nucleons are involved ; i n a. reaction, the i n t r a c t a b l e many-body problem- i s encountered. Fortunately, there i s a tendency f o r nucleons to group: themselves into alpha p a r t i c l e s arid larger c l u s t e r s ( f o r example, closed s h e l l s ) so that - 5 -many nuclear reactions roughly resemble two-body processes. This kind of approximation can usually be made whenever both the i n i t i a l and the f i n a l states involve only two p a r t i c l e s . In such cases, the i n t e r a c a t i o n between these p a r t i c l e s , and the i n t e r a c t i o n of t h e i r constituent nucleons leading to t h e i r i n t e r n a l structure,are treated separately. Such a treatment i s asymptotically exact as the separation becomes large, and allows accurate incorporation of b a r r i e r e f f e c t s as in.the Distorted Wave Born Approximation f o r d i r e c t reactions. It i s not s u r p r i s i n g , then, that the majority of data .has•> been co l l e c t e d from nuclear reactions leading to two-body: f i n a l states. However, with the high bombarding energies a v a i l a b l e from modern machines, or i n cases .* 3 3 ; where H or He are used as bombarding p a r t i c l e s , there i s enough energy ava i l a b l e i n the dpmpound system f o r m u l t i p a r t i c l e breakup. The study of f i n a l states involving more than two p a r t i c l e s i s complicated not only by involved kinematics and phase space geometry, but also by the m u l t i p l i c i t y of possible c o r r e l a t i o n s between pairs (or larger sub-groups) of the f i n a l state p a r t i c l e s . In recent years the advent of two-dimensional pulse height analysis and on-line computers has o f f s e t the e f f e c t of these complications enough that considerable knowledge of the reaction mechanisms can be gained by a study of m u t l i p a r t i c l e breakup reactions. When a reaction leads to three p a r t i c l e s i n the f i n a l state, two a l t e r n a t i v e s can be considered. The case where the three p a r t i c l e s come o f f independently with no two-body co r r e l a t i o n s provides a background or continuum i n the energy spectra of the f i n a l state p a r t i c l e s . The character-i s t i c s of spectra due to such an instantaneous breakup are determined purely by the action of the conservation l a w s and by the available phase space, and - 6 -can be predicted by computation. The postulated lack of correlations implies that no nuclear information can be obtained from observation of this continuum alone. The case where two of the three final particles interact strongly, that is via the nuclear force,after the third has ceased to influence them, produces a modified distribution of the particles in phase space. The interpretation of this can lead to information about the nature of the final state interaction producing i t . This type of process is the logical extension from reactions with two to those wit\i three particles in the final state. Consider a typical reaction a + A B + b (1.1) where B is the recoil nucleus. If this nucleus is unstable and later breaks up according to B — - c + d (1.2) then the incidence of a upon target A results in a time sequence of two reactions leaving three particles in the final state, written a + A - * - B + b - * - b + c + d .(1.3) Thus the system (a+A) undergoes a sequential breakup into the state b + c + d, proceeding through an intermediate state B. Clearly, i f the lifetime of B is sufficiently long, (1.1) and (1.2) take place as individual processes each with two particles in the final state. The process (1.3) is then purely sequential, This concept has been discussed in detail by Phillips et al (7). On the other hand, in the limit as the lifetime of B decreases, the middle step in (1,3) is no longer meaningful and instantaneous three-particle breakup occurs. - 7 -The intermediate region where the lifetime of B is bracketed by these extremes is very important to the study of nuclear reaction mechanisms. It also provides simple examples for elementary particle physics where similar effects are found and even less is understood about the forces involved. The formation and decay of such wide states exhibit some of the properties associated with the compound-nueloar model, For instance, there is often evidence of nuclear properties such as energy level structure, and there is typically a 2 1 time delay of the order of 10"" second, an order of magnitude longer than the time associated with direct reactions. On the other hand, the great width of the intermediate system implies a relatively simple structure not typical of the compound nucleus. The supposition of a simple structure is confirmed by the reduced widths which are found to be near i. .> the Wigner limit:,; implying a strong two-body parentage. These ideas have led to the cluster-compound model discussed by Phillips (8), This model describes the intermediate system B entirely in terms of the relative motion of two nuclear ^clusters, the constituent particles C and c. Levels of B observed in three-body breakup reactions are thus thought to be well represented by pure cluster-model wave functions. Regardless of how pronounced the individual characteristics of the state B may appear, i t has become customary to refer to its formation in multiparticle reactions as a final state interaction.between C and c. In the region where the lifetime of B is short, the nature of the two-body final state interaction may be strongly influenced by the presence of the third final state particle b, so that even i f B has distinguishable nuclear properties, these may differ considerably from those observed for the same nucleus in other reactions. Alternatively the energetics of the sequence may be such that one of the particles from the breakup of B could catch up with - 8 -the p a r t i c l e b. This could lead to interference i n the case of i d e n t i c a l p a r t i c l e s , or an a d d i t i o n a l modification of the d i s t r i b u t i o n of the f i n a l p a r t i c l e s i n phase space due to r e s c a t t e r i n g . These p o s s i b i l t i i e s are described by Kacser and Aitchison ( 9 ) . The outcome of such intermediate processes i s d i f f i c u l t to predict and the experimental spectra obtained by t h e i r observation-are hard to analyse. It i s from the c o r r e l a t i o n of theory with experiment i n t h i s area of nuclear physics that much may yet be learned about the mechanism of nuclear reactions at low energies, about the nuclear force i t s e l f , and about the analysis ofj.;coigj§H$)ondingprocesses i n elementary p a r t i c l e physics. This thesis i s based upon the experimental i n v e s t i g a t i o n of a p a r t i c u l a r f i n a l state i n t e r a c t i o n i n the p,oc, of system r e s u l t i n g from the 6 3 bombardment of L i with He . The reaction achieves the three-body f i n a l / 8 5 state predominantly by sequential processes involving states of Be or Li as intermediate systems. The possible channels, are discussed in d e t a i l in Chapter II I . The experimental technique has enabled attention to be ' d i r e c t e d to the sequential process invplving. the ground state of L i ^ as the 'intermediate system. This i s a wide ( 1 MeV) state with a l i f e t i m e of-*7 x 10~ 2 2 sec. The d i s t r i b u t i o n i n phase space of the f i n a l p a rticles corresponding 'to the breakup of*'this state v"was found to involve a s u b s t a n t i a l asymmetry which i s discussed i n Chapters V and VI, The existence of t h i s e f f e c t and i t s observed energjij dependence are strong i n d i c a t i o n s of the reaction mechanism involved, and t h e i r further study in t h i s and other sequential processes may prove Useful f o r examining ''the wave functions of intermediate states and f o r a better understanding of the mechanics of f i n a l state i n t e r a c t i o n s . - 9 -Chapter II REVIEW OF PREVIOUS WORK The term " f i n a l s t a t e i n t e r a c t i o n " was introduced by Watson ( i o ) i n 1952, i n co n s i d e r i n g the e f f e c t of continued i n t e r a c t i o n between p a r t i c l e s produced i n a nuclear r e a c t i o n a f t e r the r e a c t i o n i t s e l f i s completed. In cases where the p a r t i c l e s produced have low r e l a t i v e momentum, they can i n t e r a c t with each other f u r t h e r , independently of the primary r e a c t i o n mechanism, before they get outside the range of t h e i r mutual f o r c e s , Both Watson and Migdal (11) considered how such a f i n a l s t a t e i n t e r a c t i o n might modify both the energy d i s t r i b u t i o n of the products and the d i f f e r e n t i a l cross s e c t i o n f o r th© whole,reaction, Wation considered the inverse process, i n which the i n t e r a c t i o n between the o r i g i n a l r e a c t i o n products takes .place f i r s t , followed', by the i n v e r i e of the .primary r e a c t i o n . The p r o b a b i l i t y of the' inverse primary r e a c t i o n t a k i n g place then depends upon that o f the i n t e r a c t i n g p a r t i c l e s being s c a t t e r e d i n t o the primary i n t e r a c t i o n volume, This i§ d i r e c t l y r e l a t e d t© the s c a t t e r i n g cross s e c t i o n f o r the p a r t i c l e s concerned> and "when the ©riginai r e a c t i o n cress s e c t i o n i s obtained -from the inverse using the p r i n c i p l e of r e c i p r o c i t y j f a c t o r p r o p o r t i o n a l t o the S c a t t e r i n g cross; s e c t i o n of the p a r t i c l e s i n the f i n a l State remains, • fhi§ treatment Was taken f u r t h e r .by..Phillips §t a l ( 7 ) , Who introduced the " g e n g r a l i z i d density ©f s t a t e s function 1' i n @©hfie@ti©n With j s e q u e n t i a l processes l e a d i n g t o m u l t i p a r t i c l e f i n a l s t a t e s , This f u n c t i o n l a r g e l y determines the cross s e c t i o n f o r the emission of the f i r s t p a r t i c l e i n a s e q u e n t i a l process, and i s a f u n c t i o n of the energy eigenvalues of the intermediate system which decays to give the remaining p a r t i c l e s . These eigen-- 10 -values correspond to broad, s h o r t - l i v e d states i n the type of process being considered, and they are e x p l i c i t l y represented i n the density of states function i n terms of the phase s h i f t s appropriate to the mutual scattering of the p a r t i c l e s c o n s t i t u t i n g the intermediate system. The successful use of scatteringjdata to predict the s p e c t r a l shape of the f i r s t emitted p a r t i c l e (12) i n terms of previously known phase s h i f t s suggests that conversely sequential processes might be used as a t o o l f o r measuring phase s h i f t s , p a r t i c u l a r l y f o r systems where other methods are inadequate, such as that of two neutrons. Jarmie and A l l e n (13) attempted to extract the n-n s c a t t e r i n g length and information about the v i r t u a l state of the two-neutron system from the alpha spectrum of the T(.t,<x) nn reaction, but did not obtain any u s e f u l information because the alpha y i e l d due to the neutron c o r r e l a t i o n s was only of the order of l % o f the t o t a l alphas. The favoured channel was the sequential process T(t,n)He^ fallowed by the d i s s o c i a t i o n of He^. Their work and that of others (12,14) i n which one-p a r t i c l e energy spectra were f i t t e d with c a l c u l a t i o n s based on a sequential breakup model implied the prevalence of sequential processes i n m u l t i p a r t i c l e breakup, but the computations were d i f f i c u l t and the conclusions often ambiguous because of the overlap of spectra due to a l l the f i n a l state p a r t i c l e s , and of a l l open reaction channels, on the experimental spectrum. I t was also possible to f i t s i n g l e - p a r t i c l e energy spectra assuming only small contributions from sequential processes. This was shown by Dorenbusch and Browne (15) who f i t t e d the a l p h a - p a r t i c l e spectrum obtained from the reaction Be^(He 3,of)2He 4 12 by assuming predominance of instantaneous breakup of the C compound nucleus into three alpha p a r t i c l e s . - 11 -Early work on f i n a l state i n t e r a c t i o n s established the importance of sequential processes i n m u l t i p a r t i c l e f i n a l state reactions, and indicated the r e l a t i o n s h i p between the cross sections and the relevant s c a t t e r i n g parameters. The main impediment to obtaining quantitative information from experimental r e s u l t s was ambiguity r e s u l t i n g from kinematic degeneracy. This was an in e v i t a b l e consequence of c o l l e c t i n g s i n g l e - p a r t i c l e energy spectra from a system with more than two p a r t i c l e s i n the f i n a l state. When three p a r t i c l e s are,involved f i v e kinematical variables must be known to determine the f i n a l state exactly. One of these i s usually the t o t a l energy of the system and the other four the energy and d i r e c t i o n of each of two p a r t i c l e s . These and a l l other kinematic variables are i n t e r r e l a t e d by the conservation of energy and momentum. Other combinations of these variables are discussed by Zupancic (16). When the di r e c t i o n s of two of three f i n a l state p a r t i c l e s are determined, t h e i r respective energies are kinematically r e s t r i c t e d to a contour which expresses one energy as a function of the other. Such a contour on an energy-energyplot i s e l l i p t i c a l , and varies with the positions chosen f o r the detectors. The s p e c i f i c kinematic contours of i n t e r e s t i n the present work are given i n Chapter I I I . The p o s i t i o n of a given event on the appropriate contour i s determined by the p a r t i c u l a r d i s t r i b u t i o n of the available energy among,the .-three p a r t i c l e s so that a sequential process, which determines the energy d i s t r i b u t i o n uniquely, appears as a point. Sequential processes going through s h o r t - l i v e d intermediate states appear as,'line segments on. the contour due to the natural, width of these states. In a l l cases of three-body breakup the kinematic contours and positions on. them corresponding to various possible f i n a l 12 -state interactions can be computed in a straightforward manner from the conservation laws. In the past few years, a considerable volume of experimental work has been done on multiparticle final state interactions using two-dimensional analysis. This is a two detector coincidence technique which records events on an energy-energy grid thereby placing the experimental points on the appropriate kinematic contours. Two of the earliest published applications of this technique were by Etter et al (17) and Moazed et al (18). 9 3 4 The latter report a further investigation of the reaction Be (He ,p)2He previously observed by Dorenbusch and Browne.(15). This time two solid state detectors were used at fixed angles. Moazed et al concluded that sequential processes through states in Be definitely predominate in this reaction, The complete^determination of the kinematics by two-dimensional §naly§i§ resulted In a reversal of th© interpretation placed by Dorenbuech and Browne upon their single- particle energy spectra. The bulk of recent work using two-dimensional (analysis was reported at the Gatlinburg Conference on the correlation of particles emitted in nuclear reactions (1964). The conclusions reached were that sequential processes dominate over relatively small instantaneous multiparticle breakup backgrounds, and that the intermediate states could be identified, in the majority of cases. Total cross sections for these reactions were found to be large, indicating strong-cluster model configurations for the final state interactions. Also reported at the Gatlinburg Conference was the work of . Young et al (5f of the University of Maryland, who used two-dimensional 6 3 4 analysis to investigate the Li (He ,p<*)He reaction at a bombarding energy - 13 -of 2.7 MeV. This reaction leads t o a three-particle final state comprising a proton and two alpha particles. The Maryland group calculated the appropriate kinematic contours and observed the distribution of events onto these at a number of positions for two surface barrier detectors. They were able to conclude that two-body interactions between the three final state particles strongly affect this distribution. No quantitative information is given in their report beyond identification of the intermediate states contributing 6 3 4 to their spectra. A more detailed look at certain aspects of the L i (He ,pof)He reaction was the aim of the work reported in this thesis. An example of more recent results obtained from two-dimensional analysis of three-body breakup reactions is the work of Blackmore and Warren (20). 3 3 4 They report analysis of the He (He ,2p)He reaction in a plane perpendicular to the beam. In addition to confirming previous conclusions that the reaction proceeds principally through the ground and first excited states of L i ^ , they observed a correlation between the two protons in the final state, indicating the existence of a singlet two-proton interaction. This is evident only for small angular separation of the detectors observing the protons in coincidence. Work is in progress (21) to review the same reaction in the plane of the beam using proton-alpha coincidences. Detector angles are optimized for observing events corresponding to the proton-proton correlation. The analysis of events i s being c a r r i e d out on a PDP 8 computer. It is hoped that a direct comparison w i l l be p o s s i b l e with spectra obtained from the He (T,pri)Hc- r e a c t i o n with the same geometry. This w i l l l e a d , in t u r n , t o a comparison of the extent t o which the d i p r o t o n s t a t e and the s i n g l e t s t a t e of the deuteron occur as f i n a l s t a t e interactions in these two reactions. - 14 -Chapter III THE KINEMATICS OF THE Li6(He3,pop He4 REACTION The capture of a neutron into the s-shell 3 of He to form an alpha particle results in an energy release of 20.577 MeV (24). This amount of energy is large when considered.in relation to the interaction of light i • . 3 nuclei at small relative rrtatoentum. Reactions induced by He therefore tend to be highly exoergic with many MeV of excess energy to be dissipated in the final state. As a result there are' typically several open channels, and the final state kinematics are relatively insensitive to the bombarding energy until the latter becomes comparable in magnitude to the energy released by the reaction. Reaction Channels 3 6 ' The interaction of He with L i results in a. threes-particle final.state comprising a proton and two alpha particles, accompanied by the _ release of 17.071 MeV (22). The final state is achieved predominantly by way 5 8 of sequential processes (5) going through states of L i or Be . The channels leading to the three-body final state which are open at very low bombarding •energies are shown schematically in Figure 1. Thresholds for sequential processes through some high excited states of the intermediate nuclei- are given in Table. 1. Three Particle Ffoal State Kinematics Two-dimensional analysis involves the experimental determination of four kinematic quantities. These are the energies and the respective directions of motion of two of the final state particles observed in coincidence. The directions are fixed by the positions of the two particle detectors, and the energies for each coincident pair of particles are recorded by pulse 6 5 L l + He ® © © © © ® _ p + o c + cc 4' 17-071 MeV Li5(0,3/z~) or + 15-01 MeV J — - Of+-p + 1 -97MeV. Li* 5, Yz) + or + ~ t O M e V a + p + MeV B e ( 0 , 0 * ) + p 4 16- 9 8 M e V l _ a + or + O-094-MeV B e (2.9 > a + ) + p + 14-OS MeV ~ T - 0 f + or4- 2.-9 9 MeV '.Be* (»'•*•» + p -+S-57 MeV — or + or «+ 11-4-9 MeY B e 8 ( l 6 . 6 a > 2 , + ) 4- p 4- 0 - 3 $ M e V or -s-or -4- IG-71 M e V Be S ( l6 .9a. ,Z + ,0 + ) '+ p+ 0 - 0 6 MeV —'or+-cr4 i 7 - 0 ) M e V F i g u r e 1. Schematic diagram o f open channe ls l e a d i n g t o the p,oC,0C f i n a l s t a t e f o r the bombardment o f L i ^ by H e 3 a t v e r y low energy . The numbers i n b r a c k e t s a r e the e x c i t a t i o n energy i n M e V . i n t h e i n t e r m e d i a t e nuc leus and J T T r e s p e c t i v e l y . Q - v a l u e s a r e c a l c u l a t e d from the t a b l e s o f E v e r l i n g e t a l (24). TABLE 1 Thresholds for sequential processes induced by the bombardment of Li^ with He3. Bombarding Energy Threshold Intermediate State 850 keV 1.62 MeV 2.30 MeV Be8(17.64, 1+) Be8(l8.15, 1+) Li 5 ( l6 .4 , 3/2+) - 1 5 ^ height analysis. The fifth of the five kinematic variables needed to determine the f inal state exactly, is the total centre of mass energy. This- is the sum of the kinetic energy introduced by the bombarding particle and the energy released by the refection. Three of the kinematic variables are determined explicitly. These are the detector positions and the centre of mass energy which are constants set by the experimental conditions. On the other handy the energies of the two final state particles observed in coincidence share onedegree of freedom. These energies are relat'ed. in that' they must l ie on the appropriate kinematic contour on the two-dimensional energy plane. A program,has been written for the PDP 8 computer to calculate the kinematic contours for coincident observation of two particlesbelonging to a three-particle final state.' The program treats only the special case in which the momenta of a l l the final state particles are coplanar (and hence coplanar with, the beam). This restriction simplifies the computation, and is applicable ,to the observations reported in this theBis. Two complete sets of kinematic contours for the p,or,ox system resulting from 6 3 4 ' the reaction L i (He ,p<x)He + 17.071 MeV with a bombarding energy of 1.25 MeV are given in Figures 2 and 3. . These contours are for laboratory detector angles of 95° and -30° (Figure 2) and 95° and -110° (Figure 3). The angles are measured counter-clockwise from'the beam direction as shown by the figure inserts. The degree of freedom in'the energy of the final state particles allows events to populate the appropriate kinematic contour over its entire length. When a sequential process through an intermediate state takes place this degree of freedom is removed and events populate line segments of the contours (see Chapter II). These segments are diffused in proportion to the solid angles subtended by the detectors, since these allow an uncertainty in .16 J S M .13 JZ 2) A L P H A AT 95°, PROTON AT -30 ° g ) PROTON AT 95°, A L P H A A T -30° 2) A L P H A A T S5°, A L P H A A T -30° S5" BEAM - 3 0 ' © © J j t 8 9 IO II 12. 13 \A- 15 P A R T I C L E E N E R G Y A T 8 5 , M e V Figure 2. Kinematic contours calculated for Li6(He3,por<x) for bombarding energy 1.25 MeV. Two detectors operated in coincidence are at laboratory angles of 95° and -30°-. . 1 6 PARTICLE ENERGY AT 95°, M©V Figure 3. Kinematic contours calculated f o r L i 6 ( H e ,porof) f o r bombarding energy 1.25 MeV. Two detectors operated i n coincidence are at laboratory angles of 95° and -110°. - 16 -the angle of emission of the observed p a r t i c l e s . The extent of t h i s e f f e c t i s evident from the experimental two-dimensional energy spectra presented i n Chapter V. Choice of Detector Positions Previous work on the L i ^ H e ^ p c ^ H e 4 reaction has shown that considerable complication a r i s e s from overlap on the kinematic contours of l i n e segments corresponding to d i f f e r e n t sequential processes (5). Furthermore, f o r a given set of two detector positions there are three possible contours, depending on which p a t t i c l e i s observed by each detector. These contours are included i n Figures 2 and 3. With; c e r t a i n choices of geometry and bombarding energy the contours may i n t e r s e c t , as i l l u s t r a t e d i n Figure 1 of Reference (5). When the contours cross or come close together there i s a d d i t i o n a l d i f f i c u l t y i n i n t e r p r e t i n g the r e s u l t s . I f a d e t a i l e d , quantitative i n v e s t i g a t i o n i s to be made of the sequential breakup through one p a r t i c u l a r intermediate state, i t i s c l e a r l y e s s e n t i a l that the contribution of these types of overlap to the experimental spectrum can be reduced to an acceptable l e v e l i n the region of i n t e r e s t . In the present work t h i s reduction of ambiguity has been accomplished by using a t r i p l e coincidence technique i n conjunction with c a r e f u l choice of the detector po s i t i o n s . The use of three detectors i n coincidence represents an overd'^termination..6fvtbe kinematic variables i n that the energy and d i r e c t i o n of the t h i r d p a r t i c l e i s established by observation of the other two. However, i t allows unambiguous p a r t i c l e i d e n t i f i c a t i o n , thereby eliminating two out of the three contours, and i t reduces the randomr.'.boincidence, rate to a n e g l i g i b l e value. The count rate f o r the experimental points is unaffected, provided that the s o l i d angle subtended by the t h i r d detector i s large enough that i t - 17 -always contains the t h i r d p a r t i c l e i f the other two are observed. Unambiguous p a r t i c l e i d e n t i f i c a t i o n i n the p,a,c< f i n a l state was obtained by appropriate choice of surface b e r r i e r detectors. For observing alpha p a r t i c l e s up to 10 MeV, detectors were used with a depletion depth of 60 jum. In these, the maximum energy l o s t by a proton i s l e s s than 3 MeV. A low l e v e l discriminator was used i n conjunction with these detectors to suppress signals corresponding to p a r t i c l e s of energy l e s s than 3.5 MeV, thus ensuring that they were ex c l u s i v e l y alpha-p a r t i c l e detectors. For the proton detector, one of depletion depth 380 |xm was used, capable of stopping protons up to 7 MeV. Since t h i s was the only detector capable of observing protons i t s use i n t r i p l e coincidence with the alpha detectors made i t exclus i v e l y a proton counter. Optimizing the detector positions f o r the experimental i n v e s t i g a t i o n of a p a r t i c u l a r breakup sequence requires a d d i t i o n a l kinematic c a l c u l a t i o n s . The sequential process of i n t e r e s t to t h i s work i s that proceeding through the ground state of L i 5 (channel 2 of Figure 1). For t h i s process i t i s necessary to distinguish, the two alpha p a r t i c l e s . Since the d i s t r i b u t i o n of these between the two alpha detectors w i l l be shown to be unambiguous, the alphas are conveniently;' I d e n t i f i e d by the detectors which observe them. Thus the f i r s t alpha p a r t i c l e , which leaves L i ^ as a r e c o i l i n g system, i s recorded by detector a l and the other, from the decay of L i ^ , i s recorded by of2. If the p o s i t i o n of a l i s f i x e d , a r e c o i l axis i s determined f o r the L i ^ system. The d i r e c t i o n of t h i s i n the laboratory, as well as 5 the energies of both the alpha and the L i can be obtained from the standard kinematic equations (23) f o r two-body f i n a l state processes. Since 15.104 - 18 -MeV are released i n t h i s f i r s t step of the sequential process, the k i n e t i c energy of the L i ^ r e c o i l i s large (~7 MeV) compared to the energy available for i t s decay (1.967 MeV). As a r e s u l t the second alpha p a r t i c l e i s only observed within l e s s than 20° of the L i ^ r e c o i l d i r e c t i o n . The proton acquires enough v e l o c i t y from the secondary breakup to emerge i n any d i r e c t i o n , although at 180° to the L i ^ r e c o i l d i r e c t i o n i t has only a few keV of energy. By a r b i t r a r y choice of a p o s i t i o n f o r the proton detector, the f i n a l state becomes determined. The energy released by the secondary breakup i s known, so that the conservation of energy and momentum can be used to calculate the proton energy, and the d i r e c t i o n and energy i n the laboratory of the second alpha p a r t i c l e . These c a l c u l a t i o n s are c a r r i e d out on the assumption that the distance t r a v e l l e d by the intermediate state during i t s l i f e t i m e i s n e g l i g i b l e , and that i t does not lose any energy due to c o l l i s i o n before i t decays. This i s j u s t i f i a b l e because a 7 MeV p a r t i c l e of mass f i v e only t r a v e l s —2 2 5 --/10 fm i n 6.6 x 10 sec, the approximate h a l f - l i f e of L i . A computer program, ("SEQUEBREAK I") has been written f o r the PDP 8 computer to! carry out these c a l c u l a t i o n s . The program was run over a 120° range of proton detector angles f o r each p o s i t i o n of CXl. The procedure was duplicated f or several p o s i t i o n s . o f c*l and f o r several d i f f e r e n t bombarding energies. The r e s u l t s i n each case were plotted to give a kinematic phase diagram showing p a r t i c l e energy as a function of laboratory angle of the detector f o r both the proton and (X2. Thus each diagram corresponds to a f i x e d p o s i t i o n of o(l and a f i x e d bombarding energy. Also included i n the • 5 program i s allowance f o r the natural width of the L i ground state. Upon being given a width T for the ground state, the computer broadens the curve on the phase diagram into a band whose outer edges represent the FWHM of the state. - 19 -Another program ("SEQUEBREAK II") causes the computer to calculate a d d i t i o n a l curves on the phase diagram corresponding to sequential processes going through states of Be . The procedure i s analogous to that f o r SEQUEBREAK I. In t h i s case the proton i s the f i r s t emitted p a r t i c l e , and i t s d i r e c t i o n i s defined by the p o s i t i o n of the" 'proton detector. This 8 establishes a r e c o i l d i r e c t i o n f or the Be . When the l a t t e r decays, one alpha p a r t i c l e must enter detector cyl f o r the event to be recorded. The f i n a l state i s thus determined and computation y i e l d s the energies of a l l three p a r t i c l e s and the d i r e c t i o n of motion of the remaining alpha. This g program allows the e x c i t a t i o n i n Be as an input v a r i a b l e . The completed phase diagram f o r oCl at 95° and for a bombarding energy of 1.25 MeV i s shown i n Figure 4. From, t h i s phase'diagram the following features of the kinematics become apparent: a) For the sequential process through the ground state ' ' t f L i ^ the proton energy i s a. maximum at the laboratory angle corresponding to the d i r e c t i o n of motion of the L i ^ r e c o i l ( - 6 7 . 1 ° ) . .This i s an axis of symmetry f o r the proton curve, as expected. On e i t h e r side of t h i s axis the proton energy drops sharply as.a function of the laboratory angle, so that u s e f u l measurements can only be made within a proton detector angle of 60° to the L i ^ r e c o i l d i r e c t i o n . Because the associated alpha from the L i ^ breakup i s only seen within a 32° cone centred on the L i ^ axis, t h i s means that the proton must be observed at an angle of less than 90° to the detectorcX2. This makes the angle between the IO F i g u r e 4. Kinematic phase diagram f o r a l at 95 and a bombarding energy o f 1.25 MoV. Lovjer case l e t t e r s and numerals i n d i c a t e corresponding p o i n t s on proton and alpha curves. 8 :! 7 o r m z rn s . o < < 3 I O A B c D E F 6 H A L P H A S , 1 3 - 5 M E V IN D e A L P H A S , I 6 - 9 Z S T A T E B e A L P H A S , 1 6 - 6 2 . S T A T E B e 8 8 8 P R O T O N S , 15 -5 M E V IN B e P R O T O N S , \&eZ S T A T E IN Bi 8 8 P R O T O N S , 1 6 - 9 2 S T A T E . IN Be A L P H A S , L t 5 G R O U N D STATE P R O T O N 8 , Li 5 G R O U N D S T A T E - I O -ZO - 3 0 - •50 - 6 0 - 7 0 - S O - 9 0 -IOO -110 L - A B O R A T O R . Y A N G L E , D E G R E E S -IZO - ISO - 1 4 0 -150 - 20 -proton detector and 0X1 always greater than 100°. There is therefore no ambiguity as to the distribution of the alpha particles between the alpha detectors i f a coincidence is observed. 8 • The states in Be which contribute to a coincidence spectrum recording events due to the formation of the L i ^ ground state are'the two.-.narrow states at .16.62 and 16.92 MeV, and •the broad state at 11.4 MeV. The curve on the phase diagram representing an excitation in Be of 13.5 MeV is hypothetical and is meant to,'represent. the half maximum of the 11.4 MeV state on the assumption of a FWHM of.4 MeV for this state. •The [Contribution from, this state is thought to be small below this curve, because of the large angular momentum transfer needed to form the. 4+- state-, and because lihose events which do involve i t are distributed over a large energy range due to its width. ,.The proton curves corresponding to states In 8 o 1 Be are symmetrical about 0 as expected, since for these the proton is the first emitted particle. The optimum positions for the proton detector to observe events corresponding to the sequential process through the ground state of L i^ can be seen on the phase diagram. The obvious choices would be -20° or -120° . However, in order to achieve a compromise with the chamber geometry and the position of the target, the angles actually chosen were -30° and - 1 1 0 ° . , - 21 -The kinematic phase diagram given as Figure 4 shows th a t two p o s i t i o n s of the proton detector ( w i t h corresponding p o s i t i o n s f o r c*2) are a v a i l a b l e f o r observing the breakup of the L i ^ ground s t a t e . These may be chosen t o be symmetrical about the L i ^ r e c o i l a x i s i n a l l respects except g f o r a s m a l l d i f f e r e n c e i n the energies of the protons from the! Be s t a t e s , and a s m a l l d i f f e r e n c e i n the volume of phase space seen by the detector i n the two p o s i t i o n s due to centre of mass motion. I f p o s i t i o n s are chosen which are not q u i t e symmetrical, such as -30° and -110°, the r e s u l t w i l l be a r e l a t i v e d i f f e r e n c e i n observed proton energies and a s l i g h t l y d i f f e r e n t phase space f a c t o r . These are both c a l c u l a b l e geometric c o r r e c t i o n s . i i i Other Detector P o s i t i o n s As has been pointed out above, the i n f l u e n c e of bombarding energy or the f i n a l s t a t e * k i n e m a t i c s i s s m a l l , because of the r e l a t i v e l y l a r g e amount of energy released by the r e a c t i o n . The aharige i n the phase diagrams as a f u n c t i o n of bombarding energy i s t h e r e f o r e s l i g h t . For the same angle of o f l the optimum angles f o r the proton detector only vary a few degrees with an increase i n i n c i d e n t energy of 50% (from 1.0 t o 1.5 MeV). D i f f e r e n t p o s i t i o n s f o r a l at a constant^bombarding energy appear mainly as a l a t e r a l s h i f t of the a b s c i s s a i n the phase diagram. The optimum choice was 95° because other angles introduce e i t h e r an i n t e r f e r e n c e with the beam at one of the detector p o s i t i o n s , or r e s u l t i n a geometry r e q u i r i n g unfavourable t a r g e t p o s i t i o n s , An angle of greater than 60° between any detector a x i s or the beam and the normal to the t a r g e t , was considered excessive. - 21a -Residual Ambiguity When the detector positions have been chosen to optimize the experimental arrangement f o r observation of events proceeding through the L i ^ ground state as described above, i t i s s t i l l k i nematically possible, i n s p ite of the t r i p l e coincidence requirement, to observe protons from the breakup of the f i r s t excited state of L i ^ . This state i s several MeV wide and occurs at an e x c i t a t i o n between 5 and 10 MeV i n Li^(37, 38). The e n t i r e energy spectrum of protons in coincidence with alphas selected to correspond to events going through the L i ^ ground state may also be populated by events corresponding to the excited state f o r which the f i r s t alpha p a r t i c l e i s detected in the detector c>\2 and the second alpha, from t h e / L i ^ excited state breakup, i s detected in (Xl. Since t h i s p o s s i b i l i t y e n t a i l s a r e v e r s a l between the actual and 'intended d i s t r i b u t i o n of the two alpha p a r t i c l e s among the alpha detectors, we r e f e r to t h i s p a r t i c u l a r type of event as a "reverse process". Since the detector positions have already been selected according to the previous considerations, counts from the reverse process constitute an a d d i t i o n a l "background" that must be considered. The c a l c u l a t i o n s discussed i n Appendix D indicate the extent of t h i s contribution to be l e s s than 10% and further indicate that interference e f f e c t s should be unimportant, thus allowing i t to be subtracted from the experimental spectrum as a simple background. - 22 -Chapter IV EXPERIMENTAL TECHNIQUE Introduction Experimental spectra were obtained by bombarding an i s o t o p i c a l l y enriched L i F target (95.6% L i ^ ) evaporated onto a t h i n f i l m 3+ carbon backing with He ions at 1.00, 1.25 and 1.50 MeV. The reaction products were observed by surface b a r r i e r detectors operated i n t r i p l e coincidence with a resolving time of 50 nsec. Random coincidences constituted l e s s than 5% of the experimental count rate, and were not observed to occur i n the region of i n t e r e s t . Two-dimensional coincidence spectra of the energies of the f i r s t alpha p a r t i c l e and the proton from the reaction He 3 + L i 6 L i 5 + OC I oc2 + P were accumulated, i n a 64 x 64 array using a dual parameter pulse height analyser gated by the t r i p l e coincidence pulse. Observations were made i n a plane containing the incident beam, and the po s i t i o n i n g of the detectors was governed by the calculated kinematics (Chapter I I I ) . The reaction y i e l d was compared to the e l a s t i c 3 19 sca t t e r i n g y i e l d of He on F ,. which was monitored during, each run using a f i x e d detector at a laboratory angije of 143°. This chapter discusses the t e c h n i c a l aspects of the experimental arrangement. I t begins with discussion of target preparation, outlines the techniques used f o r cross section measurement and gives the d e t a i l s of the - 23 -target chamber geometry including the method of lining up. The electronics used in the experiment are also discussed while the actual experimental procedure is left for the next chapter, io be combined with presentation of the results. Target Preparation The requirement was for a L i^ target thick enough to give a reasonable yield, yet self-supporting and thin enough that the reaction products could.be observed through the target without appreciable energy loss. Its great affinity for both oxygen and water vapour leads to difficulties in using free lithium metal as a target material. When lithium metal targets are desired, the usual procedure Is to evaporate them onto a target backing within the target chamber (24). This method is useful for making thick targets on thick backings, but gives poor control over the thickness, which must be measured by a separate technique. These targets tend to deteriorate with time, even in a vacuum of 10~^  Torr. Furthermore i t is particularly undesirable to evaporate materials inside a target chamber containing surface barrier detectors which are easily ruined by contamination of their sensitive surfaces,. For these reasons it was decided to use a lithium salt as the target material. No-lithium. s:alt; is presently known from which self-supporting targets can be made. Therefore, a:strong,, thin backing material had to be 'chosen. Possible choices were aluminum oxide (A1203) or carbon. A useful technique for making ;A1203 foils is described by Bruckshaw of the University of Manitoba Physics Department (25). ; .These foils have the disadvantage of being non-conducting, so that precautions are.needed to prevent their charging - 24 -up and t o avoid overheating i n the beam. The u s u a l procedure i s t o add a conducting f i l m of copper, aluminum or gold. A f t e r some experimental t e s t s i t was decided t h a t carbon f o i l s were more s u i t a b l e , t h e i r main advantages being t h a t they are adequately conducting both t h e r m a l l y and e l e c t r i c a l l y , and t h a t they .contain only two 12 13 isotopes (C and C ) t o c o n t r i b u t e t o background nuclear r e a c t i o n s . Carbon f o i l s have been widely used f o r these reasons. The p r e p a r a t i o n of carbon f o i l s was c a r r i e d out i n the manner described by Dearnaley . A current . of approximately 100 amps flows along two \ i n c h diameter carbon rods i n .s e r i e s . Both are sharpened to a p o i n t where they touch, so t h a t the point of contact reaches a s u f f i c i e n t l y high temperature t h a t an arc i s produced i n which the carbon evaporates. The rods are f r e e t o s l i d e i n copper bushings and are h e l d i n contact by l i g h t t e n s i o n springs during, the evaporation, which * i s c a r r i e d under a vacuum of 10~^ Torr. The carbon i s evaporated onto standard glass microscope s l i d e s p o s i t i o n e d at ~10.cm from the arc. These must be extremely c l e a n , and are t h i n l y coated with a h i g h l y s o l u b l e detergent to ,enable the carbon deposit t o be f l o a t e d o f f the glass onto the surface of a water bath. As described by Dearnaley, four to seven evaporations l a s t i n g a i few seconds each gave e x c e l l e n t carbon f o i l s , w h i c h were picked up on t a r g e t holders c o n s i s t i n g of a 3 i n c h s t r i p of 1/32 inch copper, \ i n c h wide, with two \ i n c h holes d r i l l e d 3/4 inch apart near one end. This arrangement thus 3 provided two t a r g e t backings per holder. E l a s t i c s c a t t e r i n g of He from 2 s e v e r a l t a r g e t backings showed them t o have a t h i c k n e s s between 8 and 30 pg/cm . The experimental spectra given below were obtained using "a t a r g e t whose carbon backing was 10 + 2 Ug/cm t h i c k ( — 4.6 x 10 cm). This corresponds t o p a r t i c l e energy l o s s e s of the order shown i n Table 2. TABLE 2 Energy loss of particles in 10 ug/cm carbon fo i l Particle Particle Energy Energy Loss proton 1.0 MeV 2.1 keV He 1.25 MeV 13.6 keV alpha 1.0 MeV 16.2 keV alpha 5.0 MeV 7.2 keV Some attempts were made to use LiH as a target material f o r evaporation onto the carbon f o i l , but t h i s s a l t turned out to be almost as se n s i t i v e to moist a i r as l i t h i u m metal and i t tended to produce the c r i n k l i n g and weakening of the f o i l noted by Dearnaley (26) f o r a l k a l i metals. Whether t h i s was due to p a r t i a l decomposition of the L i F to give free l i t h i u m during the evaporation was d i f f i c u l t to determine. The f i n a l choice of s a l t was L i F . A sample was obtained / from AECL at Chalk River, i s o t o p i c a l l y enriched to 95.6% L i ^ . This s a l t was evaporated onto the carbon f o i l s under vacuum from a boat made by hollowing out one side of a \ inch diameter carbon rod,of the same type used to make the carbon f o i l s . In : t h i s way i t was hoped to avoid evaporation of foreign materials onto the targets. The problem observed by Dearnaley (26) that s a l t s such as L i F tend to decrepitate and puncture the f o i l s during evaporation, was solved by s h i e l d i n g the targets from the boat u n t i l the L i F was molten and i n the process of evaporating. The targets were then exposed f o r a period of ^45 sec and shielded again. With the targets at a height of 10 cm above the boat containing a ~2mm diameter p e l l e t of molten L i F at a temperature of ~1000 GC, t h i s exposure produced a L i F deposit of 10-20 pg/cm'V The energy loss of p a r t i c l e s i n t h i s i s of the same order.as f o r the carbon backing. For the complete target the energy losses were thus i n the range of two to three times the values shown i n Table 2. These losses are not s i g n i f i c a n t i n the r e s u l t s of t h i s work. L i F has several advantages;.; as a target material, I t i s easy to evaporate using standard equipment, and i t can be stored i n a i r without Jdesiccants f o r long periods without i l l e f f e c t . I t qan also stand r e l a t i v e l y high temperatures during bombardment (the melting point is^870°C), and the - 26 -one to one ratio of lithium to fluorine atoms can be utilized to monitor the target thickness as described below. Measurement of the Reaction Cross Section The determination of a nuclear reaction cross section is accomplished by measurement of the reaction yield. This is related to the cross section by Y = crnN where Y is the. yield, cr the cross section and n number of bombarding nuclei which have crossed an area populated by N target nuclei. Whether O" in this case represents the total, a differential or other partial cross section depends upon how Y is determined. The most difficult quantities to measure experimentally are n and N. The quantity n is usually determined by monitoring the total charge collected by stopping the incident beam^  either in the target backing,, or in the case of thin targets, in a Faraday cage designed for this purpose. Accurate charge monitoring is difficult because the beam knocks electrons out of collimators, the target and the beam catcher. If electrons ejected from the target and collimator are captured in the Faraday cage the beam current appears to be too small. If electrons are knocked out of the cage and not recaptured, the current appears to be too large. Considerable errors are possible unless care is taken to suppress these electron effects. The quantity N is even more difficult to determine. The only absolute'way is to weigh the target material present in a given area. This is feasible where metallic foils are used, as targets, but is out of - 27 -i the question in eases where the target is :inseparable from a holder weighing orders of magnitude more than the target itself. This problem can sometimes be solved i f a highly reproduceable technique is available §QT evaporating the target material onto a backing. In such a case a very light backing of large area can be used to weigh the target material deposited per unit area. The difficulties entailed in this procedure aire obvious.1. . A more common way of measuring N is by observing ithe yield for a reaction of known cross section. In. a l l cases, additional and often unknown errors arise from deterioration of the target during the experimental measurement, and from non-uniformity of the target thickness. The.latter can be serious i f the beam spot issmall and illuminates,different parts of the target during the measurements. The uncertainties in measuring n and N outlined above are completely avoided by the technique used for this experiment. A surface barrier detector fixed at a laboratory angle of 143°, and of small, well defined solid angle, was used to continuously monitor the elastically 3 scattered He flux from the target while the yield from.the reaction was being measured. Since both the reaction and elastic yields were obtained simultaneously, the ratio.of the two yields'is equal to the ratio of the respective cross,sections (except for geometrical factors). The. beam current and target thickness, and a l l the errors inherent in their measurement, cancel in this calculation, and therefore need no longer be considered. In practice, the reaction-yield was compared to the elastic scattering yield 3 2.9 of He from F . This was done on the assumption of a one to one ratio of fluorine to lithium nuclei in the target (a correction was of course made for the 4.4% of Li present). j i - 28 -The elastic, or Rutherford, scattering differential cross section is well known, and is given by where m, z and E are the mass, charge number and laboratory energy of the projectile, M and Z are the mass and charge numbers of the target, and© is the laboratory angle at which the scattered particle is observed. A computer program has been written to calculate this, differential cross section a s required. The Rutherford formula is exact provided the scattering ,. 3 is not influenced by nuclear f o r c e s . The Coulomb barrier for He incident 19 on F is over 4 MeV high,so that the Rutherford cross section for this ' ' 3 target nucleus was considered reliable for He bombarding energies of less 3 than 2 MeV. The classical closest distance of approach of .1.25 MeV He o 19 scattered at 143 from F is <—7 times the nuclear radius of the target. The assumption that the nuclear force does not significantly affect the elastic scattering is therefore reasonable. Experimental work has been initiated to measure the deviation from the Rutherford' formula of the 3 19 elastic scattering differential cross section for He • on F , at backward angles, as a function of bombarding energy. This:is to be done by bombarding several fluoride targets involving heavier nuclei, also in a chemically fixed ratio, such as^rubidium,cerium and barium fluorides. Elastic scattering from the largejr nuclei should/be .purely Rutherford to ai muchc higher bombarding enjrgy than that front the fluorine. This work is being done by Helmer (27). - 29 -The Target Chamber A simple angular d i s t r i b u t i o n chamber, seven inches i n diameter and seven inches deep, was constructed of brass. This i s mounted on i t s own vacuum table equipped with a 100 L/sec, l i q u i d nitrogen b a f f l e d o i l d i f f u s i o n pump. The chamber i s equipped with three coplanar detector holders, which enable detectors t o be positioned at any angles i n the reaction plane a^id at distances from 2 to 8 cm from the target. The target holder allows e i t h e r of two targets to be accurately placed i n the beam at any desired angle., The beam i s defined by two tantalum collimators 15 cm 2 apart so that an area o f ~ 4 mm on the target i s illuminated. The beam passes through the chamber into a Faraday cage with arrangements f o r electron suppression. The chamber has a long side-arm whose^axisnintersectSlithe^beam axis' at the target p o s i t i o n i n the plane of the detectors. I t makes an angle of 143° with the forward1,beam d i r e c t i o n , and has a detector holder at. it's end. The distance of t h i s detector from the •far get can be varied from 25 to 100 cm. The chamber i s equipped with a glass viewing port. The alignment,of the chamber was done o p t i c a l l y , using the beam from a continuous helium-neon gas l a s e r . The l a s e r was put into the beam l i n e close to the analyzing magnet, and the chamber adjusted u n t i l the beam passed through the collimators and illuminated the t i p of a removable pointed sp'd\idle i n d i c a t i n g the centre of the chamber. . A t h i n glass mirror was mounted i n the target holder with s i l v e r e d surface at the target holder axis. The external angular vernier on the target holder was zeroed, by s e t t i n g the mirror to r e f l e c t the l a s e r beam back on i t s e l f . This adjustment was believed correct to - \ degree. The angle between the axis of the side-arm and the forward beam d i r e c t i o n was measured to be 143 i 0.5°, by reading - 30 -the target holder vernier when the laser beam was reflected along the axis of the side-arm. The other detector holders were adjusted to be coplanar with the side-arm and beam axes by centering them on the reflected laser beam. Changing the angles of these detector holders did not cause them to move out of this plane by an observable amount. One detector holder was fixed, and, by use of the laser, was related to an angular vernier allowing the entire detector holder assembly to-be rotated in the reaction plane from outside the chamber. The laser was also used to calibrate a linear vernier to position the targets in the beam from outside the chamber. Relative angles between the detectors can be set to within i \ degree using a specifically constructed device involving a protractor automatically centred on the detector holder axis. The distance from the target axis to the collimator defining the solid angle of the elastic scattering detector was measured using a rigid rod and a meter stick. Distances from the target axis to the collimators defining the solid angles of the other detectors were measured with a vernier caliper. The pertinent measurements, derived quantities, and estimated errors are given in Table 3. A l l the detectors except the scattering monitor were protected from the elastically scattered beam by nickel foils. The thickness of these was measured by determining i" 241 the shift in the energy spectrum obtained from a 5.48 MeV Am alpha source when the fo i l was Introduced. Energy losses for other particles and energies .could then be calculated using a computer programme written for this purpose (21), The Detectors A l l the charged- particle detectors used were silicon surface barrier semiconductor diodes. These are p-n junction diodes consisting of Name of detector Type of detector Nominal Foi l Thickness Measured Fo i l Thickness Distance (cm) from target Collimator Area (cm^) Solid Angle (ster) Angle" Subtended at target 0C1 ORTEC SBCJ 050-60 45 | A n _ ,52 uin + 5 3.48 ± 0.02 0.1564 ±0.001 0.0129 +0.0001 6.5° ORTEC SBFJ 200-60 45 p. in 51 uin t 5 . 5.01 ± 0.02 1.95 ±0.1 .0.777 ±0.004 17.9° P ORTEC SBFJ 200-300 45 jiin 49 pin 1 5 4.61 ± 0.02 0.2646 +0.001 0.9125 +0.0001 6.4° R ORTEC SBEG 007-300 — — 51.9 ± 0.2 0.02500 ± 00015 9.28 ±0.07 (x IO"6) 0.17° Notes: R is the elastic (Rutherford) scattering counter at 143 t 0.-5° 0(2 has no collimator, the area given being that of the sensitive area. TABLE 3 Details of Detector Geometry. - 31 -an extremely t h i n p-type layer on the s e n s i t i v e side of a high p u r i t y , n-type s i l i c o n wafer. E l e c t r i c a l contact i s made to the p-type l a y e r through a very t h i n (^150 A) gold f i l m evaporated onto the surface, and t<5 the n-type wafer through a n o n - r e c t i f y i n g contact on the back surface. Depletion occurs at the p-n junction and extends into the n-type material to a distance known as the depletion depth. This depth varies with the square root of the applied bias voltage and, i n conjunction with the surface area, determines the s e n s i t i v e volume of the detector. The theory of surface b a r r i e r detectors has been treated by several authors, f o r instance Dearnaley and Northrop (28), and w i l l not.be discussed further. The -detectors used in t h i s experiment were provided by the Oak Ridge Technical Enterprises Corporation (ORTEC). They t y p i c a l l y have an alpha p a r t i c l e energy r e s o l u t i o n of l e s s than 60 keV FWHM, measured 241 with a t h i n Am (5.477 MeV) alpha source and a low noise («~ 5 keV) amplifying system. Surface b a r r i e r detectors have a deadlayer due to the gold f i l m on t h e i r surface, but t h i s i s very t h i n , amounting to an energy loss of ^ 1 5 keV f o r a.5 MeV alpha. E l e c t r o n i c s ' With the'exception of the preamplifiers, the e l e c t r o n i c units used i n t h i s experiment are a l l standard commercial items. The preamplifiers comprise a two nuyistor f i r s t stage, followed by four t r a n s i s t o r stages. The o r i g i n a l design, due to Alexander (29) was modified by Jones (30). The f i n a l design was incorporated into a printed c i r c u i t by the author. The c i r c u i t diagram and a d e s c r i p t i o n of operation are given by Whalen (31) and Blackmore (32). The important features of t h i s a m p l i f i e r are i t s f a s t risetime ( • — T 5 nsec) i n conjunction with low noise. The e l e c t r o n i c noise - 32 -of these units was measured as a function of the input (detector) capacitance using, a wideband VTM with the results shown in Figure 5. A block diagram of the experimental arrangement is shown in Figure 6,.and details of the commercial units are given in Table 3 the pulses from the- huvistor preamplifier corresponding to detection of a particle are double delay-line clipped by the main amplifier, whose prompt output triggers the single channel analyser at zero cross-over .(provided the pulse height falls within its window). The pulses produced by the single channel analysers trigger pulse generators with a continously variable internal delay. This delay enables the three units to be synchronized when a test pulse is applied in parallel to the inputs.of the three preamplifiers. The outputs of the pulse generators are fed to the triple coincidence.unit, which can be operated in triple% double or single coincidence modes. The resolving time of the coincidence system is determined.by the duration of the pulses from the pulse generators. This was variable down to 20 nsec giving a triple coincidence resolving time of 40 nsec. The resolving-} time used in the experiment was 50 nsec. The output pulse from the triple coincidence unit was used to trigger a pulse generator, which produced the correct gating pulse for •the dual parameter p\ilse height analyser. A variable delay in this pulser allowed the gate to be synchronized with' the delayed (2 jjisec) output of the main amplifiers. The' gate-pulse generator also drove a.scaler to count coincidences during the experiment. The delayed outputs representing particles stopped by detectors CXI and P were respectively fed to the X and Y inputs of the dual parameter analyser, where they were accumulated in a 64 x 64 array. Pulses from the elastic scattering detector were Figure 5. Preamplifier noise as a function of detector capacitance. (X2 PRE A M P P R E A M P P — * r P R E A M P L l l P R E " A M P ( § ) BEAM POSITION M A I N © A M P L I F I E R I PULSE (§) HEIGHT A N A L Y S E R SCALER PULSE GENERATOR Fi g u r e 6. Block diagram of experimental arrangement The detectors are named as i n Table 3, and .the minor r e f e r t o the l i s t o f commercial u n i t s , Table 4. MAIN (§ ) AMPLIFIER MAIN (5 ) AMPLIFIER MAIN (S) AMPLIFIER DUAL PARAMETER PULSE H E I G H T A N A L Y S E R (g) AMPLIFIER P R O M P T P R O M P T • Pfil-ftYE-P. PROMPT p E U A f E p TRIPLE COINC. UNIT S I N G L E C H A N N E L A N A L Y S E R ' © S I N G L E C H A N N E L A N A L Y S E R © S I N G L E C H A N N E A N A L Y S E " R. VL'LSE <? ENERATOR^ PULS!?. C;ENERATOt@ P U L S E : G E N E R A T O R TABLE 4 Commercial Electronic Units used in the Experimental Arrangement. ORTEC Model 201 Low Noise Amplifier (Oak Ridge Technical Enterprises, Oakridge, Tennessee). ORTEC Model 101 Low Noise Preamplifier ; (Oak Ridge Technical Enterprises Corpt, Oakridge, Tennessee). Cosmic Model 901 A Linear Amplifier (Cosmic Radiation Labs Inc., Bellport, N.Y.). Cosmic Model 901 SCA Single Chanrjel Analyser (Cosmic Radiation Labs, Inc., Bellport N.Y.) Datapulse 106A Pulse Generator (Datapulse Inc., Inglewood, California). 'LRS Model III B Three Fold Logic Unit (LeCroy Research Systems Corp., Irvington, N.Y.) ORTECModel 410 Multimode Amplifier (Oak Ridge Technical Enterprises Corp., Oakridge, Tennessee), Datapulse 101 Pulse Generator (Datapulse Inc., Inglewood, California). Nuclear Data ND 120 512 Channel Analyser (Nuclear Data, Inc, Palatine, Il l inois). 10) Nuclear Data ND 160 Dual Parameter Analyser : (Nuclear Data Inc., Palatine, Illinois). Note: The numbers refer to those on the block diagram, Figure 6. - 33 -amplified by the low noise, slow risetime ORTEC amplifying units and were fed to a 128 channel pulse height analyser. A l l the analysing equipment could be started and stopped synchronously by remote control. Not shown in the block diagram are an ORTEC detector bias unit and the test pulse generator. - 34 -Chapter V EXPERIMENTAL PROCEDURE AND RESULTS The technical details of the experiment, including the' method of aligning the target chamber, target production and electronic arrangement have been discussed in the previous chapter. The kinematic calculations governing choice of detector positions for a triple coincidence measurement investigating breakup of the p,of,(X system through the ground state of Li^ have been given in Chapter III. The following is an outline of the experimental procedure and a statement of the results. Procedure Init ial experimental work involved accumulating single-particle energy spectra using a surface barrier detector of small angular aperture. Single-particle spectra were obtained at several laboratory angles and over a range of bombarding energies from 1.00 to 1.60 MeV. The individual spectra reproduced the features of the alpha spectrum obtained by.Almqvist (2). Figure 7 is such a spectrum obtained at 1.0 MeV and a laboratory angle of 50°. The depletion depth of the detector used was 175 1^11, so that the "proton edge" (corresponding to the maximum energy which a proton.can lose in the depleted layer)1 occurs at —4.5 MeV, as indicated, on the spectrum. The spectra a l l show the broad high energy alpha group, but its structure appeared to vary somewhat with bombarding energy and angle. The difficulty of interpreting single-particle energy spectra of three-botiy breakup reactions has already been discussed. The only measurement considered useful in view of the more advanced techniques -C , 5(He 5 ,0c)c i z 00 Z D 0 o UL o CO Q lu Q: o 2 D 2 P R O T O N E D G E CP co o ^ ' o 0 OOO oo o © csS) o°oo o o - 4 - 5 G 7 S PARTICLE ENERGY , MeV IO Figure 7. Single particle energy spectrum taken with the detector at a laboratory angle of 50° and a bombarding energy of 1.00 MeV. The depletion depth of the detector was 175 pm. available was an excitation function for the.yield of alphas in the high energy group at a fixed angle. This was determined at a laboratory angle of 95°, and the relative yield was obtained by comparison with the elastic 3 19 scattering peak from He on F , as described previously. A typical elastic scattering spectrum taken at the laboratory angle of 143° and at a bombarding energy of 1.0 MeV is shown in Figure 8. The excitation function is given in Figure 9. It shows a large background with the energy dependence expected from penetrability of the Coulomb barrier, and a 9 superimposed peak attributed to formation of the 17.63 MeV state in B . The excitation function does not lead one to expect important resonant effects in the reaction. Figure 10 shows the single particle spectrum taken with the detector #2, whose angular aperture (Table 3) is 35 .8° . This spectrum was used, only to locate the proton edge. By monitoring this spectrum with the electronics in the "single coincidence" mode the lower level discriminator on the (X2 single channel analyser could be raised until only counts above the proton edge were recorded. In a coincidence mode, this detector then became exclusively an alpha-particle counter. A similar procedure was followed for setting the lower level discriminator for detector 0\1. This was done before each experimental run in a l l the coincidence work. The asymmetry in the breakup of the Li^ intermediate state 6 3 4 in the L i (He ,prx)He reaction was discovered during a preliminary measurement of the angular distribution of the first emitted salpha particle (observed by detector o(l) at a bombarding energy of 1.00 MeV. This distribution was investigated by placing o^ l at eight different-, angles and then using the appropriate computed kinematic phase diagrams (Chapter III) to determine the S C A T T E R E D H e 5 E N E R G Y , K e V Figure 8. Elastic scattering energy spectrum for ne6 scattered from a I LiF (96.5% L i 6 ) target.on a carbon backing at a laboratory angle of 143 and at a bombarding energy of 1.00 MeV. 1 4 | t 1 | 1 ! — 1 j . O I-l \'Z 1-5- I-+ 1-5 1-6 B O M B A R D I N G E N E R G Y , M e V Figure 9. Excitation function for alpha particles from the reaction Li?(He3,ppcoc) at a laboratory angle of 95° . The peak is attributed to the formation of Hie 17.63 MeV level in B . Figure 10. Single p a r t i c l e energy spectrum taken with large detector (c*2) at -81° i n the laboratory, and at bombarding energy 1.25 MeV. O 0 o 0 ° 0 o CL ° o °o° y e g , L U s o °9? z o UJ J 0 h UJ Q& Ul O u LL z r& e—-a. - 36 -optimum position for the proton detector P, and the second alpha detector, o<2. The three detectors were operated in triple coincidence and the yield derived from the output of detector ril. Energy windows were applied to the other two detectors with the help of the phase diagrams, to restrict observed coincidences to only those events representing the desired intermediate state. This technique was not very satisfactory because of the time involved in setting the energy windows, and the strong dependence of the yield on these settings. The method used to set the windows was, however, consistent, and the striking nature of the asymmetry discovered can be seen in Figure 11, which shows the measured distribution. The differential cross section given in this figure was calculated on the assumption that the breakup of Li^ in its centre of mass is isotropic, and incorporates the geometrical correction discussed below. The error bars represent uncertainty due to counting statistics. It is difficult to estimate the error on the absolute differential cross section, particularly since the breakup of Li^ is clearly not isotropic. The discontinuity in the cross section curve occurs when the proton detector is positioned on opposite sides of the Li^ recoil direction. The inset in Figure 11 illustrates this. For the proton detector iforward of the Li^ recoil axis the differential cross.section is larger. With cXl near 90° the geometry allows the proton detector to.be either forward, or backward of the Li^ axis, so that for the 90° measurement there are two points on the curve. The original intention of measuring this angular distribution was to compare the result with the predictions of a statistical model for the compound system. The angular distribution so predicted is also given in Figure 11; clearly the experimental cross section is considerably larger, indicating relatively simgle structure, for the compound system (10). The statistical model 0) \Z II 10 9 8 7 6 5 4 3 2 9 PROTON D E T E C T O R POSITIONS F O R 0C1 A T 90° 123° BEAM A F T 0 0 F O R W A R D 6 oci orz\ L i 5 AXIS F O R W A R D A F T 0 10 2 0 3 0 4 0 5 0 6 0 TO 8 0 9 0 100 UO 120 130 140 C E N T E R OF M A S S A N G L E Figure 11. Experimental points shown with p r e d i c t i o n of s t a t i s t i c a l model ( s o l i d curve). Detector positions are shown i n the i n s e t . Bombarding energy 1.00 MeV. _ - 37 -calculations are outlined in Appendix A. ;• In order to examine the nature of the asymmetry seen in the Li^ breakup in more detail, it was decided to leave the detector ctL fixed at 95° in the laboratory and to measure the double differential cross section for the: process at different bombarding energies, in each case comparing the yield with the proton counter on alternate sides of the Li^ recoil axis. This was to be done using the two-dimensional spectrum of the energies of the first alpha and the proton, with the triple coincidence pulse gating the dual parameter pulse height analyser. More information could be obtained in this way than by isolating the desired group with energy windows. The measurements were made at bombarding energies of 1.00, 1.25 and 1,50 MeV. The spectra obtained are similar and the asymmetry is evident in a l l three cases. Th'e two-dimensional energy spectra taken at 1.25 MeV bombarding energy are presented in Figures 12 and 13, and Figure 14 shows the positions of the detectors in these cases. The method used to analyse the spectra is discussed below. In order to show how effective the triple coincidence gate is in removing unwanted background from the spectra, short runs were done under identical conditions but with only a double coincidence gate. Thus Figures 15 and 16 show two-dimensional spectra obtained as were those given in Figures 12 and 13, but with the detector o<2 disconnected and the coincidence unit in the double mode. In Figures 15 and 16, the solid lines represent segments of the calculated kinematic contours, corrected for energy loss of the particles in the nickel" foils shielding the detectors'. The groups along the contburs A in both the -30° and -110° spectra are quite clear, but no counts are seen along rv ••'.>; '„'r.-.V.': - 3 0 ° 6 >-a: ui z: LU o o or a o o. ° 8 8 8 ° oo O !~ © : 6- »5 ©•" 16-50 : O©© OO oe £88888 ©CO .o d©o o _0®o O S C & Q o © o O Q O o © o o o o o o ©o ©OO^' O QO O OOO OO© • C O o o O O O o o o o o o 8 Q O OOO ooo oo oooooo oooo o oo o oo 150 100 5 0 C O U N T S 5 6 7 8 (XI E N E R G Y M e V 8 Figure 12. Two-dimensional energy spectrum taken with the t r i p l e coincidence gate. The proton detector was at -30o and oCl was at 95° i n the laboratory. The spectrum i s s h i f t e d toward lower proton energies by »^50 keV and lower alpha energies by /v/400 keV by f o i l s ij protecting the detectors from scattered beam. h-4 LU O o CL O 1-5 © 6- iS © 16-30 500 OCD0OO O OC o 8©ocpo 8©<Soo <D c o o •;. O O O O ' CDO®Q 8 8 8 S S 00©d>00 OCDOdOO o o © o o ° ^81880 OOOO O O O Q o©d© o o © o 8 8 § § ° 08008 OOOO O O O O X 150 100 5 0 C O O N T S 5 6 7 8 3 o(l ENERGY, MeV Figure 13. _ Two-dimensional energy spectrum taken with the triple coincidence gate. The proton detector was at -110° and c(l was at 95° i n the laboratory. The spectrum is shifted toward lower proton energies by >«50 keV and lower alpha energies by /~400 keV by foils protecting the detectors from scattered beam. F I R S T A L P H A D E T E C T O R , (co) B A C K W A R D P R O T O N L l 5 R-ECOIL. D E T E C T O R . POS IT ION D I R . E E C T I O N Figure 14. Schematic diagram of detector positions. The second alpha detector is not shown; for the proton detector at -30° and -110 this alpha detector is at -81° and- -51° respectively © ® ... » » » « » i • s 3 4- 5 G 7 8 S IO ' PART 1 C LE. E N E R ! ^ ^ ^ _ 9 5 ! ^ M e V ^ Figure 15. Two-dimensional energy spectrum taken with the proton detector at -30° and <rtl at 95°. The solid curves are segments of the computed kinematic contours corrected for particle energy loss in the foils protecting the detectors from scattered beam. A - p at -30°, 0(at 9S6;., B - p at 95° , ocat -30°; C - of'at .-SO0, <* at 95°. 3 4- 5 6 7 8 3 io ^ A R T I C L E ENERGY„.'AT - S S ^ M e V . Figure 16. Two-dimensional energy spectrum taken with the proton I detector at -110° and rtl at 95°. The solid curves are segments of | the computed kinematic contours corrected for particle energy loss \ in the f o i l s protecting the detectors from scattered beam. A - p at j -110° , « a t 95°; B - (group D) p at 95° , eC at - 110°; C - « a t - 1 1 0 ° , j OC at 95°. j - 38 -contours B, because these correspond to protons observed by the detector dl, which has a lower level discriminator to exclude proton counts. These counts appear in a group at the abscissa energy corresponding to the proton edge i f the lower level discriminator is set below this. Such is the origin of group D in Figure 16. Alpha-alpha coincidences appear along the contour C in Figure 15, but not in Figure 16 because the 0(~ OC contour does not fa l l in the observed region. Other counts in the double coincidence spectra are attributed to the background reactions Li (He ,o()Li + 13.32 MeV, C (He ,of)C" 1 Q Q A Q.C + 15.63 MeV and C (He ,oC)3He + 8.35 MeV. The latter may also count for background counts seen in the triple coincidence spectra. Since this reaction has a four-particle final state, its products are not restricted to a contour in two-dimensional energy space. The random coincidence rate was measured in both double and triple coincidence modes. This was done by introducing an arbitrary delay for the pulses from one of the detectors. In the double coincidence mode the proton pulse was delayed, giving a rate of f^ur counts in 15 minutes. For comparison, the splectra shown in Figures 15 and 16 were accumulated in 15 and 30 minutes respectively. In the triple coincidence mode, all . three counters were given an arbitrary delay, one at a time (the other two remaining in coincidence). The highest random rate was four counts in half an hour, when of2 was '.delayed. The triple coincidence spectra were accumulated in three and six hours respectively. In no case did the observed random coincidences appear in the region of interest on the spectra. Phase Space Corrections Before comparing the yields obtained with the proton detector in its positions forward and aft of the Li^ recoil direction, i t is necessary - 39 -t o make a o p r r e c t i o n owing t o the unequal volume of phase space observed by the detector i n these a l t e r n a t i v e p o s i t i o n s . The c o r r e c t i o n a r i s e s from the centre of mass motion and depends on the exact p o s i t i o n of the detector i n each case. In order t o give an absolute value t o the measured d o u b l e - d i f f e r e n t i a l cross s e c t i o n , a c a l c u l a t i o n must be.made of the p r o b a b i l i t y t h a t a t r i p l e coincidence i s observed w i t h the given detector geometry: i f an event of the type being considered takes place. This p r o b a b i l i t y depends not only upon the s o l i d ap^gle c o r r e c t i o n s needed t o transform the detector aperture t o the centre of mass frame of the system which i s breaking up t o give the observed p a r t i c l e , but a l s o upon the momentum d i s t r i b u t i o n of the emitted p a r t i c l e . In the case considered here the p r o b a b i l i t y of observing the f i r s t alpha p a r t i c l e i s p r o p o r t i o n a l t o the s o l i d angle subtended by the detector (Xl, transformed t o the centre of mass frame of the e n t i r e system. The formula f o r o b t a i n i n g t h i s i s w e l l known (23). No assumption need be made f o r the angular d i s t r i b u t i o n of the f i r s t alpha p a r t i c l e because only a double d i f f e r e n t i a l cross s e c t i o n i s being determined. In order t o determine the volume of phase space f o r the breakup of L i ^ seen by the proton counter i t i s necessary t o assume a l i n e shape f o r the L i ^ ground s t a t e t o o b t a i n the energy d i s t r i b u t i o n o f protons from i t s decay. Although i t i s known from p - C\ s c a t t e r i n g measurements (33) t h a t t h i s l i n e shape i s q u i t e asymmetric, a Breit-Wigner form was taken as an approximation t o s i m p l i f y computation. The absolute e r r o r e n t a i l e d by t h i s assumption may be as large as 25% and even l a r g e r f o r higher energy protons where overlap of the ground and f i r s t e x c i t e d s t a t e s i n c r e a s e s , but the r e l a t i v e e r r o r when the proton detector p o s i t i o n s are compared - 40 -should be q u i t e s m a l l . The c a l c u l a t i o n s were c a r r i e d out using the PDP 8 computer doing Simpson's Rule i n t e g r a t i o n over the l i n e shape. This program i s known as "PROCAL", and a d e s c r i p t i o n of the c a l c u l a t i o n involved i s given as Appendix B. 5 The Asymmetry i n the Breakup of L i No d e t a i l e d a n a l y s i s need be done t o see t h a t an asymmetry e x i s t s . The -30° and -110° spec t r a (Figures 12 and 13) were obtained by using the same t a r g e t at approximately the same beam c u r r e n t , but the count • r a t e at -30° was over twice t h a t at -110°. Such a d i f f e r e n c e was already obvious during the course of the experiment. That such a l a r g e e f f e c t cannot be geometrical i s already i n t u i t i v e l y obvious from the f a c t t h a t the bombarding energy i s so much l e s s than the energy released by the r e a c t i o n . This means t h a t t o t a l centre of mass motion e f f e c t s must be r e l a t i v e l y small. Since the geometry i s n e a r l y symmetric about the d i r e c t i o n o f motion of the L i r e c o i l , the centre of mass motion o f t h i s cannot c o n t r i b u t e much t o the observed e f f e c t e i t h e r . This reasoning i s supported by the c a l c u l a t i o n s described i n Appendix B which show t h a t the d i f f e r e n c e between -30° and -110° i s l e s s than 10% (see columns 4 and 8 of Table 6 ) . For a d e t a i l e d comparison o f the -30° and -110° s p e c t r a , histograms r e p r e s e n t i n g the proton energy d i s t r i b u t i o n have been p r o j e c t e d onto the ordin a t e axes i n Figures 12 and 13. Because the proton detector angles are not e x a c t l y symmetric about the L i ^ a x i s , the centre of the proton group corresponding to the ground s t a t e of L i ^ occurs at a d i f f e r e n t energy i n each of the two cases. These energies c o i n c i d e f a i r l y w e l l w i t h the p r e d i c t i o n s o f the kinematic phase diagram, Figure 4,.which are i n d i c a t e d - 41 -on the histograms after correction for energy loss in the nickel fo i l . It can be seen from these predictions that the -30° histogram appears shifted toward higher proton energy from the expected position. The histograms group the counts into bins of four channels each giving~10% statistics for each bin. The error bars shown are due to counting statistics Only. For a bin to bin comparison the histograms at -30° and -110° are aligned on the energy scale so that the predicted centres of the proton groups f a l l into corresponding bins. Compared bins should therefore represent protons emitted with the same median energy in the centre of mass frame of L i 5 . Kinematic calculation using SEQUEBREAK I shows this to be the case, as shown in Table 5. The median bin energies have been corrected for proton energy loss in the nickel fo i l . Table 6 shows the bin to bin comparison in detail. Columns 3 and.7 contain the number of counts in the bins; divided by the number of counts in the appropriate elastic scattering spectrum, thus normalizing the respective yields. Columns 4 and 8 contain.the probability with which an event corresponding to breakup through the L i 5 ground state is observed within the given bin according to PROCAL. Comparison of these two columns for.each bin shows the difference due to the volumes of phase space seen by the proton detector in its alternative positions. Columns 5 and 9 are thus proportional to the double differential cross section for the reaction as deduced from observing events i n the given bin. If the line shape used in the PROCAL calculation had been correct, one would expect the numbers in columns 5 and 9 to be independent of the bin chosen, within statistical fluctuations. The fact that this i s not so is in part due to the non-Lorentziah line shape for the L i 5 ground state, and in part due to the TABLE 5 Division of Proton Energy Spectra (Figures 12 and 13) into Energy Bins 30° 110° 1 2 3 4 5 6 1 2.20 0. 85 1. 52 0.85 0.85 i 2 2.67 0.98 1.98 0.96 0.97 3 3.14 1.15 2.45 1.15 1.15 i 4 3.61 1.37 2. 93 1.34 1.36 5 4.07 1.61 3.39 1.60 1.61 6 4.53 1.84 3.85 1.85 1.85 7 4. 99 2.10 4.32 2.12 2.11 8 5.47 2.38 4. 80 2.40 2.39 9 5. 93 2.66 5.27 2.74 2.70 10 6.41 2.97 5. 73 3.05 3.01 1) Identifying number for bins 2')., 4) Median proton energy for bins, including f o i l loss correction 3), 5) Proton energy converted to centre of mass of L i 5 . 6) Average of columns 3) and 5). Note: Energies in MeV. 1 - 30° - 1 1 0 ° 2 3 4 5 6 7 . 8 9 10 1 1.97 - 2.43 3.00 i 0.125 24.0 1.30 - 1.74 0. 88 0.103 8. 54 2.81 2 2.43 - 2.90 2.70 0.186 14. 5 1.74 - 2 .21 1.66 0.159 10.44 1.39 3 2.90 - 3.37 3.28 0.294 11.15 2.21 - 2.69 2.54 0.255 9.96 1.12 4 3.37 - 3 .84 4.93 0.470 10.49 2.69 - 3.16 3.31 0.408 8.11 1.29 5 3. 84 - 4.30 6.31 0.548 11. 51 3.16 - 3 . 6 2 3.39 0. 507 6.68 1.72 6 4.30 - 4.75 7. 90 0.379 20. 8 3.62 - 4 .08 2.95 0.381 7.74 2.69 7 4.75 - 5.23 6.47 0.219 29.5 3.08 - 4.56 2.90 0.216 13.4 2.20 8 5.23 - 5.70 5.94 0.116 51.2 4.56 - 5.03 1.96 0.114 17 .2 2.98 9 5.70 - 6.17 4.19 0.069 60.7 5.03 - 5.50 1.26 0.067 18. 8 3.23 10 6.17 - 6.64 4.35 0.045 96.6 5.50 - 5.96. 0. 57 0.042 13.6 7.1 1) I d e n t i f y i n g number f o r b i n s 2) ,6) P r o t o n energy l i m i t s d e f i n i n g b i n s 19 •3) ,7) (Number o f counts i n b i n ) x 103/(number o f counts i n F e l a s t i c peak) = N 4) ,8) P r o b a b i l i t y of o b s e r v i n g recorded event a c c o r d i n g t o PROCAL, x 10^ = W 5) ,9) N/W 10) Asymmetry r a t i o , column 5) d i v i d e d by column 9 ) . Note: Energies i n MeV TABLE 6 B i n t o B i n Y i e l d Comparison - 42 -nature of the observed asymmetry. The comparison of columns 5 and 9 gives the ratio, column 10, which represents the asymmetry in the breakup of L i 5 . This asymmetry is plotted as a function of the proton energy in the centre of mass of Li for each bin in Figure 17. The error bars shown are relative and only represent counting statistics. The'important feature of this curve is the increase in the asymmetry as a function of proton energy. This wil l be discussed further in the next chapter. The mean asymmetry can be taken as the average over those bins representing the central region of the proton group. Averaging over bins 2 to 8 inclusive gives an asymmetry ratio of 1.9 and i f bin 9 is included the ratio is 2.2. . Analysis of the runs at 1.00 and 1.50 bombarding energy was done in a similar way. The average asymmetry ratios were 2.3 ± 0.6 and 2.1 - 0.6 respectively. I The Total Cross Section In order to obtain a .crude estimate of the absolute cross section, the numbers in columns 5 and 9 of Table 6 must be averaged and multiplied by the geometric factors arising from the solid angles of the elastic scattering detector and the detector o(l. A small correction is 7 also needed to correct for the amount, of L i in the target. The main source of error in the, figure thus obtained for the double differential cross section arises from the wrong line shape used in PR0CAL. This could be improved i f necessary, but since only the relative yield is of interest K here, this was not done. Further errors are introduced when the double differential cross section is integrated to give the total cross section. The result, however, should be well within a factor of 4 of being correct. To get i t , we take the average value of the double differential cross 0 V=-50,W=-5, R=&-5, A - 0-3, H= O - l 0 S A M E A S 1 , 0 - 0 6 @ S A M E A S 1 , ^=-0-0G£5 E ( M e v ) * 0-2A-(§) SAME A S 4 - , p ~ O-J -6 P R O T O N E N E R G Y , M e V , IN UB C M F R A M E Figure 17. The asymmetry as a function of proton energy in the centre ot mass frame of L i 5 . The experimental points are shown with curves calculated according to Appendix C. V and W are in MeV, R and A in fm and pi is in 1C-22 Sec-1. - 43 -section between the -30° and -110° measurements and take that as constant for integration over the solid angle subtended by the proton detector. The resulting differential cross section is then taken to be constant for integration over the solid angle subtended by 0(1. - 44 -Chapter VI DISCUSSION There are three outstanding features characterising the experimental results which must be fitted by any viable explanation of the reaction mechanism. These are the large magnitude of the cross section, the pronounced asymmetry about the recoil direction of the intermediate state, and the increase of this asymmetry as a function of the proton energy. The Total Cross Section 3 6 If we assume a radius of 1.5 fm for He and 2.7 fm for L i , • the geometric cross section for their interaction is 550 mb, which is of the same order as the measured total cross section for the reaction channel we are considering. As indicated in the previous chapter, a statistical model calculation was done (Appendix A) to obtain both the averaged total and the differential cross sections for the same process. These calculations were done for a bombarding energy of 1.00 rather than 1.25 MeV, but the result for the total cross section, even allowing a factor of 2 increase for the higher energy, is s t i l l an order of magnitude smaller than the observed value. The large total cross section implies that a simple, peripheral process is taking'place. This means that there is no thorough mixing of target and projectile nucleons into a compound system, but rather that nuclear clusters are involved which retain their identities and interact without mutual penetration. This aspect of the reaction mechanism is in f u l l accordance with Phillips' cluster-compound model (8) introduced in - 45 -Chapter I. The supposition that the formation of a compound nucleus does not contribute s i g n i f i c a n t l y to the process i s supported by the lack of strong resonant structure i n the e x c i t a t i o n function, Figure 9. The Structure-Dependent Asymmetry In order to account for the observed asymmetry, we assume that the sequential process taking place i s of the direct-delayed type. The ground state of L i ^ comprises two neutrons and two protons i n the 3 s - s h e l l with one neutron and one proton i n the p - s h e l l , while He has a neutron vacancy which must be f i l l e d to complete i t s s - s h e l l and convert i t to an alpha p a r t i c l e . The d i r e c t part of the reaction i s taken to be 6 3 the adiabatic t r a n s f e r of the p - s h e l l neutron from L i to He . . This part of the reaction releases 15.104'MeV, accounting f o r most of the excess energy i n the final, state. The highly exoergic nature of the neutron tr a n s f e r j u s i f i e s the assumption, that i t takes place very r a p i d l y , i n a -22 time commonly associated with d i r e c t reactions (10 sec. or l e s s ) , The neutron t r a n s f e r i s followed by the delayed part of the t r a n s i t i o n to the f i n a l state, which i s the decay of the five-nucleon system, L i 5 . This process i s described as delayed, because the states of L i 5 have a measurable width (33) associated with a conjugate l i f e t i m e . 5 ' -22 For the ground state of L i t h i s l i f e t i m e i s —'7 x 10 sec. so that i t l a s t s an order of magnitude longer than the duration of the d i r e c t part of the reaction. This l i f e t i m e , however, i s s u f f i c i e n t l y short i n r e l a t i o n to the internal, motion, of the nucleons c o n s t i t u t i n g the state, that memory of i t s structure at the time of i t s formation might p e r s i s t to influence the decay. - 46 -In order to describe this direct-delayed reaction we propose a simple, geometric model for the neutron transfer process, which is illustrated in the centre of mass frame by Figure 18. We adopt the conclusion reached from the large total cross section, that the reaction is mainly a peripheral process involving no penetration of the projectile into the target core. The latter is essentially an alpha particle and participates in the reaction in cluster form. Transfer of the p-shell 6 3 neutron from Li to He occurs i f , at the time of closest approach, the neutron is located on the side of the Li^ core facing the projectile along "the transfer axis. The rapidity of the.neutron transfer compared to the 6 motion of the L i p-shell proton allows us to consider the proton as stationary or localized during the primary reaction. A simple analogy is the "stopping" action of a high-speed camera photographing a moving object. Since l i t t l e is known about the correlation between the motions of the p—shelVnucleons in L i ^ , we must consider i t as arbitrary. Accordingly, to determine what happens to the proton in the primary reaction, we assign equal probability td each of two possibilities defined in terms of the angle between the proton position vector and the direction of neutron transfer at 6 the Li centre of mass. If this angle exceeds TC/2 the proton is localized in a coordinate hemisphere shown,i as'the shaded part of A in Figure 18. If the angle is less than-Tt/2, the proton is localized on1 the side of the L i 5 core facing the alpha particle produced by the interaction of the neutron with 3 the He . This 'loosely-bound proton is thus- between two-alpha particles at the time when the large momentum transfer associated with the primary reaction is taking place. We therefore assume equal likelihood of the proton being finally attached to either fragment. The two possibilities in the case of D I R E C T I O N OF M O T I O N C F A L P H A PART ICLE -OF MOTION OF U*5 Figure 18. Schematic diagram of primary reaction in the centre of mass frame showing neutron pickup (solid curves) and two-particle pickup (dashed curves) for fixed L i 5 recoil direction. - 47 -the proton having been localized on the side of the L i core facing the direction of neutron transfer are shown as B and C of Figure 18. Based upon the foregoing assumptions the cases B and C each represent 25% of the events involving the formation of L i 5 as an intermediate state. Diagram B illustrates the case when both p-shell particles are transferred from the target to the projectile, so that the target core becomes the alpha particle observed by the detector #1. For this case the trajectories describing the collision are represented by the dashed curves in Figure 18. The different trajectories apply because the diagram represents a fixed direction of the L i 5 recoil, set by the position of CXI. If the proton is transferred along the axis of two-particle transfer shown in the diagram, then it is init ial ly localized in the coordinate hemisphere facing the direction from which it came, the shaded region of B. If we assume that the proton remains localized within the region in which it is located at the moment of neutron transfer until the L i 5 system breaks up, then the small arrows on A, B and C of Figure 18 indicate the most probable direction for proton emission. This is true in spite of the fact that the breakup must involve one unit of angular momentum, because the proton does not penetrate the alpha particle. On this assumption the only contribution made toward .proton emission in a direction aft of the L i 5 recoil axis can be made by the case illustrated by C, which represents 25% of the events being considered, and\ of ::these perhaps only half come out aft of the recoil axis.. If we note that the high velocity of the L i 5 recoil reduces the centre of mass angle by a factor close to 0.5 to give the laboratory angle between the proton emission axis and the L i 5 recoil axis, then this simple nipdel could result in the forward proton detector position being favoured over the backward one by a factor.as high as •7.' - 48 -This result is an extreme, based on a very simple, classical picture, but i t serves as a guide as to the type of mechanism which might be responsible for an asymmetry of the type observed. So far we have assumed that the proton localization is remembered until the L i 5 state breaks UP- The extent to which such memory can be expected to persist must be investigated further. This depends primarily upon the velocity of thfe proton within the L i 5 system. The proton1is in fact unbound by 1.967 MeV, but, i f we represent the alpha core by a potential well, the proton wave function has a definite amplitude inside the, well and must, leak but before the state can be said to have decayed. The proton kinetic energy is therefore mostly determined by its value outside the well, at least to a factor of 2 or 3. If we assume the proton to have a kinetic energy of I 2 MeV (the value in the centre of mass of L i 5 after breakup is 1.57 MeV) then we can determine the semi-classical value for the radius by requiring r x p = + l)'fv. to give the required angular momentum. This gives a value 4,6 fm for the -22 p-shell proton orbital radius, and in 7 x 10 sec at that energy the proton would only be able to move half way around the corie. This value of the radius may be quite a bit too big. Certainly it is nearly a factor of 2 too large for the p-shell proton radius in L i ^ , which is a tightly bound system. When L i 5 is formed by removal of the neutron, the $rpton must s t i l l \ 6 be at the radius it has in L i , and its transition to any larger distance may be regarded as a, part o'f/jthe L i 5 decay. Thlp smaller radius does not necessarily imply that the'proton' moves around the core any faster. At a small radius the influence of the slow moving proton upon the core nucleons - 49 -may well be such that surface waves or liquid drop type distortions result in an increase in the effective mass of the proton and slow it down. If we represent the proton by a wave packet: i t would "spread" due to the uncertainty in momentum resulting from whatever localization we impose upon it . Since we only have required it to be localized in a hemisphere to which we assign a radius of fm then the uncertainty in momentum is A P , -•Tcr -22 and in 7 x 10 sec.it spreads by an amount 7 x I O " 2 2 TT. , , . 0* 5 fm. rc m r which is quite small and can be neglected in comparison to the orbital motion. Because of its crudeness, oxir model does not deserve any more detailed calculations. Those above are sufficient to show that i t is reasonable to suppose that the proton localization does persist with -22 significant probability until the state decays. Since 7 x 10 sec. is approximately the time taken for the p-wave proton to go half way around the system, we can take this as an approximation to the half-life for the persistence of the localization. The half-lives of this and the L i 5 state itself are thus comparable, and the observed asymmetry should be reduced from the theoretical one accordingly. This is compatible with our observed asymmetry ratio of 2.0. Variation of the Asymmetry with Proton Energy If the foregoing explanation of the origin of the asymmetry is basically correct, then the degre*e of asymmetry observed should be a - 50 -function of how long the L i state l a s t s . Because of the large natural width of the state i t i s a c t u a l l y possible to observe t h i s e f f e c t . As can be seen from the histograms, Figures 12 and 13, the energy spread of the protons from the breakup of the L i 5 ground state i s ^  2 MeV. Converting to the centre of mass frame of the L i 5 we,\et a FWHM of r^> 1 MeV which i s the width of the state. We therefore observe protons which d i f f e r i n t h e i r centre of mass energy by a large percentage. Since the b a r r i e r p e n e t r a b i l i t y increases' r a p i d l y with energy (except f o r energies f a r above the b a r r i e r , where i t s increase i s proportional to v e l o c i t y ) , the more energetic protons tend to escape from the L i 5 system f i r s t and with a higher p r o b a b i l i t y of remembering t h e i r i n i t i a l l o c a l i z a t i o n . A quantitative c a l c u l a t i o n of the observed asymmetry as a function of the proton energy i n the L i 5 centre of mass system i s given i n Appendix C,, The parameters a f f e c t i n g the mumerical r e s u l t s of the c a l c u l a t i o n are the o p t i c a l model p e n e t r a b i l i t i e s ( calculated from the computer program ABACUS-2 ( 3 4 ) ) , the decay constant f o r the decay of the proton l o c a l i z a t i o n i n L i ^ , and the maximum asymmetry (assuming no decay of the l o c a l i z a t i o n ) . In Figure 17 the experimental asymmetry r a t i o i s shown as a function of centre of mass proton energy using the bin. to b i n comparison described i n the previous chapter. Values calculated i n Appendix C for two reasonable sets of o p t i c a l model parameters and f o r various choices of JLA , the l o c a l i z a t i o n decay constant, are shown as the curves 1.-5 i n the f i g u r e . Curves 1, 2, 4 and 5 show an increase of the asymmetry with proton energy, but not by as much as the experimental points would ind i c a t e . • Increasing the slope of these curves would require a greater v a r i a t i o n of the p e n e t r a b i l i t i e s - 51 -with proton energy, or a larger value for the maximum asymmetry. The parametersrequired to f i t the experimental points in this way are not realistic. Instead it is more reasonable to assume that there is a variation of the decay constant with proton energy. When more energy is available for its breakup, a particular L i 5 system is not as tightly bound. The amplitude of the proton wave function would therefore represent a greater probability density for the proton to be found near the diffuse edge of the well than otherwise. The proton thus has less velocity inside the well, and a smaller probability of becoming unlocalized in a given time. We might therefore expect the decay constant p. to decrease with increasing proton energy. To show the effect of this we vary |x as a linear function of the energy in curve 3. This clearly produces a better f i t to the data. In .i • order to arrive at a more meaningful f i t from which an estimate of A0and |L>I might be obtained, more experimental points are desirable, over a greater range of proton energy. It is hoped that this can be done by observing the - 5 decay of the \~ f irst excited state of L i from the same reaction. The observed sharp increase in the asymmetry at low proton energies we attribute to the contribution to the experimental spectra of 8 sequential breakup involving the 16.62 and 16.92 MeV states in Be . For these events we expect the proton to be emitted predominantly in the forward direction. The forward peaking of the protons emitted as f irst particles 8 in the sequential breakup involving Be levels as intermediate states follows from a similar model to the orie used to describe the sequence through L i 5 . Again the reaction only proceeds i f the p-shell neutron is on the side of the target facing the projectile at its closest approach. If"the proton is on - 52 -the opposite side it is free to escape directly as the neutron is absorbed 3 6 by the He , which then interacts with the L i core to resemble the two-8 alpha system Be . This gives a probable distribution for emitted protons peaked forward and symmetric about the beam direction. If the proton is on the same side of the Li^ as the neutron, i t is caught between the two alpha particles and can only be emitted in a sideways direction. Future Work A suggested program for further studying the type of structure dependent asymmetry observed in this experiment includes a detailed study 8 of the sequential breakup involving states of Be . The formation of the 11.4 MeV 4 state might be difficult to distinguish because it is very wide (^4 MeV), The existing kinematic phase diagrams could be used to 8 select detector positions which favour the higher states in Be and minimize interference from the L i 5 states. Of primary interest to corroborate our explanation of the 5 asymmetry observed in the breakup of the L i ground state, is an experiment to observe the breakup of the first excited state of this system. This could be done with a similar technique, but requires a proton counter of greater depletion depth so that more energetic protons.can be observed. Since this state is believed to be considerably wider than the ground state, and hence shorter lived,the asymmetry should be greater. It should be possible to continue the curve of asymmetry vs. centre of mass proton energy considerably further, and thus to obtain a better indication of the half-life of the localization, as well as an indication of the maximum value of the asymmetry. If this can be established, and Used in conjunction with a more - 53 -rigorous model than the simple one discussed above, it may also become possible to deduce something about the correlation between the p-shell particles in the Li^ ground state. Appendix A STATISTICAL MODEL CALCULATION FOR L i 6 ( H e 3 , o Q L i 5 6 3 5 The reaction L i (He ,o()Li i s the f i r s t step of a sequential breakup leading to the three-body f i n a l state, p,0\,tX . In t h i s step an energy of 15.104 MeV i s released, which appears as k i n e t i c energy shared between the L i 5 system and the alpha p a r t i c l e . This energy i s s u f f i c i e n t that a separation between these p a r t i c l e s of the order of 25 fm i s achieved —22 •* 5 i n a time 7 x 10 sec, the approximate l i f e t i m e of the L i ground state. This emphasizes the sequential nature of the process, and suggests that the f i r s t step of the reaction may be treated independently. The compound-nuclear model of reactions postulates as independent processes the formation of the compound nucleus and i t s decay into the available e x i t channels on a s t a t i s t i c a l b a s i s . For t h i s model the average t o t a l cross section f or a reaction i s given by (35) TC k 0 (EI+l)(2L + l) ur / { l c - T t - ( c " ) J CJTC \ S I t where 0( stands f o r the entrance channel, with i the i n t r i n s i c spin of the bombarding p a r t i c l e , I that of the target, L the o r b i t a l angular momentum, S the channel spin and J,TC the spin and par i t y of compound system. The primed quantities r e f e r to the e x i t channel, c refer s to the t o t a l i t y of a l l the channel quantum numbers, and the double prime indicates summation over a l l open e x i t channels. The.T L are o p t i c a l model transmission functions of the correct energy f o r the channel being considered, and k^ i s the wave number of the incident channel. A l l quantities are calculated i n the centre of mass frame. - 55 -3 6 This cross section was calculated for He incident on L i with L i 5 + oc in the exit channel, for a bombarding energy of 1.00 MeV. The appropriate optical model transmission functions were calculated using the computer program ABACUS-2 (34). The calculated transmission functions and the parameters used in their calculation are given in Tables A l , A2 and A3. The open exit channels are those given in Figure 1 but including the entrance channel instead of the direct three-body breakup. Sample Calculation and Results l) For 1.00 MeV bombarding energy the centre of mass wave number for He (0.6667 MeV in centre of mass) is 19 i = J2mE = 2.546 x 10 cm 'Vkl = 502.66 mb 6 + 3 + 2) For the ground state of Lij Itt = 1 , and for He ,liT = \ . Therefore the summation over JTt for the compound nucleus, considering only L = 0, 1, 2, 3, includes 9 possibilities: Jrr = \+, \~, |'+, I " | + , f- ST Each must be weighted by the number of ways in which it can be formed in the channel in question. Consider the case for He + L i , JTC = \ • Remembering that a l l the spins are vectors we have s = (I + i) = \ + or \f J = (L + s) = 1 / . L - 0,2 j T r = h+ = o + 2 - | - 5 6 -0.07171 + 0.00223 = 0.07394 For p + Be , JTC - \ , 2.9 MeV 2 state 8 = (I + i) = (2 + \) = t or £ JTC=( (. + s) = \ + . ' . 1=2 only ^ T?. = 0.7057 + 0.7057 = 1.4114 s ' L ' 3) These calculations must be done for each value of Jif. The results are combined according to the formula for the average total cross section. The results are given in Table A4. The Angular Distribution The differential cross section for the statistical process is given by (37) ^ _ 4 k 2 / '(ZZ + l)(Zi 4-1 ) S T X T i ' ( o f / ) . ( Z(L7,L,T; SL) X ( S L) (-)S"S P L (cos 9) where symbols have the meanings given above, with L = [_+ L , L = L x = t 0^L^=2L or 21.', whichever is smaller. - 57 -The Z coefficients are products of Clebsch-Gordan and Racah coefficients which have been computed and tabulated.; (see for instance (36)). The calculation is straight forward but laborious. Using the penetrabilities in Tables A l , A2 and A3 and the appropriate Z coefficients we obtain the differential cross section j^**'= 40 To.0575 + 0.0033 P (cos9) + 0.0003 P (cos9)] mb/sr This is plotted in Figure 11. Discussion The tables of penetrabilities show that the reaction is predominantly s-wave. This is reflected in the angular distribution, which is almost isotropic. That the major contribution should be s-wave is expected, because the bombarding energy is well below the'.'Gbulomb barrier ( — 2 MeV), A larger total cross section could be obtained from this model by increasing the radius of the system and hence lowering the barrier height. The radius of the L i 5 ground state could well be larger than that assumed in this calculation, but that would only increase the branching ratio, and hence would not give more than a factor of two. Increasing the radius in the entrance channel by 30% would increase the total cross section by a factor of 3, but this would require R.= 1.6 (A, / 3 + Az^s) which is certainly/too large. Changes in the optical model parameters only;'produce minor changes in the transmission functions except where giant resonance effects occur. These might be evident 8 in system p + Be at higher excitation, thus influencing the cross section for 8 the channels through the 16.62 and 16.92 MeV levels in Be . When a compound-nuclear process such as the one to which this calculation applies takes place in competition with a direct reaction, the - 58 -cross section is expected to be less than the theoretical one by a considerable amount. This follows because the direct reaction, whose cross section is nearly the geometric one, removes a significant portion of the beam which might otherwise enter the target nucleus to produce a compound nucleus. The above calculation is therefore not likely to give a true representation of the statistical process contribution to thevobserved reaction, although the branching ratio it predicts should hold for that part which does go through a compound nucleus. Table A l 3 6 Transmission Functions f o r He + L i Energy i n centre of mass = 0.6667 MeV R = 1.25 (A* + A ^ 3 ) .=. 4,07 fm Vo = -50 MeV W = -10 MeV a = 0.5 fm L 0 1 2 3 4 Ti 0.07171 0.03566 0.00223 0,00015 0.000002 Table A2 4 5 Transmission Functions f o r He + L i Energy i n centre of mass = 15.771 (|"), 10.771 d") R = 1.25 ( A ^ + A2'/3 ) = 4.12 fm Vo = -50 MeV W = -10 MeV a = 0. 5 f m I T L(!') • T L ( i - ) 0 0,9770 0.9904 1 0,9675 . 0.9560 2 0.9729 0.9907 3 0,9696 0.9366 4 0.9413 0.9469 5 0.9943 0,7922 6 0.7434 0.3571 7 0.6343 0.0660 8 0.0793 0.0059 Table A3 Transmission Functions for Be + p 8 Energies in centre of mass for (MeV) state in Be , MeV (0) : 17.64 (2.9) 1 14.-74 (11.4) 6.24 (16.62) 1.02 ; , (16.92) 0.72 R .= 1.25 A^ = 2.50 fm Vo = -55 MeV W = -5 MeV a - 0.5 fm L TL(P) TL(2.9) TL(11.4) Tt(16.62) \ (16.92) 0 0.5555 0.5810 0.6586 0.2352 0.1296 l 0.4108 0.3968 0.2974 0.0369 0.0168 2 0.6090 0.7057 0.3167 0.0006 0.0002 3 0.1382 0.0795: 0.0043 4 0.0121 0.0064 0.0002 5 0.0016 0.0008 6 0.0002 0.0001 Table A4 Average Total Cross Sections at a Bombarding Energy of 1.00 MeV Exit Channel Cross'section, mb Branching ratio percent He3 + L i 6 1.6 1.7 L i 5 ( r ) + 29.1 30. 7 i i 5 ( r ) + 24.4 25.7 Be8(0) + p 9,9 10.4 Be8(2.9) + p 24.2 25.5 Be8(11.4) + p 2.2 2.3 Be8(l6.62) + p 2.2 2.3 Be8(l6.92) + p 1.2 1.3 Total 94. 8 99.9 - 59 -Appendix B SOLID ANGLE CALCULATION Particles confined to a spacial volume dxdydz have an uncertainty in their momentum dp dp dp such that r x r y r z 3 dp^dpydp^xdydz <^ t i Let us keep the particle i n a well defined physical Volume U. Then we can write ft3 dp dp dp > rs- = ft'3  r x ry r z U We take the best possible case and replace the inequality with the equality. We then consider a momentum space comprising d i s c r e t e points, each occupying a cel l of vplume I i ' 3 . The density of points in this space is constant at 1 / n ' 3 and is invariant under transformations of the type p / = p + q. Thus, i f we consider a system of mass (m + M) moving in the laboratory with velocity V, the density of points in momentum space available to one of its constituents, say m, when i t breaks up is the same in the CM frame as it is in the laboratory. Let us consider now the breakup of (m + M) into m and M. Primed quantities refer to the CM frame of reference, unprimed ones to the laboratory frame. The velocity vector diagram is given in Figure BI where v and v' are velocities for the mass m. V" Figure BI C M motion - 60 -Consider an element of volume in momentum space in the laboratory, using cylindrical coordinates dH_ = p2sin0ded$dp. This contains d-fl/ ,^3 points in the momentum space, each of which corresponds uniquely to /a point in the momentum space transformed to the CM frame by p '= p - q where p' = muf; ,p = rrtU ahd q.= mV. We now calculate the probability of this point being occupied as a result of the breakup. Let the energy distribution of the particle m in the centre of mass system correspond to a Breit-Wigrier shape, so that the probability that the energy of the particle is E' within dE' is given by r / / 2 r r f(E')dE' - " (E-Eo)' which is normalized so that f(E')dE' Converting this to a function of momentum using the relations E' = p'^/2m .2 and dET = p T dp T / m w e get f(p')dp ( 2mr,p t ^ ( P ' 2 - P 0 2 ) 2 + m 2 r ? 2 r where f(p')dp' = 1 1 -co - 6 1 -In general the energy of m is MQ where Q is the total energy available m + M M . j — for the breakup. If the width of the state is T, then we have f = m '+ M this being the FWHM for the energy distribution of m in the centre of mass. We then have the probability that m occupies a point in momentum space within 2 a spherical shell "ft' thick at p' as f(pT)'h ,. This shell has a volume 4np' ~ftT 4TTD '2 and hence contains — * - T J — points. The probability that each point is occupied, upon the assumption of an isotropic angular distribution, is then given by 4 T p ' We now return to our volume element d-Q.in the laboratory frame, which 3 contained d-fl/h' points. The probability that dJTlwiir . b e occupied is therefore or d l ( p ' 9 f p ) = \ £ - f(p')sin©d9d<t>dp Tfpr where 6 and <t> are the usual angular coordinates in the cylindrical system with* polar axis in the beam direction, in the laboratory frame. We must now evaluate this for a finite solid angle and momentum window in the laboratory. Since both p and dp are functions of p' and 6 the only integration which can be done is over in the range AcJ), to give: f -- 62 -writing this in fu l l 2 T T To evaluate this double integral we shall divide ^9 and Ap into m and n intervals respectively and integrate using Simpson's rule. For the breakup of Li 5m = 1, M = 4 . p = O.SfJaiid 2 o-4-r Act d l A / = w-r A computer program "PROCAL" was written to carry out this integration. The limits on 8 and the angle fcfyare derived from the size and position in the laboratory of the rectangular collimator defining the aperture of the proton detector. The momenta p and p' are calculated from the kinematics and the known Q-value. For the calculations used here the width Twas taken to be 1.0 MeV. The pertinent results are given in columns 4 and 8 of Table 6. Appendix C DEPENDENCE OF THE ASYMMETRY ON PROTON ENERGY The short l i f e of the L i ground state implies through the uncertainty p r i n c i p l e that there i s a considerable spread, i n the centre of mass energy released i n i t s decay. The probable l i f e t i m e of a p a r t i c u l a r 5 *\ L i system can be given i n terms of a decay constant,/^, i n the usual .manner fo r radioactive decay, but because the energy released varies between i n d i v i d u a l L i ^ systems within the natural width of the state, we expect J\ to be a function of t h i s energy. • Such an energy dependence of the decay constant follows from the energy v a r i a t i o n of the b a r r i e r p e n e t r a b i l i t y The width f o r the decay of an i n d i v i d u a l L i 5 i s given by 2 . where ft i s the reduced width, P(E) the p e n e t r a b i l i t y calculated f o r the p -o( system using o p t i c a l model parameters and E i s the proton energy i n the centre of mass frame. The width obtained from the proton energy spectra i s ~ - l MeV, which may be taken as P(E) where E has i t s mean value f o r the L i ^ ground state, 1.574 MeV. P e n e t r a b i l i t i e s have been calculated f o r various o p t i c a l model parameters using the computer program ABACUS-2 (34) (see Table C l ) , so that y can-, be obtained using P (1.574 MeV). The values f o r Jj"^  are well within an order of magnitude of the Wigner l i m i t as i s expected from the sing l e p a r t i c l e model of L i ^ . The decay constant 'X(E) i s related to F(E) through the uncertainty p r i n c i p l e by - 64 -so that X can be obtained as a function of the proton energy in the centre of mass system of the L i 5 . Let us consider a given individual L i 5 system, formed at some time t = 0. The probability that it s t i l l exists after a time t is w(t)=. e If we assume that the p-shell proton is localized when the system is formed, and that the localization has a probability of vanishing governed by a decay constant u, then the probability that the original system s t i l l exists and that the proton is s t i l l localized after a time t is 4 , , N ~(XLE) + u ) t The probability that it wil l decay into an alpha and a proton in an interval dt at time t is d w = X ( E ) w ( t ) d t and the probability that such a decay takes place while the proton is s t i l l localized is <W = A(e)w/tt)at hence the total probability that the system wil l decay while the proton is s t i l l localized is J .00 to f f (A ( E )+ p ) t dw / ( t ) = A ( E ) e hence the probability that it decays-after the proton localization is lost is - 65 -If we suppose that the decay of systems with their protons localized gives an experimental asymmetry A , then the actual asymmetry observed will be A ( E ) = ( J ^ T ) Ate) + j H-The quantity A ( E ) has been calculated for various values of A q and p. for two sets of penetrabilities. The results are given in Figure 17 with the experimental points. Table Cl P-wave Proton-Alpha Penetrabilities as a Function of the Centre of Mass  Proton Energy for Various Optical Model Parameters Optical Model Parameters Penetrabilities Vo(MeV) W(MeV) R (fm) 0 A(fm) E(MeV) 0.64 1.28 1.92 2.56 3.20 -55 -5 2. 5 0. 5 4- 0.058 0.140 0.199 0.240 0.269 -55 -5 2.7 0. 5 Ti 0.036 0.097 0.150 0.193 0.227 -55 -5 2.7 0.4 T i 0.029 0.081 0.126 0.163 0.192 -55 -5 2.7 ' 0.3 T, 0.024 0.066 0.105 0.137 0.162 -50 -5 2. 7 0. 5 T l 0.055 0.135 0.196 0.240 0.271 -50 -5 2.5 0.4 Ti - 0.106 0.222 0.288 0.325 0. 346 -50 -5 2. 5 0.3 T * 0.091 0.210 0.265 0. 309. 0. 325 Note: Vo is the real well depth, W is the imaginary well depth, RGis the mean well radius and A is the diffuseness, where the optical model potential used is of the Saxon Woods-type, 1 + exp (5^) * used for curves 1, 2 and 3 Figure 18 ** used for curves 4 and 5 Figure 18 - 66 -APPENDIX D  REVERSE CONTRIBUTION CALCULATION  Phase Space Considerations Appendix B describes the phase space c a l c u l a t i o n which was ca r r i e d out i n order to determine the p r o b a b i l i t y of observing an event corresponding to the L i 5 ground state with the proton i n a given energy i n t e r v a l , with the experimental'1, detector geometry, The r e s u l t s of t h i s c a l c u l a t i o n may be plotted as histograms of p r o b a b i l i t y vs proton energy f o r the two experimental configurations. These are shown i n Figure Dl as A. The pertinent d e t a i l s are given i n Table 6. In t h i s case the c a l c u l a t i o n only involves the s o l i d angles of the detectors 0\1 and P because 0C2 was chosen to have a s u f f i c i e n t l y large angular aperture that any event corresponding to the L i 5 ground state within a width of 5 MeV giving o(l-P coincidences would r e s u l t i n c^2 detecting the second alpha p a r t i c l e . A completely s i m i l a r phase space c a l c u l a t i o n was c a r r i e d out for the p r o b a b i l i t y of observing the proton i n a given energy region as a r e s u l t of a t r a n s i t i o n through the f i r s t excited state of L i 5 , but with the f i r s t alpha p a r t i c l e observed by detector 0(2 and the second i n OQ_ 5 the "reverse process". This c a l c u l a t i o n i s somewhat more complex i n that the f i r s t alpha p a r t i c l e enters the detector with the 1 8 ° angular aperture. The e f f e c t i v e area of t h i s detector i s , however, severely l i m i t e d by the t r i p l e coincidence requirement. Because the other two detectors are considerably smaller and are, of course, i n the reaction plane, the e f f e c t i v e area of 0(2 i s confined to a narrow s t r i p centered on i t s i n t e r s e c t i o n with v t h i s plane. The width of t h i s s t r i p depends upon the angle at which the f i r s t alpha emerges in the reaction plane (within the allowed 1 8 ° ) and upon the e x c i t a t i o n i n the L i 5 system. The angle subtended by the e f f e c t i v e area 0 - 6 r-> 0-4-h < OQ o o-a 0-o - 1 i 0 i r 30< 1__ 2 3 4 e ~ ' 7 6 ' 9 tO PROTON E N E R G Y BIN 0 - 6 „ 2 0 - 5 (0 0 3 < g o - * CL e to" L £ 3 4 5 6 7 P R . O T O N E N E R G Y B I N 8 IO Figure Dl Probability histograms"(probability of observing a given event vs proton energy identified by bin numbers as in Table 6. A, L i 5 ground state; B, reverse process, L i 5 ; C, B x 10. TABLE Dl P r o b a b i l i t y of Detecting Events - 30 0 - 11 0° 1 2 3 4 5 1 .129 - .106 -2 .192 .0137 .164 .0232 3 .303 . 0148 .263 .0304 4 .484 .0169 i ,420 . 0343 5 .564 .0184 .522 . 0362 6 .390 .0198 .392 . 035:3 7 .226 .0198 i-222 . 03l3 8 .119 .0190 .117 .0265 9 1 .071 .0166 .069 .0207 10 U .046 .0124 .043 .0137 Notes 1) I d e n t i f y i n g number f o r proton energy bins (see Table 6). 2) , 4) P r o b a b i l i t y x 10^ f o r observing proton from L i 5 ground state (PROCAL). These figures d i f f e r from those i n columns 4 and 8 of Table 6 because the s o l i d angle f a c t o r f o r o(l i s included here. 6 5* 3) ,;j5) P r o b a b i l i t y x 10 f o r observing proton from L i (reverse process). - 67 -of 0(2 i n a plane normal to the react ion plane i s t y p i c a l l y less than 3 ° , and i s calculated by the computer during the phase space integrat ion for each i t e r a t i o n . An a d d i t i o n a l complication ar ises because events g iv ing coincidences i n detectors P and o(2 do not necessari ly r e s u l t in the second alpha p a r t i c l e being detected by the detector o&. This i s because oil i s a smaller detector than o(2 and only represents a small volume i n the phase space avai lable to the second alpha p a r t i c l e . During each i t e r a t i o n the computer therefore checks i f the event w i l l give a t r i p l e coincidence before inc luding i t i n the integrated p r o b a b i l i t y . The re ject ion of many events i n t h i s manner leads to a considerable reduction i n the p r o b a b i l i t y of observing the reverse process. The p r o b a b i l i t y of observing contributions from the reverse process i s also great ly reduced by the large width of the L i 5 excited state. The computations were done with a Breit-Wigner shape using a FWHM of 5 MeV and an e x c i t a t i o n of 7 MeV i n the L i 5 system. The actual ca lcu lat ions were done on the U n i v e r s i t y of B r i t i s h Columbia IBM 7040, using 100 i n t e r v a l s to integrate between 3 and 11 MeV e x c i t a t i o n i n L i 5 and d i v i d i n g the angular apertures of (X2 and P into 10 and 4 in terva ls respect ive ly , for a t o t a l of 4000 i t e r a t i o n s . The resul ts are given together with those from PROCAL i n Table D l . Two conclusions can be drawn from inspection of Table D l . F i r s t l y , phase space factors alone indicate that the contr ibut ion from the reverseiprocess should be i n the range 5% to 10% of the t r i p l e coincidence y i e l d in the proton energy spectra . . Secondly, the contr ibut ion for the proton detectors at 110° i s greater than for the proton detector at 30° by a factor of ~1.8. - 68 -Cross Section The ground (3/2 ) and f i r s t excited {\ ) states of L i 5 d i f f e r i n spin. In the primary reaction, the angular momentum of the L i 5 system i s determined by the spin of the neutron which i s transferred from the'.'iLi 3 target to the He . Since a large amount of energy i s av a i l a b l e f o r t h i s t r a n s f e r , the r e s i d u a l spin o r b i t i n t e r a c t i o n i n the L i 5 should not exert a s i g n i f i c a n t influence. I t i s therefore reasonable to assume that the cross sections f o r formation of the ground and f i r s t excited states i n L i 5 by the primary reaction are approximately equal except f o r the s t a t i s t i c a l : fa c t o r , which enhances the t r a n s i t i o n v i a the ground state i n the r a t i o 2:1. Most direct: reaction theories which could be used to describe the primary reaction would predict a fore and a f t peaking f o r the neutron pickup process. This i s consistent with the observations at 1 MeV bombarding energy shown i n Fig.11. I f there i s such a fore and a f t peaking i n the angular d i s t r i b u t i o n of the f i r s t alpha p a r t i c l e , the contribution from the reverse process would be enhanced at the 110° proton detector p o s i t i o n where the f i r s t alpha comes o f f at 51° i n the lab as compared to the 30° p o s i t i o n where the f i r s t alpha comes at 81°. Such a difference might amount to 10-15% according to Figure 11. Interference A more serious contribution from the reverse process might a r i s e i f interference occurs and gives a background which i s not purely ad d i t i v e . Such interference is r-discussed by P h i l l i p s (8). In the present case interference might a r i s e i n the manner i l l u s t r a t e d i n Figure D2, which shows the geometry f o r the proton detector at 110° i n the laboratory. Let us assume that the ground and f i r s t excited states of L i 5 are sharp states - 69 -and that the angular apertures of the detectors are very small. In the case of the reaction through the L i 5 ground state the f i r s t alpha p a r t i c l e i s detected i n o(l and the L i 5 r e c o i l proceeds along OA and breaks up at A i n such a way that the proton i s observed at P and the alpha at <y2. The reverse process takes place when the f i r s t alpha enters o(2 and the L i 5 r e c o i l breaks up at B to give a proton i n P and an alpha i n 0(1. These two processes are coherent so that we have two point sources A and B i l l u m i n a t i n g P with waves representing the proton. These waves are of the same wavelength to the observer at P because the proton energy spectra are s t i p u l a t e d to overlap. The number of wavelengths contained i n the distance A'B' then determines the r e l a t i v e phase at P and thus whether interference i s constructive or destructive. In the case shown in Figure D2 f o r 4 MeV protons assuming 5 -22 l i f e t i m e s f o r the L i ground and f i r s t excited states to be 7 x 10 and -22 3 x 10 seconds r e s p e c t i v e l y , the distance A'B' i s very close to one wavelength f o r the proton, so that constructive interference might r e s u l t . However, we must new include the e f f e c t of the width of the states as well as the large angular aperture of the detector 0(2. The angle PC0(2 can vary by 18° with a corresponding change i n the angle of emission of the L i 5 r e c o i l f o r the reverse process. This gives a range of values f o r OB' varying by~20%. Also, the L i 5 r e c o i l energy varies by ± 50% with a corresponding e f f e c t on OB'. S i m i l a r l y , f o r a ground state width of ~1 MeV f o r L i 5 the recoil, energy of the l a t t e r varies by - 5% and the angle POA can vary 6° e i t h e r way due to the angular width of the detectors. These factors amount to a v a r i a t i o n of ± 25% i n the d i s t ance OA'. There r e s u l t s an uncertainty i n the distance A'B' amounting to - 35% over the proton energy range of i n t e r e s t . This i s s u f f i c i e n t to change the phase of the i n t e r f e r i n g waves at l e a s t by TT-Figure D2. Geometry showing p o s s i b i l i t y f o r interference i n proton detector at 110° i n the laboratory. - 70 -At the same time we must consider interference of a s i m i l a r nature in the alpha detectors. Since the alphas carry more momentum, t h e i r wavelength i s smaller. Furthermore the angle between them and the L i 5 r e c o i l axis i s also smaller, and so the number of alpha wavelengths t r a v e l l e d by the L i 5 before i t breaks up has a correspondingly greater e f f e c t upon the phase of the i n t e r f e r i n g alpha waves at the detectors. In these cases the v a r i a t i o n i s s u f f i c i e n t to change the phase by over 27T within the region of i n t e r e s t . In order to get t o t a l l y constructive interference contributing to the experimental energy spectrum one would have to have constructive interference i n a l l three detectors simultaneously. While t h i s might be a p o s s i b i l i t y , i t i s f e l t that such e f f e c t s w i l l be averaged out because of the widths of the L i 5 states and the large angular apertures of the detectors, as discussed above. Conclusion In view of the foregoing considerations we conclude that when the phase space factors are coupled with the s t a t i s t i c a l f a c t o r enhancing the t r a n s i t i o n through the ground state, the background produced by the 5* , . reverse process in v o l v i n g contributions from L i i s of the order .of 5%, and may be subtracted from the experimental r e s u l t s because interference e f f e c t s are h i g h l y improbable. An asymmetry of the type observed in the breakup of the L i 5 ground state would r e s u l t i n an enhancement of the reverse e f f e c t contribution to the spectrum taken with the proton detector at 30°. This would be l a r g e l y compensated by the smaller phase space f a c t o r at that angle, and so the background i s of the same order f o r both positions of the proton detector. REFERENCES 3 1. D.A. Bromley and E. Almqvist, He Induced Reactions (Atomic Energy of Canada Ltd. , Report CRP, AECL, No.950, 1959). 2. E. Almqvist, Ph.D. Thesis, University of Liverpool (1954). 3. E. Almqvist, K.W. Allen and C.B. Bigham, Phys. Rev. 99, 631 (1954). 4. J.R. Erskine and CP. Browne, Phys. Rev. 123, 958 (1961). 5. F.C. Young, K.S. Jayaraman, J .E . Etter, H.D. Holmgren and M.A. Waggoner, Rev. Mod. Phys. 37, 362 (1965). 6. M.A. Reimann, P.W. Martin and E.W. Vogt, Phys. Rev. Letters (in press). 7. G.C. Phillips, T.A. Griffy and L.C. Biedenharn, Nuclear Physics 21, 327 (1960), 8. G.C. Phillips, Rev. Mod. Phys. 36, 1085 (1964). 9. C. Kacser and I.J.R. Aitchison, Rev. Mod. Phys.. 37, 350 (1965). 10. K.M. Watson, Phys. Rev. 88, 1163 (1952). 11. A.B. Migdal, Soviet Phys. JETP 1, 2 (1955). 12. E.H. Beckner, CM. Jones and G.C. Phillips, Phys. Rev. 123, 255 (1961). 13. N. Jarmie and R.C. Allen, Phys. Rev. I l l , 1121 (1958). 14. D.B. Smith, N. Jarmie and A.M. Lockett, Phys. Rev. 129f 785' (1963). 15. W.E. Dorenbuseh and CP. Browne, Phys. Rev. 132, 1759 (1963). 16. C Zupancic, Rev. Mod. Phys. 37, 330 (1965). 17. J .E . Etter, M.A. Waggoner, H.D. Holmgren, C. Moazed and A.A. Jaffe, Phys.Letters 12, 42 (1964). 18. C Moazed, J .E . Etter, H.D. Holmgren and M.A. Waggoner, Phys. Letters 12, 45 (1964). 19. Minutes of the Topical Conference on Correlation of Particles Emitted in Nuclear Reactions, Gatlinburg, Tennessee, 1964, Rev.Mod. Phys. 37_ (1965). 20. E.W. Blackmore and J.B. Warren, Phys. Rev. Letters 16, 520 (1966). 21. E.W. Blackmore, private communication. 22. F. Everling, L.A. KBnig, J.H.E. Mattauch and A.H. Wapstra, Nuclear Phys. 18. 529 (i960). References continued 23. Nuclear Data Tables, Part 3 (i960) edited by J.B. Marion (National Academy of Sciences - National Research Council, Washington, D.C.). 24. M.A. Reimann, M.A.Sc. Thesis, University of British Columbia (1964). 25. J . Bruckshaw, Private Communication. 26. G. Dearnaley, Rev. Sci. Instr. 31^ 147 (i960). 27. R. L. Helmer, Brivate Communication. 28. G. Dearnaley ard D.C. Northrop, Semiconductor Counters for Nuclear Radiation (John Wiley Inc., New York, 1963). 29. T.K. Alexander, Private Communication. 30. G. Jones, Private Communication. 31. B.A. Whalen, Ph.D. Thesis, University of British Columbia, 1965. 32. E.W. Blackmore, M.A.Sc. Thesis, University of British Columbia, 1965. 33. F.Adjzenberg-Selove and T. Lauritsen, Nuclear Physics 11, 1 (1959). 34. •E.H. Auerbach, Brookhaven National Laboratory Report, BNL 6562, 1962. 35. E.W. Vogt, Phys. Rev. 136, B99 (1964). 36. W.T. Sharp, J.M. Kennedy, B.J. Sears and M.G. Hoyle, Table of Coefficients for Angular Distribution Analysis, Chalk River Report AECL No. 97 (1954). 37. H.D. Holmgren, Private Communication, 38. F. Adjz'pnberg-Selove and T. Lauritsen, Nuclear Physics 78_ 1 (1966). 

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