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Reaction He# (He3, 2p)He4 and the diproton state Blackmore, Ewart William 1965

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THE REACTION He 3(He 3,2p)He 4 AND THE DIPROTON STATE by MART WILLIAM A. BLACKMORE B.Sc.(Eng.), Queen's U n i v e r s i t y , 1963 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE i n the Department of PHYSICS We accept t h i s t h e s i s as conforming to the r e q u i r e d standard THE UNIVERSITY OF BRITISH COLUMBIA . A p r i l j , 1965 In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of Bri t i sh Columbia, I agree that the Library shall make i t freely available for reference and study* I further agree that per-mission for extensive copying of this thesis for scholarly purposes may be granted by the Head of my Department or by his representatives. It is understood that,copying or publi-cation of this thesis for financial gain shall not be allowed without my written, permission^. . . . Department of Physics The University of Bri t i sh Columbia, Vancouver 8 5 Canada Date March 12, 1965 - i . ABSTRACT . The processes by which the three particle f i n a l stat® i s formed i n the He (He-j2p)He reaction were investigated by observing the angular distribution of coincidence events between the two protons as a function of the angle between the protons. The reaction mechanism was determined by comparing the experimental distribution with those predicted for the various possible processes obtained from kinemati® and phase space arguments. The reaction was found to proceed predominantly by sequential decays through unbotand intermediate states and to a lessor extent by a dir@et instamtaneoms tfara® b©dy breakup, Th® majority of the two stag® 5 decays passed through the ground state ©f L i , The mean lifetime of this stat® was measured and fotmd to be ( l o 0 ±. .3) x 10° s e c There was also good evidence ©f a sequential decay through the diproton stat®„ In order to f i t the shape ©f the observed distribution It was necessary to assume that a diproton system exists which i s tmbound by 600 keV and has a mean -22 lifetime of 1„5 x 10 s e c However another possible interpretation i s that a direst breakup ©©OTTS and the angular distribution ©f the protons i s distorted by an attractive f i n a l state two proton interaction similar to th© scattering interaetion,although whether this interaction weald be strong en®Hgh to produce the observed distribution i s not known. A m©re quantitative three body desay theory i s therefore necessary i n order to draw any firm (Sonclusions about th© existence ©f the diproton stat©0 ACKNOWLEDGEMENTS I wish t o express my s i n c e r e g r a t i t u d e t o Dr. J 0 B 0 Warrea f o r h i s k i n d s u p e r v i s i o n of the work described i n t h i s t h e s i s . I wotdd a l s o l i k e t o thank Dr 0 M0 McMillan f o r h i s h e l p f u l d i s c u s s i o n s on the t h e o r e t i c a l aspects o f the problem,, I am ©specially g r a t e f a l t o Mr 0 B„ A. Whalen f o r h i s assistane© i n s e t t i n g up the e l e c t r o n i c s and i n perf®rmiiag the measurements Th© help o f the other members ©f the Van de G r a a f f group I s a l s o g r e a t l y appreciated„ I am deeply indebted t o the N a t i o n a l Research C o u n c i l £®T th© two s c h o l a r s h i p s h e l d d a r i n g the course o f t h i s work and f©r their @©at-inued' f i n a n c i a l -support. - i i -TABLE OF CONTENTS Chapter 1 ~ INTRODUCTION .... Page eooooooouooooooo'ooooQQoooDoooooooooooo Chapter 2 - CHOICE OF TARGET - DETECTOR ARRANGEMENT .. „». 0,,»o o 0 o o. 9 2-1, P r e l i m i n a r y C onsiderations ... „ .. 0»»o o o..«o..... 9 2«=-2d D i s c u s s i o n o f He Targets .o,.ooooooo.oooooooooooo.ooo. 10' 2*^3o P r o p e r t i e s o f Thi n I*oils? 4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 9 0 0 0 . 0 9 0 0 0 0 0 0 IX 2- =4-o D e t e c t i o n o f R e a c t i o n Products .. „ „ .. 0.. .....».o. o...., 12 Chapter 3 - EXPERIMENTAL ARRANGEMENT 0 0 0 0 0 , 0 0 0 0 0 0 0 0 0 0 0 0 = 0 0 0 0 0 0 . 0 0 0 0 16 3- lo General O u t l i n e o f Experiment .. 0»»»»».. 0». „ 0 o o«o o o»»o. 16 3- 2. D e t a i l s o f Gas Target Chamber . . „. „ „ 0» 0..« ... ° „.«».»»». 16 3C33 O) *^l@vS'fe3r,,@2i5j©S 0 ) 9 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 3°4o Adjustment ©f S.C.A. and Delays f o r Coincidences 0 0 0 0 0 0 20 Chapter 4 - THEORETICAL ANGULAR DISTRIBUTION OF COINCIDENCE EVENTS BETWEEN THE TWO PROTONS ...... o.. o». o. „ 0»o o o... o o <> o * o o» 22 4- "Xo TWO Bod^jT Kill©infl"fci©S O 0 0 O 0 0 O O O O O 0 O O O O 0 O O O » O 0 O O O O O O O O O O O O 1^,C'JLO Tliiic1®® Bociy Kin©ni&"feics 0 0 0 0 0 0 0 * 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 2^ Chapter 5 - EXPERIMENTAL RESULTS ..„,„..,,«,. „ e .„...... o o. o. o o. o'.. 32 5"=lo Energy Spectrum of, Protons 0 o „». o ....». o... o o»......o., 32 5=2. L i f e t i m e o f the L i ^ Ground Stat© .„.„.„„. .o..o«»o. o. o 34 5- 3. Angular D i s t r i b u t i o n o f Coincidence Events Ooooooo000oo 3® Chapter 6 •= CONCLUSIONS aoo.o..«o,...ooooooo.oo.oo«.«..ooo.oooo6ooo 42 Appendix A - PHASE SPACE CALCULATIONS ................. .-.o... „....„. 46 ~ i i i -A~2<, General Phase Space Formula <,„„„„o„<,o„oooo»o,»<>ooooo<> 46 A-3. Two P a r t i c l e F i n a l State ....„„„„. „ „ „„ „°.. <>»° <>*»».»„ 49 A-4o Three P a r t i e l e F i n a l State „„»..„„<,0.„»..„<>»<,.0.»»•«. 50 Appendix B Computer Programs f o r th© Phase Spaea C a l c u l a t i o n s „ 55 BibXiOff ^ S p l i ^ r 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 o'ooooooooooooooooo ^0 - i f -LIST OF FIGURES to foil©* page 2-1. Total Reaction Gross Section for the H©^(H®3p2p)He^ Reaction 0 0 9 3-1. Sehemati© Diagram ©f Experimental. Arrangement ... .„...,..,. O o o 16 3-2, D e t a i l s of Gas Target Chamber ............................. 0 • O 16 3-3, Block Diagram of Amplification - Coincident© System .„„„... o o o 18 3-4. Nwistor Preamplifier Circuit 0 • O 18 3-5.' 19 3-S„ G'^ l^jJlSi^ jGL^ XXO® 0 JLS^'SllJL'fe o o 9 o o o o a < t o o o o o u o o o o o o o o o o o o o o o o o o o o o o o o o o o 19 3-7. 211 Calibration of Detection System with Am ^  alpha partdeles • O 0 9 20 4-—1« Th@or@ti®al Angular Distribution for the Sequential Decays 0 0 0 26 4 - 2 . Tw© Dimensional Energy Spectra ., „ „........................ 0 o 0 29 4-3. Theoretical Angular Distribution for the Direct 3-Body Break up 31 5-1. Protoa Spectrum from the He^(He 9^2p)He^ Reaction . . „ . „ o . „ 0 . 0 o o 32 5-2. Experimental Angular Distribution of Coincidence Exeats ... o o a 39 6-1. Prodiet®d 1%® Dimensional Energy Spectra f©r the He (HeJ,!>2p)H<9r*' 0 o o 44 A-l. Single Partial© Spectrum from a 3~B©dy Breakup ............ 0 o 0 53 / => V ~ LIST OF TABLES t® t&Hm pag® l a L i s t of Contaminant Reactions ............. .<,<>.. i n t e x t p. 9 2„ P r o p e r t i e s ©f N i c k e l F o i l s , „»««. <> <><><.<><.„ . „ „. 0.«»<> <>«„.„»»»*.»„» 11 3„ R-taeSieMS Gssitribiatlng t o Pr©t« Speotram <,<,..<, «o.<. o..«, 0 0 .o. <>» 32 H.©0iDLJL^@ JL^ OJS H."t21l ?7^X oooeoooooooooodooooooooooooooooooooooooooo 3^  - 1 = CHAPTER 1 I n t r o d u c t i o n In most n u c l e a r r e a c t i o n s o c c u r r i n g a t low energies o n l y two n u c l e a r p a r t l e l e s ©merge as f i n a l products as a l l ether f i n a l states ar@ forb i d d e n by energy ©onserwfcion,, However i n some eases,, i n p a r t i e u l a r f@y 3 3 some H© and H induced r e a c t i o n s 9 the compound system has a s t a f f i s l e n t l y h i g h ©xoitatioa ©aergy t h a t breakup int® %hsm e>r m©re p a r t i a l i s i s @a®3fg®t^  i e a l l y p o s s i b l e . I» the past few yaars ther® has been aa iaeireased tet®r@st i n th® study., both e x p e r i m e n t a l l y and t h e o r e t i c a l l y p o f th®s® wtltiparti©!® breakup reaeti©»s0 Of p a r t i c u l a r i n t e r e s t i n these r e a c t i o n s i s "Uie r e a c t i o n mechanism^ t h a t i s the pr©<3@ss or pr©<3@ss©s by widish th® maittpajfti&l© f i n a l state I s farmed, Th® problam i s to determine wh<9*h<§r th® rsaeti®» proceeds v i a a s i n g l e - s t a g e 9 Instantaneous breakup a by a s e r i e s ©f tw© b@Sy deeays @r by a ©ombiaation o f both these processes. I f a l l the Intermediate systems whieh w©uld r e s u l t fr©m the s e q u e n t i a l processes are unbound tfe©'© t h i s becomes a v e r y d i f f i c u l t problem t© s©iw experimentally„ . I n this mm the study o f s i n g l e partial© s p e c t r a i s usually inadequate as b&th breafciap processes would r e s u l t i n s i m i l a r ©©ntinua i n th® eaergy s p e c t r a . Iffi many ©f the early «perfja©3ats i n v o l v i n g multipartial© f i a a l states th@s® ®mkim& wer© u s u a l l y ©xplainod as due t o a d i r e c t instantaneous breakups as tin® ©lliptieal shape pr®dieted using phase spa©© arguments i&r this pr&&m® adequately f i t t e d th® observed eontinua. Asay p<aaks i n tk@ ©ontiraa were - 2 -attributed to f i n a l stat© interactions between two &f the particles,, In 1960„ P h i l l i p s and Tombrell© (i960) Introduced the cluster model which sugg©st®d that these Warn® bciy decays could be treated as a time sequence <ef two body interactions. For instance 'the reaction a 4- A —*- D* b -+- G +- e could b@ e©nsid©r®d as pr@c@ading via th® tew© reaeteiens B* B*+ b B* G + e Th© assttmption that th® reaction caa be treated as a time sequence ®f tw® "body ctetsays requires that the decay ®f D ®®&ots to such a manner that th® eossteituents ef B s ©+ G9 remain together as an eae©ite®d localised system at least slightly l©nger than the time 'required for th® ©missis© @f particl© b „ Using the assumptions ©f this model ill© cross section f&r ©mission of particle b as a fusctie® of eaergy ©an b© writt@n l a terms ©f th© scattering phase shifts f©r the reaction e +- G —*» B*—- C +• e and i n t@rms ®f th® interaction Hamiltoniaa f@r the rt&etiem a H- A —»- D * — b -t- B* Ehcperiments have beea carried out t-s» c«ifirm the assumptions of this medal, B©ckner? Jones and Phillips (1961) km® invest-Igate@d th© three b@dy decay ®f G x aad B by observing the alpha assd deut®ron particl© spectra produced i n th© reactions B (p>, cx )2H© and a I Be ( p s d)2He^„ I n both cases the spec t r a e x h i b i t e d a lew energy anomaly i n the eontinua which c o u l d o n l y be e x p l a i n e d by assuming.that the dominant mechanism i n the r e a c t i o n was & sequence o f two body decays through s t a t e s o <©f Be „ T h i s anomaly I s present a t the corresponding energy i n c< - CX phase s h i f t s and therefor© enters i n t o the energy spectrum o f th© breakup p a r t i c l e s i f th© c l u s t e r model.is used. However the o v e r a l l snap© of th© continuum i n th© alpha p a r t i c l e spectrum from the B©^(H©^S CX )2H©^ r e a c t i o n has been f i t t e d by Dorsnbasch and Brown© (1963) by assuming t h a t th© r e a c t i o n proceeds p r i m a r i l y 12 by th® instantaneous breakup o f the compound nucleus C* i n t o t h ree alpha particles„ Th@s© res-alts ar© i n common w i t h m@st of th© r e s u l t s @f f i t t i n g singi® p a r t i c l e s p e c t r a i n th@s® multiparticl© f i n a l stat® r e a c t i o n s . Th© o v e r a l l continuum can u s u a l l y b© adequately f i t t e d by assuming aa i n s t a n t a n -eous breakup and modulations ©n the continuum r e q u i r e th© assumption ©f s e q u e n t i a l decays. I t i s evident t h a t a d i f f e r e n t experimental approach i s r e q u i r e d t o determine th© r e a c t i o n mechanism mor© p r e c i s e l y . Moaz®ds E t t e r j Holmgren and Waggoner (1964) have found t h a t . I n order t o determine u n i q u e l y the r e a c t i o n mechanism i n & t h r e e p a r t i c l e f i n a l s t a t e 9 i t i s necessary t o measure th© energies E^ p Eg and d i r e c t i o n s S^ 9 Q of two @f th© outgoing p a r t i c l e s i n co i n c i d e n c e . Th© outputs o f th© two p a r t i c l e d e t ectors ar© analysed i n a two dimensional kieks©rt®r. Coaservatloa o f «a«rgy asad momentum r e s t r i c t a l l p o s s i b l e ©vents t o a kinematic curve E (E ) which B A i s d©t@rmia®d by th© tw© angles and <9g and th© r e a c t i o n Q=valu®s0 Any r e a c t i o n s which proceed by s e q u e n t i a l pr&sesses through d i s c r e t e s t a t e s of som© intermediate system would then appear as p o i n t s ®n t h i s curve or as - u -segments o f the curve i n the case o f broad s t a t e s . This technique was used by Moazed et a l t o i n v e s t i g a t e the mechanism o f the Be 9(He^, CX )2He^  r e a c t i o n . The energies of two of the alpha p a r t i c l e s detected i n coincidence were observed u s i n g two s o l i d s t a t e counters mounted at f i x e d angles. T h e i r c o n c l u s i o n from the r e s u l t s o f these measurements was t h a t the r e a c t i o n proceeds almost e n t i r e l y by s e q u e n t i a l processes through the s t a t e s o f Be , An extension o f t h i s technique would be t o measure both the angular d i s t r i b u t i o n and the energies o f two of the outgoing p a r t i c l e s detected i n coinci d e n c e . I t i s p o s s i b l e i n some cases t h a t t h e angular d i s t r i b u t i o n s alone would be s u f f i c i e n t t o g i v e c o n c l u s i v e evidence about the r e a c t i o n mechanism. I f t h i s i s the case the use of a two dimensional k i c k -s o r t e r would not be necessary, A t h e o r e t i c a l angular d i s t r i b u t i o n o f coincidence events between two of the f i n a l s t a t e p a r t i c l e s can be determined f o r both an instantaneous breakup and f o r the v a r i o u s p o s s i b l e s e q u e n t i a l breakups i f a l l f i n a l s t a t e i n t e r a c t i o n s are neglected. These c a l c u l a t i o n s then depend o n l y on kinematic and phase space arguments. The d i s t r i b u t i o n can be mo d i f i e d t o account f o r coulomb i n t e r a c t i o n s by c o n s i d e r i n g the e f f e c t o f the coulomb b a r r i e r on the energy d i s t r i b u t i o n s o f the outgoing p a r t i c l e s . The n u c l e a r i n t e r a c t i o n s o r processes which predominate i n the p a r t i c u l a r r e a c t i o n can then be determined by comparing the p r e d i c t e d d i s t r i b u t i o n s f o r the v a r i o u s p o s s i b l e processes w i t h the experimental angular d i s t r i b u t i o n . I t was decided t o use t h i s technique i n the i n v e s t i g a t i o n o f the r e a c t i o n mechanism and intermediate s t a t e s present i n the £fer(H@ , 2p)He' r e a c t i o n . I n t h i s case the angular d i s t r i b u t i o n o f coincidence ©vents between the two protons was measured. The main reason f o r choosing t h i s p a r t i c u l a r r e a c t i o n was t h a t two of the f i n a l products are protons. Sine© - 5 -i t has been found t h a t i n most cases three p a r t i c l e f i n a l s t a t e r e a c t i o n s proceed v i a s e q u e n t i a l decays through intermediate states i t was h@p©d that i n t h i s r e a c t i o n one of the intermediate states might b© the d i p r o t a n stat©» The e x i s t e n c e ©f th© d i p r o t o n s t a t e has been th© t o p i c ©f much interest and s p e c u l a t i o n i n n u c l e a r p h y s i c s mainly because i t i s on© o f th© three p o s s i b l e n u c l e a r systems c o n s i s t i n g of two nucleons, A knowledge o f the p r o p e r t i e s o f th©s© n u c l e i , th© deuteronj, d i p r o t o n and dineutron i s o f fundamental importance i n n u c l e a r p h y s i c s as thes© p r o p e r t i e s a r a th© s t a r t i n g p o i n t i n th® d@v®l©pment of a nu c l e a r theory. Of th®s@. awlear systems ©nly th® t r i p l e t stat® ( s p i n s p a r a l l e l ) ©f th© deuteran i s s t a b l e ^ being b©uad by abaat 2,22 M@V,, Th© d i p r o t o n and dineutron are forbidden from having a t r i p l e t stat® because o f th® Pauli E x c l u s i o s Principle. Th© s i n g l e t states ( s p i n s a n t i p a r a l l e l ) ©f th® three systems are unbemd and therofor© d© mt ©xist in nature although i t i s quit© possible t h a t they might ©xist f@r a wary short time i f produced during a nuclear raaetiea. The f i r s t experiments c a r r i e d out t© i n v e s t i g a t e th© exist-ence o f th® d i p r o t o n stat® were low energy p = p s c a t t e r i n g s t u d i e s . Th© s c a t t e r i n g r e s u l t s shewed c o n c l u s i v e l y that th© diproton stat® was unbsuad but did not y i e l d any Inf@rmati©n about th© lifetime of th© stat© or i t s binding energy, M@ra r e c e n t l y th© p + D and n -I- D r e a c t i o n s hav® rec@Iv<sd c o n s i d e r a b l e attention i n an effort t® l e a r n m&re about th© p - p and n •= a i n t e r a c t i o n s , Wong et a l (1959) ©bs©rv@d th© neutran spectra from the p 4- D r e a c t i o n and Fried®s and Brass©! (1963) observed the proton spectra from the sans© r e a c t i o n , Th© high energy peak in th© n©utr« spectra and th© low energy d i p i n th© proton s p e c t r a were e x p l a i n e d by ass-aming that some f i n a l stat© p - p i n t e r a c t i o n was present. However the e f f e c t ©f this ao £j » interaction was not thought sufficient to warrant calling the interaction the diproton state. Likewise, Ilakovac et al (1961) observed the proton spectra from the n + D reaction and found a similar peaking indicating the presence of a final state n - n interaction. They stated that a more thorough theoretical analysis was necessary to draw any conclusions about the existence ©f the dineutron state. The formation of the diproton state in the p -+• D reaction and in proton - proton scattering is not particularily favoured as the state mast be formed during the reaction and the coulomb barrier has to be penetrated by one of the protons from the outside. A more favour-able situation would be to have the two protons i n i t i a l l y together along with another nucleon or stable nucleus and to have this extra particle removed by the reaction. This would be the case in the He3(d, t)He 2 or He3(He3,CK)He' reactions e Since the He3 nucleus consists part of the time of two closely lying paired protons plus a neutron, i t is conceivable that this neutron will be picked up by the incident deuteron or He3 nucleus leaving the two paired protons together in their paired configuration. The lifetime of this system wil l determine whether i t is purposeful to speak of a diproton state. If the lifetime is of the order of a nuclear transit time then i t would be more appropriate to consider the system in terms of final state interactions. In order to determine whether the diproton state can be formed in this way Bilaniuk and Slobodrian (1963) observed the triton spectra 3 ? from the He^(dj, t)He reaction and concluded that the broad peak they found was evidence of a quasi-stable diproton state. In a later experiment Artjomov et al (1964) studied the single particle spectra of both the He3(d, t)He 2 and He3(He3,0<)He2 reactions but could not find definite - 7 -evidence of a diproton state and concluded that more precise measurements were necessary. After a more careful investigation ©f the He3(d, t)He 2 reaction Conzett, Shield', Slobodrian and Tamabe (I964.) announced that i t was perhaps more proper to explain the broad triton peak in terms of a final state interaction between the two protons. By using the scattering length for the p - p interaction found from scattering experiments they were able to f i t the shape of the triton peak very accurately. In the light of this uncertainty in the interpretation of the energy spectra from the neutron pickup reactions on He3 i t was hoped that the technique of measuring the angular distribution of the coincidence events between the two outgoing protons would provide more definite evidence about the two proton interaction. Also as this experiment was carried out at lower energies than that used in the previous experiments i t might be expected that the probability of the formation of the two proton cluster would be more favourable. The study of the reaction using the coincidence technique is made somewhat easier by the fact that the outgoing particles are sufficiently energetic to make detection and Identification simple. Also they are not complex nuclei so that only ground states have to be considered in the analysis of the reaction mechanism. The He3(He3,2p)He^ reaction has not been studied to any great extent. Almquist, Allen, Dewan and; Pepper (1953) and Good, Kunz and Moak (1954.) found that the proton spectrum at 90° to the incident beam consisted of a wide high energy peak corresponding to the transition to the ground state of Idr superimposed on a broad continuum which they attributed to the three body breakup. There was also a low energy peak from the decay in flight of the Li^ nucleus into a proton and - 8 -an alpha particle. There was no evidence of well defined proton groups corresponding to transitions to an excited state of L i 5 . The total reaction cross section was found to increase monotonically with incident energy from a value of 2.5 f-tb. at 200 keV to 2 mb. at 800 keV. The present investigation of the He3(He3,2p)He^ reaction was carried out using a He3 beam from the U.B.C. Van de Graaff generator. This thesis describes the design of the He3 gas target and solid state counter assembly and the results obtained from an observation of the angular distribution of coincidence events between the two final state protons. - 9 -CHAPTER 2 Choice of Target - Detector Arrangement 2-1. Pre,i,-iTn|riftTy Considerations The bombardment of He3 with He3 leads to a three particle final state consisting of two protons and one alpha particle. The overall Q-value for the reaction is 12,859 MeV and therefore the maximum proton energy is about 11 MeV for low incident beam energies. The total reaction cross section for this reaction as found by Good, Kunz and Moak (1954) is shown in Figure 2-1, Possible contaminant reactions which could be present at low bombarding energies and their cross sections at representative energies are given in the following table. TABLE 1 Reaction Q-value MeV Cross section mb. at EHq3 MeV D(He3, piHe4-. 18.351 700 .650 H?(He3J>np)He4„ 12,095 180 .800 C 1 2(He 3„ p)HL^ 4.767 0.5 1.30 N^He 3, pjo 1 6 15.242 0.5 2.00 As the cross sections for the first two contaminant reactions are considerably larger than the reaction to be studied, i t is necessary to ensure that they are not present in the target chamber to any great extent. This requirement must be taken into consideration in the choice of the target arrangement to be used. - 10 -3 2 - 2 . Discussion of He Targets Since H©^ is available only as a gas unless extremely low temperatures are used, there are two target arrangements which are possible. The simplest technique is to produce a He3 target by bombarding a thin •a metal backing of copper or aluminum with the beam of He nuclei from the ( accelerator. This method requires high beam currents and therefore cooling i» of the metal backing is necessary to allow the He buildup to take place. There are several disadvantages in using this target arrangementj a. The possibility of contaminant reactions being present is very high, 3 b. The number of target atoms cannot be determined accurately as the He distribution with thickness is not known. e. Angular distribution measurements are difficult, i f not impossible, because of the target backing. The second method involves the use of a gas target. Gener-ally i f the desired target nuclide is only available as a gas i t is worth the additional difficulties to use a gas target. There are many advantages to be gained in using such an arrangement: a. The target thickness is easily adjustable and the number of target nuclei 3 per cm is directly available from the pressure and temperature measure-ments, b. Accurate cross section measurements and angular distributions can be carried out, c. Target uniformity is inherent except for beam heating effects (Robertson et a l , 1961). d. Contamination can be held to a minimum by using collimators to determine the active volume of the target, e. If solid state detectors are used to monitor the reaction products they - 11 -can be immersed in the gas target chamber. Gas targets usually are of two types? those In which the beam enters from the high vacuum region through a thin window and those in which the beam enters through a differentially pumped capillary. The second type is preferable i f accurate measurements are required as the window introduces larger energy straggling in the beam. However differential pumping requires a larger amount of target gas and a complex vacuum system so that unless a large supply of target gas Is available, as is not the case 3 with He% i t is necessary to use the thin f o i l for the beam entrance window. 2-3. Properties of Thin Foils The choice of a suitable window material is very important as i t usually is the limiting factor in gas targets as far as maximum beam current and target pressure are concerned. Possible choices are aluminum, nickelp molybdenum, tantalum and zirconium. Other materials than metals may also be used such as SiO or carbon flirts but the extra difficulties in preparing such films limits their application even though they can withstand higher beam currents. Nickel is the most commonly used window material because of its high mechanical strength and the availability of high quality, vacuum tight foils of thicknesses down to 0.010 mils. The nickel foils used in the target chamber in the experiment described here were obtained from Chromium Corporation of America, Waterbury, Conn. In Table 2 the properties of nickel foils when used as a 3 thin window for a beam of He particles are listed. The energy loss in the nickel f o i l was obtained from the stopping cross section data of Whaling (1958). The root mean square scattering angles were determined from Fermi's formula for multiple coulomb scattering, Fermi (1950). The maximum beam current and maximum pressure differential across the f o i l were obtained from TABLE 2 Nickel foil thickness 10~^in mg/cm2 Energy of He-* beam MeV Energy loss in foil keV " R.M.S. scat, angle degrees Max safe beam curr. /Ua. Max press, diff. mm of Hg. .01 .226 .5 175 3.5 .25 200 1.0 190 1.5 1.5 175 1.0 2.0 150 0.7 .02 .452 .5 315 6.0 .50 400 1.0 390 2.4 1.5 365 1.3 2.0 300 1.0 .025 .563 .5 360 7.1 .75 500 1.0 455 3.2 1.5 480 1.7 2.0 380 1.3 .03 .678 .5 390 8.5 1.0 600 1.0 570 3.5 1.5 560 2.0 2.0 460 1.5 .04 .904 *5 440 12.5 1.0 800 1.0 740 5.1 1.5 760 2.6 2.0 590 1.7 - 12 -earlier experimental data and by tests carried out on the foils. These figures apply to the case of nickel foils mounted on 3/16 inch diameter holes and the incident beam limited to 0.10 inches in diameter. The nickel foils were soft soldered to their brass holders using a high tenacity flux (Kester Soldering Salts ). The technique found mo*t satisfactory was first to put a thin layer of solder on the face of the brass holder, allow the holder to cool, and then apply a small amount of the flux to the soldered face. The f o i l was then carefully placed on the face of the holder and a large aluminum block was used to hold the f o i l in place and also to act as a heat sink. The aluminum block was then heated until the solder underneath the f o i l melted. When the f o i l and its holder were cool again, the excess flux was removed by soaking in methanol. 2-U, Setection of Reaction Products Although there are many possible methods of detecting q q / the charged reaction products from the He (He , 2p)He reaction, the special properties of the solid state detector make It the most obvious choice in this particular experiment. The small size of these detectors makes i t possible to mount the counters inside the reaction chamber. The detectors produce very fast rising pulses which are necessary for accurate timing in coincidence measurements. The requirements on the detector for this particular use are a linear response for protons over a range of energies from 1 MeV to 12 MeV and a reasonably good intrinsic resolution ( < 35 keV). In order to stop the higher energy protons a depletion thickness of 1000 microns is reqired. This depletion thickness can be conveniently obtained in "surface barrier" detectors. These counters are made by evaporating a thin layer of - 13 -gold (100 - 2000 A.U.) onto high resistivity (10?000 ohm-em) n-type silicon. A reverse bias of 300 - 4.00 volts gives the require depletion thickness. These counters are available with an active area ranging 2 2 from a few mm to several hundred mm . Th© choice of the active area is governed mainly by th© desired counting rate and th© required resolution of the angular distribution measurements. An approximate estimate of the yield to be expected from this reaction is necessary before the choice of detector size can be made. The yield of emitted particles T is given by Y = N n L n f d ^ particles /sec 2-1 where N is the number of Incident particles per second? . n is the number of target nuclei per crn^ j L Is the thickness of the target in cm.j J~L is th© effective solid angle the detector subtends and (jjfc^) l g 2 laboratory cross section in cm averaged over th© solid angl© and over th© spread in incident beam energy. For a gas target in which the target volume is fixed by a pair of defining slits of width 'a' at a distance 'r' from the beam centre line, the geometrical factor LfZ can be expressed approximately by rR for a detector of area 'A* placed a distance 'R' from th© beam and perpendicular to the incident beam direction. The effect of the beam width is assumed to be negligible. Assuming isotropy of the reaction products the reaction yield can be written V=r 16.7 PIG CTj. par-t,cle5/5eC.. ^ where P is the target pressure in mm of Hg| I is the beam current in jlXamps} G is the geometrioal factor determined from expression 2-2 and (7~t is the total reaction cross section in mb. The product PIGGjT should be made as large as possible but the parameters are adjustable only over a restricted range of values as other conditions must satisfied as well. The incident beam energy should be high to increase the reaction cross section and to decrease the percentage of energy lost in the entrance window. However i t is desirable to have as small centre of mass motion as possible in order to simplify the kinematic calculations. The beam energy chosen was 1.60 MeV which meant that about 1.15 MeV was left after the beam passed through the entrance f o i l . The cross section corresponding to this energy is about 5 mb. The gas pressure should be high to give a high counting rate and good cooling of the entrance window but i t is limited by the stren-3 gth of the f o i l . The He gas pressures used were between 50\ and 100 mm of Hg. The beam current likewise was limited by the entrance f o i l and had to be kept below about .25 |J-a, The detectors chosen had an active area of 78,5 mm2 corresponding to a diameter of 10 mm. By placing the counter at a distance of 50 mm from the beam centre line and using a collimator of width 5 mm placed a distance of 12,5 mm from the centre line of the beam i t was possible to obtain a single particle counting rate of - 15 -=r 400 coun-fcs/sec/nflL.. for a target pressure of 75 mm of Hg. This gave a reasonable coincidence counting rate of about 10 counts per minute. - 16 -CHAPTER 3 Experimental Arrangement 3-1. General Outline of Experiment A schematic diagram of the experimental arrangement is shown in Figure 3-1. Two solid state detectors were required for the coinc-idence measurements9 both counters mounted at 90° to the beam direction. One counter was fixed and the other could be rotated at various angles with respect to the fixed counter in the plane perpendicular to the beam. In order to observe coincidence events between the two protons only i t was necessary to discriminate against th® alpha particles. This was done by placing an aluminum f o i l in front of each counter to stop completely the alpha particles but to allow most of the protons to pass through. The counters were mounted inside the He gas target and the incident He beam entered the target chamber through a thin nickel f o i l . The experiment consisted of measuring the number of coincidence events between the two protons for various angles between the two counters from 180° to the minimum possible angle. The coincidence measurements were a l l normalized to the total number of proton counts in both counters. The energy spectrum of the protons were also observed. 3-2, Details of Gas Target Chamber The gas target chamber is shown in Figure 3-2. The cylindrical slits which, together with the active area of the detector, define the target volume were arranged so that only particles from reactions occurring in the target gas could be detected. This eliminated any contamin-A 4~ a ' BEAN\ DEFINING COLL. I MATORS FIGURE 3-1, Schematic Diagram of Experimental Arrangement FIXED SOLID STATE DETECTOR 9 C Y L I N D R I C A L SLIT TO "DEFfNE ACTIVE V O L U M E OF TARGET G A S ROTATtKiG SOLID STATE DETECTOR K . O V A R S S A L To 'DETECTORS I CYLINDRICAL 3UIT TO -DEFINE TARSET VoUlhAE F iKETD S O L I D S T A T E " D E T E C T O R 4M KO/AR SEAL TO DETECTOR # 2 . • ^ S X A N G U L A R - G A L E B E A H A I N — J » H A N D L E T o R O T A T E D E T E C T O R * 2 . B E A M T U B E O - R l N O I N S U L A T O R S C U R R E N T C O L L E C T O R C U P • R O T A T I N G S O L I D S T A T E D E T F C T O R # 2 FIGURE 3-2. D e t a i l s o f Gas T a r g e t Chamber S C A L E 1/2 - [. - 17 -ant reactions from the nickel f o i l and the chamber walls. The nickel foils were typically 0.025 mils thick. The brass holder on which the f o i l was mounted also served to collimate the incident beam. The soldered connection helped to provide a good thermal contact with the rest of the chamber. As the intended beam ctirrents were less than 0.5 f>a. i t was not necessary to cool the nickel f o i l . The heat resulting from the energy loss of the incident beam in the f o i l was dissipated partly by conduction through the f o i l to Its mounting and partly by conduction through the target gas. The target gas pressure was continuously monitored by a Wallace and Tiernan absolute pressure indicator Type FA - 160 (0 - 410 mm), 3 The He gas used in the target chamber and in the ion source for the Van de Graaff generator was obtained from Monsanto Research Corporation, Mound 3 Laboratories. The specified purity of the gas was greater than 99,8 % He in He^„ greater than 99,99 $ total helium and less than 3.6 x 10"11 % tritium. This tritium content was considerably lower than that which would have given a significant contribution from the H^(He^, np)He^ reaction. After passing through the active volume of the target gas the beam entered a collector cup so that the beam current could be monitored. As the Faraday cup was surrounded by the target gas It was found necessary to apply a negative bias ©f 90 volts to the cup to prevent collection of electrons from the ionization of the gas by the beam. The current was measured with an Eldorado Current Integrator Model CI - 110. The magnitude of the current was not considered reliable but i t did provide a means of monitoring the current for focussing purposes. An accurate knowledge of the beam current was not necessary as a l l measurements were normalized to the total number of protons detected in both counters. - 18 -The solid state detectors used to detect the protons were Type PH 8-25-10 obtained from Nuclear Diodes. These counters had the required depletion thickness at an operating voltage of 300 volts. Their active diameter was 10 mm and their resolution was about 25 keV. The detectors were biased using the ORTEC Model 201, 0 - 1000 V Detector Control Unit. This bias supply also allowed the leakage current to be continuously monitored in either counter. The counters were mounted at a distance of 5 om from the centre line of the incident beam and the width of the counters o prevented measurements from being taken at angles less than 23 . However by tilting both counters at an angle to the direction of the incident particles i t was possible to observe the coincidence events at lower angles. As this resulted in a dcrease in the solid angle subtended by the detectors the angle could only be redueed to about 8° before the coincidence rate became prohibitively low. 3-3. Electronics The block diagram of the amplification and coincidence system is shown in Figure 3=3. The pulses from the solid state counters were first amplified by similar charge sensitive preamplifiers. The circuit diagram for these preamplifiers, which used nuvistors in the input stage, is shown in Figure 3-4-. These preamplifiers combined low noise (resolution less than 10 keV) with a fast risetime ( < 10 nseo.) and a slow decay time (— 100 f^sec). This shape of pulse is desirable for use In delay line clipped amplifiers. The main amplifiers were Cosmic Double Delay Lin© Clipped Linear Amplifiers, Model 901, These amplifiers could be operated in either the single clipped or double clipped mod© and were compatible with the Cosmic Single Channel Analysers, Model 901 SCA, The single SjUVISTo'R PREAMP COSMIC "DD.L CUPPED AMPLIFIER ANALY2LER VARIABLE "be L A Y CoiKicitae>jce circuit <* t PftffAJMP C O S M I C T>.feL C u P P e b A . N | P L I R S R SINGLE CHANNEL Analyzer VARtAgUE T>£LAVec> OUTPOT FIGURE 3-3. Block Diagram of Amplification - Coincidence System S c a l e r 1 S c a l e r GATING "PULSg NOCUEAR TYPE IZO KtCKSO^TEl? OUTPUT N U V I S T O R P R E A M P L I F I E R FIGURE 3-4-. Nuvistor Preamplifier Circuit - 19 -channel analysers were used as discriminators to allow only pulses of a desired voltage range to produce a timing pulse coincident in time with the zero cross over point of the D.D.L. clipped pulse. The outputs of both single channel analysers were then fed thorough variable delays and into the coincidence circuit. The timing pulses put out by the S.G.A. were about 200 n S e c . long. In order to obtain better time resolution these pulses were shortened to 10 - 30 nsee. width at the input of the coincidence circuit using a delay line clipping technique (see Figure 3-5.). The coincidence circuit i s shown in Figure 3-6. In this circuit the input pulses were transformed into current pulses passing through a tunnel diode in its low voltage state. Whenever the two pulses arrived simultaneously the current was sufficient to switch the tunnel diode to its high voltage state producing a coincidence pulse at the output. The resolut-ion time of the circuit was set at about 30 nsec. for this experiment. The probability of a random pulse was negligible with the counting rates used for this resolution time. The output of the coincidence circuit and the positive outputs of both single channel analysers were fed into similar scalers (UBC - NP11). Thus the number of coincidence events between the two protons for a given number of counts in both counters could be obtained. The delayed output of one of the Cosmic D.D.L; clipped amplifiers was attenuated and put into the Nuclear Data Type 120 Kicksorter. The output of the coincidence circuit was used as a gating pulse in the kicksorter. When the coincidence circuit was operated in the singles mode, the single particle proton spectrum could be obtained. When operated in the doubles mode, only protons from coincidence events were analysed. Q < j | ' * - -<*>T TO CO.NC «N \30i7. F R O M S . C A , -If O.OOl p. T\1Z 2.K 114-3 •2.2 K 2 2 Ii O.OO^ V^VvV*— 20 ma. T.D. < — - 0 C I R C U I T -O -IOV FIGURE 3-5. Pulse Shaper for Coincidence Circuit FIGURE 3-60 Coincidence C i r c u i t - 20 -3=4-. Adjustment o f S.C.A. and Delays f o r Coincidences Figure 3-7 shows the/ response ©f the d e t e c t o r - a m p l i f i e r system to the alpha p a r t i c l e s from an A m 2 ^ source. The major group of alpha p a r t i c l e s from t h i s source has an energy o f 5.4-77 MeV0 The sharp peaks correspond t o pulses from a Datapulse 106A Pulse Generator a p p l i e d t o the t e s t i n p u t . The inp u t p u l s e height has been increased i n equal i n t e r v a l s and the equal spacing ©f the output pulses i n d i c a t e s the l i n e a r i t y ©f the a m p l i f i c a t i o n system. The i n s e t shows a g r e a t l y a m p l i f i e d response to th© Oil alpha particles indicating th® presence of other alpha groups from th© Aur* source.. This spectrum was obtained u s i n g a biased amplifier {ORTEC Model 201 Lew Nois© B i a s e d Amplifier). It i s p o s s i b l e t o determine the puis© height necessary to simulate a 5.477 MeV alpha p a r t i c l e and also the corresponding pulse height to simulate an 11 M@V proton (th© maximum expected energy of protons from the He^(H©\ 2p)H©^ reaction). Using th© pialser to simulate the protos pulsesj, th© gains ©f the two D.D.L. c l i p p e d amplifiers were adjusted to giva an output 9..-0 v o l t s f o r an in p u t corresponding to the maximum expected protoa energy. Th© b a s e l i n e of th© two S.G.A. was set at 0.5 volts and the window was set a t 9.5 v o l t s , thereby allowing the protons of energy between 600 k©V and 12 MeV t o produce an output pulse from the S.C.A.- Th© output o f two p u l s e r s p one synchronized from the output ©f the other to produce simultaneous p o i s e s , was then fed into th© twoS.C.A. ihr©agh the amplifiers. Th© delays wer© adjusted to obtain a coincidence output from the coincidence circ u i t operating i n th® doubles mod©. Th@s© delay s e t t i n g s had to b© readjusted slightly to obtain a coincidence output when running the actual experiment to esmp@n-110=1 u. o K CO 2 FIGURE 3-7, Calibration of Detection System w i t h Am 2^ alpha p a r t i c l e s . 3 0 mv. INPUT 5.477 ft 60 mv. S O mv. !2o nr»v-9 o IO 20 30 40 50 CHANNEL 6O NUMBER SO «2>o loo - 21 -sate for unequal cable lengths or small differences in the response of th® two detector-amplifier systems to the protons. The curve of number of coincidences vs 0 delay time should be a flat plateau &f width equal to t!s@ resolution, time of th® eoiacidenee circuit» The delays were s®t at the centre of this plateau. - 22 -CHAPTER U THEORETICAL ANGULAR DISTRIBUTION OF COINCIDENCE EVENTS BETWEEN THE TWO PROTONS 4-lo Two Body Kinematics A r e a c t i o n l e a d i n g t o a three p a r t i c l e f i n a l s t a t e a -h A —*• D* —*• b -f e + C + Q MeV can be considered as a sequence of two body decays D*-~ b -t- B* -t- Q 1 B*—^ c + C •+- Q 2 i f the decay o f D occurs i n such a manner t h a t the intermediate system B remains t o g e t h e r s u f f i c i e n t l y long t o s a t i s f y the cons e r v a t i o n o f energy and momentum,, The momentum diagram f o r t h i s process i s shown below. I f the r e a c t i o n Q-values f o r the s e q u e n t i a l decays ar® and r e s p e c t i v e l y then the energy and momentum r e l a t i o n s are as f o l l o w s % - 23 -4-1 E 3 + h 4 - Q a + E a 4=2 R + ft = 0 4 - 3 P3 +• p^. = P2 4 - 4 From these r e l a t i o n s i t f o l l o w s t h a t E:, = 0, _m*_ ^ m, -+- r n a = Q, — ^ 6 E l i m i n a t i n g E and s o l v i n g f o r E g i v e s 4 J -r E 2 no, - Qzrr^ — 2/rr^ E Z E 3 co5 a no 2 n o 2 v o n z 4-7 Squaring g i v e s a q u a d r a t i c equation from which E^ can be determined as a f u n c t i o n o f cos & „ T© complete the energy determinatloE E can be found from the r e l a t i o n 4 £ 4 = Bz •+• Q a - E 3 4-8 From Appendix A, the two p a r t i c l e d e n s i t y ©f s t a t e s fast®? - 24 -which determines the probability of particle c being emitted into solid angle dS~l^ at an angle Q to the direction of B* i s given by In the He3 +- He"' reaction the following sequential decays are energetically possible. I a„ He3.fr- He 3—* p •+• L i 5 CK Q 1 = 10.892 MeV Q 2 = 1.967 MeV mj = m^  m m^  = 5m P 4 p -t- H©3 —*> p -t- Li^"^-*-Qi = 10,892 - MeV Q 2 , 1 . 9 6 7 + E E X m l = m3 = mp ~ 5 nip m 4= 4mp - 25 -where E e x i s the excitation energy ©f the f i r s t excited state i n Li5 which i s thought to be 5 - 10 MeV. The width of this state i s about 3 MeV„ The calculations w i l l be carried out for several values ©f Ea_,„ I I He^-h He^ — C X +- He2 CX + p •+• p MeV Q x = 12.859 - Q 2 p Q 2 - Q 2 p MeV m^  =• 4"ip 1&2 — 2iBp m3 = "^"."p where Q 2 p i i the binding energy of the diproton. As the binding energy i s not known the calculations w i l l be carried out for several values ©f Qgp . In reaction I particles b and c are protons. If 0 the angle between these protons the energy equation U=H can be written Z. E 3 [ Ez( I - 2 cos V) - * Q Z + J _ ( E 2 - 4 0 2 j = In terms of the energies Eg and E^ the phase space factor, equation 4"9, becomes where E^(@) i s the solution ©f the energy equation c©rr@sponding t© the angle Q . The probability that the two protons w i l l be detected i - 26 -coincidence in two detectors separated by an angle Q is found by integrating this expression over the solid angles subtended by the two counters. This integration cannot be done explicitly and requires a Simpson's rule integrat-ion. The program for this calculation is listed in Appendix B„ These calculations were carried out using the U.B.C. IBM 7040 computer. The theoretical angular distribution for the sequential decay through the ground state and 1st excited state of Li-^ is shown in Figure 4=la. The momentum diagram for this process is shown in the inset. The solid angles used in the integration correspond to detectors of diameter 10 mm positioned a distance of 50 mm from the centre line of the incident beam. The aluminum f o i l in front of ©aoh counter (thickness 3.0 mg/cm2), to prevent alpha particles from being detected and the lower level setting on the single channel analysers to disciminate against the electronic noise, result in the loss of coincidence events between the two protons in which one of the protons has an energy less than 1.5 MeV. The correction to the theoretical curves to take account of this effect is shewn by the dotted line in Figure 4-la. This correction is only required for the K distribution corresponding to the transition to the ground state ©f Li . In reaction II particles c and C are protons. This is the sequential decay which proceeds through the diproton state. If i s the angle between the two protons the energy equation 4=6 becomes \Z cos U?J and the phase space factor becomes O 3 0 60 90 ISO 150 180 A N G L E BETWeEKi I ^ T E G T D R S 1-25 3 0 <£,Q S O ISO 150 ISO A N G L E B e n u e e w toSTe.c-roRs FIGURE 4 - 1 . T h e o r e t i c a l Angular D i s t r i b u t i o n f o r the Sequential Decays - 27 -4. PZ(S) = where E^( ) and E^( i$ ) are the s o l u t i o n s o f the energy equation f o r E ^ ( ^ ) ^ E^(up). T n e r e s u l t s o f the i n t e g r a t i o n of t h i s e xpression (see Appendix B f o r a l i s t i n g o f -the computer program) over the s o l i d angles of the two c o u n t e r s 5 normalized t o the p r o b a b i l i t y o f the event o c c u r r i n g w i t h Q = 0°5 are shown i n F i g u r e 4=1D<> The d i f f e r e n t curves correspond t o the d i f f e r e n t Q~values f o r the breakup o f the diproton„ The momentum diagram f o r t h i s s e q u e n t i a l decay i s shown i n the i n s e t . The c o r r e c t i o n f o r the l o s s of coincidence events f o r proton energies kess than 1„5 MeV i s i n d i c a t e d by the dotted l i n e . The peak i n the t h e o r e t i c a l d i s t r i b u t i o n occurs a t the angle a t which the two protons are emitted w i t h equal energies. This angle i n c r e a s e s as the energy a v a i l a b l e t o the breakup protons increases„ I f the d i p r o t o n o system remains bound the peak would be expected t o occur at 0 , 4=2 „ Three Body Kinematics The three p a r t i c l e f i n a l s t a t e produced by a r e a c t i o n o f the type a 4- A —»- D —B»b •+- c •+- C can a l s o be considered t o be formed by a s i n g l e - s t a g e instantaneous breakup of the compound nucleus D , The momentum diagram for t h i s process i s shown i n the f o l l o w i n g diagram. 28 If the compound nucleus id assumed to be at rest when the breakup occurs,, which is a reasonable assumption for low bombarding energies and for a high Q reaction, then the energy and momentum relations may be written as follows H, ••+• Ez + E 3 = Q 4-11 pr +• H + K = ° In the case of the breakup of the Be° system into two protons and one alpha particle^ two of the final particles have equal masses so that m-^= m2 = m. From the energy and momentum relations i m+-m3 rvn-nn 3 This is the equation of a family of ellipses in the two-dimensional energy space - Eg. The energy calculations can be simplified by making the following substitutions. a — 29 A, Equation 4-13 becomes on s u b s t i t u t i o n o f the above expressions H ~ f - -k£ ( I - 2 ^ ? ^ Z 4~U The l i m i t s on x become > t < O s i i < ^ < _^J=> > ^ > O < .X ^ c a b For the He^(He^j,2p)He^ r e a c t i o n the o v e r a l l Q°valae i s 12,859 Me?, The two-dimensional energy s p e c t r a f o r v a r i o u s angles between the tw© outgoing protons are shown i n Fi g u r e 4=2, The p r o b a b i l i t y ©f as energy d i v i s i o n o f p a r t i c l e s 1 and 2 i n a three p a r t I©le'breakup int® energy i n t e r v a l s d E i and dEg i s given i n Appendix A by Using the s u b s t i t u t i o n s l i s t e d p r e v i o u s l y t h i s d i f f e r e n t i a l phase spae® f a c t o r becomes FIGURE 4-2. Two Dimensional Energy S p e c t r a - 30 -The probability of particle 1 being emitted at an angle between Q and to the direction of emission of particle 2 is determined from the expression 4? * fu^ dDc die, v J cLu_ by substituting for <^M- from equation 4-H and then integrating over x. cL •3M-0n making the substitution the integral becomes it . . . \ The integrations give the f o l l o w i n g results§ - 31 -The probability of a coincidence event between the two protons occurring for an angle & between the detectors is found by integrat-ing the above expressions over the solid angles subtended by the counters. This also was done by a Simpson's rule integration (see Appendix B)„ The results of the integrations normalized to the probability of the coincidence o event occurring with & = 180 is shown on Figure 4.~3a. The correction for the loss of coincidence events for proton energies less than 1„5 MeV is shown by the dotted line assuming that the single particle energy distribution is elliptical,, The theoretical distribution for a three particle breakup i n which the masses of the three final state particles are equal is shown in Figure 4°3b„ This would be the distribution expected from the photodisintegration ©f th© •a Ear nucleus into two protons and one neutron assuming no final state interact" ions. These distributions do not depend on the energy available to the breakup particles but only on their relative masses. The momentum diagrams for both breakups are shown in the insets of Figure 4-~3. FIGURE 4-3. Theoretical Angular D i s t r i b u t i o n for the Direct 3-Body Breakup - 32 -CHAPTER 5 EXPERIMENTAL RESULTS 5~1* E n e r g y S pectrum o f P r o t o n s A beam e n e r g y o f 1,60 MeV was u s e d f o r a l l measure** ments. The e n e r g y l o s s I n t h e e n t r a n c e f o i l c o r r e s p o n d i n g t o t h i s i n c i d e n t 3 He e n e r g y i s 450 keV s o t h a t t h e beam e n e r g y i n s i d e t h e gas t a r g e t was 1.15 MeV, T h i s e n e r g y was f o u n d t o be h i g h enough t o g i v e a r e a s o n a b l e c o i n c i d e n c e r a t e w h i l e t h e c e n t r e o f mass m o t i o n due t o t h e i n c i d e n t e n e r g y o f t h e He^ beam was s u f f i c i e n t l y s m a l l t h a t i t c o u l d be i g n o r e d s making t h e c o m p a r i s o n w i t h t h e t h e o r e t i c a l a n g u l a r d i s t r i b u t i o n s e a s i e r . The s i n g l e p a r t i c l e p r o t o n s p e c t r u m o b s e r v e d a t 90° t o t h e i n c i d e n t beam d i r e c t i o n i s shown i n F i g u r e 5-1. The e n e r g y s c a l e has been drawn u s i n g t h e p r o t o n peak f r o m t h e He^(dj,p)He^ r e a c t i o n a s a c a l i b r a t -i o n p o i n t . The c r o s s s e c t i o n f o r t h i s r e a c t i o n i s s u f f i c i e n t l y h i g h t h a t t h e s m a l l amount o f HD"** i n t h e He^ ^  beam, p r e s e n t i n t h e i o n s o u r c e b e c a u s e o f p r e v i o u s r u n n i n g w i t h a d e u t e r o n beam, p r o d u c e d a s i g n i f i c a n t p r o t o n peak f o r c a l i b r a t i o n p u r p o s e s . T h i s r e a c t i o n p r o d u c e s p r o t o n s o f e n e r g y 14.7 MeV, The s p e c t r u m was t a k e n w i t h t h e f r o n t f a c e o f t h e s o l i d s t a t e c o u n t e r mounted a t a n a n g l e o f 60° to t h e d i r e c t i o n of t h e p r o t o n s b e i n g d e t e c t e d . T h i s d o u b l e d t h e e f f e c t i v e d e p l e t i o n t h i c k n e s s of t h e c o u n t e r and t h e r e f o r e i t h ad a l i n e a r r e s p o n s e up t o about 16 MeV, T a b l e 3 l i s t s t h e r e a c t i o n s c o n t r i b u t i n g t o t h e sp e c t r u m and t h e p r o t o n e n e r g i e s e x p e c t e d f r o m e a c h r e a c t i o n . The Q-values f o r t h e r e a c t i o n s have been t a k e n f r o m L a u r i & s e n and A j z e n b e r g - S e l o v e (1962)„ The p r o t o n e n e r g i e s a t 90° t o t h e beam d i r e c t i o n c o r r e c t e d f o r t h e i n c i d e n t 14 { 2 M 2 0 0 to FIGOEE 5-1. Proton Spectrum from the He3(Ie3,2p)He^ Reaction t-5 M e V U ( p ) He 9 - 1 M e V 3, s 5 H e ( K e , p V L ' i > al S > a: 6 2 Hi 1 - 8 0 0 . t-T D o o 4 0 0 3 3 4-H e ( H e , 2 P ^ H e o o ^ O o° ~ ° ~ O D „ ? O _ Q_ o o . o o 14-7 M e V 3 4 too TABLE 3 Reactions C o n t r i b u t i n g t o Proton Spectrum Re a c t i o n Q MeV E ( C 0 M j *MeV E (LAB) PMeV E (MEAS) pMeV - He 3 (dpp) He4" , 18.352 14.681 15,14 15.04 H e 3 ( H e 3 s p ) L i 5 . , 100892 9,077 9.46 9»30 He3(He3 p2p)He/|-. 12.859 10,716 11.10 10,97 He 3(He 3,He 2)He 4' 12.859 8,544 8.92 8.75 L i 5 ( p ) He^ 1,967 1,573 3o87 3.57 - 33 -beam energy are listed in the column E (LAB) and the laboratory energies p corrected for the energy loss in the aluminum f o i l in front of the counter are listed in the column Ep(MEAS). The thickness of the aluminum f o i l used was 4,0 mg/cm2. The proton energy listed for the three particle breakup He3(He3,2p)He^ is the maximum expected energy. The spectrum for this instant-aneous breakup would have an elliptical shape as shown by the dotted line in Figure 5-1. However the broad continuum found in the experimental spectrum is not necessarily due to this process. The protons from a. sequential decay through either th© 3 - 5 MeV wide first excited stat© of Li-5 or a broad stat© of th© diproton would produce two wide overlapping peaks and therefore yield a similar continuum. The high energy peak superimposed on this continuum lies at approximately 9,3 MeV and therefore corresponds to protons from th© sequential decay through the ground state of Li**, The lifetime ©f this stat© can be determined from the excess broadening of the proton peak (see Section 5-2)s Immediately below this peak there are several hundred counts statist-ically above the continuum which are due to the two final state protons entering one counter simultaneously. This process would produce a peak at 8,75 MeV, the same energy as that expected from a bound diproton. Better stefistics ar© necessary in order to separate this peak from the peak at 9.3 MeV. The lowest proton energy analysed by the kioksorter was 1.5 MeV, This was determined by the thickness of the aluminum f o i l in front of the counter, Th© Q~value for the breakup ©f th® Li** nucleus into a proton and an alpha particle is 1.967 MeV, Th© expected proton energy from this breakup would therefore be - 3 4 « Ep = g : ( / . ^ 7 ) = I.157 MeV However because of the r e c o i l energy of the Li- 5 nucleus before breakup th© protons can be emitted with energies up to E where E i s the solution , . pmax pmax of th© energy equation 4-7 for 0 — 180°. Substituting for E^5 and Q and solving the quadratic equation gives E = 3 . 8 7 MeV. This would correspond to a measured energy of 3 , 5 7 M©V • JMHSOt • . . . . , . - ... v.. -1 because of the aluminum f o i l . The rise i n the number ©f counts at th© low • • • • • energy end of th© spectrum is therefore du© to the breakup protons from L i „ 5=2. Lifetime of the Ld/> Ground State The ground state of Li" 5 i s unstable to breakup into a proton and an alpha particle. The lifetime of this state can be determined from the excess broadening of the proton peak at 9 . 3 MeV. The width of this peak i s du© to th© following effects? a. S t a t i s t i c a l fluctuations i n the incident He3 energy due.'.t© the ©iatrance f o i l . b„ S t a t i s t i c a l fluctuations i n the proton energy due to energy straggling In th© aluminum f o i l i n front of th© detector, a. Geometry of th© target = detector arrangement. d. Resolution of the so l i d state detector and the amplification system. e„ Short lifetime of th© L i ^ ground state. - 35 -The magnitude of effects a ?b ?c sd, w i l l determine how much of the broadening arises from the lifetime of the I i / 5 ground state. These effects w i l l now be calculated. The energy loss of the incident beam i n the entrane® f o i l was 450 keV. For a RV beam of energy 1,5 MeV the energy straggling Is approximately 3% (Marion, I960), As the He3 incident energy has a 1$ effect on the proton energy the contribution to the width due to the nickel f o i l i s A p . ' 4 5 0 x . 0 3 « . 0 4 ~ . 5 keV A Nl fad The energy loss of the protons i n the aluminum f o i l was 160 keV. Assuming an energy straggling of 5% (Sternheimer? 1960)s the contribution to the width due to this effect i s The geometrical width i s due t© the f i n i t e size ©f the active target volume as determined by the width of the cylindrical s l i t and the area of the detector. This width arises because th® contribution to th® measured proton energy due to theiincideht beam energy i s angularly dependent«, The proton energy E^ at an angle © to the direction of the incident beam of energy E^ i s given by the expression E3(Q) - E 3 t e d ) ^llm^t E ^ f c c e e 5 - 1 M a -+-M* ^ ^ J where E 3 ( S 0 0 ) ~ Q M 4 + E,( / V M , N Q i s the corresponding Q=value - 36 -Is the mass of the Incident particle. My are the masses of the reaction products. For angles near 9 0 ° , i.e. cos © small, expression 5=1 can be approximated by The detector - s l i t geometry is shown in the following diagram. 5=2 Incident b e a m direction approximated by x TV — * -b Slit .. M^-i V R -—^ - J t o b . * ^ _ 1 t \ a b r R 5 mm 2.75 mm 12.5 mm 55 mm The geometrical factor for this arrangement can be G — 7 T G ? 2 b R(R-r) and the angle © is given by The probability of a proton being detected at an angle © to the direction of the incident beam relative to the probability of a proton being detected at an angle of & = 9 0 ° i s very difficult to determine as the protons are not produced by a point source. An approximate calculation yields the values of 8 6 ° and 9 4 ° lor the half-width angles. Th© proton - 37 -energy at 90°,for the He3(He3, p)Li 5 reaction is E3(eoo V) = ^ (io.se 2) H - - 9.4-6 MeV and the proton width as determined from the half-width angles and expression 5-2 is approximately The measured resolution of the solid state detector and the amplification system is approximately The total width of the proton peak due to these combined effects is there-fore the square root of the sum of the squares of the statistical effects pips the geometrical effect - 3 2 1 keV This inevitable width resulting from geometry and straggling can also be determined from the broadening of the proton peak from the He3(djp)He^ reaction. As this reaction proceeds through the ground state of He^ - the width due to nuclear lifetimes is negligible.. From Figwe 5-1 th® width of this proton peak is of the order of 3 channels or 4 5 0 k©?0 However the geometrical contribution to the width must be e©rreet©d f&r the higher proton energy. This results in a total width for protons ®f energy 9<.<46 MeV - 38 -of Thus the two methods of determining the width due to the geometrical and s t a t i s t i c a l effects i n the detection system give' approx-imately th© same result. In order to determine the t o t a l width of the proton peak at 9,3 MeV In Figure 5-1 du© to th© He3(He3„ p)LI^ reaction, th© contrite ution to this peak from the other possible processes must be evaluated. Protons In this energy rang© could result from either the instantaneous breakup of from th© ether possible sequential decays, the maximum contribution resulting i f the instantaneous breakup predominates. Assuming this i s the case th© width of the proton peak would be 5 channels of 720 keV*, I f i t is assumed that a l l the protons i n the region Of 9»3 MeV ar© from the transition to th© ground stat® @f Li" 5 th© width of th© peak would b© 8 channels or 1150 keV. By averaging these two extremes and by correcting the resultant width for the contributions from the geometrical and s t a t i s t i c a l effects, the width of the ground state of Li- 5 was found to be P s = (oGO ( I ±--3) keV U 1* This corresponds to a mean lifetime of t a* 6 . 5 8 * I Q 2 * ^ (j.O ± .33 K Id21 sec, H .(a GO 5=3. Angular Distribution of Coincidence Events The results of th© measurement of the number ©f of coincidence events between the two protons as a function of the angle - 39 -between the two detectors are l i s t e d i n Tables 4 and 5, The f i r s t run was made using an angular width of the detectors of 11,5° and th® see©nd run wErlsh was made to measure the angular distribution at lower angles used an angular width ©f 4-°*allowing the distribution to be observed for angles dowa t© 8®,, The t o t a l number of proton counts i n both counters was used to normalize the number of coincidence events observed at each angle. The ratio of the normalized number of coincidence events to the number at an angle of 180° i s l i s t e d i n $he last column for each angle. For the second run the results were normalized to the f i r s t run results at an angle of The distribution of coincidence events i s shown i n Figure 5-2 as a function ©f the angle between the detectors. The circles correspond to the measurements taken i n the f i r s t run and the triangles t® the.measurements taken i n the second run. The s t a t i s t i c s on th© points indicated by the circles are better than 10$ and on the points indicated by th© triangles s better than 15$. These s t a t i s t i c s were obtained with approx-imately 20 minute runs per angle i n the f i r s t ease and one hour runs per angle i n the second case, A comparison of this experimental distribution to the predicted distributions (Chapter 4» Figures 4"lj> 4-3) indicates that th® eotosidence events i n the region of 180° are m©stly due t© the sequential decay through the ground state of Li- 5, The fact that the experimental di s t r i b -ution rises considerably more sharply at 180° than the phase space calculations predict, indicates the importance ©f f i n a l state interactions i n determining the angular distribution of the reaction products. If the effect ©f the coulomb barrier on the energy distribution of the outgoing particles i s considered (see Appendix A) both the angular distributita for the instantaneous TABLE 4 RUN # 1 Beam energy 1 „ 6 0 MeV Beam current , 2 5 a. Target gas pressure 7 5 mm Hg„ Angular width of detectors 1 1 . 5 ° . 'Angle Number of Number of Number of ^coinc^® ^  between' coincidence protons protons counters events counter # 1 counter # 2 N~~7 " ( 1 8 0 ) coinc 1 8 0 2 0 6 2 3 3 9 0 2 5 0 6 7 l o 0 0 0 1 7 0 4 1 7 7 0 4 6 3 7 5 5 5 1 ,700 1 6 0 2 5 9 8 2 4 0 6 8 8 1 1 1 , 3 5 7 150 1 1 2 6 4 5 3 2 6 7 8 2 3 .199 1 4 0 1 2 5 9 0 2 8 8 9 5 1 8 1 . 1 5 9 1 3 0 9 3 1 0 3 1 6 5 1 0 9 4 5 5 , 1 0 3 1 2 0 2 0 0 2 2 4 6 8 0 2 3 6 4 9 5 , 1 0 2 i i © 8 0 94978 100079 , 0 9 6 1 0 0 1 0 4 9 0 6 8 8 9 2 2 7 5 .134 9 0 1 1 9 1 0 0 1 0 0 1 0 4 8 0 5 ,137 8 0 1 3 2 1 0 0 9 2 2 1 0 5 6 9 9 ,150 7 0 2 0 0 1 5 9 5 3 2 1 6 7 7 3 1 , i u 6 0 J 2 6 1 0 0 4 8 4 1 0 5 4 8 9 , 1 4 4 -5 0 1 0 5 79592 8 0 8 8 0 , 1 5 3 4 0 1 1 3 8 6 8 1 1 9 0 6 7 6 , 1 5 0 3 0 7 9 6 2 7 4 8 6 4 2 7 7 o 1 4 6 2 7 2 1 3 1 5 5 9 1 8 1 6 3 4 4 0 ,157 2 5 7 1 '49504 ' '51249 , 1 6 6 23 2 1 0 1 3 5 0 9 8 142475 ,178 TABLE 5 Run #2 Beam energy 1„60 MeV Beam current .25 a. Target gas pressure 78 mm Hg„ Angular width of detectors 4.0° Angle Number'of Number of Number of Ncoinc<©> between"" coincidence protons protons counters events counter #1 counter #2 • N J (180) • coinc 60 54 96480 70839 ,150 45 54 110962 82722 ,130 40 44 80978 62271 ,140 35 81 125010 92600 ,173 30 71 111672 94876 ,160 25 72 111602 84617 .171 20 91 142198 116037 .164 15 77 127469 93139 .162 10 61 120490 89473 .135 8 95 170088 123616 .150 Uo 0.8 h % lU > (ii i iii u u § IU 0 at - / D Z Ui > _ ^ « « « 85"""^ < iNSI^Nf AWfi&t© ^ B G O V = 4 = = ==== - BREAKUP* 6 c i i __ - i . i 1 D 36 66 9d l2o J5o ISO ANGLE BETWEEN DETECTORS FIGURE 5=2i Experimental Angular D i s t r i b u t i o n of Coincidence Events breakup and for the sequential decay through the states of Id? would rise more sharply at higher angles. It is extremely difficult to determine this effect exactly but i t does not seem likely that its magnitude would be sufficient to correct th© phase space curves to f i t the experimental rise at 180°. This would mean that the p; <* 6( final state Interaction causes the protons of than phase space theory with coulomb inieiraetions predicts,, fho sequential decay through the ground state of Li^ eaaset ©©ntribut© to the coincidence events observed at angles below 75® a s kinematics restricts the second proton in the decay to energies less than 1,5 MeV/ This can be determined by substituting E p«, 1,5 MeV in the energy equation 4-7 which oan be written and solving for © . However kinematics does not rule out the possibility that the events at small angles could be due to either the. insjmntaneous breakup ©r the sequential decay through th© excited stat© ©f Li- 5, Th© pred-icted angular distribution for both the®® processes ar© similar In shape. The maximum possible contribution from the instantaneous breakup Is shown by the dotted line In Figure 5-2. It is assumed that a l l the coincidence events at the minimum i n the experimental distribution are due t© this process. Since the predicted distributions for both processes mentioned above decrease monotonically as the angle decreases 0 i t i s evident that they cannot produce the observed rise in th© distribution as th® angle - 41 -decreases below 9 0 ° . The modifications to these distributions to account for coulomb interactions result in the distributions decreasing even more at lower angles „ Also i t was noted previously that the effect of a final state p •= o\ interaction must be to cause the protons to be emitted even more preferentially in opposite directions. Therefore this increase in the number of coincidence events with decreasing angle between the protons must he due to either a sequential decay through a broad diproton state or, in other terms, a strong final state p - p interaction which causes the two protons from the single stage breakup to be emitted preferentially together. By comparing the experimental distribution to the predicted distributions for the various values of the binding energy of the diproton (Figure 4.-lb,)j i t is possible to determine the lifetime and the binding energy of the diproton system. The distribution rises to a maximum at 25 - 30 and drops ©ff at lower angles. Assuming that this maximum corresponds to the maximum predicted by the phase space curves, the diproton state must be unbound by approximately 600 keV. The rise in the distribution begins at 1 1 0 ° indicating that the diproton state must be very wide. The width ne@essary t© f i t the general shape of $»he rig® is P 4 - 5 M e V corresponding to a mean lifetime of —• 2- 2. 4 : 5 <= 42 = CI-IAPTER 6 Conclusions The technique of measuring the angular distribution of c o i n c i d e ence events between two of the particles in a three particle fiaal state has proved partially successful in identifying the various processes which take place in the He3(He3,2p)He^ reaction. The reaction has been found to proceed predominantly by sequential processes through unbound Intermediate states. This is in agreement with recent-work by Beckner et a l (196I) and. Moased et al (1964) wh§-found that,|h® dominant meehaniim in the mmltipartlele breakup reactions Be^(p,d)2He^ and Be^(He3so<)2He^ is the sequential decay through the states of Be**, The intermediate state which is most evident from the exper-imental data is the ground state of Li^, This state was found to have a width ©f P= 660 (1,0 ±. ,3) keV corresponding to a mean lifetime of f . * - * ( 1 . 0 ± ,3) x 1 0 ° 2 1 sec. This width is significantly less than the width of P= 1,5 MeV quoted in Ajzenberg-Selove and Lauritsen (1959), This figure is an average ©f several values found previously for the width of this state (Likely, 1955 and Frost and Hannas 1958), However none of these results were obtained using detectors of resolution comparable t© that ©f a solid state detector and the widths quoted were not corrected for broadening from 5 effects other t h a n the short lifetime of the Li ground state. The smaller valu® ©f th© width is in. better agreement with th© width ©f the ground ©tat© ©f the mirrar nucleus He^  which is P« 550 k©V, It would be expected that the extra proton in would increase the width of this state slightly. There is also g©od evidence that a sequential decay through - 43 -a broad diproton state takes place. This state was found to be unbound by about 600 keV, The singlet state scattering length for the p - p interaction i s a g = • 7,7 x H T 1 3 cm. This would correspond to a value of i i " = ? 0 0 keV for th© energy of th© '"Virtual" singlet state of the diproton as defined for proton-prston scatterings Tha negative sign of th© scattering length indicates that th© stat©' Is unbound, Therefore the experimental result of 600 ke? for the binding energy of the diproton stat© formed in. the He%e3^p)M©^ 3?§aoti©a i i i n good agreement with tho •wttorlng length for the p * p iat§M§ti©n» Ag tha lifetime- of th© dipff§t©& gysttm aieissafgr to f i t th® oxporinontal dttft i§ approximately" *V&i s i o s S l m§B er about Ip HUOIO&F transit times i t might be eoniidered questionable to speak of this s y i t e l a§ a state. Because of this short lifetime the diproton system l i e s i n a rather nebulous region between a "virtual" state and a strong f i n a l state interaction. It might prove useful to try to f i t the shape of the angular distribution at low angles by assuming some form for the proton-proton interaction based on scattering theory and using this to modify the phase space predictions for the three particle breakup, Domett, Shield, Slobodrian and Yamabe (I964) found that they could adequately f i t the shape of th© triton spectrum from th© He ( d 9 t ) 2 p reaction by using a 2p f i n a l stat© interaction The choice of th© method ©f considering the diproton system, either as a virtu a l stat© or a f i n a l state interaction depends mainly on which method yields the most useful information about the nuclear fore© involved. The other processes which can take place i n th© H© 3(He 3 s2p)H©^ reaction are the instantaneous three particle breakup and the sequential decay through the first excited state of Li . As the contribution t© the angular distribution from both these processes is very similar i t is not possible at present to' determine whether these processes both occur t© some extent in the reaction or whether one ©r the ©ther is not present. Two extensions to the experimental technique are possible 1B order to determine the reaction mechanism more completely. They should also give further proof that the sequential decay through the diproton state does occur and the relative probabilities with which the different processes take ' place» One mtthtd involves the use of a two dimensional kicksorter to analyse the energies of both protons taking part i i the coincidence event for various angles between the detectors, .This would give a two dimensional energy spectrum of Ep^ as a function of Ep 2 . The instantaneous breakup would appear as a curve in this spectrum and the sequential pr©e®ss@s w®uld appear as segments of this curve.. The two dimensional spectra predicted f®r counters placed 180® apart and 30° apart are sh®wn in Figure 6=1 far the various processes which take place. The size ©f the points for the sequent-i a l decays is proportional t@ the number of counts expected in the correspon-ding channel. F@r the instantaneous breakup a uniform distribution ©f -counts along the kinematieal curves is expected. The other method involves the me ©f a conventional kicksorter to determine the energy spectrum ®f one ©f the coincident proton® for various angles between the detectors. At an angle of 180° where the sequential decay through the ground state ©f l&r predominates9 tw@ proton groups would b® expected, one corresponding to the proton from th® transition t© the Li'' and the ©ther to the proton fr©m the breakup ©f Li**. Similarly P O I N T S CORRESPOND T o COINCIIbENC£ EVE15VTS f??OM ~ ^ 0 S E Q U E N T I A L . D E C A Y T H R O U G H R ^ ^- S R O U N D S T A T E O F L-LS S^fe 0 1 SEQUENTIAL DECAY THROUGH ®O S ISt EXCITED STATE OF U . S + + . © = I 8 O 0 x S E Q U E N T I A L , S ^ E C A Y T H R O U G H + + OI PROTON STATE 4-X N S T A N T A N E O U S T H R E E ^BOCsV B R E A K U P . \ \ FIGURE 6-1 „ Predicted Two Dimensional "N^ \ Energy Spectra for the \ \ ' He3(He3„2p)He^ Reaction | i 6 8 IO i a ^ P R O T O N I ^ V - - 45 -at an angle of 25° = 30° the two wide proton groups from the decay through the diproton state would be expected to overlap,, The processes occurring at the minimum in the distribution at 120° would be determined from the energy spectrum at this angle between the detectors. These experiments should be carried out for several reason®. A knowledge ©f the r@acti®n meehanism in these multlparti@le breakup reactions is very useful in our understanding of the farces between nucleons and the formulation ©f a nuolear reaction theory, Th® ftmdamenta'l nature of th© proton-proioa interaction certainly makes i t w®rthwhil© t© Xeara mor® abouib the virtual diproton system. Moreover it should be possible to relate this system directly t© -th® appropriate p - p phas® shifts and hence determine • Independently a value ©f the p - p singlet state scattering length. - 46 -APPENDIX A Phase Space C a l c u l a t i o n s A°l» I n t r o d u c t i o n In the study o f strong i n t e r a c t i o n s o f elementary p a r t i c l e s a great v a r i e t y o f d i f f e r e n t f i n a l s t a t e I n t e r a c t i o n s have been detected by the observation t h a t r e l a t i v e y i e l d s , momentum and e f f e c t i v e mas® d i s t r i b u t i o n s o f the outgoing p a r t i c l e s d e v i a t e from the expe c t a t i o n s from phase space arguments. S i m i l a r i l y i n low energy phy s i c s the existence o f short l i v e d resonances o r o f f i n a l s t a t e i n t e r a c t i o n s can be determined by comparing the experimental energy and angular d i s t r i b u t i o n s w i t h those p r e d i c t e d u s i n g phase space c o n s i d e r a t i o n s . This method i s e s p e c i a l l y u s e f u l i n r e a c t i o n s l e a d i n g t@ mm than two f i n a l products and can be a p p l i e d t o determining the mechanism o f these r e a c t i o n s . relativist!® p a r t i c l e s by B l o c k (1956), Williams (1961), and Skjeggested (1964), Th© purpose o f t h i s appendix w i l l be t o d e r i v e the u s e f u l »n-relativistie exp r e s s i o n s . General Phase Space Formula c a l c u l a t i o n ©f t r a n s i t i o n r a t e s . From p e r t u r b a t i o n t h e o r y the ge n e r a l formula f o r th® t r a n s i t i o n r a t e , ©r th© p r o b a b i l i t y t h a t a t r a n s i t i @ a Srm am i n i t i a l stat® t© a f i n a l s t a t e w i l l oeeur i n u n i t t i m e , i s g i v e s by Th© ge n e r a l phase space formula have been determined f o r The concept o f "phase space" i s c l o s e l y connected t© the where f H'I'ifi is the matrix element for-the transition caused by a perturbation ©f the Hamiltonian H:j ^ (E) Is a function-of th® t©tal eaergy ©f th® system and the masses ©f th® individual particles in the final state and is usually ©ailed the "density ©f states faet®rn or phase space factor. As the matrix element is in general completely unknown^  i t is n©@@ssary t© make certain assumptions about i t s behaviour in @rder t o calculate the t r a a s l t i m rat<s 0 The simplest assumption is that the m a t r i x element is a ©©nstantj, independent o f the individual momenta of the particles in the final state. In this ease the transition rates as w e l l as th® indiv-idual momenta in the final state are determined by th® phase space factor For ©ne p a r i i o l e a d e f i n i t e s t a t e o f motion, i . e . s p e c i f i e d (x^ypss) and Momentum {vX!>'Py!>Pz) t> caa be represented as a p o i n t l a a 6 dimensional phase space. C l a s s i c a l mechanics plaees no r e s t r i c t i o n s m. th© d e n s i t y o f the r e p r e s e n t a t i o n p o i n t s . However quantum mechanics,, khr&agl th® U n c e r t a i n t y Prin©ipl@p r s q u i r e s t h a t th® r e p r e s e n t a t i o n point® be separ-a t e d by . f i n i t a distaaeesp ©&©h p o i n t being ©onfined t®> aa elementary ©@il ©f sis© (2rrn*)^.- Th© number o f f i n a l s t a t e s availabl® t® one p a r t i c l e w i l l therefor© be f i n i t e and equal t o the t o t a l volume o f phase space d i v i d e d by th® &im ©f m ®l«a»ntary c e l l o = r — ~ r I <^3P A-2 { z ? m 3 J 1 The number o f f i n a l s t a t e s f o r a system o f n p a r t i c l e s w i l l b® th© produst o f the number ©f f i n a l s t a t e s f o r ©ach partis!©„ •= 48 ° n TT d \ d 3 P i A-3 This formula a p p l i e s o n l y f o r a system of n s p i n l e s s p a r t i c l e s . I f the p a r t i c l e i has s p i n the above expression should b® m u l t i p l i e d by fr (2sLti) i-1 HoMsver s i E c e these s p i n f a c t o r s can be i n c l u d e d i n the f i n a l n o r m a l i z a t i o n of the phase space i n t e g r a l these f a c t o r s w i l l be neglected here. Sine© a l l th® p a r t i c l e s are confined t o the same g e o m e t r i c a l volume V equation A=3 can be w r i t t e n N = UL I TT d 3 Pc A-4 Given the t o t a l number o f s t a t e s the d e n s i t y i n phase space I s de f i n e d as the number ®f s t a t e s per u n i t energy i n t e r v a l . A=5 V w i l l b® dropped t o s i m p l i f y The constant faet@r handling o f the expressions. I n order t o eons@rve t o t a l momentum th® p a r t i c l e momenta eaBja©t be independent but are ©oastrained by the equation - 49 -It i s usual to indicate this restriction by putting A-6 The calculation of the density of states factor i s Impossible for a l l but the simplest systems. The non°r®lativistic calculations for n =3 2, 3 are ®f use i n low energy nuclear physic®. Two Particle Final State The energy and momentum relations for a two particle f i n a l state are given by Using equation A-6 for a = 2 A-7 In polar coordinates this becomes A=8 Since cLE and Pi PS= the expression A-8 can be written *2 or In terms of the total momentum p^ this becomes sL ^( E) = ^ ™* P 3  A-9 A=1Q For th® ease of p J o O the angular distribution is isotropic as would be expeotad for a two particle final state. Three EarMele JJffil State The energy and momentum relations extended to a three particle final state are given by K - Ff Tt * ft Using equation A-6 for n s 3 A - l l Th® overall phase space factor is determined by evaluating a l l - 51 -the integrals in expression A-ll. The integration proceeds from the right with momentum p^ held fixed during the integration over p2. However for low energy nuclear reactions the differential phase space factors with respect to energy or angle of one of the final state particles are more useful as they play a considerable role in determining the energy and angul-ar distributions of the reaction products. In the evaluation of these differential phase space factors the appropriate integrations are omitted. The phase space factor which determines the probability of particle 1 being emitted into solid angle dft ^  and momentum interval dp^  and particle 2 into solid angle &Slz is given by 4l ft(.E) Pi2 fp^dp A-12 Since . el_ _ <d — _ L _ - — cj dfk df% df% and "E^  Is fixed so that '4= - 1 cL A-13 For the ease ©f zero initial momentum this simplifies to d ^ f l z d P l ( n n ^ 3 ) P z i m p o s e « 52 = where & is the angle between p^ and *p^ . The probability of a certain energy division among the reaction products can be found from equation A-14. by transforming from momen* turn intervals to energy intervals. Upon making the following substitutions d B , _£L dp, <^Ez^£± clp-equation A-14 becomes The energy distribution of particle 1 can be found by Integrating this expression over Eg. From the energy and momentum relations + b £ z + ciE^ - E where _ rn.-i-m. ™ 3 - 53 -Squaring the above expression gives , max min. The difference (Eg - Eg ) is the difference between the solutions of this quadratic equation at cos© = - 1 and cos© = +• 1. 2 \ w Substituting for a,bp and c gives 4-<?3(E) = ^ wCm^m^ ( E , ( E - ( A ^ n n ^ F ^ The energy distribution is therefore of the form 12+T13 f (E,)dt, = N (E,f (E- to±^r>j) ff, Vs- A. 1 6 This distribution is shown in Figure A«=l, A more accurate determination o f the single particle spectrum f o r a three particle breakup requires the addition of the final state interactions. The energy distrib-ution can be approximately corrected for the coulomb barrier by adding the a p p r o p r i a t e coulomb penetrabilities t o th© distribution p r e d i c t e d by phase space calculations. This m o d i f i e d d i s t r i b u t i o n would be ©f th® form X n - N f ( E n ) Ff I f f (l) where P^ i s the p e n e t r a b i l i t y f o r p a r t i c l e 1 of 'energy A-17 DISTRIBUTION P R E D I C T E D FROM P H A S E S P A C E V C A L C U L A T I O N S MODIFIED DISTRIBUTION TO A C C O U N T F O R COULOMB B A R R I E R FIGURE A=l. S i n g l e Particle Spectrum from a 3- Body Breakup O E N E R G Y O F P A R T I C L E 1 m a +m 3 EI m,+ m 2 + m 3 - 54 -The effect of this modification i s to cause th® distribution to be peaked at higher energies as shown in Figure A-l. It i s also possible to qualitatively predict the effect that the coulomb interaction will have on the angular distributions of the final state particles. Since the probabil-ity of a particle penetrating the coulomb barrier increases with energy i t is obvious that final state momentum configurations in which the particles have the highest possible energies will be preferred. Likewise final state configurations in which ©ne of the particles has a very low energy would not be favoured. This is very similar to the effect that allowable phase space has on the distribution of the final state momentum configurations» Therefore It would be expected that any peaks in the angular distributions predicted by phase space arguments would be accentuated by the effect of the coulomb interaction. 55 -APPENDIX B Computer Programs for the Phase Space Calculations The integration of the phase space functions over the solid angle subtended by the counters has to be done numerically because of the complexity of the functions involved. The most common method of numerical integration using a computer is the Simpson's Rule integration, A subroutine was written for the IBM 7040 computer for performing the necessary integrate ions. This subroutine, called by SIMP(ASB,RES), evaluated the integral of a function FUN(U) ever the limits A to B and stored the result in RES. The function FUN(U) was evaluated in a second subroutine. A listing of the Simpson's Rule integration subroutine is given below SUBROUTINE SIMP(A,B,RES) N = 1 FODD - 0.0 FEVEN » FUN( (A -+-' B)/2„0) FAB = FUN(A) -I- FUN(B) 11 N - 2 * N. EN = N H = (B - A)/(2.0*EN) FEVEN = FEVEN + FODD •FODD = Qo0 DO 10 I = ljN DI a 2 * I - 1 10 FODD - FODD + FUN(A + DI * H) - 56 -8 IF (N - 8) 11,11,8 RES - (H/3.0) * (FAB + 4.0 *F0DD 2.0 * FEVEN) 9 RETURN END For each of the various possible reaction processes a main program to read in the necessary data, evaluate the limits of integration and print the results, had to be written as well as the subroutine to evaluate the phase space function. For the sequential decays the input data consisted of the masses of the four particles present in the reaction denoted by AMI, AM2, AM3, AM4 and the Q-values for the sequential decays denoted by Ql and Q2. For the instantaneous breakup only one parameter AA was necessary in the function subroutine corresponding to the variable 'a' in Section 4-2. The main program for evaluating the angular distribute ion for the sequential decay through the states of Li^ is listed below. The programs for the other processes are similar except for the input data and the function subroutine. The subroutines FUN(U) for evaluating the phase space functions corresponding to each process are also listed. The output is printed in three columns ANGLE, INT. RES, and RATIO. In the last column the ratio of the Integration result corresponding to the angle given in the first column divided by the result at Q « 0° Is listed. C ANGULAR DISTRIBUTION FOR SEQUENTIAL DECAY THROUGH LI5 PRINT 1 READ 3, AMI, AM2, AM3, AM4, Ql, Q2 COMMON AMI, AM2, AM3, AM4, Ql, Q2, El, E2 - 57 -DTHETA = 0 . 2 0 G DTHETA I S T p ANGULAR WIDTH OF COUNTER IN RADIANS E I = Q1*AM2/(AM1 -|- AH2) E2 • Q2 * AMI/ (AMI •+• AM2) THETA = 0,0 8 A S3 eOS(DTHETA) B = 1.00 CALL SIMP(A,B,RES) PZERO m RES #4.0 PTHETA m PZERO GO TO 15 10 I F (THETA - 180.0) 13,12,16 12 A = - 1.00 B = - COS(DTHETA) CALL SIMP(A,B,RES) PTHETA as RES * 4.0 GO TO 15 13 TRAD s THETA* .017453 A s COS(TRAD 4 DTHETA) B = COS(TRAD - DTHETA) CALL SIMP(A,B,RES) ST - SIN(TRAD) DFI - 2,O*ATAN(DTHETA/(2 O0*ST)) PTHETA • RES * DFI 15 RAT m PTHETA/PZERO PRINT 2, THETA, RES, RAT - 58 -THETA = THETA 10,0 GO TO 10 STOP FORMAT (30H ANGLE INT. RES RATIO /) FORMAT (I X , FlO.Oj, 2F10 .4) FORMAT (4F5.0, 2F1Q.Q) END FUNCTION FUN(U) PHASE SPACE FUNCTION EVALUATION FOR L I 5 DECAY COMMON AMI, AM2-, AM3, AM4, 01* 02, E l , E2 C = (AM3*-E2/AM2 - AM4 * Q2/A12) * * 2 B = E 2 * AM3/AM2* (1.0 - 2.0*U**2) - AM4/AM2*Q2 IF (H) 5,5,6 E3 = - SQRT(B**2 - C) - B GO TO 7 E3 = SQRT(B**2 - G) - B FUN = E3/(SQRT(E3/AM3) - SQRT(E2/AM2)* U) RETURN END FUNCTION FUN(U) PHASE SPACE FUNCTION EVALUATION FOR DIPROTON DECAY COMMON AMI, AM2, AM3, AM4P Ql» Q2, E l , E2 SUM - E2 Q2 DIFF o E2 - Q2 IF (U) 5 , 5 , 4 - 59 -4 IF (SUM - DIFF/U) 5,6,6 5 FUN ss 0.0 RETURN 6 E3 = 0.5* (SUM •+• SQRT(SUM**2 - (DIFF/U)** 2)) E4 ss SUM * E3 FUN B E3/(SQRT(E3) - SQRT(E4.)*U) RETURN END FUNCTION FUN(U) C PHASE SPACE FUNCTION EVALUATION FOR 3 BODY BREAKUP COMMON AA UA = U * AA UA2 ss UA*UA FACT s SQRTU.O - UA2) FACT5 = FACT « * 5 IF (UA) 5,6,7 5 UA 8 3 ABS(UA) FUN s ((2.0 + UA2) * (3oU159 - ATAN(FACT/UA)) -h 1.5 *UA*FACT)/ 1 FACT5 GO TO 8 6 FUN ss 3.U159/32 GO TO 8 7 FUN = ((2.Q+- UA2)* ATAN(FACT/UA) - 1.5*UA* FACT)/FACT5 8 RETURN END - 60 -BIBLIOGRAPHY Ajzenberg-Selove, F.A., and Lauritsen, T. (1959), Nuclear Physics 11, 1. Almquist, E., Allen, K.W., Dewan, J.T., and Pepper, T.P, (1953), Phys. Rev. 91, 1022. Artjomov, K.P., Chuev, V.J., Goldberg, V.Z., Ogloblin, A.A., Rudakov, V.P., and Serikov, J.N, (1964), Phys. Letters 12, 53. Beckner, E.H., Jones, C.M., and Phillips, G.G. (1961), Phys. Rev. 123., 255. Bilaniuk, O.M., and Slobodrian, R.J. (1963), Phys, Letters Z, 77. Block, M. (1956), Phys. Rev. 10£, 796. Conzett, H.E., Shield, E.J., Slobodrian, R.J., and Yamabe, S. (1964), Phys. Rev. Letters 13., 625. Dorenbusch, W.E., and Browne, C.F. (1963), Phys. Rev. 122, 1759. Fermi, E. (1950), Nuclear Physics, University of Chicago Press, Chicago, p.37. Friedes, J.L., and Brussel, M.K. (1963), Phys. Rev. 13JL, 1194. Frost, R.T., and Hanna, S.S, (1958), Phys. Rev. 110, 939. Good, W.M., Kunz, W.E., and Moak, C.D. (1954), Phys. Rev. 87. Ilakovac, K., Kuo, L., Petravic, M., Slaus, I,, and Thomas, P. (1961), Phys. Rev. Letters 6, 356. Lauritsen, T., and Ajzeriberg-Selove, F. (1962), Energy Levels of Light Nuclei, National Academy of Sciences - National Research Council. Likely, J.G. (1955), Phys. Rev. 98, 1538A. Marion, J.B, (i960), Nuclear Data Tables Part 3, p.24, National Academy of Sciences - National Research Council. Moazed, C., Etter, J.E., Holmgren, H.D., and Waggoner, M.A. (1964), Phys. Letters 12, 45. Phillips, G.C.., and Tombrello, T.A. (i960), Nuclear Phys. 2&9 525. Robertson, L.P., White, B.L,, and Erdman, K.L. (1961), Rev. Sei, Inst. 32, U05, - 61 -Skjeggested, 0. (1964)/ Proceedings of the 1964 Easter School for Physicists, CERN 64-13, Volume II. Sternheimer, R.M. ( i 9 6 0 ) , Phys. Rev. 117, 485. Whaling, W. (1958), Handbuch der Physik, vol. 3U9 p. 193. ed. by S. Flugge, (Springer - Verlag, Berlin). Williams, W.S.G. (1961), An Introduction to Elementary Particles, Academic Press, New York and London. Wong, C., Anderson, J., Gardner, G., McClure, J., and Nakada, M. (1959) Phys. Rev. 116. 164. 

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